Radiative Heat Transfer [4 ed.] 9780128181430

Citation preview



Radiative Heat Transfer

Radiative Heat Transfer Fourth Edition

Michael F. Modest The University of California at Merced Merced, CA, United States

Sandip Mazumder The Ohio State University Columbus, OH, United States

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright
©

Elsevier
Inc.
All
rights
reserved. MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-323-98406-5 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Cathleen Sether Acquisitions Editor: Maria Convey Editorial Project Manager: Alice Grant Publishing Services Manager: Shereen Jameel Production Project Manager: Kamatchi Madhavan Designer: Vicky Pearson-Esser Typeset by VTeX Printed in the United States of America Last digit is the print number: 9 8 7

6 5 4

3

2

1

About the Authors Michael F. Modest was born in Berlin and spent the first 25 years of his life in Germany. After receiving his Dipl.-Ing. degree from the Technical University in Munich, he came to the United States, and in 1972 obtained his M.S. and Ph.D. in Mechanical Engineering from the University of California at Berkeley, where he was first introduced to theory and experiment in thermal radiation. Since then, he has carried out many research projects in all areas of radiative heat transfer (measurement of surface, liquid, and gas properties; theoretical modeling for surface transport and within participating media). Since many laser beams are a form of thermal radiation, his work also encompasses the heat transfer aspects in the field of laser processing of materials. For several years he has taught at Rensselaer Polytechnic Institute and the University of Southern California, and for 23 years was a professor of Mechanical Engineering at the Pennsylvania State University. Dr. Modest spent the remainder of his professional career as the Shaffer and George Professor of Engineering at the University of California, Merced, the 10th campus of the University of California system, and the first newly established research university of the 21st century. Dr. Modest is also the author of a research monograph (with D. C. Haworth) Radiative Heat Transfer in Turbulent Combustion Systems (Springer, 2015). He is the recipient of many awards, among them the ASME Heat Transfer Memorial Award (2005), Germany’s Humboldt Research Award (2007), and the AIAA Thermophysics Award (2008); he is an honorary member of the American Society of Mechanical Engineers, and an associate fellow of the American Institute of Aeronautics and Astronautics. Dr. Modest and his wife Monika now mostly reside in Aptos, CA, with frequent visits to Merced, CA.

Sandip Mazumder was born in Calcutta (Kolkata), India. Following his bachelor’s degree in Mechanical Engineering from the Indian Institute of Technology, Kharagpur, he started his graduate education in the autumn of 1991. In 1997, he graduated with a Ph.D. in Mechanical Engineering from the Pennsylvania State University. After graduation, he joined CFD Research Corporation, where he was one of the architects and early developers of the commercial computational fluid dynamics code CFD-ACE+. In 2004, he joined the Ohio State University, where he is currently a full professor. His research in radiation has primarily involved developing efficient methods for solving the radiative transfer equation and coupling it to other modes of heat transfer for practical applications. Dr. Mazumder is the author of a graduate-level textbook titled Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods (Academic Press, 2016). He is the recipient of the McCarthy award for teaching and the Lumley award for research from the Ohio State College of Engineering among other awards and is also a Fellow of the American Society of Mechanical Engineers (ASME). He resides in Columbus, Ohio, with his wife Srirupa.

v

To the m&m’s in my life, Monika, Mara, and Michelle –mfm– To Srirupa and Abhik, who make me smile! –sm–

Contents Preface to the Fourth Edition List of Symbols

xv xix

1. Fundamentals of Thermal Radiation 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11

Introduction The Nature of Thermal Radiation Basic Laws of Thermal Radiation Emissive Power Solid Angles Radiative Intensity Radiative Heat Flux Radiation Pressure Visible Radiation (Luminance) Radiative Intensity in Vacuum Introduction to Radiation Characteristics of Opaque Surfaces 1.12 Introduction to Radiation Characteristics of Gases 1.13 Introduction to Radiation Characteristics of Solids and Liquids 1.14 Introduction to Radiation Characteristics of Particles 1.15 The Radiative Transfer Equation 1.16 Outline of Radiative Transport Theory Problems References

1 2 4 5 10 12 14 16 17 19 20 21 23 23 25 26 26 28

2. Radiative Property Predictions from Electromagnetic Wave Theory 2.1 Introduction 2.2 The Macroscopic Maxwell Equations 2.3 Electromagnetic Wave Propagation in Unbounded Media 2.4 Polarization 2.5 Reflection and Transmission 2.6 Theories for Optical Constants Problems References

31 31 32 37 41 55 58 58

3. Radiative Properties of Real Surfaces 3.1 Introduction 3.2 Definitions 3.3 Predictions from Electromagnetic Wave Theory 3.4 Radiative Properties of Metals 3.5 Radiative Properties of Nonconductors 3.6 Effects of Surface Roughness 3.7 Effects of Surface Damage, Oxide Films, and Dust 3.8 Radiative Properties of Semitransparent Sheets 3.9 Special Surfaces 3.10 Earth’s Surface Properties and Climate Change 3.11 Experimental Methods Problems References

59 60 69 72 80 85 89 90 97 102 105 116 121

4. View Factors 4.1 Introduction 4.2 Definition of View Factors 4.3 Methods for the Evaluation of View Factors 4.4 Area Integration 4.5 Contour Integration 4.6 View Factor Algebra 4.7 The Crossed-Strings Method 4.8 The Inside Sphere Method 4.9 The Unit Sphere Method 4.10 View Factor Between Arbitrary Planar Polygons Problems References

127 128 131 132 135 140 143 148 150 151 154 158

5. Radiative Exchange Between Gray, Diffuse Surfaces 5.1 Introduction 5.2 Radiative Exchange Between Black Surfaces

161 161 ix

x Contents

5.3 Radiative Exchange Between Gray, Diffuse Surfaces (Net Radiation Method) 5.4 Electrical Network Analogy 5.5 Radiation Shields 5.6 Solution Methods for the Governing Integral Equations Problems References

166 174 177 179 188 196

6. Radiative Exchange Between Nondiffuse and Nongray Surfaces 6.1 Introduction 6.2 Enclosures with Partially Specular Surfaces 6.3 Radiative Exchange in the Presence of Partially Specular Surfaces 6.4 Semitransparent Sheets (Windows) 6.5 Radiative Exchange Between Nongray Surfaces 6.6 Directionally Nonideal Surfaces 6.7 Analysis for Arbitrary Surface Characteristics Problems References

199 199 204 211 214 219 226 227 233

235 239 240 242 243 249 251 256 258

8. Surface Radiative Exchange in the Presence of Conduction and Convection 8.1 Introduction 8.2 Challenges in Coupling Surface-to-Surface Radiation with Conduction/Convection 8.3 Coupling Procedures 8.4 Radiative Heat Transfer Coefficient 8.5 Conduction and Surface Radiation—Fins 8.6 Convection and Surface Radiation—Tube Flow Problems References

9.1 Introduction 9.2 Attenuation by Absorption and Scattering 9.3 Augmentation by Emission and Scattering 9.4 The Radiative Transfer Equation 9.5 Formal Solution to the Radiative Transfer Equation 9.6 Boundary Conditions for the Radiative Transfer Equation 9.7 RTE for a Medium with Graded Refractive Index 9.8 Radiation Energy Density 9.9 Radiative Heat Flux 9.10 Divergence of the Radiative Heat Flux 9.11 Integral Formulation of the Radiative Transfer Equation 9.12 Overall Energy Conservation 9.13 Solution Methods for the Radiative Transfer Equation Problems References

285 285 287 289 291 293 297 298 298 299 302 303 305 306 308

10.Radiative Properties of Molecular Gases

7. The Monte Carlo Method for Surface Exchange 7.1 Introduction 7.2 Numerical Quadrature by Monte Carlo 7.3 Heat Transfer Relations for Radiative Exchange Between Surfaces 7.4 Surface Description 7.5 Random Number Relations for Surface Exchange 7.6 Ray Tracing 7.7 Efficiency Considerations Problems References

9. The Radiative Transfer Equation in Participating Media (RTE)

261

261 264 272 273 276 279 281

10.1 10.2 10.3 10.4 10.5 10.6 10.7

Fundamental Principles Emission and Absorption Probabilities Atomic and Molecular Spectra Line Radiation Nonequilibrium Radiation High-Resolution Spectroscopic Databases Spectral Models for Radiative Transfer Calculations 10.8 Narrow Band Models 10.9 Narrow Band k-Distributions 10.10 Wide Band Models 10.11 Total Emissivity and Mean Absorption Coefficient 10.12 Gas Properties of Earth’s Atmosphere and Climate Change 10.13 Experimental Methods Problems References

311 312 315 322 330 331 333 335 344 356 369 377 381 386 391

11.Radiative Properties of Particulate Media 11.1 Introduction 11.2 Absorption and Scattering from a Single Sphere 11.3 Radiative Properties of a Particle Cloud

401 402 407

Contents

11.4 Radiative Properties of Small Spheres (Rayleigh Scattering) 11.5 Rayleigh–Gans Scattering 11.6 Anomalous Diffraction 11.7 Radiative Properties of Large Spheres 11.8 Absorption and Scattering by Long Cylinders 11.9 Approximate Scattering Phase Functions 11.10 Radiative Properties of Irregular Particles and Aggregates 11.11 Radiative Properties of Combustion Particles 11.12 Experimental Determination of Radiative Properties of Particles Problems References

411 413 414 414 420 422 425 426 438 443 445

12.Radiative Properties of Semitransparent Media 12.1 Introduction 12.2 Absorption by Semitransparent Solids 12.3 Absorption by Semitransparent Liquids 12.4 Radiative Properties of Porous Solids 12.5 Experimental Methods Problems References

453 453 455 457 460 463 463

13.Exact Solutions for One-Dimensional Gray Media 13.1 Introduction 13.2 General Formulation for a Plane-Parallel Medium 13.3 Plane Layer of a Nonscattering Medium 13.4 Plane Layer of a Scattering Medium 13.5 Plane Layer of a Graded Index Medium 13.6 Radiative Transfer in Spherical Media 13.7 Radiative Transfer in Cylindrical Media 13.8 Numerical Solution of the Governing Integral Equations Problems References

467 467 471 478 480 483 487 490 491 493

14.Approximate Solution Methods for One-Dimensional Media 14.1 The Optically Thin Approximation 14.2 The Optically Thick Approximation (Diffusion Approximation) 14.3 The Schuster–Schwarzschild Approximation 14.4 The Milne–Eddington Approximation (Moment Method) 14.5 The Exponential Kernel Approximation

497 498 503 505 507

Problems References

xi

509 510

15.The Method of Spherical Harmonics (PN -Approximation) 15.1 Introduction 15.2 General Formulation of the PN -Approximation 15.3 The PN -Approximation for a One-Dimensional Slab 15.4 Boundary Conditions for the PN -Method 15.5 The P1 -Approximation 15.6 P3 - and Higher-Order Approximations 15.7 Simplified PN -Approximation 15.8 Other Methods Based on the P1 -Approximation 15.9 Comparison of Methods Problems References

513 513 514 516 519 526 539 543 553 556 558

16.The Method of Discrete Ordinates (SN -Approximation) 16.1 16.2 16.3 16.4

Introduction General Relations The One-Dimensional Slab One-Dimensional Concentric Spheres and Cylinders 16.5 Multidimensional Problems 16.6 The Finite Angle Method (FAM) 16.7 The Modified Discrete Ordinates Method 16.8 Even-Parity Formulation 16.9 Other Related Methods 16.10 Concluding Remarks Problems References

563 563 566 571 576 593 602 603 604 606 606 608

17.The Zonal Method 17.1 Introduction 17.2 Surface Exchange — No Participating Medium 17.3 Radiative Exchange in Gray Absorbing/Emitting Media 17.4 Radiative Exchange in Gray Media with Isotropic Scattering 17.5 Radiative Exchange through a Nongray Medium 17.6 Accuracy and Efficiency Considerations Problems References

617 617 622 628 634 636 637 638

18.Collimated Irradiation and Transient Phenomena 18.1 Introduction

641

xii Contents

18.2 Reduction of the Problem 18.3 The Modified P1 -Approximation with Collimated Irradiation 18.4 Short-Pulsed Collimated Irradiation with Transient Effects Problems References

643 646 649 652 653

19.Solution Methods for Nongray Extinction Coefficients 19.1 19.2 19.3 19.4 19.5 19.6

Introduction The Mean Beam Length Method Semigray Approximations The Stepwise-Gray Model (Box Model) General Band Model Formulation The Weighted-Sum-of-Gray-Gases (WSGG) Model 19.7 The Spectral-Line-Based Weighted-Sum-of-Gray-Gases (SLW) Model 19.8 Outline of k-Distribution Models 19.9 The Narrow Band and Wide Band k-Distribution Methods 19.10 The Full Spectrum k-Distribution (FSK) Method for Homogeneous Media 19.11 The FSK and SLW Methods for Nonhomogeneous Media 19.12 Evaluation of k-Distributions and ALBDFs 19.13 Higher Order k-Distribution Methods Problems References

657 658 664 667 673 677

683 686 688 690 696 711 720 727 729

784 790 799 806 808

22.1 Introduction 22.2 Coupling Considerations 22.3 Combined Radiation and Laminar Combustion 22.4 Combined Radiation and Turbulent Combustion 22.5 Comparison of RTE Solvers for Reacting Systems 22.6 Radiation in Concentrating Solar Energy Systems References

819 819 821 824 838 844 848

23.Inverse Radiative Heat Transfer 23.1 Introduction 23.2 Solution Methods 23.3 Regularization 23.4 Gradient-Based Optimization 23.5 Metaheuristics 23.6 Summary of Inverse Radiation Research Problems References

859 859 865 867 874 877 879 881

24.Nanoscale Radiative Transfer 737 737 738 743 749 750 757 760 766 766 769 770

21.Radiation Combined with Conduction and Convection 21.1 Introduction

775

22.Radiation in Chemically Reacting Systems

20.The Monte Carlo Method for Participating Media 20.1 Introduction 20.2 Heat Transfer Relations for Participating Media 20.3 Random Number Relations for Participating Media 20.4 Treatment of Spectral Line Structure Effects 20.5 Overall Energy Conservation 20.6 Discrete Particle Fields 20.7 Backward Monte Carlo 20.8 Efficiency/Accuracy Considerations 20.9 Media with Variable Refractive Index 20.10 Example Problems Problems References

21.2 Combined Radiation and Conduction 21.3 Melting and Solidification with Internal Radiation 21.4 Combined Radiation and Convection 21.5 General Formulations for Coupling Problems References

24.1 Introduction 24.2 Coherence of Light 24.3 Evanescent Waves 24.4 Radiation Tunneling 24.5 Surface Waves (Polaritons) 24.6 Fluctuational Electrodynamics 24.7 Heat Transfer Between Parallel Plates 24.8 Experiments on Nanoscale Radiation 24.9 Applications Problems References

A. Constants and Conversion Factors B. Tables for Radiative Properties of Opaque Surfaces References

775

887 887 887 889 891 892 894 898 899 900 900

C. Blackbody Emissive Power Table

920

Contents

D. View Factor Catalogue References

F. Computer Codes 934

E. Exponential Integral Functions References

xiii

943

References Author Index Index

951 955 975

Preface to the Fourth Edition Ten more years have passed since the third edition of “Radiative Heat Transfer” came out. Thermal radiation remains a relatively young field, with basic relations dating back a bit more than 100 years, and serious heat transfer models only starting to appear in the 1950s. Since the last edition new seminal contributions have appeared in all areas of the science. However, the most important tools of greatest interest to the practicing engineer appear to be slowly reaching maturity, and many of the modern developments pertain to ancillary applications. Nevertheless, substantial new developments in the core sciences of radiative properties and solution methods make it important to give the book another significant upgrade, and to further improve its general readability and usefulness. Thirty years have gone by since the publication of the first edition and, alas, the primary author has become correspondingly older and has now retired from teaching and much of his research. We are fortunate that Professor Sandip Mazumder is bringing his energy to the book, with his extensive expertise in numerical analysis and combined-mode heat transfer nicely complementing the senior author’s background. The objectives of this book remain the same and are more extensive than to provide a standard textbook for a one-semester core course on thermal radiation, since it does not appear possible to cover all important topics in the field of radiative heat transfer in a single graduate course. A number of important areas that would not be part of a “standard” one-semester course have been treated in some detail. It is anticipated that the engineer who may have used this book as his or her graduate textbook will be able to master these advanced topics through self-study. By including all important advanced topics, as well as a large number of references for further reading, the book is also intended as a reference book for the practicing engineer. On the other hand, a few advanced topics are only presented at an introductory level; these include the section on solar thermal and thermochemical energy conversion, and chapters on inverse radiation and radiative transfer at the micro- and nanoscales. These should be understood as introductions to extensive new fields, giving the engineer a basic understanding of these research areas, and a good foundation to embark on further reading of the pertinent literature. The largest change in the fourth edition is a much greater emphasis on combined-mode heat transfer. As a result the chapter dealing with combined conduction and/or convection with surface radiation has been greatly expanded, and the chapter on combined modes with participating media has been split into two, devoting an entire chapter to radiation in the presence of chemical reactions. Radiative transfer in media with variable refractive index (“graded media”) has received increasing attention in recent years, and short discussions of the topic have been added in appropriate places. Also, with the impacts of climate change now felt on a daily basis, we have added material in several chapters that help elucidate the role of the radiative properties of Earth’s surface and its atmosphere on its energy balance. The appendix describing a number of computer programs has been retained, and the codes may be downloaded from a dedicated web site located at https://www.elsevier.com/books-and-journals/book-companion/ 9780323984065. Some of the codes are very basic and are entirely intended to aid the reader with the solution to the problems given at the end of the early chapters on surface transport. A few new codes are intended to aid engineers to deal with combined-modes heat transfer. Others were born out of research, some basic enough to aid a graduate student with more complicated assignments or a semester project, and a few so sophisticated in nature that they will be useful only to the practicing engineer conducting his or her own research. Recognizing that many graduate students no longer learn compiler languages, such as Fortran and C++, the more basic programs are now also available in Matlab . Many smaller changes have also been made, such as omission of some obsolete material, inclusion of many new small developments, and restructuring of material between chapters to aid readability. And, of course, a xv

xvi

Preface to the Fourth Edition

comprehensive literature update has been provided, and many new homework problems have been added at the end of the chapters. As in the previous three editions, each chapter shows the development of all analytical methods in substantial detail, and contains a number of examples to show how the developed relations may be applied to practical problems. At the end of each chapter a number of exercises are included to give the student additional opportunity to familiarize him- or herself with the application of analytical methods developed in the preceding sections. The breadth of the description of analytical developments is such that any scientist with a satisfactory background in calculus and differential equations will be able to grasp the subject through self-study—for example, the heat transfer engineer involved in furnace calculations, the architectural engineer interested in lighting calculations, the oceanographer concerned with solar penetration into the ocean, or the meteorologist who studies atmospheric radiation problems. An expanded Instructor’s Solutions Manual is available for adopting instructors who register at http://educate.elsevier.com/9780323984065. The book is again divided into 24 chapters, covering the four major areas in the field of radiative heat transfer. After the Introduction, there are two chapters dealing with theoretical and practical aspects of radiative properties of opaque surfaces, including a brief discussion of experimental methods. These are followed by four chapters dealing with purely radiative exchange between surfaces in an enclosure without a “radiatively participating” medium, and one more chapter examining the interaction of conduction and convection with surface radiation. The rest of the book deals with radiative transfer through absorbing, emitting, and scattering media (or “participating media”). After a detailed development of the radiative transfer equation, radiative properties of gases, particulates, and semitransparent media are discussed, again including brief descriptions of experimental methods. The next seven chapters cover the theory of radiative heat transfer through participating media, separated into a number of basic problem areas and solution methods. And, finally, the book ends with two chapters on combined-modes heat transfer as well as chapters introducing the emerging fields of inverse and nanoscale radiative heat transfer. We have attempted to keep the book in modular form as much as possible. Chapter 2 is a fairly detailed (albeit concise) treatment of electromagnetic wave theory, which can (and will) be skipped by most instructors for a first course in radiative heat transfer. The chapter on opaque surface properties is self-contained and is not required reading for the rest of the book. The four chapters on surface transport (Chapters 4 through 8) are also self-contained and not required for the study of radiation in participating media. Similarly, the treatment of participating medium properties is not a prerequisite to studying the solution methods. Along the same line, any of the different solution aspects and methods discussed in Chapters 13 through 20 may be studied in any sequence (although Chapter 20 requires knowledge of Chapter 7). Whether any of the last four chapters are covered or skipped will depend entirely on the instructor’s preferences or those of his or her students. We have not tried to mark those parts of the book that should be included in a one-semester course on thermal radiation, since we feel that different instructors will, and should, have different opinions on that matter. Indeed, the relative importance of different subjects may not only vary with different instructors, but also depend on student background, location, or the year of instruction. Our personal opinion is that a one-semester course should touch on all four major areas (surface properties, surface transport, properties of participating media, and transfer through participating media) in a balanced way. For the average US student who has had very little exposure to thermal radiation during his or her undergraduate heat transfer experience, we suggest that about half the course be devoted to Chapters 1, 3, 4, 5, plus parts of Chapters 6, 7, and/or 8, leaving out the more advanced features. While the Monte Carlo method of Chapter 7 may be considered an “advanced feature,” we have found it to be immensely popular with students, and at the same time gives exposure to an engineering tool of fast-growing importance. The second half of the course should be devoted to Chapters 9, 10, and 11 (again omitting less important features); some coverage of Chapter 13; and a thorough discussion of Chapter 14. If time permits (primarily, if surface and/or participating media properties are treated in less detail than indicated above), we suggest to cover the P1 -approximation (which may be studied by itself, as outlined in the beginning of Chapter 15), the basic ideas behind the discrete ordinates method, and/or a portion of Chapter 19 (solution methods for nongray media). The senior author would like to express his gratitude to a number of his former graduate students and postdocs, who contributed to Chapters 10 (Tao Ren, Shanghai and Somesh Roy, Milwaukee), 19 (Chaojun Wang, Beijing), 20 (S. Roy), 22 (S. Roy and Wenjun Ge, Oak Ridge), as well as 23 (T. Ren). Further thank yous are owed to

Preface to the Fourth Edition

xvii

Hadi Bordbar (Aalto, Finland) for aid with Fig. 19.6, Zhuomin Zhang (Atlanta) (Fig. 24.7), Pascal Boulet (Nancy, France) (Fig. 3.30), and Nehal Jajal (Columbus) (Fig. 16.10). Michael F. Modest Sandip Mazumder February 2021

List of Symbols The following is a list of symbols used frequently in this book. A number of symbols have been used for several different purposes. Alas, the Roman alphabet has only 26 lowercase and another 26 uppercase letters, and the Greek alphabet provides 34 more different ones, for a total of 86, which is, unfortunately, not nearly enough. Hopefully, the context will always make it clear which meaning of the symbols is to be used. We have used what we hope is a simple and uncluttered set of variable names. This usage, of course, comes at a price. For example, the subscript “λ” is often dropped (meaning “at a given wavelength,” or “per unit wavelength”), assuming that the reader recognizes the variable as a spectral quantity from the context. Whenever applicable, units have been attached to the variables in the following table. Variables without indicated units have multiple sets of units. For example, the units for total band absorptance A depend on the spectral variable used (λ, η, or ν), and on the absorption coefficient (linear, density- or pressure-based), for a total of nine different possibilities. a a a a ak an , bn A A∗ A, An A, Ap Am Ai j , Bi j b b B Bi Bo c, c0 c C1 , C2 , C3 C1 , C2 , C3 dnij , Dnij D D, D∗ Df ê E, Eb E E E(m) En f

semimajor axis of polarization ellipse, [N/C] plane-polarized component of electric field, [N/C] particle radius, [m] weight function for full-spectrum k-distribution methods, [−] weight factors for sum-of-gray-gases, [−] Mie scattering coefficients, [−] total band absorptance (or effective band width) nondimensional band absorptance = A/ω, [−] slab absorptivity (of n parallel sheets), [−] area, projected area, [m2 ] scattering phase function coefficients, [−] Einstein coefficients self-broadening coefficient, [−] semiminor axis of polarization ellipse, [N/C] rotational constant convection-to-conduction parameter (Biot number), [−] convection-to-radiation parameter (Boltzmann number), [−] speed of light, (in vacuum), [m/s] specific heat capacity, [J/kg K] constants for Planck function and Wien’s displacement law wide band parameters for outdated model Wigner-D functions, [−] diameter, [m] detectivity (normalized), [1/W] ([cm Hz1/2 /W]) mass fractal dimension, [−] unit vector into local coordinate direction, [−] emissive power, blackbody emissive power molecular energy level, [J] electric field vector, [N/C] refractive index function, [−] exponential integral of order n, [−] k-distribution, [cm] xix

xx List of Symbols

f fv , fs , fl f (nλT) F F Fi− j Fi−s j Fi→ j gk g g gi s j , gi gk gs, gg G Gi S j , Gi Gk G h h, hR H H H H H i î I I I Ib Il , Ilm I0 , I1  j ˆj J J k, kR k k k kf kˆ K K l, m, n L L L Le L0 , Lm L m m m˙

probability density function volume, solid, liquid fractions, [−] fractional blackbody emissive power, [−] objective function wide band k-distribution, [cm] (diffuse) view factor, [−] specular view factor, [−] radiation exchange factor, [−] degeneracy, [−] nondimensional incident radiation, [−] cumulative k-distribution, [−] direct exchange areas in zonal method, [cm2 ] direct exchange area matrix, [cm2 ] incident radiation = direction-integrated intensity total exchange areas in zonal method, [cm2 ] dyadic Green’s function Planck’s constant, = 6.6261 × 10−34 J s heat transfer coefficient, convective or radiative, [W/m2 K] irradiation onto a surface Heaviside’s unit step function, [−] nondimensional heat transfer coefficient, [−] nondimensional irradiation onto a surface, [−] magnetic field vector, [C/m s] nondimensional polarized intensity, [−] unit vector into the x-direction, [−] intensity of radiation first Stokes’ parameter for polarization, [N2 /C2 ] moment of inertia, [kg cm2 ] blackbody intensity (Planck function) position-dependent intensity functions modified Bessel functions, [−] imaginary part of complex number rotational quantum number, [−] unit vector into the y-direction, [−] radiosity, [W/m2 ] nondimensional radiosity, [−] conductivity, thermal or radiative, [W/m K] Boltzmann’s constant, = 1.3807 × 10−23 J/K absorptive index in complex index of refraction, [−] absorption coefficient variable, [cm−1 ] fractal prefactor, [−] unit vector into the z-direction, [−] kernel function luminous efficacy, [lm/W] direction cosines with x-, y-, z-axis, [−] length, [m] latent heat of fusion, [J/kg] luminance mean beam length, [m] geometric, or average mean beam length, [m] Laplace transform, or differential operator mass, [kg] complex index of refraction, [−] mass flow rate, [kg/s]

List of Symbols xxi

M n n n nˆ N, Nc NT Nu O{} p p P Pl , Plm Pr q, q qR qlum qsol Q Q Q˙  r r r R Ru R R R, Rn  Re s sˆ si s j , si gk ss, sg S S S S S St Ste Si Sj , Si Gk SS, SG t t t ˆt T T, Tn u u u u

molecular weight, [kg/kmol] self-broadening exponent, [−] refractive index, [−] number distribution function for particles, [cm−4 ] unit surface normal (pointing away from surface into the medium), [−] conduction-to-radiation parameter (for a fin), [−] number of particles per unit volume, [m−3 ] Nusselt number, [−] order of magnitude, [−] pressure, [bar]; radiation pressure, [N/m2 ] parameter vector probability function, [−] (associated) Legendre polynomials, [−] Prandtl number, [−] heat flux, heat flux vector, [W/m2 ] radiative flux, [W/m2 ] luminous flux, [lm/m2 = lx] solar constant, = 1366 W/m2 heat rate, [W] second Stokes’ parameter for polarization, [N2 /C2 ] heat production per unit volume, [W/m3 ] radial coordinate, [m] reflection coefficient, [−] position vector, [m] radius, [m] universal gas constant, = 8.3145 J/mol K random number, [−] radiative resistance, [cm−2 ] slab reflectivity (of n parallel sheets), [−] real part of complex number Reynolds number, [−] geometric path length, [m] unit vector into a given direction, [−] direct exchange areas in zonal method, [cm2 ] direct exchange area matrix, [cm2 ] distance between two zones, or between points on enclosure surface, [m] line-integrated absorption coefficient = line strength radiative source function Poynting vector, [W/m2 ] solid angle vector, [sr] Stanton number, [−] Stefan number, [−] total exchange areas in zonal method, [cm2 ] total exchange area matrix, [cm2 ] time, [s] transmission coefficient, [−] fin thickness, [m] unit vector in tangential direction, [−] temperature, [K] slab transmissivity (of n parallel sheets), [−] internal energy, [J/kg] radiation energy density velocity, [m/s] scaling function for absorption coefficient, [−]

xxii

List of Symbols

uk U v v v V V w wi W W x, y, z x x x X X X Y Ylm z α α α α α, β, γ β β β∗ γ γ γ γ γE δ δ δ δi j δk Δnij



ε η η η ηlum θ θ Θ Θ κ λ λm

nondimensional transition wavenumber, [−] third Stokes’ parameter for polarization, [N2 /C2 ] vibrational quantum number, [−] velocity, [m/s] velocity vector, [m/s] volume, [m3 ] fourth Stokes’ parameter for polarization, [N2 /C2 ] wave vector, [cm−1 ] quadrature weights, [−] equivalent line width weighting matrix, [−] Cartesian coordinates, [m] particle size parameter, [−] line strength parameter, [−] mole fraction, [−] optical path length interface location, [m] sensitivity matrix mass fraction, [−] spherical harmonics, [−] nondimensional spectral variable, [−] absorptance or absorptivity, [−] band-integrated absorption coefficient = band strength parameter opening angle, [rad] thermal diffusivity, [m2 /s] Euler rotation angles, [−] extinction coefficient line overlap parameter, [−] line overlap parameter for dilute gas, [−] complex permittivity, [C2 /N m2 ] azimuthal rotation angle for polarization ellipse, [rad] oscillation damping factor, [Hz] line half-width Euler’s constant, = 0.57221. . . line spacing Dirac-delta function, [−] polarization phase angle, [rad] Kronecker’s delta, [−] vibrational transition quantum step = Δv, [−] rotation matrix, [−] emittance or emissivity, [−] electrical permittivity, [C2 /N m2 ] complex dielectric function, or relative permittivity, = ε − iε , [−] wavenumber, [cm−1 ] direction cosine, [−] nondimensional (similarity) coordinate, [−] luminous efficiency, [−] polar angle, [rad] nondimensional temperature, [−] scattering angle, [rad] Planck oscillator, [J] absorption coefficient wavelength, [μm] overlap parameter, [cm−1 ]

List of Symbols xxiii

λ μ μ μ ν ν ξ ξ ρ ρ ρf σ σs σe , σdc σh σl τ τ φ φ φ Φ Φ Φ Φ χ χ ψ ψ Ψ Ψ ω ω ω Ω Subscripts 0 1, 2 ∞ a av b B c C D e E f g h i

regularization parameter, [−] dynamic viscosity, [kg/m s] magnetic permeability, [N s2 /C2 ] direction cosine (of polar angle), cos θ, [−] frequency, [Hz] kinematic viscosity, [m2 /s] direction cosine, [−] nondimensional coordinate, [−] reflectance or reflectivity, [−] density, [kg/m3 ] charge density, [C/m3 ] Stefan–Boltzmann constant, = 5.670 × 10−8 W/m2 K4 scattering coefficient electrical conductivity, dc-value, [C2 /N m2 s = 1/Ω m] root-mean-square roughness, [cm] correlation length, [cm] transmittance or transmissivity, [−] optical coordinate, optical thickness, [−] phase angle, [rad] normalized line shape function composition variable vector (T, p, x) scattering phase function, [sr−1 ] nondimensional medium emissive power function temperature function for line overlap β, [−] dissipation function, [J/kg m2 ] line shape correction factor Eddington factor azimuthal angle, [rad] stream function, [m2 /s] temperature function for band strength α, [−] nondimensional heat flux single scattering albedo, [−] angular frequency, [rad/s] relaxation parameter, [−] solid angle, [sr]

reference value, or in vacuum, or at length = 0 in medium, or at location, “1” or “2” far from surface absorbing, or apparent average blackbody value band integrated value at band center, or at cylinder, or critical value, or denoting a complex quantity, or cold collision Doppler, or based on diameter effective value, or at equilibrium, or emission point Earth fluid gas, or at a given cumulative k-distribution value hot incoming, or dummy counter

xxiv List of Symbols

j k L m n o p p P r ref R s S sol t u v w W x, y, z, r θ, ψ η λ ν  ⊥

at a rotational state, or dummy counter at a given value of the absorption coefficient variable at length = L modified Planck value, or medium value, or mean (bulk) value in normal direction outgoing, or from outside related to pressure, or polarizing value plasma Planck mean reflected component reference value Rosseland-mean, or radiation, or at r = R along path s, or at surface, or at sphere, or at source, or solid Stark solar transmitted component upper limit at a vibrational state, or at constant volume wall value value integrated over spectral windows in a given direction in a given direction at a given wavenumber, or per unit wavenumber at a given wavelength, or per unit wavelength at a given frequency, or per unit frequency polarization component, or situated in plane of incidence polarization component, or situated in plane perpendicular to plane of incidence

Superscripts  real and imaginary parts of complex number, or directional values, or dummy variables  hemispherical value ∗ complex conjugate, or obtained by P1 -approximation, or from previous iteration +, − into “positive” and “negative” directions d diffuse s specular ¯ average value ˜ complex number, or scaled value (for nonisothermal path), or Favre average ˆ unit vector 

Chapter 1

Fundamentals of Thermal Radiation 1.1 Introduction The terms radiative heat transfer and thermal radiation are commonly used to describe the science of the heat transfer caused by electromagnetic waves. Obvious everyday examples of thermal radiation include the heating effect of sunshine on a clear day, the fact that—when one is standing in front of a fire—the side of the body facing the fire feels much hotter than the back, and so on. More subtle examples of thermal radiation are that the clear sky is blue, that sunsets are red, and that, during a clear winter night, we feel more comfortable in a room whose curtains are drawn than in a room (heated to the same temperature) with open curtains. All materials continuously emit and absorb electromagnetic waves, or photons, by lowering or raising their molecular energy levels. The strength and wavelengths of emission depend on the temperature of the emitting material. As we shall see, for heat transfer applications wavelengths between 10−7 m and 10−3 m (ultraviolet, visible, and infrared) are of greatest importance and are, therefore, the only ones considered here. Before embarking on the analysis of thermal radiation we want briefly to compare the nature of this mode of heat transfer with the other two possible mechanisms of transferring energy, conduction and convection. In the case of conduction in a solid, energy is carried through the atomic lattice by free electrons or by phonon–phonon interactions (i.e., excitation of vibrational energy levels for interatomic bonds). In gases and liquids, energy is transferred from molecule to molecule through collisions (i.e., the faster molecule loses some of its kinetic energy to the slower one). Heat transfer by convection is similar, but many of the molecules with raised kinetic energy are carried away by the flow and are replaced by colder fluid (low-kinetic-energy molecules), resulting in increased energy transfer rates. Thus, both conduction and convection require the presence of a medium for the transfer of energy. Thermal radiation, on the other hand, is transferred by electromagnetic waves, or photons, which may travel over a long distance without interacting with a medium. The fact that thermal radiation does not require a medium for its transfer makes it of great importance in vacuum and space applications. This so-called “action at a distance” also manifests itself in a number of everyday thermodynamic applications. For example, on a cold winter day in a heated room we feel more comfortable when the curtains are closed: our bodies exchange heat by convection with the warm air surrounding us, but also by radiation with walls (including cold window panes if they are without curtains); we feel the heat from a fire a distance away from us, and so on. Another distinguishing feature between conduction and convection on the one hand and thermal radiation on the other is the difference in their temperature dependencies. For the vast majority of conduction applications heat transfer rates are well described by Fourier’s law as qx = −k

∂T , ∂x

(1.1)

where qx is conducted heat flux1 in the x-direction, T is temperature, and k is the thermal conductivity of the medium. Similarly, convective heat flux may usually be calculated from a correlation such as q = h(T − T∞ ),

(1.2)

where h is known as the convective heat transfer coefficient and T∞ is a reference temperature. While k and h may depend on temperature, this dependence is usually not very strong. Thus, for most applications, conductive and convective heat transfer rates are linearly proportional to temperature differences. As we shall see, radiative heat transfer rates are generally proportional to differences in temperature to the fourth (or higher) power, i.e., 4 q ∝ T 4 − T∞ .

(1.3)

1. In this book we shall use the term heat flux to denote the flow of energy per unit time and per unit area and the term heat rate for the flow of energy per unit time (i.e., not per unit area). Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00009-2 Copyright © 2022 Elsevier Inc. All rights reserved.

1

2 Radiative Heat Transfer

Therefore, radiative heat transfer becomes more important with rising temperature levels and may be totally dominant over conduction and convection at very high temperatures. Thus, thermal radiation is important in combustion applications (fires, furnaces, rocket nozzles, engines, etc.), in nuclear reactions (such as in the sun, in a fusion reactor, or in nuclear bombs), during atmospheric reentry of space vehicles, etc. As modern technology strives for higher efficiencies, this will require higher and higher temperatures, making thermal radiation ever more important. Other applications that are increasing in importance include solar energy collection and the greenhouse effect (both due to emission from our high-temperature sun). And, finally, one of the most pressing issues for mankind today are the effects of global warming, caused by the absorption of solar energy by man-made carbon dioxide released into the Earth’s atmosphere. The same reasons that make thermal radiation important in vacuum and high-temperature applications also make its analysis more difficult, or at least quite different from “conventional” analyses. Under normal conditions, conduction and convection are short-range local phenomena: The average distance between molecular collisions (mean free path for collision) is generally very small, e.g., around 70 nm for air at standard temperature and pressure. If it takes, say, 10 collisions until a high-kinetic-energy molecule has a kinetic energy similar to that of the surrounding molecules, then any external influence is not directly felt over a distance larger than 10−6 m. Thus we are able to perform an energy balance on an “infinitesimal volume,” i.e., a volume negligibly small in comparison with overall dimensions, but very large in comparison with the mean free path for collision. The principle of conservation of energy then leads to a partial differential equation to describe the temperature field and heat fluxes for both conduction and convection. This equation may have up to four independent variables (three space coordinates and time) and is linear in temperature for the case of constant properties. Thermal radiation, on the other hand, is generally a long-range phenomenon. The mean free path for a photon (i.e., the average distance a photon travels before interacting with a molecule) may be as short as 10−10 m (e.g., absorption in a metal), but can also be as long as 10+10 m or larger (e.g., the sun’s rays hitting Earth). Thus, conservation of energy cannot be applied over an infinitesimal volume, but must be applied over the entire volume under consideration. This leads to an integral equation in up to seven independent variables (the frequency of radiation, three space coordinates, two coordinates describing the direction of travel of photons, and time). The analysis of thermal radiation is further complicated by the behavior of the radiative properties of materials. Properties relevant to conduction and convection (thermal conductivity, kinematic viscosity, density, etc.) are fairly easily measured and are generally well behaved (isotropic throughout the medium, perhaps with relatively weak temperature dependence). Radiative properties are usually difficult to measure and often display erratic behavior. For liquids and solids the properties normally depend only on a very thin surface layer, which may vary strongly with surface preparation and often even from day to day. All radiative properties (in particular for gases) may vary strongly with wavelength, adding another dimension to the governing equation. Rarely, if ever, may this equation be assumed to be linear. Because of these difficulties inherent in the analysis of thermal radiation, a good portion of this book has been set aside to discuss radiative properties and different approximate methods to solve the governing energy equation for radiative transport.

1.2 The Nature of Thermal Radiation Thermal radiative energy may be viewed as consisting of electromagnetic waves (as predicted by electromagnetic wave theory) or as consisting of massless energy parcels, called photons (as predicted by quantum mechanics). Neither point of view is able to describe completely all radiative phenomena that have been observed. It is, therefore, customary to use both concepts interchangeably. In general, radiative properties of liquids and solids (including tiny particles), and of interfaces (surfaces) are more easily predicted using electromagnetic wave theory, while radiative properties of gases are more conveniently obtained from quantum mechanics. All electromagnetic waves, or photons, are known to propagate through any medium at a high velocity. Since light is a part of the electromagnetic wave spectrum, this velocity is known as the speed of light, c. The speed of light depends on the medium through which it travels, and may be related to the speed of light in vacuum, c 0 , by the formula c0 c= , c 0 = 2.998 × 108 m/s, (1.4) n where n is known as the refractive index of the medium. By definition, the refractive index of vacuum is n ≡ 1. For most gases the refractive index is very close to unity, for example, air at room temperature has n = 1.00029 over the

Fundamentals of Thermal Radiation Chapter | 1 3

visible spectrum. Therefore, light propagates through gases nearly as fast as through vacuum. Electromagnetic waves travel considerably slower through dielectrics (electric nonconductors), which have refractive indices between approximately 1.4 and 4, and they hardly penetrate at all into electrical conductors (metals). Each wave may be identified either by its frequency, ν

(measured in cycles/s = s−1 = Hz);

wavelength, λ

(measured in μm = 10−6 m or nm = 10−9 m);

wavenumber, η

(measured in cm−1 ); or

angular frequency, ω

(measured in radians/s = s−1 ).

All four quantities are related to one another through the formulae ν=

ω c = = cη. 2π λ

(1.5)

Each wave or photon carries with it an amount of energy, , determined from quantum mechanics as

= hν,

h = 6.626 × 10−34 J s,

(1.6)

where h is known as Planck’s constant. The frequency of light does not change when light penetrates from one medium to another since the energy of the photon must be conserved. On the other hand, wavelength and wavenumber do, depending on the values of the refractive index for the two media. Sometimes electromagnetic waves are characterized in terms of the energy that a photon carries, hν, using the energy unit electron volt (1 eV = 1.6022 × 10−19 J). Thus, light with a photon energy (or “frequency”) of a eV has a wavelength (in vacuum) of λ=

6.626 × 10−34 J s × 2.998 × 108 m/s 1.240 hc = μm. = hν a a 1.6022 × 10−19 J

(1.7)

Since electromagnetic waves of vastly different wavelengths carry vastly different amounts of energy, their behavior is often quite different. Depending on their behavior or occurrence, electromagnetic waves have been grouped into a number of different categories, as shown in Fig. 1.1. Thermal radiation may be defined to be those

FIGURE 1.1 Electromagnetic wave spectrum (for radiation traveling through vacuum, n = 1).

4 Radiative Heat Transfer

FIGURE 1.2 Kirchhoff’s law.

electromagnetic waves which are emitted by a medium due solely to its temperature [1]. As indicated earlier, this definition limits the range of wavelengths of importance for heat transfer considerations to between 0.1 μm (ultraviolet) and 20 μm (midinfrared).

1.3 Basic Laws of Thermal Radiation When an electromagnetic wave traveling through a medium (or vacuum) strikes the surface of another medium (solid or liquid surface, particle or bubble), the wave may be reflected (either partially or totally), and any nonreflected part will penetrate into the medium. While passing through the medium the wave may become continuously attenuated. If attenuation is complete so that no penetrating radiation reemerges, it is known as opaque. If a wave passes through a medium without any attenuation, it is termed transparent, while a body with partial attenuation is called semitransparent.2 Whether a medium is transparent, semitransparent, or opaque depends on the material as well as on its thickness (i.e., the distance the electromagnetic wave must travel through the medium). Metals are nearly always opaque, although it is a common high school physics experiment to show that light can penetrate through extremely thin layers of gold. Nonmetals generally require much larger thicknesses before they become opaque, and some are quite transparent over part of the spectrum (for example, window glass in the visible part of the spectrum). An opaque surface that does not reflect any radiation is called a perfect absorber or a black surface: When we “see” an object, our eyes absorb electromagnetic waves from the visible part of the spectrum, which have been emitted by the sun (or artificial light) and have been reflected by the object toward our eyes. We cannot see a surface that does not reflect radiation, and it appears “black” to our eyes.3 Since black surfaces absorb the maximum possible amount of radiative energy, they serve as a standard for the classification of all other surfaces. It is easy to show that a black surface also emits a maximum amount of radiative energy, i.e., more than any other body at the same temperature. To show this, we use one of the many variations of Kirchhoff’s law:∗ Consider two identical black-walled enclosures, thermally insulated on the outside, with each containing a small object—one black and the other one not—as shown in Fig. 1.2. After a long time, in accordance with the Second Law of Thermodynamics, both entire enclosures and the objects within them will be at a single uniform temperature. This characteristic implies that every part of the surface (of the enclosure as well as the objects) emits precisely as much energy as it absorbs. Both objects in the different enclosures receive exactly the same amount of radiative energy. But since the black object absorbs more energy (i.e., the maximum possible), it must also emit more energy than the nonblack object (i.e., also the maximum possible). By the same reasoning it is easy to show that a black surface is a perfect absorber and emitter at every wavelength and for any direction (of incoming or outgoing electromagnetic waves), and that the radiation field 2. A medium that allows a fraction of light to pass through, while scattering the transmitted light into many different directions, for example, milky glass, is called translucent. 3. Note that a surface appearing black to our eyes is by no means a perfect absorber at nonvisible wavelengths and vice versa; indeed, many white paints are actually quite “black” at longer wavelengths. ∗

Gustav Robert Kirchhoff (1824–1887) German physicist. After studying in Berlin, Kirchhoff served as professor of physics at the University of Heidelberg for 21 years before returning to Berlin as professor of mathematical physics. Together with the chemist Robert Bunsen, he was the first to establish the theory of spectrum analysis.

Fundamentals of Thermal Radiation Chapter | 1 5

within an isothermal black enclosure is isotropic (i.e., the radiative energy density is the same at any point and in any direction within the enclosure).

1.4 Emissive Power Every medium continuously emits electromagnetic radiation randomly into all directions at a rate depending on the local temperature and on the properties of the material. This is sometimes referred to as Prévost’s law (after Pierre Prévost, an early 19th century Swiss philosopher and physicist). The radiative heat flux emitted from a surface is called the emissive power, E. We distinguish between total and spectral emissive power (i.e., heat flux emitted over the entire spectrum, or at a given frequency per unit frequency interval), so that spectral emissive power, Eν ≡ emitted energy/time/surface area/frequency, total emissive power, E ≡ emitted energy/time/surface area. Here and elsewhere we use the subscripts ν, λ, or η (depending on the choice of spectral variable) to express a spectral quantity whenever necessary for clarification. Thermal radiation of a single frequency or wavelength is sometimes also called monochromatic radiation (since, over the visible range, the human eye perceives electromagnetic waves to have the colors of the rainbow). It is clear from their definitions that the total and spectral emissive powers are related by  E(T) =

∞

Eν (T, ν) dν.

(1.8)

0

Blackbody Emissive Power Spectrum Scientists had tried for many years to theoretically predict the sun’s emission spectrum, which we know today to behave very nearly like a blackbody at approximately 5777 K [2]. The spectral solar flux falling onto Earth, or solar irradiation, is shown in Fig. 1.3 for extraterrestrial conditions (as measured by high-flying balloons and satellites) and for unity air mass (air mass is defined as the value of 1/ cos θS , where the zenith angle θS is the angle between the local vertical and a vector pointing toward the sun) [3,4]. Solar radiation is attenuated significantly as it penetrates through the atmosphere by phenomena that will be discussed in Sections 1.12 and 1.14, and again

FIGURE 1.3 Solar irradiation onto Earth.

6 Radiative Heat Transfer

in Section 10.12. Lord Rayleigh (1900) [5]† and Sir James Jeans (1905) [6]‡ independently applied the principles of classical statistics with its equipartition of energy to predict the spectrum of the sun, with dismal results. Wilhelm Wien (1896) [7]§ used some thermodynamic arguments together with experimental data to propose a spectral distribution of blackbody emissive power that was very accurate over large parts of the spectrum. Finally, in 1901 Max Planck [8]¶ published his work on quantum statistics: Assuming that a molecule can emit photons only at distinct energy levels, he found the spectral blackbody emissive power distribution, now commonly known as Planck’s law, for a black surface bounded by a transparent medium with refractive index n, as Ebν (T, ν) =

c20

2πhν3 n2 ,  hν/kT −1 e

(1.9)

where k = 1.3807 × 10−23 J/K is known as Boltzmann’s constant.4 While frequency ν appears to be the most logical spectral variable (since it does not change when light travels from one medium into another), the spectral variables wavelength λ (primarily for surface emission and absorption) and wavenumber η (primarily for radiation in gases) are also frequently (if not more often) employed. Equation (1.9) may be readily expressed in terms of wavelength and wavenumber through the relationships     η dn c0 c0 λ dn c0 c0 = η, dν = − 2 1 + dλ = 1− dη, (1.10) ν= nλ n n dλ n n dη nλ and



∞

Eb (T) = 0

 Ebν dν =

∞

 Ebλ dλ =

0

∞

Ebη dη,

(1.11)

0

or Ebν dν = −Ebλ dλ = Ebη dη.

(1.12)

Here λ and η are wavelength and wavenumber for the electromagnetic waves within the medium of refractive index n (while λ0 = nλ and η0 = η/n would be wavelength and wavenumber of the same wave traveling through vacuum). Equation (1.10) shows that equation (1.9) gives convenient relations for Ebλ and Ebη only if the refractive index is independent of frequency (or wavelength, or wavenumber). This is certainly the case for †

John William Strutt, Lord Rayleigh (1842–1919) English physical scientist. Rayleigh obtained a mathematics degree from Cambridge, where he later served as professor of experimental physics for five years. He then became secretary, and later president, of the Royal Society. His work resulted in a number of discoveries in the fields of acoustics and optics, and he was the first to explain the blue color of the sky (cf. the Rayleigh scattering laws in Chapter 11). Rayleigh received the 1904 Nobel Prize in Physics for the isolation of argon.

‡

Sir James Hopwood Jeans (1877–1946) English physicist and mathematician, whose work was primarily in the area of astrophysics. He applied mathematics to several problems in thermodynamics and electromagnetic radiation.

§

Wilhelm Wien (1864–1928) German physicist, who served as professor of physics at the University of Giessen and later at the University of Munich. Besides his research in the area of electromagnetic waves, his interests included other rays, such as electron beams, X-rays, and α-particles. For the discovery of his displacement law he was awarded the Nobel Prize in Physics in 1911.

¶

Max Planck (1858–1947) German physicist. Planck studied in Berlin with H. L. F. von Helmholtz and G. R. Kirchhoff, but obtained his doctorate at the University of Munich before returning to Berlin as professor in theoretical physics. He later became head of the Kaiser Wilhelm Society (today the Max Planck Institute). For his development of the quantum theory he was awarded the Nobel Prize in Physics in 1918. An extensive account of Planck’s professional life and his part of the discovery of quantum theory can be found in [9].

4. Equation (1.9) is valid for emission into a medium whose absorptive index (to be introduced in Chapter 2) is much less than the refractive index. This includes semitransparent media such as water, glass, quartz, etc., but not opaque materials. Emission into such bodies is immediately absorbed and is of no interest.

Fundamentals of Thermal Radiation Chapter | 1 7

FIGURE 1.4 Blackbody emissive power spectrum.

vacuum (n = 1) and ordinary gases (n 1), and may be of acceptable accuracy for some semitransparent media over large parts of the spectrum (for example, for quartz 1.52 < n < 1.68 between the wavelengths of 0.2 and 2.4 μm). Thus, with the assumption of constant refractive index, 2πhc20 ,  n2 λ5 ehc 0 /nλkT − 1 2πhc20 η3 Ebη (T, η) = 2  hc η/nkT , n e 0 −1

Ebλ (T, λ) =

(n = const),

(1.13)

(n = const).

(1.14)

Figure 1.4 is a graphical representation of equation (1.13) for a number of blackbody temperatures. As one can see, the overall level of emission rises with rising temperature (as dictated by the Second Law of Thermodynamics), while the wavelength of maximum emission shifts toward shorter wavelengths. The blackbody emissive power is also plotted in Fig. 1.3 for an effective solar temperature of 5777 K. This plot is in good agreement with extraterrestrial solar irradiation data. It is customary to introduce the abbreviations C1 = 2πhc20 = 3.7418 × 10−16 W m2 , C2 = hc 0 /k = 14,388 μm K = 1.4388 cm K, so that equation (1.13) may be recast as Ebλ C1 = , n3 T 5 (nλT) 5 [eC2 /(nλT) − 1]

(n = const),

(1.15)

which is seen to be a function of (nλT) only. Thus, it is possible to plot this normalized emissive power as a single line vs. the product of wavelength in vacuum (nλ) and temperature (T), as shown in Fig. 1.5, and a detailed tabulation is given in Appendix C. The maximum of this curve may be determined by differentiating equation (1.15),   d Ebλ = 0, d(nλT) n3 T 5 leading to a transcendental equation that may be solved numerically as (nλT) max = C3 = 2898 μm K.

(1.16)

8 Radiative Heat Transfer

FIGURE 1.5 Normalized blackbody emissive power spectrum.

Equation (1.16) is known as Wien’s displacement law since it was developed independently by Wilhelm Wien [10] in 1891 (i.e., well before the publication of Planck’s emissive power law). It is important to recognize that the location of maximum emissive power depends on the chosen spectral variable; for example, for Ebη the maximum is at (η/nT) max = 1.9610 cm−1 /K,

(1.17)

which corresponds to a wavelength roughly 60% longer than the one given by equation (1.16). A good discussion of these maxima has been given by Stewart [11]. Example 1.1. At what wavelength has the sun its maximum emissive power? At what wavelength Earth? Solution From equation (1.16), with the sun’s surface at Tsun 5777 K and bounded by vacuum (n = 1), it follows that λmax,sun =

2898 μm K C3 = 0.50 μm, = Tsun 5777 K

which is near the center of the visible region. Apparently, evolution has caused our eyes to be most sensitive in that section of the electromagnetic spectrum where the maximum daylight is available. In contrast, Earth’s average surface temperature may be in the vicinity of TEarth = 290 K, or λmax,Earth

2898 μm K = 10 μm, 290 K

that is, Earth’s maximum emission occurs in the midinfrared, leading to infrared cameras and detectors for night “vision.”

It is of interest to look at the asymptotic behavior of Planck’s law for small and large wavelengths. For very small values of hc 0 /nλkT (large wavelength, or small frequency), the exponent in equation (1.13) may be approximated by a two-term Taylor series, leading to Ebλ =

2πc 0 kT , nλ4

hc 0

1. nλkT

(1.18)

The same result is obtained if one lets h → 0, i.e., if one allows photons of arbitrarily small energy content to be emitted, as postulated by classical statistics. Thus, equation (1.18) is identical to the one derived by Rayleigh and Jeans and bears their names. The Rayleigh–Jeans distribution is also included in Fig. 1.5. Obviously, this formula is accurate only for very large values of (nλT), where the energy of the emissive power spectrum is negligible. Thus, this formula is of little significance for engineering purposes.

Fundamentals of Thermal Radiation Chapter | 1 9

For large values of (hc 0 /nλkT), the −1 in the denominator of equation (1.13) may be neglected, leading to Wien’s distribution (or Wien’s law), Ebλ

2πhc20 n2 λ 5

e−hc 0 /nλkT =

C1 −C2 /nλT e , n2 λ 5

hc 0  1, nλkT

(1.19)

since it is identical to the formula first proposed by Wien, before the advent of quantum mechanics. Examination of Wien’s distribution in Fig. 1.5 shows that it is very accurate over most of the spectrum, with a total energy content of the entire spectrum approximately 8% lower than for Planck’s law. Thus, Wien’s distribution is frequently utilized in theoretical analyses in order to facilitate integration.

Total Blackbody Emissive Power The total emissive power of a blackbody may be determined from equations (1.11) and (1.13) as  ∞  ∞ d(nλT) 2 4 Eb (T) = Ebλ (T, λ) dλ = C1 n T   5 (nλT) eC2 /(nλT) − 1 0 ⎡0  ∞ ⎤ ⎢⎢ C1 ξ3 dξ ⎥⎥⎥ 2 4 = ⎢⎢⎣ 4 ⎥ n T , (n = const). C2 0 eξ − 1 ⎦

(1.20)

The integral in this expression may be evaluated by complex integration, and is tabulated in many good integral tables: Eb (T) = n2 σT 4 ,

σ=

W π4 C1 = 5.670 × 10−8 , 15C42 m2 K4

(1.21)

where σ is known as the Stefan–Boltzmann constant. If Wien’s distribution is to be used then the −1 is absent from the denominator of equation (1.20), and a corrected Stefan–Boltzmann constant should be employed, evaluated as W 6C1 σW = 4 = 5.239 × 10−8 , (1.22) C2 m2 K 4 indicating that Wien’s distribution underpredicts total emissive power by about 7.5%. Historically, the “T 4 radiation law,” equation (1.21), predates Planck’s law and was found through thermodynamic arguments. A short history may be found in [12]. It is often necessary to calculate the emissive power contained within a finite wavelength band, say between λ1 and λ2 . Then   λ2 C1 C2 /nλ1 T ξ3 dξ 2 4 nT . Ebλ dλ = 4 (1.23) C2 C2 /nλ2 T eξ − 1 λ1 It is not possible to evaluate the integral in equation (1.23) in simple analytical form. Therefore, it is customary to express equation (1.23) in terms of the fraction of blackbody emissive power contained between 0 and nλT, λ f (nλT) =

 0∞ 0



Ebλ dλ Ebλ dλ



nλT

= 0



  ξ3 dξ Ebλ 15 ∞ , d(nλT) = n3 σT 5 π4 C2 /nλT eξ − 1

(1.24)

Josef Stefan (1835–1893) Austrian physicist. Serving as professor at the University of Vienna, Stefan determined in 1879 that, based on his experiments, blackbody emission was proportional to temperature to the fourth power. Ludwig Erhard Boltzmann (1844–1906) Austrian physicist. After receiving his doctorate from the University of Vienna he held professorships in Vienna, Graz (both in Austria), Munich, and Leipzig (in Germany). His greatest contributions were in the field of statistical mechanics (Boltzmann statistics). He derived the fourth-power law from thermodynamic considerations in 1889.

10 Radiative Heat Transfer

so that



λ2 λ1

  Ebλ dλ = f (nλ2 T) − f (nλ1 T) n2 σT 4 .

(1.25)

Equation (1.24) can be integrated only after expanding the denominator into an infinite series, resulting in f (nλT) =

∞  15  e−mζ  2 3 (mζ) (mζ) (mζ) 6 + 6 + 3 , + π4 m=1 m4

ζ=

C2 . nλT

(1.26)

The fractional emissive power is a function of a single variable, nλT, and is therefore easily tabulated, as has been done in Appendix C. For computer calculations a little Fortran routine of equation (1.26), bbfn, is given in Appendix F, as well as a stand-alone program, planck, which, after inputting wavelength (or wavenumber) and temperature, returns Ebλ , Ebη , and f . Example 1.2. What fraction of total solar emission falls into the visible spectrum (0.4 to 0.7 μm)? Solution With n = 1 and a solar temperature of 5777 K it follows that for λ1 = 0.4 μm, nλ1 Tsun = 1 × 0.4 × 5777 = 2310.8 μm K; and for λ2 = 0.7 μm, nλ2 Tsun = 4043.9 μm K. From Appendix C we find f (nλ1 Tsun ) = 0.12220 and f (nλ2 Tsun ) = 0.48869. Thus, from equations (1.21) and (1.25) the visible fraction of sunlight is f (nλ2 Tsun ) − f (nλ1 Tsun ) = 0.48869 − 0.12220 = 0.36649. (Writing a one-line program bbfn(4043.9)-bbfn(2310.8) returns the slightly more accurate value of 0.36661.) Therefore, with a bandwidth of only 0.3 μm the human eye responds to approximately 37% of all emitted sunlight!

We could repeat this example using wavenumber. Employing the data from Appendix C with f (η/nT) = 1 − f (nλT) leads to the identical result: unlike the peak value of spectral emissive power, fractions of emissive power between two spectral locations are unaffected by the choice of spectral variable.

1.5 Solid Angles When radiative energy leaves one medium and enters another (i.e., emission from a surface into another medium), this energy flux usually has different strengths in different directions. Similarly, the electromagnetic wave, or photon, flux passing through any point inside any medium may vary with direction. It is customary to describe the direction vector in terms of a spherical or polar coordinate system. Consider a point P on an opaque surface dA radiating into another medium, say air, as shown in Fig. 1.6. It is apparent that the surface can radiate into infinitely many directions, with every ray penetrating through a hemisphere of unit radius as indicated in the

FIGURE 1.6 Emission direction and solid angles as related to a unit hemisphere.

Fundamentals of Thermal Radiation Chapter | 1 11

figure. The total surface area of this hemisphere, 2π 12 = 2π, is known as the total solid angle above the surface. An arbitrary emission direction from the surface is specified by the unit direction vector sˆ , which may be expressed ˆ and the azimuthal angle ψ (measured between in terms of the polar angle θ (measured from the surface normal n) an arbitrary axis on the surface and the projection of sˆ onto the surface). If the surface lies in the x-y-plane, and ψ is measured from the x-axis, this may be expressed as ˆ sˆ = sin θ cos ψî + sin θ sin ψˆj + cos θk,

(1.27)

where î, jˆ, kˆ are unit vectors along the x-, y-, z-axes, and nˆ = kˆ in this configuration. It is seen that, for a hemisphere, 0 ≤ θ ≤ π/2 and 0 ≤ ψ ≤ 2π. The unit direction vector, sˆ , is an important quantity that will be used throughout this book. The solid angle with which an infinitesimal surface dA j is seen from a point P is defined as the projection of the surface onto a plane normal to the direction vector, divided by the square of the distance S between dA j and P, as also shown in Fig. 1.6. If the surface is projected onto the unit hemisphere above the point, the solid angle is equal to the projected area itself, or dΩ =

dA jp S2

=

cos θj dA j S2

= dAj .

(1.28)

Thus, an infinitesimal solid angle is simply an infinitesimal area on a unit sphere, or dΩ = dAj = (1 × sin θ dψ)(1 × dθ) = sin θ dθ dψ.

(1.29)

Integrating over all possible directions we obtain 

2π ψ=0



π/2

θ=0

sin θ dθ dψ = 2π,

(1.30)

for the total solid angle above the surface, as already seen earlier. The solid angle, with which a finite surface A j is seen from point P, follows immediately from equation (1.28) as    dA jp cos θj dA j = = dAj = Aj , (1.31) Ω= 2 S2 A jp S A Aj i.e., the projection of A j onto the hemisphere above P. While a little unfamiliar at first, solid angles are simply two-dimensional angular space: Similar to the way a one-dimensional angle can vary between 0 and π (measured in dimensionless radians, equivalent to length along a semicircular line), the solid angle may vary between 0 and 2π (measured in dimensionless steradians, sr, equivalent to surface area on a hemisphere). Example 1.3. Determine the solid angle with which the sun is seen from Earth. Solution The area of the sun projected onto a plane normal to the vector pointing from Earth to the sun (or, simply, the image of the sun that we see from Earth) is a disk of radius RS 6.96 × 108 m (i.e., the radius of the sun), at a distance of approximately SES 1.496 × 1011 m (averaged over Earth’s yearly orbit). Thus the solid angle of the sun is ΩS =

(πR2S ) S2ES

=

π × (6.955 × 108 )2 = 6.79 × 10−5 sr. (1.496 × 1011 )2

This solid angle is so small that we may generally assume that solar radiation comes from a single direction, i.e., that all the light beams are parallel. Example 1.4. What is the solid angle with which the narrow strip shown in Fig. 1.7 is seen from point “0”? Solution Since the strip is narrow we may assume that the projection angle for equation (1.31) varies only in the x-direction as indicated in Fig. 1.7, leading to

12 Radiative Heat Transfer

FIGURE 1.7 Solid angle subtended by a narrow strip.



L

Ω=w 0

and



L

Ω=w 0

cos θ0 dx , r2 h dx = wh r3

cos θ0 =



L 0

h , r

r2 = h2 + x2 ,

L  x dx w wL = . √  = √ h h2 + x2 0 h h2 + L2 (h2 + x2 )3/2

1.6 Radiative Intensity While emissive power appears to be the natural choice to describe radiative heat flux leaving a surface, it is inadequate to describe the directional dependence of the radiation field, in particular inside an absorbing/emitting medium, where photons may not have originated from a surface. Therefore, very similar to the emissive power, we define the radiative intensity I, as radiative energy flow per unit solid angle and unit area normal to the rays (as opposed to surface area). Again, we distinguish between spectral and total intensity. Thus, spectral intensity, Iλ ≡ radiative energy flow/time/area normal to rays/solid angle/wavelength, total intensity, I ≡ radiative energy flow/time/area normal to rays/solid angle. Again, spectral and total intensity are related by 

∞

I(r, sˆ ) =

Iλ (r, sˆ , λ) dλ.

(1.32)

0

Here, r is a position vector fixing the location of a point in space and sˆ is a unit direction vector as defined in the previous section. While emissive power depends only on position and wavelength, the radiative intensity depends, in addition, on the direction vector sˆ . The emissive power can be related to intensity by integrating over all the directions pointing away from the surface. Considering Fig. 1.8, we find that the emitted energy from dA into the direction sˆ , and contained within an infinitesimal solid angle dΩ = sin θ dθ dψ is, from the definition of intensity, I(r, sˆ ) dAp dΩ = I(r, sˆ ) dA cos θ sin θ dθ dψ,

FIGURE 1.8 Relationship between blackbody emissive power and intensity.

Fundamentals of Thermal Radiation Chapter | 1 13

FIGURE 1.9 Kirchhoff’s law for the directional behavior of blackbody intensity.

where dAp is the projected area of dA normal to the rays (i.e., the way dA is seen when viewed from the −ˆs direction). Thus, integrating this expression over all possible directions gives the total energy emitted from dA, or, after dividing by dA 

2π



E(r) = 0

π/2

 I(r, θ, ψ) cos θ sin θ dθ dψ =

I(r, sˆ ) nˆ · sˆ dΩ.

(1.33)

2π

0

This expression is, of course, also valid on a spectral basis. The directional behavior of the radiative intensity leaving a blackbody is easily obtained from a variation of Kirchhoff’s law: Consider a small, black surface suspended at the center of an isothermal spherical enclosure, as depicted in Fig. 1.9. Let us assume that the enclosure has a (hypothetical) surface coating that reflects all incoming radiation totally and like a mirror everywhere except over a small area dAs , which also reflects all incoming radiation except for a small wavelength interval between λ and λ + dλ. Over this small range of wavelengths dAs behaves like a blackbody. Now, all radiation leaving dA, traveling to the sphere (with the exception of light of wavelength λ traveling toward dAs ), will be reflected back toward dA where it will be absorbed (since dA is black). Thus, the net energy flow from dA to the sphere is, recalling the definitions for intensity and solid angle,   dAs Ibλ (T, θ, ψ, λ)(dA cos θ) dΩs dλ = Ibλ (T, θ, ψ, λ)(dA cos θ) dλ, R2 where dΩ s is the solid angle with which dAs is seen from dA. On the other hand, also by Kirchhoff’s law, the sphere does not emit any radiation (since it does not absorb anything), except over dAs at wavelength λ. All energy emitted from dAs will eventually come back to itself except for the fraction intercepted by dA. Thus, the net energy flow from the sphere to dA is   dA cos θ dλ, Ibnλ (T, λ) dAs dΩ dλ = Ibnλ (T, λ) dAs R2 where the subscript n denotes emission into the normal direction (θs = 0, ψs arbitrary) and dΩ is the solid angle with which dA is seen from dAs . Now, from the Second Law of Thermodynamics, these two fluxes must be equal for an isothermal enclosure. Therefore, Ibλ (T, θ, ψ, λ) = Ibnλ (T, λ). Since the direction (θ, ψ), with which dAs is oriented, is quite arbitrary we conclude that Ibλ is independent of direction, or Ibλ = Ibλ (T, λ) only.

(1.34)

14 Radiative Heat Transfer

Substituting this expression into equation (1.33) we obtain the following relationship between blackbody intensity and emissive power: Ebλ (r, λ) = π Ibλ (r, λ).

(1.35)

This equation implies that the intensity leaving a blackbody (or any surface whose outgoing intensity is independent of direction, or diffuse) may be evaluated from the blackbody emissive power (or outgoing heat flux) as Ibλ (r, λ) = Ebλ (r, λ)/π.

(1.36)

In the literature the spectral blackbody intensity is often referred to as the Planck function. The directional behavior of the emission from a blackbody is found by comparing the intensity (energy flow per solid angle and area normal to the rays) and directional emitted flux (energy flow per solid angle and per unit surface area). The directional heat flux is sometimes called directional emissive power, and Ebλ (r, λ, θ, ψ) dA = Ibλ (r, λ) dAp , or Ebλ (r, λ, θ, ψ) = Ibλ (r, λ) cos θ,

(1.37)

that is, the directional emitted flux of a blackbody varies with the cosine of the polar angle. This is sometimes referred to as Lambert’s law∗∗ or the cosine law.

1.7 Radiative Heat Flux Consider the surface shown in Fig. 1.10. Let thermal radiation from an infinitesimal solid angle around the direction sˆ i impinge onto the surface with an intensity of Iλ (ˆsi ). Such radiation is often called a “pencil of rays” since the infinitesimal solid angle is usually drawn looking like the tip of a sharpened pencil. Recalling the definition for intensity we see that it imparts an infinitesimal heat flow rate per wavelength on the surface in the amount of dQλ = Iλ (ˆsi ) dΩ i dAp = Iλ (ˆsi ) dΩ i (dA cos θi ), where heat rate is taken as positive in the direction of the outward surface normal (going into the medium), so that the incoming flux going into the surface is negative since cos θi < 0. Integrating over all 2π incoming directions and dividing by the surface area gives the total incoming heat flux per unit wavelength, i.e.,    Iλ (ˆsi ) cos θi dΩ i . (1.38) qλ in = cos θi 0

If the surface is black ( λ = 1), there is no energy reflected from the surface and Iλ = Ibλ , leading to (qλ ) out = Ebλ . If the surface is not black, the outgoing intensity consists of contributions from emission as well as reflections. The outgoing heat flux is positive since it is going into the medium. The net heat flux from the surface may be calculated by adding both contributions, or        Iλ (ˆs) cos θ dΩ, (1.40) qλ net = qλ in + qλ out = 4π

∗∗

Johann Heinrich Lambert (1728–1777) German mathematician, astronomer, and physicist. Largely self-educated, Lambert did his work under the patronage of Frederick the Great. He made many discoveries in the areas of mathematics, heat, and light. The lambert, a measurement of diffusely reflected light intensity, is named in his honor (see Section 1.9).

Fundamentals of Thermal Radiation Chapter | 1 15

FIGURE 1.10 Radiative heat flux on an arbitrary surface.

where a single direction vector sˆ was used to describe the total range of solid angles, 4π. It is readily seen from ˆ Fig. 1.10 that cos θ = nˆ · sˆ and, since the net heat flux is evaluated as the flux into the positive n-direction, one gets    Iλ (ˆs) nˆ · sˆ dΩ. (1.41) qλ net = qλ · nˆ = 4π

In order to obtain the total radiative heat flux at the surface, equation (1.41) needs to be integrated over the spectrum, and  ∞  ∞ q = q · nˆ = qλ · nˆ dλ = Iλ (ˆs) nˆ · sˆ dΩ dλ. (1.42) 0

4π

0

Example 1.5. A solar collector mounted on a satellite orbiting Earth is directed at the sun (i.e., normal to the sun’s rays). Determine the total solar heat flux incident on the collector per unit area. Solution The total heat rate leaving the sun is Q˙ S = 4πR2S Eb (TS ), where RS 6.96 × 108 m is the radius of the sun. Placing an imaginary spherical shell around the sun of radius SES = 1.496 × 1011 m, where SES is the distance between the sun and Earth, we find the heat flux going through that imaginary sphere (which includes the solar collector) as qsol =

4πR2S Eb (TS ) 2

4πSES

= Ib (TS )

πR2S S2ES

= Ib (TS ) Ω S ,

where we have replaced the sun’s emissive power by intensity, Eb = πIb , and Ω S = 6.79 × 10−5 sr is the solid angle with which the sun is seen from Earth, as determined in Example 1.3. Therefore, with Ib (TS ) = σTS4 /π and TS = 5777 K, 1 qin = −(σTS4 /π)(Ω S ) = − 5.670 × 10−8 × 57774 × 6.79 × 10−5 W/m2 π = −1366 W/m2 , where we have added a minus sign to emphasize that the heat flux is going into the collector. The total incoming heat flux may, of course, also be determined from equation (1.38) as  I(ˆsi ) cos θi dΩ i . qin = cos θi 0), the source function in equation (1.67) contains the radiative intensity at every point along the path, for all possible directions (not just sˆ ): the radiative transfer equation, equation (1.63), is an integro-differential equation (intensity appears, both, as a derivative and also inside the integral on the right-hand side) in five dimensions (three space dimensions and two directional coordinates). This makes the RTE extremely difficult to solve, and much of this book will be devoted to describing the various methods of solution that have been devised over the years (Chapters 13–20).

1.16 Outline of Radiative Transport Theory Thermal radiation calculations are always performed by making an energy balance for an enclosure bounded by opaque walls (some of which may be artificial to account for radiation penetrating through openings in the enclosure). If the enclosure is evacuated or filled with a nonabsorbing, nonscattering medium (such as air at low to moderate temperatures), we speak of surface radiation transport. If the enclosure is filled with an absorbing gas or a semitransparent solid or liquid, or with absorbing and scattering particles (or bubbles), we refer to it as radiative transport in a participating medium. Of course, radiation in a participating medium is always accompanied by surface radiation transport. When considering heat transfer by conduction and/or convection within a medium, we require knowledge of a number of material properties, such as thermal conductivity k, thermal diffusivity α, kinematic viscosity ν, and so on. This knowledge, together with the law of conservation of energy, allows us to calculate the energy field within the medium in the form of the basic variable, temperature T. Once the temperature field is determined, the local heat flux vector may be found from Fourier’s law. The evaluation of radiative energy transport follows a similar pattern: Knowledge of radiative properties is required (emittance , absorptance α, and reflectance ρ, in the case of surfaces, as well as absorption coefficient κ and scattering coefficient σs for semitransparent media), and the law of conservation of energy is applied to determine the energy field. Two major differences exist between conduction/convection and thermal radiation that make the analysis of radiative transport somewhat more complex: (i) Unlike their thermophysical counterparts, radiative properties may be functions of direction as well as of wavelength, and (ii) the basic variable appearing in the law of conservation of radiative energy, the radiative transfer equation introduced in the previous section, is not temperature but radiative intensity, which is a function not only of location in space (as is temperature), but also of direction. Only after the intensity field has been determined can the local temperatures (as well as the radiative heat flux vector) be calculated.

Problems 1.1 Solar energy impinging on the outer layer of Earth’s atmosphere (usually called the “solar constant”) has been measured as 1366 W/m2 . What is the solar constant on Mars? (Distance from Earth to sun = 1.496 × 1011 m, Mars to sun = 2.28 × 1011 m.) 1.2 Assuming Earth to be a blackbody, what would be its average temperature if there was no internal heating from the core of Earth? 1.3 Assuming Earth to be a black sphere with a surface temperature of 300 K, what must Earth’s internal heat generation be in order to maintain that temperature (neglect radiation from the stars, but not the sun) (radius of the Earth RE = 6.37 × 106 m)? 1.4 To estimate the diameter of the sun, one may use solar radiation data. The solar energy impinging onto the Earth’s atmosphere (called the “solar constant”) has been measured as 1366 W/m2 . Assuming that the sun may be approximated to have a black surface with an effective temperature of 5777 K, estimate the diameter of the sun (distance from sun to Earth SES 1.496 × 1011 m). 1.5 Solar energy impinging on the outer layer of Earth’s atmosphere (usually called the “solar constant”) has been measured as 1366 W/m2 . Assuming the sun may be approximated as having a surface that behaves like a blackbody, estimate its effective surface temperature (distance from sun to Earth SES 1.496 × 1011 m, radius of sun RS 6.96 × 108 m). 1.6 A rocket in space may be approximated as a black cylinder of length L = 20 m and diameter D = 2 m. It flies past the sun at a distance of 140 million km such that the cylinder axis is perpendicular to the sun’s rays. Assuming that (i) the sun is a blackbody at 5777 K and (ii) the cylinder has a high conductivity (i.e., is essentially isothermal), what is the temperature of the rocket? (Radius of sun RS = 696,000 km; neglect radiation from Earth and the stars.)

Fundamentals of Thermal Radiation Chapter | 1 27

1.7 A black sphere of very high conductivity (i.e., isothermal) is orbiting Earth. What is its temperature? (Consider the sun but neglect radiation from the Earth and the stars.) What would be the temperature of the sphere if it were coated with a material that behaves like a blackbody for wavelengths between 0.4 μm and 3 μm, but does not absorb and emit at other wavelengths? 1.8 Derive an expression for the solid angle subtended by the curved surface of a cylinder of height H and radius R on the center of its base. If the cylinder is very tall compared to its radius, what do you expect this solid angle to be? Verify your expectation mathematically using the expression you have derived. 1.9 An interior space of a building, with dimensions Lx , L y , and Lz , has a skylight on the roof, as shown in the figure. The skylight is of width L y /4. In order to properly design the interior lighting, an architect is interested in knowing the solid angle subtended on any point, (x0 , y0 , z0 ), inside this space by the skylight. Derive an expression for this solid angle.

1.10 A 100 W lightbulb may be considered to be an isothermal black sphere at a certain temperature. If the light flux (i.e., visible light, 0.4 μm < λ < 0.7 μm) impinging on the floor directly (2.5 m) below the bulb is 42.6 mW/m2 , and assuming conduction/convection losses to be negligible, what is the lightbulb’s effective temperature? What is its efficiency? 1.11 When a metallic surface is irradiated with a highly concentrated laser beam, a plume of plasma (i.e., a gas consisting of ions and free electrons) is formed above the surface that absorbs the laser’s energy, often blocking it from reaching the surface. Assume that a plasma of 1 cm diameter is located 1 cm above the surface, and that the plasma behaves like a blackbody at 20,000 K. Based on these assumptions calculate the radiative heat flux and the total radiation pressure on the metal directly under the center of the plasma. 1.12 Solar energy incident on the surface of the Earth may be broken into two parts: a direct component (traveling unimpeded through the atmosphere) and a sky component (reaching the surface after being scattered by the atmosphere). On a clear day the direct solar heat flux has been determined as qsun = 1000 W/m2 (per unit area normal to the rays), while the intensity of the sky component has been found to be diffuse (i.e., the intensity of the sky radiation hitting the surface is the same for all directions) and Isky = 70 W/m2 sr. Determine the total solar irradiation onto Earth’s surface if the sun is located 60◦ above the horizon (i.e., 30◦ from the normal). 1.13 If the average absorptance of the Earth’s surface is denoted by αe , and its average emittance is denoted by e , derive an expression for the equilibrium surface temperature of the Earth, Te , in terms of these quantities and the solar constant, qsol . Assume that the atmosphere is absent and there is negligible heat generation within the Earth’s core. Using the value of the solar constant, given by equation (1.43), αe = 0.7, and e = 1.0, estimate the Earth’s equilibrium surface temperature. Comment on your result. Would the result change if the Earth’s surface was gray? Would the Earth’s surface temperature change if its radius was five times larger? 1.14 A window (consisting of a vertical sheet of glass) is exposed to direct sunshine at a strength of 1000 W/m2 . The window is pointing due south, while the sun is in the southwest, 30◦ above the horizon. Estimate the amount of solar energy that (i) penetrates into the building, (ii) is absorbed by the window, and (iii) is reflected by the window. The window is made of (a) plain glass, (b) tinted glass, whose radiative properties may be approximated by ρλ = 0.08 ⎧ ⎪ ⎪ ⎨0.90 τλ = ⎪ ⎪ ⎩0 ⎧ ⎪ ⎪ ⎨0.90 τλ = ⎪ ⎪ ⎩0

for all wavelengths (both glasses), for 0.35 μm < λ < 2.7 μm for all other wavelengths

(plain glass),

for 0.5 μm < λ < 1.4 μm for all other wavelengths

(tinted glass).

(c) By what fraction is the amount of visible light (0.4 μm < λ < 0.7 μm) reduced, if tinted rather than plain glass is used? How would you modify this statement in the light of Fig. 1.11?

28 Radiative Heat Transfer

1.15 On an overcast day the directional behavior of the intensity of solar radiation reaching the surface of the Earth after being scattered by the atmosphere may be approximated as Isky (θ) = Isky (θ = 0) cos θ, where θ is measured from the surface normal. For a day with Isky (0) = 100 W/m2 sr determine the solar irradiation hitting a solar collector, if the collector is (a) horizontal, (b) tilted from the horizontal by 30◦ . Neglect radiation from the Earth’s surface hitting the collector (by emission or reflection). 1.16 A 100 W lightbulb is rated to have a total light output of 1750 lm. Assuming the lightbulb to consist of a small, black, radiating body (the light filament) enclosed in a glass envelope (with a transmittance τg = 0.9 throughout the visible wavelengths), estimate the filament’s temperature. If the filament has an emittance of f = 0.7 (constant for all wavelengths and directions), how does it affect its temperature? 1.17 A pyrometer is a device with which the temperature of a surface may be determined remotely by measuring the radiative energy falling onto a detector. Consider a black detector of 1 mm × 1 mm area that is exposed to a 1 cm2 hole in a furnace located a distance of 1 m away. The inside of the furnace is at 1500 K and the intensity escaping from the hole is essentially blackbody intensity at that temperature. (a) What is the radiative heat rate hitting the detector? (b) Assuming that the pyrometer has been calibrated for the situation in (a), what temperature would the pyrometer indicate if the nonabsorbing gas between furnace and detector were replaced by one with an (average) absorption coefficient of κ = 0.1 m−1 ? 1.18 Consider a pyrometer, which also has a detector area of 1 mm × 1 mm, which is black in the wavelength range 1.0 μm ≤ λ ≤ 1.2 μm, and perfectly reflecting elsewhere. In front of the detector is a focusing lens ( f = 10 cm) of diameter D = 2 cm, and transmissivity of τl = 0.9 (around 1 μm). In order to measure the temperature inside a furnace, the pyrometer is focused onto a hot black surface inside the furnace, a distance of 1 m away from the lens. (a) How large a spot on the furnace wall does the detector see? (Remember that geometric optics dictates 1 1 1 = + ; f u v

M=

v h (detector size)   = , u H spot size

where u = 1 m is the distance from lens to furnace wall, and v is the distance from lens to detector.) (b) If the temperature of the furnace wall is 1200 K, how much energy is absorbed by the detector per unit time? (c) It turns out the furnace wall is not really black, but has an emittance of = 0.7 (around 1 μm). Assuming there is no radiation reflected from the furnace surface reaching the detector, what is the true surface temperature for the pyrometer reading of case (b)? (d) To measure higher temperatures pyrometers are outfitted with filters. If a τf = 0.7 filter is placed in front of the lens, what furnace temperature would provide the same pyrometer reading as case (b)?

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

E.M. Sparrow, R.D. Cess, Radiation Heat Transfer, Hemisphere, New York, 1978. NASA web page on the solar system and the sun, http://solarsystem.nasa.gov/planets, 2011. M.P. Thekaekara, The solar constant and spectral distribution of solar radiant flux, Solar Energy 9 (1) (1965) 7–20. M.P. Thekaekara, Solar energy outside the earth’s atmosphere, Solar Energy 14 (1973) 109–127. L. Rayleigh, The law of complete radiation, Philosophical Magazine 49 (1900) 539–540. J.H. Jeans, On the partition of energy between matter and the ether, Philosophical Magazine 10 (1905) 91–97. W. Wien, Über die Energieverteilung im Emissionsspektrum eines schwarzen Körpers, Annalen der Physik 58 (1896) 662–669. M. Planck, Distribution of energy in the spectrum, Annalen der Physik 4 (3) (1901) 553–563. D. Hoffmann, Historical review: “... you can’t say to anyone to their face: your paper is rubbish. ” Max Planck as editor of the Annalen der Physik, Annalen der Physik 17 (5) (2008) 269–271. W. Wien, Temperatur und Entropie der Strahlung, Annalen der Physik 52 (1894) 132–165. S.M. Stewart, Spectral peaks and Wien’s Displacement Law, Journal of Thermophysics and Heat Transfer 26 (2012) 689–691. J. Crepeau, A brief history of the T4 radiation law, ASME paper no. HT2009-88060, 2009. C. Fröhlich, J. Lean, Solar radiative output and its variability: evidence and mechanisms, The Astronomy and Astrophysics Review 3 (2004) 273–320. World Radiation Center solar constant web page, http://www.pmodwrc.ch/pmod.php?topic=tsi/composite/SolarConstant, 2010. P. Moon, Scientific Basis of Illuminating Engineering, Dover Publications, New York, 1961, originally published by McGraw-Hill, New York, 1936. R.B. Hopkinson, P. Petherbridge, J. Longmore, Daylighting, Pitman Press, London, 1966. J.E. Kaufman (Ed.), IES Lighting Handbook, Illuminating Engineering Society of North America, New York, 1981. F.M. White, Heat Transfer, Addison-Wesley, Reading, MA, 1984.

Fundamentals of Thermal Radiation Chapter | 1 29

[19] D.K. Edwards, Radiation interchange in a nongray enclosure containing an isothermal CO2 –N2 gas mixture, ASME Journal of Heat Transfer 84C (1962) 1–11. [20] N. Neuroth, Der Einfluss der Temperatur auf die spektrale Absorption von Gläsern im Ultraroten, I (Effect of temperature on spectral absorption of glasses in the infrared, I), Glastechnische Berichte 25 (1952) 242–249. [21] L. Rayleigh, Philosophical Magazine 12 (1881). [22] G.A. Mie, Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen, Annalen der Physik 25 (1908) 377–445.

Chapter 2

Radiative Property Predictions from Electromagnetic Wave Theory 2.1 Introduction The basic radiative properties of surfaces forming an enclosure, i.e., emissivity, absorptivity, reflectivity, and transmissivity, must be known before any radiative heat transfer calculations can be carried out. Many of these properties vary with incoming direction, outgoing direction, and wavelength, and must usually be found through experiment. However, for pure, perfectly smooth surfaces these properties may be calculated from classical electromagnetic wave theory.1 These predictions make experimental measurements unnecessary for some cases, and help interpolating as well as extrapolating experimental data in many other situations. The first important discoveries with respect to light were made during the seventeenth century, such as the law of refraction (by Snell in 1621), the decomposition of white light into monochromatic components (by Newton in 1666), and the first determination of the speed of light (by Römer in 1675). However, the true nature of light was still unknown: The corpuscular theory (suggested by Newton) competed with a rudimentary wave theory. Not until the early nineteenth century was the wave theory finally accepted as the correct model for the description of light. Young proposed a model of purely transverse waves in 1817 (as opposed to the model prevalent until then of purely longitudinal waves), followed by Fresnel’s comprehensive treatment of diffraction and other optical phenomena. In 1845 Faraday proved experimentally that there was a connection between magnetism and light. Based on these experiments, Maxwell presented in 1861 his famous set of equations for the complete description of electromagnetic waves, i.e., the interaction between electric and magnetic fields. Their success was truly remarkable, in particular because the theories of quantum mechanics and special relativity, with which electromagnetic waves are so strongly related, were not discovered until half a century later. To this day Maxwell’s equations remain the basis for the study of light.∗

2.2 The Macroscopic Maxwell Equations The original form of Maxwell’s equations is based on electrical experiments available at the time, with their very coarse temporal and spatial resolution. Thus any of these measurements were spatial averages taken over many layers of atoms and temporal averages over many oscillations of an electromagnetic wave. For this reason the original set of equations is termed macroscopic. Today we know that electromagnetic waves interact with matter at the molecular level, with strong field fluctuations over each wave period. Therefore, more detailed treatises on optics and electromagnetic waves now generally start with a microscopic description of the wave equations, for example, the book by Stone [1]. While there is little disagreement in the literature on the microscopic equations, the macroscopic equations often differ somewhat from book to book, depending on assumptions made and constitutive relations used. Following the development of Stone [1], we may state the macroscopic Maxwell 1. The National Institute of Standards and Technology (NIST, formerly NBS) has recommended to reserve the ending “-ivity” for radiative properties of pure, perfectly smooth materials (the ones discussed in this chapter), and “-ance” for rough and contaminated surfaces. Most real surfaces fall into the latter category, discussed in Chapter 3. While we will follow this convention throughout this book, the reader should be aware that many researchers in the field employ endings according to their own personal preference. ∗

James Clerk Maxwell (1831–1879) Scottish physicist. After attending the University of Edinburgh he obtained a mathematics degree from Trinity College in Cambridge. Following an appointment at Kings College in London he became the first Cavendish Professor of Physics at Cambridge. While best known for his electromagnetic theory, he made important contributions in many fields, such as thermodynamics, mechanics, and astronomy.

Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00010-9 Copyright © 2022 Elsevier Inc. All rights reserved.

31

32 Radiative Heat Transfer

equations as ∇ · ( E) = ρ f ,

(2.1)

∇ · (μH) = 0,

(2.2)

∂H ∇ × E = −μ , ∂t ∂E ∇×H=

+ σe E, ∂t

(2.3) (2.4)

where E and H are the electric field and magnetic field vectors, respectively, is the electrical permittivity, μ is the magnetic permeability, σe is the electrical conductivity, and ρ f is the charge density due to free electrons, which is generally assumed to be related to the electric field by the equation ∂ρ f ∂t

= −∇ · (σe E).

(2.5)

The phenomenological coefficients σe , μ, and depend on the medium under consideration, but may be assumed independent of the fields (for a linear medium) and independent of position and direction (for a homogeneous and isotropic medium); they may, however, depend on the wavelength of the electromagnetic waves [2].

2.3 Electromagnetic Wave Propagation in Unbounded Media We seek a solution to the above set of equations in the form of a wave. The most general form of a time-harmonic field (i.e., a wave of constant frequency or wavelength) is F = A cos ωt + B sin ωt = A cos 2πνt + B sin 2πνt,

(2.6)

where ω is the angular frequency (in radians/s) and ν = ω/2π is the frequency in cycles per second. While a little less convenient, we will use the cyclical frequency ν in the following development in order to limit the number of different spectral variables employed in this book. When it comes to the time-harmonic solution of linear partial differential equations, it is usually advantageous to introduce a complex representation of the real field. Thus, setting Fc = Fc e2πiνt ,

Fc = A − iB,

(2.7)

where Fc is the time-average of the complex field, results in F = {Fc },

(2.8)

where the symbol  denotes that the real part of the complex vector Fc is to be taken. Since the Maxwell equations are linear in the fields E and H, one may solve them for their complex fields, and then extract their real parts after a solution has been found. Therefore, setting E = {Ec } = {Ec e2πiνt }, H = {Hc } = {Hc e

2πiνt

},

(2.9) (2.10)

results in ∇ · (γEc ) = 0,

(2.11)

∇ · Hc = 0,

(2.12)

∇ × Ec = −2πiνμHc ,

(2.13)

∇ × Hc = 2πiνγEc ,

(2.14)

σe 2πν

(2.15)

where γ= −i

Radiative Property Predictions from Electromagnetic Wave Theory Chapter | 2 33

FIGURE 2.1 Phase propagation of an electromagnetic wave.

is the complex permittivity. If γ  0, then it can be shown that the solution to the above set of equations must be plane waves, i.e., the electric and magnetic fields are transverse to the direction of propagation (have no component in the direction of propagation). Thus, the solution of equations (2.11) through (2.14) will be of the form E = {Ec e2πiνt } = {E0 e−2πi(w·r−νt) },

(2.16)

H = {Hc e2πiνt } = {H0 e−2πi(w·r−νt) },

(2.17)

where r is a vector pointing to an arbitrary point in space, w is known as the wave vector,2 and E0 and H0 are constant vectors. In general w is a complex vector, w = w − iw ,

(2.18)

where w turns out to be a vector whose magnitude is the wavenumber and w is known as the attenuation vector. Employing equation (2.18), equations (2.16) and (2.17) may be rewritten as Ec = E0 e−2πw



·r −2πi(w ·r−νt)

e

,

−2πw ·r −2πi(w ·r−νt)

Hc = H0 e

e

(2.19)

.

(2.20) 



Thus, the complex electric and magnetic fields have local amplitude vectors E0 e−2πw ·r and H0 e−2πw ·r and an  oscillatory part e−2πi(w ·r−νt) with phase angle φ = 2π(w · r − νt). The position vector r may be considered to have two components: one parallel to w and the other perpendicular to it. The vector product w · r is constant for all vectors r that have the same component parallel to w , i.e., on planes normal to the vector w ; these planes are known as planes of equal phase. To see how the wave travels let us look at the phase angle at two different times and locations (Fig. 2.1). First, consider the point r = 0 at time t = 0 with a zero phase angle. Second, consider another point a distance z away into the direction of w ; we see that the phase angle is zero at that point when t = |w |z/ν. Thus, the phase velocity with which the wave travels from one point to the other is c = z/t = ν/w . We conclude that the wave propagates into the direction of w , and that the vector’s magnitude, w , is equal to the wavenumber η. Examining the amplitude vectors we see that w · r = const are planes of equal amplitude, and that the amplitude of the fields diminishes into the direction of w . If planes of equal phase and equal amplitude coincide (i.e., if w and w are parallel) we say the wave is homogeneous, otherwise the wave is said to be inhomogeneous. Since E0 and w are independent of position, we can substitute equation (2.19) into equation (2.11) and, assuming γ to be also invariant with space, find that     ∇ · (γEc ) = γ∇ · E0 e−2πi(w·r−νt) = γE0 · ∇ e−2πi(w·r−νt) = γE0 e−2πi(w·r−νt) · ∇ (−2πiw · r) = −2πiγw · E0 e−2πi(w·r−νt) = 0.

(2.21)

2. The present definition of the wave vector differs by a factor of 2π and in name from the definition k = 2πw in most optics texts in order to conform with our definition of wavenumber.

34 Radiative Heat Transfer

FIGURE 2.2 Electric and magnetic fields of a homogeneous wave.

Similarly, substituting equation (2.19) into equation (2.13) results in     ∇ × Ec = ∇ × E0 e−2πi(w·r−νt) = ∇ e−2πi(w·r−νt) × E0 = −2πiw e−2πi(w·r−νt) × E0 = −2πiνμH0 e−2πi(w·r−νt) .

(2.22)

Thus, the partial differential equations (2.11) through (2.14) may be replaced by a set of algebraic equations, w · E0 w · H0 w × E0 w × H0

= 0, = 0, = νμH0 , = −νγE0 .

(2.23) (2.24) (2.25) (2.26)

It is clear from equations (2.23) and (2.24) that both E0 and H0 are perpendicular to w, and it follows then from equations (2.25) and (2.26) that they are also perpendicular to each other.3 If the wave is homogeneous, then w points into the direction of wave propagation, and the electric and magnetic fields lie in planes perpendicular to this direction, as indicated in Fig. 2.2. It remains to relate the complex wave vector w to the properties of the medium. Taking the vector product of equation (2.25) with w and recalling the vector identity derived, for example, in Wylie [3], A × (B × C) = B(A · C) − C(A · B),

(2.27)

which leads to w × (w × E0 ) = w(w · E0 ) − E0 w · w = νμw × H0 = −ν2 μγE0 , or

w · w = ν2 μγ.

(2.28)

If the wave travels through vacuum there can be no attenuation (w = 0) and μ = μ0 , γ = 0 . We thus obtain the speed of light in vacuum as √ 1 c 0 = ν/w = ν/ w · w = √ .

0 μ0 3. Remember that all three vectors are complex and, therefore, the interpretation of “perpendicular” is not straightforward.

(2.29)

Radiative Property Predictions from Electromagnetic Wave Theory Chapter | 2 35

It is customary to introduce the complex index of refraction m = n − ik into equation (2.28) such that



w · w = ν μγ = ν 0 μ0 2

2

(2.30)



μ σe μ = η20 m2 , −i

0 μ0 2πν 0 μ0

(2.31)

where η0 = ν/c 0 is the wavenumber of a wave with frequency ν and phase velocity c 0 , i.e., of a wave traveling through vacuum. This definition of m demands that

μ = μc20 ,

0 μ0 σe μ σe μλ0 c 0 , nk = = 4πν 0 μ0 4π

n2 − k 2 =

(2.32) (2.33)

where λ0 = 1/η0 = c 0 /ν is the wavelength for the wave in vacuum. Equations (2.32) and (2.33) may be solved for the refractive index n and the absorptive index4 k as ⎤ ⎡   2  2 ⎥⎥ ⎢⎢

σ

1 λ ⎥⎥ ⎢ 0 e ⎥⎥ , (2.34) + n2 = ⎢⎢⎢⎢ + 2 ⎣ 0

0 2πc 0 0 ⎥⎦ ⎤ ⎡   2  2 ⎥⎥ ⎢⎢

σ

1 λ ⎥⎥ ⎢ 0 e ⎥⎥ , k2 = ⎢⎢⎢⎢− + (2.35) + 2 ⎣ 0

0 2πc 0 0 ⎥⎦ where we have assumed the material to be nonmagnetic, or μ = μ0 . These relations do not reveal the frequency (wavelength) dependence of the complex index of refraction, since the phenomenological coefficientss and σe may depend on frequency. If the wave is homogeneous the wave vector may be written as w = (w −iw )ˆs, where sˆ is a unit vector in the direction of wave propagation, and it follows from equation (2.31) that w −iw = η0 (n−ik), so that the electric and magnetic fields reduce to Ec = E0 e−2πη0 kz e−2πiη0 n(z−c 0 t/n) ,

(2.36)

−2πη0 kz −2πiη0 n(z−c 0 t/n)

(2.37)

Hc = H0 e

e

,

where z = sˆ · r is distance along the direction of propagation. For a nonvacuum, the phase velocity c of an electromagnetic wave is5 c0 (2.38) c= . n Further, the field strengths decay exponentially for nonzero values of k; thus, the absorptive index gives an indication of how quickly a wave is absorbed within the medium. Inspection of equation (2.35) shows that a large absorptive index k corresponds to a large electrical conductivity σe : Electromagnetic waves tend to be attenuated rapidly in good electrical conductors, such as metals, but are often transmitted with weak attenuation in media with poor electrical conductivity, or dielectrics, such as glass. The magnitude and direction of the transfer of electromagnetic energy is given by the Poynting vector, i.e., a vector of magnitude EH pointing into the direction of propagation (cf. Fig. 2.2),6 S = E × H = {Ec } × {Hc }.

(2.39)

4. The absorptive index is often referred to as extinction coefficient in the literature. Since the term extinction coefficient is also employed for another, related property we will always use the term absorptive index in this book to describe the imaginary part of the index of refraction. 5. Since there are materials that have n < 1 it is possible to have phase velocities (i.e., the velocity with which the amplitude of continuous waves penetrates through a medium) larger than c 0 ; these should be distinguished from the signal velocities (i.e., the velocity with which the energy contained in the waves travels), which can never exceed the speed of light in vacuum. The difference between the two may be grasped more easily by visualizing the movement of ocean waves: The wave crests move at a certain speed across the ocean surface (phase velocity), while the actual velocity of the water (signal velocity) is relatively slow. 6. Note that, since the vector cross-product is a nonlinear operation, the Poynting vector may not be calculated from S = {Ec × Hc }.

36 Radiative Heat Transfer

The instantaneous value for the Poynting† vector is a rapidly varying function of time. Of greater value to the engineer is a time-averaged value of the Poynting vector, say 1 S= δt



t+δt

S(t) dt,

(2.40)

t

where δt is a very small amount of time, but significantly larger than the duration of a period, 1/ν; since S repeats itself after each period (if no attenuation occurs) a δt equal to any multiple of 1/ν will give the same result for S, namely S = 12 {Ec × H∗c },

(2.41)

where H∗ denotes the complex conjugate of H, and the factor of 1/2 results from integrating over cos2 (2πη0 c 0 t) and sin2 (2πη0 c 0 t) terms. Thus using equation (2.25) and the vector identity (2.27), the Poynting vector may be expressed as 1 1 {Ec × (w∗ × E∗c )} = {w∗ (Ec · E∗c )} 2νμ 2νμ n = |E0 |2 e−4πη0 kz sˆ . 2c 0 μ

S=

(2.42)

The vector S points into the direction of propagation, and—as the wave traverses the medium—its energy content is attenuated exponentially, where the attenuation factor κ = 4πη0 k

(2.43)

is known as the absorption coefficient of the medium. Example 2.1. A plane homogeneous wave propagates through a perfect dielectric medium (n = 2) in the direction of sˆ = 0.8î + 0.6kˆ with a wavenumber of η0 = 2500 cm−1 and an electric field amplitude vector of E0 = E0 [(6 + 3i)î + (2 − √ ˆ 154, where E0 = 600 N/C, and the î, ˆj, and kˆ are unit vectors in the x-, y-, and z-directions. Determine 5i)ˆj − (8 + 4i)k]/ the magnetic field amplitude vector and the energy contained in the wave, assuming that the medium is nonmagnetic. Solution Since w = w is colinear with sˆ , we find from equation (2.31) that w = wˆs = η0 nˆs and, from equation (2.25), 1 1 n w × E0 = sˆ × E0 w × E0 = νμ νμ0 c 0 μ0     ˆj î kˆ nE0   = √ 0.0 0.6   0.8  c 0 μ0 154  6 + 3i 2 − 5i −8 − 4i  nE0 ˆ [(−6 + 15i)î + (50 + 25i)ˆj + (8 − 20i)k] = √ c 0 μ0 5 154 H0 ˆ = √ [(−6+15i)î + (50+25i)ˆj + (8−20i)k], 3850

H0 =

where H0 =

nE0 2 × 600 N/C = = 3.185 C/m s, c 0 μ0 2.998×108 m/s×4π×10−7 N s2 /C2

and it is assumed that, for a nonmagnetic medium, the magnetic permeability is equal to the one in vacuum, μ = μ0 (from Table A.1). The energy content of the wave is given by the Poynting vector, either equation (2.41) or equation (2.42). †

John Henry Poynting (1852–1914) British physicist. He served as professor of physics at the University of Birmingham from 1880 until his death. His discovery that electromagnetic energy is proportional to the product of electric and magnetic field strength is known as Poynting’s theorem.

Radiative Property Predictions from Electromagnetic Wave Theory Chapter | 2 37

FIGURE 2.3 Vibration ellipse for a monochromatic wave.

Choosing the latter, we get S=

n E2 sˆ = Sˆs, 2c 0 μ0 0

S=

2 × 6002 N2 /C2 2×2.2998×10−8 m/s×4π×10−7 N s2 /C2

= 955.6 W/m2 .

2.4 Polarization Knowledge of the frequency, direction of propagation, and the energy content [i.e., the magnitude of the Poynting vector, equation (2.42)] does not completely describe a monochromatic (or time-harmonic) electromagnetic wave. Every train of electromagnetic waves has a property known as the state of polarization. Polarization effects are generally not very important to the heat transfer engineer since emitted light generally is randomly polarized. In some applications partially or fully polarized light is employed, for example, from laser sources; and the engineer needs to know (i) how the reflective behavior of a surface depends on the polarization of incoming light, and (ii) how reflection from a surface tends to alter the state of polarization. We shall give here only a very brief introduction to polarization, based heavily on the excellent short description in Bohren and Huffman [2]. More detailed accounts on the subject may be found in the books by van de Hulst [4], Chandrasekhar [5], and others. Consider a plane monochromatic wave with wavenumber η propagating through a nonabsorbing medium (k ≡ 0) in the z-direction. When describing polarization, it is customary to relate parameters to the electric field (keeping in mind that the magnetic field is simply perpendicular to it), which follows from equation (2.36) as E = {Ec } = {(A − iB) e−2πiηn(z−ct) } = A cos 2πηn(z − ct) − B sin 2πηn(z − ct),

(2.44)

where the vector E0 and its real components A and B are independent of position and lie, at any position z, in the plane normal to the direction of propagation. At any given location, say z = 0, the tip of the electric field vector traces out the curve E(z = 0, t) = A cos 2πνt + B sin 2πνt.

(2.45)

This curve, shown in Fig. 2.3, describes an ellipse that is known as the vibration ellipse. The ellipse collapses into a straight line if either A or B vanishes, in which case the wave is said to be linearly polarized (sometimes also called plane polarized). If A and B are perpendicular to one another and are of equal magnitude, the vibration ellipse becomes a circle and the wave is known as circularly polarized. In general, the wave in equation (2.44) is elliptically polarized. At any given time, say t = 0, the curve described by the tip of the electric field vector is a helix (Fig. 2.4), or E(z, t = 0) = A cos 2πnηz − B sin 2πnηz.

(2.46)

Equation (2.46) describes the electric field at any one particular time. As time increases the helix moves into the direction of propagation, and its intersection with any plane z = const describes the local vibration ellipse.

38 Radiative Heat Transfer

FIGURE 2.4 Space variation of electric field at fixed times.

The state of polarization, which is characterized by its vibration ellipse, is defined by its ellipticity, b/a (the ratio of the length of its semiminor axis to that of its semimajor axis, as shown in Fig. 2.3), its azimuth γ (the angle between an arbitrary reference direction and its semimajor axis), and its handedness (i.e., the direction with which the tip of the electric field vector traverses through the vibration ellipse, clockwise or counterclockwise). These three parameters together with the magnitude of the Poynting vector are the ellipsometric parameters of a plane wave. Example 2.2. Calculate the ellipsometric parameters a, b, and γ for the wave considered in Example 2.1. Solution From equation (2.44) we find √ ˆ A = E0 (6î + 2ˆj − 8k)/ 154,

√ ˆ B = −E0 (3î − 5ˆj − 4k)/ 154,

and at any given location, say z = 0, the electric field vector may be written as   √ E = E0 (6 cos 2πνt − 3 sin 2πνt)î + (2 cos 2πνt + 5 sin 2πνt)ˆj − (8 cos 2πνt − 4 sin 2πνt)kˆ / 154. The time-varying magnitude |E| at this location then is |E|2 = E · E =

E20

(36 cos2 2πνt − 36 cos 2πνt sin 2πνt + 9 sin2 2πνt 154 + 4 cos 2πνt + 20 cos2 2πνt sin 2πνt + 25 sin2 2πνt + 64 cos 2πνt − 64 cos2 2πνt sin 2πνt + 16 sin2 2πνt)

= E20 (50 − 80 cos 2πνt sin 2πνt + 54 cos2 2πνt)/154. The maximum (a) and minimum (b) of |E| may be found by differentiating the last expression with respect to t and setting the result equal to zero. This operation leads to −80(cos2 2πνt − sin2 2πνt) = 108 sin 2πνt cos 2πνt −80 cos 4πνt = 54 sin 4πνt or

  80 . 2πνt = 0.5 tan−1 − 54

Radiative Property Predictions from Electromagnetic Wave Theory Chapter | 2 39

This function is double-valued, leading to (2πνt) 1 = −27.99◦ and (2πνt) 2 = 62.01◦ . Substituting these values into the expression for E gives ˆ E1 = E0 (0.5404î − 0.0468ˆj − 0.7205k),

|E| = a = 0.9009E0

ˆ E2 = E0 (0.0134î + 0.4314ˆj − 0.0179k),

|E| = b = 0.4339E0 .

and

The evaluation of the azimuth depends on the choice of a reference axis in the plane of the vibration ellipse. In the present problem the y-axis lies in this plane and is, therefore, the natural choice. Thus, cos γ =

E · ˆj 0.0468 =− = −0.0519, |E| 0.9009

γ = 92.97◦ .

While the ellipsometric parameters completely describe any monochromatic wave, they are difficult to measure directly (with the exception of the Poynting vector). In addition, when two or more waves of the same frequency but different polarization are superposed, only their strengths are additive: The other three ellipsometric parameters must be calculated anew. For these reasons a different but equivalent description of polarized light, known as Stokes’ parameters, is usually preferred. The Stokes’ parameters are defined by separating the wave train into two perpendicular components: Ec = E0 e−2πiηn(z−ct) ;

E0 = E eˆ  + E⊥ eˆ ⊥ ,

(2.47)

where eˆ  and eˆ ⊥ are real orthogonal unit vectors in the plane normal to wave propagation, such that eˆ  lies in an arbitrary reference plane that includes the wave propagation vector, and eˆ ⊥ is perpendicular to it.7 The parallel (E ) and perpendicular (E⊥ ) polarization components are generally complex and may be written as E = a e−iδ ,

E⊥ = a⊥ e−iδ⊥ ,

(2.48)

where a is the magnitude of the electric field and δ is the phase angle of polarization. Waves with parallel polarization (i.e., with electric field in the plane of incidence, and magnetic field normal to it) are also called transverse magnetic (TM) waves; and perpendicular polarization is transverse electric (TE). Substitution into equation (2.44) leads to E = {a e−iδ −2πiηn(z−ct) eˆ  + a⊥ e−iδ⊥ −2πiηn(z−ct) eˆ ⊥ } = a cos[δ + 2πηn(z − ct)]ˆe + a⊥ cos[δ⊥ + 2πηn(z − ct)]ˆe⊥ .

(2.49)

Thus, the arbitrary wave given by equation (2.44) has been decomposed into two linearly polarized waves that are perpendicular to one another. The four Stokes’ parameters I, Q, U, and V are defined by I = E E∗ + E⊥ E∗⊥ = a2 + a2⊥ , Q= U= V=

E E∗ − E⊥ E∗⊥ = E E∗⊥ + E⊥ E∗ = i(E E∗⊥ − E⊥ E∗ )

a2

−

a2⊥ ,

(2.50) (2.51)

2a a⊥ cos(δ − δ⊥ ),

(2.52)

= 2a a⊥ sin(δ − δ⊥ ),

(2.53)

where the asterisks again denote complex conjugates. It can be shown that these four parameters may be determined through power measurements either directly (I ), using a linear polarizer (arranged in the parallel and perpendicular directions for Q, rotated 45◦ for U ), or a circular polarizer (V ) (see, for example, Bohren and Huffman [2]). It is clear that only three of the Stokes’ parameters are independent, since I 2 = Q2 + U 2 + V 2 .

(2.54)

Since the Stokes’ parameters of a wave train are expressed in terms of the energy contents of its component waves [which can be seen by comparison with equation (2.42)], it follows that the Stokes’ parameters for a collection of waves are additive. 7. In the literature subscripts p and s are also commonly used, from the German words “parallel” and “senkrecht” (perpendicular).

40 Radiative Heat Transfer

TABLE 2.1 Stokes’ parameters for several cases of polarized light. Linearly Polarized 0◦

90◦

+45◦

−45◦

↔ ⎛ ⎞ ⎜⎜1⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜1⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜0⎟⎟ ⎜⎜ ⎟⎟ ⎝ ⎠ 0



 ⎛ ⎞ ⎜⎜1⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜0⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜1⎟⎟ ⎜⎜ ⎟⎟ ⎝ ⎠ 0



⎛ ⎞ ⎜⎜ 1 ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜−1⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ 0 ⎟⎟ ⎜⎝ ⎟⎠ 0

⎛ ⎞ ⎜⎜ 1 ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ 0 ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜−1⎟⎟ ⎜⎝ ⎟⎠ 0

γ ⎛ ⎞ ⎜⎜ 1 ⎟⎟ ⎜⎜ ⎟ ⎜⎜cos 2γ⎟⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟ ⎜⎜ sin 2γ ⎟⎟⎟ ⎜⎝ ⎟⎠ 0

Circularly Polarized Right

Left

 ⎛ ⎞ ⎜⎜1⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜0⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜0⎟⎟ ⎜⎜ ⎟⎟ ⎝ ⎠ 1

 ⎛ ⎞ ⎜⎜ 1 ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ 0 ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ 0 ⎟⎟ ⎜⎝ ⎟⎠ −1

The Stokes’ parameters may also be related to the ellipsometric parameters by I = a2 + b2 ,

(2.55)

Q = (a − b ) cos 2γ,

(2.56)

U = (a − b ) sin 2γ, V = ±2ab,

(2.57) (2.58)

2 2

2 2

where the azimuth γ is measured from eˆ  , and the sign of V specifies the handedness of the vibration ellipse. The sets of Stokes’ parameters for a few special cases of polarization are shown—normalized, and written as column vectors—in Table 2.1 (from [2]). The parameters Q and U show the degree of linear polarization (plus its orientation), while V is related to the degree of circular polarization. The above definition of the Stokes’ parameters is correct for strictly monochromatic waves as given by equation (2.47). Most natural light sources, such as the sun, lightbulbs, fires, and so on, produce light whose amplitude, E0 , is a slowly varying function of time (i.e., in comparison with a full wave period, 1/ν), or E0 (t) = E (t)ˆe + E⊥ (t)ˆe⊥ .

(2.59)

Such waves are called quasi-monochromatic. If, through their slow respective variations with time, E and E⊥ are uncorrelated, then the wave is said to be unpolarized. In such a case the vibration ellipse changes slowly with time, eventually tracing out ellipses of all shapes, orientations, and handedness. All waves discussed so far had a fixed relationship between E and E⊥ , and are known as (completely) polarized. If some correlation between E and E⊥ exists (for example, a wave of constant handedness, ellipticity, or azimuth), then the wave is called partially polarized. For quasi-monochromatic waves the Stokes’ parameters are defined in terms of time-averaged values, and equation (2.54) must be replaced by I 2 ≥ Q2 + U 2 + V 2 ,

(2.60)

where the equality sign holds only for polarized light. For unpolarized light one gets Q = U = V = 0, while for partially polarized light the magnitudes of Q, U, and V give the following: % degree of polarization = Q2 + U2 + V 2 /I, % degree of linear polarization = Q2 + U2 /I, degree of circular polarization = V/I. Example 2.3. Reconsider the plane wave of the last two examples. Decompose the wave into two linearly polarized waves, one in the x-z-plane, and the other perpendicular to it. What are the Stokes’ coefficients, the phase differences between the two polarizations, and the different degrees of polarization?

Radiative Property Predictions from Electromagnetic Wave Theory Chapter | 2 41

Solution With sˆ = 0.8î + 0.6kˆ and the knowledge that eˆ  must lie in the x-z-plane, i.e., eˆ  · ˆj = 0, and that eˆ  must be normal to sˆ , or eˆ  · sˆ = 0, and finally that eˆ ⊥ must be perpendicular to both of them, we get ˆ eˆ  = 0.6î − 0.8k,

eˆ ⊥ = ˆj,

where the choice of sign for both vectors is arbitrary (and we have chosen to let eˆ  , eˆ ⊥ , and sˆ form a right-handed coordinate system). Thus, from equation (2.47) and √ ˆ 154 E0 = E0 [(6 + 3i)î + (2 − 5i)ˆj − (8 + 4i)k]/ it follows immediately that  √  √ ˆ 154 = 5/ 154 (2 + i)E0 eˆ  , E =E0 (2 + i)(3î − 4k)/   √ √ E⊥ =E0 (2 − 5i)ˆj/ 154 = (2 − 5i)/ 154 E0 eˆ ⊥ , or 

 E = 5/ 154 (2 + i)E0 = √

&

125 E0 e−iδ , 154 &   √ 29 E0 e−iδ⊥ , E⊥ = (2 − 5i)/ 154 E0 = 154

with

  1 = −26.565◦ , 2   5 δ⊥ = − tan−1 − = 68.199◦ , 2

δ = − tan−1

and a phase difference between the two polarizations of δ − δ⊥ = −94.76◦ (since tan−1 is a double-valued function, the correct value is determined by checking the signs of the real and imaginary parts of E). The Stokes’ parameters can be calculated either directly from equations (2.50) through (2.53), or from equations (2.55) through (2.58) (using the ellipsometric parameters calculated in the last example). We use here the first approach so that we get I = (125 + 29)E20 /154 = E20 , Q = (125 − 29)E20 /154 = 48E20 /77, U = 5(4 + 2i + 10i − 5 + 4 − 2i − 10i − 5)E20 /154 = −5E20 /77, V = 5i(4 + 2i + 10i − 5 − 4 + 2i + 10i + 5)E20 /154 = −60E20 /77. % % Finally, the degrees of polarization follow as Q2 + U2 + V 2 /I = 100% total polarization, Q2 + U2 /I = 62.7% linear polarization, and |V|/I = 77.9% circular polarization.

In general, the state of polarization of an electromagnetic wave train is changed when it interacts with an optical element (which may be a polarizer or reflector, but can also be a reflecting surface in an enclosure, or a scattering element, such as suspended particles). While a polarized beam is characterized by its four-element Stokes vector, it is possible to represent the effects of an optical element by a 4 × 4 matrix, known as the Mueller matrix, which describes the relations between incident and transmitted Stokes vectors. Details can be found, e.g., in Bohren and Huffman [2].

2.5 Reflection and Transmission When an electromagnetic wave is incident on the interface between two homogeneous media, the wave will be partially reflected and partially transmitted into the second medium. We will limit our discussion here to plane interfaces, i.e., to cases where the local radius of curvature is much greater than the wavelength of the incoming light, λ, for which the problem may be reduced to algebraic equations. Some discussion on strongly

42 Radiative Heat Transfer

FIGURE 2.5 Geometry for derivation of interface conditions.

curved surfaces in the form of small particles will be given in Chapter 11, which deals with radiative properties of particulate clouds. In the following, after first establishing the general conditions for Maxwell’s equations at the interface, we shall consider a wave traveling from one nonabsorbing medium into another nonabsorbing medium, followed by a short discussion of a wave incident from a nonabsorbing onto an absorbing medium.

Interface Conditions for Maxwell’s Equations To establish boundary conditions for E and H at an interface between two media, we shall apply the theorems of Gauss and Stokes to Maxwell’s equations. Both theorems convert volume integrals to surface integrals and are discussed in detail in standard mathematical texts such as Wylie [3]. Given a vector function F, defined within a volume V and on its boundary Γ, the theorems may be stated as Gauss’ theorem:   ∇ · F dV = F · dΓ, (2.61) Γ

V

Stokes’ theorem:



 ∇ × F dV = − V

Γ

F × dΓ,

(2.62)

where dΓ = nˆ dΓ and nˆ is a unit surface normal pointing out of the volume. Now consider a thin volume element δV = A δs containing part of the interface as shown in Fig. 2.5. Applying Gauss’ theorem to the first of Maxwell’s equations, equation (2.11) yields    ˆ + (γEc ) 2 · n] ˆ dA = 0, ∇· (γEc ) dV = γEc · dΓ ≈ [(γEc ) 1 · (−n) (2.63) δV

Γ

A

where Γ is the total surface area of δV, and contributions to the surface integral come mainly from the two sides parallel to the interface since δs is small. Also, shrinking A to an arbitrarily small area, we conclude that, everywhere along the interface, ˆ m21 Ec1 · nˆ = m22 Ec2 · n,

(2.64)

where equation (2.31) has been used, together with assuming nonmagnetic media, to eliminate the complex permittivity γ. Similarly, from equation (2.12) ˆ Hc1 · nˆ = Hc2 · n.

(2.65)

Thus, the normal components of m2 Ec and Hc are conserved across a plane boundary. Stokes’ theorem may be applied to equations (2.13) and (2.14), again for the volume element shown in Fig. 2.5. For example,     ∇ × Hc dV = − Hc × dΓ ≈ (Hc1 −Hc2 ) × nˆ dA = 2πiνγEc dV, (2.66) δV

Γ

A

V

or, after shrinking δs → 0 and A to a small value, Ec1 × nˆ = Ec2 × nˆ and

(2.67)

Radiative Property Predictions from Electromagnetic Wave Theory Chapter | 2 43

FIGURE 2.6 Transmission and reflection of a plane wave at the interface between two nonabsorbing media.

ˆ Hc1 × nˆ = Hc2 × n.

(2.68)

Therefore, the tangential components of both Ec and Hc are conserved across a plane boundary. Given the incident wave, it is possible to find the complete fields from Maxwell’s equations and the above interface conditions. However, it is obvious that there will be a reflected wave in the medium of incidence, and a transmitted wave in the other medium. We may also assume that all waves remain plane waves. A consequence of having guessed the solution to this point is that conditions (2.67) and (2.68) are sufficient to specify the reflected and transmitted waves, and it turns out that conditions (2.64) and (2.65) are automatically satisfied (Stone [1]).

The Interface between Two Nonabsorbing Media The reflection and transmission relationships become particularly simple if homogeneous plane waves reach the plane interface between two nonabsorbing media. For such a wave train the planes of equal phase and equal amplitude coincide and are normal to the direction of propagation, as shown in Fig. 2.6. This plane, also called the wavefront, moves at constant speed c 1 = c 0 /n1 through Medium 1, and at a constant speed c 2 = c 0 /n2 through Medium 2. If n2 > n1 then, as shown in Fig. 2.6, the wavefront will move more slowly through Medium 2, lagging behind the wavefront traveling through Medium 1. This is readily put in mathematical terms by looking at points A and B on the wavefront at a certain time t. At time t + Δt the part of the wavefront initially at A will have reached point A on the interface while the wavefront at point B, traveling a shorter distance through Medium 2, will have reached point B , where Δt =

AA BB = . c1 c2

(2.69)

Using geometric relations for AA and BB and substituting for the phase velocities, we obtain Δt =

BA sin θi BA sin θ2 BA sin θr = = , c 0 /n1 c 0 /n2 c 0 /n1

(2.70)

where the last term pertains to reflection, for which a similar relationship must exist (but which is not shown to avoid overcrowding of the figure). Thus we conclude that θr = θ i = θ 1 ,

(2.71)

that is, according to electromagnetic wave theory, reflection of light is always purely specular. This is a direct consequence of a “plane” interface, i.e., a surface that is not only flat (with infinite radius of curvature) but

44 Radiative Heat Transfer

also perfectly smooth. Equation (2.70) also gives a relationship between the directions of the incoming and transmitted waves as sin θ2 n1 = , (2.72) sin θ1 n2 which is known as Snell’s law.‡ The angles θ1 = θi and θ2 = θr are called the angles of incidence and refraction. The present derivation of Snell’s law was based on geometric principles and is valid only for plane homogeneous waves, which limits its applicability to the interface between two nonabsorbing media, i.e., two perfect dielectrics. A more rigorous derivation of a generalized version of Snell’s law is given when incidence on an absorbing medium is considered. Besides the directions of reflection and transmission we should like to be able to determine the amounts of reflected and transmitted light. From equations (2.19) and (2.20) we can write expressions for the electric and magnetic fields in Medium 1 (consisting of incident and reflected waves) by setting w = 0 for a nonabsorbing medium as 



Ec1 = E0i e−2πi(wi ·r−νt) + E0r e−2πi(wr ·r−νt) , −2πi(wi ·r−νt)

−2πi(wr ·r−νt)

Hc1 = H0i e

+ H0r e

(2.73) .

(2.74)

Similarly for Medium 2, 

Ec2 = E0t e−2πi(wt ·r−νt) ,

(2.75)

−2πi(wt ·r−νt)

(2.76)

Hc2 = H0t e

.

For convenience we place the coordinate origin at that point of the boundary where reflection and transmission are to be considered. Thus, at that point of the interface, with r = 0, using boundary conditions (2.67) and (2.68), ˆ (E0i + E0r ) × nˆ = E0t × n, ˆ (H0i + H0r ) × nˆ = H0t × n.

(2.77) (2.78)

To evaluate the tangential components of the electric and magnetic fields at the interface, it is advantageous to break up the fields (which, in general, may be unpolarized or elliptically polarized) into two linearly polarized ˆ waves, one parallel to the plane of incidence (formed by the incident wave vector wi and the surface normal n), and the other perpendicular to it, or E0 = E eˆ  + E⊥ eˆ ⊥ ,

H0 = H eˆ  + H⊥ eˆ ⊥ .

(2.79)

This is shown schematically in Fig. 2.7. It is readily apparent from the figure that, in the plane of incidence, the ˆ and tangential to the interface (ˆt) may be expressed as unit vectors normal to the interface (n) nˆ = sˆ i cos θ1 − eˆ i sin θ1 = −ˆsr cos θ1 + eˆ r sin θ1 = sˆ t cos θ2 − eˆ t sin θ2 , ˆt = sˆ i sin θ1 + eˆ i cos θ1 = sˆ r sin θ1 + eˆ r cos θ1 = sˆ t sin θ2 + eˆ t cos θ2 .

(2.80a) (2.80b)

As defined in Fig. 2.7 the unit vectors eˆ  , eˆ ⊥ , and sˆ form right-handed coordinate systems for the incident and transmitted waves, i.e., eˆ  = eˆ ⊥ × sˆ ,

eˆ ⊥ = sˆ × eˆ  ,

sˆ = eˆ  × eˆ ⊥ ,

(2.81)

and a left-handed coordinate system for the reflected wave (leading to opposite signs for the above cross-products of unit vectors).8 Therefore, from equation (2.80) eˆ  × nˆ = ±ˆe × sˆ cos θ = −ˆe⊥ cos θ, eˆ ⊥ × nˆ = ±ˆe⊥ × sˆ cos θ ∓ eˆ ⊥ × eˆ  sin θ = eˆ  cos θ + sˆ sin θ = ˆt, ‡

Willebrord van Snel van Royen (1580–1626) Dutch astronomer and mathematician, who discovered Snell’s law in 1621.

8. This is necessary for consistency, i.e., for normal incidence there should not be any difference between parallel and perpendicular polarized waves.

Radiative Property Predictions from Electromagnetic Wave Theory Chapter | 2 45

FIGURE 2.7 Orientation of wave vectors at an interface.

where the top sign applies to the incident and transmitted waves, while the lower sign applies to the reflected component. The second of these relations can also be obtained directly from Fig. 2.7. Using these relations, equations (2.77) and (2.78) may be rewritten in terms of polarized components as 

 Ei + Er cos θ1 = Et cos θ2 ,

(2.82)

Ei⊥ + Er⊥ = Et⊥ ,   Hi + Hr cos θ1 = Ht cos θ2 ,

(2.83)

Hi⊥ + Hr⊥ = Ht⊥ .

(2.85)

(2.84)

The magnetic field may be eliminated through the use of equation (2.25): With w = η0 mˆs = (ν/c 0 )mˆs from equation (2.31) we have m m sˆ × E0 = ± (nˆ ± eˆ  sin θ) × (E eˆ  + E⊥ eˆ ⊥ ) c0μ c 0 μ cos θ   m m =± (E eˆ ⊥ − E⊥ eˆ  ). E cos θˆe⊥ − E⊥ (ˆt − sˆ sin θ) = ± c 0 μ cos θ c0μ

H0 =

(2.86)

Again, the upper sign applies to incident and transmitted waves, and the lower sign to reflected waves. The last two conditions may now be rewritten in terms of the electric field. Assuming the magnetic permeability to be the same in both media, and setting m = n (nonabsorbing media), this leads to (Ei⊥ − Er⊥ ) n1 cos θ1 = Et⊥ n2 cos θ2 ,   Ei − Er n1 = Et n2 .

(2.87) (2.88)

From this one may calculate the reflection coefficient r and the transmission coefficient t as Er n1 cos θ2 − n2 cos θ1 = , Ei n1 cos θ2 + n2 cos θ1 Er⊥ n1 cos θ1 − n2 cos θ2 r⊥ = = , Ei⊥ n1 cos θ1 + n2 cos θ2 Et 2n1 cos θ1 t = = , Ei n1 cos θ2 + n2 cos θ1 Et⊥ 2n1 cos θ1 t⊥ = = . Ei⊥ n1 cos θ1 + n2 cos θ2 r =

(2.89) (2.90) (2.91) (2.92)

46 Radiative Heat Transfer

For an interface between two nonabsorbing media these coefficients turn out to be real, even though the electric field amplitudes are complex. The reflectivity ρ is defined as the fraction of energy in a wave that is reflected and must, therefore, be calculated from the Poynting vector, equation (2.42), so that 

Er = ρ = Ei Si Sr

2 = r2

(2.93)

gives the reflectivity of that part of the wave whose electric field vector lies in the plane of incidence (with its magnetic field normal to it), and   Sr⊥ Er⊥ 2 ρ⊥ = = = r2⊥ (2.94) Ei⊥ Si⊥ is the reflectivity for the part whose electric field vector is normal to the plane of incidence. In terms of these polarized components the overall reflectivity may be stated as “reflected energy for both polarizations, divided by the total incoming energy,” or ρ=

Ei E∗i ρ + Ei⊥ E∗i⊥ ρ⊥ Ei E∗i + Ei⊥ E∗i⊥

.

For unpolarized and circularly polarized light Ei = Ei⊥ , and the reflectivity for the entire wave train is      1 n1 cos θ2 − n2 cos θ1 2 n1 cos θ1 − n2 cos θ2 2 1 + ρ= . ρ + ρ⊥ = 2 2 n1 cos θ2 + n2 cos θ1 n1 cos1 +n2 cos θ2 From this relationship the refractive indices may be eliminated through Snell’s law, giving   1 tan2 (θ1 − θ2 ) sin2 (θ1 − θ2 ) , + ρ= 2 tan2 (θ1 + θ2 ) sin2 (θ1 + θ2 )

(2.95)

(2.96)

(2.97)

which is known as Fresnel’s relation.§ Subroutine fresnel in Appendix F is a generalized version of Fresnel’s relation for an interface between a perfect dielectric and an absorbing medium (see following section), where n = n2 /n1 , k = k2 /n1 , and th = θ1 . The overall transmissivity τ may similarly be evaluated from the Poynting vector, equation (2.42), but the different refractive indices and wave propagation directions in the transmitting and incident media must be considered, so that τ=

n2 cos θ2 2 t = 1 − ρ. n1 cos θ1

(2.98)

An example for the angular reflectivity at the interface between two dielectrics (with n2 /n1 = 1.5) is given in Fig. 2.8. It is seen that, at an angle of incidence of θ1 = θp , r passes through zero resulting in a zero reflectivity for the parallel component of the wave. This angle is known as the polarizing angle or Brewster’s angle,¶ since light reflected from the surface—regardless of the incident polarization—will be completely polarized. Brewster’s angle follows from equations (2.72) and (2.89) as tan θp =

n2 . n1

(2.99)

§

Augustin-Jean Fresnel (1788–1827) French physicist, and one of the early pioneers for the wave theory of light. Serving as an engineer for the French government he studied aberration of light and interference in polarized light. His optical theories earned him very little recognition during his lifetime.

¶

Sir David Brewster (1781–1868) Scottish scientist, entered Edinburgh University at age 12 to study for the ministry. After completing his studies he turned his attention to science, particularly optics. In 1815, the year he discovered the law named after him, he was elected Fellow of the Royal Society.

Radiative Property Predictions from Electromagnetic Wave Theory Chapter | 2 47

FIGURE 2.8 Reflection coefficients and reflectivities for the interface between two dielectrics (n2 /n1 = 1.5).

FIGURE 2.9 Reflection coefficients and reflectivities for the interface between two dielectrics (n1 /n2 = 1.5).

Different behavior is observed if light travels from one dielectric into another, optically less dense medium (n1 > n2 ),9 shown in Fig. 2.9. Examination of equation (2.72) shows that θ2 reaches the value of 90◦ for an angle of incidence θc , called the critical angle, sin θc =

n2 . n1

(2.100)

It is left as an exercise for the reader to show that, for θ1 > θc , light of any polarization is reflected, and nothing is transmitted into the second medium. It is important to realize that upon reflection a wave changes its state of polarization, since E and E⊥ are attenuated by different amounts. If the incident wave is unpolarized (e.g., emission from a hot surface), E and E⊥ are unrelated and will remain so after reflection. If the incident wave is polarized (e.g., laser radiation), the relationship between E and E⊥ will change, causing a change in polarization. Example 2.4. The plane homogeneous wave of the previous examples encounters the flat interface with another dielectric (n2 = 8/3) that is described by the equation z = 0 (i.e., the x-y-plane at z = 0). Calculate the angles of incidence, reflection, and refraction. What fraction of energy of the wave is reflected, and how much is transmitted? In addition, determine the state of polarization of the reflected wave. Solution ˆ From sˆ = 0.8î + 0.6kˆ Since the interface is described by z = 0, the surface normal (pointing into Medium 2) is simply nˆ = k. ◦ and nˆ · sˆ = cos θ1 = 0.6, it follows that the angle of incidence is θ1 = 53.13 off normal, which is equal to the angle of 9. The optical density of a medium is related to the number of atoms contained over a distance equal to the wavelength of the light and is proportional to the refractive index.

48 Radiative Heat Transfer

reflection, while the angle of refraction follows from Snell’s law, equation (2.72), as n1 2 × 0.8 = 0.6, sin θ1 = n2 8/3

sin θ2 =

θ2 = 36.87◦ .

It follows that cos θ2 = 0.8 and the reflection coefficients are calculated from equations (2.89) and (2.90) as 2 × 0.8 − (8/3) × 0.6 1.6 − 1.6 = = 0, 2 × 0.8 + (8/3) × 0.6 3.2 2 × 0.6 − (8/3) × 0.8 3.6 − 6.4 r⊥ = = = −0.28, 2 × 0.6 + (8/3) × 0.8 10.0 r =

and the respective reflectivities follow as ρ = 0 and ρ⊥ = (−0.28)2 = 0.0784. For the present wave and interface, the wave impinges on the surface at Brewster’s angle, i.e., the component of the wave that is linearly polarized in the plane of incidence is totally transmitted. In general, to calculate the overall reflectivity, the wave must be decomposed into two linear polarized components, vibrating within the plane of incidence and perpendicular to it. Fortunately, this was already done in Example 2.3. From √ √ equation (2.95), together with the values of Ei = [5(2 + i)/ 154]E0 and Ei⊥ = [(2 − 5i)/ 154]E0 from the previous example, we obtain ρ=

Ei E∗i ρ + Ei⊥ E∗i⊥ ρ⊥ Ei E∗i + Ei⊥ E∗i⊥

125 × 0 + 29 × 0.0784 = 0.0148, 154

=

and the overall transmissivity τ follows as τ = 1 − ρ = 0.9852. To determine the polarization of the reflected beam, we first need to determine the reflected electric field amplitude vector. From the definition of the reflection coefficient we have Er = r Ei = 0,

2 − 5i E0 Er⊥ = r⊥ Ei⊥ = −0.28 × √ 154

and, from equations (2.50) through (2.53), 0.282 29 E20 = 0.01476 E20 , 154

I = −Q = Er⊥ E∗r⊥ = U = V = 0.

Therefore, the wave remains 100% polarized, but the polarization is not completely linear. Indeed, any polarized radiation reflecting off a surface at Brewster’s angle will become linearly polarized with only a perpendicular component.

The Interface between a Perfect Dielectric and an Absorbing Medium The analysis of reflection and transmission at the interface between two perfect dielectrics is relatively straightforward, since an incident plane homogeneous wave remains plane and homogeneous after reflection and transmission. However, if a plane homogeneous wave is incident upon an absorbing medium, then the transmitted wave is, in general, inhomogeneous. If a beam travels from one absorbing medium into another absorbing medium, then the wave is usually inhomogeneous in both, making the analysis somewhat cumbersome. Fortunately, the interface between two absorbers is rarely important: A wave traveling through an absorbing medium is usually strongly attenuated, if not totally absorbed, before hitting a second absorber. In this section we shall consider a plane homogeneous light wave incident from a perfect dielectric on an absorbing medium. The incident, reflected, and transmitted waves are again described by equations (2.73) through (2.76), except that the wave vector for transmission, wt , may be complex. Thus using equations (2.67) and (2.68), the interface condition may be written as 







E0i × nˆ e−2πiwi ·r + E0r × nˆ e−2πiwr ·r = E0t × nˆ e−2πi(wt ·r−iwt ·r) , −2πiwi ·r

H0i × nˆ e

−2πiwr ·r

+ H0r × nˆ e

−2πi(wt ·r−iw t ·r)

= H0t × nˆ e

,

(2.101) (2.102)

Radiative Property Predictions from Electromagnetic Wave Theory Chapter | 2 49

FIGURE 2.10 Transmission and reflection at the interface between a dielectric and an absorbing medium.

where r is left arbitrary here in order to derive formally the generalized form of Snell’s law although, for convenience, we still assume that the coordinate origin lies on the interface. We note that none of the amplitude vectors, E0i , H0i , etc., depends on location, and that r is a vector to an arbitrary point on the interface, which may be varied independently. Thus, in order for equations (2.101) and (2.102) to hold at any point on the interface, we must have wi · r = wr · r = wt · r,

(2.103)

0 = w t · r,

(2.104)

that is, since r is tangential to the interface, the tangential components of the wave vector w must be continuous   ˆ across the interface, while the tangential component of the attenuation vector w t must be zero, or wt = wt n. Thus, within the absorbing medium, planes of equal amplitude are parallel to the interface, as indicated in Fig. 2.10. Since wr has the same tangential component as wi as well as the same magnitude [cf. equation (2.31)], it follows again that the reflection must be specular, or θr = θi . The continuity of the tangential component for the transmitted wave vector indicates that wi sin θ1 = η0 n1 sin θ1 = wt sin θ2 .

(2.105)

The wave vector for transmission, wt , may be eliminated from equation (2.105) by using equation (2.31): 2   2 2 2 2 2 wt · wt = wt 2 − w t − 2iwt · wt = η0 m2 = η0 (n2 − k2 − 2in2 k2 ),

(2.106a)

2 2 2 2 wt 2 − w t = η0 (n2 − k2 ),

(2.106b)

w t

(2.106c)

or wt

·

=

wt w t

cos θ2 =

η20 n2 k2 .

Thus, equations (2.105) and (2.106) constitute three equations in the three unknowns θ2 , wt , and w t . This system of equations may be solved to yield 

 ) wt cos θ2 2 1 ' ( 2 p = = (n2 − k22 − n21 sin2 θ1 )2 + 4n22 k22 + (n22 − k22 − n21 sin2 θ1 ) , η0 2   2 ( ' ) wt 1 q2 = = (n22 − k22 − n21 sin2 θ1 )2 + 4n22 k22 − (n22 − k22 − n21 sin2 θ1 ) , η0 2 2

(2.107a) (2.107b)

50 Radiative Heat Transfer

and the refraction angle θ2 may be calculated from equation (2.105) as p tan θ2 = n1 sin θ1 .

(2.108)

Equation (2.108) together with equations (2.107) is known as the generalized Snell’s law. The reflection coefficients are calculated in the same fashion as was done for two dielectrics (left as an exercise). This leads to * r = * r⊥ =

2  Er n21 (wt cos θ2 − iw t ) − m2 wi cos θ1 = 2  , 2  Ei n1 (wt cos θ2 − iw t ) + m2 wi cos θ1

(2.109a)

   Er⊥ wi cos θ1 − (wt cos θ2 − iwt ) , =   Ei⊥ wi cos θ1 + (wt cos θ2 − iw t )

(2.109b)

where the tilde has been added to indicate that the reflection coefficients are now complex. From equations (2.106) through (2.107) we find m22 =

p2 − q2 − 2ipq = p2 (1 + tan2 θ2 ) − q2 − 2ipq = p2 − q2 + n21 sin2 θ1 − 2ipq. cos2 θ2

(2.110)

Eliminating the wave vectors, the reflection coefficients may be written as * r =

n1 (p − iq) − (p2 − q2 + n21 sin2 θ1 − 2ipq) cos θ1

n1 (p − iq) + (p2 − q2 + n21 sin2 θ1 − 2ipq) cos θ1 n1 cos θ1 − p + iq * . r⊥ = n1 cos θ1 + p − iq

,

(2.111a) (2.111b)

The expression for * r may be simplified by dividing the numerator (and denominator) of * r by cos θ1 times the numerator (or denominator) of * r⊥ . This operation leads to * r =

p − n1 sin θ1 tan θ1 − iq * r⊥ . p + n1 sin θ1 tan θ1 − iq

(2.112)

Finally, the reflectivities are again calculated as ρ = * r∗ = r* ρ⊥ = * r⊥* r∗⊥ =

(p − n1 sin θ1 tan θ1 )2 + q2 ρ⊥ , (p + n1 sin θ1 tan θ1 )2 + q2

(2.113a)

(n1 cos θ1 − p)2 + q2 . (n1 cos θ1 + p)2 + q2

(2.113b)

Subroutine fresnel in Appendix F calculates ρ , ρ⊥ , and ρ = (ρ +ρ⊥ )/2 from this generalized version of Fresnel’s relation for an interface between a perfect dielectric and an absorbing medium, where n = n2 /n1 , k = k2 /n1 , and th = θ1 . We note that for normal incidence θ1 = θ2 = 0, resulting in p = n2 , q = k2 and ρ = ρ ⊥ =

(n1 − n2 )2 + k22 (n1 + n2 )2 + k22

.

(2.114)

The directional behavior of the reflectivity for a typical metal with n2 = 4.46 and k2 = 31.5 (corresponding to the experimental values for aluminum at 3.1 μm [6]) exposed to air (n1 = 1) is shown in Fig. 2.11. Example 2.5. Redo Example 2.4 for a metallic interface, i.e., the plane homogeneous wave of the previous examples encounters the flat interface with a metal (n2 = k2 = 90), which again is described by the equation z = 0. Calculate the incidence, reflection, and refraction angles. What fraction of energy of the wave is reflected, and how much is transmitted?

Radiative Property Predictions from Electromagnetic Wave Theory Chapter | 2 51

FIGURE 2.11 Directional reflectivity for a metal (aluminum at 3.1 μm with n 2 = 4.46, k 2 = 31.5) in contact with air (n1 = 1).

Solution If n2 and k2 are much larger than n1 it follows from equations (2.107) that p ≈ n2 and q ≈ k2 and, from equation (2.105), n1 sin θ1 ≈ n2 tan θ2 ≈ n2 sin θ2 (i.e., as long as n2  n1 , Snell’s law between dielectrics holds) and it follows that θ2 = 1.02◦ . With n2 = k2 equations (2.113) reduce to ρ⊥ = ρ =

(n1 cos θ1 − n2 )2 + n22 (n1 cos θ1 + n2 )2 + n22

=

(1.2 − 90)2 + 902 = 0.9737, (1.2 + 90)2 + 902

(n2 −n1 sin θ1 tan θ1 )2 +n22 (n2 +n1 sin θ1 tan θ1 )2 +n22

ρ⊥ =

(90 − 2×0.82 /0.6)2 +902 ×0.9737 = 0.9286, (90+2×0.82 /0.6)2 +902

and the total reflectivity is again evaluated from equation (2.95) as ρ=

Ei E∗i ρ + Ei⊥ E∗i⊥ ρ⊥ Ei E∗i + Ei⊥ E∗i⊥

=

125 × 0.9286 + 29 × 0.9737 = 0.9371. 154

Thus, nearly 94% of the radiation is being reflected (and even more would have been reflected if the metal was surrounded by air with n ≈ 1), and only 6% is transmitted into the metal, where it undergoes total attenuation after a very short distance because of the large value of k2 : equation (2.42) shows that the transmission reaches its 1/e value at 4πη0 k2 z = 1,

or

z = 1/(4π × 2500 × 90) = 3.5 × 10−7 cm = 0.0035 μm.

Reflection and Transmission by a Thin Film or Slab As a final topic we shall briefly consider the reflection and transmission by a thin film or slab of thickness d and complex index of refraction m2 = n2 − ik2 , embedded between two media with indices of refraction m1 and m3 , as illustrated in Fig. 2.12. While the theory presented in this section is valid for slabs of arbitrary thickness, it is most appropriate for the study of interference wave effects in thin films or coatings. When an electromagnetic wave is reflected by a thin film, the waves reflected from both interfaces have different phases and interfere with one another (i.e., they may augment each other for small phase differences, or cancel each other for phase differences of 180◦ ). For thick slabs, such as window panes, geometric optics provides a much simpler vehicle to determine overall reflectivity and transmissivity. However, for an antireflective coating on a window, thin film optics should be considered.

52 Radiative Heat Transfer

FIGURE 2.12 Reflection and transmission by a slab.

Normal Incidence Since the computations become rather cumbersome, we shall limit ourselves to the simpler case of normal incidence (θ = 0). For more detailed discussions, including oblique incidence angles, the reader is referred to books on the subject such as the one by Knittl [7] or to the very readable monograph by Anders [8]. Consider the slab shown in Fig. 2.12: The wave incident at the left interface is partially reflected, and partially transmitted toward the second interface. At the second interface, again, the wave is partially reflected and partially transmitted into Medium 3. The reflected part travels back to the first interface where a part is reflected back toward the second interface, and a part is transmitted into Medium 1, i.e., it is added to the reflected wave, etc. Therefore, the reflected wave Er and the transmitted wave Et consist of many contributions, and ˆ respectively. Thus, the inside Medium 2 there are two waves E+2 and E−2 traveling into the directions nˆ and −n, boundary conditions, equations (2.67) and (2.68), may be written for the first interface, similar to equations (2.82) through (2.85), as Ei + Er = E+2 + E−2 ,

z = r · nˆ = 0 :

Hi + Hr =

H2+

+

(2.115)

H2− ,

(2.116)

where polarization of the beam does not appear since at normal incidence E = E⊥ . The magnetic field may again be eliminated using equation (2.25), as well as wi = −wr = η0 m1 nˆ and w+ = −w− = η0 m2 nˆ [from equation (2.31)], or (Ei − Er )m1 = (E+2 − E−2 )m2 .

(2.117)

The boundary condition at the second interface follows [similar to equations (2.101) and (2.102)] as E+2 e−2πiη0 m2 d + E−2 e+2πiη0 m2 d = Et e−2πiη0 m3 d

z = r · nˆ = d :

(E+2 e−2πiη0 m2 d

−

E−2 e+2πiη0 m2 d )m2

−2πiη0 m3 d

= Et e

(2.118) m3 .

(2.119)

Equations (2.115), (2.117), (2.118), and (2.119) are four equations in the unknowns Er , E+2 , E−2 , and Et , which may be solved for the reflection and transmission coefficients of a thin film. After some algebra one obtains * rfilm =

* r23 e−4πiη0 dm2 Er r12 + * = , Ei r23 e−4πiη0 dm2 1 +* r12*

(2.120)

Radiative Property Predictions from Electromagnetic Wave Theory Chapter | 2 53

* t12* t23 e−2πiη0 dm2 Et e−2πiη0 dm3 * tfilm = = , Ei r23 e−4πiη0 dm2 1 +* r12*

(2.121)

where * ri j and * ti j are the complex reflection and transmission coefficients of the two interfaces, m1 − m2 , m1 + m2 2m1 * t12 = , m1 + m2

m2 − m3 ; m2 + m3 2m2 * t23 = . m2 + m3

* r12 =

* r23 =

(2.122a) (2.122b)

To evaluate the thin film reflectivity and transmissivity from the complex coefficients, it is advantageous to write the coefficients in polar notation (cf., for example, Wylie [3]), * ri j = ri j eiδi j ,

ri j = |* ri j |,

tan δi j =

* ti j = ti j ei i j ,

ti j = |* ti j |,

tan i j =

(* ri j ) (* ri j ) (* ti j ) (* ti j )

,

(2.123a)

,

(2.123b)

where ri j and ti j are the absolute values, and δi j and i j the phase angles of the coefficients. Care must be taken in the evaluation of phase angles, since the tangent has a period of π, rather than 2π: The correct quadrant for δi j and i j is found by inspecting the signs of the real and imaginary parts of * ri j and * ti j , respectively. This calculation leads, after more algebra, to the reflectivity, Rfilm , and transmissivity, Tfilm , of the thin film as Rfilm = * r* r∗ = Tfilm =

r212 + 2r12 r23 e−κ2 d cos(δ12 − δ23 + ζ2 ) + r223 e−2κ2 d 1 + 2r12 r23 e−κ2 d cos(δ12 + δ23 − ζ2 ) + r212 r223 e−2κ2 d

,

n3 **∗ τ12 τ23 e−κ2 d , tt = n1 1 + 2r12 r23 e−κ2 d cos(δ12 + δ23 − ζ2 ) + r212 r223 e−2κ2 d

(2.124)

(2.125)

where r2i j = ρi j =

(ni − nj )2 + (ki − k j )2 (ni + nj )2 + (ki + k j )2

,

4(n2i + ki2 ) ni , ni nj (ni + nj )2 + (ki + k j )2 2(ni k j − nj ki ) tan δi j = 2 , ni + ki2 − (nj2 + k2j ) nj

t2i j = τi j =

κi = 4πη0 ki ,

ζi = 4πη0 ni d.

(2.126a) (2.126b) (2.126c) (2.126d)

The correct quadrant for δi j is found by checking the sign of both the numerator and denominator in equation (2.126c) (which, while different from the real and imaginary parts of * ri j , carry their signs). If both adjacent media, i and j, are dielectrics then * ri j = ri j is real. In that case we set δi j = 0 and let ri j carry a sign. The definition of the thin film transmissivity includes the factor (n3 /n1 ), since it is the magnitude of the transmitted and incoming Poynting vector, equation (2.42), that must be compared. Example 2.6. Determine the reflectivity and transmissivity of a 5 μm thick manganese sulfide (MnS) crystal (n = 2.68, k 1), suspended in air, for the wavelength range between 1 μm and 1.25 μm. Solution Assuming n1 = n3 = 1, k1 = k2 = k3 = 0, and n2 = 2.68 and substituting these into equations (2.126) leads to n2 − 1 2 2n2 ; t12 = , t23 = ; n2 + 1 n2 + 1 n2 + 1 0 0 tan δ12 = = 0. = 0; tan δ23 = 2 2 1 − n2 n2 − 1

r12 = r23 =

54 Radiative Heat Transfer

FIGURE 2.13 Normal reflectivity of a thin film with interference effects.

Since the real part of * r12 is negative, i.e., 1 − n22 < 0, it follows that δ12 = π. By similar reasoning δ23 = 0. Alternatively, since all media are dielectrics, we could have set δ12 = δ23 = 0 and r12 = −r23 . Thus, with κ2 = 0, the reflectivity and transmissivity of a dielectric thin film follow as Rfilm = Tfilm =

2ρ12 (1 − cos ζ2 ) 1 − 2ρ12 cos ζ2 + ρ212 τ212 1 − 2ρ12 cos ζ2 + ρ212

,

(2.127)

.

(2.128)

It is a simple matter to show that τ12 = τ23 = 1 − ρ12 and, therefore, Rfilm + Tfilm = 1 for a dielectric medium. Substituting numbers for MnS gives ρ12 = 0.2084 and Rfilm =

0.3995(1 − cos ζ2 ) , 1 − 0.3995 cos ζ2

Tfilm =

0.6005 , 1 − 0.3995 cos ζ2

with ζ2 = 4πn2 dη0 = 168.4 μm η0 = 168.4 μm/λ0 . Rfilm and Tfilm are periodic with a period of Δη0 = 2π/168.4 μm = 0.0373 μm−1 . At λ0 = 1 μm this fact implies Δλ0 = λ20 Δη0 = 0.0373 μm. The reflectivity of the dielectric film in Fig. 2.13 shows a periodic reflectivity with maxima of 0.5709 (at ζ2 = π, 3π, . . .). For values of ζ2 = 2π, 4π, . . ., the reflectivity of the layer vanishes altogether. Also shown is the case of a slightly absorbing film, with k2 = 0.01. Maximum and minimum reflectivity (as well as transmissivity) decrease and increase somewhat, respectively. This effect is less pronounced at larger wavelengths, i.e., wherever the absorption coefficient κ2 is smaller [cf. equation (2.126d)].

While equations (2.124) through (2.126) are valid for arbitrary slab thicknesses, their application to thick slabs becomes problematic as well as unnecessary. Problematic because (i) for d  λ0 the period of reflectivity oscillations corresponds to smaller values of Δλ0 between extrema than can be measured, and (ii) for d  λ0 it becomes rather unlikely that the distance d remains constant within a fraction of λ0 over an extended area. Thick slab reflectivities and transmissivities may be obtained by averaging equations (2.124) and (2.125) over a period through integration, which results in Rslab = ρ12 + Tslab =

ρ23 (1 − ρ12 )2 e−2κ2 d , 1 − ρ12 ρ23 e−2κ2 d

(1 − ρ12 )(1 − ρ23 ) e−κ2 d , 1 − ρ12 ρ23 e−2κ2 d

(2.129)

(2.130)

where for Tslab use has been made of the fact that k1 and k2 must be very small, if an appreciable amount of energy is to reach Medium 3. The same relations for thick sheets without wave interference will be developed in the following chapter through geometric optics.

Radiative Property Predictions from Electromagnetic Wave Theory Chapter | 2 55

Oblique Incidence Knittl [7] has shown that equations (2.124) and (2.125) remain valid for each polarization for oblique incidence if the interface reflectivities, ρi j , and transmissivities, τi j , are replaced by their directional values; see, for example, equations (2.113). We will state the final result here, mostly following the development of Zhang [9]. The field reflection and transmission coefficients are then expressed as * t12* r23 e−2iβ t21* , r23 e−2iβ 1 −* r21* * t23 e−iβ t12* * , t= 1 −* r21* r23 e−2iβ

* r =* r12 +

(2.131a) (2.131b)

which are known as Airy’s formulae. Here the interface reflectivity and transmissivity coefficients are given by equations (2.89) through (2.92) for dielectrics, and by equations (2.111) and (2.112) for absorbing media, and the phase shift in Medium 2 is, for a dielectric film, calculated from β = 2πη0 ni d cos θ2 .

(2.131c)

The overall reflectivity of the film follows from Rfilm

2  *  t12* r23 e−2iβ  t21* =* r* r = * r +  ,  12 1 − * r23 e−2iβ  r21* ∗

and, if Media 1 and 3 are dielectrics, the film transmissivity is evaluated as   t23 e−iβ  n3 cos θ3 **∗ n3 cos θ3  * t12* Tfilm = tt =  . n1 cos θ1 n1 cos θ1  1 − * r21* r23 e−2iβ 

(2.132)

(2.133)

As for single interfaces, for random polarization equations (2.132) and (2.133) are evaluated independently for parallel and perpendicular polarizations, followed by averaging.

2.6 Theories for Optical Constants If the radiative properties of a surface—absorptivity, emissivity, and reflectivity—are to be theoretically evaluated from electromagnetic wave theory, the complex index of refraction, m, must be known over the spectral range of interest. A number of classical and quantum mechanical dispersion theories have been developed to predict the phenomenological coefficients (electrical permittivity) and σe (electrical conductivity) as functions of the frequency (or wavelength) of incident electromagnetic waves for a number of different interaction phenomena and types of surfaces. While the complex index of refraction, m = n − ik, is most convenient for the treatment of wave propagation, the complex dielectric function (or relative permittivity), ε = ε − iε , is more appropriate when the microscopic mechanisms are considered that determine the magnitude of the phenomenological coefficients. The two sets of parameters are related by the expression ε = ε − iε =

σe −i = m2

0 2πν 0

(2.134)

[compare equations (2.31) through (2.35)] and, therefore,

= n2 − k 2 ,

0 σe = 2nk, ε = 2πν 0  1   √ 2 n2 = ε + ε + ε2 , 2  1   √ 2 k2 = −ε + ε + ε2 , 2 ε =

where we have again assumed the medium to be nonmagnetic (μ = μ0 ).

(2.135a) (2.135b) (2.136a) (2.136b)

56 Radiative Heat Transfer

FIGURE 2.14 Electron energy bands and band gaps in a solid (shading indicates amount of electrons filling the bands) [2].

Any material may absorb or emit radiative energy at many different wavelengths as a result of impurities (presence of foreign atoms) and imperfections in the ionic crystal lattice. However, a number of phenomena tend to dominate the optical behavior of a substance. In the frequency range of interest to the heat transfer engineer (ultraviolet to midinfrared), electromagnetic waves are primarily absorbed by free and bound electrons or by change in the energy level of lattice vibration (converting a photon into a phonon, i.e., a quantum of lattice vibration). Since electricity is conducted by free electrons, and since free electrons are a major contributor to a solid’s ability to absorb radiative energy, there are distinct optical differences between conductors and nonconductors of electricity. Every solid has a large number of electrons, resulting in a near-continuum of possible energy states (and, therefore, a near-continuum of photon frequencies that can be absorbed). However, these allowed energy states occur in bands. Between the bands of allowed energy states may be band gaps, i.e., energy states that the solid cannot attain. This is schematically shown in Fig. 2.14. If a material has a band gap between completely filled and completely empty energy bands, the material is a nonconductor, i.e., an insulator (wide band gap), or a semiconductor (narrow band gap). If a band of electron energy states is incompletely filled or overlaps another, empty band, electrons can be excited into adjacent energy states resulting in an electric current, and the material is called a conductor. Electronic absorption by nonconductors is likely only for photons with energies greater than the band gap, although sometimes two or more photons may combine to bridge the band gap. An intraband transition occurs when an electron changes its energy level, but stays within the same band (which can only occur in a conductor); if an electron moves into a different band (i.e., overcomes the band gap) the movement is termed an interband transition (and can occur in both conductors and nonconductors). This difference between conductors and nonconductors causes substantially different optical behavior: Insulators tend to be transparent and weakly reflecting for photons with energies less than the band gap, while metals tend to be highly absorbing and reflecting between the visible and infrared wavelengths [2]. During the beginning of the century Lorentz [10] developed a classical theory for the evaluation of the dielectric function by assuming electrons and ions are harmonic oscillators (i.e., springs) subjected to forces from interacting electromagnetic waves. His result was equivalent to the subsequent quantum mechanical development, and may be stated, as described by Bohren and Huffman [2], as ε(ν) = 1 +





ν2p j

j

νj2 − ν2 + iγj ν

,

(2.137)

Hendrik Anton Lorentz (1853–1928) Dutch physicist. Lorentz studied at Leiden University, where he subsequently served as professor of mathematical physics for the rest of his life. His major work lay in refining the electromagnetic theory of Maxwell. For his theory that the oscillations of charged particles inside atoms were the source of light, he and his student Pieter Zeeman received the 1902 Nobel Prize in Physics. Lorentz is also famous for his Lorentz transformations, which describe the increase of mass of a moving body. These laid the foundation for Einstein’s special theory of relativity.

Radiative Property Predictions from Electromagnetic Wave Theory Chapter | 2 57

FIGURE 2.15 Lorentz model for (a) the dielectric function, (b) the index of refraction, and normal, spectral reflectivity.

where the summation is over different types of oscillators, νp j is known as the plasma frequency (and ν2p j is proportional to the number of oscillators of type j), νj is the resonance frequency, and γj is the damping factor of the oscillators. Thus, the dielectric function may have a number of bands centered at νj , which may or may not overlap one another. Inspecting equation (2.137), we see that for ν  νj the contribution of band j to ε vanishes, while for ν νj it goes to the constant value of (νp j /νj )2 . Therefore, for any nonoverlapping band i, we may rewrite equation (2.137) as ε(ν) = ε0 +

ν2pi ν2i − ν2 + iγi ν

,

(2.138)

where ε0 incorporates the contributions from all bands with νj > νi . Equation (2.138) may be separated into its real and imaginary components, or 

ε = ε0 + ε =

ν2pi (ν2i − ν2 ) (ν2i − ν2 )2 + γ2i ν2 ν2pi γi ν

(ν2i − ν2 )2 + γ2i ν2

.

,

(2.139a) (2.139b)

The frequency dependence of the real and imaginary parts of the dielectric function for a single oscillating band is shown qualitatively in Fig. 2.15; also shown are the corresponding curves for the real and imaginary parts of the complex index of refraction as evaluated from equation (2.136), along with the qualitative behavior of the normal, spectral reflectivity of a surface from equation (2.114). A strong band with k  0 results in a region with strong absorption around the resonance frequency and an associated region of high reflection: Incoming photons are mostly reflected, and those few that penetrate into the medium are rapidly attenuated. On either side outside the band the refractive index n increases with increasing frequency (or decreasing wavelength); this is called normal dispersion. However, close to the resonance frequency, n decreases with increasing frequency; this decrease is known as anomalous dispersion. Note that ε may become negative, resulting in spectral regions with n < 1. All solids and liquids may absorb photons whose energy content matches the energy difference between filled and empty electron energy levels on separate bands. Since such transitions require a substantial amount of energy, they generally occur in the ultraviolet (i.e., at high frequency). A near-continuum of electron energy levels results in an extensive region of strong absorption (and often many overlapping bands). It takes considerably less energy to excite the vibrational modes of a crystal lattice, resulting in absorption bands in the midinfrared (around 10 μm). Since generally few different vibrational modes exist in an isotropic lattice, such transitions can often be modeled by equation (2.137) with a single band. In the case of electrical conductors photons may also be absorbed to raise the energy levels of free electrons and of bound electrons within partially filled or partially overlapping electron bands. The former, because of the nearly arbitrary energy levels that a free electron may

58 Radiative Heat Transfer

assume, results in a single large band in the far infrared; the latter causes narrower bands in the ultraviolet to infrared.

Problems 2.1 Show that for an electromagnetic wave traveling through a dielectric (m1 = n1 ), impinging on the interface with another, optically less dense dielectric (n2 < n1 ), light of any polarization is totally reflected for incidence angles larger than θc = sin−1 (n2 /n1 ). Hint: Use equations (2.105) and (2.106) with k2 = 0. 2.2 Derive equations (2.109) using the same approach as in the development of equations (2.89) through (2.92). ˆ this implies that E0 is not a vector Hint: Remember that within the absorbing medium, w = w − iw = w sˆ − iw n; normal to sˆ . It is best to assume E0 = E eˆ  + E⊥ eˆ ⊥ + Es sˆ . 2.3 Find the normal spectral reflectivity at the interface between two absorbing media. Hint: Use an approach similar to the one that led to equations (2.89) and (2.90), keeping in mind that all wave vectors will be complex, but that the wave will be homogeneous in both media, i.e., all components of the wave vectors are colinear with the surface normal. 2.4 A circularly polarized wave in air is incident upon a smooth dielectric surface (n = 1.5) with a direction of 45◦ off normal. What are the normalized Stokes’ parameters before and after the reflection, and what are the degrees of polarization? 2.5 A circularly polarized wave in air traveling along the z-axis is incident upon a dielectric surface (n = 1.5). How must the dielectric–air interface be oriented so that the reflected wave is a linearly polarized wave in the y-z-plane? 2.6 A polished platinum surface is coated with a 1 μm thick layer of MgO. (a) Determine the material’s reflectivity in the vicinity of λ = 2 μm (for platinum at 2 μm mPt = 5.29 − 6.71 i, for MgO mMgO = 1.65 − 0.0001 i). (b) Estimate the thickness of MgO required to reduce the average reflectivity in the vicinity of 2 μm to 0.4. What happens to the interference effects for this case?

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

J.M. Stone, Radiation and Optics, McGraw-Hill, New York, 1963. C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley & Sons, New York, 1983. C.R. Wylie, Advanced Engineering Mathematics, 5th ed., McGraw-Hill, New York, 1982. H.C. van de Hulst, Light Scattering by Small Particles, John Wiley & Sons, New York, 1957, also Dover Publications, New York, 1981. S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960, originally published by Oxford University Press, London, 1950. R.C. Weast (Ed.), CRC Handbook of Chemistry and Physics, 68th ed., Chemical Rubber Company, Cleveland, OH, 1988. Z. Knittl, Optics of Thin Films, John Wiley & Sons, New York, 1976. H. Anders, Thin Films in Optics, The Focal Press, New York, London, 1967. Z.M. Zhang, Nano/Microscale Heat Transfer, McGraw-Hill, New York, 2007. H.A. Lorentz, Collected Papers, vol 8, Martinus Nijhoff, The Hague, 1935.

Chapter 3

Radiative Properties of Real Surfaces 3.1 Introduction Ideally, electromagnetic wave theory may be used to predict all radiative properties of any material (reflectivity and transmissivity at an interface, absorption and emission within a medium). For a variety of reasons, however, the usefulness of the electromagnetic wave theory is extremely limited in practice. For one, the theory incorporates a large number of assumptions that are not necessarily good for all materials. Most importantly, electromagnetic wave theory neglects the effects of surface conditions on the radiative properties of these surfaces, instead assuming optically smooth interfaces of precisely the same (homogeneous) material as the bulk material—conditions that are very rarely met in practice. In the real world surfaces of materials are generally coated to varying degree with contaminants, oxide layers, and the like, and they usually have a certain degree of roughness (which is rarely even known on a quantitative basis). Thus, the greatest usefulness of the electromagnetic wave theory is that it provides the engineer with a tool to augment sparse experimental data through intelligent interpolation and extrapolation. Still, it is important to realize that radiative properties of opaque materials depend exclusively on the makeup of a very thin surface layer and, thus, may, for the same material, change from batch to batch and, indeed, overnight. This behavior is in contrast to most other thermophysical properties, such as thermal conductivity, which are bulk properties and as such are insensitive to surface contamination, roughness, and so on. The National Institute of Standards and Technology (NIST, formerly NBS) has recommended to reserve the ending “-ivity” for radiative properties of pure, perfectly smooth materials (the ones discussed in the previous chapter), and “-ance” for rough and contaminated surfaces. Most real surfaces fall into the latter category, discussed in the present chapter. Consequently, we will use the ending “-ance” for the definitions in the following section, and for most surface properties throughout this chapter (and the remainder of this book), unless the surface in question is optically smooth and the property is obtained from electromagnetic wave theory. Note that there will be occasions when either term could be used (“almost smooth” surfaces, comparing experimental data with electromagnetic wave theory, etc.). In the present chapter we shall first develop definitions of all radiative properties that are relevant for real opaque surfaces. We then apply electromagnetic wave theory to predict trends of radiative properties for metals and for dielectrics (electrical nonconductors). These theoretical results are compared with a limited number of experimental data. This is followed by a brief discussion of phenomena that cannot be predicted by electromagnetic wave theory, such as the effects of surface roughness, of surface oxidation and contamination, and of the preparation of “special surfaces” (i.e., surfaces whose properties are customized through surface coatings and/or controlled roughness). Most experimental data available today were taken in the 1950s and 1960s during NASA’s “Golden Age,” when considerable resources were directed toward sending a man to the moon. Interest waned, together with NASA’s funding, during the 1970s and early 1980s. More recently, because of the development of hightemperature ceramics and high-temperature applications, there has been renewed interest in the measurement of radiative surface properties. No attempt is made here to present a complete set of experimental data for radiative surface properties. Extensive data sets of such properties have been collected in a number of references, such as [1–8], although all of these surveys are somewhat outdated. Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00011-0 Copyright © 2022 Elsevier Inc. All rights reserved.

59

60 Radiative Heat Transfer

FIGURE 3.1 Directional variation of surface emittances (a) for several nonmetals and (b) for several metals [9].

3.2 Definitions Emittance The most basic radiative property for emission from an opaque surface is its spectral, directional emittance, defined as

λ (T, λ, sˆ o ) ≡

Iλ (T, λ, sˆ o ) cos θo dΩ o Iλ (T, λ, sˆ o ) , = Ibλ (T, λ) cos θo dΩ o Ibλ (T, λ)

(3.1)

which compares the actual spectral, directional emissive power with that of a black surface at the same conditions. We have added a prime to the letter to distinguish the directional emittance from the hemispherical (i.e., directionally averaged) value, and the subscript λ to distinguish the spectral emittance from the total (i.e., spectrally averaged) value. The direction vector is denoted by sˆ o to emphasize that, for emission, we are considering directions away from a surface (outgoing). Finally, we have chosen wavelength λ as the spectral variable, since this is the preferred variable by most authors in the field of surface radiation phenomena. Expressions identical to equation (3.1) hold if frequency ν or wavenumber η are employed. Some typical trends for experimentally determined directional emittances for actual materials are shown in Fig. 3.1a,b, as given by Schmidt and Eckert [9] (all emittances in these figures have been averaged over the entire spectrum; see the definition of the total, directional emittance below). For nonmetals the directional emittance varies little over a large range of polar angles but decreases rapidly at grazing angles until a value of zero is reached at θ = π/2. Similar trends hold for metals, except that, at grazing angles, the emittance first increases sharply before dropping back to zero (not shown). Note that emittance levels are considerably higher for nonmetals. A spectral surface whose emittance is the same for all directions is called a diffuse emitter, or a Lambert surface [since it obeys Lambert’s law, equation (1.37)]. No real surface can be a diffuse emitter since electromagnetic

Radiative Properties of Real Surfaces Chapter | 3 61

wave theory predicts a zero emittance at θ = π/2 for all materials. However, little energy is emitted into grazing directions, as seen from equation (1.33), so that the assumption of diffuse emission is often a good one. The spectral, hemispherical emittance, defined as

λ (T, λ) ≡

Eλ (T, λ) , Ebλ (T, λ)

(3.2)

compares the actual spectral emissive power (i.e., emission into all directions above the surface) with that of a black surface. The spectral, hemispherical emittance may be related to the directional one through equations (1.33) and (1.35),  2π  π/2 Iλ (T, λ, θ, ψ) cos θ sin θ dθ dψ

λ (T, λ) = 0 0 π Ibλ (T, λ)  2π  π/2 

λ (T, λ, θ, ψ)Ibλ (T, λ) cos θ sin θ dθ dψ , (3.3) = 0 0 π Ibλ (T, λ) which may be simplified to

λ (T, λ) =

1 π



2π



0

π/2

0

λ (T, λ, θ, ψ) cos θ sin θ dθ dψ,

(3.4)

since Ibλ does not depend on direction. For an isotropic surface, i.e., a surface that has no different structure, composition, or behavior for different directions on the surface (azimuthal angle), equation (3.4) reduces to  π/2

λ (T, λ) = 2

λ (T, λ, θ) cos θ sin θ dθ. (3.5) 0

We note that the hemispherical emittance is an average over all solid angles subject to the weight factor cos θ (arising from the directional variation of emissive power). For a diffuse surface, λ does not depend on direction and we find

λ (T, λ) = λ (T, λ).

(3.6)

The total, directional emittance is a spectral average of λ , defined by

 (T, sˆ ) =

I(T, sˆ ) cos θ dΩ I(T, sˆ ) = , Ib (T) cos θ dΩ Ib (T)

or, from equations (1.32) and (1.36),  ∞  ∞  ∞ 1 1 1

 (T, sˆ ) = Iλ dλ =

λ Ibλ dλ = 2 4

 (T, λ, sˆ ) Ebλ (T, λ) dλ. Ib 0 Ib 0 n σT 0 λ

(3.7)

(3.8)

Finally, the total, hemispherical emittance is defined as

(T) =

E(T) , Eb (T)

and may be related to the spectral, hemispherical emittance through ∞  ∞ Eλ (T, λ) dλ 1 0

(T) = = 2 4

λ (T, λ) Ebλ (T, λ) dλ. Eb (T) n σT 0

(3.9)

(3.10)

It is apparent that the total emittance is a spectral average with the spectral blackbody emissive power as a weight factor. If the spectral emittance is the same for all wavelengths then equation (3.10) reduces to

(T) = λ (T).

(3.11)

62 Radiative Heat Transfer

FIGURE 3.2 Directional irradiation onto a surface.

Such surfaces are termed gray. If we have the very special case of a gray, diffuse surface, this implies

(T) = λ =  = λ .

(3.12)

While no real surface is truly gray, it often happens that λ is relatively constant over that part of the spectrum where Ebλ is substantial, making the simplifying assumption of a gray surface warranted. Example 3.1. A certain surface material has the following spectral, directional emittance when exposed to air: ⎧ ⎪ ⎪ ⎨0.9 cos θ, 0 < λ < 2 μm, 

λ (λ, θ) = ⎪ ⎪ ⎩0.3, 2 μm < λ < ∞. Determine the total hemispherical emittance for a surface temperature of T = 500 K. Solution We first determine the hemispherical, spectral emittance from equation (3.5) as ⎧  π/2 ⎪ ⎪ ⎪ ⎨2 × 0.9 0 cos2 θ sin θ dθ = 0.6, 0 < λ < 2 μm,

λ (λ) = ⎪  π/2 ⎪ ⎪ ⎩ 2 × 0.3 0 cos θ sin θ dθ = 0.3, 2 μm < λ < ∞. The total, hemispherical emittance follows from equation (3.10) as   2μm    ∞ 0.6 − 0.3 2μm 1 Ebλ dλ + 0.3 Ebλ dλ = 0.3 + 2 4 Ebλ dλ

(T) = 2 4 0.6 n σT n σT 0 2μm 0   = 0.3 1 + f (1×2 μm×500 K) = 0.3 × (1 + 0.00032) 0.3, where the fractional blackbody emissive power f (nλT) is as defined in equation (1.24). For a temperature of 500 K the spectrum below 2 μm is unimportant, and the surface is essentially gray and diffuse.

Absorptance Unlike emittance, absorptance (as well as reflectance and transmittance) is not truly a surface property, since it depends on the external radiation field, as seen from its definition, equation (1.54). As for emittance we distinguish between directional and hemispherical, as well as spectral and total absorptances. The radiative heat transfer rate per unit wavelength impinging onto an infinitesimal area dA, from the direction of sˆ i over a solid angle of dΩ i is, as depicted in Fig. 3.2, Iλ (r, λ, sˆ i )(cos θi dA) dΩ i ,

Radiative Properties of Real Surfaces Chapter | 3 63

FIGURE 3.3 Kirchhoff’s law for the spectral, directional absorptance.

where we have used the definition of intensity as radiative heat transfer rate per unit area normal to the rays, and per unit solid angle. Iλ is the local radiative intensity at location r (just above the surface). This incoming heat transfer rate, when evaluated per unit surface area dA and per unit incoming solid angle dΩ i , is known as spectral, directional irradiation, Hλ (r, λ, sˆ i ) = Iλ (r, λ, sˆ i ) cos θi .

(3.13)

Irradiation is a heat flux always pointing into the surface. Thus, there is no need to attach a sign to its value, and it is evaluated as an absolute value (in contrast to the definition of net heat flux in Chapter 1). The spectral, directional absorptance at surface location r is then defined as 

αλ (r, λ, sˆ i ) 

≡

Hλ,abs Hλ

,

(3.14)



where Hλ,abs is that part of Hλ that is absorbed by dA. If local thermodynamic equilibrium prevails, the fraction αλ will not change if Hλ increases or decreases. Under this condition we find that the spectral, directional absorptance does not depend on the external radiation field and is a surface property that depends on local temperature, wavelength, and incoming direction. To determine its magnitude, we consider an isothermal spherical enclosure shown in Fig. 3.3, similar to the one used in Section 1.6 to establish the directional isotropy of blackbody intensity. The enclosure coating is again perfectly reflecting except for a small area dAs , which is also perfectly reflecting except over the wavelength interval between λ and λ + dλ, over which it is black. However, the small surface dA suspended at the center is now nonblack. Following the same arguments as for the development of equation (1.34), augmenting the emitted flux by λ and the absorbed flux by αλ , we find immediately αλ (T, λ, θ, ψ) = λ (T, λ, θ, ψ).

(3.15)

Therefore, if local thermodynamic equilibrium prevails, the spectral, directional absorptance is a true surface property and is equal to the spectral, directional emittance. The spectral radiative heat flux incident on a surface per unit wavelength from all directions, i.e., from the hemisphere above dA, is  Hλ (r, λ) = 2π

 Hλ (r, λ, sˆ i ) dΩ i =

Iλ (r, λ, sˆ i ) cos θi dΩ i . 2π

(3.16)

64 Radiative Heat Transfer

Of this the amount absorbed is, from equation (3.14),  αλ (T, λ, sˆ i )Iλ (r, λ, sˆ i ) cos θi dΩ i . 2π

Thus, we define the spectral, hemispherical absorptance as  αλ (T, λ, sˆ i )Iλ (r, λ, sˆ i ) cos θi dΩ i Hλ,abs αλ (r, λ) ≡ = 2π  . Hλ I (r, λ, sˆ i ) cos θi dΩ i 2π λ

(3.17)

Since the incoming radiation, Iλ , depends on the radiation field of the surrounding enclosure, the spectral, hemispherical absorptance normally depends on the entire temperature field and is not a surface property. However, if the incoming radiation is approximately diffuse (i.e., if Iλ is independent of sˆ i ), then the Iλ may be moved outside the integrals in equation (3.17) and cancelled. Then αλ (T, λ) =

1 π

 0

2π



π/2

0

αλ (T, λ, θi , ψi ) cos θi sin θi dθi dψi ,

(3.18)

or, using equations (3.4) and (3.15), αλ (T, λ) = λ (T, λ)

(diffuse irradiation).

(3.19)

This equality also holds if αλ = λ are independent of direction, in which case αλ can be removed from the integral. Therefore, spectral hemispherical absorptances and emittances are equal if (and only if) either the irradiation and/or the spectral, directional absorptance are diffuse (i.e., do not depend on incoming direction). On the other hand, energy incident from a single distant source results in (near-) parallel rays from a unique direction sˆ i , such as irradiation from the sun or from a laser. This is known as collimated irradiation, and leads to Hλ (r, λ) = Hλ (r, λ, sˆ i ) δΩ i = Iλ (r, λ, sˆ i ) cos θi δΩ i

(3.20)

and αλ (T, λ) = αλ (T, λ, sˆ i ) = λ (T, λ, sˆ i )

(collimated irradiation).

(3.21)

Thus, for collimated irradiation there is no difference between directional and hemispherical absorptances. The total irradiation per unit area and per unit solid angle, but over all wavelengths, is  ∞  Iλ (r, λ, sˆ i ) cos θi dλ. (3.22) H (r) = 0

Thus, we may define a total, directional absorptance as ∞ αλ (T, λ, sˆ i )Iλ (r, sˆ i ) dλ ∞ α (r, sˆ i ) ≡ 0 , Iλ (r, sˆ i ) dλ 0

(3.23)

where the factor cos θi has cancelled out since it does not depend on wavelength. Again, α is not normally a surface property but depends on the entire radiation field. However, if the irradiation may be written as Iλ (r, λ, sˆ i ) = C(ˆs i )Ibλ (T, λ),

(3.24)

where C(ˆs) is an otherwise arbitrary function that does not depend on wavelength, i.e., if the incoming radiation is gray (based on the local surface temperature T), then, from equations (3.8) and (3.15), α (T, θ, ψ) =  (T, θ, ψ). Of course, this relation also holds if the surface is gray (i.e., αλ = λ do not depend on wavelength).

(3.25)

Radiative Properties of Real Surfaces Chapter | 3 65

Finally, the total irradiation per unit area from all directions and over the entire spectrum is  ∞ Iλ (r, λ, sˆ i ) cos θi dΩ i dλ. H(r) = 0

(3.26)

2π

Therefore, the total, hemispherical absorptance is defined as ∞  ∞ α (r, λ)H (r, λ) dλ αλ (T, λ, sˆ i )Iλ (r, λ, sˆ i ) cos θi dΩ i dλ λ λ Habs α(r) ≡ = 0 ∞ = 0 2π ∞  . H Hλ (r, λ) dλ I (r, λ, sˆ i ) cos θi dΩi dλ 2π λ 0 0

(3.27)

This absorptance is related to the total hemispherical emittance only for the very special cases of a gray, diffuse surface, equation (3.12), and/or diffuse and gray irradiation, i.e., if Iλ (r, λ, sˆ i ) = CIbλ (T, λ),

(3.28)

where T is the temperature of the surface and C is a constant. Under those conditions we find, again using equation (3.15), α(T) = (T).

(3.29)

Example 3.2. Let the surface considered in the previous example be irradiated by the sun from a 30◦ off-normal direction (i.e., a vector pointing to the sun from the surface forms a 30◦ angle with the outward surface normal). Determine the relevant surface absorptance. Solution Since the sun irradiates the surface from only one direction, but over the entire spectrum, we need to find the total, directional absorptance. From the last example, with θi = 30◦ , we have ⎧ √  ⎪  ⎪ π ⎨0.45 3, 0 < λ < 2 μm,  =⎪ αλ λ, θi = ⎪ ⎩0.3, 6 2 μm < λ < ∞. Since we know that the sun behaves like a blackbody at a temperature of Tsun = 5777 K, we also know the spectral behavior of the sunshine falling onto our surface, or Iλ (λ, θi ) = CIbλ (Tsun , λ),

(3.30)

where C is a proportionality constant independent of wavelength.1 Substituting this into equation (3.23) leads to ∞  

λ (λ, θi )Ibλ (Tsun , λ) dλ π = 0 ∞ α θi = 6 Ibλ (Tsun , λ) dλ 0    2μm  ∞ √ 1 = 2 4 0.45 3 Ebλ (Tsun , λ) dλ + 0.3 Ebλ (Tsun , λ) dλ n σTsun 0 2μm √ = 0.3 + (0.45 3 − 0.3) f (1×2×5777) = 0.3 + (0.779 − 0.3) × 0.93962 = 0.750. In contrast to the previous example we find that at a temperature of 5777 K the spectrum above 2 μm is of very little importance, and the surface is again essentially gray.

We realize from this example that (i) if a surface is irradiated from a gray source (i.e., spectrally proportional to a blackbody) at temperature Tsource , and (ii) if the spectral, directional emittance of the surface is independent of temperature (as it is for most surfaces with good degree of accuracy), then the total absorptance is equal to its total emittance evaluated at the source temperature, or α = (Tsource ). This relation holds on a directional basis, and also for hemispherical values if the irradiation is diffuse. 1. As we have seen in Section 1.7, this constant is equal to unity.

(3.31)

66 Radiative Heat Transfer

FIGURE 3.4 The bidirectional reflection function.

Reflectance The reflectance of a surface depends on two directions: the direction of the incoming radiation, sˆ i , and the direction into which the reflected energy travels, sˆ r . Therefore, we distinguish between total and spectral values, and between a number of directional reflectances. The heat flux per unit wavelength impinging on an area dA from a direction of sˆ i over a solid angle of dΩ i was given by equation (3.13) as Hλ dΩ i = Iλ (r, λ, sˆ i ) cos θi dΩ i .

(3.32)

Of this, the finite fraction αλ will be absorbed by the surface (assuming it to be opaque), and the rest will be reflected into all possible directions (total solid angle 2π). Therefore, in general, only an infinitesimal fraction will be reflected into an infinitesimal cone of solid angle dΩ r around direction sˆ r , as shown in Fig. 3.4. Denoting (r, λ, sˆ i , sˆ r ) dΩ r we obtain the reflected energy within the cone dΩ r as this fraction by ρ λ ˆ i , sˆ r ) dΩ r . dIλ (r, λ, sˆ i , sˆ r ) dΩ r = (Hλ dΩ i )ρ λ (r, λ, s

(3.33)

(r, λ, sˆ i , sˆ r ) The spectral, bidirectional reflection function, or bidirectional reflection distribution function (BRDF)2 ρ λ is directly proportional to the magnitude of reflected light that travels into the direction of sˆ r , ˆ i , sˆ r ) = ρ λ (r, λ, s

dIλ (r, λ, sˆ i , sˆ r ) . Iλ (r, λ, sˆ i ) cos θi dΩ i

(3.34)

Equation (3.34) is the most basic of all radiation properties: All other radiation properties of an opaque surface can be related to it. However, experimental determination of this function for all materials, temperatures, wavelengths, incoming directions, and outgoing directions would be a truly Herculean task, limiting its practicality. One may readily show that the law of reciprocity holds for the spectral, bidirectional reflection function (cf. McNicholas [10] or Siegel and Howell [11]), ˆ i , sˆ r ) = ρ ρ s r , −ˆs i ), λ (r, λ, s λ (r, λ, −ˆ

(3.35a)

or  ρ λ (r, λ, θi , ψi , θr , ψr ) = ρλ (r, λ, θr , ψr , θi , ψi ).

(3.35b)

This is done with another variation of Kirchhoff’s law by placing a surface element into an isothermal black enclosure and evaluating the net heat transfer rate—which must be zero—between two arbitrary, infinitesimal 2. ρ is sometimes referred to as a bidirectional reflectance; we avoid this nomenclature since the bidirectional reflectance function is not a λ fraction (i.e., constrained to values between 0 and 1), but may be larger than unity.

Radiative Properties of Real Surfaces Chapter | 3 67

FIGURE 3.5 Normalized bidirectional reflection function for magnesium oxide [12].

surface elements on the enclosure wall. The sign change on the right-hand side of equation (3.35) emphasizes < ∞. that sˆ i points into the surface, while sˆ r points away from it. Examination of equation (3.34) shows that 0 ≤ ρ λ  → ∞ implies that a finite fraction of H is reflected into an infinitesimal cone of solid Reaching the limit of ρ λ λ angle dΩ r . Such ideal behavior is achieved by an optically smooth surface, resulting in specular reflection (perfect = 0 for all sˆ r except the specular direction θr = θi , ψr = ψi + π, for mirror). For a specular reflector we have ρ λ → ∞ (see Fig. 3.4). which ρ λ Some measurements by Torrance and Sparrow [12] for the bidirectional reflection function are shown in Fig. 3.5 for magnesium oxide, a material widely used in radiation experiments because of its diffuse reflectance, as defined in equation (3.38) below, in the near infrared (discussed in the last part of this chapter). The data in Fig. 3.5 are for an average surface roughness of 1 μm and are normalized with respect to the value in the specular direction. It is apparent that the material reflects rather diffusely at shorter wavelengths, but displays strong specular peaks for λ > 2 μm. A property of greater practical importance is the spectral, directional–hemispherical reflectance, which is defined as the total reflected heat flux leaving dA into all directions due to the spectral, directional irradiation Hλ . With the reflected intensity (i.e., reflected energy per unit area normal to sˆ r ) given by equation (3.33), we have, after multiplying with cos θr ,  dIλ (r, λ, sˆ i , sˆ r ) cos θr dΩ r  , (3.36) ρλ (r, λ, sˆ i ) ≡ 2π Hλ (r, λ, sˆ i ) dΩ i or  ρλ (r, λ, sˆ i )

 = 2π

ˆ i , sˆ r ) cos θr dΩ r , ρ λ (r, λ, s

(3.37)

where the (Hλ dΩ i ) cancels out since it does not depend on outgoing direction sˆ r . Here we have temporarily  added the superscript “ ” to distinguish the directional–hemispherical reflectance (ρ ) from the hemispherical–  directional reflectance (ρ , defined below). If the reflection function is independent of both sˆ i and sˆ r , then the surface reflects equal amounts into all directions, regardless of incoming direction, and  ρλ (r, λ) = πρ λ (r, λ).

(3.38)

68 Radiative Heat Transfer

Such a surface is called a diffuse reflector. Comparing the definition of the spectral, directional–hemispherical reflectance with that of the spectral, directional absorptance, equation (3.14), we also find, for an opaque surface,  ρλ (r, λ, sˆ i ) = 1 − αλ (r, λ, sˆ i ).

(3.39)

Sometimes it is of interest to determine the amount of energy reflected into a certain direction, coming from all possible incoming directions. Equation (3.33) gives the reflected intensity due to a single incoming direction. Integrating this expression over the entire hemisphere of incoming directions leads to     ˆ i , sˆ r ) Iλ (r, λ, sˆ i ) cos θi dΩ i . Iλ (r, λ, sˆ r ) = ρλ (r, λ, sˆ i , sˆ r ) Hλ (r, λ, sˆ i ) dΩ i = ρ (3.40) λ (r, λ, s 2π

2π

On the other hand, the spectral, hemispherical irradiation is  Iλ (r, λ, sˆ i ) cos θi dΩ i . Hλ (r, λ) =

(3.41)

2π

If the surface were a perfect reflector, it would reflect all of Hλ , and it would reflect it equally into all outgoing directions. Thus, for the ideal case, the outgoing intensity would be, from equation (1.36), Hλ /π. Consequently, the spectral, hemispherical–directional reflectance is defined as  ρ (r, λ, sˆ i , sˆ r ) Iλ (r, λ, sˆ i ) cos θi dΩ i Iλ (r, λ, sˆ r )  λ = 2π 1  ρλ (r, λ, sˆ r ) ≡ . (3.42) Hλ (r, λ)/π ˆ π 2π Iλ (r, λ, s i ) cos θi dΩ i For the special case of diffuse irradiation (i.e., the incoming intensity does not depend on sˆ i ) equation (3.42) reduces to   ˆ i , sˆ r ) cos θi dΩ i , ρλ (r, λ, sˆ r ) = ρ (3.43) λ (r, λ, s 2π

which is identical to equation (3.37) if the reciprocity of the bidirectional reflection function, equation (3.35), is invoked. Thus, for diffuse irradiation,   ρλ  (r, λ, sˆ r ) = ρλ (r, λ, sˆ i ), or

sˆ i = −ˆs r ,

  ρλ  (r, λ, θr , ψr ) = ρλ (r, λ, θi = θr , ψi = ψr ),

(3.44a) (3.44b)

that is, reciprocity exists between the spectral directional–hemispherical and hemispherical–directional reflectances for any given irradiation/reflection direction. Use of this fact is often made in experimental measurements: While the directional–hemispherical reflectance is of great practical importance, it is very difficult to measure; the hemispherical–directional reflectance, on the other hand, is not very important but readily measured (see Section 3.11). Finally, we define a spectral, hemispherical reflectance as the fraction of the total irradiation from all directions reflected into all directions. From equation (3.36) we have the heat flux reflected into all directions for a single direction of incidence, sˆ i , as  ρλ (r, λ, sˆ i ) Hλ (r, λ, sˆ i ) dΩ i . Integrating this expression as well as Hλ itself over all incidence angles gives  ρλ (r, λ) =

2π

   ρλ (r, λ, sˆ i ) Hλ (r, λ, sˆ i ) dΩi ρ (r, λ, sˆ i ) Iλ (r, λ, sˆ i ) cos θi dΩ i 2π λ   = . H (r, λ, sˆ i ) dΩ i I (r, λ, sˆ i ) cos θi dΩ i 2π λ 2π λ

(3.45)

Radiative Properties of Real Surfaces Chapter | 3 69

If the incident intensity is independent of direction (diffuse irradiation), then equation (3.45) may be simplified again, and 1 ρλ (r, λ) = π

 2π

 ρλ (r, λ, sˆ i ) cos θi dΩ i .

(3.46)

Also, comparing the definitions of spectral, hemispherical absorptance and reflectance, we obtain, for an opaque surface, ρλ (r, λ) = 1 − αλ (r, λ).

(3.47)

Finally, as for emittance and absorptance we need to introduce spectrally-integrated or “total” reflectances. This is done by integrating numerator and denominator independently over the full spectrum for each of the spectral reflectances, leading to the following relations: Total, bidirectional reflection function ∞ ρ (r, sˆ i , sˆ r ) =

0

ρ (r, λ, sˆ i , sˆ r ) Iλ (r, λ, sˆ i ) dλ λ ∞ ; Iλ (r, λ, sˆ i ) dλ 0

(3.48)

 ρλ (r, λ, sˆ i ) Iλ (r, λ, sˆ i ) dλ ∞ ; ˆ I (r, λ, s ) dλ λ i 0

(3.49)

Total, directional–hemispherical reflectance ∞



ρ (r, sˆ i ) =

0

Total, hemispherical–directional reflectance  ρ  (r, sˆ r ) =

 ∞  ρλ  (r, λ, sˆ r ) 2π Iλ (r, λ, sˆ i ) cos θi dΩ i dλ 0  ∞ ; ˆ I (r, λ, s ) cos θ dΩ dλ λ i i i 2π 0

(3.50)

 ρλ (r, λ) 2π Iλ (r, λ, sˆ i ) cos θi dΩi dλ  ∞ . I (r, λ, sˆ i ) cos θi dΩ i dλ 2π λ 0

(3.51)

Total, hemispherical reflectance ∞ ρ(r) =

0

The reciprocity relations in equations (3.35) and (3.44) also hold for total reflectances (subject to the same restrictions), as do the relations between reflectance and absorptance, equations (3.39) and (3.47). The rather confusing array of radiative property definitions and their interrelationships have been summarized in Table 3.1 (property definitions) and Table 3.2 (property interrelations).

3.3 Predictions from Electromagnetic Wave Theory In Chapter 2 we developed in some detail how the spectral, directional–hemispherical reflectivity of an optically smooth interface (specular reflector) can be predicted by the electromagnetic wave and dispersion theories. Before comparing such predictions with experimental data, we shall briefly summarize the results of Chapter 2. Consider an electromagnetic wave traveling through air (refractive index = 1), hitting the surface of a conducting medium (complex index of refraction m = n − ik) at an angle of θ1 with the surface normal (cf. Fig. 3.6).

70 Radiative Heat Transfer

TABLE 3.1 Summary of definitions for radiative properties of surfaces. Property

Symbol

Equation

Comments

Spectral, directional

λ (T, λ, θ, ψ)

(3.1)

hemispherical

λ (T, λ)

(3.4)

directional average of λ (over outgoing directions)

 (T, θ, ψ)

(3.8)

spectral average of λ (with Ibλ as weight factor)

(T)

(3.10)

directional and spectral average of λ

Spectral, directional

αλ (T, λ, θ, ψ)

(3.14)

hemispherical

αλ (Iλ,in , T, λ)

(3.17)

directional average of αλ (over incoming directions)

Total, directional

α (Iin , T, θ, ψ)

(3.23)

spectral average of αλ (with Iλ,in as weight factor)

α(Iin , T)

(3.27)

directional and spectral average of αλ

ρ (T, λ, θi , ψi , θr , ψr ) λ

(3.34)

Emittance

Total, directional hemispherical

depends on incoming intensity Iin

Absorptance

hemispherical

depends on incoming intensity Iin

Reflectance Spectral, bidirectional

reflection function, 0 ≤ ρ ≤∞ λ

directional–hemispherical

 ρ (I

λ,in , T, λ, θi , ψi )

(3.37)

integral of ρ over outgoing direcλ tions

hemispherical–directional

ρλ (Iλ,in , T, λ, θr , ψr )

(3.42)

directional average of ρ over inλ coming directions

hemispherical

ρλ (Iλ,in , T, λ)

(3.45)

directional average of ρλ (incoming and outgoing directions)

Total, bidirectional

ρ (Iin , T, θi , ψi , θr , ψr )

(3.48)

spectral average of ρ λ (with Iλ,in as weight factor)

(3.49)

integral of ρ over outgoing directions

λ





directional–hemispherical

ρ (Iin , T, θi , ψi )

hemispherical–directional

ρ  (Iin , T, θr , ψr )

(3.50)

directional average of ρ over incoming directions

ρ(Iin , T)

(3.51)

directional and spectral average of  ρλ



hemispherical

TABLE 3.2 Summary of relations between radiative properties of surfaces. Property Spectral, directional

Relation αλ (T, λ, θ, ψ)

= =



1 − ρλ (T, λ, θ, ψ)

λ (T, λ, θ, ψ)

αλ (T, λ) = 1 − ρλ (T, λ)

Spectral, hemispherical

= λ (T, λ)

Total, directional

 α (T, θ, ψ) = 1 − ρ (T, θ, ψ) =  (T, θ, ψ) α (T

Total, hemispherical

s , T, θ, ψ)

=

 (T

s , θ, ψ)

Restrictions opaque surfaces (θ, ψ = incoming directions) none (θ, ψ = outgoing directions) opaque surfaces (values depend on directional distribution of source) irradiation and/or λ independent of direction (diffuse) opaque surfaces (values depend on spectral distribution of source)

λ independent of wavelength (gray)

source is gray with source temperature Ts , and λ is independent of T, or Ts = T

α(T) = 1 − ρ(T)

opaque surfaces (values depend on spectral and directional distribution of source)

= (T)

λ independent of wavelength and direction (gray and diffuse)

α(Ts , T) = (Ts )

source is gray and diffuse with source temperature Ts , and λ is independent of T, or Ts = T

Radiative Properties of Real Surfaces Chapter | 3 71

FIGURE 3.6 Transmission and reflection at an interface between air and an absorbing medium.

Fresnel’s relations predict the reflectivities for parallel- and perpendicular-polarized light from equations (2.107) through (2.113)3 as

where

ρ =

(p − sin θ1 tan θ1 )2 + q2 ρ⊥ , (p + sin θ1 tan θ1 )2 + q2

(3.52)

ρ⊥ =

(cos θ1 − p)2 + q2 , (cos θ1 + p)2 + q2

(3.53)

 − − sin θ1 + + (n − k − sin θ1 ) , (  1 2 2 2 2 2 2 2 2 2 2 q = (n − k − sin θ1 ) + 4n k − (n − k − sin θ1 ) . 2

1 p = 2 2

(

(n2

2

k2

)2

4n2 k2

2

2

2

(3.54) (3.55)

Nonreflected light is refracted into the medium, traveling on at an angle of θ2 with the surface normal, as predicted by the generalized Snell’s law, from equation (2.108), p tan θ2 = sin θ1 .

(3.56)

For normal incidence θ1 = θ2 = 0, and equations (3.52) through (3.55) simplify to p = n, q = k, and ρnλ = ρ = ρ⊥ =

(n − 1)2 + k2 . (n + 1)2 + k2

(3.57)

If the incident radiation is unpolarized, the reflectivity may be calculated as an average, i.e., ρ = 12 (ρ + ρ⊥ ).

(3.58)

For a dielectric medium (k = 0), p2 = n2 − sin2 θ1 , and Snell’s law becomes n sin θ2 = sin θ1 . 

(3.59)

3. For simplicity of notation we shall drop the superscripts  for the directional–hemispherical reflectivity whenever there is no possibility of confusion.

72 Radiative Heat Transfer

Therefore, p = n cos θ2 and, with q = 0, Fresnel’s relations reduce to  cos θ2 − n cos θ1 2 ρ = , cos θ2 + n cos θ1   cos θ1 − n cos θ2 2 ρ⊥ = . cos θ1 + n cos θ2 

(3.60a) (3.60b)

Except for the section on semitransparent sheets, in this chapter we shall be dealing with opaque media. For such media ρ + α = 1 and, from Kirchhoff’s law,

λ = αλ = 1 − ρλ .

(3.61)

To predict radiative properties from electromagnetic wave theory, the complex index of refraction, m, must be known, either from direct measurements or from dispersion theory predictions. In the dispersion theory the complex dielectric function, ε = ε − iε , is predicted by assuming that the surface material consists of harmonic oscillators interacting with electromagnetic waves. The complex dielectric function is related to the complex index of refraction by ε = m2 , or  √ ε2 + ε2 ,   √ k2 = 12 −ε + ε2 + ε2 ,

n2 =

1 2



ε +

(3.62a) (3.62b)

where ε =

,

0

ε =

σe ; 2πν 0

is the electrical permittivity, 0 is its value in vacuum, and σe is the medium’s electrical conductivity. Both

and σe are functions of the frequency of the electromagnetic wave ν. For an isolated oscillator (nonoverlapping band) ε is predicted by the Lorentz model, equation (2.139), as ε = ε0 + 

ε =

ν2pi (ν2i − ν2 ) (ν2i − ν2 )2 + γ2i ν2 ν2pi γi ν

(ν2i − ν2 )2 + γ2i ν2

,

,

(3.63a) (3.63b)

where ε0 is the contribution to ε from bands at shorter wavelengths, νi is the resonance frequency, νpi is called the plasma frequency, and γi is an oscillation damping factor. If these three constants can be determined or measured, then n and k can be predicted for all frequencies (or wavelengths) from equation (3.62), and the radiative properties can be calculated for all frequencies (or wavelengths) and all directions from equations (3.52) through (3.55).

3.4 Radiative Properties of Metals In this section we shall briefly discuss how the radiative properties of clean and smooth metallic surfaces (i.e., electrical conductors) can be predicted from electromagnetic wave theory and dispersion theory, and how these predictions compare with experimental data. The variation of the spectral, normal reflectance with wavelength and total, normal properties will be examined, followed by a discussion of the directional dependence of radiative properties and the evaluation of hemispherical reflectances (and emittances). Finally, we will look at the temperature dependence of spectral as well as total properties.

Radiative Properties of Real Surfaces Chapter | 3 73

FIGURE 3.7 Spectral, normal reflectivity at room temperature for aluminum, copper, and silver.

Wavelength Dependence of Spectral, Normal Properties Metals are in general excellent electrical conductors because of an abundance of free electrons. Drude [13] developed an early theory to predict the dielectric function for free electrons that is essentially a special case of the Lorentz model: Since free electrons do not oscillate but propagate freely, they may be modeled as a “spring” with a vanishing spring constant leading to a resonance frequency of νi = 0. Thus the Drude theory for the dielectric function for free electrons follows from equation (3.63) as ε (ν) = ε0 − ε (ν) =

ν2p ν2 + γ2

ν2p γ ν(ν2 + γ2 )

,

.

(3.64a) (3.64b)

Figure 3.7 shows the spectral, normal reflectivity of three metals—aluminum, copper, and silver. The theoretical lines are from Ehrenreich and coworkers [14] (aluminum) and Ehrenreich and Phillip [15] (copper and silver), who semiempirically determined the values of the unknowns ε0 , νp , and γ in equation (3.64). The experimental reflectance data are taken from Shiles and coworkers [16] (aluminum) and Hagemann and coworkers [17] (copper and silver). The agreement between experiment and theory in the infrared is very good. For wavelengths λ > 1 μm the Drude theory has been shown to represent the reflectivity of many metals accurately, if samples are prepared with great care. Discrepancies are due to surface preparation methods and the limits of experimental accuracy. Aluminum has a dip in reflectivity centered at ∼ 0.8 μm; this is due to bound electron transitions that are not considered by the Drude model. Since γ νp always, there exists for each metal a frequency in the vicinity of the plasma frequency, ν νp , where ε = 1 and ε 1 or n 1, k 1: This fact implies that many metals neither reflect nor absorb radiation in the ultraviolet near νp , but are highly transparent! For extremely long wavelengths (very small frequency ν), we find from equations (3.64) and (2.134) that ε =

ν2p νγ

=

σe , 2πν 0

ν γ,

(3.65)

where σe is the (in general, frequency-dependent) electrical conductivity, and σe = 2π 0 ν2p /γ = const = σdc .

(3.66)

74 Radiative Heat Transfer

TABLE 3.3 Inverse relaxation times and dc electrical conductivities for various metals at room temperature [18]. γ, Hz

σdc , Ω−1 cm−1

ν2p = σdc γ/2π0 , Hz2

Lithium

1.85 × 1013

1.09 × 105

3.62 × 1030

Sodium

5.13 × 1012

2.13 × 105

1.96 × 1030

Potassium

3.62 × 1012

1.52 × 105

9.88 × 1029

Cesium

7.56 × 1012

0.50 × 105

6.78 × 1029

Copper

5.89 × 1012

5.81 × 105

6.14 × 1030

Silver

3.88 × 1012

6.29 × 105

4.38 × 1030

Gold

5.49 × 1012

4.10 × 105

4.04 × 1030

Nickel

1.62 × 1013

1.28 × 105

3.72 × 1030

Cobalt

1.73 × 1013

1.02 × 105

3.17 × 1030

Iron

6.63 × 1012

1.00 × 105

1.19 × 1030

Palladium

1.73 × 1013

0.91 × 105

2.83 × 1030

Platinum

1.77 × 1013

1.00 × 105

3.18 × 1030

Metal

Note that at the long-wavelength limit the electrical conductivity becomes independent of wavelength and is known as the dc-conductivity. Since the dc-conductivity is easily measured it is advantageous to recast equation (3.64) as σdc γ/2π 0 , ν(ν + iγ) σdc γ/2π 0 ε = ε0 − 2 , ν + γ2

ε(ν) = ε0 −

ε =

σdc γ2 /2π 0 . ν(ν2 + γ2 )

(3.67a) (3.67b) (3.67c)

Room temperature values for electrical resistivity, 1/σdc , and for electron relaxation time, 1/2πγ, have been given by Parker and Abbott [18] for a number of metals. They have been converted and are reproduced in Table 3.3. Note that these values differ appreciably from those given in Fig. 3.7. No values for ε0 are given; however, the influence of ε0 is generally negligible in the infrared. Extensive sets of spectral data for a large number of metals have been collected by Ordal and coworkers [19] (for a smaller number of metals they also give the Drude parameters, which are also conflicting somewhat with the data of Table 3.3), while a listing of spectral values of the complex index of refraction for a large numbers of metals and semiconductors has been given in a number of handbooks [20–23]. For long wavelengths equation (3.62) may be simplified considerably, since for such case, ε  |ε |, and it follows that n2 ≈ k2 ≈ ε /2 =

σdc σdc λ0 =  1, 4πν 0 4πc 0 0

(3.68)

where λ0 is the wavelength in vacuum. Substituting values for the universal constants c 0 and 0 , equation (3.68) becomes % n k 30λ0 σdc , λ0 in cm, σdc in Ω−1 cm−1 , (3.69) which is known as the Hagen–Rubens relation [24]. For comparison, results from equation (3.69) are also included in Fig. 3.7. It is commonly assumed that the Hagen–Rubens relation may be used for λ0 > 6 μm, although this assumption can lead to serious errors, in particular as far as evaluation of the index of refraction is concerned. While equation (3.69) is valid for the metal being adjacent to an arbitrary material, we will—for notational simplicity—assume for the rest of this discussion that the adjacent material has a refractive index of unity

Radiative Properties of Real Surfaces Chapter | 3 75

(vacuum or gas), that is, λ0 = λ. Substituting equation (3.69) into equation (3.57) leads to 2n2 − 2n + 1 , 2n2 + 2n + 1

ρnλ =

nλ = 1 − ρnλ =

(3.70)

4n . 2n2 + 2n + 1

(3.71)

Since n  1 equation (3.71) may be further simplified to

nλ =

2 2 − 2 + ··· , n n

(3.72a)

and, with equation (3.69), to

nλ √

2 30λ σdc

−

1 , 15λ σdc

λ in cm,

σdc in Ω−1 cm−1 .

(3.72b)

√ This 1/ λ dependence is not predicted by the Drude theory (except for the far infrared), nor is it observed with optically smooth surfaces. However, it often approximates the behavior of polished (i.e., not entirely smooth) surfaces. Example 3.3. Using the constants given in Fig. 3.7 calculate the complex index of refraction and the normal, spectral reflectivity of silver at λ = 6.2 μm, using (a) the Drude theory and (b) the Hagen–Rubens relation. Solution (a) From Fig. 3.7 we have for silver ε0 = 3.4, νp = 2.22 × 1015 Hz, and γ = 4.30 × 1012 Hz. Substituting these into equation (3.64) with ν = c 0 /λ = 2.998 × 108 m/s × (106 μm/m)/6.2 μm = 4.84 × 1013 Hz, we obtain (2.22 × 1015 )2 = 3.4 − 2087 = −2084, (4.84 × 1013 )2 + (4.30 × 1012 )2 + ε = 2087 × 4.30 × 1012 4.84 × 1013 = 185.1. ε = 3.4 −

The complex index of refraction follows from equation (3.62) as  √ 1 −2084 + 20842 + 185.12 = 4.102, 2  √ 1 2 k = 2084 + 20842 + 185.12 = 2088, 2

n2 =

or n = 2.03 and k = 45.7. Finally, the normal reflectivity follows from equation (3.57) as ρnλ =

(1 − 2.03)2 + 45.72 = 0.996. (1 + 2.03)2 + 45.72

(b) Using the Hagen–Rubens relation we find, from equation (3.66), that σdc = 2π × 8.8542 × 10−12 = 6.376×107

 2 + C2 × 2.22 × 1015 Hz 4.30 × 1012 Hz 2 Nm

C2 = 6.376×107 Ω−1 m−1 = 6.376×105 Ω−1 cm−1 . N m2 s

Substituting this value into equation (3.69) yields √ n = k = 30 × 6.2 × 10−4 × 6.376 × 105 = 108.9, and ρnλ = 1 − nλ = 1 −

2 2 2 2 + + =1− = 0.982. n n2 108.9 108.92

The two sets of results may be compared with experimental results of n = 2.84, k = 45.7, and ρnλ = 0.995 [17]. At first glance the Hagen–Rubens prediction for ρnλ appears very good because, for any k  1, ρnλ ≈ 1. The values for n and k show that the Hagen–Rubens relation is in serious error even at a relatively long wavelength of λ = 6.2 μm.

76 Radiative Heat Transfer

FIGURE 3.8 Total, normal emittance of various polished metals as a function of temperature [18].

Total Properties for Normal Incidence The total, normal reflectance and emittance may be evaluated from equation (3.8), with spectral, normal properties evaluated from the Drude theory or from the simple Hagen–Rubens relation. While the Hagen–Rubens relation is not very accurate, it does predict the emittance trends correctly in the infrared, and it does allow an explicit evaluation of total, normal emittance. Substituting equation (3.72) into equation (3.8) leads to an integral that may be evaluated in a similar fashion as for the total emissive power, equation (1.20), and, retaining the first three terms of the series expansion

n = 0.578 (T/σdc )1/2 − 0.178 (T/σdc ) + 0.0584 (T/σdc )3/2 ,

T in K, σdc in Ω−1 cm−1 .

(3.73)

Of course, equation (3.73) is only valid for small values of (T/σdc ), i.e., the temperature of the surface must be such that only a small fraction of the blackbody emissive power comes from short wavelengths (where the Hagen–Rubens relation is not applicable). For pure metals, to a good approximation, the dc-conductivity is inversely proportional to absolute temperature, or σdc = σref

Tref . T

(3.74)

Therefore, for low enough temperatures, the total, normal emittance of a pure metal should be approximately linearly proportional to temperature. Comparison with experiment (Fig. 3.8) shows that this nearly linear relationship holds for many metals up to surprisingly high temperatures; for example, for platinum (T/σdc )1/2 = 0.5 corresponds to a temperature of 2700 K. It is interesting to note that spectral integration of the Drude model results in 30% to 70% lower total emissivities for all metals and, thus, fails to follow experimental trends. Such integration was carried out by Parker and Abbott [18] in an approximate fashion. They attributed the discrepancy to imperfections in the molecular lattice induced by surface preparation and to the anomalous skin effect [25], both of which lower the electrical conductivity in the surface layer.

Directional Dependence of Radiative Properties The spectral, directional reflectivity at the interface between an absorber and a nonabsorber is given by Fresnel’s relations, (3.52) through (3.55). Since, in the infrared, n and k are generally fairly large for metals, one may with little error neglect the sin2 θ1 in equations (3.54) and (3.55), leading to p n and q k. Then, from equations (3.52) and (3.53) the reflectivities for parallel- and perpendicular-polarized light are evaluated from4 ρ =

(n cos θ − 1)2 + (k cos θ)2 , (n cos θ + 1)2 + (k cos θ)2

(3.75a)

4. The simple form for ρ used here is best obtained from the reflection coefficient given by equation (2.111) by neglecting sin2 θ1 and canceling m = n − ik from both numerator and denominator.

Radiative Properties of Real Surfaces Chapter | 3 77

FIGURE 3.9 Spectral, directional reflectance of platinum at λ = 2 μm.

ρ⊥ =

(n − cos θ)2 + k2 . (n + cos θ)2 + k2

(3.75b)

The directional, spectral emissivity (unpolarized) follows as

λ = 1 − 12 (ρ + ρ⊥ ),

(3.76)

and is shown (as reflectance) in Fig. 3.9 for platinum at λ = 2 μm. The theoretical line for room temperature has been calculated with n = 5.29, k = 6.71 from [23]. Comparison with experimental emittances of Brandenberg [26], Brandenberg and Clausen [27], and Price [28] demonstrates the validity of Fresnel’s relations.5 Equation (3.75) may be integrated analytically over all directions to obtain the spectral, hemispherical emissivity from equation (3.5). This was done by Dunkle [29] for the two different polarizations, resulting in     (n2 −k2 ) 8n n 2 2 −1 k

 = 2 2 1 − 2 2 ln (n+1) +k + tan , (3.77a) n+1 n +k n +k k(n2 +k2 )   (n+1)2 +k2 (n2 −k2 ) k

⊥ = 8n 1 − n ln , (3.77b) tan−1 + 2 2 k n +k n(n+1)+k2 1

λ = (  + ⊥ ). (3.77c) 2 Figure 3.10, from Dunkle [30], is a plot of the ratio of the hemispherical and normal emissivities, λ / nλ . For the case of k/n = 1 the dashed line represents results from equation (3.77), while the solid lines were obtained by numerically integrating equations (3.52) through (3.55). For k/n > 1 the two lines become indistinguishable. Hering and Smith [31] reported that equation (3.77) is accurate to within 1–2% for values of n2 + k2 larger than 40 and 3.25, respectively. In view of the large values that n and, in particular, k assume for metals, equation (3.77) is virtually always accurate to better than 2% for metals in the visible and infrared wavelengths. For the reader’s convenience the function emmet is included in Appendix F for the evaluation of equation (3.77). Example 3.4. Determine the spectral, hemispherical emissivity for room-temperature nickel at a wavelength of λ = 10 μm, using (a) the Drude theory and (b) the Hagen–Rubens relation. 5. In the original figure of Brandenberg and Clausen [27] older values for n and k were used that gave considerably worse agreement with experiment.

78 Radiative Heat Transfer

FIGURE 3.10 Ratio of hemispherical and normal spectral emissivity for electrical conductors as a function of n and k [30].

Solution We first need to determine the optical constants n and k from either theory, then calculate the hemispherical emissivity from equation (3.77) or read it from Fig. 3.10. (a) Using values for nickel from Table 3.3 in equation (3.64), we find with ν = c 0 /λ = 2.998 × 108 m/s/10−5 m = 2.998 × 1013 Hz, +  ε = 1.0 − 3.72 × 1030 (2.998 × 1013 )2 + (1.62 × 1013 )2 = 1 − 3204 = −3203, + ε = 3204 × 1.62 × 1013 2.998 × 1013 = 1731,   √ n2 = 0.5 × −3208 + 32082 + 17312 = 219,   √ k2 = 0.5 × 3208 + 32082 + 17312 = 3422, and n = 14.8,

k = 58.5,

k/n = 58.5/14.8 = 3.95.

To use Fig. 3.10, we first determine ρnλ as ρnλ =

13.82 + 58.52 = 0.984, 15.82 + 58.52

and

nλ = 1 − ρnλ = 0.016. From Fig. 3.10 λ / nλ 1.29 and, therefore, λ 0.021. (b) Using the Hagen–Rubens relation we find, from equation (3.72),

nλ = √

2 30 ×

10−3

× 1.28 ×

105

−

1 = 0.032. 15 × 10−3 × 1.28 × 105

√

Further, with n k 30 × 10−3 × 1.28 × 105 = 62.0, we obtain from Fig. 3.10 λ / nλ 1.275 and λ 0.041. The answers from both models differ by a factor of ∼2. This agrees with the trends shown in Fig. 3.7.

Theoretical values for total, directional emissivities are obtained by (numerical) integration of equations (3.75) and (3.76) over the entire spectrum. The directional behavior of total emissivities is similar to that of spectral emissivities, as shown by the early measurements of Schmidt and Eckert [9], as depicted in Fig. 3.1b in a polar diagram (as opposed to the Cartesian representation of Fig. 3.9). The emittances were determined from total radiation measurements from samples heated to a few hundred degrees Celsius.

Radiative Properties of Real Surfaces Chapter | 3 79

FIGURE 3.11 Total, hemispherical emittance of various polished metals as a function of temperature [18].

Total, Hemispherical Emittance Equation (3.77) may be integrated over the spectrum using equation (3.10), to obtain the total, hemispherical emittance of a metal. Several approximate relations, using the Hagen–Rubens limit, have been proposed, notably the ones by Davisson and Weeks [32] and by Schmidt and Eckert [9]. Expanding equation (3.77) into a series of powers of 1/n (with n = k  1), Parker and Abbott [18] were able to integrate equation (3.77) analytically, leading to

(T) = 0.766(T/σdc )1/2 − [0.309 − 0.0889 ln(T/σdc )] (T/σdc ) − 0.0175(T/σdc )3/2 ,

T in K,

σdc in Ω−1 cm−1 . (3.78)

Like the total, normal emittance the total, hemispherical emittance is seen to be approximately linearly proportional to temperature (since σdc ∝ 1/T) as long as the surface temperature is relatively low (so that only long wavelengths are of importance, for which the Hagen–Rubens relation gives reasonable results). Emittances calculated from equation (3.78) are compared with experimental data in Fig. 3.11. Parker and Abbott also integrated the series expansion of equation (3.77) with n and k evaluated from the Drude theory. As for normal emissivities, the Drude model predicts values 30–70% lower than the Hagen–Rubens relations, contrary to experimental evidence shown in Fig. 3.11. Again, the discrepancy was attributed to lattice imperfections and to the anomalous skin effect.

Effects of Surface Temperature The Hagen–Rubens relation, equation (3.72), predicts that the spectral, normal emittance of a metal should be √ proportional to 1/ σdc . Since the electrical conductivity is approximately inversely proportional to temperature, the spectral emittance should, therefore, be proportional to the square root of absolute temperature for long enough wavelengths. This trend should also hold for the spectral, hemispherical emittance. Experiments have shown that this is indeed true for many metals. A typical example is given in Fig. 3.12, showing the spectral dependence of the hemispherical emittance for tungsten for a number of temperatures [33]. Note that the emittance for tungsten tends to increase with temperature beyond a crossover wavelength of approximately 1.3 μm, while the temperature dependence is reversed for shorter wavelengths. Similar trends of a single crossover wavelength have been observed for many metals. The total, normal or hemispherical emittances are calculated by integrating spectral values over all wavelengths, with the blackbody emissive power as weight function. Since the peak of the blackbody emissive power shifts toward shorter wavelengths with increasing temperature, we infer that hotter surfaces emit a higher fraction of energy at shorter wavelengths, where the spectral emittance is higher, resulting in an increase in total emittance as demonstrated in Figs. 3.8 and 3.11. Since the crossover wavelength is fairly short for many metals, the Hagen–Rubens temperature relation often holds for surprisingly high temperatures.

80 Radiative Heat Transfer

FIGURE 3.12 Temperature dependence of the spectral, hemispherical emittance of tungsten [33].

FIGURE 3.13 Spectral, normal reflectivity of α-SiC at room temperature [34].

3.5 Radiative Properties of Nonconductors Electrical nonconductors have few free electrons and, thus, do not display the high reflectance and opaqueness behavior across the infrared as do metals. Semiconductors, as their title suggests, have some free electrons and are usually discussed together with nonconductors; however, they display some of the characteristics of a metal. The radiative properties of pure nonconductors are dominated in the infrared by photon–phonon interaction, i.e., by the photon excitation of the vibrational energy levels of the solid’s crystal lattice. Outside the spectral region of strong absorption by vibrational transitions there is generally a region of fairly high transparency (and low reflectance), where absorption is dominated by impurities and imperfections in the crystal lattice. As such, these spectral regions often show irregular and erratic behavior.

Wavelength Dependence of Spectral, Normal Properties The spectral behavior of pure, crystalline nonconductors is often well described by the single oscillator Lorentz model of equation (3.63). One such material is the semiconductor α-SiC (silicon carbide), a high-temperature ceramic of ever increasing importance. The spectral, normal reflectivity of pure, smooth α-SiC at room temperature is shown in Fig. 3.13, as given by Spitzer and coworkers [34]. The theoretical reflectivity in Fig. 3.13 is evaluated from equations (3.63), (3.62), and (3.57) with ε0 = 6.7, νpi = 4.327 × 1013 Hz, νi = 2.380 × 1013 Hz, and γi = 1.428 × 1011 Hz. Agreement between theory and experiment is superb for the entire range between 2 μm and 22 μm. Inspection of equations (3.63) and (3.62) shows that outside the spectral range 10 μm < λ < 13 μm (or 2.5 × 1013 Hz > ν > 1.9 × 1013 Hz), α-SiC is essentially transparent (absorptive index k 1) and weakly reflecting. Within the range of 10 μm < λ < 13 μm α-SiC is not only highly reflecting but also opaque (i.e.,

Radiative Properties of Real Surfaces Chapter | 3 81

FIGURE 3.14 Spectral, normal reflectivity of MgO at room temperature [36].

FIGURE 3.15 Spectral, normal reflectance of silicon at room temperature [7].

any radiation not reflected is absorbed within a very thin surface layer, since k > 1). The reflectivity drops off sharply on both sides of the absorption band. For this reason materials such as α-SiC are sometimes used as bandpass filters: If electromagnetic radiation is reflected several times by an α-SiC mirror, the emerging light will nearly exclusively lie in the spectral band 10 μm < λ < 13 μm. This effect has led to the term Reststrahlen band (German for “remaining rays”) for absorption bands due to crystal vibrational transitions. Bao and Ruan [35] have demonstrated that the dielectric function for semiconductors can be calculated through density functional theory, resulting in good agreement with experiment for GaAs. Not all crystals are well described by the single oscillator model since two or more different vibrational transitions may be possible and can result in overlapping bands. Magnesium oxide (MgO) is an example of material that can be described by a two-oscillator model (two overlapping bands), as Jasperse and coworkers [36] have shown (Fig. 3.14). The theoretical reflectivities are obtained with the parameters for the evaluation of equation (3.63) given in the figure. Note that for the calculation of ε and ε , equation (3.63) needs to be summed over both bands, i = 1 and 2. From a quantum viewpoint, the second, weaker oscillator is interpreted as the excitation of two phonons by a single photon [37]. Since the radiative properties outside a Reststrahlen band depend strongly on defects and impurities they may vary appreciably from specimen to specimen and even between different points on the same sample. For example, the spectral, normal reflectance of silicon at room temperature is shown in Fig. 3.15 (redrawn from

82 Radiative Heat Transfer

FIGURE 3.16 Refractive indices for various semitransparent materials [20].

data collected by Touloukian and DeWitt [7]). Strong influence of different types and levels of impurities is clearly evident. Therefore, looking up properties for a given material in published tables is problematic unless a detailed description of surface and material preparation is given. Equation (3.63) demonstrates that—outside a Reststrahlen band—ε and, therefore, the absorptive index k of a nonconductor are very small; typically k < 10−6 for a pure substance. While impurities and lattice defects can increase the value of k, it is very unlikely to find values of k > 10−2 for a nonconductor outside Reststrahlen bands. At first glance it might appear, therefore, that all nonconductors must be highly transparent in the near infrared (and the visible). That this is not the case is readily seen from equation (1.58), which relates transmissivity to absorption coefficient. This, in turn, is related to the absorptive index through equation (2.42): τ = e−κs = e−4πks/λ0 .

(3.79)

For a 1 mm thick layer of a material with k = 10−3 at a wavelength (in vacuum) of λ0 = 2 μm, equation (3.79) translates into a transmissivity of τ = exp(−4π × 10−3 × 1/2 × 10−3 ) = 0.002, i.e., the layer is essentially opaque. Still, the low values of k allow us to simplify Fresnel’s relations considerably for the reflectivity of an interface. With k2 (n − 1)2 the nonconductor essentially behaves like a perfect dielectric and, from equation (3.57), the spectral, normal reflectivity may be evaluated as ρnλ =



n−1 n+1

2

,

k 2 n2 .

(3.80)

Therefore, for optically smooth nonconductors the radiative properties may be calculated from refractive index data. Refractive indices for a number of semitransparent materials at room temperature are displayed in Fig. 3.16

Radiative Properties of Real Surfaces Chapter | 3 83

FIGURE 3.17 Spectral, directional reflectivity of glass at room temperature, for polarized light [26].

as a function of wavelength [20]. All these crystalline materials show similar spectral behavior: The refractive index drops rapidly in the visible region, then is nearly constant (declining very gradually) until the midinfrared, where n again starts to drop rapidly. This behavior is explained by the fact that crystalline solids tend to have an absorption band, due to electronic transitions, near the visible, and a Reststrahlen band in the infrared: The first drop in n is due to the tail end of the electronic band, as illustrated in Fig. 2.15b;6 the second drop in the midinfrared is due to the beginning of a Reststrahlen band. Listings of refractive indices for various glasses, water, inorganic liquids, and air are also available [23].

Directional Dependence of Radiative Properties For optically smooth nonconductors experiment has been found to follow Fresnel’s relations of electromagnetic wave theory closely. Figure 3.17 shows a comparison between theory and experiment for the directional reflectivity of glass (blackened on one side to avoid multiple reflections) for polarized, monochromatic irradiation [26]. Because k2 n2 , the absorptive index may be eliminated from equations (3.52) and (3.53), and the relations for a perfect dielectric become valid. Thus, for unpolarized light incident from vacuum (or a gas), from equations (3.59) and (3.60)

λ

√ ⎞2 ⎡⎛  1 1 ⎢⎢⎢⎜⎜⎜ n2 cos θ − n2 − sin2 θ ⎟⎟⎟ = 1 − ρ + ρ⊥ = 1 − ⎢⎣⎜⎝ ⎟ + √ 2 2 n2 cos θ + n2 − sin2 θ ⎠

√ ⎞2 ⎤ ⎛ ⎜⎜ cos θ − n2 − sin2 θ ⎟⎟ ⎥⎥⎥ ⎟⎟ ⎥⎥ . ⎜⎜ √ ⎠⎦ ⎝ cos θ + n2 − sin2 θ

(3.81)

Of course, the spectral, directional reflectivity for a dielectric can also be calculated from subroutine fresnel in Appendix F by setting k equal to zero. The directional variation of the spectral emissivity of dielectrics is shown in Fig. 3.18. Comparison with Fig. 3.1 demonstrates that experiment agrees well with electromagnetic wave theory for a large number of nonconductors, even for total (rather than spectral) directional emittances. The spectral, hemispherical emissivity of a nonconductor may be obtained by integrating equation (3.81) with equation (3.5). While tedious, such an integration is possible, as shown by Dunkle [30]: 4(2n + 1) , 3(n + 1)2   16n4 (n4 +1) ln n 4n3 (n2 +2n−1) 2n2 (n2 −1)2 n+1

⊥ = 2 − + ln , n−1 (n2 +1)3 (n +1)(n4 −1) (n2 +1)(n4 −1)2 1

λ = (  + ⊥ ). 2

 =

6. Note that the abscissa in Fig. 2.15b is frequency ν, i.e., wavelength increases to the left.

(3.82a) (3.82b) (3.82c)

84 Radiative Heat Transfer

FIGURE 3.18 Directional emissivities of nonconductors as predicted by electromagnetic wave theory.

FIGURE 3.19 Normal and hemispherical emissivities for nonconductors as a function of refractive index.

The variation of normal and hemispherical emissivities with refractive index may be calculated with functions emdiel ( λ ) and emdielr ( λ / nλ ) from Appendix F and is shown in Fig. 3.19. While for metals the hemispherical emittance is generally larger than the normal emittance (cf. Fig. 3.10), the opposite is true for nonconductors. The reason for this behavior is obvious from Fig. 3.1: Metals have a relatively low emittance over most directions, but display a sharp increase for grazing angles before dropping back to zero. Nonconductors, on the other hand, have a (relatively high) emittance for most directions, which gradually drops to zero at grazing angles (without a peak). Example 3.5. The directional reflectance of silicon carbide at λ = 2 μm and an incidence angle of θ = 10◦ has been measured as ρλ = 0.20 (cf. Fig. 3.13). What is the hemispherical emittance of SiC at 2 μm?

Radiative Properties of Real Surfaces Chapter | 3 85

FIGURE 3.20 Variation of the spectral, normal reflectance of MgO with temperature [36].

Solution Since at θ = 10◦ the directional reflectance does not deviate substantially from the normal reflectance (cf. Fig. 3.18), we have nλ = 1 − ρnλ 1 − 0.20 = 0.80. Then, from Fig. 3.19, n 2.6 and λ 0.76.

Effects of Surface Temperature The temperature dependence of the radiative properties of nonconductors is considerably more difficult to quantify than for metals. Infrared absorption bands in ionic solids due to excitation of lattice vibrations (Reststrahlen bands) generally increase in width and decrease in strength with temperature, and the wavelength of peak reflection/absorption shifts toward higher values. Figure 3.20 shows the behavior of the MgO Reststrahlen band [36]; similar results have been obtained for SiC [38]. The reflectance for shorter wavelengths largely depends on the material’s impurities. Often the behavior is similar to that of metals, i.e., the emittance increases with temperature for the near infrared, while it decreases with shorter wavelengths. As an example, Fig. 3.21 shows the normal emittance for zirconium carbide [39]. On the other hand, the emittance of amorphous solids (i.e., solids without a crystal lattice) tends to be independent of temperature [40].

3.6 Effects of Surface Roughness Up to this point, our discussion of radiative properties has assumed that the material surfaces are optically smooth, i.e., that the average length scale of surface roughness is much less than the wavelength of the electromagnetic wave. Therefore, a surface that appears rough in visible light (λ 0.5 μm) may well be optically smooth in the intermediate infrared (λ 50 μm). This difference is the primary reason why the electromagnetic wave theory ceases to be valid for very short wavelengths. In this section we shall very briefly discuss some fundamental aspects of how surface roughness affects the radiative properties of opaque surfaces. Detailed discussions have been given in the books by Beckmann and Spizzichino [41] and Bass and Fuks [42], and in a review article by Ogilvy [43]. The character of roughness may be very different from surface to surface, depending on the material, method of manufacture, surface preparation, and so on, and classification of this character is difficult. A common measure of surface roughness is given by the root-mean-square roughness σh , defined as (cf. Fig. 3.22)   1/2 , -1/2 1 2 2 σh = (z − zm ) = (z − zm ) dA , A A

(3.83)

86 Radiative Heat Transfer

FIGURE 3.21 Temperature dependence of the spectral, normal emittance of zirconium carbide [39].

FIGURE 3.22 Topography of a rough surface: (a) roughness with gradual slopes, (b) roughness with steep slopes. Both surfaces have similar root-mean-square roughness.

where A is the surface to be examined and |z − zm | is the local height deviation from the mean. The root-meansquare roughness can be readily measured with a profilometer (a sharp stylus that traverses the surface, recording the height fluctuations). Unfortunately, σh alone is woefully inadequate to describe the roughness of a surface as seen by comparing Fig. 3.22a and b. Surfaces of identical σh may have vastly different frequencies of roughness peaks, resulting in different average slopes along the rough surface; in addition, σh gives no information on second order (or higher) roughness superimposed onto the fundamental roughness. A first published attempt at modeling was made by Davies [44], who applied diffraction theory to a perfectly reflecting surface with roughness distributed according to a Gaussian probability distribution. The method neglects shading from adjacent peaks and, therefore, does poorly for grazing angles and for roughness with steep slopes (Fig. 3.22b). Comparison with experiments of Bennett [45] shows that, for small incidence angles, Davies’ model predicts the decay of specular peaks rather well (e.g., Fig. 3.14 for MgO). , for the specular reflection Davies’ model predicts a sharp peak in the bidirectional reflection function, ρ λ direction, as has been found to be true experimentally for most cases as long as the incidence angle was not too large (e.g., Fig. 3.5). For large off-normal angles of incidence, experiment has shown that the bidirectional reflectance function has its peak at polar angles greater than the specular direction. An example is given in Fig. 3.23 for magnesium oxide with a roughness of σh = 1.9 μm, illuminated by radiation with a wavelength of

Radiative Properties of Real Surfaces Chapter | 3 87

FIGURE 3.23 Normalized bidirectional reflection function (in plane of incidence) for magnesium oxide ceramic; σh = 1.9 μm, λ = 0.5 μm [47].

λ = 0.5 μm. Shown is the bidirectional reflection function (normalized with its value in the specular direction) for the plane of incidence (the plane formed by the surface normal and the direction of the incoming radiation). We see that for small incidence angles (θi = 10◦ ) the reflection function is relatively diffuse, with a small peak in the specular direction. For comparison, diffuse reflection with a direction-independent reflection function is indicated by the dashed line. For larger incidence angles the reflection function displays stronger and stronger off-specular peaks. For example, for an incidence angle of θi = 45◦ , the off-specular peak lies in the region of θ = 80◦ to 85◦ . Apparently, these off-specular peaks are due to shadowing of parts of the surface by adjacent peaks. The effects of shadowing have been incorporated into the model by Beckmann [46] and Torrance and Sparrow [47]. With the appropriate choice for two unknown constants, Torrance and Sparrow found their model agreed very well with their experimental data (Fig. 3.23). The above models assumed that the surfaces have a certain root-mean-square roughness, but that they were otherwise random—no attempt was made to classify roughness slopes, secondary roughness, etc. Berry and coworkers [48,49] considered diffraction of radiation from fractal surfaces. The behavior of fractal surfaces is such that the enlarged images appear very similar to the original surface when the surface roughness is repeatedly magnified (Fig. 3.22b). Majumdar and colleagues [50,51] carried out roughness measurements on a variety of surfaces and found that both processed and unprocessed surfaces are generally fractal. Majumdar and Tien [52] extended Davies’ theory to include fractal surfaces, resulting in good agreement for experiments with different types of metallic surfaces [53,54]. However, since shadowing effects have not been considered, the model is again limited to near-normal incidence. Buckius and coworkers [55–58] have investigated various one-dimensionally rough surfaces (i.e., where surface height is a function of one coordinate only, z = z(x) in Fig. 3.23), including the effects of roughness peak frequency (or slopes). For a randomly rough surface peak-to-peak spacing is usually characterized by a correlation length σl in a Gaussian correlation function C(L), where L is the length over which the correlation diminishes by a factor of e, or C(L) =

 1 , 2 z(x) − z z(x+L) − z = e−(L/σl ) . m m 2 σh

(3.84)

They first considered triangular grooves with roughnesses σh , σl and wavelength λ all of the same order, finding the bidirectional reflectance by solving an integral form of Maxwell’s equations. They found that these exact solutions predict the same scattering peaks as found from optical grating theory. They then applied their model to randomly rough surfaces described by equations (3.83) and (3.84), and compared their electromagnetic

88 Radiative Heat Transfer

FIGURE 3.24 Domains of validity for the geometric optics and the statistical rough surface reflection models, constructed for incidence angles between −45◦ and +45◦ from the surface normal.

wave theory results with those from the simple Kirchhoff approximation [41]. In the Kirchhoff approximation a simplified set of electromagnetic wave equations is considered, assuming that at every point on the surface the electromagnetic field is equal to the field that would exist on a local tangent plane, and multiple reflections between local peaks are neglected. This approximation has been applied by a number of researchers to oneand two-dimensionally rough surfaces, and domains of validity have been constructed [56,59–61]. It is generally understood that the Kirchhoff approximation gives satisfactory results when surface geometric parameters (σh , σl ) are less than or comparable to the wavelength and the slope of the roughness is small (σh /σl  0.3). In more recent work Buckius and coworkers have concentrated on geometric optics (i.e., assuming Fresnel’s relations to hold at every point on the surface), noting that Kirchhoff’s approximation results in considerably larger numerical effort without significant improvement over the specular approximation. They considered one- and two-dimensionally uncoated rough surfaces [58,62,63], and surfaces coated with a thin film [64] (together with thin film theory). A map was constructed, shown in Fig. 3.24, depicting under what conditions geometric optics gives satisfactory results as compared to exact electromagnetic wave theory calculations, using the criterion  π/2  .  π/2   Ed =  (I − I ) cos θ dθ I cos θ dθ < 0.2, (3.85)  −π/2 e a  −π/2 e where Ie and Ia are exact and approximate reflected intensities, respectively. In general, geometric optics requires generation of statistical surfaces together with ray tracing, a relatively time-consuming task. Along the same line Zhang and coworkers investigated scattering from rough silicon surfaces and wafers [65–67]. Surface topographic data obtained with an atomic force microscope showed the surface roughness to be significantly non-Gaussian and anisotropic. Nevertheless, the use of two-dimensional slope distributions and statistical ray tracing recovered experimental bidirectional reflection very accurately. Tang and Buckius [68] also introduced a statistical geometric optics model that does not require ray tracing. The resulting closedform expressions were found to be satisfactory for σh /σl  1, as also indicated in Fig. 3.24. Comparison of geometric optics calculations with experiment (Al2 O3 film on aluminum) showed good agreement, corroborating the applicability of their model [64]. Figure 3.24 was further confirmed (and augmented somewhat) by Fu and Hsu [69], who compared statistical ray tracing results with numerical solutions of Maxwell’s equations. Carminati and colleagues [70] used Kirchhoff’s approximation to provide an expression for the spectral, directional emittance (polarized or unpolarized) of a one-dimensionally randomly rough surface as 

λ (θ) =



∞ −∞



   1 − ρλ (θ − tan−1 p) 1 − p tan θ P(p) dp,

(3.86)

Radiative Properties of Real Surfaces Chapter | 3 89

where ρλ (θ) is the reflectivity as given by Fresnel’s relations, equations (3.52) through (3.55), and P(p) is a slope probability derived from the correlation function as P(p) =

σl 2 e−(pσl /2σh ) . √ σh 4π

(3.87)

Calling this a “small slope emission model” (since, similar to the conclusions of Fig. 3.24, its validity—in particular for parallel polarization—is limited to σh /λ  0.3), they extended this formula to a “large slope emission model,” using Ishimaru and Chen’s [71] shadowing function and assuming secondary reflection fields to be isotropic.

3.7 Effects of Surface Damage, Oxide Films, and Dust Even optically smooth surfaces have a surface structure that is different from the bulk material, due to either surface damage or the presence of thin layers of foreign materials, such as oxide films and dust. Surface damage is usually caused by the machining process, particularly for metals and semiconductors, which distorts or damages the crystal lattice near the surface. Thin foreign coats may be formed by chemical reaction (mostly oxidation), adsorption (e.g., coats of grease or water), or electrostatics (e.g., dust particles). All of these effects may have a severe impact on the radiation properties of metals, and may cause considerable changes in the properties of semiconductors. Other materials are usually less affected, because metals have large absorptive indices, k, and thus high reflectances. A thin, nonmetallic layer with small k can significantly decrease the composite’s reflectance (and raise its emittance). Dielectric materials, on the other hand, have small k’s and their relatively strong emission and absorption take place over a very thick surface layer. The addition of a thin, different dielectric layer cannot significantly alter their radiative properties. A minimum amount of surface damage is introduced during sample preparation if (i) the technique of electropolishing is used [45], (ii) the surface is evaporated onto a substrate within an ultra-high vacuum environment [72], or (iii) the metal is evaporated onto a smooth sheet of transparent material and the reflectance is measured at the transparent medium–metal interface [73]. Figure 3.25 shows the spectral, normal emittance of aluminum for a surface prepared by the ultra-high vacuum method [72], and for several other aluminum surface finishes [74]. While ultra-high vacuum aluminum follows the Drude theory for λ > 1 μm (cf. Fig. 3.7), polished aluminum (clean and optically smooth for large wavelengths) has a much higher emittance over the entire spectrum. Still, the overall level of emittance remains very low, and the reflectance remains rather specular. Similar results have been obtained by Bennett [45], who compared electropolished and mechanically polished copper samples. As Fig. 3.25 shows, the emittance is much larger still when off-the-shelf commercial aluminum is tested, probably due to a combination of roughness, contamination, and slight atmospheric oxidation. Bennett and colleagues [75] have shown that deposition of a thin oxide layer on aluminum (up to 100 Å) appreciably increases the emittance only for wavelengths less than 1.5 μm. This statement clearly is not true for thick oxide layers, as evidenced by Fig. 3.25: Anodized aluminum (i.e., electrolytically oxidized material with a thick layer of alumina, Al2 O3 ) no longer displays the typical trends of a metal, but rather shows the behavior of the dielectric alumina. The effects of thin and thick oxide layers have been measured for many metals, with similar results. A good collection of such measurements has been given by Wood and coworkers [3]. As a rule of thumb, clean metal exposed to air at room temperature grows oxide films so thin that infrared emittances are not affected appreciably. On the other hand, metal surfaces exposed to high-temperature oxidizing environments (furnaces, etc.) generally have radiative properties similar to those of their oxide layer. Dust deposits significantly alter the radiative properties of most surfaces. Lin et al. [76] measured the effect of dust particles of various chemical composition, density, and size distribution on the effective spectral absorptance of dust-coated surfaces. Three different substrates, i.e., one type of bulk material painted with three different paints whose properties are well known, were considered. In most cases (substrate–dust particle combination), a decrease in effective absorptance was observed around 2 μm and 4 μm. A model based on geometrical optics that performed ray tracing though the dust particle bed to determine the effective spectral absorptance was also proposed and validated. The model was then explored to predict the effect of moisture content, and chemical composition of the dust particles on the effective spectral absorptance. While most severe for metallic surfaces, the problem of surface modification is not unknown for nonmetals. For example, it is well known that silicon carbide (SiC), when exposed to air at high temperature, forms a silica (SiO2 ) layer on its surface, resulting in a reflection band around 9 μm [77]. Nonoxidizing chemical reactions can also significantly change the radiative properties of dielectrics. For example, the strong ultraviolet radiation in

90 Radiative Heat Transfer

FIGURE 3.25 Spectral, normal emittance for aluminum with different surface finishes [72,74].

FIGURE 3.26 Effects of ultraviolet and gamma ray irradiation on a titanium dioxide/epoxy coating [79].

outer space (from the sun) as well as gamma rays (from inside the Earth’s van Allen belt) can damage the surface of spacecraft protective coatings like white acrylic paint [78] or titanium dioxide/epoxy coating [79], as shown in Fig. 3.26. In summary, radiative properties for opaque surfaces, when obtained from figures in this chapter, from the tables given in Appendix B, or from other tabulations and figures of [1–8,80,81], should be taken with a grain of salt. Unless detailed descriptions of surface purity, preparation, treatment, etc., are available, the data may not give any more than an order-of-magnitude estimate. One should also keep in mind that the properties of a surface may change during a process or overnight (by oxidation and/or contamination).

3.8 Radiative Properties of Semitransparent Sheets The properties of radiatively participating media will be discussed in Chapters 10 through 12; i.e., semitransparent media that absorb and emit in depth and whose temperature distribution is, thus, strongly affected by thermal radiation. There are, however, important applications where thermal radiation enters an enclosure through semitransparent sheets, and where the temperature distribution within the sheet is unimportant or not significantly affected by thermal radiation. Applications include solar collector cover plates, windows in connec-

Radiative Properties of Real Surfaces Chapter | 3 91

FIGURE 3.27 Reflectivity and transmissivity of a thick semitransparent sheet.

tion with light level calculations within interior spaces, and so forth. We shall, therefore, briefly present here the radiative properties of window glass, for single and multiple pane windows with and without surface coatings. Glass and other amorphous solids tend to have extremely smooth surfaces, allowing for accurate predictions of interface reflectivities from electromagnetic wave theory (and the relevant surface properties, therefore, have the ending -ivity).

Properties of Single Pane Glasses For an optically smooth window pane of a thickness d substantially larger than the wavelength of incident light, d  λ, the radiative properties are readily determined through geometric optics and ray tracing. Consider the sheet of semitransparent material depicted in Fig. 3.27. The sheet has a complex index of refraction m2 = n2 − ik2 with k2 1, so that the transmission through the sheet (not counting surface reflections), τ = e−κ2 d/cos θ2 = e−4πk2 d/λ0 cos θ2 ,

(3.88)

is appreciable [cf. equation (2.42)]. Here κ2 = 4πk2 /λ0 is the absorption coefficient, λ0 is the wavelength of the incident light in vacuum, and d/cos θ2 is the distance a light beam of oblique incidence travels through Medium 2 in a single pass. The semitransparent sheet is surrounded by two dielectric materials with refractive indices n1 and n3 . To calculate the reflectivity at the interfaces 1–2 and 2–3 it is sufficient to use Fresnel’s relations for dielectric media, since k2 1. Interchanging n1 and n2 , as well as θ1 and θ2 , in equation (2.96) shows that the reflectivity at the 1–2 interface is the same, regardless of whether radiation is incident from Medium 1 or Medium 2, i.e., ρ12 = ρ21 and ρ23 = ρ32 . Now consider radiation of unit strength to be incident upon the sheet from Medium 1 in the direction of θ1 . As indicated in Fig. 3.27 the fraction ρ12 is reflected at the first interface, while the fraction (1 − ρ12 ) is refracted into Medium 2, according to Snell’s law. After traveling a distance d/cos θ2 through Medium 2 the attenuated fraction (1 − ρ12 )τ arrives at the 2–3 interface. Here the amount (1 − ρ12 )τρ23 is reflected back to the 1–2 interface, while the fraction (1 − ρ12 )τ(1 − ρ23 ) leaves the sheet and penetrates into Medium 3 in a direction of θ3 . The internally reflected fraction keeps bouncing back and forth between the interfaces, as indicated in the figure, until all energy is depleted by reflection back into Medium 1, by absorption within Medium 2, and by transmission into Medium 3. Therefore, the slab reflectivity, Rslab , may be calculated by summing over all contributions, or   Rslab = ρ12 + ρ23 (1 − ρ12 )2 τ2 1 + ρ12 ρ23 τ2 + (ρ12 ρ23 τ2 )2 + · · · . Since ρ12 ρ23 τ2 < 1 the series is readily evaluated [82], and Rslab = ρ12 +

ρ23 (1 − ρ12 )2 τ2 ρ12 + (1 − 2ρ12 )ρ23 τ2 = . 1 − ρ12 ρ23 τ2 1 − ρ12 ρ23 τ2

(3.89)

92 Radiative Heat Transfer

FIGURE 3.28 Spectral, normal slab transmissivity and reflectivity for panes of five different types of glasses at room temperature; data from [7].

Similarly, the slab transmissivity, Tslab , follows as   Tslab = (1 − ρ12 )(1 − ρ23 )τ 1 + ρ12 ρ23 τ2 + (ρ12 ρ23 τ2 )2 + · · · =

(1 − ρ12 )(1 − ρ23 )τ . 1 − ρ12 ρ23 τ2

(3.90)

These relations are the same as the ones evaluated for thick sheets by the electromagnetic wave theory, equations (2.129) and (2.130). From conservation of energy Aslab + Rslab + Tslab = 1, and the slab absorptivity follows as Aslab =

(1 − ρ12 )(1 + ρ23 τ)(1 − τ) . 1 − ρ12 ρ23 τ2

If Media 1 and 3 are identical (say, air), then ρ12 = ρ23 = ρ and equations (3.89) through (3.91) reduce to   (1 − ρ)2 τ2 Rslab = ρ 1 + , 1 − ρ2 τ2

(3.91)

(3.92)

Tslab =

(1 − ρ)2 τ , 1 − ρ2 τ2

(3.93)

Aslab =

(1 − ρ)(1 − τ) . 1 − ρτ

(3.94)

Figure 3.28 shows typical slab transmissivities and reflectivities of several different types of glasses for normal incidence and for a pane thickness of 12.7 mm. Most glasses have fairly constant and low slab reflectivity in the spectral range from 0.1 μm up to about 9 μm (relatively constant refractive index n, small absorptive index k). Beyond 9 μm the reflectivity increases because of two Reststrahlen bands [83] (not shown). Glass transmissivity tends to be very high between 0.4 μm and 2.5 μm. Beyond 2.5 μm the transmissivity of window glass diminishes rapidly, making windows opaque to infrared radiation. This gives rise to the so-called “greenhouse” effect: Since the sun behaves much like a blackbody at 5777 K, most of its energy (≈ 95%) falling onto Earth lies in the spectral range of high glass transmissivities. Therefore, solar energy falling onto a window passes readily into the space behind it. The spectral variation of solar irradiation, for extraterrestrial and unity air mass conditions, was given in Fig. 1.3. On the other hand, if the space behind the window is at low to moderate temperatures (300 to 400 K), emission from such surfaces is at fairly long wavelengths, which is absorbed by the glass and, thus, cannot escape.

Radiative Properties of Real Surfaces Chapter | 3 93

FIGURE 3.29 Spectral, normal slab transmissivity and reflectivity of soda–lime glass at room temperature, for a number of pane thicknesses; data from [7].

The influence of pane thickness on reflectivity and transmissivity is shown in Fig. 3.29 for the case of soda– lime glass (i.e., ordinary window glass). As the pane thickness increases, transmissivity decreases due to the increasing absorption. Since the absorption coefficient is small for λ < 2.7 μm (see Fig. 1.17), the effect is rather minor (and even less so for the other glasses shown in Fig. 3.28). In some high-temperature applications the emission from hot glass surfaces becomes important (e.g., in the manufacture of glass). Gardon [84] has calculated the spectral, hemispherical and total, hemispherical emissivity of soda–lime glass sheets at 1000◦ C based on the data of Neuroth [85]. Spectral emissivities beyond 2.7 μm do not depend strongly on temperature since the absorption coefficient is relatively temperature-independent (see Fig. 1.17). For all but the thinnest glass sheets the material becomes totally opaque, and the hemispherical emissivity is evaluated as λ = 1 − ρλ 0.91.7 Another semitransparent material that has found widespread use in many engineering applications, and commonly known as plexiglass, is polymethyl methacrylate (PMMA). Boulet et al. [86] measured the effective spectral reflectivity and transmissivity of PMMA slabs of various thicknesses using FTIR spectroscopy. As shown in Fig. 3.30, unlike glass slabs, strong oscillations are observed in both the reflectivity and the transmissivity, especially in the visible and near-infrared parts of the spectrum. At thicknesses larger than about 15 mm, PMMA becomes fairly opaque over the shorter wavelengths of visible light. However, all samples are fairly transmissive in the infrared. In the same study, the spectral absorptive and refractive indices of PMMA were also inferred (not shown) from the measured data using an inverse methodology.

Coatings Glass sheets and other transparent solids often have coatings on them for a variety of reasons: to eliminate transmission of ultraviolet radiation, to decrease or increase transmission over certain spectral regions, and the like. We distinguish between thick coatings (d  λ, no interference effects) and thin film coatings (d = O(λ), with wave interference, as discussed in Chapter 2). The effects of a thick dielectric layer (with refractive index n2 , and absorptive index k2 0) on the reflectivity of a thick sheet of glass (n3 and k3 0) is readily analyzed with the two-interface formula given by equation (3.89). With τ 1 and, for normal incidence, ρ12 =



n1 − n2 n1 + n2

2 and

ρ23 =



n2 − n3 n2 + n3

2

,

7. The hemispherical emissivity is evaluated by first evaluating ρnλ : With n 1.5 (for λ > 2.7 μm), from Fig. 3.16 ρnλ = 0.04 and nλ = 0.96; finally, from Fig. 3.19 λ 0.91.

94 Radiative Heat Transfer

FIGURE 3.30 Measured effective spectral properties of PMMA (plexiglass) slabs: (a) reflectivity and (b) transmissivity; reproduced from [86].

the coating reflectivity becomes ρ12 + ρ23 − 2ρ12 ρ23 (1 − ρ12 )(1 − ρ23 ) =1− 1 − ρ12 ρ23 1 − ρ12 ρ23 (4n1 n2 )(4n2 n3 ) =1− , (n1 + n2 )2 (n2 + n3 )2 − (n1 − n2 )2 (n2 − n3 )2

Rcoat =

which is readily simplified to Rcoat = 1 −

(n22

4n1 n2 n3 . + n1 n3 )(n1 + n3 )

(3.95)

If the aim is to minimize the overall reflectivity of the semitransparent sheet, then a value for the refractive index of the coating must be chosen to make Rcoat a minimum. Thus, setting dRcoat /dn2 = 0 leads to √ (3.96) n2,min = n1 n3 . Substituting equation (3.96) into (3.95) results in a minimum coated-surface reflectivity of √ 2 n1 n3 . Rcoat,min = 1 − n1 + n3

(3.97)

The slab reflectivity for a thin dielectric coating on a dielectric substrate, d = O(λ), is subject to wave interference effects and has been evaluated in Chapter 2, from equation (2.124), with δ12 = π and δ23 = 0 (cf. Example 2.6), as Rcoat = r12 =

r212 + 2r12 r23 cos ζ + r223 1 + 2r12 r23 cos ζ + r212 r223 n1 − n2 , n1 + n2

r23 =

,

(3.98a)

n2 − n3 , n2 + n3

ζ=

4πn2 d . λ

(3.98b)

Equation (3.98) has an interference minimum when ζ = π (i.e., if the film thickness is a quarter of the wavelength inside the film, d = 0.25λ/n2 ). For this interference minimum the reflectivity of the coated surface becomes Rcoat =



r12 − r23 1 − r12 r23

2

.

(3.99)

Radiative Properties of Real Surfaces Chapter | 3 95

FIGURE 3.31 Spectral, normal reflectivity and transmissivity of a 0.35 μm thick Sn-doped In2 O3 film deposited on Corning 7059 glass [88].

√ Clearly, this equation results in a minimum (or zero) reflectivity if r12 = r23 , or n2,min = n1 n3 , which is the same as for thick films, equation (3.96). To obtain minimum reflectivities for glass (n3 1.5) facing air (n1 1) would require a dielectric film with n2 1.22. Dielectric films of such low refractive index do not appear possible. However, Yoldas and Partlow [87] showed that a porous film (pore size λ) can effectively lower the refractive index, and they obtained glass transmissivities greater than 99% throughout the visible. In other applications a strong reflectivity is desired. An example of experimentally determined reflectivity and transmissivity of a coated dielectric is given in Fig. 3.31 for a 0.35 μm thick layer of Sn-doped In 2 O 3 film on glass [88]. The oscillating properties clearly demonstrate the effects of wave interference at shorter wavelengths. At wavelengths λ > 1.5 μm the material has a strong absorption band, making it highly reflective and opaque. Thus, this coated glass makes a better solar collector cover plate than ordinary glass, since internally emitted infrared radiation is reflected back into the collector (rather than being absorbed), keeping the cover glass cool and reducing losses. Similar behavior was obtained by Yoldas and O’Keefe [89], who deposited thin (20 to 50 nm) triple-layer films (titanium dioxide–silver–titanium dioxide) on soda–lime glass. It is also possible to tailor the directional reflection behavior using special, obliquely deposited films [90].

Multiple Parallel Sheets To minimize convection losses, two or more parallel sheets of windows are often employed, as illustrated in Fig. 3.32a. To find the total reflectivity and transmissivity of n layers, we break the system up into a single layer and the remaining (n − 1) layers. Then ray tracing (see Fig. 3.32b) results in   Rn = R1 + T12 Rn−1 1 + R1 Rn−1 + (R1 Rn−1 )2 + · · · = R1 +

T12 Rn−1 1 − R1 Rn−1

,

(3.100)

and, similarly, Tn =

T1 Tn−1 , 1 − R1 Rn−1

(3.101)

where Rn−1 and Tn−1 are the net reflectivity and transmissivity of (n − 1) layers. The net absorptivity of the n layers can be calculated directly either from An = A1 + A1 T1 Rn−1 (1 + R1 Rn−1 + · · · ) + An−1 T1 (1 + R1 Rn−1 + · · · ) T1 (A1 Rn−1 + An−1 ) = A1 + , 1 − R1 Rn−1

(3.102)

96 Radiative Heat Transfer

FIGURE 3.32 Reflectivity and transmissivity of multiple sheets: (a) geometric arrangement, (b) ray tracing for interaction between a single layer and the remainder of the sheets.

or from conservation of energy, i.e., An + Rn + Tn = 1. In the development of equation (3.100) we have assumed that R1 is the same for light shining onto the top or the bottom of the sheet (ρ12 = ρ23 ), in other words, that equation (3.92) is valid. The above recursion formulae were first derived by Edwards [91] without the restriction of ρ12 = ρ23 . In a later paper Edwards [92] expanded the method to include wave interference effects for stacked thin films. Multiple sheets subject to mixed diffuse and collimated irradiation, but without interference effects, were analyzed by Mitts and Smith [93]. Example 3.6. Determine the normal transmissivity of a triple-glazed window for visible wavelengths. The window panes are thin sheets of soda–lime glass, separated by layers of air. Solution The reflectivity R1 and transmissivity T1 of a single sheet are readily calculated from equations (3.92) and (3.93). For thin sheets (e.g., curve 1 in Fig. 3.29) we have τ 1, and with n 1.5 (cf. Fig. 3.16), ρ = [(1.5 − 1)/(1.5 + 1)]2 = 0.04. Therefore,   2ρ (1 − ρ)2 2 × 0.04 = = 0.0769, = R1 = ρ 1 + 1 − ρ2 1 + ρ 1 + 0.04 T1 =

(1 − ρ)2 1−ρ = 1 − R1 = 0.9231 = 1 − ρ2 1+ρ

(and A1 = 0, since we assumed τ 1). For two panes, from equations (3.100) and (3.101) with n = 2,  0.92312 = 0.1429, R2 = R1 + = 0.0769 1 + 1 − 0.07692 1 − R21 

T12 R1

T2 =

T12 1 − R21

= 0.8571

(and, again A2 = 0). Finally, for three panes T12 R2

0.92312 × 0.1429 = 0.2000, 1 − R1 R2 1 − 0.0769 × 0.1429 T1 T2 0.9231 × 0.8571 T3 = = 0.8000. = 1 − R1 R2 1 − 0.0769 × 0.1429

R3 = R1 +

= 0.0769 +

Assuming negligible absorption within the glass, 80% of visible radiation is transmitted through the triple-pane window (at normal incidence), while 20% is reflected back.

Although they are valid, equations (3.89) and (3.90) are quite cumbersome for oblique incidence, in particular, if absorption cannot be neglected. Some calculations for nonabsorbing (for n = 1.5 [94] and for n = 1.526 [95]) and absorbing [95] (n = 1.526) multiple sheets of window glass have been carried out. Note that, for oblique

Radiative Properties of Real Surfaces Chapter | 3 97

FIGURE 3.33 Transmissivities of 1, 2, 3, and 4 sheets of glass (n = 1.526) for different optical thicknesses per sheet, κd [95].

incidence, the overall reflectivity and transmissivity are different for parallel- and perpendicular-polarized light. Even for unpolarized light the polarized components must be determined before averaging, as Rn =

1 (Rn⊥ + Rn ), 2

Tn =

1 (Tn⊥ + Tn ). 2

(3.103)

The results of the calculations by Duffie and Beckman [95] are given in graphical form in Fig. 3.33.

3.9 Special Surfaces For many engineering applications it would be desirable to have a surface material available with very specific radiative property characteristics. For example, the net radiative heat gain of a solar collector is the difference between absorbed solar energy and radiation losses due to emission by the collector surface. While a black absorber plate would absorb all solar irradiation, it unfortunately would also lose a maximum amount of energy due to surface emission. An ideal solar collector surface has a maximum emittance for those wavelengths and directions over which solar energy falls onto the surface, and a minimum emittance for all other wavelengths and directions. On the other hand, a radiative heat rejector, such as the ones used by the U.S. Space Shuttle to reject excess heat into outer space, should have a high emittance at longer wavelengths, and a high reflectance for those wavelengths and directions with which sunshine falls onto the heat rejector. To a certain degree the radiative properties of a surface can be tailored toward desired characteristics. Surfaces that absorb and emit strongly over one wavelength range, and reflect strongly over the rest of the spectrum are called spectrally selective, while surfaces with tailored directional properties are known as directionally selective. An ideal, spectrally selective surface would be black (αλ = λ = 1) over the wavelength range over which maximum absorption (or emission) is desired, and would be totally reflective (αλ = λ = 0) beyond a certain cutoff wavelength λc , where undesirable emission (or absorption) would occur. Of course, in practice such behavior can only be approximated. The performance of a selective surface is usually measured by the “α/ -ratio,” where α is the total, directional absorptance of the material for solar irradiation, while is the total, hemispherical emittance for infrared surface

98 Radiative Heat Transfer

FIGURE 3.34 Spectral, hemispherical reflectances of several spectrally selective surfaces [101].

FIGURE 3.35 Solar irradiation on and emission from a solar collector plate.

emission. Consider a solar collector plate (Fig. 3.35), irradiated by the sun at an off-normal angle of θs . Making an energy balance (per unit area of the collector), we find 4 − αqsun cos θs , qnet = σTcoll

(3.104)

where the factor cos θs appears since qsun is solar heat flux per unit area normal to the sun’s rays. The total, hemispherical emittance may be related to spectral, hemispherical values through equation (3.10), while the total, directional absorptance is found from equation (3.23). Thus  ∞ 1

λ (Tcoll , λ) Ebλ (Tcoll , λ) dλ, (3.105a)

= 4 σTcoll 0  ∞  ∞ 1 1 αλ (Tcoll , λ, θs ) qsun,λ dλ = αλ (Tcoll , λ, θs ) Ebλ (Tsun , λ) dλ, (3.105b) α= 4 qsun 0 σTsun 0 where we have made use of the fact that the spectral distribution of qsun is the same as the blackbody emission from the sun’s surface. Clearly, for optimum performance of a collector the solar absorptance should be maximum, while the infrared emittance should be minimum. Therefore, a large α/ -ratio indicates a better performance for a solar collector. On the other hand, for radiative heat rejectors a minimum value for α/ is desirable. Most selective absorbers are manufactured by coating a thin nonmetallic film onto a metal. Over most wavelengths the nonmetallic film is very transmissive and incoming radiation passes straight through to the metal interface with its very high reflectance. However, many nonconductors have spectral regions over which they do absorb appreciably without being strongly reflective (usually due to lattice defects or contaminants). The result is a material that acts like a strongly reflecting metal over most of the spectrum, but like a strongly absorbing nonconductor for selected wavelength ranges. A few examples of such selective surfaces are also given in Fig. 3.34. Black chrome (chrome-oxide coating) and black nickel (nickel-oxide coating) are popular

Radiative Properties of Real Surfaces Chapter | 3 99

solar collector materials, while epoxy paint may be used as an efficient solar energy rejector. If the coatings are extremely thin, interference effects can also be exploited to improve selectivity. For example, Martin and Bell [96] showed that a three-layer coating of SiO2 –Al–SiO2 on metallic substrates has a solar absorptance greater than 90%, but an infrared emittance of < 10%. Fan and Bachner [88] produced a coating for glass that raised its reflectance to > 80% for infrared wavelengths, without appreciably affecting solar transmittance (Fig. 3.31). The advantages of spectrally selective surface properties were first recognized by Hottel and Woertz [97]. With the growing interest in solar energy collection during the 1950s and 1960s, a number of selective coatings were developed, and the subject was discussed by Gier and Dunkle [98] and Tabor and coworkers [99,100]. There are several compilations for radiative properties of selective absorbers [3,8,101]. A somewhat more detailed discussion about spectrally selective surface properties has been given by Duffie and Beckman [95]. A significant body of literature is also available on the radiative properties of pigmented coatings [102–105]. Maruyama and coworkers [102,103], using a combination of experimental characterization and models based on ray tracing, have developed a procedure to optimize and design pigmented coatings with embedded TiO2 and CuO particles that are visually appealing and stay cool under strong solar irradiation. Such coatings are used in building thermal management applications. The same methods have also been used to design ultraviolet barrier coatings with embedded TiO2 and ZnO particles [104]. Huang and Ruan [105] have developed a TiO2 nanoparticle based double-layer coating for passive radiative cooling. Yang and Zhao [106,107] measured the spectral reflectance and absorptance of plasma-deposited thermal barrier coatings in the range 0.1–16 μm. In other studies, Sun et al. [108] measured the spectral reflectance of sixteen different sand and soil samples for remote sensing applications. Example 3.7. Let us assume that it is possible to manufacture a diffusely absorbing/emitting selective absorber with a spectral emittance λ = s = 0.05 for 0 < λ < λc and λ = c = 0.95 for λ > λc , where the cutoff wavelength can be varied through manufacturing methods. Determine the optimum cutoff wavelength for a solar collector with an absorber plate at 350 K that is exposed to solar irradiation of qsun = 1000 W/m2 at an angle of θs = 30◦ off-normal. What is the net radiative energy gain for such a collector? Solution A simple energy balance on the surface, using equations (3.9) and (3.41) leads to  (θs ) = Eb − α (θs ) H (θs ) qnet = E − Habs

where qnet > 0 if a net amount of energy leaves the surface and qnet < 0 if energy is collected. Total, hemispherical emittance follows from equation (3.10) while total, directional absorptance is determined from equation (3.23). For our diffuse absorber we have αλ (λ, θ) = λ (λ) and   λc   ∞  ( c − s ) ∞ 1

E (T , λ) dλ +

E (T , λ) dλ =

+ Ebλ (Tcoll , λ) dλ, s bλ coll c bλ coll s 4 4 σTcoll σTcoll 0 λc λc    λc  ∞  ( c − s ) ∞ 1 E (T , λ) dλ +

E (T , λ) dλ =

+ Ebλ (Tsun , λ) dλ. α=

s bλ sun c bλ sun s 4 4 σTsun σTsun 0 λc λc

=

Substituting these expressions into our energy balance leads to   ∞ qsun cos θs 4 Ebλ (Tcoll , λ) − − qsun cos θs ) + ( c − s ) E (T , λ) dλ. qnet = s (σTcoll bλ sun 4 σTsun λc Optimizing the value of λc implies finding a maximum for qnet . Therefore, from Leibniz’s rule (see, e.g., [82]), which states that  b(x)  b df d da db f (x, b) − f (x, a) + (x, y) dy, (3.106) f (x, y) dy = dx a(x) dx dx a dx we find

  dqnet qsun cos θs = −( c − s ) Ebλ (Tcoll , λc ) − E (T , λ ) = 0, bλ sun c 4 dλc σTsun

100 Radiative Heat Transfer

or Ebλ (Tcoll , λc ) =

qsun cos θs 4 σTsun

Ebλ (Tsun , λc ).

Note that the cutoff wavelength does not depend on the values for c and s . Using Planck’s law, equation (1.13), with n = 1 (surroundings are air), the last expression reduces to exp(C2 /λc Tcoll ) − 1 =

4 σTsun   exp(C2 /λc Tsun ) − 1 . qsun cos θs

This transcendental equation needs to be solved by iteration. As a first guess one may employ Wien’s distribution, equation (1.19) (dropping two ‘−1’ terms), exp(C2 /λc Tcoll ) or

' exp

4 σTsun exp(C2 /λc Tsun ) qsun cos θs

 ) 4 σTsun 1 C2 1 − , λc Tcoll Tsun qsun cos θs

 4 σTsun 1 1 + − ln Tcoll Tsun qsun cos θs   + 5.670 × 10−8 × 57774 1 1 − μm ln = 14,388 = 3.45 μm. 350 5777 1000 × cos 30◦

λc C2



Iterating the full Planck’s law leads to a cutoff wavelength of λc = 3.69 μm. Substituting these values into the expressions for emittance and absorptance,  

= s + ( c − s ) 1 − f (λc Tcoll ) = 0.95 − 0.90 + 0.90 f (3.69 × 350) = 0.05 + 0.90 × 0.00413 = 0.054,   α = s + ( c − s ) 1 − f (λc Tsun ) = 0.05 + 0.90 × f (3.69 × 5777) = 0.05 + 0.90 × 0.98785 = 0.939. The net heat flux follows then as qnet = 0.054×5.760×10−8 ×3504 − 0.939×1000×cos 30◦ = −767 W/m2 . Actually, neither f (λc Tcoll ) 0 nor f (λc Tsun ) 1 is particularly sensitive to the exact value of λc , because there is very little spectral overlap between solar radiation (95% of which is in the wavelength range8 of λ < 2.2 μm) and blackbody emission at 350 K (95% of which is at λ > 5.4 μm).

Surfaces can be made directionally selective by mechanically altering the surface finish on a microscale (microgrooves) or macroscale. For example, large V-grooves (large compared with the wavelengths of radiation) tend to reflect incoming radiation several times for near-normal incidence, as indicated in Fig. 3.36 (from Trombe and coworkers [110]) for an opening angle of γ = 30◦ , each time absorbing a fraction of the beam. The number of reflections decreases with increasing incidence angle, down to a single reflection for incidence angles θ > 90◦ − γ (or 60◦ in the case of Fig. 3.36). Hollands [111] has shown that this type of surface has a significantly higher normal emittance, which is important for collection of solar irradiation, than hemispherical emittance, which governs emission losses. A similarly shaped material, with flat black bottoms, was theoretically analyzed by Perlmutter and Howell [112]. Their analytical values for directional emittance were experimentally confirmed by Brandenberg and Clausen [27], as illustrated in Fig. 3.37. In outer space, radiation is the only mode of heat transfer. Radiative fins, shields and collectors used in space applications have to perform the dual task of minimizing heat gain when sunlight is available and minimizing heat loss when sunlight is absent. This is difficult to attain with static surfaces and with fixed radiative surface properties. In recognition of this challenge, recent research by Iverson and coworkers [113–116] has delved into designing complex tesselated surface textures that are origami inspired and move according to the orientation 8. Based on a blackbody at 5777 K. This number remains essentially unchanged for true, extraterrestrial solar irradiation [109], while the 95% fraction moves to even shorter wavelengths if atmospheric absorption is taken into account (cf. Fig. 1.3).

Radiative Properties of Real Surfaces Chapter | 3 101

FIGURE 3.36 Directional absorption and reflection of irradiation by a V-grooved surface [110].

FIGURE 3.37 Directional emittance of a grooved surface with highly reflective, specular sidewalls and near-black base. Results are for plane perpendicular to groove length. Theory (ρsides = base = 1) from [112], experiment [taken at λ = 8 μm with aluminum sidewalls and black paint base with λ (8 μm) = 0.95] from [27].

and availability of sunlight. The aforementioned V-groove structure served as the starting point for these studies. The total apparent hemispherical absorptance and emittance of the V-groove and other more complex structures were measured and predicted using ray tracing for both diffuse and specular surfaces and a combination thereof. Example 3.8. Collimated solar irradiation of qsun = 1000 W/m2 is incident on a sphere. The surface of the sphere is gray and coated with a material whose directional emittance is given by ⎧ ⎪ ⎪ ⎨0.95,

(θ) = ⎪ ⎪ ⎩0, 

0 ≤ θ < π/6, π/6 ≤ θ ≤ π/2,

where θ is the angle between the outward-pointing normal from the surface of the sphere and the line of sight from the Earth to the sun. If the sphere has a radius of 0.5 m, calculate the net energy absorbed by the sphere. How would the result change if the surface was diffuse with an emittance of 0.95? Solution We choose a polar coordinate system, such that the z-axis points toward the sun, as shown in Fig. 3.38. Thus, the polar angle is the same as the angle between the surface normal and the line of sight from the Earth to the sun, denoted by θ. From Fig. 3.38, it follows that θS = π − θ. By Kirchhoff’s law, since α (π − θ) =  (θ), the solar energy absorbed by the

102 Radiative Heat Transfer

FIGURE 3.38 Polar coordinate and the sun’s orientation for Example 3.8.

sphere is given by  Qabs =



 As /2

α (θS ) qsun cos θS dA =

As /2

 (θ) qsun cos(π − θ) dA = −qsun

As /2

 (θ) cos θ dA,

where dA is a differential area on the surface of the sphere, and As /2 represents half the surface area of the sphere, i.e., the surface area facing the sun. Following equation (1.29), the differential area may be expressed in polar coordinates as dA = R2s sin θ dθ dψ, where Rs is the radius of the sphere. Substituting this expression into the above equation, and setting the appropriate limits for the polar and azimuthal angles to cover a hemisphere, we obtain  Qabs = −qsun R2s

2π ψ=0



π/2

θ=0

 (θ) cos θ sin θ dθ dψ.

Substituting the given directional emittance values, we get  Qabs = −qsun R2s × 0.95 ×

2π

ψ=0



π/6 θ=0

 cos θ sin θ dθ dψ = −qsun (2πR2s ) × 0.95 ×

π/6

θ=0

1 cos θ sin θ dθ = −qsun (2πR2s ) × 0.95 × , 8

which, upon using the values provided above, results in Qabs = −186.53 W. The negative sign is consistent with the sign convention that any emitted radiation is positive, while absorbed radiation is negative. If the surface is diffuse with α = = 0.95, then  Qabs = −qsun (2πR2s ) × 0.95 ×

π/2 θ=0

cos θ sin θ dθ = −qsun (2πR2s ) × 0.95 ×

1 = −qsun (πR2s ) × 0.95, 2

resulting in Qabs = −746.12 W, which, in this particular case, is 4 times the absorbed radiation for the nondiffuse surface. Since πR2s is the projected area of the sphere facing the sun, for a diffuse surface, the irradiation heat rate (in W) on the sphere is simply the product of the collimated incident radiation flux (in W/m2 ) and the projected area facing the collimated beam.

3.10 Earth’s Surface Properties and Climate Change The radiative properties of the Earth’s surface exhibit strong variation with wavelength [117,118] and play a major role in dictating its energy balance. Figure 3.39 shows the spectral reflectance of the main components of the Earth’s surface. These data were compiled from the ECOSTRESS [119] and ASTER [120] spectral libraries by Kääb [118], and represent average values over seasons and geographic locations. Furthermore, the data represent average values over several different types (or forms) of the same component. For example, the data shown for “Snow” are an average representation of snow of various degrees of compaction and granularity. Likewise, the data for “Vegetation” represent average values for various types of nonchlorophyllic vegetation. Regardless, a few important observations are in order: (1) snow is highly reflective in the visible part of the spectrum, (2) sea water is a poor reflector at all wavelengths, and (3) both vegetation and rock are poorer reflectors than snow in the visible and near-infrared parts of the spectrum. These trends have important repercussions on the Earth’s energy balance and its average temperature.

Radiative Properties of Real Surfaces Chapter | 3 103

FIGURE 3.39 Spectral reflectance of the major components of the Earth’s surface, as compiled in [118].

The role of the Earth’s surface radiative properties on its heating or cooling can be partly understood by performing a simple energy balance on the Earth’s surface without considering the effects of the atmosphere. In the absence of the atmosphere, other heat and mass transfer phenomena, such as convection and evaporation, cannot occur, thereby making such analysis relatively straightforward. Although the radiative energy balance, locally, of any point on the Earth’s surface is transient by nature, averaged over a day or several days, half the Earth’s surface always sees the sun (incident energy from the sun), and the whole surface always loses energy by emission regardless of whether it is facing the sun or not. Since it is known that all components of the Earth’s surface exhibit almost diffuse behavior [117], the net energy per unit time absorbed by the Earth follows immediately from Example 3.8 as [see equation (1.44)] 

∞

Qabs = −αE (πRE ) 2

Hsλ dλ = −αE (πR2E ) qsol ,

(3.107)

0

where αE is the effective total absorptance of the Earth’s surface, Hsλ is the spectral solar flux, and RE is the radius of the Earth. The net energy emitted by the entire surface of the Earth depends on its own temperature, and is written as  ∞ Qem = E AE Ebλ (TE ) dλ = E (4πR2E ) σTE4 , (3.108) 0

where E is the total emittance of the Earth’s surface. The effective total absorptance of the Earth’s surface can be approximated by ∞ ∞ ∞ αwλ Hsλ dλ αsλ Hsλ dλ αlλ Hsλ dλ 0 0 αE = aw  ∞ + as  ∞ + al 0 ∞ , Hsλ dλ Hsλ dλ Hsλ dλ 0 0 0

(3.109)

where aw , as , and al = 1 − aw − as are the fractions of the Earth’s surface covered by water, snow (or clean ice sheet), and exposed land, respectively. Likewise, αwλ , αsλ , and αlλ are the spectral absorptances of water, snow, and exposed land, respectively. Using αλ = 1 − ρλ for an opaque surface, and equation (1.44), equation (3.109) reduces to ∞ ∞ ∞ (1 − ρ ) E (T (1 − ρ ) E (T (1 − ρlλ ) Ebλ (TS ) dλ wλ S ) dλ sλ S ) dλ bλ bλ αE = aw 0  ∞ + as 0  ∞ + al 0  ∞ . (3.110) Ebλ (TS ) dλ Ebλ (TS ) dλ Ebλ (TS ) dλ 0 0 0

104 Radiative Heat Transfer

Likewise, the effective total emittance of the Earth’s surface may also be calculated from the spectral reflectance by making use of Kirchhoff’s law, namely λ = αλ , to yield ∞ ∞ ∞ (1 − ρwλ ) Ebλ (TE ) dλ (1 − ρsλ ) Ebλ (TE ) dλ (1 − ρlλ ) Ebλ (TE ) dλ 0 0 ∞ ∞

E = a w + as + al 0  ∞ , E (T E (T E (T E ) dλ E ) dλ E ) dλ bλ bλ bλ 0 0 0

(3.111)

where TE is the (unknown) surface temperature of the Earth. Assuming that conduction from the hot core of the Earth is negligible, its surface temperature will reach an equilibrium (steady state) if the absorbed energy balances the emitted energy. Thus, combining equations (3.107) and (3.108), we get σTE4 =

1 αE qsol 4 E

(3.112)

In order to calculate the total absorptance and emittance of the surface of the Earth from spectral reflectance data, one must first split the spectrum into a number of spectral intervals (or bands), Nλ , with constant reflectances within each band. The integrals in equations (3.109) and (3.111) can then be replaced by summations over bands, resulting in αE =

Nλ 

[aw (1 − ρw,i ) + as (1 − ρs,i ) + al (1 − ρl,i )] [ f (λi,u TS ) − f (λi,l TS )],

(3.113a)

[aw (1 − ρw,i ) + as (1 − ρs,i ) + al (1 − ρl,i )] [ f (λi,u TE ) − f (λi,l TE )].

(3.113b)

i=1

E =

Nλ  i=1

Here, ρw,i denotes the reflectance of water in the i-th band, and so on. Furthermore, equation (1.25) has been used to express the energy within each band. The lower and upper wavelengths of the i-th band are denoted by λi,l and λi,u , respectively. Equation (3.112) can be solved iteratively to determine the equilibrium surface temperature of the Earth, TE . Starting with an initial guess for TE , the fractions of blackbody radiation, namely f (λi,l TE ) and f (λi,u TE ) are first determined, and the total emittance of the Earth’s surface is calculated using equation (3.113). This value is then substituted into equation (3.112), and a new value of TE is determined. This new value of TE is then used to recalculate the blackbody radiation fractions, and the procedure is repeated until convergence. The total absorptance remains unchanged during the iterative process. In order to use the data shown in Fig. 3.39 for calculation of the equilibrium surface temperature of the Earth, the spectral range between 0.1 μm and 100 μm is chosen since more than 99.5% of the energy radiated by either the sun or the Earth is in this range. In the databases [119,120] that were used to generate Fig. 3.39, the wavelengths at which the reflectance values are reported are given by λi,l = 10−1+3(i−1)/Nλ , where Nλ = 3000 is the total number of bands. Here, the same spectral intervals are used for calculations; in practice, fewer bands will suffice. The spectral reflectance is assumed to remain constant at the respective end value for wavelengths beyond the range shown in Fig. 3.39. The fraction of blackbody radiation in each band is computed using equation (1.26). It is assumed that 71% of the Earth’s surface is covered by the ocean, while snow/ice covers 12% [117]. No data for “Vegetation” and “Rock” are given in [117]: thus, an average of the two is employed for “exposed land.” With an initial guess of TE = 288 K, convergence up to 2 decimal places in the computed temperature is attained after 3 iterations, which yields TE = 267.93 K (–5.22 ◦ C). The effective total absorptance and emittance are found to be 0.817 and 0.955, respectively. The computed equilibrium surface temperature of TE is considerably lower than the recorded average temperature of the Earth’s surface, which is about 14.9 ◦ C (as of 2019), due to the fact that the role of the atmosphere has been ignored in the present estimate. Atmospheric scientists have also made estimates of what the surface temperature of the Earth may be without the atmosphere. In such estimates, it is usually assumed that the Earth’s surface has a total absorptance of 0.7, while its total emittance is taken as 1.0 [117], yielding TE = 254.81 K. The actual values of effective total absorptance and emittance of the Earth’s surface also depend on a number of other factors, such as seasonal snow cover, foliage, the type of snow cover (fresh vs. dirty), the salinity of the sea water, etc.

Radiative Properties of Real Surfaces Chapter | 3 105

If we assume the Earth’s surface to be gray, αE = E by definition, and equation (3.112) reduces to σTE4 =

qsol , 4

which yields TE = 278.58 K (5.43◦ C). This temperature is considerably higher than what was calculated earlier with nongray properties. Most importantly, according to the above gray equation, the temperature of the Earth’s surface would be completely unaffected by any radiative property. Likewise, the relative fractional areas of water, snow, or exposed land would have no bearing on its temperature according to the gray model. From these observations, one may conclude that use of a gray model to estimate the temperature of the Earth’s surface is unrealistic. The nongray model [equation (3.112)], on the other hand, can be used to answer pertinent exploratory questions pertaining to the Earth’s climatological changes. For example, if half the surface area covered by snow/ice were to be exposed, as would change from 0.12 to 0.06. Repeating the above calculation, one obtains TE = 269.69 K. In other words, the surface temperature of the Earth is expected to rise by almost 2 degrees if half the snow-covered landmass is laid bare. Physically, since the reflectance of snow/ice is higher than that of exposed land (see Fig. 3.39), especially for wavelengths pertinent to solar irradiation, if snow/ice is replaced by bare land, the Earth will absorb more solar radiation. This trend, among other reasons, explains the growing concern within the scientific community with regard to increased glacial melting.

3.11 Experimental Methods It is quite apparent from the discussion in the preceding sections that, although electromagnetic wave theory can be used to augment experimental data, it cannot replace them. While the spectral, bidirectional reflection function, equation (3.34), is the most basic radiation property of an opaque surface, to which all other properties can be related, it is rarely measured. Obtaining the bidirectional reflection function is difficult because of the low achievable signal strength. It is also impractical since it is a function of both incoming and outgoing directions and of wavelength and temperature. A complete description of the surface requires enormous amounts of data. In addition, the use of the bidirectional reflection function complicates the analysis to such a point that it is rarely attempted. If bidirectional data are not required it is sufficient, for an opaque material, to measure one of the following, from which all other ones may be inferred: absorptance, emittance, directional–hemispherical reflectance, and hemispherical–directional reflectance. Various different measurement techniques have been developed, which may be separated into three loosely-defined groups: calorimetric emission measurements, radiometric emission measurements, and reflection measurements. The interest in experimental methods was at its peak during the 1960s as a result of the advent of the space age. Compilations covering the literature of that period have been given in two NASA publications [121,122]. Interest waned during the 1970s and 1980s but has recently picked up again because of the development of better and newer materials operating at higher temperatures. Sacadura [123] has given an updated review of experimental methods. While measurement techniques vary widely from method to method, most of them employ similar optical components, such as light sources, monochromators, and detectors. Therefore, we shall begin our discussion of experimental methods with a short description of important optical components.

Instrumentation Radiative property measurements generally require a light source, a monochromator, a detector, and the components of the optical path, such as mirrors, lenses, beam splitters, optical windows, and so on. Depending on the nature of the experiment and/or detector, other accessories, such as optical choppers, may also be necessary. LIGHT SOURCES. Light sources are required for the measurement of absorption by, or reflection from, an opaque surface, as well as for the alignment of optical components in any spectroscopic system. In addition, light sources are needed for transmission and scattering measurements of absorbing/scattering media, such as gases, particles, semitransparent solids, and liquids (to be discussed in later chapters). We distinguish between monochromatic and polychromatic light sources. Monochromatic sources. These types of sources operate through stimulated emission, producing light over an extremely narrow wavelength range. Their monochromaticity, low beam divergence, coherence, and high

106 Radiative Heat Transfer

FIGURE 3.40 Spectral irradiation on a distant surface from various incandescent light sources.

power concentration make lasers particularly attractive as light sources. While only invented some 30 years ago, there are today literally dozens of solid-state and gas lasers covering the spectrum between the ultraviolet and the far infrared. Although lasers are generally monochromatic, there are a number of gas lasers that can be tuned over a part of the spectrum by stimulating different transitions. For example, dye lasers (using large organic dye molecules as the lasing medium) may be operated at a large number of wavelengths in the range 0.2 μm < λ < 1 μm, while the common CO2 laser (usually operating at 10.6 μm) may be equipped with a movable grating, allowing it to lase at a large number of wavelengths in the range 9 μm < λ < 11 μm. Even solid-state lasers can be operated at several wavelengths through frequency-doubling. For example, the Nd-YAG laser, the most common solid-state laser, can be used at 1.064 μm, 0.532 μm, 0.355 μm, and 0.266 μm. Of particular importance for radiative property measurements is the helium–neon laser because of its low price and small size and because it operates in the visible at 0.633 μm (making it useful for optical alignment). A different kind of monochromatic source is the low-pressure gas discharge lamp, in which a low-density electric current passes through a low-pressure gas. Gas atoms and molecules become ionized and conduct the current. Electrons bound to the gas atoms become excited to higher energy levels, from which they fall again, emitting radiation over a number of narrow spectral lines whose wavelengths are characteristic of the gas used, such as zinc, mercury, and so on. Polychromatic sources. These usually incandescent light sources emit radiation by spontaneous emission due to the thermal excitation of source atoms and molecules, resulting in a continuous spectrum. The spectral distribution and total radiated power depend on the temperature, area, and emittance of the surface. Incandescent sources may be of the filament type (similar to an ordinary light bulb) or of the bare-element type. The quartz– tungsten–halogen lamp has a doped tungsten filament inside a quartz envelope, which is filled with a rare gas and a small amount of a halogen. Operating at a filament temperature greater than 3000 K, this lamp produces a nearblackbody spectrum with maximum emission below 1 μm. However, because of the transmission characteristics of quartz (which is the same as fused silica, Fig. 3.28), there is no appreciable emission beyond 3 μm. Bareelement sources are either rods of silicon carbide, called globars, or heating wires embedded in refractory oxides, called Nernst glowers. Globars operate at a temperature of 1000 K and produce an almost-gray spectrum with a maximum around 2.9 μm. Nernst glowers operate at temperatures up to 1500 K, with a somewhat less ideal spectral distribution. The irradiation onto a distant surface from different incandescent sources is shown in Fig. 3.40. None of the light sources shown in Fig. 3.40 has a truly “black” spectral distribution, since their output is influenced by their spectral emittance. In most experiments this is of little importance since, in general, sample and reference signals (coming from the same spectral source) are compared. If a true blackbody source is required (primarily for calibration of instruments) blackbody cavity sources are available from a number of manufacturers. In these sources a cylindrical and/or conical cavity, made of a high-temperature, high-emittance material (such as silicon carbide) is heated to a desired temperature. Radiation leaving the cavity, also commonly called Hohlraum (German for “hollow space”), is essentially black (cf. Table 5.1).

Radiative Properties of Real Surfaces Chapter | 3 107

FIGURE 3.41 Schematic of spectral separation with (a) a transparent prism, (b) a diffraction grating.

The brightest conventional source of optical radiation is the high-pressure gas discharge lamp, which combines the characteristics of spontaneous and stimulated emission. The lamp is similar to a low-pressure gas discharge source, but with high current density and gas pressure. This configuration results in an arc with highly excited atoms and molecules forming a plasma. While the hot plasma emits as an incandescent source, ionized atoms emit over substantially broadened spectral lines, resulting in a mixed spectrum (Fig. 3.40). Commonly used gases for such arc sources are xenon, mercury, and deuterium. SPECTRAL SEPARATORS. Spectral radiative properties can be measured over part of the spectrum in one of two ways: (i) Measurements are made using a variety of monochromatic light sources, which adequately represent the desired part of the spectrum, or (ii) a polychromatic source is used together with a device that allows light of only a few select wavelengths to reach the detector. Such devices may consist of simple optical filters, manually driven or motorized monochromators, or highly sophisticated FTIR (Fourier Transform InfraRed) spectrometers. Optical filters. These are multilayer thin-film devices that selectively transmit radiation only over desired ranges of wavelengths. Bandpass filters transmit light only over a finite, usually narrow, wavelength region, while edge filters transmit only above or below certain cutoff or edge wavelengths. Bandpass filters consist of a series of thin dielectric films that, at each interface, partially reflect and partially transmit radiation (cf. Fig. 2.13). The spacing between layers is such that beams of the desired wavelength are, after multiple reflections within the layers, in phase with the transmitted beam (constructive interference). Other wavelengths are rejected because they destructively interfere with one another. Bandpass filters for any conceivable wavelength between the ultraviolet and the midinfrared are routinely manufactured. Edge filters operate on the same principle, but are more complex in design. Monochromators. These devices separate an incoming polychromatic beam into its spectral components. They generally consist of an entrance slit, a prism or grating that spreads the incoming light according to its wavelengths, and an exit slit, which allows only light of desired wavelengths to escape. If a prism is used, it is made of a highly-transparent material with a refractive index that varies slightly across the spectrum (cf. Fig. 3.16). As shown in Fig. 3.41a, the incoming radiant energy is separated into its constituent wavelengths since, by Snell’s law, the prism bends different wavelengths (with different refractive index) by different amounts. Rotating the prism around an axis allows different wavelengths to escape through the exit slit. Instead of a prism one can use a diffraction grating to separate the wavelengths of incoming light, employing the principle of constructive and destructive interference [124], as schematically indicated in Fig. 3.41b. Until a few years ago all monochromators employed salt prisms, while today almost all systems employ diffraction gratings, since they are considerably cheaper and simpler to handle (salt prisms tend to be hygroscopic, i.e., they are attacked by the water vapor in the surrounding air). However, diffraction gratings have the disadvantages that their spectral range is more limited (necessitating devices with multiple gratings), and they may give erroneous readings due to higher-order signals (frequency-doubling). FTIR spectrometers. These instruments collect the entire radiant energy (i.e., comprising all FTIR spectrometer wavelengths) after reflection from a moving mirror. The measured intensity depends on the position of the moving mirror owing to constructive and destructive interference. This signal is converted by a computer through an inverse Fast Fourier Transform into a power vs. wavelength plot. The spectral range of FTIRs is limited only by the choice of beam splitters and detectors, and is comparable to that of prism monochromators. However, while monochromators generally require several minutes to collect data over their entire spectral range, the FTIR

108 Radiative Heat Transfer

FIGURE 3.42 Schematic of (a) a pyroelectric detector, (b) a photoconductive detector.

is able to do this in a fraction of a second. Detailed descriptions of the operation of FTIRs may be found in books on the subject, such as the one by Griffiths and de Haseth [125]. DETECTORS. In a typical spectroscopic experiment, the detector measures the intensity of incoming radiation due to transmission through, emission from, or reflection by, a sample. This irradiation may be relatively monochromatic (i.e., covers a very narrow wavelength range after having passed through a filter or monochromator), or may be polychromatic (for total emittance measurements, or if an FTIR is used). In either case, the detector converts the beam’s power into an electrical signal, which is amplified and recorded. The performance of detectors is measured by certain criteria, which are generally functions of several operating conditions, such as wavelength, temperature, modulating frequency, bias voltage, and gain of any internal amplifier. The response time (τ) is the time for a detector’s output to reach 1 − 1/e = 63% of its final value, after suddenly being subjected to constant irradiation. The linearity range of a detector is the range of input power over which the output signal is a linear function of the input. The noise equivalent power (NEP) is the radiant energy rate in watts that is necessary to give an output signal equal to the rms noise output from the detector. More widely used is the reciprocal of NEP, the detectivity (D). The detectivity is known to vary inversely with the square root of the detector area, AD , while the signal noise is proportional to the square root of the amplifier’s noise-equivalent bandwidth Δ f (in Hz). Thus, a normalized detectivity (D*) is defined to allow comparison between different types of detectors regardless of their detector areas and amplifier bandwidths as D∗ = (AD Δ f )1/2 D.

(3.114)

Depending on how the incoming radiation interacts with the detector material, detectors are grouped into thermal and photon (or quantum) detectors. Thermal detectors. These devices convert incident radiation into a temperature rise. This temperature change is measured either through one or more thermocouples, or by using the pyroelectric effect. A single, usually blackened (to increase absorptance) thermocouple is the simplest and cheapest of all thermal detectors. However, it suffers from high amplifier noise and, therefore, limited detectivity. One way to increase output voltage and detectivity is to connect a number of thermocouples in series (typically 20 to 120), constituting a thermopile. Thermopiles can be manufactured economically through thin-film processes. Pyroelectric detectors are made of crystalline materials that have permanent electric polarization. When heated by irradiation, the material expands and changes its polarization, which causes a current to flow in a circuit that connects the detector’s top and bottom surfaces, as shown in the schematic of Fig. 3.42a. Since the change in temperature produces the current, pyroelectric detectors respond only to pulsed or chopped irradiation. They respond to changes in irradiation much more rapidly than thermocouples and thermopiles, and are not affected by steady background radiation. Photon detectors. These absorb the energy of incident radiation with their electrons, producing free charge carriers (photoconductive and photovoltaic detectors) or even ejecting electrons from the material (photoemissive detectors). In photoconductive and photovoltaic detectors the production of free electrons increases the electrical conductivity of the material. In the photoconductive mode an applied voltage, or reverse bias, causes a current that is proportional to the strength of irradiation to flow, as schematically shown in Fig. 3.42b. In the photovoltaic mode no bias is applied and, closing the electric circuit, a current flows as a result of the excitation of electrons (as in the operation of photovoltaic, or solar, cells). Photovoltaic detectors have greater detectivity, while photoconductive

Radiative Properties of Real Surfaces Chapter | 3 109

FIGURE 3.43 Typical spectral ranges and normalized detectivities for various detectors.

detectors exhibit extremely fast response times. For optimum performance each mode requires slightly different design, although a single device may be operated in either mode. Typical semiconductor materials used for photovoltaic and photoconductive detectors are silicon (Si), germanium (Ge), indium antimonide (InSb), mercury cadmium telluride (HgCdTe),9 lead sulfide and selenide (PbS and PbSe), and cadmium sulfide (CdS). While most semiconductor detectors have a single detector element, many of them today are also available as linear arrays and surface arrays (up to 512 × 512 elements), which—when combined with a monochromator—allows for ultra-fast data acquisition at many wavelengths. The most basic photoemissive device is a photodiode, in which high-energy photons (ultraviolet to near infrared) cause emission of electrons from photocathode surfaces placed in a vacuum. Applying a voltage causes a current that is proportional to the intensity of incident radiation to flow. The signal of a vacuum photodiode is amplified in a photomultiplier by fitting it with a series of anodes (called dynodes), which produce secondary emission electrons and a current. The latter is an order-of-magnitude higher than the original photocurrent. Thermal detectors generally respond evenly across the entire spectrum, while photon detectors have limited spectral response but higher detectivity and faster response times. The normalized detectivity of several detectors is compared in Fig. 3.43. The spectral response of photon detectors can be tailored to a degree by varying the relative amounts of detector material components. The response time of thermal detectors is relatively slow, normally in the order of milliseconds, while the response time of photon detectors ranges from microseconds to a few nanoseconds. The detectivity is often increased by cooling the detector thermoelectrically (to −30◦ C), with dry ice (195 K), or by attaching it to a liquid-nitrogen Dewar flask (77 K). OTHER COMPONENTS. In a spectroscopic experiment light from a source and/or sample is guided toward the detector by a number of mirrors and lenses. Plane mirrors are employed to bend the beam path while curved mirrors are used to focus an otherwise diverging beam onto a sample, the monochromator entrance slit, or the detector. Today’s optical mirrors provide extremely high reflectivities (> 99.5%) over the entire spectrum of interest. While focusing mirrors are generally preferable for a number of reasons, sometimes lenses need to be used for focusing. The most important drawbacks of lenses are that they tend to have relatively large reflection losses and their spectral range (with high transmissivity) is limited. While antireflection coatings can be applied, these coatings are generally only effective over narrow spectral ranges as a result of interference effects. Common lens materials for the infrared are zinc selenide (ZnSe), calcium fluoride (CaF2 ), germanium (Ge), and others. Sometimes it is necessary to split a beam into two portions (e.g., to create a reference beam that does not pass over the sample) using a beam splitter. Beam splitters are made of the same material as lenses, exploiting their 9. Mercury–Cadmium–Telluride detectors are also commonly referred to as MCT detectors.

110 Radiative Heat Transfer

FIGURE 3.44 Typical setup for calorimetric emission measurements [127].

reflecting and transmitting tendencies. It is also common to chop the beam using a mechanical chopper, which consists of a rotating blade with one or more holes or slits. Chopping may be done for a variety of reasons, such as to provide an alternating signal for a pyroelectric detector, to separate background radiation from desired radiation, to decrease electronic noise by using a lock-in amplifier tuned into the chopper frequency, and so on.

Calorimetric Emission Measurement Methods If only knowledge of the total, hemispherical emittance of a surface is required, this is most commonly determined by measuring the net radiative heat loss or gain of an isolated specimen [126–142]. Figure 3.44 shows a typical experimental setup, which was used by Funai [127]. The specimen is suspended inside an evacuated test chamber, the walls of which are coated with a near-black material. The chamber walls are cooled, while the specimen is heated electrically, directly (metallic samples), through a metal substrate (nonconducting samples), or by some other means. Temperatures of the specimen and chamber wall are monitored by thermocouples. The emittance of the sample can be determined from steady-state [126–134,143] or transient measurements [128,135–142]. In the steady-state method the sample is heated to, and kept at, a desired temperature by passing the appropriate current through the heating element. The total, hemispherical emittance may then be calculated by equating electric heat input to the specimen with the radiative heat loss from the specimen to the surroundings, or

(T) =

I2 R , As σ(Ts4 − Tw4 )

(3.115)

where I2 R is the dissipated electrical power, As is the exposed surface area of the specimen, and Ts and Tw are the temperatures of specimen and chamber walls, respectively. As will be discussed in Chapter 5, equation (3.115) assumes that the surface area of the chamber is much larger than As and/or that the emittance of the chamber wall is near unity [cf. equation (5.36)]. In the transient calorimetric technique the current is switched off when the desired temperature has been reached, and the rates of loss of internal energy and radiative heat loss are equated, or

(T) = −

ms cs dTs /dt , As σ(Ts4 − Tw4 )

where ms and cs are mass and specific heat of the sample, respectively.

(3.116)

Radiative Properties of Real Surfaces Chapter | 3 111

FIGURE 3.45 Emissometer with separate reference blackbody and two optical paths [144].

Radiometric Emission Measurement Methods High-temperature, spectral, directional surface emittances are most often determined by comparing the emission from a sample with that from a blackbody at the same temperature and wavelength, both viewed by the same detector over an identical or equivalent optical path. Under those conditions the signal from both measurements will be proportional to emitted intensity (with the same proportionality constant), and the spectral, directional emittance is found by taking the ratio of the two signals, or

λ (T, λ, θ, ψ) =

Iλ (T, λ, θ, ψ) . Ibλ (T, λ)

(3.117)

The comparison blackbody may be a separate blackbody kept at the same temperature, or it may be an integral part of the sample chamber. The latter is generally preferred at high temperatures, where temperature control is difficult, and for short wavelengths, where small deviations in temperatures can cause large inaccuracies. Separate reference blackbody. In this method a blackbody, usually a long, cylindrical, isothermal cavity with an L/D-ratio larger than 4, is kept separate from the sample chamber, while both are heated to the same temperature. Radiation coming from this Hohlraum is essentially black (cf. Table 5.1). The control system keeps the sample and blackbody at the same temperature by monitoring temperature differences with a differential thermocouple and taking corrective action whenever necessary. To monitor sample and blackbody emission via an identical optical path, either two identical paths have to be constructed, or sample and blackbody must be alternately placed into the single optical path. In the former method, identical paths are formed either through two sets of optics [144], or by moving optical components back and forth [39]. Figure 3.45 shows an example of a system with two different optical paths [144], while Fig. 3.46 is an example of a linearly actuated blackbody/sample arrangement [145]. It is also possible to combine blackbody and sample, and the device is rotated or moved back-and-forth inside a single furnace [146]. Markham and coworkers [147] mounted sample/reference blackbody individually on a turntable, heated them with a torch, and measured the directional, spectral emittance of sandblasted aluminum (up to 750 K), alumina (1300 to 2200 K), fused quartz (900 K), and sapphire (1000 K) with an FTIR spectrometer. Other modern devices employing FTIRs with wide spectral ranges include del Campo et al. [148], who constructed a rotatable sample holder inside a stainless steel chamber to control ambient gases to measure spectral, directional emittances, and Zhang and coworkers [149], who use a water-cooled chamber to measure normal emittances at low temperatures (323–373 K). Other materials measured with the separate reference blackbody technique include the normal, spectral emittance of solid and liquid silicon just below and above the melting point [150], and of a collection of 30 metals and alloys at temperatures up to 1200◦ C [151,152]. Integrated reference blackbody. At high temperatures it is preferable to incorporate the reference blackbody into the design of the sample furnace. If the sample rests at the bottom of a deep isothermal, cylindrical cavity,

112 Radiative Heat Transfer

FIGURE 3.46 Emissometer with separate reference blackbody and linearly actuated sample/blackbody arrangement [145].

the radiation leaving the sample (by emission and reflection) corresponds to that of a black surface. If the hot side wall is removed or replaced by a cold one, radiation leaving the sample is due to emission only. Taking the ratio of the two signals then allows the determination of the spectral, directional emittance from equation (3.117). Removing the reflection component from the signal may be achieved in one of two ways. Several researchers have used a tubular furnace with the sample mounted on a movable rod [153–155]. When the sample is deep inside the furnace the signal corresponds to a blackbody. The sample is then rapidly moved to the exit of the furnace and the signal is due to emission alone. Disadvantages of the method are (i) maintaining isothermal conditions up to close to the end of the tube, (ii) keeping the sample at the same temperature after displacement, and (iii) stress on the high-temperature sample due to the rapid movement. In the approach of Vader and coworkers [156] and Postlethwait et al. [157], reflection from the sample is suppressed by freely dropping a cold tube into the blackbody cavity. A schematic of the apparatus of Postlethwait et al. is shown in Fig. 3.47. Once the cold tube has been dropped, measurements must be taken rapidly (in a few seconds’ time), before substantial heating of the drop tube (and cooling of the sample). Vader and coworkers obtained spectral measurements by placing various filters in front of their detector, performing a number of drops for each sample temperature. Postlethwait employed an FTIR spectrometer, allowing them to measure the entire spectral range from 1 μm to 9 μm in a single drop. In a method more akin to the separate blackbody technique, Havstad and colleagues [146] incorporated a small blackbody cavity into a tungsten crucible (holding liquid metal samples). The entire assembly is then moved to have the optics focus on sample or blackbody, respectively.

Reflection Measurements Reflection measurements are carried out to determine the bidirectional reflection function, the directional– hemispherical reflectance, and the hemispherical–directional reflectance. The latter two provide indirect means to determine the directional absorptance and emittance of opaque specimens, in particular, if sample temperatures are too low for emission measurements. BIDIRECTIONAL REFLECTION MEASUREMENTS. If the bidirectional reflection behavior of a surface is of interest, the bidirectional reflection function, ρ , must be measured directly, by irradiating the sample with λ a collimated beam from one direction and collecting the reflected intensity over various small solid angles. A sketch of an early apparatus used by Birkebak and Eckert [158] and Torrance and Sparrow [159] is shown in Fig. 3.48. Radiation from a globar A travels through a diaphragm to a spherical mirror SM, which focuses it onto the test sample S. A pencil of radiation reflected from the sample into the desired direction is collected by another spherical mirror and focused onto the entrance slit of the monochromator, in which the wavelengths are separated by the rock salt prism P, and the signal is recorded by the thermopile T. The test sample is

Radiative Properties of Real Surfaces Chapter | 3 113

FIGURE 3.47 Schematic of a drop-tube emissometer [157].

FIGURE 3.48 Schematic of the bidirectional reflection measurement apparatus of Birkebak and Eckert [158].

mounted on a multiple-yoke apparatus, which allows independent rotation around three perpendicular axes. The resulting measurements are relative (i.e., absolute values can only be obtained by calibrating the apparatus with a known standard in place of the test sample). Example measurements for magnesium oxide are shown in Fig. 3.5 [12]. More recently built devices use sophisticated, multiple-degree-of-freedom sample mounts as well as FTIR spectrometers, such as the one of Ford and coworkers [160], who measured the bidirectional reflectances of diffuse gold and grooved nickel. The main problem with bidirectional reflection measurements is the low level of reflected radiation that must be detected (particularly in off-specular directions), even with the advent of FTIR spectrometers and highly sensitive detectors. Consequently, a number of designs have employed strong monochromatic laser sources to overcome this problem, for example, [161–165]. An overview of the different methods to determine directional–hemispherical and hemispherical–directional reflectances has been given by Touloukian and DeWitt [6]. The different types of experiments may be grouped into

114 Radiative Heat Transfer

FIGURE 3.49 Schematic of a heated cavity reflectometer [166].

three categories, heated cavity reflectometers, integrating sphere reflectometers, and integrating mirror reflectometers, each having their own ranges of applicability, advantages, and shortcomings. HEATED CAVITY REFLECTOMETERS. The heated cavity reflectometer [6,166–168] (sometimes known as the Gier–Dunkle reflectometer after its inventors [168]) consists of a uniformly heated enclosure fitted with a watercooled sample holder and a viewport, as schematically shown in Fig. 3.49. Since the sample is situated within a more or less closed isothermal enclosure, the intensity striking it from any direction is essentially equal to the blackbody intensity Ibλ (Tw ) (evaluated at the cavity-wall temperature, Tw ). Images of the sample and a spot on the cavity wall are alternately focused onto the entrance slit of a monochromator. The signal from the specimen corresponds to emission (at the sample’s temperature, Ts ) plus reflection of the cavity-wall’s blackbody intensity, Ibλ (Tw ). Since the signal from the cavity wall is proportional to Ibλ (Tw ), the ratio of the two signals corresponds to  ρλ  (ˆs) Ibλ (Tw ) + λ (ˆs) Ibλ (Ts ) Is . (3.118) = Iw Ibλ (Tw ) If the sample is relatively cold (Ts Tw ), emission may be neglected and the device simply measures the hemispherical–directional reflectance. For higher specimen temperatures, and for an opaque surface with diffuse irradiation, from equations (3.42), (3.39), and (3.44),   (3.119) ρ λ (ˆs) = ρλ (ˆs) = 1 − αλ (ˆs) = 1 − λ (ˆs), and   Ibλ (Ts ) Is = 1 − λ (ˆs) 1 − . (3.120) Iw Ibλ (Tw ) The principal source of error in this method is the difficulty in making the entire cavity reasonably isothermal and (as a consequence) making the reference signal proportional to a blackbody at the cavity-wall temperature. To make these errors less severe the method is generally only used for low sample temperatures. INTEGRATING SPHERE REFLECTOMETERS. These devices are most commonly employed for reflectance measurements [167,169–179] and are available commercially in a variety of forms, either as separate instruments or already incorporated into spectrophotometers. A good early discussion of different designs was given by Edwards and coworkers [174]. The integrating sphere may be used to measure hemispherical–directional or directional–hemispherical reflectance, depending on whether it is used in indirect or direct mode. Schematics of integrating spheres operating in the two modes are shown in Fig. 3.50. The ideal device is coated on its inside

Radiative Properties of Real Surfaces Chapter | 3 115

FIGURE 3.50 Typical integrating sphere reflectometers: (a) direct mode, (b) indirect mode.

with a material of high and perfectly diffuse reflectance. The most common material in use is smoked magnesium oxide, which reflects strongly and very diffusely up to λ 2.6 μm (cf. Fig. 3.5). Other materials, such as “diffuse gold” [175–178], have been used to overcome the wavelength limitations. The strong, diffuse reflectance, together with the spherical geometry, assures that any external radiation hitting the surface of the sphere is converted into a perfectly diffuse intensity field due to many diffuse reflections. In the direct method the sample is illuminated directly by an external source, as shown in Fig. 3.50a. All of the reflected radiation is collected by the sphere and converted into a diffuse intensity field, which is measured by a detector. Similar readings are then taken on a comparison standard of known reflectance, under the same conditions. The sample may be removed and replaced by the standard (substitution method); or there may be separate sample and standard holders, which are alternately irradiated by the external source (comparison method), the latter being generally preferred. In the indirect method a spot on the sphere surface is irradiated while the detector measures the intensity reflected by the sample (or the comparison standard) directly. Errors in integrating sphere measurements are primarily caused by imperfections of the surface coating (imperfectly diffuse reflectance), losses out of apertures, and unwanted irradiation onto the detector (direct reflection from the sample in the direct mode, direct reflection from the externally-irradiated spot on the sphere in the indirect mode). Because of temperature sensitivity of the diffuse coatings, integrating-sphere measurements have mostly been limited to moderate temperature levels. However, for monochromatic and high-speed FTIR measurements it is possible to rapidly heat up only the sample by a high-power source, such as a laser, as was done by Zhang and Modest [180]. INTEGRATING MIRROR REFLECTOMETERS. An alternative to the integrating sphere is a similar design utilizing an integrating mirror. Mirrors in general have high reflectivities in the infrared and are much more efficient than integrating spheres and, hence, are highly desirable in the infrared where the energy of the light source is low. On the other hand, it is difficult to collect the radiant energy, reflected by the sample into the hemisphere above it, into a parallel beam of small cross-section. For this reason, an integrating mirror reflectometer requires a large detector area. There are three types of integrating mirrors: hemispherical [181], paraboloidal [166,182], and ellipsoidal [183–189]. Schematics of the three different types are shown in Fig. 3.51. The principle of operation of all three is the same, only the shape of the mirror is different. Each of these mirrors has two conjugate focal points, i.e., if a point source of light is placed at one focal point, all radiation will, after reflection off the mirror, fall onto the second focal point. Thus, in the integrating mirror technique an external beam is focused onto the sample, which is located at one of the focal points, through a small opening in the mirror. Radiation reflected from the sample into any direction will be reflected by the integrating mirror and is then collected by the detector located at the other focal point. This technique yields the directional–hemispherical reflectance of the sample, after comparison with a reference signal. Alternatively, one of the focal points can hold a blackbody source, with the ellipsoidal mirror focusing the energy onto the sample at the second focal point. Radiation leaving the sample is then probed through a small hole in the mirror and spectrally resolved and detected by a monochromator or FTIR spectrometer, yielding the hemispherical–directional reflectance of the sample [189]. Sources for error in the integrating mirror method are absorption by the mirror, energy lost

116 Radiative Heat Transfer

FIGURE 3.51 Design schematics of several integrating mirror reflectometers, using (a) a hemispherical, (b) a paraboloidal, and (c) an ellipsoidal mirror.

through the entrance port, nonuniform angular response of detectors, and energy missing the detector owing to mirror aberrations. To minimize aberrations, ellipsoids are preferable over hemispheres. The method has generally been limited to relatively large wavelengths, > 2.5 μm (because of mirror limitations), and to moderate temperatures. Designs allowing sample temperatures up to about 1000◦ C have been reported by Battuello and coworkers [187], Ravindra and colleagues [190], and by Freeman et al. [191], while the torch-heated sample of Markham and coworkers’ design [189] allows sample temperatures up to 2000◦ C. In general, integrating mirrors are somewhat less popular than integrating spheres because mirrors are more sensitive to flux losses and misalignment errors.

Problems 3.1 A diffusely emitting surface at 500 K has a spectral, directional emittance that can be approximated by 0.5 in the range 0 < λ < 5 μm and 0.3 for λ > 5 μm. What is the total, hemispherical emittance of this surface surrounded by (a) air and (b) a dielectric medium of refractive index n = 2? 3.2 A certain material at 600 K has the following spectral, directional emittance: ⎧ ⎪ ⎪ ⎨0.9 cos θ, λ < 1 μm, 

λ = ⎪ ⎪ ⎩0.2, λ > 1 μm. (a) What is the total, hemispherical emittance of the material? (b) If the sun irradiates this surface at an angle of θ = 60◦ off-normal, what is the relevant total absorptance? (c) What is the net radiative energy gain or loss of this surface (per unit time and area)? 3.3 For optimum performance a solar collector surface has been treated so that, for the spectral, directional emittance /

λ = =

0.9 cos 2θ, 0.0 0.1,

θ < 45◦ θ > 45◦ all θ,

0 ,

λ < 2 μm, λ > 2 μm.

For solar incidence of 15◦ off-normal and a collector temperature of 400 K, what is the relevant ratio of absorptance to emittance? 3.4 A long, cylindrical antenna of 1 cm radius on an Earth-orbiting satellite is coated with a material whose emittance is ⎧ ⎪ ⎪ λ < 1 μm, ⎨0, 

λ = ⎪ ⎪ ⎩cos θ, λ ≥ 1 μm.

Radiative Properties of Real Surfaces Chapter | 3 117

Find the absorbed energy per meter length. (Assume irradiation is from the sun only, and in a direction normal to the antenna’s axis; neglect the Earth and stars.) 3.5 The spectral, hemispherical emittance of a (hypothetical) metal may be approximated by the relationship ⎧ ⎪ ⎪ ⎨0.5,

λ = ⎪ ⎪ ⎩0.5λc /λ,

λ < λc = 0.5 μm, λ > λc

(independent of temperature). Determine the total, hemispherical emittance of this material using (a) Planck’s law and (b) Wien’s distribution, for a surface temperature of (i) 300 K and (ii) 1000 K. How accurate is the prediction using Wien’s distribution? 3.6 A treated metallic surface is used as a solar collector material; its spectral, directional emittance may be approximated by

λ

⎧ ⎪ ⎪ ⎨0.5 μm/λ, =⎪ ⎪ ⎩0,

θ < 45◦ , θ > 45◦ .

What is the relevant α/ -ratio for near normal solar incidence if Tcoll 600 K? 3.7 A surface sample with ⎧ ⎪ ⎪ ⎨0.9 cos θ,

λ = ⎪ ⎪ ⎩0.2,

λ < 2 μm, λ > 2 μm,

is irradiated by three tungsten lights as shown. The tungsten lights may be approximated by black spheres at T = 2000 K fitted with mirrors to produce parallel light beams aimed at the sample. Neglecting background radiation, determine the absorptance of the sample. 3.8 An antenna of a satellite may be approximated by a long half cylinder, which is exposed to sunshine as shown in the sketch. The antenna has a high conductivity (i.e., is isothermal), and is coated with the material of Fig. 3.37, i.e., the material may be assumed to be gray with the following directional characteristics: ⎧ ⎪ ⎪ ⎨0.9, 0 ≤ θ < 40◦ , 

λ = ⎪ ⎪ ⎩0, θ > 40◦ . Determine the equilibrium temperature of the antenna, assuming it exchanges heat only with the sun (and cold outer space). 3.9 A large isothermal plate (temperature T = 400 K) is exposed to a long monochromatic (λ = 1 μm) line source as shown. The strength of the line source is Q (W/m length of source) = hσT 4 , spreading equally into all directions. The plate has a spectral, directional emittance of ⎧ ⎪ ⎪ π ⎨0.9 cos2 θ, λ < 2.5 μm,  0≤θ< .

λ = ⎪ ⎪ ⎩0.1, 2 λ > 2.5 μm, For a general location, x, determine relevant absorptance, emittance, and the net local heat flux qnet (x), which must be supplied to/removed from the plate to keep it isothermal at T. 3.10 A large isothermal plate (temperature T = 400 K) is exposed to a long tungsten–halogen line source as shown in the sketch next to Problem 3.9. The strength of the line source is Q = 1000 W/m length of source, spreading equally into all directions, and it has the spectral distribution of a blackbody at 4000 K. The plate has a spectral, directional emittance of / π 0.8 cos θ, λ < 3μm, 0≤θ
3μm.

118 Radiative Heat Transfer

For a general location, x, give an expression for local irradiation H, determine the relevant absorptance and emittance, and give an expression for the net local heat flux qnet (x) that must be supplied to/removed from the plate to keep it isothermal at T. 3.11 An isothermal disk (temperature T = 400 K) is exposed to a small black spherical source (temperature Ts = 4000 K) as shown. The strength of the source is Q (W), spreading equally into all directions. The plate has a spectral, directional emittance of / π 0.9 cos θ, λ < 4μm,

λ = 0≤θ< 2 0.3, λ > 4μm. For a general location, r, determine relevant absorptance, relevant emittance, and the net local heat flux qnet (r) that must be supplied to/removed from the plate to keep it isothermal at T. 3.12 A conical cavity is irradiated by a defocused CO2 laser (wavelength = 10.6 μm) as shown. The conical surface is maintained at 500 K. For cavity coating with a spectral, directional emittance ⎧ ⎪ ⎪ ⎨0.15 cos θ, λ < 6 μm, 

λ (λ, θ) = ⎪ ⎪ ⎩0.8 cos2 θ, λ > 6 μm, determine the relevant total absorptance and emittance.

3.13 A metal (m2 = 50 − 50 i) is coated with a dielectric (m1 = 2 − 0 i), which is exposed to vacuum. (a) What is the range of possible directions from which radiation can impinge on the metal? (b) What is the normal reflectance of the dielectric–metal interface? (c) What is the (approximate) relevant hemispherical reflectance for the dielectric–metal interface? 3.14 For a certain material, temperature, and wavelength the spectral, hemispherical emittance has been measured as λ . Estimate the refractive index of the material under these conditions, assuming the material to be (a) a dielectric with

λ = 0.8, (b) a metal in the infrared with λ = 0.2 (the Hagen–Rubens relation being valid). 3.15 It can be derived from electromagnetic wave theory that

λ 4 1 − nλ

nλ 3 4

for

nλ 1.

Determine λ for metals with nλ 1 as a function of wavelength and temperature. 3.16 A solar collector surface with emittance

λ

⎧ ⎪ ⎪ ⎨0.9 cos θ, λ < 2 μm, =⎪ ⎪ ⎩0.2, λ > 2 μm,

is to be kept at Tc = 500 K. For qsol = 1300 W/m2 , what is the range of possible sun positions with respect to the surface for which at least 50% of the maximum net radiative energy is collected? Neglect conduction and convection losses from the surface. 3.17 On one of those famous clear days in Central Pennsylvania (home of PennState), a solar collector is irradiated by direct sunshine and by a diffuse atmospheric radiative flux. The magnitude of the solar flux is qsun = 1000 W/m2 (incident at θsun = 45◦ ), and the effective blackbody temperature for the sky is Tsky = 244 K. The absorber plate is isothermal at 320 K and is covered with a nongray, nondiffuse material whose spectral, directional emittance may be approximated by ⎧ ⎪ ⎪ ⎨0.9, λ < 2.2 μm, 

λ (λ, θ) = nλ cos θ,

nλ = ⎪ ⎪ ⎩0.1, λ > 2.2 μm, where nλ is the normal, spectral emittance. Determine the net radiative flux on the collector.

Radiative Properties of Real Surfaces Chapter | 3 119

3.18 A small plate, insulated at the bottom, is heated by irradiation from a defocused CO2 laser beam (wavelength 10.6 μm) with an incidence angle of 30◦ off-normal. The radiative properties of the surface are ⎧ ⎪ ⎪ ⎨0.2 cos2 θ,

λ = ⎪ ⎪ ⎩0.8 cos θ,

λ < 3 μm, λ > 3 μm.

The strength of the laser beam is 1300 W/m2 . Neglecting losses due to natural convection, determine the temperature of the plate. Note: For such weak laser irradiation levels the heating effect is relatively small. 3.19 A thin disk, insulated at the bottom, is irradiated by a CO2 laser (λ = 10.6 μm) as shown. The top surface is exposed to a low temperature (300K) environment. Assume that the entire disk surface is uniformly irradiated with qL = 5 MW/m2 and that the specific heat/area of the disk is ρcp δ = 2 kJ/m2 K. The disk is at ambient temperature when the laser is turned on. The emittance of the disk surface is ⎧ ⎪ ⎪ λ < 6 μm, ⎨0.2, 

λ = ⎪ ⎪ ⎩0.9 cos θ, λ > 6 μm. (a) Indicate how to calculate the temperature history of the disk. (b) Determine the initial heating rate (in K/s) at t = 0. (c) What is the steady state temperature of the disk? (This is expected to be very high, say > 3000 K.) 3.20 Determine the total, normal emittance of copper, silver, and gold for a temperature of 1500 K. Check your results by comparing with Fig. 3.8. 3.21 Determine the total, hemispherical emittance of copper, silver, and gold for a temperature of 1500 K. Check your results by comparing with Fig. 3.11. 3.22 A polished platinum sphere is heated until it is glowing red. An observer is stationed a distance away, from where the sphere appears as a red disk. Using the various aspects of electromagnetic wave theory and/or Fig. 3.9 and Table 3.3, explain how the brightness of emitted radiation would vary across the disk, if observed with (a) the human eye, (b) an infrared camera. 3.23 Two aluminum plates, one covered with a layer of white enamel paint, the other polished, are directly facing the sun, which is irradiating the plates with 1000 W/m2 . Assuming that convection/conduction losses of the plates to the environment at 300 K can be calculated by using a heat transfer coefficient of 10 W/m2 K, and that the back sides of the plates are insulated, estimate the equilibrium temperature of each plate. 3.24 Consider a metallic surface coated with a dielectric layer. (a) Show that the fraction of energy reflected at the vacuum–dielectric interface is negligible (n1 = 1.2; k1 = 0). (b) Develop an expression for the normal, spectral emittance for the metal substrate, similar to the Hagen–Rubens relationship. (c) Develop an approximate relation for the directional, spectral emittance of the metal substrate for large wavelengths and moderate incidence angles, say θ < 75◦ . 3.25 A plate of metal with n2 = k2 = 100 is covered with a dielectric as shown. The dielectric has an absorption band such that n1 = 2, and k1 = 1 for 0.2 μm < λ < 2 μm and k1 = 0 elsewhere. The dielectric is thick enough, such that any light traveling through it of wavelengths 0.2 μm < λ < 2 μm is entirely absorbed before it reaches the metal. (a) What is the total, normal emittance of the composite if its temperature is 400 K? (b) What is the total, normal absorptance if the sun shines perpendicularly onto the composite? 3.26 Estimate the total, normal emittance of α-SiC for a temperature of (i) 300 K, (ii) 1000 K. You may assume the spectral, normal emittance to be independent of temperature. 3.27 Estimate the total, hemispherical emittance of a thick slab of pure silicon at room temperature. 3.28 Estimate and compare the total, normal emittance of room temperature aluminum for the surface finishes given in Fig. 3.25.

120 Radiative Heat Transfer

3.29 A satellite orbiting Earth has part of its (flat) surface coated with spectrally selective “black nickel,” which is a diffuse emitter and whose spectral emittance may be approximated by ⎧ ⎪ ⎪ λ < 2 μm, ⎨0.9,

λ = ⎪ ⎪ ⎩0.25, λ > 2 μm. Assuming the back of the surface to be insulated, and the front exposed to solar irradiation of qsol = 1367 W/m2 (normal to the surface), determine the relevant α/ -ratio for the surface. What is its equilibrium temperature? What would be its equilibrium temperature if the surface is turned away from the sun, such that the sun’s rays strike it at a polar angle of θ = 60◦ ? 3.30 Repeat Problem 3.29 for white paint on aluminum, whose diffuse emittance may be approximated by ⎧ ⎪ ⎪ ⎨0.1, λ < 2 μm,

λ = ⎪ ⎪ ⎩0.9, λ > 2 μm. 3.31 Estimate the spectral, hemispherical emittance of the grooved materials shown in Fig. 3.37. Repeat Problem 3.29 for these materials, assuming them to be gray. 3.32 Repeat Problem 1.7 for a sphere covered with the grooved material of Fig. 3.37, whose directional, spectral emittance may be approximated by ⎧ ⎪ ⎪ ⎨0.9, 0 ≤ θ < 40◦ , 

λ = ⎪ ⎪ ⎩0.0, 40◦ < θ < 90◦ . Assume the material to be gray. 3.33 A solar collector consists of a metal plate coated with “black nickel.” The collector is irradiated by the sun with a strength of qsol = 1000 W/m2 from a direction that is θ = 30◦ from the surface normal. On its top the surface loses heat by radiation and by free convection (heat transfer coefficient h1 = 10 W/m2 K), both to an atmosphere at Tamb = 20◦ C. The bottom surface delivers heat to the collector fluid (h2 = 50 W/m2 K), which flows past the surface at Tfluid = 20◦ C. What is the equilibrium temperature of the collector plate? How much energy (per unit area) is collected (i.e., carried away by the fluid)? Discuss the performance of this collector. Assume black nickel to be a diffuse emitter. 3.34 Make a qualitative plot of temperature vs. the total hemispherical emittance of: (a) a 3 mm thick sheet of window glass, (b) polished aluminum, and (c) an ideal metal that obeys the Hagen–Rubens relation. 3.35 Using a two-band representation of the reflectances of various components of the Earth’s surface shown in the table below, as well as the surface cover fractions given in Section 3.10, calculate the equilibrium surface temperature of the Earth in the absence of the atmosphere. What would happen if half the snow-cover was exposed as land, and the other half turned to sea surface? Band 1 (λ ≤ 0.7 μm)

Band 2 (λ > 0.7 μm)

Sea water

0.07

0.02

Snow/Ice

0.92

0.05

Exposed land

0.3

0.1

3.36 A horizontal sheet of 5 mm thick glass is covered with a 2 mm thick layer of water. If solar radiation is incident normal to the sheet, what are the transmissivity and reflectivity of the water/glass layer at λ1 = 0.6 μm and λ2 = 2 μm? For water mH2 O (0.6 μm) = 1.332 − 1.09 × 10−8 i, mH2 O (2 μm) = 1.306 − 1.1 × 10−3 i [192]; for glass mglass (0.6 μm) = 1.517 − 6.04 × 10−7 i, mglass (2 μm) = 1.497 − 5.89 × 10−5 i [83]. 3.37 A solar collector plate of spectral absorptivity αcoll = 0.90 is fitted with two sheets of 5 mm thick glass as shown in the adjacent sketch. What fraction of normally incident solar radiation is absorbed by the collector plate at a wavelength of 0.6 μm? At 0.6 μm mglass = 1.517 − 6.04 × 10−7 i [83].

Radiative Properties of Real Surfaces Chapter | 3 121

References [1] A. Goldsmith, T.E. Waterman, Thermophysical properties of solid materials, Technical Report WADC TR 58-476, Armour Research Foundation, 1959. [2] G.G. Gubareff, J.E. Janssen, R.H. Torborg, Thermal radiation properties survey, Honeywell Research Center, Minneapolis, MI, 1960. [3] W.D. Wood, H.W. Deem, C.F. Lucks, Thermal Radiative Properties, Plenum Publishing Company, New York, 1964. [4] D.I. Svet, Thermal Radiation: Metals, Semiconductors, Ceramics, Partly Transparent Bodies, and Films, Plenum Publishing Company, New York, 1965. [5] D.K. Edwards, I. Catton, Radiation characteristics of rough and oxidized metals, in: S. Gratch (Ed.), Adv. Thermophys. Properties Extreme Temp. Pressures, ASME, 1965, pp. 189–199. [6] Y.S. Touloukian, D.P. DeWitt (Eds.), Thermal Radiative Properties: Metallic Elements and Alloys, Thermophysical Properties of Matter, vol. 7, Plenum Press, New York, 1970. [7] Y.S. Touloukian, D.P. DeWitt (Eds.), Thermal Radiative Properties: Nonmetallic Solids, Thermophysical Properties of Matter, vol. 8, Plenum Press, New York, 1972. [8] Y.S. Touloukian, D.P. DeWitt, R.S. Hernicz (Eds.), Thermal Radiative Properties: Coatings, Thermophysical Properties of Matter, vol. 9, Plenum Press, New York, 1973. [9] E. Schmidt, E.R.G. Eckert, Über die Richtungsverteilung der Wärmestrahlung von Oberflächen, Forschung auf dem Gebiete des Ingenieurwesens 7 (1935) 175. [10] H.J. McNicholas, Absolute methods of reflectometry, Journal of Research of the National Bureau of Standards 1 (1928) 29–72. [11] R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, 3rd ed., Hemisphere, New York, 1992. [12] K.E. Torrance, E.M. Sparrow, Biangular reflectance of an electric nonconductor as a function of wavelength and surface roughness, ASME Journal of Heat Transfer 87 (1965) 283–292. [13] P. Drude, Annalen der Physik 39 (1890) 530. [14] H. Ehrenreich, H.R. Phillip, B. Segall, Optical properties of aluminum, Physical Review 132 (5) (1963) 1918–1928. [15] H. Ehrenreich, H.R. Phillip, Optical properties of Ag and Cu, Physical Review 128 (1) (1962) 1622–1629. [16] E. Shiles, T. Sasaki, M. Inokuti, D.Y. Smith, Self-consistency and sum-rule tests in the Kramers–Kronig analysis of optical data: applications to aluminum, Physical Review B 22 (1980) 1612–1628. [17] H.J. Hagemann, W. Gudat, C. Kunz, Optical constants from the far infrared to the x-ray region: Mg, Al, Cu, Ag, Au, Bi, C and Al2 O3 , Journal of the Optical Society of America 65 (1975) 742–744. [18] W.J. Parker, G.L. Abbott, Theoretical and experimental studies of the total emittance of metals, in: S. Katzoff (Ed.), Symposium on Thermal Radiation of Solids, 1965, pp. 11–28, NASA SP-55. [19] M.A. Ordal, L.L. Long, R.J. Bell, S.E. Bell, R.W. Alexander, C.A. Ward, Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared, Applied Optics 22 (7) (1983) 1099–1119. [20] D.E. Gray (Ed.), American Institute of Physics Handbook, 3d ed., McGraw-Hill, New York, 1972, ch. 6. [21] E.D. Palik (Ed.), Handbook of Optical Constants of Solids, vol. I, Academic Press, 1985. [22] E.D. Palik (Ed.), Handbook of Optical Constants of Solids, vol. II, Academic Press, 1991. [23] D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics, 78th ed., Chemical Rubber Company, Cleveland, OH, 1998. [24] E. Hagen, H. Rubens, Metallic reflection, Annalen der Physik 1 (2) (1900) 352–375. [25] G.E.H. Reuter, E.H. Sondheimer, The theory of the anomalous skin effect in metals, Proceedings of the Royal Society (London) A 195 (1042) (1948) 336–364. [26] W.M. Brandenberg, The reflectivity of solids at grazing angles, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 75–82, NASA SP-31. [27] W.M. Brandenberg, O.W. Clausen, The directional spectral emittance of surfaces between 200 and 600 °C, in: S. Katzoff (Ed.), Symposium on Thermal Radiation of Solids, 1965, pp. 313–319, NASA SP-55. [28] D.J. Price, The emissivity of hot metals in the infrared, Proceedings of the Physical Society 59 (331) (1947) 118–131. [29] R.V. Dunkle, Thermal radiation characteristics of surfaces, in: J.A. Clark (Ed.), Theory and Fundamental Research in Heat Transfer, Pergamon Press, New York, 1963, pp. 1–31. [30] R.V. Dunkle, Emissivity and inter-reflection relationships for infinite parallel specular surfaces, in: S. Katzoff (Ed.), Symposium on Thermal Radiation of Solids, 1965, pp. 39–44, NASA SP-55. [31] R.G. Hering, T.F. Smith, Surface radiation properties from electromagnetic theory, International Journal of Heat and Mass Transfer 11 (1968) 1567–1571. [32] C. Davisson, J.R. Weeks, The relation between the total thermal emissive power of a metal and its electrical resistivity, Journal of the Optical Society of America 8 (5) (1924) 581–605. [33] R.C. Weast (Ed.), CRC Handbook of Chemistry and Physics, 68th ed., Chemical Rubber Company, Cleveland, OH, 1988. [34] W.G. Spitzer, D.A. Kleinman, C.J. Frosch, D.J. Walsh, Optical properties of silicon carbide, in: J.R. O’Connor, J. Smiltens (Eds.), Silicon Carbide — A High Temperature Semiconductor, Proceedings of the 1959 Conference on Silicon Carbide, Boston, Massachusetts, Pergamon Press, 1960, pp. 347–365. [35] H. Bao, X. Ruan, Ab initio calculations of thermal radiative properties: the semiconductor GaAs, International Journal of Heat and Mass Transfer 53 (2010) 1308–1312.

122 Radiative Heat Transfer

[36] J.R. Jasperse, A. Kahan, J.N. Plendl, S.S. Mitra, Temperature dependence of infrared dispersion in ionic crystals LiF and MgO, Physical Review 140 (2) (1966) 526–542. [37] C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley & Sons, New York, 1983. [38] S. Roy, S.Y. Bang, M.F. Modest, V.S. Stubican, Measurement of spectral, directional reflectivities of solids at high temperatures between 9 and 11 μm, Applied Optics 32 (19) (1993) 3550–3558. [39] T.R. Riethof, V.J. DeSantis, Techniques of measuring normal spectral emissivity of conductive refractory compounds at high temperatures, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 565–584, NASA SP-31. [40] S.S. Mitra, S. Nudelman, Far-Infrared Properties of Solids, Plenum Press, New York, 1970. [41] P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Macmillan, New York, 1963. [42] F.G. Bass, I.M. Fuks, Wave Scattering from Statistically Rough Surfaces, Pergamon Press, Oxford, 1979. [43] J.A. Ogilvy, Wave scattering from rough surfaces, Reports on Progress in Physics 50 (1987) 1553–1608. [44] H. Davies, The reflection of electromagnetic waves from a rough surface, Proceedings of the IEEE, part IV 101 (1954) 209–214. [45] H.E. Bennett, Specular reflection of aluminized ground glass and the height distribution of surface irregularities, Journal of the Optical Society of America 53 (1963) 1389–1394. [46] P. Beckmann, Shadowing of random rough surfaces, IEEE Transactions on Antennas and Propagation AP-13 (1965) 384–388. [47] K.E. Torrance, E.M. Sparrow, Theory for off-specular reflection from roughened surfaces, Journal of the Optical Society of America 57 (9) (1967) 1105–1114. [48] M.V. Berry, Diffractals, Journal of Physics A: Mathematical and General 12 (1979) 781–797. [49] M.V. Berry, T.M. Blackwell, Diffractal echoes, Journal of Physics A: Mathematical and General 14 (1981) 3101–3110. [50] A. Majumdar, C.L. Tien, Reflection of radiation by rough fractal surfaces, in: Radiation Heat Transfer: Fundamentals and Applications, vol. HTD-137, ASME, June 1990, pp. 27–35. [51] A. Majumdar, B. Bhushan, Role of fractal geometry in roughness characterization and contact mechanics of surfaces, ASME Journal of Tribology 112 (1990) 205–216. [52] A. Majumdar, C.L. Tien, Fractal characterization and simulation of rough surfaces, Wear 136 (1990) 313–327. [53] A.F. Houchens, R.G. Hering, Bidirectional reflectance of rough metal surfaces, in: G.B. Heller (Ed.), Thermophysics of Spacecrafts and Planetary Bodies, in: AIAA, Progress in Astronautics and Aeronautics, vol. 20, MIT Press, 1967, pp. 65–89. [54] T.F. Smith, R.G. Hering, Comparison of bidirectional reflectance measurements and model for rough metallic surfaces, in: Proceedings of the Fifth Symposium on Thermophysical Properties, ASME, Boston, 1970, pp. 429–435. [55] R.A. Dimenna, R.O. Buckius, Electromagnetic theory predictions of the directional scattering from triangular surfaces, ASME Journal of Heat Transfer 116 (3) (1994) 639–645. [56] R.A. Dimenna, R.O. Buckius, Quantifying specular approximations for angular scattering from perfectly conducting random rough surfaces, Journal of Thermophysics and Heat Transfer 8 (3) (1994) 393–399. [57] Y. Yang, R.O. Buckius, Surface length scale contributions to the directional and hemispherical emissivity and reflectivity, Journal of Thermophysics and Heat Transfer 9 (4) (1995) 653–659. [58] K. Tang, R.A. Dimenna, R.O. Buckius, Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces, International Journal of Heat and Mass Transfer 40 (1) (1997) 49–59. [59] E. Thorsos, The validity of the Kirchhoff approximation for rough surface scattering using Gaussian roughness spectrum, The Journal of the Acoustical Society of America 83 (1988) 78–92. [60] M.F. Chen, A.K. Fung, A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models, Radio Science 23 (1988) 163–170. [61] M. Nieto-Vesperinas, J.A. Sanchez-Gil, Light transmission from a randomly rough dielectric diffuser: theoretical and experimental results, Optics Letters 15 (1990) 1261–1263. [62] K. Tang, R.O. Buckius, The geometric optics approximation for reflection from two-dimensional random rough surfaces, International Journal of Heat and Mass Transfer 41 (13) (1998) 2037–2047. [63] K. Tang, Y. Yang, R.O. Buckius, Theory and experiments on scattering from rough interfaces, in: C.L. Tien (Ed.), Annual Review of Heat Transfer, vol. X, Begell House, 1999, pp. 101–140. [64] K. Tang, P. Kawka, R.O. Buckius, Geometric optics applied to rough surfaces coated with an absorbing thin film, Journal of Thermophysics and Heat Transfer 13 (2) (1999) 169–176. [65] Y. Zhou, Z.M. Zhang, Radiative properties of semitransparent silicon wafers with rough surfaces, ASME Journal of Heat Transfer 125 (3) (2003) 462–470. [66] Q.Z. Zhu, Z.M. Zhang, Anisotropic slope distribution and bidirectional reflectance of a rough silicon surface, ASME Journal of Heat Transfer 126 (6) (2004) 985–993. [67] H.J. Lee, Y.B. Chen, Z.M. Zhang, Directional radiative properties of anisotropic rough silicon and gold surfaces, International Journal of Heat and Mass Transfer 49 (2006) 4482–4495. [68] K. Tang, R.O. Buckius, A statistical model of wave scattering from random rough surfaces, International Journal of Heat and Mass Transfer 44 (21) (2001) 4059–4073. [69] K. Fu, P.-F. Hsu, New regime map of the geometric optics approximation for scattering from random rough surfaces, Journal of Quantitative Spectroscopy and Radiative Transfer 109 (2) (2008) 180–188.

Radiative Properties of Real Surfaces Chapter | 3 123

[70] R. Carminati, J.-J. Greffet, A model for the radiative properties of opaque rough surfaces, in: Proceedings of the 11th International Heat Transfer Conference, vol. 7, Kyongju, Korea, 1998, pp. 427–432. [71] A. Ishimaru, J.S. Chen, Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing, The Journal of the Acoustical Society of America 88 (4) (1990) 1877–1883. [72] H.E. Bennett, M. Silver, E.J. Ashley, Infrared reflectance of aluminum evaporated in ultra-high vacuum, Journal of the Optical Society of America 53 (9) (1963) 1089–1095. [73] L.G. Schulz, The experimental study of the optical properties of metals and the relation of the results to the Drude free electron theory, Advances in Physics 6 (1957) 102–144. [74] R.V. Dunkle, J.T. Gier, Snow characteristics project progress report, Technical report, University of California at Berkeley, June 1953. [75] H.E. Bennett, J.M. Bennett, E.J. Ashley, Infrared reflectance of evaporated aluminum films, Journal of the Optical Society of America 52 (1962) 1245–1250. [76] Q. Lin, Y. Xuan, Y. Han, Prediction of the radiative properties of surfaces covered with particulate deposits, Journal of Quantitative Spectroscopy and Radiative Transfer 196 (2017) 112–122. [77] W.G. Spitzer, D.A. Kleinman, D.J. Walsh, Infrared properties of hexagonal silicon carbide, Physical Review 113 (1) (January 1959) 127–132. [78] R.E. Gaumer, E.R. Streed, T.F. Vajta, Methods for experimental determination of the extra-terrestrial solar absorptance of spacecraft materials, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 135–146, NASA SP-31. [79] G.F. Pezdirtz, R.A. Jewell, A study of the photodegradation of selected thermal control surfaces, in: S. Katzoff (Ed.), Symposium on Thermal Radiation of Solids, 1965, pp. 433–441, NASA SP-55. [80] H.C. Hottel, Radiant heat transmission, in: W.H. McAdams (Ed.), Heat Transmission, 3rd ed., McGraw-Hill, New York, 1954, ch. 4. [81] D.K. Edwards, A.F. Mills, V.E. Denny, Transfer Processes, 2nd ed., Hemisphere/McGraw-Hill, New York, 1979. [82] C.R. Wylie, Advanced Engineering Mathematics, 5th ed., McGraw-Hill, New York, 1982. [83] C.K. Hsieh, K.C. Su, Thermal radiative properties of glass from 0.32 to 206 μm, Solar Energy 22 (1979) 37–43. [84] R. Gardon, The emissivity of transparent materials, Journal of the American Ceramic Society 39 (8) (1956) 278–287. [85] N. Neuroth, Der Einfluss der Temperatur auf die spektrale Absorption von Gläsern im Ultraroten, I (Effect of temperature on spectral absorption of glasses in the infrared, I), Glastechnische Berichte 25 (1952) 242–249. [86] P. Boulet, J. Gérardin, G. Parent, A. Collin, Y. Pizzo, B. Porterie, Optical and radiative properties of clear PMMA samples exposed to a radiant heat flux, International Journal of Thermal Sciences 82 (2014) 1–8. [87] B.E. Yoldas, D.P. Partlow, Wide spectrum antireflective coating for fused silica and other glasses, Applied Optics 23 (9) (1984) 1418–1424. [88] J.C.C. Fan, F.J. Bachner, Transparent heat mirrors for solar-energy applications, Applied Optics 15 (4) (1976) 1012–1017. [89] B.E. Yoldas, T. O’Keefe, Deposition of optically transparent IR reflective coatings on glass, Applied Optics 23 (20) (1984) 3638–3643. [90] G.B. Smith, S. Dligatch, R. Sullivan, Thin film angular selective glazing, Solar Energy 62 (3) (1998) 229–244. [91] D.K. Edwards, Solar absorption by each element in an absorber-coverglass array, Solar Energy 19 (1977) 401–402. [92] D.K. Edwards, Finite element embedding with optical interference, in: Presentation of the Twentieth Joint ASME/AIChE National Heat Transfer Conference, vol. 81-HT-65, 1981. [93] S.J. Mitts, T.F. Smith, Solar energy transfer through semitransparent systems, Journal of Thermophysics and Heat Transfer 1 (4) (1987) 307–312. [94] W.A. Shurcliff, Transmittance and reflection loss of multi-plate planar window of a solar-radiation collector: formulas and tabulations of results for the case of n = 1.5, Solar Energy 16 (1974) 149–154. [95] J.A. Duffie, W.A. Beckman, Solar Energy Thermal Processes, John Wiley & Sons, New York, 1974. [96] D.C. Martin, R.J. Bell, The use of optical interference to obtain selective energy absorption, in: Proceedings of the Conference on Coatings for the Aerospace Environment, vol. WADD-TR-60-TB, 1960. [97] H.C. Hottel, B.B. Woertz, The performance of flat-plate solar-heat collectors, Transactions of ASME, Journal of Heat Transfer 64 (1942) 91–104. [98] J.T. Gier, R.V. Dunkle, Selective spectral characteristics as an important factor in the efficiency of solar collectors, in: Transactions of the Conference on the Use of Solar Energy, vol. 2, University of Arizona Press, Tucson, AZ, 1958, p. 41. [99] H. Tabor, J.A. Harris, H. Weinberger, B. Doron, Further studies on selective black coatings, in: Proceedings of the UN Conference on New Sources of Energy, vol. 4, 1964, p. 618. [100] H. Tabor, Selective surfaces for solar collectors, in: Low Temperature Engineering Applications of Solar Energy, ASHRAE, 1967. [101] D.K. Edwards, K.E. Nelson, R.D. Roddick, J.T. Gier, Basic studies on the use and control of solar energy, Technical Report 60-93, The University of California, Los Angeles, CA, 1960. [102] M. Baneshi, H. Gonome, A. Komiya, S. Maruyama, The effect of particles size distribution on aesthetic and thermal performances of polydisperse TiO2 pigmented coatings: comparison between numerical and experimental results, Journal of Quantitative Spectroscopy and Radiative Transfer 113 (2012) 594–606. [103] H. Gonome, M. Baneshi, J. Okajima, A. Komiya, S. Maruyama, Controlling the radiative properties of cool black-color coatings pigmented with CuO submicron particles, Journal of Quantitative Spectroscopy and Radiative Transfer 132 (2014) 90–98. [104] H. Gonome, J. Okajima, A. Komiya, S. Maruyama, Experimental evaluation of optimization method for developing ultraviolet barrier coatings, Journal of Quantitative Spectroscopy and Radiative Transfer 133 (2014) 454–463.

124 Radiative Heat Transfer

[105] Z. Huang, X. Ruan, Nanoparticle embedded double-layer coating for daytime radiative cooling, International Journal of Heat and Mass Transfer 104 (2017) 890–896. [106] G. Yang, C.Y. Zhao, A comparative experimental study on radiative properties of EB-PVD and air plasma sprayed thermal barrier coatings, ASME Journal of Heat Transfer 137 (2015) 091024. [107] G. Yang, C.Y. Zhao, Infrared radiative properties of EB-PVD thermal barrier coatings, International Journal of Heat and Mass Transfer 94 (2016) 199–210. [108] Z. Sun, J. Zhang, Z. Tong, Y. Zhao, Particle size effects on the reflectance and negative polarization of light backscattered from natural surface particulate medium: soil and sand, Journal of Quantitative Spectroscopy and Radiative Transfer 133 (2014) 1–12. [109] M.P. Thekaekara, Solar energy outside the Earth’s atmosphere, Solar Energy 14 (1973) 109–127. [110] F. Trombe, M. Foex, V. LePhat, Research on selective surfaces for air conditioning dwellings, in: Proceedings of the UN Conference on New Sources of Energy, vol. 4, 1964, pp. 625–638. [111] K.G.T. Hollands, Directional selectivity, emittance, and absorptance properties of vee corrugated specular surfaces, Solar Energy 7 (3) (1963) 108–116. [112] M. Perlmutter, J.R. Howell, A strongly directional emitting and absorbing surface, ASME Journal of Heat Transfer 85 (3) (1963) 282–283. [113] R.B. Mulford, M.R. Jones, B.D. Iverson, Dynamic control of radiative surface properties with origami-inspired design, ASME Journal of Heat Transfer 138 (2016) 032701. [114] R.B. Mulford, N.S. Collins, M.S. Farnsworth, M.R. Jones, B.D. Iverson, Total hemispherical apparent radiative properties of the infinite V-groove with specular reflection, International Journal of Heat and Mass Transfer 124 (2018) 168–176. [115] R.B. Mulford, M.R. Jones, B.D. Iverson, Heat transfer, efficiency and turn-down ratio of a dynamic radiative heat exchanger, International Journal of Heat and Mass Transfer 143 (2019) 118441. [116] R.B. Mulford, V.H. Dwivedi, M.R. Jones, B.D. Iverson, Control of net radiative heat transfer with a variable-emissivity accordion tessellation, ASME Journal of Heat Transfer 141 (2019) 032702. [117] D.L. Hartmann, Global Physical Climatology, second ed., Elsevier, 2015. [118] A. Kääb, Remote Sensing of Mountain Glaciers and Permafrost Creep, Geographisches Institut der Universität, Zürich, 2005. [119] S.K. Meerdink, S.J. Hook, D.A. Robert, E.A. Abbott, The ECOSTRESS spectral library version 1.0, Remote Sensing of Environment 230 (2019) 11196. [120] A.M. Balridge, S.J. Hook, C.I. Grove, G. Rivera, The ASTER spectral library version 2.0, Remote Sensing of Environment 113 (2009) 711–715. [121] J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, NASA SP-31. [122] S. Katzoff (Ed.), Symposium on Thermal Radiation Properties of Solids, 1964, NASA SP-55. [123] J.-F. Sacadura, Measurement techniques for thermal radiation properties, in: Proceedings of the Ninth International Heat Transfer Conference, Hemisphere, Washington, D.C., 1990, pp. 207–222. [124] M.C. Hutley, Diffraction Gratings, Academic Press, New York, 1982. [125] P.R. Griffiths, J.A. de Haseth, Fourier Transform Infrared Spectrometry, Chemical Analysis, vol. 83, John Wiley & Sons, New York, 1986. [126] R. Sadler, L. Hemmerdinger, I. Rando, A device for measuring total hemispherical emittance, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 217–223, NASA SP-31. [127] A.I. Funai, A multichamber calorimeter for high-temperature emittance studies, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 317–327, NASA SP-31. [128] D.L. McElroy, T.G. Kollie, The total hemispherical emittance of platinum, columbium-1%, zirconium, and polished and oxidized iron-8 in the range 100◦ C to 1200◦ C, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 365–379, NASA SP-31. [129] V.S. Moore, A.R. Stetson, A.G. Metcalfe, Emittance measurements of refractory oxide coatings up to 2900◦ C, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 527–533, NASA SP-31. [130] T.W. Nyland, Apparatus for the measurement of hemispherical emittance and solar absorptance from 270◦ C to 650◦ C, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 393–401, NASA SP-31. [131] G.A. Zerlaut, An apparatus for the measurement of the total normal emittance of surfaces at satellite temperatures, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 275–285, NASA SP-31. [132] S.H.P. Chen, S.C. Saxena, Experimental determination of hemispherical total emittance of metals as a function of temperature, Industrial & Engineering Chemistry Fundamentals 12 (2) (1973) 220–224. [133] B.J. Jody, S.C. Saxena, Radiative heat transfer from metal wires: hemispherical total emittance of platinum, Journal of Physics E: Scientific Instruments 9 (1976) 359–362. [134] R.E. Taylor, Determination of thermophysical properties by direct electrical heating, High Temperatures - High Pressures 13 (1981) 9–22. [135] G.D. Gordon, A. London, Emittance measurements at satellite temperatures, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 147–151, NASA SP-31. [136] R.L. Rudkin, Measurement of thermal properties of metals at elevated temperatures, in: Temperature, Its Measurement and Control in Science and Industry, vol. 3, part 2, Reinhold Publishing Corp., New York, 1962, pp. 523–534.

Radiative Properties of Real Surfaces Chapter | 3 125

[137] R.E. Gaumer, J.V. Stewart, Calorimetric determination of infrared emittance and the α/ ratio, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 127–133, NASA SP-31. [138] C.P. Butler, R.J. Jenkins, Space chamber emittance measurements, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 39–43, NASA SP-31. [139] C.P. Butler, E.C.Y. Inn, A method for measuring total hemispherical and emissivity of metals, in: First Symposium - Surface Effects on Spacecraft Materials, John Wiley & Sons, New York, 1960, pp. 117–137. [140] R. Smalley, A.J. Sievers, The total hemispherical emissivity of copper, Journal of the Optical Society of America 68 (1978) 1516–1518. [141] K.G. Ramanathan, S.H. Yen, High-temperature emissivities of copper, aluminum and silver, Journal of the Optical Society of America 67 (1977) 32–38. [142] H. Masuda, M. Higano, Measurement of total, hemispherical emissivities of metal wires by using transient calorimetric techniques, ASME Journal of Heat Transfer 110 (1988) 166–172. [143] J. Kim, C. Jang, T.H. Song, Measurement of total hemispherical emissivity using vacuum guarded hot plate, ASME Journal of Heat Transfer 134 (2012) 114501. [144] T. Limperis, D.M. Szeles, W.L. Wolfe, The measurement of total normal emittance of three nuclear reactor materials, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 357–364, NASA SP-31. [145] W.B. Fussell, F. Stair, Preliminary studies toward the determination of spectral absorption coefficients of homogeneous dielectric material in the infrared at elevated temperatures, in: S. Katzoff (Ed.), Symposium on Thermal Radiation of Solids, 1965, pp. 287–292, NASA SP-55. [146] M.A. Havstad, W.I. McLean, S.A. Self, Apparatus for the measurement of the optical constants and thermal radiative properties of pure liquid metals from 0.4 to 10 μm, Review of Scientific Instruments 64 (1993) 1971–1978. [147] J.R. Markham, P.R. Solomon, P.E. Best, An FT-IR based instrument for measuring spectral emittance of material at high temperature, Review of Scientific Instruments 61 (12) (1990) 3700–3708. [148] L. del Campo, R.B. Pérez-Sáez, X. Esquisabel, I. Fernández, M.J. Tello, New experimental device for infrared spectral directional emissivity measurements in a controlled environment, Review of Scientific Instruments 77 (11) (2006) 113111. [149] K. Zhang, K. Yu, Y. Liu, Y. Zhao, An improved algorithm for spectral emissivity measurements at low temperatures based on the multi-temperature calibration method, International Journal of Heat and Mass Transfer 114 (2017) 1037–1044. [150] E. Takasuka, E. Tokizaki, K. Terashima, S.O. Kazutaka, Emissivity of liquid silicon in visible and infrared regions, Journal of Applied Physics 81 (1997) 6384–6389. [151] M. Kobayashi, M. Otsuki, H. Sakate, F. Sakuma, A. Ono, System for measuring the spectral distribution of normal emissivity of metals with direct current heating, International Journal of Thermophysics 20 (1) (1999). [152] M. Kobayashi, A. Ono, M. Otsuki, H. Sakate, F. Sakuma, Database of normal spectral emissivities of metals at high temperatures, International Journal of Thermophysics 20 (1) (1999) 299–308. [153] S. Knopken, R. Klemm, Evaluation of thermal radiation at high temperatures, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 505–514, NASA SP-31. [154] W.H. Bennethum, Thin film sensors and radiation sensing techniques for measurement of surface temperature of ceramic components, in: HITEMP Review, Advanced High Temperature Engine Materials Technology Program, 1989, NASA CP-10039. [155] W.H. Atkinson, M.A. Cyr, Sensors for temperature measurement for ceramic materials, in: HITEMP Review, Advanced High Temperature Engine Materials Technology Program, 1989, pp. 287–292, NASA CP-10039. [156] D.T. Vader, R. Viskanta, F.P. Incropera, Design and testing of a high-temperature emissometer for porous and particulate dielectrics, Review of Scientific Instruments 57 (1) (1986) 87–93. [157] M.A. Postlethwait, K.K. Sikka, M.F. Modest, J.R. Hellmann, High temperature normal spectral emittance of silicon carbide based materials, Journal of Thermophysics and Heat Transfer 8 (3) (1994) 412–418. [158] R.C. Birkebak, E.R.G. Eckert, Effect of roughness of metal surfaces on angular distribution of monochromatic reflected radiation, ASME Journal of Heat Transfer 87 (1965) 85–94. [159] K.E. Torrance, E.M. Sparrow, Off-specular peaks in the directional distribution of reflected thermal radiation, ASME Journal of Heat Transfer 88 (1966) 223–230. [160] J.N. Ford, K. Tang, R.O. Buckius, Fourier transform infrared system measurement of the bidirectional reflectivity of diffuse and grooved surfaces, ASME Journal of Heat Transfer 117 (4) (1995) 955–962. [161] J.J. Hsia, J.C. Richmond, A high resolution laser bidirectional reflectometer, Journal of Research of the National Institute of Standards and Technology 80A (2) (1976) 189–220. [162] A.A. De Silva, B.W. Jones, Bidirectional spectral reflectance and directional-hemispherical spectral reflectance of six materials used as absorbers of solar energy, Solar Energy Materials 15 (1987) 391–401. [163] J.-J. Greffet, Design of a fully automated bidirectional laser reflectometer; applications to emissivity measurement, in: R.P. Breault (Ed.), Proceedings of SPIE on Stray Light and Contamination in Optical Systems, vol. 967, 1989, pp. 184–191. [164] M. Al Hamwi, J.-F. Sacadura, Méthode de détermination des propriétes radiatives spectrales et directionnelles, dans le proche et moyen i.r., de surfaces opaques métalliques et non-métalliques, in: Proceedings of JITH ’89, November 1989, pp. 126–136. [165] J.R. Zaworski, J.R. Welty, M.K. Drost, Measurement and use of bi-directional reflectance, International Journal of Heat and Mass Transfer 39 (1996) 1149–1156.

126 Radiative Heat Transfer

[166] R.V. Dunkle, Spectral reflection measurements, in: First Symposium - Surface Effects on Spacecraft Materials, John Wiley & Sons, New York, 1960, pp. 117–137. [167] R.J. Hembach, L. Hemmerdinger, A.J. Katz, Heated cavity reflectometer modifications, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 153–167, NASA SP-31. [168] J.T. Gier, R.V. Dunkle, J.T. Bevans, Measurement of absolute spectral reflectivity from 1.0 to 15 microns, Journal of the Optical Society of America 44 (1954) 558–562. [169] W.B. Fussell, J.J. Triolo, F.A. Jerozal, Portable integrating sphere for monitoring reflectance of spacecraft coatings, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 103–116, NASA SP-31. [170] L.F. Drummeter, E. Goldstein, Vanguard emittance studies at NRL, in: First Symposium - Surface Effects on Spacecraft Materials, John Wiley & Sons, New York, 1960, pp. 152–163. [171] K.A. Snail, L.M. Hangsen, Integrating sphere designs with isotropic throughput, Applied Optics 28 (May 1989) 1793–1799. [172] W.G. Egan, T. Hilgeman, Integrating spheres for measurements between 0.185 μm and 12 μm, Applied Optics 14 (May 1975) 1137–1142. [173] G.J. Kneissl, J.C. Richmond, A laser source integrating sphere reflectometer, Technical Report NBS-TN-439, National Bureau of Standards, 1968. [174] D.K. Edwards, J.T. Gier, K.E. Nelson, R.D. Roddick, Integrating sphere for imperfectly diffuse samples, Journal of the Optical Society of America 51 (1961) 1279–1288. [175] R.R. Willey, Fourier transform infrared spectrophotometer for transmittance and diffuse reflectance measurements, Applied Spectroscopy 30 (1976) 593–601. [176] W. Richter, Fourier transform reflectance spectrometry between 8000 cm−1 (1.25 μm) and 800 cm−1 (12.5 μm) using an integrating sphere, Applied Spectroscopy 37 (1983) 32–38. [177] K. Gindele, M. Köhl, M. Mast, Spectral reflectance measurements using an integrating sphere in the infrared, Applied Optics 24 (1985) 1757–1760. [178] W. Richter, W. Erb, Accurate diffuse reflection measurements in the infrared spectral range, Applied Optics 26 (21) (November 1987) 4620–4624. [179] D. Sheffer, U.P. Oppenheim, D. Clement, A.D. Devir, Absolute reflectometer for the 0.8–2.5 μm region, Applied Optics 26 (3) (1987) 583–586. [180] Z. Zhang, M.F. Modest, Temperature-dependent absorptances of ceramics for Nd:YAG and CO2 laser processing applications, ASME Journal of Heat Transfer 120 (2) (1998) 322–327. [181] J.E. Janssen, R.H. Torborg, Measurement of spectral reflectance using an integrating hemisphere, in: J.C. Richmond (Ed.), Measurement of Thermal Radiation Properties of Solids, 1963, pp. 169–182, NASA SP-31. [182] R.T. Neher, D.K. Edwards, Far infrared reflectometer for imperfectly diffuse specimens, Applied Optics 4 (1965) 775–780. [183] J.T. Neu, Design, fabrication and performance of an ellipsoidal spectroreflectometer, NASA CR 73193, 1968. [184] S.T. Dunn, J.C. Richmond, J.F. Panner, Survey of infrared measurement techniques and computational methods in radiant heat transfer, Journal of Spacecraft and Rockets 3 (July 1966) 961–975. [185] R.P. Heinisch, F.J. Bradar, D.B. Perlick, On the fabrication and evaluation of an integrating hemi-ellipsoid, Applied Optics 9 (2) (1970) 483–489. [186] B.E. Wood, P.G. Pipes, A.M. Smith, J.A. Roux, Hemi-ellipsoidal mirror infrared reflectometer: development and operation, Applied Optics 15 (4) (1976) 940–950. [187] M. Battuello, F. Lanza, T. Ricolfi, Infrared ellipsoidal mirror reflectometer for measurements between room temperature and 1000◦ C, High Temperature 18 (1986) 683–688. [188] K.A. Snail, Reflectometer design using nonimaging optics, Applied Optics 26 (24) (1987) 5326–5332. [189] J.R. Markham, K. Kinsella, R.M. Carangelo, C.R. Brouillette, M.D. Carangelo, P.E. Best, P.R. Solomon, Bench top Fourier transform infrared based instrument for simultaneously measuring surface spectral emittance and temperature, Review of Scientific Instruments 64 (9) (1993) 2515–2522. [190] N.M. Ravindra, S. Abedrabbo, W. Chen, F.M. Tong, A.K. Nanda, A.C. Speranza, Temperature-dependent emissivity of silicon-related materials and structures, IEEE Transactions on Semiconductor Manufacturing 11 (1) (1998) 30–39. [191] R.K. Freeman, F.A. Rigby, N. Morley, Temperature-dependent reflectance of plated metals and composite materials under laser irradiation, Journal of Thermophysics and Heat Transfer 14 (3) (2000) 305–312. [192] G.M. Hale, M.R. Querry, Optical constants of water in the 200 nm to 200 μm wavelength region, Applied Optics 12 (1973) 555–563.

Chapter 4

View Factors 4.1 Introduction In many engineering applications the exchange of radiative energy between surfaces is virtually unaffected by the medium that separates them. Such (radiatively) nonparticipating media include vacuum as well as monatomic and most diatomic gases (including air) at low to moderate temperature levels (i.e., before ionization and dissociation occurs). Examples include spacecraft heat rejection systems, solar collector systems, radiative space heaters, chemical vapor deposition reactors used in semiconductor manufacturing, illumination problems, and so on. In the following four chapters we shall consider the analysis of surface radiation transport, i.e., radiative heat transfer in the absence of a participating medium, for different levels of complexity. It is common practice to simplify the analysis by making the assumption of an idealized enclosure and/or of ideal surface properties. The greatest simplification arises if all surfaces are black: for such a situation no reflected radiation needs to be accounted for, and all emitted radiation is diffuse (i.e., the intensity leaving a surface does not depend on direction). The next level of difficulty arises if surfaces are assumed to be gray, diffuse emitters (and, thus, absorbers) as well as gray, diffuse reflectors. The vast majority of engineering calculations are limited to such ideal surfaces, which are the topic of Chapter 5. If the reflective behavior of a surface deviates strongly from a diffuse reflector (e.g., a polished metal, which reflects almost like a mirror) one may often approximate the reflectance to consist of a purely diffuse and a purely specular component. Such surfaces are discussed in Chapter 6. If greater accuracy is desired, i.e., the reflectance cannot be approximated by purely diffuse and specular components, or if the assumption of a gray surface is not acceptable, a more general approach must be taken. A few such methods are also outlined in Chapter 6. As discussed in Chapter 1 thermal radiation is generally a long-range phenomenon. This is always the case in the absence of a participating medium, since photons will travel unimpeded from surface to surface. Therefore, performing a thermal radiation analysis for one surface implies that all surfaces, no matter how far removed, that can exchange radiative energy with one another must be considered simultaneously. How much energy any two surfaces exchange depends in part on their size, separation distance, and orientation, leading to geometric functions known as view factors. In the present chapter these view factors are developed for gray, diffusely radiating (i.e., emitting and reflecting) surfaces. However, the view factor is a very basic function that will also be employed in the analysis of specular reflectors as well as for the analysis for surfaces with arbitrary emission and reflection properties. Making an energy balance on a surface element, as shown in Fig. 4.1, we find

FIGURE 4.1 Surface energy balance. Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00012-2 Copyright © 2022 Elsevier Inc. All rights reserved.

127

128 Radiative Heat Transfer

FIGURE 4.2 (a) Irradiation from different locations in an enclosure, (b) real and ideal enclosures for radiative transfer calculations.

q = qemission − qabsorption = E − αH.

(4.1)

In this relation qemission and qabsorption are absolute values with directions as given by Fig. 4.1, while q is the net heat flux supplied to the surface, as defined in Chapter 1 by equation (1.40). According to this definition q is positive if the heat is coming from inside the wall material, by conduction or other means (q > 0), and negative if going from the enclosure into the wall (q < 0). Alternatively, the heat flux may be expressed as q = qout − qin = (qemission + qreflection ) − qirradiation = (E + ρH) − H,

(4.2)

which is, of course, the same as equation (4.1) since, for opaque surfaces, ρ = 1 − α. The irradiation H depends, in general, on the level of emission from surfaces far removed from the point under consideration, as schematically indicated in Fig. 4.2a. Thus, in order to make a radiative energy balance we always need to consider an entire enclosure rather than an infinitesimal control volume (as is normally done for other modes of heat transfer, i.e., conduction or convection). The enclosure must be closed so that irradiation from all possible directions can be accounted for, and the enclosure surfaces must be opaque so that all irradiation is accounted for, for each direction. In practice, an incomplete enclosure may be closed by introducing artificial surfaces. An enclosure may be idealized in two ways, as indicated in Fig. 4.2b: by replacing a complex geometrical shape with a few simple surfaces, and by assuming surfaces to be isothermal with constant (i.e., average) heat flux values across them. Obviously, the idealized enclosure approaches the real enclosure for sufficiently small isothermal subsurfaces.

4.2 Definition of View Factors To make an energy balance on a surface element, equation (4.1), the irradiation H must be evaluated. In a general enclosure the irradiation will have contributions from all visible parts of the enclosure surface. Therefore, we need to determine how much energy leaves an arbitrary surface element dA that travels toward dA. The geometric relations governing this process for “diffuse” surfaces (for surfaces that absorb and emit diffusely, and also reflect radiative energy diffusely) are known as view factors. Other names used in the literature are configuration factor, angle factor, and shape factor, and sometimes the term diffuse view factor is used (to distinguish from specular view factors for specularly reflecting surfaces; see Chapter 6). The view factor between two infinitesimal surface elements dAi and dA j , as shown in Fig. 4.3a, is defined as dFdAi −dA j ≡

diffuse energy leaving dAi directly toward and intercepted by dA j total diffuse energy leaving dAi

,

(4.3)

where the word “directly” is meant to imply “on a straight path, without intervening reflections.” This view factor is infinitesimal since only an infinitesimal fraction can be intercepted by an infinitesimal area. From the definition of intensity and Fig. 4.3a we may determine the heat transfer rate from dAi to dA j as

View Factors Chapter | 4 129

FIGURE 4.3 Radiative exchange between (a) two infinitesimal surface elements, (b) one infinitesimal and one finite surface element, and (c) two finite surfaces.

I(ri )(dAi cos θi ) dΩ j = I(ri ) cos θi cos θj dAi dA j /S2 ,

(4.4)

where θi (or θj ) is the angle between the surface normal nˆ i (or nˆ j ) and the line connecting dAi and dA j (of length S). The total radiative energy leaving dAi into the hemisphere above it is J = E + ρH, where J is called the radiosity. Since the surface emits and reflects diffusely both E and ρH obey equation (1.35), and the outgoing flux may be related to intensity by   J(ri ) dAi = E(ri ) + ρ(ri ) H(ri ) dAi = πI(ri ) dAi . Note that the radiative intensity away from dAi , due to emission and/or reflection, does not depend on direction. Therefore, the view factor between two infinitesimal areas is dFdAi −dA j =

cos θi cos θj πS2

dA j .

(4.5)

By introducing the abbreviation si j = rj − ri , and noting that cos θi = nˆ i · si j /|si j |, the view factor may be recast in vector form as (nˆ i · si j )(nˆ j · s ji )

dFdAi −dA j =

πS4

dA j .

(4.6)

Switching subscripts i and j in equation (4.5) immediately leads to the important law of reciprocity, dAi dFdAi −dA j = dA j dFdA j −dAi .

(4.7)

Often, enclosures are idealized to consist of a number of finite isothermal subsurfaces, as indicated in Fig. 4.2b. Therefore, we should like to expand the definition of the view factor to include radiative exchange between one infinitesimal and one finite area, and between two finite areas. Consider first the exchange between an infinitesimal dAi and a finite A j , as shown in Fig. 4.3b. The total energy leaving dAi toward all of A j is, from equation (4.4),  I(ri ) dAi Aj

cos θi cos θj S2

dA j ,

130 Radiative Heat Transfer

while the total energy leaving the dAi into all directions remains unchanged. Thus, we find  cos θi cos θj dA j , FdAi −A j = πS2 Aj

(4.8)

which is now finite since the intercepting surface, A j , is finite. Next we consider the view factor from A j to the infinitesimal dAi . The amount of radiation leaving all of A j toward dAi is, from equation (4.4) (after switching subscripts i and j),  cos θi cos θj I(r j ) dA j , dAi S2 Aj and the total amount leaving A j into all directions is  π I(r j ) dA j . Aj

Thus, we find the view factor between surfaces A j and dAi is  dFA j −dAi =

I(r j )

cos θi cos θj S2

Aj

.  dA j dAi π I(r j ) dA j ,

(4.9)

Aj

which is infinitesimal since the intercepting surface, dAi , is infinitesimal. The view factor in equation (4.9)—unlike equations (4.5) and (4.8)—is not a purely geometric parameter since it depends on the radiation field I(r j ). However, for an ideal enclosure as shown in Fig. 4.2b, it is usually assumed that the intensity leaving any surface is not only diffuse but also does not vary across the surface, i.e., I(r j ) = I j = const. With this assumption equation (4.9) becomes  cos θi cos θj 1 dA j dAi . (4.10) dFA j −dAi = A j Aj πS2 Comparing this with equation (4.8) we find another law of reciprocity, with A j dFA j −dAi = dAi FdAi −A j ,

(4.11)

subject to the restriction that the intensity leaving A j does not vary across the surface. Finally, we consider radiative exchange between two finite areas Ai and A j as depicted in Fig. 4.3c. The total energy leaving Ai toward A j is, from equation (4.4),   I(ri ) Ai

and the view factor follows as

I(ri ) Ai

Aj

S2

Aj

 

FAi −A j =

cos θi cos θj

cos θi cos θj S2

dA j dAi ,

.  dA j dAi π I(ri ) dAi .

(4.12)

Ai

If we assume again that the intensity leaving Ai does not vary across the surface, the view factor reduces to   cos θi cos θj 1 dA j dAi . (4.13) FAi −A j = A i Ai A j πS2 The law of reciprocity follows readily as Ai FAi −A j = A j FA j −Ai ,

(4.14)

View Factors Chapter | 4 131

which is now subject to the condition that the radiation intensities leaving Ai and A j must both be constant across their respective surfaces. In a somewhat more compact notation, the law of reciprocity may be summarized as dAi dFdi−d j = dA j dFd j−di , dAi Fdi− j = A j dF j−di , (I j = const), Ai Fi−j = A j F j−i , (Ii , I j = const). The different levels of view factors may be related to one another by  Fdi− j = dFdi−d j , Aj

Fi−j

1 = Ai

(4.15a) (4.15b) (4.15c)

(4.16a)

 Fdi− j dAi .

(4.16b)

Ai

If the receiving surface consists of a number of subsurfaces, we also have Fi− j =

K 

Fi−( j,k) , with A j =

K 

k=1

A( j,k) .

(4.17)

k=1

Finally, an enclosure consisting of N surfaces, each with constant outgoing intensities, obeys the summation relation, N  j=1

Fdi− j =

N 

Fi−j = 1.

(4.18)

j=1

The last two relations follow directly from the definition of the view factor (i.e., the sum of all fractions must add up to unity). Note that equation (4.18) includes the view factor Fi−i . If surface Ai is flat or convex, no radiation leaving it will strike itself directly, and Fi−i simply vanishes. However, if Ai is concave, part of the radiation leaving it will be intercepted by itself and Fi−i > 0.

4.3 Methods for the Evaluation of View Factors The calculation of a radiative view factor between any two finite surfaces requires the solution to a double area integral, or a fourth-order integration. Such integrals are exceedingly difficult to evaluate analytically except for very simple geometries. Even numerical quadrature may often be problematic because of singularities in the integrand, obstructions between surfaces, and because limits of integration are difficult to define for irregular shapes. Therefore, considerable effort has been directed toward tabulation and the development of evaluation methods for view factors. Early tables and charts for simple configurations were given by Hamilton and Morgan [1], Leuenberger and Pearson [2], and Kreith [3]. Fairly extensive tabulations were given in the books by Sparrow and Cess [4] and Siegel and Howell [5]. Siegel and Howell also give an exhaustive listing of sources for more involved view factors. The most complete tabulation is given in a catalogue by Howell [6,7], the latest version of which can also be accessed on the Internet via http://www.engr.uky.edu/rtl/Catalog/. A number of commercial and noncommercial computer programs for their evaluation are also available [8–18], and a review of available numerical methods has been given by Emery and coworkers [19]. Some experimental methods have been discussed by Jakob [20] and Liu and Howell [21]. Within the present book, Appendix D gives view factor formulae for an extensive set of geometries. A self-contained Fortran/C++/Matlab program viewfactors is included in Appendix F for the evaluation of all view factors listed in Appendix D [this program calls a function view, which may also be used from within other programs]. Several more programs for viewfactors not included in Appendix D are also given in Appendix F. Radiation view factors may be determined by a variety of methods. One possible grouping of different approaches could be: 1. Direct integration:

132 Radiative Heat Transfer

(i) analytical or numerical integration of the relations given in the previous section (surface integration); (ii) conversion of the relations to contour integrals, followed by analytical or numerical integration (contour integration). 2. Statistical determination: View factors may be determined through statistical sampling with the Monte Carlo method. 3. Special methods: For many simple shapes integration can be avoided by employing one of the following special methods: (i) view factor algebra, i.e., repeated application of the rules of reciprocity and the summation relationship; (ii) crossed-strings method: a simple method for evaluation of view factors in two-dimensional geometries; (iii) unit sphere method: a powerful method for determining view factors between one infinitesimal and one finite area; (iv) inside sphere method: a simple method to determine view factors for a few special shapes. All of the above methods will be discussed in the following pages, except for the Monte Carlo method, which is treated in considerable detail in Chapter 7.

4.4 Area Integration To evaluate equation (4.5) or to carry out the integrations in equations (4.8) and (4.13) the integrand (i.e., cos θi , cos θj , and S) must be known in terms of a local coordinate system that describes the geometry of the two surfaces. While the evaluation of the integrand may be straightforward for some simple configurations, it is desirable to have a more generally applicable formula at one’s disposal. Using an arbitrary coordinate origin, a vector pointing from the origin to a point on a surface may be written as ˆ r = xî + yˆj + zk,

(4.19)

where î, jˆ, and kˆ are unit vectors pointing into the x-, y-, and z-directions, respectively. Thus the vector from dAi going to dA j is determined (see Fig. 4.3) as ˆ si j = −s ji = r j − ri = (x j − xi )î + (y j − yi )ˆj + (z j − zi )k.

(4.20)

The length of this vector is determined as |si j |2 = |s ji |2 = S2 = (x j − xi )2 + (y j − yi )2 + (z j − zi )2 .

(4.21)

ˆ or, We will now assume that the local surface normals are also known in terms of the unit vectors î, ˆj, and k, from Fig. 4.4, ˆ nˆ = l î + mˆj + nk,

FIGURE 4.4 Unit normal and direction cosines for a surface element.

(4.22)

View Factors Chapter | 4 133

FIGURE 4.5 View factor for strips on an infinitely long groove.

ˆ i.e., l = nˆ · î = cos θx is the cosine of the angle θx where l, m, and n are the direction cosines for the unit vector n, between nˆ and the x-axis, etc. We may now evaluate cos θi and cos θj as  1 (x j − xi )li + (y j − yi )mi + (z j − zi )ni , S S  nˆ j · s ji 1 = cos θj = (xi − x j )l j + (yi − y j )m j + (zi − z j )n j . S S

cos θi =

nˆ i · si j

=

(4.23a) (4.23b)

Example 4.1. Consider the infinitely long (−∞ < y < +∞) wedge-shaped groove as shown in Fig. 4.5. The groove has sides of widths a and b and an opening angle α. Determine the view factor between the narrow strips shown in the figure. Solution After placing the coordinate system as shown in the figure, we find z1 = 0, x2 = u2 cos α, and z2 = u2 sin α, leading to S2 = (x1 − u2 cos α)2 + (y1 − y2 )2 + u22 sin2 α = (x21 − 2x1 u2 cos α + u22 ) + (y1 − y2 )2 = S20 + (y1 − y2 )2 , where S0 is the projection of S in the x-z-plane and is constant in the present problem. The two surface normals are readily determined as ˆ nˆ 1 = k,

l1 = m1 = 0, n1 = 1, ˆ nˆ 2 = î sin α − k cos α, or l2 = sin α, m2 = 0, n2 = − cos α, or

leading to cos θ1 = u2 sin α/S, cos θ2 = [(x1 −u2 cos α) sin α + u2 sin α cos α] /S = x1 sin α/S. For illustrative purposes we will first calculate dFd1−strip 2 from equation (4.8), and then dFstrip 1−strip 2 from equation (4.16). Thus   2 cos θ1 cos θ2 du2 +∞ x1 u2 sin α dy2 dA = dFd1−strip 2 = 2  2 πS π −∞ S2 + (y − y )2 2 dAstrip 2 0

x1 u2 sin2 α du2 = π

1

2

⎤+∞ ⎡ ⎢⎢ y2 − y1 y − y1 ⎥⎥⎥ 1 ⎢⎢ −1 2 ⎥⎥   + 3 tan ⎢⎢ ⎣ 2S2 S2 +(y − y )2 S0 ⎥⎦ 2S0 1 2 0 0 −∞

134 Radiative Heat Transfer

FIGURE 4.6 Two-dimensional wedge-shaped groove with projected distances.

=

du2 x1 u2 sin2 α du2 1 u2 sin α x1 sin α du2 1 = = cos θ10 cos θ20 , 2 S0 S0 S0 2 S0 2S30

where θ10 and θ20 are the projections of θ1 and θ2 in the x-z-plane. Looking at Fig. 4.6 this may be rewritten as dFd1−strip 2 =

1 2

cos φ dφ,

where φ = θ10 is the off-normal angle at which dAstrip 2 is oriented from dAstrip 1 . We note that dFd1−strip 2 does not depend on y1 . No matter where on strip 1 an observer is standing, he sees the same strip 2 extending from −∞ to +∞. It remains to calculate dFstrip 1−strip 2 from equation (4.16). Since equation (4.16) simply takes an average, and since dFd1−strip 2 does not vary along dAstrip 1 , it follows immediately that dFstrip 1−strip 2 =

1 2

cos φ dφ =

x1 sin2 α u2 du2 . 2S30

Example 4.2. Determine the view factor F1−2 for the infinitely long groove shown in Fig. 4.6. Solution Since we already know the view factor between two infinite strips, we can write 

b

Fstrip 1−2 =

dFstrip 1−strip 2 , 0

F1−2 =

1 a



a

Fstrip 1−2 dx1 . 0

Therefore, from Example 4.1,

Fstrip 1−2 =

x1 sin α 2 2



b 0

u2 du2 (x21 − 2x1 u2 cos α + u22 )3/2

⎛ ⎞ ⎜⎜ ⎟⎟ ⎟⎟ b cos α − x1 1 ⎜⎜⎜ ⎟⎟ . = ⎜⎜⎜1 + ( ⎟⎟ 2 ⎜⎝ ⎟ 2 2 x1 − 2bx1 cos α + b ⎠

b  2 cos α u − x x x1 sin α 1 2  1 = (  2 2 2 2 2 x1 sin α x1 − 2x1 u2 cos α + u2  2

0

Finally, carrying out the second integration we obtain F1−2

1 = a



a 0

⎞ ⎛   2 ⎟ a  (  ⎜⎜ ⎟⎟ b b 1 1 b 1 ⎜  ⎟⎟ . 1− Fstrip 1−2 dx1 = x21 − 2bx1 cos α + b2  = ⎜⎜⎜⎜1 + − 1 − 2 cos α + ⎟ 2 a 2⎝ a a a ⎟⎠ 0

View Factors Chapter | 4 135

FIGURE 4.7 Coordinate systems for the view factor between parallel, coaxial disks.

Example 4.3. As a final example for area integration we shall consider the view factor between two parallel, coaxial disks of radius R1 and R2 , respectively, as shown in Fig. 4.7. Solution Placing x-, y-, and z-axes as shown in the figure, and making a coordinate transformation to cylindrical coordinates, we find x1 = r1 cos ψ1 , y1 = r1 sin ψ1 , z1 = 0;

dA1 = r1 dr1 dψ1 ;

x2 = r2 cos ψ2 , y2 = r2 sin ψ2 , z2 = h;

dA2 = r2 dr2 dψ2 ;

S = (r1 cos ψ1 − r2 cos ψ2 ) + (r1 sin ψ1 − r2 sin ψ2 )2 + h2 2

2

= h2 + r21 + r22 − 2r1 r2 cos(ψ1 − ψ2 ). ˆ we also find l1 = l2 = m1 = m2 = 0, n1 = −n2 = 1, and from equation (4.23) cos θ1 = cos θ2 = h/S. Since nˆ 1 = kˆ and nˆ 2 = −k, Thus, from equation (4.13) F1−2

1 = (πR21 )π



R1



r1 =0

R2



r2 =0



2π ψ1 =0

2π ψ2 =0

h2 r1 r2 dψ2 dψ1 dr2 dr1  2 . h2 +r21 +r22 −2r1 r2 cos(ψ1 −ψ2 )

Changing the dummy variable ψ2 to ψ = ψ1 − ψ2 makes the integrand independent of ψ1 (integrating from ψ1 − 2π to ψ1 is the same as integrating from 0 to 2π, since integration is over a full period), so that the ψ1 -integration may be carried out immediately: F1−2 =

2h2 πR21



R1

r1 =0



R2 r2 =0



2π ψ=0

r1 r2 dψ dr2 dr1 (h2 +r21 +r22 −2r1 r2 cos ψ)2

.

This result can also be obtained by physical argument, since the view factor from any pie slice of A1 must be the same (and equal to the one from the entire disk). While a second integration (over r1 , r2 , or ψ) can be carried out, analytical evaluation of the remaining two integrals appears bleak. We shall abandon the problem here in the hope of finding another method with which we can evaluate F1−2 more easily.

4.5 Contour Integration According to Stokes’ theorem, as developed in standard mathematics texts such as Wylie [22], a surface integral may be converted to an equivalent contour integral (see Fig. 4.8) through  1 f · ds = (∇ × f) · nˆ dA, (4.24) Γ

A

136 Radiative Heat Transfer

FIGURE 4.8 Conversion between surface and contour integral; Stokes’ theorem.

where f is a vector function defined everywhere on the surface A, including its boundary Γ, nˆ is the unit surface normal, and s is the position vector for a point on the boundary of A (ds, therefore, is the vector describing the boundary contour of A). By convention, the contour integration in equation (4.24) is carried out in the counterclockwise sense for an observer standing atop the surface (i.e., on the side from which the normal points up). If a vector function f that makes the integrand of equation (4.24) equivalent to the one of equation (4.8) can be identified, then the area (or double) integral of equation (4.8) can be reduced to a contour (or single) integral. Applying Stokes’ theorem twice, the double area integration of equation (4.13) could be converted to a double line integral. Contour integration was first applied to radiative view factor calculations (in the field of illumination engineering) by Moon [23]. The earliest applications to radiative heat transfer appear to have been by de Bastos [24] and Sparrow [25].

View Factors from Differential Elements to Finite Areas For this case the vector function f may be identified as f=

1 s12 × nˆ 1 , 2π S2

leading to Fd1−2

1 = 2π

1 Γ2

(s12 × nˆ 1 ) · ds2 , S2

(4.25)

(4.26)

where s12 is the vector pointing from dA1 to a point on the contour of A2 (described by vector s2 ), while ds2 points along the contour of A2 . For the interested reader with some background in vector calculus we shall briefly prove that equation (4.26) is equivalent to equation (4.8). Using the identity (given, e.g., by Wylie [22]), ∇ × (ϕa) = ϕ∇ × a − a × ∇ϕ,

(4.27)

    1 1 s12 × nˆ 1 ˆ ˆ = ∇ ×(s × n )−(s × n )×∇ 2 12 1 12 1 2 2 . 2 2 S S S

(4.28)

we may write1 2π∇2 ×f = ∇2 ×

From equations (4.20) and (4.21) it follows that   2s12 1 2 2 s12 ∇2 2 = − 3 ∇2 S = − 3 =− 4 . S S S S S 1. We add the subscript 2 to all operators to make clear that differentiation is with respect to position coordinates on A2 , for example, x 2 , y 2 , and z 2 if a Cartesian coordinate system is employed.

View Factors Chapter | 4 137

We also find, using standard vector identities, (s12 × nˆ 1 ) × s12 = nˆ 1 (s12 · s12 ) − s12 (s12 · nˆ 1 ) = S2 nˆ 1 − s12 (s12 · nˆ 1 ),

(4.29a)

∇2 × (s12 × nˆ 1 ) = nˆ 1 · ∇2 s12 − s12 · ∇2 nˆ 1 + s12 ∇2 · nˆ 1 − nˆ 1 ∇2 · s12 .

(4.29b)

In the last expression the terms ∇2 nˆ 1 and ∇2 · nˆ 1 drop out since nˆ 1 is independent of surface A2 . Also, from equation (4.20) we find ∇2 · s12 = 3,

∇2 s12 = îî + ˆjˆj + kˆ kˆ = δ,

(4.30)

where δ is the unit tensor whose diagonal elements are unity and whose nondiagonal elements are zero: ⎞ ⎛ ⎜⎜1 0 0⎟⎟ ⎟⎟ ⎜⎜ δ = ⎜⎜⎜0 1 0⎟⎟⎟ . ⎟⎠ ⎝⎜ 0 0 1

(4.31)

With nˆ 1 · δ = nˆ 1 equation (4.29b) reduces to ∇2 × (s12 × nˆ 1 ) = nˆ 1 − 3nˆ 1 = −2nˆ 1 . Substituting all this into equation (4.28), we obtain 2π∇2 × f = −

 2nˆ 1 2  2 2 ˆ ˆ n + − s (s · n ) = − 4 s12 (s12 · nˆ 1 ), S 1 12 12 1 2 4 S S S

and (∇2 × f) · nˆ 2 = −

(s12 · nˆ 1 )(s12 · nˆ 2 ) cos θ1 cos θ2 = . πS2 πS4

(4.32)

Together with Stokes’ theorem this completes the proof that equation (4.26) is equivalent to an area integral over the function given by equation (4.32). For a Cartesian coordinate system, using equations (4.19) through (4.22), we have ˆ ds2 = dx2 î + dy2 ˆj + dz2 k, and equation (4.26) becomes 1

(z2 −z1 ) dy2 − (y2 − y1 ) dz2 m1 Fd1−2 + 2π S2 Γ 1 2 (y2 − y1 ) dx2 − (x2 −x1 ) dy2 n1 + . 2π Γ2 S2 l1 = 2π

1 Γ2

(x2 −x1 ) dz2 − (z2 −z1 ) dx2 S2

Example 4.4. Determine the view factor Fd1−2 for the configuration shown in Fig. 4.9. Solution With the coordinate system as shown in the figure we have S=

( x2 + y2 + c2 ,

ˆ or l1 = m1 = 0 and n1 = −1, it follows that equation (4.33) reduces to and, with nˆ 1 = −k,

(4.33)

138 Radiative Heat Transfer

FIGURE 4.9 View factor to a rectangular plate from a parallel infinitesimal area element located opposite a corner.

1

y dx − x dy S2 Γ2 ⎧   y=a   x=0   y=0  ⎫ x=b ⎪ ⎪ ⎪ y y (−x) (−x) 1 ⎪ ⎬ ⎨ =− dx + dy + dx + dy ⎪ ⎪ ⎪ ⎪ 2 2 2 2 ⎩ ⎭ 2π S S S x=0 S y=0 x=b y=a y=0 y=a x=b x=0   a  b b dy 1 a dx = + 2 +a2 +c2 2π y=0 b2 + y2 +c2 x x=0

Fd1−2 = −

1 2π

⎛ a b ⎞ ⎜⎜ y   ⎟⎟⎟ x a −1 −1 ⎜⎜ √ b tan √ tan √  + √  ⎟⎟⎠ ⎜⎝ 2 2 2 2 2 2 2 2 b +c b +c 0 a +c a +c 0   b a b 1 a −1 −1 . = tan √ + √ tan √ √ 2π b2 +c2 b2 +c2 b2 +c2 a2 +c2

1 = 2π Fd1−2

View Factors Between Finite Areas To reduce the order of integration for the determination of the view factor between two finite surfaces A1 and A2 , Stokes’ theorem may be applied twice, leading to 1 1 1 A1 F1−2 = ln S ds2 · ds1 , (4.34) 2π Γ1 Γ2 where the contours of the two surfaces are described by the two vectors s1 and s2 . To prove that equation (4.34) is equivalent to equation (4.13) we get, comparing with equation (4.24) (for surface A1 ), 1 1 f= ln S ds2 . (4.35) 2π Γ2 Taking the curl leads, by means of equation (4.27), to 1 1 1 2π∇1 × f = ∇1 × (ln S ds2 ) = ∇1 (ln S) × ds2 = Γ2

Γ2

Γ2

1 ∇1 S × ds2 , S

(4.36)

where differentiation is with respect to the coordinates of surface A1 (for which Stokes’ theorem has been applied). Forming the dot product with nˆ 1 then results in

View Factors Chapter | 4 139

1 nˆ 1 · (∇1 × f) =

Γ2

1 nˆ 1 · (∇1 S × ds2 ) = 2πS

1 Γ2

nˆ 1 × ∇1 S · ds2 , 2πS

(4.37)

where use has been made of the vector relationship u · (v × w) = (u × v) · w.

(4.38)

Again, from equations (4.20) and (4.21) it follows that ∇1 S = −s12 /S, so that 1 1 nˆ 1 × s12 s12 × nˆ 1 nˆ 1 · (∇1 × f) = − · ds2 = · ds2 2 2 2πS Γ2 Γ2 2πS  cos θ1 cos θ2 = Fd1−2 = dA2 , πS2 A2 where equation (4.26) has been employed. Finally,    A1 F1−2 = nˆ 1 · (∇1 × f) dA1 = A1

A1 A2

cos θ1 cos θ2 dA2 dA1 , πS2

which is, of course, identical to equation (4.13). For Cartesian coordinates, with s1 and s2 from equation (4.19), equation (4.34) becomes 1 1 1 A1 F1−2 = ln S (dx2 dx1 + dy2 dy1 + dz2 dz1 ). 2π Γ1 Γ2

(4.39)

(4.40)

Example 4.5. Determine the view factor between two parallel, coaxial disks, Example 4.3, by contour integration. Solution With ds = dx î + dy ˆj + dz kˆ it follows immediately from the coordinates given in Example 4.3 that ds1 = R1 dψ1 (− sin ψ1 î + cos ψ1 ˆj), ds2 = R2 dψ2 (− sin ψ2 î + cos ψ2 ˆj), ds1 · ds2 = R1 R2 dψ1 dψ2 (sin ψ1 sin ψ2 + cos ψ1 cos ψ2 ) = R1 R2 cos(ψ1 − ψ2 ) dψ1 dψ2 , where, it should be remembered, ds is along the periphery of a disk, i.e., at r = R. Substituting the last expression into equation (4.34) leads to F1−2

R1 R2 = 2π(πR21 )



2π

ψ1 =0



−2π

ψ2 =0

 1/2 ln h2 +R21 +R22 −2R1 R2 cos(ψ1 −ψ2 ) cos(ψ1 −ψ2 ) dψ2 dψ1 ,

where the integration for ψ2 is from 0 to −2π since, for an observer standing on top of A2 , the integration must be in a counterclockwise sense. Just like in Example 4.3, we can eliminate one of the integrations immediately since the angles appear only as differences, i.e., ψ1 − ψ2 : F1−2 = −

1 R2 π R1



2π 0

 1/2 ln h2 +R21 +R22 −2R1 R2 cos ψ cos ψ dψ.

Integrating by parts we obtain: F1−2 = − =

1 R2 π R1

R2 /R1 2π

 

   2π 1/2 2π  sin2 ψ dψ sin ψ ln h2 +R21 +R22 −2R1 R2 cos ψ  − R1 R2  h2 +R21 +R22 −2R1 R2 cos ψ 0 0 2π 0

sin2 ψ dψ , X − cos ψ

where we have introduced the abbreviation X=

h2 + R21 + R22 2R1 R2

.

140 Radiative Heat Transfer

FIGURE 4.10 View factor configuration for Example 4.6.

The integral can be found in better integral tables, or may be converted to a simpler form through trigonometric relations, leading to F1−2 =

  R2   √ √ R2 /R1 X − X2 − 1 . 2π X − X2 − 1 = 2π R1

4.6 View Factor Algebra Many view factors for fairly complex configurations may be calculated without any integration by simply using the rules of reciprocity and summation, and perhaps the known view factor for a more basic geometry. That is, besides one (or more) known view factor we will only use the following three basic equations: Ai Fi−j = A j F j−i ,

Reciprocity Rule:

N 

Summation Relation:

(4.15c)

Fi−j = 1,

(4.18)

j=1

Subsurface Summation A j =

K 

A( j,k) :

Fi−j =

k=1

K 

Fi−( j,k)

(4.17)

k=1

We shall illustrate the usefulness of this view factor algebra through a few simple examples. Example 4.6. Suppose we have been given the view factor for the configuration shown in Fig. 4.9, that is, Fd1−2 = F(a, b, c) as determined in Example 4.4. Determine the view factor Fd1−3 for the configuration shown in Fig. 4.10. Solution To express Fd1−3 in terms of known view factors F(a, b, c) (with the differential area opposite one of the corners of the large plate), we fill the plane of A3 with hypothetical surfaces A4 , A5 , and A6 as indicated in Fig. 4.10. From the definition of view factors, or equation (4.13), it follows that Fd1−(3+4+5+6) = Fd1−3 + Fd1−4 + Fd1−(5+6) , Fd1−4 = Fd1−(4+6) − Fd1−6 . Thus, Fd1−3 = Fd1−(3+4+5+6) − Fd1−(4+6) + Fd1−6 − Fd1−(5+6) . All four of these are of the type discussed in Example 4.4. Therefore, Fd1−3 = F(a+b, c+d, e) − F(a, c+d, e) + F(a, c, e) − F(a+b, c, e). We have successfully converted the present complex view factor to a summation of four known, more basic ones.

View Factors Chapter | 4 141

FIGURE 4.11 Configuration for Example 4.7: (a) full corner piece (b) strips on a corner piece.

Example 4.7. Assuming the view factor for a finite corner, as shown in Fig. 4.11a, is known as F1−2 = f (a, b, c), where f is a known function of the dimensions of the corner pieces (as given in Appendix D), determine the view factor F3−4 , between the two perpendicular strips as shown in Fig. 4.11b. Solution From the definition of the view factor, and since the energy traveling to A4 is the energy going to A2 plus A4 minus the energy going to A2 , it follows that F3−4 = F3−(2+4) − F3−2 , and, using reciprocity, F3−4 =

 1  (A2 + A4 )F(2+4)−3 − A2 F2−3 . A3

F3−4 =

 A2   A2 + A4  F(2+4)−(1+3) − F(2+4)−1 − F2−(1+3) − F2−1 . A3 A3

Similarly, we find

All view factors on the right-hand side are corner pieces and are, thus, known by evaluating the function f with appropriate dimensions. Example 4.8. Again, assuming the view factor is known for the configuration in Fig. 4.11a, determine F1−6 as shown in Fig. 4.12. Solution Examining Fig. 4.12, and employing reciprocity, we find   (A5 + A6 )F(5+6)−(1+2) = (A5 + A6 ) F(5+6)−1 + F(5+6)−2 = A1 (F1−5 + F1−6 ) + A2 (F2−5 + F2−6 )     = A1 F1−(3+5) − F1−3 + A2 F2−(4+6) − F2−4 + A1 F1−6 + A2 F2−5 . On the other hand, we also have   (A5 + A6 ) F(5+6)−(1+2) = (A1 + A2 ) F(1+2)−(3+4+5+6) − F(1+2)−(3+4) . In both expressions all view factors, with the exceptions of F1−6 and F2−5 , are of the type given in Fig. 4.11a. These last two view factors may be related to one another, as is easily seen from their integral forms. From equation (4.13) we have 



A2 F2−5 = A2

A5

cos θ2 cos θ5 dA5 dA2 . πS2

142 Radiative Heat Transfer

FIGURE 4.12 Configuration for Example 4.8.

With a coordinate system as shown in Fig. 4.12, we get from equations (4.21) and (4.23) S2 = x22 + (y2 − y5 )2 + z25 , cos θ2 = z5 /S, cos θ5 = x2 /S, or  A2 F2−5 =

e



x2 =0

b y2 =a



a



y5 =0

d z5 =c

x2 z5 dz5 dy5 dy2 dx2 2 .  π x22 +(y2 − y5 )2 +z25

Similarly, we obtain for F1−6  A1 F1−6 =

e x1 =0



a y1 =0



b y6 =a



d z6 =c

x1 z6 dz6 dy6 dy1 dx1 2 .  π x21 +(y1 − y6 )2 +z26

Switching the names for dummy integration variables, it is obvious that A2 F2−5 = A1 F1−6 , which may be called the law of reciprocity for diagonally opposed pairs of perpendicular rectangular plates. Finally, solving for F1−6 we obtain F1−6 =

 1  A2   A1 + A2  F(1+2)−(3+4+5+6) − F(1+2)−(3+4) − F2−(4+6) − F2−4 . F1−(3+5) − F1−3 − 2A1 2 2A1

Using similar arguments, one may also determine the view factor between two arbitrarily orientated rectangular plates lying in perpendicular planes (Fig. 4.13a) or in parallel planes (Fig. 4.13b). After considerable algebra, one finds [1]: Perpendicular plates (Fig. 4.13a): 2A1 F1−2 = f (x2 , y2 , z3 ) − f (x2 , y1 , z3 ) − f (x1 , y2 , z3 ) + f (x1 , y1 , z3 ) + f (x1 , y2 , z2 ) − f (x1 , y1 , z2 ) − f (x2 , y2 , z2 ) + f (x2 , y1 , z2 ) − f (x2 , y2 , z3 −z1 ) + f (x2 , y1 , z3 −z1 ) + f (x1 , y2 , z3 −z1 ) − f (x1 , y1 , z3 −z1 ) + f (x2 , y2 , z2 −z1 ) − f (x2 , y1 , z2 −z1 ) − f (x1 , y2 , z2 −z1 ) + f (x1 , y1 , z2 −z1 ),

(4.41)

where f (w, h, l) = A1 F1−2 is the product of area and view factor between two perpendicular rectangles with a common edge as given by Configuration 39 in Appendix D. Parallel plates (Fig. 4.13b): 4A1 F1−2 = f (x3 , y3 ) − f (x3 , y2 ) − f (x3 , y3 − y1 ) + f (x3 , y2 − y1 )   − f (x2 , y3 ) − f (x2 , y2 ) − f (x2 , y3 − y1 ) + f (x2 , y2 − y1 )   − f (x3 −x1 , y3 ) − f (x3 −x1 , y2 ) − f (x3 −x1 , y3 − y1 ) + f (x3 −x1 , y2 − y1 ) + f (x2 −x1 , y3 ) − f (x2 −x1 , y2 ) − f (x2 −x1 , y3 − y1 ) + f (x2 −x1 , y2 − y1 ),

(4.42)

View Factors Chapter | 4 143

FIGURE 4.13 View factors between generalized rectangles: (a) surfaces are on perpendicular planes, (b) surfaces are on parallel planes.

where f (a, b) = A1 F1−2 is the product of area and view factor between two directly opposed, parallel rectangles, as given by Configuration 38 in Appendix D. Equations (4.41) and (4.42) are not restricted to x3 > x2 > x1 , and so on, but hold for arbitrary values, for example, they are valid for partially overlapping surfaces. Fortran functions perpplates and parlplates are included in Appendix F for the evaluation of these view factors, based on calls to Fortran function view (i.e., calls to function view to evaluate the various view factors for Configurations 39 and 38, respectively). Example 4.9. Show that equation (4.42) reduces to the correct expression for directly opposing rectangles. Solution For directly opposing rectangles, we have x1 = x3 = a, y1 = y3 = b, and x2 = y2 = 0. We note that the formula for A1 F1−2 for Configuration 38 in Appendix D is such that f (a, b) = f (−a, b) = f (a, −b) = f (−a, −b), i.e., the view factor and area are both “negative” for a single negative dimension, making their product positive, and similarly if both a and b are negative. Also, if either a or b is zero (zero area), then f (a, b) = 0. Thus, 4A1 F1−2 = f (a, b) − 0 − 0 + f (a, −b) − [0 − 0 − 0 + 0] − [0 − 0 − 0 + 0] + f (−a, b) − 0 − 0 + f (−a, −b) =4 f (a, b).

Many other view factors for a multitude of configurations may be obtained through view factor algebra. A few more examples will be given in this and the following chapters (when radiative exchange between black, gray-diffuse, and gray-specular surfaces is discussed).

4.7 The Crossed-Strings Method View factor algebra may be used to determine all view factors in long enclosures with constant cross-section. The method is credited to Hottel [26],∗ and is called the crossed-strings method since the view factors can be determined experimentally by a person armed with four pins, a roll of string, and a yardstick. Consider the configuration in Fig. 4.14, which shows the cross-section of an infinitely long enclosure, continuing into and out of the plane of the figure: We would like to determine F1−2 . Obviously, the surfaces shown are rather irregular (partly convex, partly concave), and the view between them may be obstructed. We shudder at the thought of having to carry out the view factor determination by integration, and plant our four pins at the two ends of each ∗

Hoyte Clark Hottel (1903–1998) American engineer. Obtained his M.S. from the Massachusetts Institute of Technology in 1924, and was on the Chemical Engineering faculty at M.I.T. from 1927 until his death. While Hottel is credited with the method’s discovery, he has stated that he found it in a publication while in the M.I.T. library; but, by the time he first published it, he was unable to rediscover its source. Hottel’s major contributions have been his pioneering work on radiative heat transfer in furnaces, particularly his study of the radiative properties of molecular gases (Chapter 10) and his development of the zonal method (Chapter 17).

144 Radiative Heat Transfer

FIGURE 4.14 The crossed-strings method for arbitrary two-dimensional configurations.

surface, as indicated by the labels a, b, c, and d. We now connect points a and c and b and d with tight strings, making sure that no visual obstruction remains between the two strings. Similarly, we place tight strings ab and cd across the surfaces, and ad and bc diagonally between them, as shown in Fig. 4.14. Now assuming the strings to be imaginary surfaces Aab , Aac , and Abc , we apply the summation rule to the “triangle” abc: Aab Fab−ac + Aab Fab−bc = Aab ,

(4.43a)

Aac Fac−ab + Aac Fac−bc = Aac ,

(4.43b)

Abc Fbc−ac + Abc Fbc−ab = Abc ,

(4.43c)

where Fab−ab = Fac−ac = Fbc−bc = 0 since a tightened string will always form a convex surface. Equations (4.43) are three equations in six unknown view factors, which may be solved by applying reciprocity to three of them: Aab Fab−ac + Aab Fab−bc = Aab ,

(4.44a)

Aab Fab−ac + Aac Fac−bc = Aac ,

(4.44b)

Aac Fac−bc + Aab Fab−bc = Abc .

(4.44c)

Adding the first two equations and subtracting the last leads to the view factor for an arbitrarily shaped triangle with convex surfaces, Fab−ac =

Aab + Aac − Abc , 2Aab

(4.45)

which states that the view factor between two surfaces in an arbitrary “triangle” is equal to the area of the originating surface, plus the area of the receiving surface, minus the area of the third surface, divided by twice the originating surface. Applying equation (4.45) to triangle abd we find immediately Fab−bd =

Aab + Abd − Aad . 2Aab

(4.46)

But, from the summation rule, Fab−ac = Fab−bd + Fab−cd = 1.

(4.47)

View Factors Chapter | 4 145

Thus Aab + Aac − Abc Aab + Abd − Aad − 2Aab 2Aab (Abc + Aad ) − (Aac + Abd ) = . 2Aab

Fab−cd = 1 −

(4.48)

Inspection of Fig. 4.14 shows that all radiation leaving Aab traveling to Acd will hit surface A1 . At the same time all radiation from Aab going to A1 must pass through Acd . Therefore, Fab−cd = Fab−1 . Using reciprocity and repeating the argument for surfaces Aab and A2 , we find Fab−cd = Fab−1 =

A1 A1 F1−ab = F1−2 , Aab Aab

and, finally, F1−2 =

(Abc + Aad ) − (Aac + Abd ) . 2A1

(4.49)

This formula is easily memorized by looking at the configuration between any two surfaces as a generalized “rectangle,” consisting of A1 , A2 , and the two sides Aac and Abd . Then F1−2 =

diagonals − sides . 2 × originating area

(4.50)

Example 4.10. Calculate F1−2 for the configuration shown in Fig. 4.15. Solution From the figure it is obvious that s21 = (c − d cos α)2 + d2 sin2 α = c2 + d2 − 2cd cos α. Similarly, we have s22 = (a + c)2 + (b + d)2 − 2(a + c)(b + d) cos α, d21 = (a + c)2 + d2 − 2(a + c)d cos α, d22 = c2 + (b + d)2 − 2c(b + d) cos α, and F1−2 =

d1 + d2 − (s1 + s2 ) . 2a

For c = d = 0, this reduces to the result of Example 4.2, or F1−2 =

a+b−

√ a2 + b2 − 2ab cos α . 2a

Example 4.11. Find the view factor Fd1−2 of Fig. 4.15 for the case that A1 is an infinitesimal strip of width dx. Use the crossed-strings method. Solution We can obtain the result right away by replacing a by dx in the previous example. Throwing out differentials of second

146 Radiative Heat Transfer

FIGURE 4.15 Infinitely long wedge-shaped groove for Examples 4.10 and 4.11.

and higher order, we find that s1 and d2 remain unchanged, and % (c + dx)2 + d2 − 2(c + dx) d cos α % c2 + d2 − 2cd cos α + 2(c − d cos α) dx   √ (c − d cos α) dx dx 2 2 = s1 + (c−d cos α) c +d −2cd cos α 1+ 2 c + d2 −2cd cos α s1

d1 =

s2 =

% (c + dx)2 + (b + d)2 − 2(c + dx)(b + d) cos α

d2 +

dx [c − (b + d) cos α] . d2

Substituting this into equation (4.50), we obtain s1 + (c−d cos α) dx/s1 + d2 − s1 − d2 − [c−(b+d) cos α] dx/d2 2 dx ⎤ ⎡ c − (b+d) cos α c − d cos α 1 ⎢⎢⎢ ⎥⎥⎥ − % = ⎢⎣ √ ⎥⎦. 2 2 2 2 2 c + (b+d) − 2c(b+d) cos α c + d − 2cd cos α

Fd1−2 =

The same result could also have been obtained by letting Fd1−2 = lim F1−2 , a→0

where F1−2 is the view factor from the previous example. Using de l’Hopital’s rule to determine the value of the resulting expression leads to   1 ∂d1 ∂s2  − Fd1−2 =  , 2 ∂a ∂a a=0 and the above result.

Thus, the crossed-strings method may also be applied to strips. Example 4.1 could also have been solved this way; since the result is infinitesimal this computation would require retaining differentials up to second order. However, integration becomes simpler for strips of differential widths, while application of the crossed-strings method becomes more involved. We shall present one final example to show how view factors for curved surfaces and for configurations with floating obstructions can be determined by the crossed-strings method. Example 4.12. Determine the view factor F1−2 for the configuration shown in Fig. 4.16. Solution In the figure the end points of A1 and A2 (pin points) have been labeled a, b, c, and d, and other strategic points have been labeled with capital letters. A closed-contour surface such as a cylinder may be modeled by placing two pins right next to each other, with surface A2 being a strongly bulging convex surface between the pins. While the location of the two pins on the cylinder is arbitrary, it is usually more convenient to pick a location out of sight of A1 . Since A1 can see A2

View Factors Chapter | 4 147

FIGURE 4.16 Configuration for view factor calculation of Example 4.12; string placement (a) for Fl1−2 , (b) for Fr1−2 .

from both sides of the obstruction, F1−2 cannot be determined with a single set of strings. Using view factor algebra, we can state that F1−2 = Fl1−2 + Fr1−2 , where Fl1−2 and Fr1−2 are the view factors between A1 and A2 when considering only light paths on the left or right of the obstruction, respectively. The placement of strings for Fl1−2 is given in Fig. 4.16a, and for Fr1−2 in Fig. 4.16b. Considering first Fl1−2 , the diagonals and sides may be determined from d1 = aD + DE + Ed, s1 = aC + Cc,

d2 = bA + AB + BC + Cc, s2 = bA + AE + Ed.

Substituting these expressions into equation (4.50) and canceling those terms that appear in a diagonal as well as in a side (Ed, bA, and Cc), we obtain Fl1−2 =

aD + DE + AB + BC − (aC+AE) . 2ab

Looking at Fig. 4.16a we also notice that aC = aD and AB = AE, so that Fl1−2 =

  BC + DE αR + (π−2β−α)R 1 π = = −β . 2ab 2 × 2R 2 2

  But cot β = tan π/2 − β = R/(h + H). Thus,

Fl1−2 =

R 1 tan−1 . 2 h+H

Similarly, we find from Fig. 4.16b for Fr1−2 , d1 = aF + FI + IJ + Jd,

d2 = bG + GH + Hc,

s1 = aF + FH + Hc,

s2 = bJ + Jd,

Fr1−2 =

FI + IJ + bG + GH − (FH+bJ) . 2ab

148 Radiative Heat Transfer

FIGURE 4.17 The inside sphere method.

By inspection bG = bJ and FI = FH, leading to

Fr1−2

    π −δ−γ R + π−2β+δ− π2 −γ R IJ + GH 2 = = 2ab 2 × 2R     1 π 1 R l = −β−γ = tan−1 − tan−1 . 2 2 2 h+H h

Note that this formula only holds as long as GH > 0 (i.e., as long as the cylinder is seen without obstruction from point b). Finally, adding the left and right contributions to the view factor, F1−2 = tan−1

1 R l − tan−1 . h+H 2 h

4.8 The Inside Sphere Method Consider two surfaces A1 and A2 that are both parts of the surface of one and the same sphere, as shown in Fig. 4.17. We note that, for this type of configuration, θ1 = θ2 = θ and S = 2R cos θ. Therefore,  Fd1−2 = A2

cos θ1 cos θ2 dA2 = πS2

 A2

cos2 θ 1 dA2 = π(2R cos θ)2 4πR2

 dA2 = A2

A2 , As

(4.51)

where As = 4πR2 is the surface area of the entire sphere. Similarly, from equation (4.16), F1−2 = Fd1−2 =

A2 , As

(4.52)

since Fd1−2 does not depend on the position of dA1 . Therefore, because of the unique geometry of a sphere, the view factor between two surfaces on the same sphere only depends on the size of the receiving surface, and not on the location of either one.

View Factors Chapter | 4 149

FIGURE 4.18 View factor between coaxial parallel disks.

The inside sphere method is primarily used in conjunction with view factor algebra, to determine the view factor between two surfaces that may not necessarily lie on a sphere. Example 4.13. Find the view factor between two parallel, coaxial disks of radius R1 and R2 using the inside sphere method. Solution Inspecting Fig. 4.18 we see that it is possible to place the parallel disks inside a sphere of radius R in such a way that the entire peripheries of both disks lie on the surface of the sphere. Since all radiation from A1 to A2 travels on to the spherical cap A2 (in the absence of A2 ), and since all radiation from A1 to A2 must pass through A2 , we have F1−2 = F1−2 . Using reciprocity and applying a similar argument for A1 and spherical cap A1 , we find F1−2 = F1−2 =

A2 A2 A1 A2 F2 −1 = F2 −1 = . A1 A1 A1 As

The areas of the spherical caps are readily calculated as 

βi

Ai = 2πR2

sin β dβ = 2πR2 (1 − cos βi ),

i = 1, 2.

0

Thus, with A1 = πR21 and As = 4πR2 , this results in F1−2 =

(2πR2 )2 (1 − cos β1 )(1 − cos β2 ) πR21 4πR2

From Fig. 4.18 one finds (assuming βi ≤ π/2) cos βi = F1−2 =

(

.

R2 − R2i /R, and

( (    1 2 − R2 R − R R2 − R22 . R − 1 2 R1

It remains to find the radius of the sphere R, since only the distance between disks, h, is known. From Fig. 4.18 h=

(

R2 − R21 +

(

R2 − R22 ,

150 Radiative Heat Transfer

FIGURE 4.19 Surface projection for the unit sphere method.

which may be solved (by squaring twice), to give R2 = (X2 − 1)



R1 R2 h

2

,

X=

h2 + R21 + R22 2R1 R2

.

This result is, of course, identical to the one given in Example 4.5, although it is not trivial to show this.

4.9 The Unit Sphere Method The unit sphere method is a powerful tool to calculate view factors between one infinitesimal and one finite area. It is particularly useful for the experimental determination of such view factors, as first stated by Nusselt [27]. An experimental implementation of the method through optical projection has been discussed by Farrell [28]. To determine the view factor Fd1−2 between dA1 and A2 we place a hemisphere2 of radius R on top of A1 , centered over dA1 , as shown in Fig. 4.19. From equations (4.4) and (4.8) we may write   cos θ1 cos θ2 cos θ1 Fd1−2 = dΩ 2 . dA = (4.53) 2 2 π πS A2 Ω2 The solid angle dΩ 2 may also be expressed in terms of area dA2 (dA2 projected onto the hemisphere) as dΩ 2 = = cos θ1 dA2 . dA2 /R2 . Further, the area dA2 may be projected along the z-axis onto the plane of A1 as dA 2 Thus,    dA A cos θ1 dA2 2 2 Fd1−2 = = = , (4.54) 2 2 π R2 πR πR A2 A 2 that is, Fd1−2 is the fraction of the disk πR2 that is occupied by the double projection of A2 . Experimentally this can be measured, for example, by placing an opaque area A2 within a hemisphere, made of a translucent material, and which has a light source at the center (at dA1 ). Looking down onto the translucent hemisphere in the negative z-direction, A2 will appear as a shadow. A photograph of the shadow (and the bright disk) can be taken, showing the double projection of A2 , and Fd1−2 can be measured. Example 4.14. Determine the view factor for Fd1−2 between an infinitesimal area and a parallel disk as shown in Fig. 4.20. 2. The name unit sphere method originated with Nusselt, who used a sphere of unit radius; however, a sphere of arbitrary radius may be used.

View Factors Chapter | 4 151

FIGURE 4.20 Geometry for the view factor in Example 4.14.

Solution √ While a hemisphere of arbitrary radius could be employed, we shall choose here for convenience a radius of R = a2 + d2 , i.e., a hemisphere that includes the periphery of the disk on its surface. Then A2 = A2 = πa2 , and the view factor follows as πa2 a2 = . πR2 a2 + d2

Fd1−2 =

Obviously, only a few configurations will allow such simple calculation of view factors. For a more general case it would be desirable to have some “cookbook formula” for the application of the method. This is readily achieved by looking at the vector representation of the surfaces. Any point on the periphery of A2 may be expressed as a vector ˆ s12 = xî + yˆj + zk.

(4.55)

The corresponding point on A2 may be expressed as s12 = x î + y ˆj + z kˆ = %

R x2 + y2 + z2

s12 ,

(4.56)

as and on A 2     s 12 = x î + y ˆj = x î + y ˆj.

(4.57)

Thus, any point (x, y, z) on A2 is double-projected onto A as 2 x 2 = %

x x2 + y2 + z2

R,

y 2 = %

y x2 + y2 + z2

R.

(4.58)

Only the area formed by the projection of the periphery of A2 through equation (4.58) needs to be found. This integration is generally considerably less involved than the one in equation (4.8).

4.10 View Factor Between Arbitrary Planar Polygons The area integral method or the contour integral method for computing view factors are relatively straightforward to use when the exchanging surfaces are of regular shape. Mesh generators used in modern-day computations, however, generate control volumes (or elements) that are bounded by flat surfaces, which could be triangles, quadrilaterals, or, in general, arbitrary convex polygons. Consequently, general-purpose radiation exchange codes used for practical applications often require calculation of view factors between planar surfaces of arbitrary shape and placed at arbitrary orientation relative to each other. In such a scenario, the limits of integration using either method are difficult to define using Cartesian or cylindrical coordinates, and parametric representation of the surfaces is preferable. Calculation of view factors using parametric representation of surfaces has been demonstrated by Hollands [29] and by Schroder and Hanrahan [30]. Hollands demonstrated the method only for regular shapes and configurations. Schroder and Hanrahan, on the other hand, considered

152 Radiative Heat Transfer

FIGURE 4.21 Schematic representation of two planar polygonal surfaces showing relevant vectors used in the formulation presented in Section 4.10.

the view factor between two arbitrary polygons. Their final result, although presented in closed form, involves computation of the dilogarithm (or Spencer’s) function [31], and ultimately requires a computer program to evaluate. Recently, Narayanaswamy [32] has proposed a formulation for evaluating the view factor between two arbitrary planar triangles using the unit sphere method, in which the triangles are first projected on to a unit sphere prior to area integration. The final closed-form analytical result, once again, requires computation of the dilogarithm function. In this section, a method for calculating the view factor between two planar polygons placed at an arbitrary orientation relative to each other is presented. The method presented herein is based on the contour integral method presented in Section 4.5 and, thus, is limited to polygons without visual obstructions between them. We begin by considering two planar polygonal surfaces, labeled 1 and 2, as shown in Fig. 4.21. Surface 1 is shown to have N vertices, labeled Q1 through QN , while Surface 2 is shown to have M vertices, labeled P1 through PM . For each surface, the vertices are ordered such that the direction in which the thumb points in accordance with the right hand screw rule is the same as the direction of the surface normal. Under this general description, the contour integration formula given by equation (4.34) is directly applicable. To derive the integrands on the right-hand side of equation (4.34), we use parametric representation of the two surfaces. The vector s, shown in Fig. 4.21, may be written as s = −s1 + s2 ,

(4.59)

where, following the convention in Section 4.5, s1 and s2 are position vectors pointing to the tails of differential vectors ds1 and ds2 , respectively. Denoting q1,2 as the vector joining Q1 and Q2 , and p1,2 as the vector joining P1 and P2 , respectively, equation (4.59) may be rewritten as − → −−−→ → − s = −(μ1 q1,2 + Q1 ) + (μ2 p1,2 + P1 ) = −μ1 q1,2 + Q1 P1 + μ2 p1,2 ,

(4.60)

− → → − where Q1 and P1 denote position vectors of vertices Q1 and P1 , respectively. μ1 is the fraction of the vector q1,2 that is equal to s1 and, likewise, μ2 is the fraction of the vector p1,2 that is equal to s2 , i.e., s1 = μ1 q1,2 and s2 = μ2 p1,2 . Since q1,2 and p1,2 are vectors that are not changing, it follows that ds1 = dμ1 q1,2 ; ds2 = dμ2 p1,2 .

(4.61)

From Fig. 4.21, it also follows that 0 ≤ μ1 ≤ 1 and 0 ≤ μ2 ≤ 1 for the tail and tip of the vector s to scan from vertex 1 to 2 for both line segments. Using equation (4.60), the square of the magnitude of the vector s may be

View Factors Chapter | 4 153

written as −−−→ −−−→ S2 = s · s = (−μ1 q1,2 + Q1 P1 + μ2 p1,2 ) · (−μ1 q1,2 + Q1 P1 + μ2 p1,2 ) = μ21 |q1,2 |2 + μ22 |p1,2 |2 − 2μ1 μ2 q1,2 · p1,2 −−−→ −−−→ −−−→ − 2μ1 Q1 P1 · q1,2 + 2μ2 Q1 P1 · p1,2 + |Q1 P1 |2 .

(4.62)

Furthermore, since contours Γ1 and Γ2 are comprised of discrete line segments, equation (4.34) may be rewritten as F1−2 =

 N M  1  ln Sm,n ds1,n · ds2,m . 2πA1 Γ2,m Γ1,n

(4.63)

n=1 m=1

Generalization of equations (4.61) and (4.62) to any pair of vertices on line segments n (on Γ1 ) and m (on Γ2 ), followed by substitution into equation (4.63), yields N M   1  1 1  2 ln μ1 |qn,n+1 |2 + μ22 |pm,m+1 |2 − 2μ1 μ2 qn,n+1 · pm,m+1 4πA1 0 0 n=1 m=1 −−−−→ −−−−→ −−−−→  − 2μ1 Qn Pm · qn,n+1 + 2μ2 Qn Pm · pm,m+1 + |Qn Pm |2 (qn,n+1 · pm,m+1 ) dμ1 dμ2 .

F1−2 =

(4.64)

The coordinates of all vertices of both surfaces must be known to define the surface. Thus, all vectors shown in the integration kernel in equation (4.64) can be easily determined, and represent constants that change only with change in the outer summation indices. Therefore, the integration kernel is a nonlinear function of μ1 and μ2 only. It is possible to integrate one of the integrals in equation (4.64) analytically, which would result in inverse tangent functions. This result may be substituted into the second integral, and the resulting integration can then be performed numerically. Alternatively, both integrations may be performed numerically. One final point to note is that when the last index of the summation is reached (either N or M), the subsequent index (or vertex) is replaced by the first vertex, i.e., n = 1, and m = 1, rather than n = N + 1 or m = M + 1, since the vertices on each of the two surfaces form a closed loop. A special case arises when the two surfaces share a common edge. In such a case, the distance S is zero and, consequently, a singularity arises. In order to address such a singularity, Ambirajan and Venkateshan [33] proposed an analytical treatment, whereby the contribution to the summation in equation (4.64) by the shared edge becomes      = |qshared edge |2 32 − 12 ln |qshared edge |2 , ΔF1−2  (4.65) shared edge

where |qshared edge | is the length of the shared edge. As stated earlier, the computation of the view factor using equation (4.64) requires numerical integration. Therefore, the accuracy of the final answer depends on the numerical quadrature scheme being used. While simple numerical quadrature schemes, such as the Trapezoidal rule or the Simpson’s rule, may be easy to implement, best accuracy and efficiency is attained by using Gaussian quadrature schemes [31]. Ravishankar and Mazumder [34] have investigated six different orientations, and have shown that accuracy up to six decimal places can be attained using a 10-point Gaussian quadrature scheme. A general-purpose Fortran program, vfplanepoly, which computes the view factor between two arbitrary planar polygons with the vertex coordinates as inputs, is provided in Appendix F. Example 4.15. Determine the view factor between the following two triangular surfaces: Surface 1 with vertices at Q1 (0, 0, 0), Q2 (1, 0, 0), and Q3 (1, 1, 0), and Surface 2 with vertices at P1 (1, 0, 1), P2 (0, 1, 1), and P3 (1, 1, 1), as illustrated in Fig. 4.22. Solution The first task is to determine the vectors qn,n+1 and pm,m+1 . Using the vertex coordinates given, we obtain p1,2 = −î + ˆj, p2,3 = î, p3,1 = −ˆj, q1,2 = î, q2,3 = ˆj, and q3,1 = −î − ˆj. Next, we focus on the term qn,n+1 · pm,m+1 in the integration kernel of equation (4.64) since this term multiplies the logarithmic term. Out of the nine terms in this combination, three are zero. The remaining (nonzero) terms are as follows: p1,2 · q1,2 = −1, p1,2 · q2,3 = 1, p2,3 · q1,2 = 1, p2,3 · q3,1 = −1, p3,1 · q2,3 = −1,

154 Radiative Heat Transfer

FIGURE 4.22 Geometry considered in Example 4.15.

−−−→ ˆ the first of these six nonzero terms (i.e., for n = 1 and and p3,1 · q3,1 = 1. Using these values, and noting that Q1 P1 = î + k, m = 1) in equation (4.64) may be written as  1 1   1  ln μ21 |q1,2 |2 + μ22 |p1,2 |2 − 2μ1 μ2 q1,2 · p1,2 F1−2  = 4πA1 0 0 1,1 −−−→ −−−→ −−−→  − 2μ1 Q1 P1 · q1,2 + 2μ2 Q1 P1 · p1,2 + |Q1 P1 |2 (q1,2 · p1,2 ) dμ1 dμ2  1 1   1 = ln μ21 + 2μ22 − 2μ1 − 2μ2 + 2μ1 μ2 + 2 dμ1 dμ2 . 4πA1 0 0 The integration kernel shown above is clearly a nonlinear function of μ1 and μ2 , and the integration may be performed numerically. In vfplanepoly.f90, for example, the coordinates of the vertices of the two triangles are first provided as inputs. The program then performs the integration using 10-point Gaussian quadrature, and the summation over all values of n and m is carried out using two loops. After computing the six integrals and summing the results, we obtain F1−2 = 0.099912.

Problems 4.1 For Configuration 11 in Appendix D, find Fd1−2 by (a) area integration, and (b) contour integration. Compare the effort involved. 4.2 Using the results of Problem 4.1, find F1−2 for Configuration 33 in Appendix D. 4.3 Find F1−2 for Configuration 32 in Appendix D, by area integration. 4.4 Evaluate Fd1−2 for Configuration 13 in Appendix D by (a) area integration, and (b) contour integration. Compare the effort involved. 4.5 Using the result from Problem 4.4, calculate F1−2 for Configuration 40 in Appendix D. 4.6 Find the view factor Fd1−2 for Configuration 11 in Appendix D, with dA1 tilted toward A2 by an angle φ. 4.7 Find Fd1−2 for the surfaces shown in the figure, using (a) area integration, (b) view factor algebra, and Configuration 11 in Appendix D.

View Factors Chapter | 4 155

4.8 For the infinite half-cylinder depicted in the figure, find F1−2 .

4.9 Find Fd1−2 for the surfaces shown in the figure.

4.10 Find Fd1−2 from the infinitesimal area to the disk as shown in the figure, with 0 ≤ β ≤ π.

4.11 Determine for Configuration 51 in Appendix D, using (a) other, more basic view factors given in Appendix D, (b) the crossed-strings rule. 4.12 Find F1−2 for the configuration shown in Fig. 4.22 using the view factor between two parallel square plates (Appendix D) and view factor algebra. Hint: Use symmetry. 4.13 To reduce heat transfer between two infinite concentric cylinders a third cylinder is placed between them as shown in the figure. The center cylinder has an opening of half-angle θ. Calculate F4−2 .

156 Radiative Heat Transfer

4.14 Consider the two long concentric cylinders as shown in the figure. Between the two cylinders is a long, thin flat plate as also indicated. Determine F4−2 .

4.15 Calculate the view factor F1−2 for surfaces on a cone as shown in the figure.

4.16 Determine the view factor F1−2 for the configuration shown in the figure, if (a) the bodies are two-dimensional (i.e., infinitely long perpendicular to the paper); (b) the bodies are axisymmetric (cones).

4.17 Consider the configuration shown; determine the view factor F1−2 assuming the configuration is (a) axisymmetric (1 is conical, 2 is a disk with a hole), or (b) two-dimensional Cartesian (1 is a V-groove, 2 is comprised of two infinitely long strips).

View Factors Chapter | 4 157

4.18 Find F1−2 for the configuration shown in the figure (infinitely long perpendicular to paper).

4.19 Calculate the view factor between two infinitely long cylinders as shown in the figure. If a radiation shield is placed between them to obstruct partially the view (dashed line), how does the view factor change?

4.20 Find the view factor between spherical caps as shown in the figure, for the case of R2 R2 + ( 2 , H≥ ( 1 R21 − a21 R22 − a22 where H = distance between sphere centers, R = sphere radius, and a = radius of cap base. Why is this restriction necessary?

4.21 Find the view factor of the spherical ring shown in the figure to itself, F1−1 , using the inside sphere method.

4.22 Determine the view factor for Configuration 18 in Appendix D, using the unit sphere method. 4.23 Consider the axisymmetric configuration shown in the figure. Calculate the view factor F1−3 .

158 Radiative Heat Transfer

4.24 Consider the configuration shown (this could be a long cylindrical BBQ with a center shelf/hole; or an integrating sphere). Determine the view factors F2−2 and F2−3 assuming the configuration is (a) axisymmetric (sphere), (b) two-dimensional Cartesian (cylinder), using view factor algebra, (c) two-dimensional Cartesian (cylinder), using the string rule (F2−3 only).

4.25 In the solar energy laboratory at UC Merced parabolic concentrators are employed to enhance the absorption of tubular solar collectors as shown in the sketch. Calculate the view factor from the parabolic concentrator A1 to collecting cylinder A2 , using (a) view factor algebra, (b) Hottel’s string rule.

4.26 The interior of a right-circular cylinder of length L = 4R, where R is its radius, is to be broken up into 4 ring elements of equal width. Determine the view factors between all the ring elements, using (a) view factor algebra and the view factors of Configuration 40, (b) Configuration 9 with the assumption that this formula can be used for rings of finite widths. Assess the accuracy of the approximate view factors. What would be the maximum allowable value for ΔX to ensure that all view factors within a distance of 4R are accurate to at least 5%? (Exclude the view factor from a ring to itself, which is best evaluated last, applying the summation rule.) Use the program viewfactors or the function view in your calculations. 4.27 The inside surfaces of a furnace in the shape of a parallelepiped with dimensions 1 m × 2 m × 4 m are to be broken up into 28 1 m × 1 m subareas. Determine all necessary view factors using the functions parlplates and perpplates in Appendix F. 4.28 Using the contour integral method described in Section 4.10, compute the view factor between two parallel square plates of unit length placed directly opposite each other at a distance of 10 units. Make use of the Fortran program, vfplanepoly.f90, provided in Appendix F or write your own program. Verify your answer against the answer obtained using the analytical expression for the same configuration provided in Appendix D. 4.29 Compute the view factor between two planar quadrilaterals whose vertices are as follows: quadrilateral 1 with vertices at Q1 (0, 0.5, 0), Q2 (1, 0, 0), Q3 (1, 1, 0) and Q4 (0, 1, 0), and quadrilateral 2 with vertices at P1 (2, 0.5, 0), P2 (3, 0, 0.5), P3 (3, 2, 0.5) and P4 (2, 1.5, 0). Use the Fortran program, vfplanepoly.f90, provided in Appendix F or write your own program.

References [1] [2] [3] [4] [5] [6] [7]

D.C. Hamilton, W.R. Morgan, Radiant interchange configuration factors, NACA TN 2836, 1952. H. Leuenberger, R.A. Pearson, Compilation of radiant shape factors for cylindrical assemblies, ASME paper no. 56-A-144, 1956. F. Kreith, Radiation Heat Transfer for Spacecraft and Solar Power Design, International Textbook Company, Scranton, PA, 1962. E.M. Sparrow, R.D. Cess, Radiation Heat Transfer, Hemisphere, New York, 1978. R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, 4th ed., Taylor and Francis-Hemisphere, Washington, 2002. J.R. Howell, A Catalog of Radiation Configuration Factors, McGraw-Hill, New York, 1982. J.R. Howell, M.P. Mengüç, Radiative transfer configuration factor catalog: a listing of relations for common geometries, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (2011) 910–912. [8] R.L. Wong, User’s manual for CNVUFAC–the General Dynamics heat transfer radiation view factor program, Technical report, University of California, Lawrence Livermore National Laboratory, 1976.

View Factors Chapter | 4 159

[9] A.B. Shapiro, FACET–a computer view factor computer code for axisymmetric, 2D planar, and 3D geometries with shadowing, Technical report, University of California, Lawrence Livermore National Laboratory, August 1983, maintained by Nuclear Energy Agency under http://www.oecd-nea.org/tools/abstract/detail/nesc9578/. [10] P.J. Burns, MONTE–a two-dimensional radiative exchange factor code, Technical report, Colorado State University, Fort Collins, 1983. [11] A.F. Emery, VIEW–a radiation view factor program with interactive graphics for geometry definition (version 5.5.3), Technical report, NASA Computer Software Management and Information Center, Atlanta, 1986, available from http://www.openchannelfoundation. org/projects/VIEW. [12] T. Ikushima, MCVIEW: a radiation view factor computer program or three-dimensional geometries using Monte Carlo method, Technical report, Japan Atomic Energy Research Institute (JAERI), 1986, maintained by Nuclear Energy Agency under http://www. oecd-nea.org/tools/abstract/detail/nea-1166. [13] C.L. Jensen, TRASYS-II user’s manual–thermal radiation analysis system, Technical report, Martin Marietta Aerospace Corp., Denver, 1987. [14] G.N. Walton, Algorithms for calculating radiation view factors between plane convex polygons with obstructions, in: Fundamentals and Applications of Radiation Heat Transfer, vol. HTD-72, ASME, 1987, pp. 45–52. [15] J.H. Chin, T.D. Panczak, L. Fried, Spacecraft thermal modeling, International Journal for Numerical Methods in Engineering 35 (1992) 641–653. [16] C.N. Zeeb, P.J. Burns, K. Branner, J.S. Dolaghan, User’s manual for Mont3d – Version 2.4, Colorado State University, Fort Collins, CO, 1999. [17] G.N. Walton, Calculation of obstructed view factors by adaptive integration, Technical Report NISTIR–6925, National Institute of Standards and Technology (NIST), Gaithersburg, MD, 2002. [18] J.J. MacFarlane, VISRAD-a 3D view factor code and design tool for high-energy density physics experiments, Journal of Quantitative Spectroscopy and Radiative Transfer 81 (2003) 287–300. [19] A.F. Emery, O. Johansson, M. Lobo, A. Abrous, A comparative study of methods for computing the diffuse radiation viewfactors for complex structures, ASME Journal of Heat Transfer 113 (2) (1991) 413–422. [20] M. Jakob, Heat Transfer, vol. 2, John Wiley & Sons, New York, 1957. [21] H.P. Liu, J.R. Howell, Measurement of radiation exchange factors, ASME Journal of Heat Transfer 109 (2) (1956) 470–477. [22] C.R. Wylie, Advanced Engineering Mathematics, 5th ed., McGraw-Hill, New York, 1982. [23] P. Moon, Scientific Basis of Illuminating Engineering, Dover Publications, New York, 1961, originally published by McGraw-Hill, New York, 1936. [24] R. de Bastos, Computation of radiation configuration factors by contour integration, M.S. thesis, Oklahoma State University, 1961. [25] E.M. Sparrow, A new and simpler formulation for radiative angle factors, ASME Journal of Heat Transfer 85 (1963) 73–81. [26] H.C. Hottel, Radiant heat transmission, in: W.H. McAdams (Ed.), Heat Transmission, 3rd ed., McGraw-Hill, New York, 1954, ch. 4. [27] W. Nusselt, Graphische Bestimming des Winkelverhältnisses bei der Wärmestrahlung, VDI Zeitschrift 72 (1928) 673. [28] R. Farrell, Determination of configuration factors of irregular shape, ASME Journal of Heat Transfer 98 (2) (1976) 311–313. [29] K.G.T. Hollands, Application of parametric surface representation to evaluating form factors and like quantities, in: M.P. Mengüç, N. Selçuk (Eds.), Proceedings of the ICHMT 4th International Symposium on Radiative Transfer, Istanbul, Turkey, 2004. [30] P. Schroder, P. Hanrahan, On the form factor between two polygons, in: M.C. Whitton (Ed.), 20th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’93, ACM, Annaheim, CA, 1993, pp. 163–164. [31] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, 9th ed., Dover, New York, NY, 1970. [32] A. Narayanaswamy, An analytic expression for radiation view factor between two arbitrarily oriented planar polygons, International Journal of Heat and Mass Transfer 91 (2015) 841–847. [33] A. Ambirajan, S.P. Venkateshan, Accurate determination of diffuse view factors between planar surfaces, International Journal of Heat and Mass Transfer 36 (8) (1993) 2203–2208. [34] M. Ravishankar, S. Mazumder, General procedure for calculation of diffuse view factors between arbitrary planar polygons, International Journal of Heat and Mass Transfer 55 (2012) 7330–7335.

Chapter 5

Radiative Exchange Between Gray, Diffuse Surfaces 5.1 Introduction In this chapter we shall begin our analysis of radiative heat transfer rates within enclosures without a participating medium, making use of the view factors developed in the preceding chapter. We shall first deal with the simplest case of a black enclosure, that is, an enclosure where all surfaces are black. Such simple analysis may often be sufficient, for example, for furnace applications with soot-covered walls. This will be followed by expanding the analysis to enclosures with gray, diffuse surfaces, whose radiative properties do not depend on wavelength, and which emit as well as reflect energy diffusely. Considerable experimental evidence demonstrates that most surfaces emit (and, therefore, absorb) diffusely except for grazing angles (θ > 60◦ ), which are unimportant for heat transfer calculations (for example, Fig. 3.1). Most surfaces tend to be fairly rough and, therefore, reflect in a relatively diffuse fashion. Finally, if the surface properties vary little across that part of the spectrum over which the blackbody emissive powers of the surfaces are appreciable, then the simplification of gray properties may be acceptable. In both cases—black enclosures as well as enclosures with gray, diffuse surfaces—we shall first derive the governing integral equation for arbitrary enclosures, which is then reduced to a set of algebraic equations by applying it to idealized enclosures. At the end of the chapter solution methods to the general integral equations are briefly discussed.

5.2 Radiative Exchange Between Black Surfaces Consider a black-walled enclosure of arbitrary geometry and with arbitrary temperature distribution as shown in Fig. 5.1. An energy balance for dA yields, from equation (4.1), q(r) = Eb (r) − H(r),

(5.1)

where H is the irradiation onto dA. From the definition of the view factor, the rate with which energy leaves dA and is intercepted by dA is (Eb (r ) dA ) dFdA −dA . Therefore, the total rate of incoming heat transfer onto dA from the entire enclosure and from outside (for enclosures with some semitransparent surfaces and/or holes) is  H(r) dA = Eb (r ) dFdA −dA dA + Ho (r) dA, (5.2) A

where Ho (r) is the external contribution to the irradiation, i.e., any part not due to emission from the enclosure surface. Using reciprocity, this may be stated as   cos θ cos θ  H(r) = Eb (r ) dFdA−dA + Ho (r) = Eb (r ) (r, r ) dA + Ho (r), (5.3) πS2 A A where θ and θ are angles at the surface elements dA and dA , respectively, and S is the distance between them, as defined in Section 4.2. For an enclosure with known surface temperature distribution, the local heat flux is readily calculated as1 1. When looking at equation (5.4) one is often tempted by intuition to replace dFdA−dA by dFdA −dA . It should always be remembered that we have used reciprocity, since dFdA −dA is per unit area at r , while equation (5.4) is per unit area at r. Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00013-4 Copyright © 2022 Elsevier Inc. All rights reserved.

161

162 Radiative Heat Transfer

FIGURE 5.1 A black enclosure of arbitrary geometry.

 Eb (r ) dFdA−dA − Ho (r).

q(r) = Eb (r) −

(5.4)

A

To simplify the problem it is customary to break up the enclosure into N isothermal subsurfaces, as shown in Fig. 4.2b. Then equation (5.4) becomes qi (ri ) = Ebi −

N 

 Ebj Aj

j=1

dFdAi −dA j − Hoi (ri ),

(5.5)

or, from equation (4.16), qi (ri ) = Ebi −

N 

Ebj Fdi− j (ri ) − Hoi (ri ).

(5.6)

j=1

Even though the temperature may be constant across Ai , the heat flux is usually not since (i) the local view factor Hoi may not be uniform. We may calculate an Fdi− j nearly always varies across Ai and (ii) the external irradiation  average heat flux by averaging equation (5.6) over Ai . With A Fdi− j dAi = Ai Fi−j this leads to i

qi =

1 Ai

 qi (ri ) dAi = Ebi − Ai

N 

Ebj Fi−j − Hoi ,

i = 1, 2, . . . , N,

(5.7)

j=1

where qi and Hoi are now understood to be average values. 5 Employing equation (4.18) we rewrite Ebi as Nj=1 Ebi Fi−j , or qi =

N 

Fi− j (Ebi − Ebj ) − Hoi ,

i = 1, 2, . . . , N.

(5.8)

j=1

In this equation the heat flux is expressed in terms of the net radiative energy exchange between surfaces Ai and A j, Qi− j = qi− j Ai = Ai Fi−j (Ebi − Ebj ) = −Q j−i .

(5.9)

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 163

FIGURE 5.2 Two-dimensional black duct for Example 5.1.

Example 5.1. Consider a very long duct as shown in Fig. 5.2. The duct is 30 cm × 40 cm in cross-section, and all surfaces are black. The top and bottom walls are at temperature T1 = 1000 K, while the side walls are at temperature T2 = 600 K. Determine the net radiative heat transfer rate (per unit duct length) on each surface. Solution We may use either equation (5.7) or (5.8). We shall use the latter here since it takes better advantage of the symmetry of the problem (i.e., it uses the fact that the net radiative exchange between two surfaces at the same temperature must be zero). Thus, with no external irradiation, and using symmetry (e.g., Eb1 = Eb3 , F1−2 = F1−4 , etc.), q1 = F1−2 (Eb1 − Eb2 ) + F1−3 (Eb1 − Eb3 ) + F1−4 (Eb1 − Eb4 ) = 2F1−2 (Eb1 − Eb2 ) = q3 , q2 = q4 = 2F2−1 (Eb2 − Eb1 ). Only the view factors F1−2 and F2−1 are required, which are readily determined from the crossed-strings method as √ 30 + 40 − ( 302 + 402 + 0) 1 = , F1−2 = 2 × 40 4 A1 40 1 1 × = . F1−2 = F2−1 = A2 30 4 3 Therefore (using a prime to indicate “per unit duct length”), Q1 = Q3 = 2A1 F1−2 σ(T14 − T24 ) = 2×0.4 m×0.25×5.670×10−8 Q2 = Q4 = 2A2 F2−1 σ(T24 − T14 ) = −9870 W/m

W (10004 −6004 ) K4 = 9870 W/m m2 K 4

It is apparent from this example that the sum of all surface heat transfer rates must vanish. This follows immediately from conservation of energy: The total heat transfer rate into the enclosure (i.e., the heat transfer rates summed over all surfaces) must be equal to the rate of change of radiative energy within the enclosure. Since radiation travels at the speed of light, steady state is reached almost instantaneously, so that the rate of change of radiative energy may nearly always be neglected. Mathematically, we may multiply equation (5.7) by Ai and sum over all areas: N  i=1

(Qi + Ai Hoi ) =

N  i=1

Ai Ebi −

N  i=1

Ai

N  j=1

Ebj Fi−j =

N 

Ai Ebi −

i=1

N  j=1

A j Ebj

N 

F j−i = 0.

(5.10)

i=1

This relationship is most useful to check the correctness of one’s calculations, or their accuracy (for computer calculations). Example 5.2. Consider two concentric, isothermal, black spheres with radii R1 and R2 , and temperatures T1 and T2 , respectively, as shown in Fig. 5.3. The space between the two spheres is filled with a material of refractive index n > 1. Show how the temperature of the inner sphere can be deduced, if temperature and heat flux of the outer sphere are measured. Solution We have only two surfaces, and equation (5.8) becomes q1 = F1−2 (Eb1 − Eb2 );

q2 = F2−1 (Eb2 − Eb1 ).

164 Radiative Heat Transfer

FIGURE 5.3 Concentric black spheres for Example 5.2.

Since all radiation from Sphere 1 travels to 2, we have F1−2 = 1 and, by reciprocity, F2−1 = A1 /A2 . Thus, Q1 = −Q2 = A1 σ n2 (T14 − T24 ). Solving this for T1 we get, with Ai = 4πR2i , T14 = T24 −



R2 R1

2 q 2 . σ n2

Whenever T1 is larger than T2 , q2 is negative, and vice versa. The above equation also implies that if the space between the two spheres is filled with glass (n ≈ 1.4), for example, T1 will be closer to T2 than if the space is filled with air. The reason is that the emissive power from a surface increases with n2 , i.e., a smaller temperature difference is required to produce the same heat flux. Example 5.3. A right-angled groove, consisting of two long black surfaces of width a, is exposed to solar radiation qsol (Fig. 5.4). The entire groove surface is kept isothermal at temperature T. Determine the net radiative heat transfer rate from the groove. Solution Again, we may employ either equation (5.7) or (5.8). However, this time the enclosure is not closed; and we must close it artificially. We note that any radiation leaving the cavity will not come back (barring any reflection from other surfaces nearby). Thus, our artificial surface should be black. We also assume that, with the exception of the (parallel)

FIGURE 5.4 Right-angled groove exposed to solar irradiation, Example 5.3.

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 165

solar irradiation, no external radiation enters the cavity. Since the solar irradiation is best treated separately through the external irradiation term Ho , our artificial surface is nonemitting. Both criteria are satisfied by covering the groove with a black surface at 0 K. Even though we now have three surfaces, the last one does not really appear in equation (5.7) (since Eb3 = 0), but it does appear in equation (5.8). Using equation (5.7) we find q1 = Eb1 − F1−2 Eb2 − Ho1 = σT 4 (1 − F1−2 ) − qsol cos α, q2 = Eb2 − F2−1 Eb1 − Ho2 = σT 4 (1 − F2−1 ) − qsol sin α. From Configuration 33 in Appendix D we find, with H = 1,  √  F1−2 = 12 2 − 2 = 0.293 = F2−1 , and Q = a(q1 + q2 ) = a

√

 2σT 4 − qsol (cos α + sin α) .

These examples demonstrate that equation (5.8) is generally more convenient to use for closed configurations, since it takes advantage of the fact that the net exchange between two surfaces at the same temperature (or with itself) is zero. Equation (5.7), on the other hand, is more convenient for open configurations, since the hypothetical surfaces employed to close the configuration do not contribute (because of their zero emissive power): With this equation the hypothetical closing surfaces may be completely ignored! Equation (5.7) may be written in a third form that is most convenient for computer calculations. Using Kronecker’s delta function, defined as ⎧ ⎪ ⎪ ⎨1, i = j, (5.11) δi j = ⎪ ⎪ ⎩0, i  j, we find

N  j=1

δi j = 1 and

N 

Ebj δi j = Ebi . Thus,

j=1

qi =

N 

(δi j − Fi−j )Ebj − Hoi ,

i = 1, 2, . . . , N.

(5.12)

j=1

Let us suppose that for surfaces i = 1, 2, . . . , n the heat fluxes are prescribed (and temperatures are unknown), while for surfaces i = n + 1, . . . , N the temperatures are prescribed (heat fluxes unknown). Unlike for the heat fluxes, no explicit relations for the unknown temperatures exist. Placing all unknown temperatures on one side of equation (5.12), we may write n 

(δi j − Fi− j )Ebj = qi + Hoi +

j=1

N 

Fi−j Ebj ,

i = 1, 2, . . . , n,

(5.13)

j=n+1

where everything on the right-hand side of the equation is known. In matrix form this is written2 as A · eb = b, where

⎛ ⎜⎜ 1 − F1−1 ⎜⎜ ⎜⎜ ⎜⎜ −F2−1 ⎜⎜ A = ⎜⎜⎜ .. ⎜⎜ ⎜⎜ . ⎜⎜⎜ ⎝ −Fn−1

(5.14)

−F1−2

···

−F1−n

1 − F2−2

···

−F2−n

..

.. .

−Fn−2

.

···

1 − Fn−n

⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ , ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎠

(5.15)

2. For easy readability of matrix manipulations we shall follow here the convention that a two-dimensional matrix is denoted by a bold capitalized letter, while a vector is written as a bold lowercase letter.

166 Radiative Heat Transfer

⎛ ⎜⎜ Eb1 ⎜⎜ ⎜⎜ ⎜⎜ Eb2 ⎜⎜ eb = ⎜⎜⎜ ⎜⎜ .. ⎜⎜ . ⎜⎜ ⎜⎝ Ebn

⎛ 5 ⎞ ⎜⎜ q1 +Ho1 + Nj=n+1 F1−j Ebj ⎟⎟ ⎜ ⎜ ⎟⎟ ⎜⎜ 5 ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ q2 +Ho2 + Nj=n+1 F2−j Ebj ⎟⎟ ⎜ ⎟⎟ , b = ⎜⎜⎜ ⎟⎟ ⎜⎜ .. ⎟⎟ ⎜⎜ ⎟⎟ . ⎜⎜ ⎟⎟ ⎜⎜ ⎠ ⎜⎝ 5N qn +Hon + j=n+1 Fn−j Ebj

⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ . ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎠

(5.16)

Formally, the n × n matrix A is readily inverted on a computer, and the unknown emissive powers (from which temperatures may be extracted) are calculated as eb = A−1 · b.

(5.17)

In practice, it is generally not necessary to invert the matrix A; rather, the linear system of equations [equation (5.14)] is solved directly.

5.3 Radiative Exchange Between Gray, Diffuse Surfaces (Net Radiation Method) We shall now assume that all surfaces are gray, that they are diffuse emitters, absorbers, and reflectors. Under these conditions = λ = αλ = α = 1 − ρ. The total heat flux leaving a surface at location r is, from Fig. 4.1, J(r) = (r)Eb (r) + ρ(r)H(r),

(5.18)

which is called the surface radiosity J at location r. Since both emission and reflection are diffuse, so is the resulting intensity leaving the surface: I(r, sˆ ) = I(r) = J(r)/π.

(5.19)

Therefore, an observer at a different location is unable to distinguish emitted and reflected radiation on the basis of directional behavior. However, the observer may be able to distinguish the two as a result of their different spectral behavior. Consider Example 5.2 for the case of a black outer sphere but a gray, diffuse inner sphere. On the inner sphere the emitted radiation has the spectral distribution of a blackbody at temperature T1 , while the reflected radiation—which was originally emitted at the outer sphere—has the spectral distribution of a blackbody at temperature T2 . Thus, the spectral radiosity will behave as shown qualitatively in Fig. 5.5. An observer will be able to distinguish between emitted and reflected radiation if he has the ability to distinguish between radiation at different wavelengths. A gray surface does not have this ability, since it behaves in the same fashion toward all incoming radiation at any wavelength, i.e., it is “color blind.” Consequently, a gray surface does not “know” whether its irradiation comes from a gray, diffuse surface or from a black surface with an effective emissive power J. This fact simplifies the analysis considerably since it allows us to calculate radiative heat transfer rates between surfaces by balancing the net outgoing radiation (i.e., emission and reflection) traveling directly from surface to surface (as opposed to emitted radiation traveling to another surface directly or after any number of reflections). For this reason the following analysis is often referred to as the net radiation method.

FIGURE 5.5 Qualitative spectral behavior of radiosity for irradiation from an isothermal source.

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 167

FIGURE 5.6 Radiative exchange in a gray, diffuse enclosure.

Making an energy balance on a surface dA in the enclosure shown in Fig. 5.6 we obtain from equation (4.2) q(r) = (r)Eb (r) − α(r)H(r) = J(r) − H(r).

(5.20)

The irradiation H(r) is again found by determining the contribution from a differential area dA (r ), followed by integrating over the entire surface. From the definition of the view factor the heat transfer rate leaving dA intercepted by dA is (J(r ) dA ) dFdA −dA . Thus, similar to the black-surface case,  J(r ) dFdA −dA dA + Ho (r) dA, (5.21) H(r) dA = A

where Ho (r) is again any external radiation arriving at dA. Using reciprocity this equation reduces to  H(r) = J(r ) dFdA−dA + Ho (r).

(5.22)

A

Substitution into equation (5.20) yields  

q(r) = (r)Eb (r) − α(r)

J(r ) dFdA−dA

 + Ho (r) .

(5.23)

A

Thus, the unknown heat flux (or temperature) could be calculated if the radiosity field had been known. A governing integral equation for radiosity is readily established by solving equation (5.20) for J:   J(r) = (r)Eb (r) + ρ(r) J(r ) dFdA−dA + Ho (r) , (5.24) A

for those surface locations where the temperature is known, or  J(r ) dFdA−dA + Ho (r), J(r) = q(r) +

(5.25)

A

for those parts of the surface where the local heat flux is specified. However, in problems without participating media there is rarely a need to determine radiosity, and it is usually best to eliminate radiosity from equation (5.23). Expressing radiosity in terms of local temperature and heat flux and eliminating irradiation H from

168 Radiative Heat Transfer

equation (5.20) we have q − αq = ( Eb − αH) − α(J − H) = Eb − αJ. Up to this point we have differentiated between emittance and absorptance, to keep the relations as general as possible (i.e., to accommodate nongray surface properties if necessary). We shall now invoke the assumption of gray, diffuse surfaces, or α = . Then

(r) [Eb (r) − J(r)]. 1 − (r)

(5.26)

 1 J(r) = Eb (r) − − 1 q(r).

(r)

(5.27)

q(r) = Solving for radiosity, we get



For a black surface J(r) = Eb (r), since there is no reflective component leaving the surface. While this leads to a possible zero-over-zero division in equation (5.26), this never causes a problem with the net radiation equations developed next. Substituting this into equation (5.23), we obtain an integral equation relating temperature T and heat flux q:     q(r) 1   − − 1 q(r ) dF + H (r) = E (r) − Eb (r ) dFdA−dA . (5.28) o dA−dA b )

(r)

(r A A Note that equation (5.28) reduces to equation (5.4) for a black enclosure. However, for a black enclosure with known temperature field the local heat flux can be determined with a simple integration over emissive power. For a gray enclosure an integral equation must be solved, i.e., an equation where the unknown dependent variable q(r) appears inside an integral. This requirement makes the solution considerably more difficult. As for a black enclosure it is customary to break up a gray enclosure into N subsurfaces, over each of which the radiosity is assumed constant. Then equation (5.23) becomes  qi (ri ) = Ebi (ri ) − Jj Fdi− j (ri ) − Hoi (ri ),

i (ri ) N

i = 1, 2, . . . , N,

(5.29)

j=1

and, taking an average over subsurface Ai ,  qi = Ebi − Jj Fi−j − Hoi ,

i N

i = 1, 2, . . . , N.

(5.30)

j=1

Taking a similar average for equation (5.26) gives qi =

i [Ebi − Ji ] . 1 − i

(5.31)

Solving for J and substituting into equation (5.30) then leads to  N  N  qi  1 − − 1 Fi− j q j + Hoi = Ebi − Fi−j Ebj ,

i

j j=1

i = 1, 2, . . . , N.

(5.32)

j=1

This relation also follows directly from equation (5.28) if both (1/ − 1)q and Eb (the components of J) are assumed 5 constant across the subsurfaces. Recalling the summation rule, Nj=1 Fi−j = 1, we may also write equation (5.32) as an interchange between surfaces,  N  N  qi  1 − − 1 Fi− j q j + Hoi = Fi−j (Ebi − Ebj ),

i

j j=1

j=1

i = 1, 2, . . . , N.

(5.33)

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 169

FIGURE 5.7 Two-dimensional gray, diffuse duct for Example 5.4.

Either one of these equations, of course, reduces to equation (5.8) for a black enclosure. Equation (5.32) is preferred for open configurations, since it allows one to ignore hypothetical closing surfaces; and equation (5.33) is preferred for closed enclosures, because it eliminates transfer between surfaces at the same temperature. Sometimes one wishes to determine the radiosity of a surface, for example, in the field of pyrometry (relating surface temperature to radiative intensity leaving a surface). Depending on which of the two is unknown, elimination of qi or Ebi from equation (5.30) with the help of equation (5.31) leads to ⎞ ⎛ N ⎜⎜ ⎟⎟ ⎟ ⎜⎜ Ji = i Ebi + (1− i ) ⎜⎜ Jj Fi−j + Hoi ⎟⎟⎟ ⎠ ⎝

(5.34a)

j=1

= qi +

N 

Jj Fi−j + Hoi ,

i = 1, 2, . . . , N.

(5.34b)

j=1

These two relations simply repeat the definition of radiosity, the first stating that radiosity consists of emitted and reflected heat fluxes and the second that radiosity, or outgoing heat flux, is equal to net heat flux (with negative qin ) plus the absolute value of qin . Example 5.4. Reconsider Example 5.1 for a gray, diffuse surface material. Top and bottom walls are at T1 = T3 = 1000 K with 1 = 3 = 0.3, while the side walls are at T2 = T4 = 600 K with 2 = 4 = 0.8 as shown in Fig. 5.7. Determine the net radiative heat transfer rates for each surface. Solution Using equation (5.33) for i = 1 and i = 2, and recalling that F1−2 = F1−4 and F2−1 = F2−3 ,     q1 1 1 −2 − 1 F1−2 q2 − − 1 F1−3 q1 = 2F1−2 (Eb1 − Eb2 ),

1

2

1     q2 1 1 −2 − 1 F2−1 q1 − − 1 F2−4 q2 = 2F2−1 (Eb2 − Eb1 ).

2

1

2

i=1: i=2:

We have already evaluated F1−2 = 14 and F2−1 = 13 in Example 5.1. From the summation rule F1−3 = 1 − 2F1−2 = 12 and F2−4 = 1 − 2F2−1 = 13 . Substituting these, as well as emittance values, into the relations reduces them to the simpler form of   )   1 1 1 1 1 − −1 q1 − 2 − 1 q2 = 2 × 14 (Eb1 − Eb2 ), 0.3 0.3 2 0.8 4    )  ' 1 1 1 1 1 − 1 q1 + − −1 q2 = 2 × 13 (Eb2 − Eb1 ), −2 0.3 3 0.8 0.8 3 '

or 13 q1 − 6 14 − q1 + 9

1 1 q2 = (Eb1 − Eb2 ), 8 2 7 2 q2 = − (Eb1 − Eb2 ). 6 3

170 Radiative Heat Transfer

Thus,    1 7 2 1 13 7 14 1 × − × q1 = × − × (Eb1 − Eb2 ), 6 6 9 8 2 6 3 8 3 1 3 σ(T14 − T24 ), q1 = × (Eb1 − Eb2 ) = 7 2 14 

and     1 14 7 13 1 14 2 13 + × q2 = × − × (Eb1 − Eb2 ), − × 8 9 6 6 2 9 3 6 3 2 2 q2 = − × (Eb1 − Eb2 ) = − σ(T14 − T24 ). 7 3 7 Finally, substituting values for temperatures, W (10004 −6004 ) K4 = 4230 W/m, m2 K 4 W Q2 = −0.3 m× 27 ×5.670×10−8 2 4 (10004 −6004 ) K4 = −4230 W/m. m K Q1 = 0.4 m× 143 ×5.670×10−8

Of course, both heat transfer rates must again add up to zero. We observe that these rates are less than half the ones for the black duct. Example 5.5. Determine the radiative heat flux between two isothermal gray concentric spheres with radii R1 and R2 , temperatures T1 and T2 , and emittances 1 and 2 , respectively, as shown in Fig. 5.8a. Solution Again applying equation (5.33) for i = 1 (inner sphere) and i = 2 (outer sphere), we obtain: i=1: i=2:

   q1  1 1 − − 1 F1−1 q1 − − 1 F1−2 q2 = F1−2 (Eb1 − Eb2 ),

1

1

2    q2  1 1 − − 1 F2−1 q1 − − 1 F2−2 q2 = F2−1 (Eb2 − Eb1 ).

2

1

2

With F1−1 = 0, F1−2 = 1, F2−1 = A1 /A2 , and F2−2 = 1 − F2−1 = 1 − A1 /A2 , these two equations reduce to   1 1 q1 − − 1 q2 = σ(T14 − T24 ),

1

2 )  '    A1 1 1 A1 A1 1 q2 = − σ(T14 − T24 ). −1 q1 + − −1 1−

1 A2

2

2 A2 A2

FIGURE 5.8 Radiative transfer between (a) two concentric spheres, (b) a convex surface and a large isothermal enclosure.

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 171

This may be solved for q1 by eliminating q2 (or using conservation of energy, i.e., A1 q1 + A2 q2 = 0), or q1 =

σ(T14 − T24 )  . 1 A1 1 + −1

1 A2 2

(5.35)

We note that equation (5.35) is not just limited to concentric spheres, but holds for any convex surface A1 (i.e., with F1−1 = 0) that radiates only to A2 (i.e., F1−2 = 1) as indicated in Fig. 5.8b. This is often convenient for a convex surface Ai placed into a large, isothermal environment (Aa  Ai ) at temperature Ta , leading to qi = i σ(Ti4 − Ta4 ).

(5.36)

Surface Ai may also be a hypothetical one, closing an open configuration contained within a large environment. Example 5.6. Consider a two-dimensional enclosure, as shown in Fig. 5.9. The vertical surface, A1 , is adiabatic, while the other two surfaces have fixed temperatures: T2 = 300 K and T3 = 1000 K. All surfaces may be assumed to be gray and diffuse, with an emittance of 0.2. Determine the temperature, T1 , of the adiabatic surface. Solution We begin by applying equation (5.32) to the two perpendicular surfaces: i=1: i=2:

   q1  1 1 − − 1 F1−2 q2 − − 1 F1−3 q3 = Eb1 − F1−2 Eb2 − F1−3 Eb3 ,

1

2

3    q2  1 1 − − 1 F2−1 q1 − − 1 F2−3 q3 = Eb2 − F2−1 Eb1 − F2−3 Eb3 .

2

1

3

Note that since q1 = 0 (adiabatic surface), all terms containing 1 vanish from the above exchange equations regardless of the value of 1 , implying that the temperature of any adiabatic surface is independent of its own emittance: in the absence of other modes of heat transfer an adiabatic wall must re-emit any radiation it absorbs. And, since the surface is gray as well as diffusely reflecting (and only for such a surface), there is no distinction between reflection and emission (cf. also the definition of radiosity J). Further, using = 2 = 3 = 0.2, the above two equations may be simplified and rearranged to write i=1:

Eb1 + 4F1−2 q2 + 4F1−3 q3 = F1−2 Eb2 + F1−3 Eb3 ,

i=2:

F2−1 Eb1 + 5q2 − 4F2−3 q3 = Eb2 − F2−3 Eb3 .

The above two equations have three unknowns, namely Eb1 , q2 , and q3 . The third equation may be derived by either applying equation (5.32) to surface 3, or by applying global energy balance to the enclosure. Using the latter approach, we get A1 q1 + A2 q2 + A3 q3 = 0,

FIGURE 5.9 Triangular enclosure for Example 5.6.

172 Radiative Heat Transfer

which, upon using the appropriate area values and q1 = 0, simplifies to q2 +

√

2 q3 = 0.

From Configuration 33 in Appendix D we find, with H = 1,  √  F1−2 = 12 2 − 2 = 0.2929 = F2−1 . √

Thus, F1−3 = 1 −F1−2 = 0.7071 = F2−3 . Also, using reciprocity, F3−1 = (A1 /A3 ) F1−3 = (1/ 2) ×0.7071 = 0.5 = F3−2 . Substituting √ q2 = − 2 q3 , along with the calculated view factors into the above exchange equations, we get i=1:

Eb1 + 1.1715 q3 = 0.2929 Eb2 + 0.7071 Eb3 ,

i=2:

0.2929 Eb1 − 9.8995 q3 = Eb2 − 0.7071 Eb3 .

Multiplying the first equation by 9.8995 and the second equation by 1.1715, and adding the two resulting equations, we get 10.2426 Eb1 = 4.0711 Eb2 + 6.1715Eb3 ,

or

Eb1 = 0.3975 Eb2 + 0.6025 Eb3 .

The above equation shows the relative contributions of Eb2 (= σT24 ) and Eb3 (= σT34 ) to Eb1 , and implies that the temperature, T1 , is likely to be much closer to T3 than T2 . Substituting the given temperature values, we obtain Eb1 = 34346.141 W/m2 , which finally yields T1 = [Eb1 /σ]1/4 ≈ 882 K. Example 5.7. Consider the cavity shown in Fig. 5.10, which consists of a cylindrical hole of diameter D and length L. The top of the cavity is covered with a disk, which has a hole of diameter d. The entire inside of the cavity is isothermal at temperature T, and is covered with a gray, diffuse material of emittance . Determine the amount of radiation escaping from the cavity. Solution For simplicity, since the entire surface is isothermal and has the same emittance, we use a single zone A1 , which comprises the entire groove surface (sides, bottom, and top). Therefore, equation (5.32) reduces to '

)   1 1 − − 1 F1−1 q1 = (1 − F1−1 )Eb1 .

1

1

Since the total radiative energy rate leaving the cavity is Q1 = A1 q1 , we get Q1 =

1 − F1−1 A1 Eb1 .   1 1 − − 1 F1−1

1

1

FIGURE 5.10 Cylindrical cavity with partial cover plate, Example 5.7.

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 173

The view factor F1−1 is easily determined by recognizing that Fo−1 = 1 (and Ao is the opening at the top) and, by reciprocity, F1−1 = 1 − F1−o = 1 −

Ao Ao Fo−1 = 1 − . A1 A1

Therefore, the radiative heat flux leaving the cavity, per unit area of opening, is   A o A1 Eb1 1−1+ Q1 Eb1 A1 Ao = .  =     1 1 Ao 1 Ao Ao − −1 1− 1+ −1

1

1 A1

1 A1 Thus, if Ao /A1 1, the opening of the cavity behaves like a blackbody with emissive power Eb1 . Such cavities are commonly used in experimental methods in which blackbodies are needed for comparison. For example, a cavity with d/D = 1/2 and L/D = 2 has d2 πd2 /4 Ao = = 2 2 2 A1 2πD /4−πd /4+πDL 2D −d2 +4DL (d/D)2 1/4 1 = = = . 2−(d/D)2 +4(L/D) 2−1/4+4×2 39 For 1 = 0.5 this results in an apparent emittance of

a =

Q1 = Ao Eb1

39 1 1 = 0.975. = =    1 1 1 Ao 40 −1 1+ 1+ −1 0.5 39

1 A1 

For computer calculations the Kronecker delta is introduced into equation (5.32), as was done for a black enclosure, leading to N   δi j j=1

  N    1 − − 1 Fi−j q j = δi j − Fi−j Ebj − Hoi .

j

j 

(5.37)

j=1

If all the temperatures are known and the radiative heat fluxes are to be determined, equation (5.37) may be cast in matrix form as C · q = A · eb − ho ,

(5.38)

where C and A are matrices with elements Ci j =

δi j

j

 −

 1 − 1 Fi−j ,

j

Ai j = δi j − Fi−j , and q, eb , and ho are vectors of the unknown heat fluxes q j and the known emissive powers Ebj and external irradiations Ho j . The solution to equation (5.38) may then be formally stated as q = C−1 · [A · eb − ho ] .

(5.39)

If the emissive power is known over only some of the surfaces, and the heat fluxes are specified elsewhere, equation (5.38) may be rearranged into a similar equation for the vector containing all the unknowns. Subroutine graydiff is provided in Appendix F for the solution of the simultaneous equations (5.38), requiring surface information and a partial view factor matrix as input. The solution to a three-dimensional version of Example 5.4 is also given in the form of a program graydiffxch, which may be used as a starting point for the solution to other problems. Fortran90, C++ as well as Matlab versions are provided. Several commercial solvers are also available, usually including software for view factor evaluation, such as TRASYS [1] and TSS [2].

174 Radiative Heat Transfer

5.4 Electrical Network Analogy While equation (5.37) represents the most convenient set of governing equations for numerical calculations on today’s digital computers, a more physical interpretation of the radiative exchange problem can be given by representing it through an analogous electrical network, a method more suitable for analog computers— now nearly extinct. For completeness, we shall briefly present this electrical network method, which was first introduced by Oppenheim [3]. From equation (5.20) we have qi = Ji − Hi ,

i = 1, 2, . . . , N,

(5.40)

or, with equations (5.30) and (5.31), qi = Ji −

N 

Jj Fi− j − Hoi =

j=1

N 

(Ji − Jj )Fi−j − Hoi,

i = 1, 2, . . . , N.

(5.41)

j=1

We shall first consider the simple case of two infinite parallel plates without external irradiation. Thus, N = 2, Hoi = 0, and Q1 = A1 q1 =

J1 − J2 = −Q2 . 1 A1 F1−2

(5.42)

As written, equation (5.42) may be interpreted as follows: If the radiosities are considered potentials, 1/A1 F1−2 is a radiative resistance between surfaces, or a space resistance, and Q is a radiative heat flow “current,” then equation (5.42) is identical to the one governing an electrical current flowing across a resistor due to a voltage potential, as indicated in Fig. 5.11a. The space resistance is a measure of how easily a radiative heat flux flows from one surface to another: The larger F1−2 , the more easily heat can travel from A1 to A2 , resulting in a smaller resistance. The same heat flux is also given by equation (5.31) as Q1 =

Eb1 − J1 J2 − Eb2 = = −Q2 , 1 − 1 1 − 2 A 1 1 A 2 2

(5.43)

where (1 − i )/Ai i are radiative surface resistances. This situation is shown in Fig. 5.11b. The surface resistance describes a surface’s ability to radiate. For the maximum radiator, a black surface, the resistance is zero. This fact implies that, for a finite heat flux, the potential drop across a zero resistance must be zero, i.e., Ji = Ebi . Of course, the radiosities may be eliminated from equations (5.42) and (5.43), and Q1 =

Eb1 − Eb2 = −Q2 , 1 − 1 1 1 − 2 + + A 1 1 A1 F1−2 A 2 2

(5.44)

FIGURE 5.11 Electrical network analogy for infinite parallel plates: (a) space resistance, (b) surface resistance, and (c) total resistance.

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 175

FIGURE 5.12 Network representation for radiative heat flux between surface Ai and all other surfaces.

where the denominator is the total radiative resistance between surfaces A1 and A2 . Since the three resistances are in series they simply add up as electrical resistances do; see Fig. 5.11c. This network analogy is readily extended to more complicated situations by rewriting equation (5.41) as Qi =

N N  Ebi − Ji  Ji − Jj = − Ai Hoi = Qi−j − Ai Hoi . 1 − i 1 j=1 j=1 A i i Ai Fi−j

(5.45)

Thus, the total heat flux at surface i is the net radiative exchange between Ai and all the other surfaces in the enclosure. The electrical analog is shown in Fig. 5.12, where the current flowing from Ebi to Ji is divided into N parallel lines, each with a different potential difference and with different resistors. Example 5.8. Consider a solar collector shown in Fig. 5.13a. The collector consists of a glass cover plate, a collector plate, and side walls. We shall assume that the glass is totally transparent to solar irradiation, which penetrates through the glass and hits the absorber plate with a strength of 1000 W/m2 . The absorber plate is black and is kept at a constant temperature T1 = 77◦ C by heating water flowing underneath it. The side walls are insulated and made of a material with emittance

2 = 0.5. The glass cover may be considered opaque to thermal (i.e., infrared) radiation with an emittance 3 = 0.9. The collector is 1 m × 1 m × 10 cm in dimension and is reasonably evacuated to suppress free convection between absorber plate and glass cover. The convective heat transfer coefficient at the top of the glass cover is known to be h = 5.0 W/m2 K, and the temperature of the ambient is Ta = 17◦ C. Estimate the collected energy for normal solar incidence. Solution We may construct an equivalent network (Fig. 5.13b), leading to Q1 =

σ(T14 − Ta4 ) − A1 qs , 1 − 3 R13 + + R3a A3 3

where R13 is the total resistance between surfaces A1 and A3 , and R3a is the resistance, by radiation as well as free convection, between glass cover and environment. We note that, since A2 is insulated, there is no heat flux entering/leaving at Eb2 and, from equation (5.43), J2 = Eb2 . Thus, the total resistance between A1 and A3 comes from two parallel circuits, one with resistance 1/(A1 F1−3 ) and the other with two resistances in series, 1/(A1 F1−2 ) and 1/(A3 F3−2 ), or 1 1 1 + = R13 1/(A1 F1−3 ) 1/(A1 F1−2 ) + 1/(A3 F3−2 )   = A1 F1−3 + 12 A1 F1−2 = A1 F1−3 + 12 F1−2 , where we have used the fact that A1 F1−2 = A3 F3−2 by symmetry. From Configuration 38 in Appendix D we obtain, with X = Y = 10, F1−3 = 0.827 and F1−2 = 1 − F1−3 = 0.173, and +  R13 = 1 1 m2 × (0.827 + 0.5 × 0.173) = 1.095 m−2 .

176 Radiative Heat Transfer

FIGURE 5.13 Schematics for Example 5.8: (a) geometry, (b) network.

The resistance between glass cover and ambient is a little more complicated. The total heat loss from the cover plate, by free convection and radiation, is Q3a = 3 A3 σ(T34 − Ta4 ) + hA3 (T3 − Ta ), where we have assumed that the environment (sky) radiates to the collector with the ambient temperature Ta . To convert this to the correct form we rewrite it as   h(T3 − Ta ) , Q3a = σ(T34 − Ta4 )A3 3 + σ(T34 − Ta4 )

or

    1 1 h T3 − Ta h = A3 3 + . = A3 3 + R3a σ T34 − Ta4 σ T33 + T32 Ta + T3 Ta2 + Ta3

As a first approximation, if T3 is not too different from Ta ,     1 h 5 W/m2 K 1 2 m2. A3 3 + = 1 m 0.9+ = R3a 4σTa3 4×5.670×10−8 W/m2 K4 ×(273+17)3 K3 0.554 Finally, substituting the resistances into the expression for Q1 we get Q1 =

  5.670×10−8 W/m2 K4 (273+77)4 −(273+17)4 K4 1−0.9 1.095 m + +0.554 m−2 0.9 m2 −2

− 1 m2 × 1000 W/m2

= −744 W. Since the system could collect a theoretical maximum of −1000 W, the collector efficiency is ηcollector =

Q1 744 = 0.744 = 74.4%. = A1 qs 1000

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 177

This efficiency should be compared with an uncovered black collector plate, whose net heat flux would be   Q1 = A1 σ(T14 − Ta4 ) + h(T1 − Ta ) − qs   = 1 m2 5.670×10−8 ×(3504 −2904 )+5×(350−290) − 1000 W/m2 = −250 W. Thus, an unprotected collector at that temperature would have an efficiency of only 25%.

The electrical network analogy is a very simple and physically appealing approach for simple two- and three-surface enclosures, such as the one of the previous example. However, in more complicated enclosures with multiple surfaces the method quickly becomes tedious and intractable.

5.5 Radiation Shields In high-performance insulating materials it is common to suppress conductive and convective heat transfer by evacuating the space between two surfaces. This leaves thermal radiation as the dominant heat loss mode even for low-temperature applications such as insulation in cryogenic storage tanks. The radiation loss may be minimized by placing a multitude of closely spaced, parallel, highly reflective radiation shields between the surfaces. The radiation shields are generally made of thin metallic foils or, to reduce conductive losses further, of dielectric foils coated with metallic films. In either case radiation shields tend to be very specular reflectors. However, for closely spaced shields the directional behavior of the reflectance tends to be irrelevant and assuming diffuse reflectances gives excellent accuracy (see also discussion of shields in Section 6.3 of the following chapter). A typical arrangement for N radiation shields between two concentric cylinders (or concentric spheres) is shown in Fig. 5.14. This geometry includes the case of parallel plates for large (and nearly equal) radii. Let the inner cylinder have temperature Ti , surface area Ai , and emittance i . Similarly, each shield has temperature Tn (unknown), An , ni (on its inner surface), and no (on its outer surface). The last shield, AN , faces the outer cylinder with To , Ao , and o . The net radiative heat rate leaving Ai is, of course, equal to the heat rate going through each shield and to the one arriving at Ao . This net heat rate may be readily determined from the electrical network analogy, or by repeated application of the enclosure relations, equation (5.32). However, this is the type of problem for which the network analogy truly shines and we will use this method here. The case of concentric surfaces was already evaluated in Example 5.5, so that the net heat rate between any two of the concentric cylinders is then Q=

Ebj − Ebk R j−k

,

R j−k =

  1 1 1 + −1 .

j A j A k k

FIGURE 5.14 Concentric cylinders (or spheres) with N radiation shields between them.

(5.46)

178 Radiative Heat Transfer

Therefore, we may write QRi−1i = Ebi − Eb1 , QR1o−2i = Eb1 − Eb2 , .. . QRNo−o = EbN − Ebo . Adding all these equations eliminates all the unknown shield temperatures, and, after solving for the heat flux, we obtain Q=

Ebi − Ebo . 5N−1 Ri−1i + n=1 Rno−n+1,i + RNo−o

(5.47)

Example 5.9. A Dewar holding 4 liters of liquid helium at 4.2 K consists essentially of two concentric stainless steel ( = 0.3) cylinders of 50 cm length, and inner and outer diameters of Di = 10 cm and Do = 20 cm, respectively. The space between the cylinders is evacuated to a high vacuum to eliminate conductive/convective heat losses. Radiation shields are to be placed between the Dewar walls to reduce radiative losses to the point that it takes 24 hours for the 4-liter filling to evaporate if the Dewar is placed into an environment at 298 K. For the purpose of this example the following may be assumed: (i) end losses as well as conduction/convection losses are negligible, (ii) the wall temperatures are at Ti = 4.2 K and To = 298 K, respectively, and (iii) radiation is one-dimensional. Thin plastic sheets coated on both sides with aluminum ( = 0.05) are available as shield material. Estimate the number of shields required. The heat of evaporation for helium at atmospheric pressure is hfg,He = 20.94 J/g (which is a very low value compared with other liquids), and the liquid density is ρHe = 0.125 g/cm3 [4]. Solution The total heat required to evaporate 4 liters of liquid helium is Q = ρHe VHe hfg,He = 0.125

g J 103 cm3 × 20.94 = 10.47 kJ. × 4 liters × 3 cm liter g

If all of this energy is supplied through radial radiation over a time period of 24 hours, one infers that the heat flux ˙ i = in equation (5.47) must be held at or below Q˙ = Q/24 h = 10,470 J/24 h × (1 h/3600 s) = 0.1212 W, or qi = Q/A 2 −5 0.1212 W/(π × 10 cm × 50 cm) = 7.71 × 10 W/cm . Therefore, the total resistance must, from equation (5.47), be a minimum of Ai Rtot = |Ebi − Ebo |/qi = 5.670 × 10−12 × |4.24 − 2984 |/7.71 × 10−5 = 580.0. We note from equation (5.46) that the resistances are inversely proportional to shield area. Therefore, it is best to place the shields as close to the inner cylinder as possible. We will assume that the shields can be so closely spaced that Ai A2 . . . AN = As = πDs L, with Ds = 11 cm. Evaluating the total resistance from equations (5.46) and (5.47), we find Ai Rtot =

N−1       1 1 Ai  2 Ai 1 Ai 1 Ai + −1 + −1 + + −1 ,

w

s As n=1 s As s As

w Ao

where w = 0.3 is the emittance of the (stainless steel) walls and s = 0.05 is the emittance of the (aluminized) shields. Since the elements of the series in the last equation do not depend on n, we may solve for N as   1 1 Ai   1 1 Ai Rtot − − −1 − 0.3 − 1 10 580.0 − 0.3

w

w Ao 20   N= =   2 10 2 Ai − 1 11 0.05 −1

s As = 16.23.

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 179

Therefore, a minimum of 17 radiation shields would be required. Note from equation (5.35) that, without radiation shields, qi =

|Ebi − Ebo | 5.670 × 10−12 |4.24 − 2984 | =     1 1 1 1 1 Ai + −1 × + −1 0.3 0.3 2

w

w Ao

= 9.94 × 10−3 W/cm2 , that is, the heat loss is approximately 100 times larger!

5.6 Solution Methods for the Governing Integral Equations The usefulness of the method described in the previous sections is limited by the fact that it requires the radiosity to be constant over each subsurface. This is rarely the case if the subsurfaces of the enclosure are relatively large (as compared with typical distances between surfaces). Today, with the advent of powerful digital computers, more accurate solutions are usually obtained by increasing the number of subsurfaces, N, in equation (5.37), which then become simply a finite-difference solution to the integral equation (5.28). Still, there are times when more accurate methods for the solution of equation (5.28) are desired (for computational efficiency), or when exact or approximate solutions are sought in explicit form. Therefore, we shall give here a very brief outline of such solution methods. If radiosity J is to be determined, the governing equation that needs to be solved is either equation (5.24), if the surface temperature is given, or equation (5.25), if surface heat flux is specified. If unknown temperatures or heat fluxes are to be determined directly, equation (5.28) must be solved. In all cases the governing equation may be written as a Fredholm integral equation of the second kind,  K(r, r ) φ(r ) dA , (5.48) φ(r) = f (r) + A

where K(r, r ) is called the kernel of the integral equation, f (r) is a known function, and φ(r) is the function to be determined (e.g., radiosity or heat flux). Comprehensive discussions for the treatment of such integral equations are given in mathematical texts such as Courant and Hilbert [5] or Hildebrand [6]. A number of radiative heat transfer examples have been discussed by Özi¸sik [7]. Numerical solutions to equation (5.48) may be found in a number of ways. In the method of successive approximation a first guess of φ(r) = f (r) is made with which the integral in equation (5.48) is evaluated (analytically in some simple situations, but more often through numerical quadrature). This leads to an improved value for φ(r), which is substituted back into the integral, and so on. This scheme is known to converge for all surface radiation problems. Another possible solution method is reduction to algebraic equations by using numerical quadrature for the integral, i.e., replacing it by a series of quadrature coefficients and nodal values. This leads to a set of equations similar to equation (5.37), but of higher accuracy. This type of solution method is most easily extended to arbitrary, three-dimensional geometries, for example, as recently demonstrated by Daun and Hollands [8], who employed nonuniform rational B-splines (NURBS) to express the surfaces. A third method of solution has been given by Sparrow and Haji-Sheikh [9], who demonstrated that the method of variational calculus may be applied to general problems governed by a Fredholm integral equation. Most early numerical solutions in the literature dealt with two very basic systems. The problem of twodimensional parallel plates of finite width was studied in some detail by Sparrow and coworkers [9–11], using the variational method. The majority of studies have concentrated on radiation from cylindrical holes because of the importance of this geometry for cylindrical tube flow, as well as for the preparation of a blackbody for calibrating radiative property measurements. The problem of an infinitely long isothermal hole radiating from its opening was first studied by Buckley [12] and by Eckert [13]. Buckley’s work appears to be the first employing the kernel approximation method. Much later, the same problem was solved exactly through the method of successive approximation (with numerical quadrature) by Sparrow and Albers [14]. A finite hole, but with both ends open, was studied by a number of investigators. Usiskin and Siegel [15] considered the constant wall heat flux case, using the kernel approximation as well as a variational approach. The constant wall temperature case was studied by Lin and Sparrow [16], and combined convection/surface radiation was investigated by Perlmutter and Siegel [17,18]. Of greater importance for the manufacture of a blackbody is the isothermal cylindrical cavity

180 Radiative Heat Transfer

TABLE 5.1 Apparent emittance, a = J/σT 4 , at the bottom center of an isothermal, partially covered cylindrical cavity [21,22]. a 

Ri /R

(L/R = 2)

(L/R = 4)

(L/R = 8)

0.25

0.4

0.916

0.968

0.990

0.6

0.829

0.931

0.981

0.8

0.732

0.888

0.969

0.50

0.75

1.0

0.640

0.844

0.965

0.4

0.968

0.990

0.998

0.6

0.932

0.979

0.995

0.8

0.887

0.964

0.992

1.0

0.839

0.946

0.989

0.4

0.988

0.997

0.999

0.6

0.975

0.997

0.998

0.8

0.958

0.988

0.997

1.0

0.939

0.982

0.996

of finite depth, which was studied by Sparrow and coworkers [19,20] using successive approximations. If part of the opening is covered by a flat ring with a smaller hole, such a cavity behaves like a blackbody for very small L/R ratios. This problem was studied by Alfano [21] and Alfano and Sarno [22]. Because of their importance for the manufacture of blackbody cavities these results are summarized in Table 5.1. A detector removed from the cavity will sense a signal proportional to the intensity leaving the bottom center of the cavity in the normal direction. Thus the effectiveness of the blackbody is measured by how close to unity the ratio In /Ib (T) is. For perfectly diffuse reflectors, In = J/π, and with Ib = σT 4 /π an apparent emittance is defined as

a = In /Ib (T) = J/σT 4 .

(5.49)

To give an outline of how the different methods may be applied we shall, over the following few pages, solve the same simple example by three different methods, the first two being “exact,” and the third being the kernel approximation. Example 5.10. Consider two long parallel plates of width w as shown in Fig. 5.15. Both plates are isothermal at the (same) temperature T, and both have a gray, diffuse emittance of . The plates are separated by a distance h and are placed in a large, cold environment. Determine the local radiative heat fluxes along the plate using the method of successive approximation. Solution From equation (5.24) we find, with dFdi−di = 0, 

w

J1 (x1 ) = σT 4 + (1 − )

J2 (x2 ) dFd1−d2 , 

0 w

J2 (x2 ) = σT 4 + (1 − )

J1 (x1 ) dFd2−d1 , 0

% and, from Configuration 1 in Appendix D, with s12 = h/cos φ, s12 dφ = dx2 cos φ, and cos φ = h/ h2 + (x2 − x1 )2 , dx1 dFd1−d2 = dx2 dFd2−d1 =

cos3 φ h2 dx1 dx2 1 1 cos φ dφ dx1 = dx1 dx2 = . 2 2h 2 [h2 + (x1 − x2 )2 ]3/2

Introducing nondimensional variables W = w/h, ξ = x/h, and J(x) = J(x)/σT 4 , and realizing that, as a result of symmetry, J1 = J2 (and q1 = q2 ), we may simplify the governing integral equation to 1 2



W

J(ξ) = + (1 − ) 0

J(ξ )

dξ . [1 + (ξ − ξ)2 ]3/2

(5.50)

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 181

FIGURE 5.15 Radiative exchange between two long isothermal parallel plates.

Making a first guess of J

(1)

= we obtain a second guess by substitution, /

1 2



J (2) (ξ) = 1 + (1 − )

W

0

dξ [1 + (ξ − ξ)2 ]3/2

0

⎧ ⎡ ⎤⎫ ⎪ ⎪ ⎪ ⎢⎢ ⎥⎥⎪ W−ξ 1 ξ ⎨ ⎢ ⎥⎥⎬ 1 + (1 − ) ⎢⎣ % = ⎪ + √ ⎦⎪ ⎪. ⎪ 2 ⎩ 2 2 1+ξ ⎭ 1 + (W − ξ) Repeating the procedure we get

J

(3)

⎧ ⎡ ⎤ ⎪ ⎪ ⎢⎢ ⎥⎥ 1 ξ ⎨ ⎢⎢ % W − ξ ⎥⎥ 1 + (1 −

) (ξ) = ⎪ + √ ⎣ ⎪ 2⎦ ⎩ 2 2 1 + ξ 1 + (W − ) ⎫ ⎤  W⎡ ⎪    ⎪ ⎢⎢ ⎥⎥ W − ξ dξ 1 ξ ⎬ 2 ⎢⎢ % ⎥⎥ , + (1 − ) + √ ⎪ ⎣ ⎦ ⎪ 3/2  2 [1 + (ξ − ξ)2 ] ⎭  2 4 1 + ξ 0 1 + (W − ξ )

where the last integral becomes quite involved. We shall stop at this point since further successive integrations would have to be carried out numerically. It is clear from the above expression that the terms in the series diminish as [(1 − )W]n , i.e., few successive iterations are necessary for surfaces with low reflectances and/or w/h ratios. Once the radiosity has been (2) determined the local heat flux follows from equation (5.26). Limiting ourselves to J (single successive approximation), this yields Ψ(ξ) =

⎡ ⎤   ⎥⎥ q(ξ) W−ξ

2 ⎢⎢⎢ ξ ⎥⎥ − O 2 (1 − )W 2 , [1 − = J (ξ)] +

− + ⎢ % √ ⎣ ⎦ 4 1−

2 σT 1 + ξ2 1 + (W − ξ)2

where O(z) is shorthand for “order of magnitude z.” Some results are shown in Fig. 5.16 and compared with other solution methods for the case of W = w/h = 1 and three values of the emittance. Observe that the heat loss is a minimum at the center of the plate, since this location receives maximum irradiation from the other plate (i.e., the view factor from this location to the opposing plate is maximum). For decreasing the heat loss increases, of course, since more is emitted; however, this increase is less than linear since also more energy is coming in, of which a larger fraction is absorbed. The first successive approximation does very well for small and large as expected from the order of magnitude of the neglected terms. Example 5.11. Repeat Example 5.10 using numerical quadrature. Solution The governing equation is, of course, again equation (5.50). We shall approximate the integral on the right-hand side by a series obtained through numerical integration, or quadrature. In this method an integral is approximated by a weighted series of the integrand evaluated at a number of nodal points; or 

b a

f (ξ, ξ ) dξ (b − a)

J  j=1

c j f (ξ, ξ j ),

J  j=1

c j = 1.

(5.51)

182 Radiative Heat Transfer

FIGURE 5.16 Local radiative heat flux on long, isothermal parallel plates, determined by various methods.

Here the ξ j represent J locations between a and b, and the c j are weight coefficients. The nodal points ξ j may be equally spaced for easy presentation of results (Newton–Cotes quadrature), or their location may be optimized for increased accuracy (Gaussian quadrature); for a detailed treatment of quadrature see, for example, the book by Fröberg [23]. Using equation (5.51) in equation (5.50) we obtain

Ji = + (1 − )W

J 

c j Jj fij ,

i = 1, 2, . . . , J,

j=1

where fij =

3/2 1+ 1 + (ξ j − ξi )2 . 2

This system of equations may be further simplified by utilizing the symmetry of the problem, i.e., J(ξ) = J(W − ξ). Assuming that nodes are placed symmetrically about the centerline, ξ J+1−j = ξ j , leads to c J+1−j = c j and JJ+1−j = Jj , or J odd:

⎫ ⎧(J−1)/2 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ , Ji = + (1 − )W ⎪ c j Jj [ fij + fi,J+1−j ] + c(J+1)/2 J(J+1)/2 fi,(J+1)/2 ⎪ ⎪ ⎪ ⎭ ⎩ j=1

J even:

Ji = + (1 − )W

J/2  j=1

c j Jj ( fij + fi,J+1−j ),

i = 1, 2, . . . ,

J+1 , 2

J i = 1, 2, . . . , . 2

The values of the radiosities may be determined by successive approximation or by direct matrix inversion. In Fig. 5.16 the simple case of J = 5 (resulting in three simultaneous equations) is included, using Newton–Cotes quadrature with ξ j = W( j − 1)/4 and c1 = c5 = 7/90, c2 = c4 = 32/90, and c3 = 12/90 [23].

Exact analytical solutions that yield explicit relations for the unknown radiosity are rare and limited to a few special geometries. However, approximate analytical solutions may be found for many geometries through the kernel approximation method. In this method, the kernel K(x, x ) is approximated by a linear series of special  functions such as e−ax , cos ax , cosh ax , and so on (i.e., functions that, after one or two differentiations with respect to x , turn back into the original function except for a constant factor). It is then often possible to convert integral equation (5.48) into a differential equation that may be solved explicitly. The method is best illustrated through an example. Example 5.12. Repeat Example 5.11 using the kernel approximation method. Solution We again need to solve equation (5.50), this time by approximating the kernel. For convenience we shall choose a simple

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 183

exponential form, 1

K(ξ, ξ ) =



a e−b|ξ −ξ| .

[1 + (ξ − ξ)2 ]3/2

We shall determine “optimum” parameters a and b by letting the approximation satisfy the 0th and 1st moments. This implies multiplying the expression by |ξ − ξ| raised to the 0th and 1st powers, followed by integration over the entire domain for |ξ − ξ|, i.e., from 0 to ∞ (since W could be arbitrarily large).3 Thus, 

∞

dx =1= (1 + x2 )3/2

0th moment: 

0

∞

x dx =1= (1 + x2 )3/2

1st moment: 0

 

∞

a e−bx dx =

a , b

a e−bx x dx =

a , b2

0 x 0

leading to a = b = 1 and 

K(ξ, ξ ) e−|ξ −ξ| . Substituting this expression into equation (5.50) leads to 1 J(ξ) + (1 − ) 2



ξ

J(ξ ) e 

−(ξ−ξ )

 

dξ +

0

W

J(ξ ) e 

ξ

−(ξ −ξ)

 

dξ .

We shall now differentiate this expression twice with respect to ξ, for which we need to employ Leibniz’s rule, equation (3.106). Therefore,    ξ  W dJ 1  −(ξ−ξ )   −(ξ −ξ)  = (1 − ) J(ξ) − J(ξ ) e dξ − J(ξ) + J(ξ ) e dξ , dξ 2 0 ξ    ξ  W d2 J 1  −(ξ−ξ )   −(ξ −ξ)  = (1 − ) −J(ξ) + J(ξ ) e dξ − J(ξ) + J(ξ ) e dξ , dξ2 2 0 ξ or, by comparison with the expression for J(ξ), d2 J = J − − (1 − ) J = ( J − 1). dξ2 Thus, the governing integral equation has been converted into a second-order ordinary differential equation, which is readily solved as

J(ξ) = 1 + C1 e−

√

ξ

+ C2 e+

√

ξ

.

While an integral equation does not require any boundary conditions, we have converted the governing equation into a differential equation that requires two boundary conditions in order to determine C1 and C2 . The dilemma is overcome by substituting the general solution back into the governing integral equation (with approximated kernel). This calculation can be done for variable values of ξ by comparing coefficients of independent functions of ξ, or simply for two arbitrarily selected values for ξ. The first method gives the engineer proof that his analysis is without mistake, but is usually considerably more tedious. Often it is also possible to employ symmetry, as is the case here, since J(ξ) = J(W − ξ) or  √   √   √  √ √ √ √ C1 e− ξ − e− (W−ξ) = −C2 e ξ − e (W−ξ) = C2 e W e− ξ − e− (W−ξ) , or C1 = C2 e

√

W

.

Consequently, 

J(ξ) = 1 + C1 e−

√

ξ

+ e−

√

(W−ξ)



,

3. Using the actual W at hand will result in a better approximation, but new values for a and b must be determined if W is changed; in addition, the mathematics become considerably more involved.

184 Radiative Heat Transfer

and substituting this expression into the governing equation at ξ = 0 gives   √ J(0) = 1 + C1 1+ e− W  W6  √  7  √ 1  = + (1− ) 1+C1 e− ξ + e− (W−ξ ) e−ξ dξ 2 0  W6  7 √  1   √  = + (1− ) e−ξ + C1 e−(1+ )ξ + e−ξ − (W−ξ ) dξ 2 0  √  −(1+√ )ξ /  0W e e−ξ− (W−ξ )  1 −ξ = − (1− ) e +C1 √ + √  2 1+

1−

0 √ √  / 0 1− e−(1+ )W e− W − e−W 1 + . = + (1− ) 1−e−W +C1 √ √ 2 1+

1−

Solving this for C1 gives   √    √  √    √ √ 1− − 12 (1− )(1−e−W ) = C1 12 1− 1−e−(1+ )W + 12 1+ e− W −e−W − 1−e− W 6  √   √  √  √  7   √  √ √   1 (1− ) 1+e−W = C1 12 1− + 12 1+ e− W −1−e− W − 12 1− e− W + 12 1+ e−W , 2 or C1 = −

(1 +

√

1−

√ √

) + (1 − ) e− W

and √

√

e− ξ + e− (W−ξ) J(ξ) = 1 − (1 − ) . √ √ √ (1 + ) + (1 − ) e− W Finally, the nondimensional heat flux follows as   √ √

e− ξ + e− (W−ξ)

Ψ(ξ) = [1 − J(ξ)] = , √ √ √ 1−

(1 + ) + (1 − ) e− W which is also included in Fig. 5.16.  Note that e−|ξ −ξ| is not a particularly good approximation for the kernel, since the actual kernel has a zero first  derivative at ξ = ξ. A better approximation can be obtained by using 



K(ξ, ξ ) a1 e−b1 |ξ −ξ| + a2 e−b2 |ξ −ξ| (with a1 > 1 and a2 < 0). If W is relatively small, say < 12 , a good approximation may be obtained using K(ξ, ξ ) cos a(ξ − ξ) (since the kernel has an inflection point at |ξ − ξ| = 12 ).

We shall conclude this chapter with two examples that demonstrate that exact analytical solutions are possible for a few simple geometries for which the view factors between area elements attain certain special forms. Example 5.13. Consider a hemispherical cavity irradiated by the sun as shown in Fig. 5.17. The surface of the cavity is kept isothermal at temperature T and is coated with a gray, diffuse material with emittance . Assuming that the cavity is, aside from the solar irradiation, exposed to cold surroundings, determine the local heat flux rates that are necessary to maintain the cavity surface at constant temperature. Solution From equation (5.24) the local radiosity at position (ϕ, ψ) is determined as J(ϕ) = σT 4 + (1 − )H(ϕ)   = σT 4 + (1 − ) J(ϕ ) dFdA−dA + Ho (ϕ) , A

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 185

FIGURE 5.17 Isothermal hemispherical cavity irradiated normally by the sun, Example 5.13.

where we have already stated that radiosity is a function of ϕ only, i.e., there is no dependence on azimuthal angle ψ. The view factor between infinitesimal areas on a sphere is known from the inside sphere method, equation (4.33), as R2 sin ϕ dϕ dψ dA = . 4πR2 4πR2

dFdA−dA =

The external irradiation at dA is readily determined as Ho (ϕ) = qsun cos ϕ, and the expression for radiosity becomes 

2π



π/2

J(ϕ) = σT 4 + (1 − ) 1−

= σT + 2

0 π/2



4

J(ϕ )

0

sin ϕ dϕ dψ + qsun cos ϕ 4π



J(ϕ ) sin ϕ dϕ + (1 − )qsun cos ϕ.

0

Because of the unique behavior of view factors between sphere surface elements we note that the irradiation at location ϕ that arrives from other parts of the sphere, Hs , does not depend on ϕ. Thus, Hs =

1 2



π/2

J(ϕ ) sin ϕ dϕ = const,

0

and J(ϕ) = σT 4 + (1 − )Hs + (1 − )qsun cos ϕ. Substituting this equation into the expression for Hs leads to Hs =

1 2



π/2





σT 4 + (1 − )Hs + (1 − )qsun cos ϕ sin ϕ dϕ

0

= 12 σT 4 + 12 (1 − )Hs + 14 (1 − )qsun , or Hs =

1−

σT 4 + qsun . 1+

2(1 + )

An energy balance at dA gives q(ϕ) = σT 4 − H(ϕ) = (σT 4 − Hs − qsun cos ϕ) or

 q(ϕ) =

   1−

σT 4 − + cos ϕ qsun . 1+

2(1 + )

We observe from this example that in problems where all radiating surfaces are part of a sphere, none of the view factors involved depend on the location of the originating surface, and an exact analytical solution can

186 Radiative Heat Transfer

FIGURE 5.18 Thin radiating wire with radiating sheath, Example 5.14.

always be found in a similar fashion. Apparently, this was first recognized by Jensen [24] and reported in the book by Jakob [25]. Exact analytical solutions are also possible for such configurations where all relevant view factors have repeating derivatives (as in the kernel approximation). Example 5.14. A long thin radiating wire is to be employed as an infrared light source. To maximize the output of infrared energy into the desired direction, the wire is fitted with an insulated, highly reflective sheath as shown in Fig. 5.18. The sheath is cylindrical with radius R (which is much larger than the diameter of the wire), and has a cutout of half-angle ϕ to let the concentrated infrared light escape. Assuming that the wire is heated with a power of Q W/m length of wire, and that the sheath can lose heat only by radiation and only from its inside surface, determine the temperature distribution across the sheath. Solution From an energy balance on a surface element dA it follows from equation (5.20) that, with q(θ) = 0, σT 4 (θ) = J(θ) = H(θ), and

 J(θ ) dFdA−dA + Ho (θ).

H(θ) = A

We may treat the energy emitted from the wire as external radiation (neglecting absorption by the wire since it is so small). Since the total released energy will spread equally into all directions, we find Ho (θ) = Q /2πR = const. The view factor dFdA−dA between two infinitely long strips on the cylinder surface is given by Configuration 1 in Appendix D as FdA−dA =

1 2

cos β dβ,

where the angle β is indicated in Fig. 5.18 and may be related to θ through 2β + |θ − θ| = π. Differentiating β with respect to θ we obtain dβ = ±dθ /2, depending on whether θ is larger or less than θ. Substituting for β in the view factor, this becomes       π  θ − θ  1  1 1  θ − θ      − dθ = sin  FdA−dA = cos dθ , 2 2  2  2 4 2 

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 187

where the ± has been omitted since the view factor is always positive (i.e., |dβ| is to be used). Substituting this into the above relationship for radiosity we obtain     1 π−ϕ  θ − θ   J(θ ) sin  J(θ) = dθ + Ho 4 −π+ϕ 2    1 θ θ − θ  1 π−ϕ θ − θ  dθ + dθ + Ho . = J(θ ) sin J(θ ) sin 4 −π+ϕ 2 4 θ 2 Since the view factor in the integrand has repetitive derivatives we may convert this integral equation into a second-order differential equation, as was done in the kernel approximation method. Differentiating twice, we have dJ 1 = dθ 8



θ −π+ϕ

J(θ ) cos

d2 J 1 1 = J(θ) − 2 dθ 8 16



θ −π+ϕ

θ − θ  1 dθ − 2 8 J(θ ) sin



π−ϕ θ

J(θ ) cos

θ − θ  dθ , 2

1 θ − θ  1 dθ + J(θ) − 2 8 16



π−ϕ θ

J(θ ) sin

θ − θ  dθ . 2

Comparing this result with the above integral equation for J(θ) we find d2 J = 14 J(θ) − 14 [J(θ) − Ho ] = dθ2

1 4

Ho .

This equation is readily solved as J(θ) =

1 8

Ho θ2 + C1 θ + C2 .

The two integration constants must now be determined by substituting the solution back into the governing integral equation. However, C1 may be determined from symmetry since, for this problem, J(θ) = J(−θ) and C1 = 0. To determine C2 we evaluate J at θ = 0:     1 0 θ 1 π−ϕ θ  J(0) = C2 = dθ + dθ + Ho J(θ ) sin − J(θ ) sin 4 −π+ϕ 2 4 0 2  1 π−ϕ θ  dθ + Ho = J(θ ) sin 2 0 2  π−ϕ   θ  1 Ho  2 θ sin dθ + Ho . C2 + = 2 0 8 2 Integrating twice by parts we obtain     θ π−ϕ Ho π−ϕ  Ho  2 θ  θ cos  dθ + θ cos C2 = Ho − C2 + 8 2 0 4 0 2    π−ϕ   )  ' π ϕ Ho Ho  θ π−ϕ θ  2 (π − ϕ) cos − + C2 + θ sin  dθ = Ho − C2 + − sin 8 2 2 2 2 0 2 0     ) '  ϕ Ho θ π−ϕ Ho π ϕ (π − ϕ)2 sin + − + 2 cos  = Ho + C2 − C2 + (π − ϕ) sin 8 2 2 2 2 2 0 ) ' ϕ ϕ ϕ H Ho o (π − ϕ)2 sin + (π − ϕ) cos + Ho sin − Ho . = Ho + C2 − C2 + 8 2 2 2 2 Solving this equation for C2 we get ' ) ϕ 1 π−ϕ cos − (π − ϕ)2 . C2 = Ho 1 + 2 2 8 Therefore, T 4 (θ) =

8 9 π−ϕ ϕ 1 Q J = 1+ cos − (π − ϕ)2 − θ2 . σ 2πRσ 2 2 8

We find that the temperature has a minimum at θ = 0, since around that location the view factor to the opening is maximum, resulting in a maximum of escaping energy. The temperature level increases as ϕ decreases (since less energy can escape) and reaches T → ∞ as ϕ = 0 (since this produces an insulated closed enclosure with internal heat production).

188 Radiative Heat Transfer

The fact that long cylindrical surfaces lend themselves to exact analysis was apparently first recognized by Sparrow [26]. The preceding two examples have shown that exact solutions may be found for a number of special geometries, namely, (i) enclosures whose surfaces all lie on a single sphere and (ii) enclosures for which view factors between surface elements have repetitive derivatives. For other still fairly simple geometries an approximate analytical solution may be determined from the kernel approximation method. However, the vast majority of radiative heat transfer problems in enclosures without a participating medium must be solved by numerical methods. A large majority of these are solved using the net radiation method described in the first few sections of this chapter. If greater accuracy or better numerical efficiency is desired, one of the numerical methods briefly described in this section needs to be used, such as numerical quadrature leading to a set of linear algebraic equations (as in the net radiation method).

Problems 5.1 A firefighter (approximated by a two-sided black surface at 310 K 180 cm long and 40 cm wide) is facing a large fire at a distance of 10 m (approximated by a semi-infinite black surface at 1500 K). Ground and sky are at 0◦ C (and may also be approximated as black). What are the net radiative heat fluxes on the front and back of the firefighter? Compare these with heat rates by free convection (h = 10 W/m2 K, Tamb = 0◦ C). 5.2 A small furnace consists of a cylindrical, black-walled enclosure, 20 cm long and with a diameter of 10 cm. The bottom surface is electrically heated to 1500 K, while the cylindrical sidewall is insulated. The top plate is exposed to the environment, such that its temperature is 500 K. Estimate the heating requirements for the bottom wall, and the temperature of the cylindrical sidewall, by treating the sidewall as (a) a single zone and (b) two equal rings of 10 cm height each.

5.3 Repeat Problem 5.2 for a 20 cm high furnace of quadratic (10 cm × 10 cm) cross-section. 5.4 A small star has a radius of 100,000 km. Suppose that the star is originally at a uniform temperature of 1,000,000 K before it “dies,” i.e., before nuclear fusion stops supplying heat. If it is assumed that the star has a constant heat capacity of ρcp = 1 kJ/m3 K, and that it remains isothermal during cool-down, estimate the time required until the star has cooled to 10,000 K. Note: A body of such proportions radiates like a blackbody (Why?). 5.5 A collimated light beam of q0 = 10 W/cm2 originating from a blackbody source at 1250 K is aimed at a small target A1 = 1 cm2 as shown. The target is coated with a diffusely reflecting material, whose emittance is ⎧ ⎪ ⎪ ⎨0.9 cos θ, λ < 4 μm, 

λ = ⎪ ⎪ ⎩0.2, λ > 4 μm. Light reflected from A1 travels on to a detector A2 = 1 cm2 , coated with the same material as A1 . How much of the collimated energy q0 is absorbed by detector A2 ? 5.6 Repeat Problem 5.2 for the case that the top surface of the furnace is coated with a gray, diffuse material with emittance

3 = 0.5 (other surfaces remain black). 5.7 Repeat Problem 5.6, breaking up the sidewall into four equal ring elements. Use the view factors calculated in Problem 4.26 together with program graydiffxch of Appendix F. 5.8 Repeat Example 5.3 for a groove whose surface is gray and diffuse, with emittance , rather than black.

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 189

5.9 A long half-cylindrical rod is enclosed by a long diffuse, gray isothermal cylinder as shown. Both rod and cylinder may be considered isothermal (T1 = T2 , 1 = 2 , T3 , 3 ) and gray, diffuse reflectors. Give an expression for the heat lost from the rod (per unit length).

5.10 Consider a 90◦ pipe elbow as shown in the figure (pipe diameter = D = 1 m; inner elbow radius = 0, outer elbow radius = D). The elbow is isothermal at temperature T = 1000 K, has a gray diffuse emittance = 0.4, and is placed in a cool environment. What is the total heat loss from the isothermal elbow (inside and outside)?

5.11 For the configuration shown in the figure, determine the temperature of Surface 2 with the following data: Surface 1 :

T1 = 1000 K, q1 = −1 W/cm2 ,

1 = 0.6;

Surface 2 : 2 = 0.2; Surface 3 : 3 = 0.3, perfectly insulated. All configurations are gray and diffuse.

5.12 Two pipes carrying hot combustion gases are enclosed in a cylindrical duct as shown. Assuming both pipes to be isothermal at 2000 K and diffusely emitting and reflecting ( = 0.5), and the duct wall to be isothermal at 500 K and diffusely emitting and reflecting ( = 0.2), determine the radiative heat loss from the pipes.

5.13 A cubical enclosure has gray, diffuse walls which interchange energy. Four of the walls are isothermal at Ts with emittance s , the other two are isothermal at Tt with emittance t . Calculate the heat flux rates per unit time and area. 5.14 Consider the enclosure described in Example 5.6. The adiabatic surface is now replaced by a surface that is externally cooled by convection with a heat transfer coefficient, ho = 100 W/m2 K, and an ambient temperature, To = 300 K. You may neglect external radiation. Determine the temperature of this surface. Hint: resulting nonlinear equations may have to be solved numerically.

190 Radiative Heat Transfer

5.15 During launch the heat rejector radiative panels of the Space Shuttle are folded against the inside of the Shuttle doors. During orbit the doors are opened and the panels are rotated out by an angle ϕ as shown in the figure. Assuming door and panel can be approximated by infinitely long, isothermal quarter-cylinders of radius a and emittance = 0.8, calculate the necessary rotation angle ϕ so that half the total energy emitted by panel (2) and door (1) escapes through the opening. At what opening angle will a maximum amount of energy be rejected? How much and why?

5.16 Consider two 1 × 1 m2 , thin, gray, diffuse plates located a distance h = 1 m apart. The temperature of the top plate is maintained at T1 = 1200 K, whereas the bottom plate is initially at T2 = 300 K and insulated on the outside. In case 1, the surface of the top plate is flat, whereas in case 2 grooves, whose dimensions are indicated below, have been machined in the plate’s surface. In either case the surfaces are gray and diffuse, and the surroundings may be considered as black and having a temperature T∞ = 500 K; convective heat transfer effects may be neglected. (a) Estimate the effect of the surface preparation of the top surface on the initial temperature change of the bottom plate (dT2 /dt at t = 0). (b) Justify, then use, a lumped-capacity analysis for the bottom plate to predict the history of temperature and heating rates of the bottom plate until steady state is reached. The following properties are known: top plate: 1 = 0.6, T1 = 1200 K; bottom plate: T2 (t = 0) = 300 K, 2 = 0.5, ρ2 = 800 kg/m3 (density), cp2 = 440 J/kg K, k2 = 200 W/m K. 5.17 A row of equally spaced, cylindrical heating elements (s = 2d) is used to heat the inside of a furnace as shown. Assuming that the outer wall is made of firebrick with 3 = 0.3 and is perfectly insulated, that the heating rods are made of silicon carbide ( 1 = 0.8), and that the inner wall has an emittance of 2 = 0.6, what must the operating temperature of the rods be to supply a net heat flux of 300 kW/m2 to the furnace, if the inner wall is at a temperature of 1300 K?

5.18 A thermocouple used to measure the temperature of cold, low-pressure helium flowing through a long duct shows a temperature reading of 10 K. To minimize heat losses from the duct to the surroundings the duct is made of two concentric thin layers of stainless steel with an evacuated space in between (inner diameter di = 2 cm, outer diameter do = 2.5 cm; stainless layers very thin and of high conductivity). The emittance of the thermocouple is TC = 0.6, the convection heat transfer coefficient between helium and tube wall is hi = 5 W/m2 K, and between thermocouple and helium is hTC = 2 W/m2 K, and the emittance of the stainless steel is ss = 0.2 (gray and diffuse, all four surfaces). The free convection heat transfer coefficient between the outer tube and the surroundings at Tamb = 300 K is ho = 5 W/m2 K. To determine the actual temperature of the helium, (a) Prepare an energy balance for the thermocouple. (b) Prepare an energy balance for the heat loss through the duct wall (the only unknowns here should be THe , Ti , and To ). (c) Outline how to solve for the temperature of the helium (no need to carry out solution). (d) Do you expect the thermocouple to be accurate? (Hint: Check the magnitudes of the terms in (a).)

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 191

5.19 During a materials processing experiment on the Space Shuttle (under microgravity conditions), a platinum sphere of 3 mm diameter is levitated in a large, cold black vacuum chamber. A spherical aluminum shield (with a circular cutout) is placed around the sphere as shown, to reduce heat loss from the sphere. Initially, the sphere is at 200 K and is suddenly irradiated with a laser providing an irradiation of 100 W (normal to beam) to raise its temperature rapidly to its melting point (2741 K). Determine the time required to reach the melting point. You may assume the platinum and aluminum to be gray and diffuse ( Pt = 0.25, Al = 0.1), the sphere to be essentially isothermal at all times, and the shield to have zero heat capacity.

5.20 Two identical circular disks are connected at one point of their periphery by a hinge. The configuration is then opened by an angle φ as shown in the figure. Assuming the opening angle to be φ = 60◦ , d = 1 m, calculate the average equilibrium temperature for each of the two disks, with solar radiation entering the configuration parallel to Disk 2 with a strength of qsun = 1000 W/m2 . Disk 1 is gray and diffuse with α = = 0.5, Disk 2 is black. Both disks are insulated.

5.21 A long greenhouse has the cross-section of an equilateral triangle as shown. The side exposed to the sun consists of a thin sheet of glass (A1 ) with reflectivity ρ1 = 0.1. The glass may be assumed perfectly transparent to solar radiation, and totally opaque to radiation emitted inside the greenhouse. The other side wall (A2 ) is opaque with emittance 2 = 0.2, while the floor (A3 ) has 3 = 0.8. All surfaces reflect diffusely. For simplicity, you may assume surfaces A1 and A2 to be perfectly insulated, while the floor loses heat to the ground according to q3,conduction = U(T3 − T∞ ) where T∞ = 280 K is the temperature of the ground and U = 19.5 W/m2 K is an overall heat transfer coefficient. Determine the temperatures of all three surfaces for the case that the sun shines onto the greenhouse with strength qsun = 1000 W/m2 in a direction parallel to surface A2 . 5.22 A long, black V-groove is irradiated by the sun as shown. Assuming the groove to be perfectly insulated, and radiation to be the only mode of heat transfer, determine the average groove temperature as a function of solar incidence angle θ (give values for θ = 0◦ , 15◦ , 30◦ , 60◦ , 90◦ ). For simplicity the V-groove wall may be taken as a single zone.

5.23 Consider the conical cavity shown (radius of opening R, opening angle γ = 30◦ ), which has a gray, diffusely reflective coating ( = 0.6) and is perfectly insulated. The cavity is irradiated by a collimated beam of strength H0 and radius Rb = 0.5R. (a) Using a single node analysis, develop an expression relating H0 to the average cavity temperature T.

192 Radiative Heat Transfer

(b) For a more accurate analysis a two-node analysis is to be performed. What nodes would you choose? Develop expressions for the necessary view factors in terms of known ones (including those given in App. D) and surface areas, then relate the two temperatures to H0 . (c) Qualitatively, what happens to the cavity’s overall average temperature, if the beam is turned away by an angle α? 5.24 A (simplified) radiation heat flux meter consists of a conical cavity coated with a gray, diffuse material, as shown in the figure. To measure the radiative heat flux, the cavity is perfectly insulated. (a) Develop an expression that relates the flux, Ho , to the cavity temperature, T. (b) If the cavity is turned away from the incoming flux by an angle α, what happens to the cavity temperature? 5.25 A very long solar collector plate is to collect energy at a temperature of T1 = 350 K. To improve its performance for off-normal solar incidence, a highly reflective surface is placed next to the collector as shown in the adjacent figure. How much energy (per unit length) does the collector plate collect for a solar incidence angle of 30◦ ? For simplicity you may make the following assumptions: The collector is isothermal and gray-diffuse with emittance 1 = 0.8; the reflector is gray-diffuse with 2 = 0.1, and heat losses from the reflector by convection as well as all losses from the collector ends may be neglected.

5.26 A thermocouple (approximated by a 1 mm diameter sphere with gray-diffuse emittance 1 = 0.5) is suspended inside a tube through which a hot, nonparticipating gas at T g = 2000 K is flowing. In the vicinity of the thermocouple the tube temperature is known to be T2 = 1000 K (wall emittance 2 = 0.5). For the purpose of this problem you may assume both ends of the tube to be closed with a black surface at the temperature of the gas, T3 = 2000 K. Again, for the purpose of this problem, you may assume that the thermocouple gains a heat flux of 104 W/m2 of thermocouple surface area, which it must reject again in the form of radiation. Estimate the temperature of the thermocouple. Hints: (a) Treat the tube ends together as a single surface A3 . (b) Note that the thermocouple is small, i.e., Fx−1 1. 5.27 A small spherical heat source outputting Qs = 10 kW power, spreading equally into all directions, is encased in a reflector as shown, consisting of a hemisphere of radius R = 40 cm, plus a ring of radius R and height h = 30 cm. The arrangement is used to heat a disk of radius = 25 cm a distance of L = 30 cm below the reflector. All surfaces are gray and diffuse, with emittances of 1 = 0.8 and 2 = 0.1. Reflector A2 is insulated. Determine (per unit area of receiving surface) (a) the irradiation from heat source to reflector and to disk; (b) all relevant view factors; and (c) the temperature of the disk, if 0.4 kW of power is extracted from the disk.

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 193

5.28 A long thin black heating wire radiates 300 W per cm length of wire and is used to heat a flat surface by thermal radiation. To increase its efficiency the wire is surrounded by an insulated half-cylinder as shown in the figure. Both surfaces are gray and diffuse with emittances 2 and 3 , respectively. What is the net heat flux at Surface 3? How does this compare with the case without cylinder? Hint: You may either treat the heating wire as a thin cylinder whose radius you eventually shrink to zero, or treat radiation from the wire as external radiation (the second approach being somewhat simpler).

5.29 Consider the configuration shown, consisting of a cylindrical cavity A2 , a circular disk A1 at the bottom, and a small spherical radiation source (blackbody at 4000K) of strength Q = 10, 000 W as shown (R = 10 cm, h = 10 cm). The cylinder wall A2 is covered with a gray, diffuse material with 2 = 0.1, and is perfectly insulated. Surface A1 is kept at a constant temperature of 400 K. No other external surfaces or sources affect the heat transfer. Assuming surface A1 to be gray and diffuse with

1 = 0.3 determine the amount of heat that needs to be removed from A1 (Q1 ).

5.30 Determine F1−2 for the rotationally symmetric configuration shown in the figure (i.e., a big sphere, R = 13 cm, with a circular hole, r = 5 cm, and a hemispherical cavity, r = 5 cm). Assuming Surface 2 to be gray and diffuse ( = 0.5) and insulated and Surface 1 to be black and also insulated, what is the average temperature of the black cavity if collimated irradiation of 1000 W/m2 is penetrating through the hole as shown?

5.31 An integrating sphere (a device to measure surface properties) is 10 cm in radius. It contains on its inside wall a 1 cm2 black detector, a 1 × 2 cm entrance port, and a 1 × 1 cm sample as shown. The remaining portion of the sphere is smoked with magnesium oxide having a short-wavelength reflectance of 0.98, which is almost perfectly diffuse. A collimated beam of radiant energy (i.e., all energy is contained within a very small cone of solid angles) enters the sphere through the entrance port, falls onto the sample, and then is reflected and interreflected, giving rise to a sphere wall radiosity and irradiation. Radiation emitted from the walls is not detected because the source radiation is chopped, and the detector–amplifier system responds only to the chopped radiation. Find the fractions of the chopped incoming radiation that are (a) lost out the entrance port, (b) absorbed by the MgO-smoked wall, and (c) absorbed by the detector. [Item (c) is called the “sphere efficiency.”]

194 Radiative Heat Transfer

5.32 The side wall of a flask holding liquid helium may be approximated as a long doublewalled cylinder as shown in the adjacent sketch. The container walls are made of 1 mm thick stainless steel (k = 15 W/m K, = 0.2), and have outer radii of R2 = 10 cm and R4 = 11 cm. The space between walls is evacuated, and the outside is exposed to free convection with the ambient at Tamb = 20◦ C and a heat transfer coefficient of ho = 10 W/m2 K (for the combined effects of free convection and radiation). It is reasonable to assume that the temperature of the inner wall is at liquid helium temperature, or T(R2 ) = 4 K. (a) Determine the heat gain by the helium, per unit length of flask. (b) To reduce the heat gain a thin silver foil ( = 0.02) is placed midway between the two walls. How does this affect the heat flux? For the sake of the problem, you may assume both steel and silver to be diffuse reflectors. 5.33 The inside surfaces of a furnace in the shape of a parallelepiped with dimensions 1 m × 2 m × 4 m are to be broken up into 28 1 m × 1 m subareas. The gray-diffuse side walls (of dimension 1 m × 2 m and 1 m × 4 m) have emittances of s = 0.7 and are perfectly insulated, the bottom surface has an emittance of b = 0.9 and a temperature Tb = 1600 K, while the emittance of the top surface is t = 0.2 and its temperature is Tt = 500 K. Using the view factors calculated in Problem 4.27 and program graydiffxch of Appendix F, calculate the heating/cooling requirements for bottom and top surfaces, as well as the temperature distribution along the side walls. 5.34 For your Memorial Day barbecue you would like to broil a steak on your backyard BBQ, which consists of a base unit in the shape of a hemisphere (D= 60 cm), fitted with a disk-shaped coal rack and a disk-shaped grill, as shown in the sketch. Hot coal may be assumed to cover the entire floor of the unit, with uniform temperature Tc = 1200 K, and an emittance of c = 1. The side wall is soot covered and black on the inside, but has an outside emittance of o = 0.5. The steak (modeled as a ds = 15 cm disk, 1 cm thick, emittance s = 0.8, initially at Ts = 280 K) is now placed on the grill (assumed to be so lightweight as to be totally transparent and not participating in the heat transfer). The environment is at 300 K, and free convection may be neglected. (a) Assuming that the lid is not placed on top of the unit, estimate the initial heating rates on the two surfaces of the steak. (b) How would the heating rates change, if the lid (also a hemisphere) is put on ( i = o = 0.5)? Could one achieve a more even heating rate (top and bottom) if the emittance of the inside surface is increased or decreased? Note: Part (b) will be quite tedious, unless program graydiffxch of Appendix F is used (which, in turn, will require iteration or a little trickery). 5.35 To calculate the net heat loss from a part of a spacecraft, this part may be approximated by an infinitely long black plate at temperature T2 = 600 K, as shown. Parallel to this plate is another (infinitely long) thin plate that is gray and emits/reflects diffusely with the same emittance 1 on both sides. You may assume the surroundings to be black at 0 K. Calculate the net heat loss from the black plate.

5.36 A large isothermal surface (exposed to vacuum, temperature Tw , diffuse-gray emittance w ) is irradiated by the sun. To reduce the heat gain/loss from the surface, a thin copper shield (emittance c and initially at temperature Tc0 ) is placed between surface and sun as shown in the figure. (a) Determine the relationship between Tc and time t (it is sufficient to leave the answer in implicit form with an unsolved integral). (b) Give the steady-state temperature for Tc (i.e., for t → ∞).

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 195

(c) Briefly discuss qualitatively the following effects: (i) The shield is replaced by a moderately thick slab of styrofoam coated on both sides with a very thin layer of copper. (ii) The surfaces are finite in size. 5.37 Consider two infinitely long, parallel, black plates of width L as shown. The bottom plate is uniformly heated electrically with a heat flux of q1 = const, while the top plate is insulated. The entire configuration is placed into a large cold environment. (a) Determine the governing equations for the temperature variation across the plates. (b) Find the solution by the kernel substitution method. To avoid tedious algebra, you may leave the final result in terms of two constants to be determined, as long as you outline carefully how these constants may be found. (c) If the plates are gray and diffuse with emittances 1 and 2 , how can the temperature distribution be determined, using the solution from part (b)? 5.38 To reduce heat transfer between two infinite concentric cylinders a third cylinder is placed between them as shown in the figure. The center cylinder has an opening of half-angle θ. The inner cylinder is black and at temperature T1 = 1000 K, while the outer cylinder is at T4 = 300 K. The outer cylinder and both sides of the shield are coated with a reflective material, such that c = 2 = 3 = 4 . Determine the heat loss from the inner cylinder as function of coating emittance c , using (a) the net radiation method, (b) the network analogy.

5.39 Consider the two long concentric cylinders as shown in the figure. Between the two cylinders is a long, thin flat plate as also indicated. The inner cylinder is black and generating heat on its inside in the amount of Q1 = 1 kW/m length of the cylinder, which must be removed by radiation. The plate is gray and diffuse with emittance 2 = 3 = 0.5, while the outer cylinder is black and cold (T4 = 0 K). Determine the temperature of the inner cylinder, using (a) the net radiation method, (b) the network analogy.

5.40 An isothermal black disk at T1 = 500K is flush with the outer surface of a spacecraft and is thus exposed to outer space. To minimize heat loss from the disk a diskshaped radiation shield is placed coaxially and parallel to the disk as shown; the shield radius is R2 (which may be smaller or larger than R1 ), and its distance from the black disk is a variable h. Determine an expression for the heat loss from the black disk as a function of shield radius and distance, using (a) the net radiation method, (b) the network analogy.

196 Radiative Heat Transfer

5.41 Consider Configuration 33 in Appendix D with h = w. The bottom wall is at constant temperature T1 and has emittance

1 ; the side wall is at T2 = const and 2 . Find the exact expression for q1 (x) if 2 = 1. 5.42 An infinitely long half-cylinder is irradiated by the sun as shown in the figure, with qsun = 1000 W/m2 . The inside of the cylinder is gray and diffuse, the outside is insulated. There is no radiation from the background. Determine the equilibrium temperature distribution along the cylinder periphery, (a) using four isothermal zones of 45◦ each, (b) using the exact relations. Hint: Use differentiation as in the kernel approximation method. 5.43 Consider a gray-diffuse spherical enclosure. The upper half of the enclosure is at constant temperature T1 with emittance 1 , and the lower half is constant temperature T2 with emittance 2 . (a) Calculate local and total heat fluxes for each hemisphere using the exact relations. (b) Calculate total heat fluxes for each hemisphere using the net radiation method. Compare with the exact values.

References [1] C.L. Jensen, TRASYS-II user’s manual–thermal radiation analysis system, Technical report, Martin Marietta Aerospace Corp., Denver, 1987. [2] J.H. Chin, T.D. Panczak, L. Fried, Spacecraft thermal modeling, International Journal for Numerical Methods in Engineering 35 (1992) 641–653. [3] A.K. Oppenheim, Radiation analysis by the network method, Transactions of ASME, Journal of Heat Transfer 78 (1956) 725–735. [4] R.H. Kropschot, B.W. Birmingham, D.B. Mann (Eds.), Technology of Liquid Helium, National Bureau of Standards, Washington, D.C., 1968, Monograph 111. [5] R. Courant, D. Hilbert, Methods of Mathematical Physics, Interscience Publishers, New York, 1953. [6] F.B. Hildebrand, Methods of Applied Mathematics, Prentice Hall, Englewood Cliffs, NJ, 1952. [7] M.N. Özi¸sik, Radiative Transfer and Interactions with Conduction and Convection, John Wiley & Sons, New York, 1973. [8] K.J. Daun, K.G.T. Hollands, Infinitesimal-area radiative analysis using parametric surface representation, through NURBS, ASME Journal of Heat Transfer 123 (2) (2001) 249–256. [9] E.M. Sparrow, A. Haji-Sheikh, A generalized variational method for calculating radiant interchange between surfaces, ASME Journal of Heat Transfer 87 (1965) 103–109. [10] E.M. Sparrow, Application of variational methods to radiation heat transfer calculations, ASME Journal of Heat Transfer 82 (1960) 375–380. [11] E.M. Sparrow, J.L. Gregg, J.V. Szel, P. Manos, Analysis, results, and interpretation for radiation between simply arranged gray surfaces, ASME Journal of Heat Transfer 83 (1961) 207–214. [12] H. Buckley, On the radiation from the inside of a circular cylinder, Philosophical Magazine 4 (23) (1927) 753–762. [13] E.R.G. Eckert, Das Strahlungsverhältnis von Flächen mit Einbuchtungen und von zylindrischen Bohrungen, Archiv für Wärmewirtschaft 16 (1935) 135–138. [14] E.M. Sparrow, L.U. Albers, Apparent emissivity and heat transfer in a long cylindrical hole, ASME Journal of Heat Transfer 82 (1960) 253–255. [15] C.M. Usiskin, R. Siegel, Thermal radiation from a cylindrical enclosure with specified wall heat flux, ASME Journal of Heat Transfer 82 (1960) 369–374. [16] S.H. Lin, E.M. Sparrow, Radiant interchange among curved specularly reflecting surfaces, application to cylindrical and conical cavities, ASME Journal of Heat Transfer 87 (1965) 299–307. [17] M. Perlmutter, R. Siegel, Effect of specularly reflecting gray surface on thermal radiation through a tube and from its heated wall, ASME Journal of Heat Transfer 85 (1963) 55–62. [18] R. Siegel, M. Perlmutter, Convective and radiant heat transfer for flow of a transparent gas in a tube with a gray wall, International Journal of Heat and Mass Transfer 5 (1962) 639–660. [19] E.M. Sparrow, L.U. Albers, E.R.G. Eckert, Thermal radiation characteristics of cylindrical enclosures, ASME Journal of Heat Transfer 84 (1962) 73–81. [20] E.M. Sparrow, R.P. Heinisch, The normal emittance of circular cylindrical cavities, Applied Optics 9 (1970) 2569–2572. [21] G. Alfano, Apparent thermal emittance of cylindrical enclosures with and without diaphragms, International Journal of Heat and Mass Transfer 15 (12) (1972) 2671–2674. [22] G. Alfano, A. Sarno, Normal and hemispherical thermal emittances of cylindrical cavities, ASME Journal of Heat Transfer 97 (3) (1975) 387–390.

Radiative Exchange Between Gray, Diffuse Surfaces Chapter | 5 197

[23] C.E. Fröberg, Introduction to Numerical Analysis, Addison-Wesley, Reading, MA, 1969. [24] H.H. Jensen, Some notes on heat transfer by radiation, Matematisk-fysiske meddelelser Kongelige Danske Videnskabernes Selskab 24 (8) (1948) 1–26. [25] M. Jakob, Heat Transfer, vol. 2, John Wiley & Sons, New York, 1957. [26] E.M. Sparrow, Radiant absorption characteristics of concave cylindrical surfaces, ASME Journal of Heat Transfer 84 (1962) 283–293.

Chapter 6

Radiative Exchange Between Nondiffuse and Nongray Surfaces 6.1 Introduction In the previous two chapters it was assumed that all surfaces constituting the enclosure are—besides being gray— diffuse emitters as well as diffuse reflectors of radiant energy. Diffuse emission is nearly always an acceptable simplification. The assumption of diffuse reflection, on the other hand, often leads to considerable error, since many surfaces deviate substantially from this behavior. Electromagnetic wave theory predicts reflection to be specular for optically smooth surfaces, i.e., to reflect light like a mirror. All clean metals, many nonmetals such as glassy materials, and most polished materials display strong specular reflection peaks. Nevertheless, they all, to some extent, reflect somewhat into other directions as a result of their surface roughness. Surfaces may appear dull (i.e., diffusely reflecting) to the eye, but are rather specular in the infrared, since the ratio of every surface’s root-mean-square roughness to wavelength decreases with increasing wavelength. Similarly, assuming surfaces to be gray can also lead to large errors. For example, we noted that solar collectors did not appear to perform very well because, in our gray analysis, the reradiation losses were rather large. On the other hand, experience has shown that reradiation losses can be reduced substantially if selective surfaces (i.e., strongly nongray surfaces) are used for the collector plates. Clearly, there are a substantial number of applications for which our idealized treatment (gray, diffuse, i.e., wavelength- and direction-independent absorptance and emittance) is not sufficiently accurate. Actual surface properties deviate from our idealized treatment in a number of ways: 1. As seen from the discussion in Chapter 3, radiative properties can vary appreciably across the spectrum. 2. Spectral properties and, in particular, spectrally averaged properties may depend on the local surface temperature. 3. Absorptance and reflectance of a surface may depend on the direction of the incoming radiation. 4. Emittance and reflectance of a surface may depend on the direction of the outgoing radiation. 5. The components of polarization of incident radiation are reflected differently by a surface. Even for unpolarized radiation this difference can cause errors if many consecutive specular reflections take place. In the case of polarized laser irradiation this effect will always be important. In this chapter we shall briefly discuss how nondiffuse and nongray property effects may be incorporated into the analyses of the previous chapter. We shall also develop the governing equation for the intensity leaving the surface of an enclosure with arbitrary radiative properties (spectrally and directionally), from which heat transfer rates may be calculated. This expression will be applied to a simple geometry to show how directionally irregular surface properties may be incorporated in the analysis.

6.2 Enclosures with Partially Specular Surfaces For a surface with diffuse reflectance the reflected radiation has the same (diffuse) directional distribution as the emitted energy, as discussed in the beginning of Section 5.3. Therefore, the radiation field within the enclosure is completely specified in terms of the radiosity, which is a function of location along the enclosure walls (but not a function of direction as well). If reflection is nondiffuse, then the radiation intensities leaving any surface are functions of direction as well as surface location, and the analysis becomes immensely more complicated.1 To make the analysis tractable, one may make the idealization that the reflectance, while not diffuse, can be 1. In addition, if the irradiation is polarized (e.g., owing to irradiation from a laser source), specular reflections will change the state of polarization (because of the different values for ρ and ρ⊥ , as discussed in Chapter 2). We shall only consider unpolarized radiation. Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00014-6 Copyright © 2022 Elsevier Inc. All rights reserved.

199

200 Radiative Heat Transfer

FIGURE 6.1 (a) Subdivision of the reflectance of oxidized brass (shown for plane of incidence) into specular (shaded) and diffuse components (unshaded), from [1]; (b) equivalent idealized reflectance.

adequately represented by a combination of a diffuse and a specular component, as illustrated in Fig. 6.1 for oxidized brass [1]. Thus, for the present chapter, we assume the radiative properties to be of the form ρ = ρs + ρ d = 1 − α = 1 − = 1 − λ ,

(6.1)

where ρs and ρ d are the specular and diffuse components of the reflectance, respectively. Since the surfaces are assumed to be gray, diffuse emitters ( = λ ), it follows that neither α nor ρ depend on wavelength or on incoming direction (i.e., the magnitude of ρ does not depend on incoming direction); how ρ is distributed over outgoing directions depends on incoming direction through ρs . With this approximation, the separate reflection components may be found analytically by splitting the bidirectional reflection function into two parts, ρ (r, sˆ i , sˆ r ) = ρ s (r, sˆ i , sˆ r ) + ρ d (r, sˆ i , sˆ r ).

(6.2)

Substituting this expression into equation (3.43) and equation (3.46) then leads to ρs and ρ d . Values of ρs and ρ d may also be determined directly from experiment, as reported by Birkebak and coworkers [2], making detailed measurements of the bidirectional reflection function unnecessary.

Specular View Factors Within an enclosure consisting of surfaces with purely diffuse and purely specular reflection components, the complexity of the problem may be reduced considerably by realizing that any specularly reflected beam may be traced back to a point on the enclosure surface from which it emanated diffusely (i.e., any beam was part of an energy stream leaving the surface after emission or diffuse reflection), as illustrated in Fig. 6.2. Therefore, by redefining the view factors to include specular reflection paths in addition to direct view, the radiation field may again be described by a diffuse energy function that is a function of surface location but not of direction. To accommodate surfaces with reflectances described by equation (6.1), we define a specular view factor as

s ≡ dFdA i −dA j

diffuse energy leaving dAi intercepted by dA j , by direct travel or any number of specular reflections total diffuse energy leaving dAi

.

(6.3)

The concept of the specular view factor is illustrated in Figs. 6.2 and 6.3. Diffuse radiation leaving dAi (by emission or diffuse reflection) can reach dA j either directly or after one or more reflections. Usually only a finite number of specular reflection paths such as dAi − a − dA j or dAi − b − c − dA j (and others not indicated in the figure) will be possible. The surface at points a, b, and c behaves like a perfect mirror as far as the specular part

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 201

FIGURE 6.2 Radiative exchange in an enclosure with specular reflectors.

FIGURE 6.3 Specular view factor between infinitesimal surface elements; formation of images.

of the reflection is concerned. Therefore, if an observer stood on top of dA j looking toward c, it would appear as if point b as well as dAi were situated behind point c as indicated in Fig. 6.3; the point labeled b(c) is the image of point b as mirrored by the surface at c, and dAi (cb) is the image of dAi as mirrored by the surfaces at c and b. Therefore, as we examine Figs. 6.2 and 6.3, we may formally evaluate the specular view factor between two infinitesimal areas as s dFdA = dFdAi −dA j + ρsa dFdAi (a)−dA j + ρsb ρsc dFdAi (cb)−dA j + other possible reflection paths. i −dA j

(6.4)

Thus, the specular view factor may be expressed as a sum of diffuse view factors, with one contribution for each possible direct or reflection path. Note that, for images, the diffuse view factors must be multiplied by the specular reflectances of the mirroring surfaces, since radiation traveling from dAi to dA j is attenuated by every reflection. If all specularly reflecting parts of the enclosure are flat, then all images of dAi have the same shape and size as dAi itself. However, curved surfaces tend to distort the images (focusing and defocusing effects). In the case of only flat, specularly reflecting surfaces we may multiply equation (6.4) by dAi and, invoking the law of reciprocity for diffuse view factors, equation (4.7), we obtain s dAi dFdA = dA j dFdA j −dAi + ρas dA j dFdA j −dAi (a) + ρsb ρsc dA j dFdA j −dAi (bc) i −dA j

= dA j dFdA j −dAi + ρsa dA j dFdA j (a)−dAi + ρsb ρsc dA j dFdA j (bc)−dAi + . . . s = dA j dFdA , j −dAi

(6.5)

that is, the law of reciprocity holds for specular view factors as long as all specularly reflecting surfaces are flat. Although considerably more complicated, it is possible to show that the law of reciprocity also holds for curved

202 Radiative Heat Transfer

s and F s . FIGURE 6.4 (a) Geometry for Example 6.1, (b) ray tracing for the evaluation of F1−1 1−2

specular reflectors. If we also assume that the diffuse energy leaving Ai and A j is constant across each respective area, we have the equivalent to equation (4.15), s s dAi dFdi−d j = dA j dFd j−di , s dAi Fdi−j Ai Fi−s j

= =

s A j dFj−di , s A j Fj−i ,

(6.6a) (Jj = const),

(6.6b)

(Ji , Jj = const),

(6.6c)

where we have adopted the compact notation first introduced in Chapter 4, and Ji is the total diffuse energy (per unit area) leaving surface Ai (again called the radiosity). s s Example 6.1. Evaluate the specular view factors F1−1 and F1−2 for the parallel plate geometry shown in Fig. 6.4a.

Solution s must be the same for any dA1 on surface A1 . Since We note that, because of the one-dimensionality of the problem, Fd1−2 s s s s . It is sufficient to consider energy leaving from F1−2 is nothing but a surface average of Fd1−2 , we conclude that Fd1−2 = F1−2 an infinitesimal area (rather than all of A1 ). Examining Fig. 6.4b we see that every beam (assumed to have unity strength) leaving dA1 , regardless of its direction, must travel to surface A2 (a beam of strength “1” is intercepted). After reflection at A2 a beam of strength ρs2 returns to A1 specularly, where it is reflected again and a beam of strength ρs2 ρs1 returns to A2 specularly. After one more reflection a beam of strength (ρs2 ρ1s )ρs2 returns to A1 , and so on. Thus, the specular view factor may be evaluated as s s = F1−2 = 1 + ρs1 ρs2 + (ρs1 ρs2 )2 + (ρs1 ρs2 )3 + . . . . Fd1−2

Since ρs1 ρs2 < 1 the sum in this equation is readily evaluated by the methods given in Wylie [3], and s F1−2 =

1 s = F2−1 . 1 − ρs1 ρs2

The last part of this relation is found by switching subscripts or by invoking reciprocity (and A1 = A2 ). We notice that specular view factors are not limited to values between zero and one, but are often greater than unity because much of the radiative energy leaving a surface is accounted for more than once. All energy from A1 is intercepted by A2 after direct travel, but only the fraction (1 − ρs2 ) is removed (by absorption and/or diffuse reflection) from the specular reflection path. s that must have a The fraction ρs2 travels on specularly and is, therefore, counted a second time, etc. Thus, it is (1 − ρs2 )F1−2 value between zero and one, and the summation relation, equation (4.18), must be replaced by N 

s (1 − ρsj )Fi−j = 1.

(6.7)

j=1

Equation (6.7), formed here through intuition, will be developed rigorously in the next section. s may be found similarly as F1−1 s = ρs2 + (ρs1 ρs2 )ρs2 + (ρs1 ρs2 )2 ρs2 + . . . = F1−1

We note in passing that

ρs2 1 − ρs1 ρs2

.

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 203

FIGURE 6.5 (a) Geometry for Example 6.2, (b) repeated reflections along outer surface.

s s (1 − ρs1 )F1−1 + (1 − ρs2 )F1−2 =

(1 − ρs1 )ρs2 + 1 − ρs2 1 − ρs1 ρs2

= 1,

as postulated by equation (6.7). Example 6.2. Evaluate all specular view factors for two concentric cylinders or spheres. Solution Possible beam paths with specular reflections from inner to outer cylinders (or spheres) and vice versa are shown in Fig. 6.5a. As in the previous example a beam leaving A1 in any direction must hit surface A2 (with strength “1”). Because of the circular geometry, after specular reflection the beam (now of strength ρs2 ) must return to A1 (i.e., it cannot hit A2 again before hitting A1 ). After renewed reflections the beam keeps bouncing back and forth between A1 and A2 . Thus, as for parallel plates, s = 1 + ρs1 ρs2 + (ρs1 ρs2 )2 + . . . = F1−2

1 . 1 − ρs1 ρs2

Similarly, we have s F1−1 = ρs2 + (ρs1 ρs2 )ρs2 + . . . =

ρs2 1 − ρs1 ρs2

.

A beam emanating from A2 will first hit either A1 , and then keep bouncing back and forth between A1 and A2 (cf. Fig. 6.5a), or A2 , and then keep bouncing along A2 without ever hitting A1 (cf. Fig. 6.5b). Thus, since the fraction F2−1 of the diffuse energy leaving A2 hits A1 after direct travel, we have   A1 /A2 s F2−1 = F2−1 1 + ρs1 ρs2 + (ρs1 ρs2 )2 + . . . = , 1 − ρs1 ρs2     s F2−2 = F2−2 1 + ρs2 + (ρs2 )2 + (ρ2s )3 + . . . + F2−1 ρs1 + ρs1 (ρs1 ρs2 ) + . . . =

s 1 − A1 /A2 ρ1 A1 /A2 + , 1 − ρs2 1 − ρs1 ρs2

s where the simple diffuse view factors F2−1 and F2−2 have been evaluated in terms of A1 and A2 . Of course, F2−1 could have s s by reciprocity and F2−2 with the aid of equation (6.7). been found from F1−2

A few more examples of specular view factor determinations will be given once the appropriate heat transfer relations have been developed.

204 Radiative Heat Transfer

FIGURE 6.6 Energy balance for surfaces with partially specular reflection.

6.3 Radiative Exchange in the Presence of Partially Specular Surfaces Consider an enclosure of arbitrary geometry as shown in Fig. 6.2. All surfaces are gray, diffuse emitters and gray reflectors with purely diffuse and purely specular components, i.e., their radiative properties obey equation (6.1). Under these conditions the net heat flux at a surface at location r is, from Fig. 6.6, q(r) = qemission − qabsorption = (r)[Eb (r) − H(r)] = qout − qin = (r)Eb (r) + ρ d (r)H(r) + ρs (r)H(r) − H(r).

(6.8)

The first two terms on the last right-hand side of equation (6.8), or the part of the outgoing heat flux that leaves diffusely, we will again call the surface radiosity, J(r) = (r)Eb (r) + ρ d (r)H(r),

(6.9)

q(r) = J(r) − [1 − ρs (r)]H(r).

(6.10)

so that Eliminating the irradiation H(r) from equations (6.8) and (6.10) leads to q(r) =



(r)  [1 − ρs (r)]Eb (r) − J(r) , d ρ (r)

(6.11)

which, of course, reduces to equation (5.26) for a diffusely reflecting surface if ρs = 0 and ρ d = 1 − . For a purely specular reflecting surface (ρ d = 0) equation (6.11) is indeterminate since the radiosity consists only of emission, or J = Eb . As in Chapter 5 the irradiation H(r) is found by determining the contribution to H from a differential area dA (r ), followed by integration over the entire enclosure surface. A subtle difference is that we do not track the total energy leaving dA (multiplied by a suitable direct-travel view factor); rather, the contribution from specular reflections is subtracted and attributed to the surface from which it leaves diffusely. The more complicated path of such energy is then accounted for by the definition of the specular view factor. Thus, similar to equation (5.21),  s  s H(r) dA = J(r ) dFdA (6.12)  −dA dA + Ho (r) dA, A

Hos (r)

where is any external irradiation arriving at dA (through openings or semitransparent walls). Similar to the specular view factors, the Hos includes external radiation hitting dA directly or after any number of specular reflections. Using reciprocity, equation (6.12) becomes  s s H(r) = J(r ) dFdA−dA (6.13)  + Ho (r), A

and, after substitution into equation (6.9), an integral equation for the unknown radiosity is obtained as   d  s s J(r ) dFdA−dA + Ho (r) . (6.14) J(r) = (r)Eb (r) + ρ (r) A

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 205

For surface locations for which heat flux q(r) is given rather than Eb (r), equation (6.10) should be used rather than equation (6.9). It is usually more desirable to eliminate the radiosity, to obtain a single relationship between surface blackbody emissive powers and heat fluxes. Solving equation (6.11) for J gives ρ d (r)   q(r), J(r) = 1 − ρs (r) Eb (r) −

(r)

(6.15)

and substituting this expression into equation (6.14) leads to ρd (1 − ρ )Eb − q = (1 − ρs − ρ d )Eb + ρ d



 (1 − ρ

s

or

A



q(r)   s − Eb (r) − 1 − ρs (r ) Eb (r ) dFdA−dA  =

(r) A

 A

s

s )Eb dFdA−dA 

− A

 ρd s s q dFdA−dA + Ho ,

ρ d (r )  s s q(r ) dFdA−dA  + Ho (r).

(r )

(6.16)

We note that, for diffusely reflecting surfaces with ρs = 0, ρ d = 1 − , Fi−s j = Fi−j , and Hos = Ho , equation (6.16) reduces to equation (5.28). If the specular view factors can be calculated (and that is often a big “if”), then equation (6.16) is not any more difficult to solve than equation (5.28). Indeed, if part or all of the surface is purely specular (ρ d = 0), equation (6.16) becomes considerably simpler. As for black and gray-diffuse enclosures, it is customary to simplify the analysis by using an idealized enclosure, consisting of N relatively simple subsurfaces, over each of which the radiosity is assumed constant. Then  

J(r A

s ) dFdA−dA 



N 

 Jj Aj

j=1

s dFdA−dA j

=

N 

s Jj FdA−A , j

j=1

and, after averaging over a subsurface Ai on which dA is situated, equation (6.14) simplifies to ⎛ ⎞ N ⎜⎜ ⎟⎟ ⎜ ⎟ Ji = i Ebi + ρid ⎜⎜⎜ Jj Fi−s j + Hois ⎟⎟⎟ , i = 1, 2, . . . , N. ⎝ ⎠

(6.17)

j=1

Eliminating radiosity through equation (6.15) then simplifies equation (6.16) to Ebi −

N 

qi  ρ j s = − F q j + Hois ,

i

j i− j N

(1 −

ρsj )Fi−s j Ebj

j=1

d

i = 1, 2, . . . , N.

(6.18)

j=1

The summation relation, equation (6.7), is easily obtained from equation (6.18) by considering a special case: In s s an isothermal enclosure (Eb1 = Eb2 = · · · = EbN ) without external irradiation (Ho1 = Ho2 = · · · = 0), according to the Second Law of Thermodynamics, all heat fluxes must vanish (q1 = q2 = · · · = 0). Thus, canceling emissive powers, N 

(1 − ρsj )Fi−s j = 1,

i = 1, 2, . . . , N.

(6.19)

j=1

Since the Fi−s j are geometric factors and do not depend on temperature distribution, equation (6.19) is valid for arbitrary emissive power values. Finally, for computer calculations it may be advantageous to write the emissive power and heat fluxes in matrix form. Introducing Kronecker’s delta equation (6.18) becomes N   j=1

δi j − (1 − ρsj )Fi−s j



⎛ ⎞ d N ⎜  ⎜⎜ δi j ρ j s ⎟⎟⎟ ⎜⎜ − Fi− j ⎟⎟ q j + Hois , Ebj = ⎝ j ⎠

j j=1

i = 1, 2, . . . , N,

(6.20)

206 Radiative Heat Transfer

or2 A · eb = C · q + hso ,

(6.21)

where C and A are matrices with elements Ai j = δi j − (1 − ρsj )Fi−s j , Ci j =

δi j

j

−

ρ dj

j

Fi−s j ,

and q, eb , and hso are vectors for the surface heat fluxes, emissive powers, and external irradiations, respectively. If all temperatures and external irradiations are known, the unknown heat fluxes may be formally expressed as   q = C−1 · A · eb − hso . (6.22) It is worth noting that if a surface, say Ak , is a purely specular reflector (ρkd = 0), qk appears only in a single equation, i.e., when i = k in equation (6.20). Thus, for an enclosure consisting of N surfaces, of which n are purely specular with known temperature, only N − n simultaneous equations need to be solved. While this fact simplifies specular enclosure analysis as compared with diffuse enclosures, one should remember that, in general, specular view factors are considerably more difficult to evaluate. If the emissive power is only known over some of the surfaces, and the heat fluxes are specified elsewhere, equation (6.21) may be rearranged into a similar equation for the vector containing all the unknowns. Subroutine graydifspec is provided in Appendix F for the solution of the simultaneous equations (6.21), requiring surface information and a partial view factor matrix as input. The solution to a sample problem is also given in the form of a program grspecxch, which may be used as a starting point for the solution to other problems. Fortran90, C++ as well as Matlab versions are provided. Example 6.3. Two large parallel plates are separated by a nonparticipating medium as shown in Fig. 6.4a. The bottom surface is isothermal at T1 , with emittance 1 and a partially specular, partially diffuse reflectance ρ1 = ρ1d + ρs1 . Similarly, the top surface is isothermal at T2 with 2 and ρ2 = ρ2d + ρs2 . Determine the radiative heat flux between the surfaces. Solution s From equation (6.18) we have, for i = 1, with Ho1 = 0, s s Eb1 − (1 − ρs2 )F1−2 Eb2 = Eb1 − (1 − ρs1 )F1−1

ρd s q1 ρ1d s − F1−1 q1 − 2 F1−2 q2 .

1

1

2

While we could apply i = 2 to equation (6.18) to obtain a second equation for q1 and q2 , it is simpler here to use overall conservation of energy, or q2 = −q1 . Thus,   s s 1 − (1 − ρs1 )F1−1 Eb1 − (1 − ρs2 )F1−2 Eb2 q1 = . s s 1 − 1 − ρ1 s 1 − 2 − ρ2 s 1 − F1−1 + F1−2

1

1

2 s Using the results from Example 6.1 and dividing both numerator and denominator by F1−2 , we obtain

q1 = 

s (1 − ρs2 )F1−2 (Eb1 − Eb2 ) (1 − ρs2 )(1)(Eb1 − Eb2 ) Eb1 − Eb2 =  = ,   1 1 1 1 1 1 s s s (1 − ρs2 )F1−2 (1 − ρs2 )(1) + ρs2 − 1 + + F1−1 − F1−2 + + −1

1 2

1 2

1 2

(6.23)

which produces the same result whether we have diffusely or specularly reflecting surfaces. Indeed, equation (6.23) is valid for the radiative transfer between two isothermal parallel plates, regardless of the directional behavior of the reflectance (i.e., it is not limited to the idealized reflectances considered in this chapter). Any beam leaving A1 must hit surface A2 and vice versa, regardless of whether the reflectance is diffuse, specular, or neither of the two; the surface locations will be different but the directional variation of reflectance has no influence on the heat transfer rate since the surfaces are isothermal. 2. Again, for easy readability of matrix manipulations we shall follow here the convention that a two-dimensional matrix is denoted by a bold capitalized letter, while a vector is written as a bold lowercase letter.

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 207

FIGURE 6.7 Geometry for Example 6.5.

Example 6.4. Repeat the previous example for concentric spheres and cylinders. Solution Again, from equation (6.18) with i = 1 and Hois = 0, we obtain s s Eb1 − (1 − ρs2 )F1−2 Eb2 = Eb1 − (1 − ρs1 )F1−1

ρd s q1 ρ1d s − F1−1 q1 − 2 F1−2 q2 .

1

1

2

In this case conservation of energy demands q2 A2 = −q1 A1 , and   s s s 1 − (1 − ρs1 )F1−1 Eb1 − (1 − ρ2s )F1−2 Eb2 (1 − ρs2 )F1−2 (Eb1 − Eb2 ) =  . q1 =  s s 1 − 1 − ρ1 s 1 1 A1 A1 s 1 − 2 − ρ2 A1 s 1 s s s (1 − ρ + )F + F − F − F1−1 + F 2 1−2 1−1

1 2 A2 A2 1−2

1

1

2 A2 1−2 s s and F1−2 are the same as in the previous example (cf. Example 6.2), leading to The specular view factors F1−1

q1 =

Eb1 − Eb2 s . 1 1 A1 A1 /A2 − ρ2 + −

1 2 A2 1 − ρ2s

(6.24)

We note that equation (6.24) does not depend on ρs1 : Again, any radiation reflected off surface A1 must return to surface A2 , regardless of the directional behavior of its reflectance. If surface A2 is purely specular (ρs2 = 1 − 2 ), all radiation from A1 bounces back and forth between A1 and A2 , and equation (6.24) reduces to equation (6.23), i.e., the heat flux between these concentric spheres or cylinders is the same as between parallel plates. On the other hand, if A2 is diffuse (ρs2 = 0) equation (6.24) reduces to the purely diffuse case since the directional behavior of ρ1 is irrelevant. Example 6.5. A very long solar collector plate is to collect energy at a temperature of T1 = 350 K. To improve its performance for off-normal solar incidence, a highly reflective surface is placed next to the collector as shown in Fig. 6.7. For simplicity you may make the following assumptions: The collector is isothermal and gray-diffuse with emittance

1 = 1 − ρ1d = 0.8; the mirror is gray and specular with 2 = 1 − ρs2 = 0.1, and heat losses from the mirror by convection as well as all losses from the collector ends may be neglected. How much energy (per unit length) does the collector plate collect for solar irradiation of qsun = 1000 W/m2 at an incidence angle of 30◦ ? Solution Applying equation (6.20) to the absorber plate (i = 1) as well as the mirror (i = 2) we obtain    ρd s ρd s 1 s s s Eb2 = − 1 F1−1 q2 + Ho1 , q1 − 2 F1−2 Eb1 − (1 − ρs2 )F1−2 1 − (1 − ρs1 )F1−1

1

1

2     ρd s ρd s 1 s s s q2 + Ho2 Eb2 = − 1 F2−1 Eb1 + 1 − (1 − ρs2 )F2−2 q1 + − 2 F2−2 . −(1 − ρs1 )F2−1

1

2

2 

208 Radiative Heat Transfer

s s s s = F2−2 = 0 and also F1−2 = F1−2 , F2−1 = F2−1 . For this configuration no specular reflections Since ρs1 = 0, it follows that F1−1 from one surface to another surface are possible (radiation leaving the absorber plate, after specular reflection from the mirror, always leaves the open enclosure). Thus, with q2 = 0,

q1 s + Ho1 ,

1   1 s −F2−1 Eb1 + Eb2 = − − 1 F2−1 q1 + Ho2 .

1

Eb1 − 2 F1−2 Eb2 =

Eliminating Eb2 , by multiplying the second equation by 2 F1−2 and adding, leads to (1 − 2 F1−2 F2−1 )Eb1 =

)   1 1 s s − −1 2 F1−2 F2−1 q1 +Ho1 + 2 F1−2 Ho2 .

1 1

'

The external fluxes are evaluated as follows: The mirror receives solar flux only directly (no specular reflection off the s absorber plate is possible), i.e., Ho2 = qsun sin ϕ. The absorber plate receives a direct contribution, qsun cos ϕ, and a second contribution after specular reflection off the mirror. This second contribution has the strength of ρs2 qsun cos ϕ per unit area. However, only part of the collector plate (l2 tan ϕ) receives this secondary contribution, which, for our crude two-node description, must be averaged over l1 . Thus, s = qsun cos ϕ + ρs2 qsun cos ϕ Ho1

Therefore,

  l2 tan ϕ l2 = qsun cos ϕ + (1 − 2 ) sin ϕ . l1 l1

  (1− 2 F1−2 F2−1 )Eb1 − cos ϕ+(1− 2 ) sin ϕ(l2 /l1 )+ 2 F1−2 sin ϕ qsun . q1 =   1 1 − 2 − 1 F1−2 F2−1

1

1

The view factors are readily evaluated by the crossed-strings method as F1−2 = (80 + 60 − 100)/(2 × 80) = F2−1 = 80 × 14 /60 = 13 . Substituting numbers, we obtain

1 4

and

   √ 1−0.1× 14 × 13 5.670×10−8 ×3504 − 23 +0.9× 12 × 60 +0.1× 14 × 12 1000 80   q1 = = −298 W/m2 . 1 1 1 1 − 0.1 − 1 × × 0.8 0.8 4 3 Under these conditions, therefore, the collector is about 30% efficient. This result should be compared with a collector without a mirror (l2 = 0 and F1−2 = 0), for which we get q1,no mirror =

√   Eb1 − qsun cos ϕ 3 = −12 W/m2 . = 0.8 × 5.670 × 10−8 × 3504 − 1000 × 1/ 1 2

This absorber plate collects hardly any energy at all (indeed, after accounting for convection losses, it would experience a net energy loss). If the mirror had been a diffuse reflector the heat gain would have been q1,diffuse mirror = −172 W/m2 , which is significantly less than for the specular mirror (cf. Problem 5.25). We conclude from this example that (i) mirrors can significantly improve collector performance and (ii) infrared reradiation losses from near-black collectors are very substantial. Of course, reradiation losses may be significantly reduced by using selective surfaces or glass-covered collectors (cf. Chapter 3).

As in the case for diffusely reflecting surfaces, the net radiation method of this section requires the radiosity to be constant over each subsurface, a condition rarely met in practice. More accurate results may be obtained by solving the governing integral equation, either equation (6.14) (to determine radiosity J) or equation (6.16) (to determine the unknown heat flux and/or surface temperature directly), by any of the methods outlined in Chapter 5. Such calculations were first done for two long parallel plates by Eckert and Sparrow [4]. In general, equation (6.16) is actually easier to solve than its diffuse-reflection counterpart if some or all of the surfaces are purely specular. However, the necessary specular view factors are generally much more difficult—if not impossible—to evaluate. Such a case arises, for example, for curved surfaces with multiple specular reflections.

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 209

Curved Surfaces with Specular Reflection Components In all our examples we have only considered idealized enclosures consisting of flat surfaces, for which the mirror images necessary for specular view factor calculations are relatively easily determined. If some or all of the reflecting surfaces are curved then equations (6.16) and (6.18) remain valid, but the specular view factors tend to be much more difficult to obtain. Analytical solutions can be found only for relatively simple geometries, such as axisymmetric surfaces, but even then they tend to get very involved. The very simple case of cylindrical cavities (with and without specularly reflecting end plate) has been studied by Sparrow and coworkers [5–7] and by Perlmutter and Siegel [8]. The more involved case of conical cavities has been treated by Sparrow and colleagues [6,7,9] as well as Polgar and Howell [10], while spherical cavities have been addressed by Tsai and coworkers [11,12] and Sparrow and Jonsson [13,14]. Somewhat more generalized discussions on the determination of specular view factors for curved surfaces have been given by Plamondon and Horton [15] and by Burkhard and coworkers [16]. In view of the complexity involved in these evaluations, specular view factors for curved surfaces are probably most conveniently calculated by a ray tracing method, such as the Monte Carlo method, which will be discussed in detail in Chapter 7. A considerably more detailed discussion of thermal radiation from and within grooves and cavities is given in the book by Sparrow and Cess [17].

Electrical Network Analogy The electrical network analogy, first introduced in Section 5.4, may be readily extended to allow for partially specular reflectors. This possibility was first demonstrated by Ziering and Sarofim [18]. Expressing equations (6.10) and (6.13) for an idealized enclosure [i.e., an enclosure with finite surfaces of constant radiosity, exactly as was done in equation (6.17)], we can evaluate the nodal heat fluxes as ⎡ ⎤ N ⎢⎢ ⎥⎥ ⎢ ⎥ Jj Fi−s j + Hois ⎥⎥⎥ , qi = Ji − (1 − ρsi ) ⎢⎢⎢ ⎣ ⎦

i = 1, 2, . . . , N.

(6.25)

j=1

Using the summation rule, equation (6.19), this relation may also be written as the sum of net radiative interchange between any two surfaces, qi =

N  

 (1 − ρsj )Ji − (1 − ρsi )Jj Fi−s j − (1 − ρsi )Hois

j=1

⎤ N ⎡  Jj ⎥⎥ ⎢⎢ Ji ⎢ ⎥⎥ (1 − ρs )(1 − ρs )F s − (1 − ρs )H s . = − ⎢⎣ i j i− j i oi 1 − ρsi 1 − ρsj ⎦

(6.26)

j=1

Similarly, from equation (6.11), qi =

(1 − ρsi ) i ρid



 Ji Ebi − . 1 − ρsi

(6.27)

After multiplication with Ai these relations may be combined and written in terms of potentials [Ebi and Ji /(1−ρsi )] and resistances as Ebi − Qi =

Ji 1 − ρsi

ρid (1 − ρsi ) i Ai

=

N  j=1

Jj Ji s − 1 − ρi 1 − ρjs 1 (1 − ρsi )(1 − ρjs )Ai Fi−s j

− (1 − ρsi ) Ai Hois .

(6.28)

Of course, this relation reduces to equation (5.45) for the case of purely diffuse surfaces (ρsi = 0, i = 1, 2, . . . , N). Note that, unlike diffuse reflectance, the specular reflectance is not irrelevant for insulated surfaces.

210 Radiative Heat Transfer

Radiation Shields As noted in Section 5.5 radiation shields tend to be made of specularly reflecting materials, such as polished metals or dielectric sheets coated with a metallic film. We would like, therefore, to extend the analysis to partly specular surfaces, i.e., (referring to Fig. 5.14) k = 1 − ρsk − ρkd for all surfaces (inside and outside wall, all shield surfaces). Again, the analysis is most easily carried out using the electrical network analogy, and the resistance between any two layers has already been evaluated in Example 6.4, equation (6.24), as   ρsk 1 1 1 1 R j−k = + − − . (6.29)

j A j k Ak 1 − ρks Ak A j The resistances given in equation (6.29) may be simplified somewhat if surface Ak is either a purely diffuse reflector (ρsk = 0), or a purely specular reflector (1 − ρsk = k ):   1 1 1 + −1 , (6.30a) Ak diffuse : R j−k =

jA j

k Ak   1 1 1 Ak specular : R j−k = + −1 . (6.30b)

j k Aj Following the procedure of Section 5.5, equation (5.47) still holds, i.e., Q=

Ebi − Ebo . 5N−1 Ri−1i + n=1 Rno−n+1,i + RNo−o

(6.31)

In most applications shields are very closely spaced, and the influence of specularity is usually very minor. To show this is left as an exercise (Problem 6.20).

General Observations Regarding Specular Analysis It is worthwhile to understand under what circumstances the assumption of a partly diffuse, partly specular reflector is appropriate. The analysis for such surfaces is generally considerably more involved than for diffusely reflecting surfaces, as a result of the more difficult evaluation of specular view factors. On the other hand, the analysis is substantially less involved than for surfaces with more irregular reflection behavior (as will be discussed later in this chapter). Example 6.3 showed that for infinitely large parallel plates the nature of reflectance has no influence on the heat transfer rates. In general, it may be stated that, in fully closed configurations (without external irradiation), the heat fluxes show very little dependence on specularity. Showing that is left as an exercise (i.e., several problems at the end of the chapter, e.g., Problems 6.5, 6.6, 6.7, and others). This is true for all closed configurations as long as there are no long and narrow channels separating surfaces of widely different temperatures (cf. Problems 6.2 and 6.3). Therefore, for most practical enclosures it should be sufficient to evaluate heat fluxes assuming purely diffuse reflectors—even though a number of surfaces may be decidedly specular. On the other hand, in open configurations (e.g., Example 6.5), in long and narrow channels, in configurations with collimated irradiation—whenever there is a possibility of beam channeling—the influence of specularity can be very substantial and must be accounted for. Also, it is tempting to think of diffuse and specular reflection as not only extreme but also limiting cases: This leads to the thought that—if heat fluxes have been determined for purely diffuse reflection, and again for purely specular reflection—the heat flux for a surface with more irregular reflection behavior must always lie between these two limiting values. This consideration is true in most cases, in particular since most real surfaces tend to have a reflectance maximum near the specular direction. However, there are cases when the actual heat flux is not bracketed by the diffuse and specular reflection models, particularly for directionally selective surfaces. As an example consider the local radiative heat flux from an isothermal groove, such as the one investigated by Toor [20], who analyzed such grooves for diffuse reflectors, for specular reflectors, and for three different types of surface roughnesses analyzed with the Monte Carlo method, and his results are shown in Fig. 6.8. It is quite apparent that, near the vertex of the groove, diffuse and specular reflectors both seriously overpredict the heat loss. The reason is that, at grazing angles, rough surfaces tend to reflect strongly back into the direction of incidence.

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 211

FIGURE 6.8 Local radiative heat flux from the surface of an isothermal V-groove for different reflection behavior; for all surfaces 2γ = 90◦ and = 0.1; σo is root-mean-square optical roughness and a is a measure [19] for average distance between roughness peaks [20].

6.4 Semitransparent Sheets (Windows) When we developed the governing relations for radiative heat transfer in an enclosure bounded by diffusely reflecting surfaces (Chapter 5) or by partially diffuse/partially specular reflectors (this chapter), we made allowance for external radiation to penetrate into the enclosure through holes and/or semitransparent surfaces (windows). While we have investigated some examples with external radiation entering through holes, only one (Example 5.8) has dealt with a simple semitransparent surface. Radiative heat transfer in enclosures with semitransparent windows occurs in a number of important applications, such as solar collectors, externally irradiated specimens kept in a controlled atmosphere, furnaces with sight windows, and so on. We shall briefly outline in this section how such enclosures may be analyzed with equation (6.16) or (6.20). To this purpose we shall assume that properties of the semitransparent window are wavelength-independent (gray), that equation (6.1) describes the reflectance (facing the inside of the enclosure), and that the transmittance of the window also has specular (light is transmitted without change of direction) and diffuse (light leaving the window is perfectly diffuse) components.3 Thus, ρ + τ + α = ρs + ρ d + τs + τ d + α = 1,

= α.

(6.32)

Further, we shall assume that radiation hitting the outside of the window has a collimated component qoc (i.e., parallel rays coming from a single direction, such as sunshine) and a diffuse component qod (such as sky radiation coming in from all directions with equal intensity). Making an energy balance for the net radiative heat flux from the semitransparent window into the enclosure leads to (cf. Fig. 6.9): q(r) = qem + qtr,in − qabs − qtr,out = (r)Eb (r) + τ d (r)qoc (r) + τ(r)qod (r) − α(r)H(r) − τ(r)H(r),

(6.33)

where the specularly transmitted fraction of the collimated external radiation, τs qoc , has not been accounted for since it enters the enclosure in a nondiffuse fashion; it is accounted for in Hos (r ) as part of the irradiation at another enclosure location r (traveling there directly, or after any number of specular reflections). Using equation (6.32), equation (6.33) may also be written as   q(r) = qout − qin = Eb + τ d qoc + τqod + ρ d H + ρs H − H,

(6.34)

3. It is unlikely that a realistic window has both specular and diffuse transmittance components; rather its transmittance will either be specular (clear windows) or diffuse (milky windows, glass blocks, etc.). We simply use the more general expression to make it valid for all types of windows.

212 Radiative Heat Transfer

FIGURE 6.9 Energy balance for a semitransparent window.

where qin is the energy falling onto the inside of the window coming from within the enclosure. The first four terms of qout are diffuse and may be combined to form the radiosity J(r) = Eb + τ d qoc + τqod + ρ d H.

(6.35)

Examination of equations (6.32) through (6.35) shows that they may be reduced to equations (6.8) through (6.10) if we introduce an apparent emittance a and an apparent blackbody emissive power Eb,a as

a (r) = + τ = 1 − ρ,

(6.36a)

a Eb,a (r) = Eb + τ qoc + τqod .

(6.36b)

d

Thus, the semitransparent window is equivalent to an opaque surface with apparent emittance a and apparent emissive power Eb,a (if the radiative properties are gray). Therefore, equations (6.16) and (6.20) remain valid as long as the emittance and blackbody emissive powers of semitransparent surfaces are understood to be apparent values. Example 6.6. A long hallway 3 m wide by 4 m high is lighted with a skylight that covers the entire ceiling. The skylight is double-glazed with an optical thickness of κd = 0.037 per window plate. The floor and sides of the hallway may be assumed to be gray and diffuse with = 0.2. The outside of the skylight is exposed to a clear sky, so that diffuse visible light in the amount of qsky = 20,000 lm/m2 is incident on the skylight. Direct sunshine also falls on the skylight in the amount of qsun = 80,000 lm/m2 (normal to the rays). For simplicity assume that the sun angle is θs = 36.87◦ as indicated in Fig. 6.10. Determine the amount of light incident on a point in the lower right-hand corner (also indicated in the figure) if (a) the skylight is clear and (b) the skylight is diffusing (with the same transmittance and reflectance). Solution From Fig. 3.33 for double glazing and κd = 0.037 we find a hemispherical transmittance (i.e., directionally averaged) of τ 0.70, while for solar incidence with θ = 36.87◦ we have τθ 0.75. The hemispherical reflectance of the skylight may be estimated by assuming that the reflectance is the same as the one of a nonabsorbing glass. Then, from Fig. 3.32 ρ1 = ρs1 = 1 − τ(κd = 0) 1 − 0.75 = 0.25. From equation (6.36) we find 1,a = 1 − ρ1 = 0.75 and, for a clear skylight,

1,a Eb1,a = 0 + 0 + τqsky since τ d = 0, and since there is no luminous emission from the window (or from any of the other walls, for that matter). Because of the special sun angle, direct sunshine falls only onto surface A2 , filling the entire wall, s = τθ qsun sin θs . i.e., Ho2 To determine the illumination at the point in the corner, we need to calculate the local irradiation H (in terms of lumens). This calculation, in turn, requires knowledge of the radiosity for all the surfaces of the hallway (for the skylight it is already known as J1 = 1,a Eb1,a = τqsky , since ρ1d = 0). To this purpose we shall approximate the hallway as a four-surface enclosure for which we shall calculate the average radiosities. Based on these radiosities we may then calculate the local irradiation for a point from equation (6.13). While equation (6.20) is most suitable for heat transfer calculations, we shall

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 213

FIGURE 6.10 Geometry for a skylit hallway (Example 6.6).

use equation (6.17) for this example since radiosities are more useful in lighting calculations.4 Therefore, for i = 2, 3, and 4,   s s s s s + J2 F2−2 + J3 F2−3 + J4 F2−4 , J2 = ρ2 J1 F2−1 + Ho2   s s s s + J2 F3−2 + J3 F3−3 + J4 F3−4 J3 = ρ3 J1 F3−1 ,   s s s s J4 = ρ4 J1 F4−1 + J2 F4−2 + J3 F4−3 + J4 F4−4 . The necessary view factors are readily calculated from the crossed-strings method: s = F2−1 = F2−1 s F2−3

3+4−5 = 0.25, 2×4

= F2−3 + ρ1 F2(1)−3

s F2−4 = F2−4 + ρ1 F2(1)−4

s F2−2 = 0,

√ 8+5−(4+ 73) = 0.25(1+0.05700) = 0.26425, = 0.25 + 0.25 × 2×4 √ 3+ 73−2 × 5 = 0.5 + 0.25 × 0.19300 = 0.54825, = 0.5 + 0.25 × 2×4

2×5 − 2×4 = 0.33333, 2×3 A2 s 4 = F = × 0.26425 = 0.35233, A3 2−3 3 √ 2× 73−2×8 = 0.25 × 0.18133 = 0.04533, = ρ1 F3(1)−3 = 0.25 × 2×3 s = F3−2 = 0.35233,

s F3−1 = F3−1 = s F3−2 s F3−3 s F3−4

s s F4−1 = F2−1 = 0.2500, s s F4−3 = F2−3 = 0.26425,

s s F4−2 = F2−4 = 0.54825, s F4−4 = 0.

s /J1 , and with ρ2 = ρ3 = ρ4 = 1 − 0.2 = 0.8, Therefore, after normalization with Ji = Ji /J1 and H = Ho2

J2 = 0.8(0.25 + 0 + 0.26425 J3 + 0.54825 J4 ) + H, J3 = 0.8(0.33333 + 0.35233 J2 + 0.04533 J3 + 0.35233 J4 ), J4 = 0.8(0.25 + 0.54825 J2 + 0.26425 J3 + 0), 4. If equation (6.20) is used the resulting heat fluxes are converted to radiosities using equation (6.11), or J = −ρ d q/ (since Eb = 0).

214 Radiative Heat Transfer

or

J2 − 0.21140 J3 − 0.43860 J4 = H + 0.2,

−0.28186 J2 + 0.96374 J3 − 0.28186 J4 = 0.26667, −0.43860 J2 − 0.21140 J3 + J4 = 0.2.

Omitting the details of solving these three simultaneous equations, we find

J2 = 1.48978H + 0.59051, J3 = 0.66812H + 0.62211, J4 = 0.79466H + 0.59051. The irradiation onto the corner point is, from equation (6.13), Hp =

4 

  s s s s s , Jj Fp−j = J1 Fp−1 + J2 Fp−2 + J3 Fp−3 + J4 Fp−4

j=1

where the view factors may be determined from Configurations 10 and 11 in Appendix D (with b → ∞, and multiplying by 2 since the strip tends to infinity in both directions): a 1 1 3 = × = 0.3, √ 2 a2 + c2 2 5   = Fp−2 + ρ1 Fp(1)−2 = Fp−2 + ρ1 Fp(1)−2+2(1) − Fp(1)−2(1) ,     3 1 1 c 1− = 0.2, Fp−2 = = 1− √ 2 2 5 a2 + c2   3 1 Fp(1)−2(1) = Fp−2 = 0.2, Fp(1)−2+2(1) = = 0.32444, 1− √ 2 73

s = Fp−1 = Fp−1 s Fp−2

s Fp−2 = 0.2 + 0.25 × (0.32444 − 0.2) = 0.23111, s = ρ1 Fp(1)−3 = 0.25 × Fp−3

3 1 × √ = 0.04389, 2 73

s = 0.5. Fp−4

Therefore,

Hp =

Hp = 0.3+0.23111×(1.48978H +0.59051) + 0.04389×(0.66812H +0.62211)+0.5×(0.79466H +0.59051) J1 = 0.77096H + 0.75903.

s Finally, for a clear window, J1 = τ1 qsky = 0.7 × 20,000 = 14,000 lx, and Ho2 = τθ qsun sin 36.87◦ = 0.75 × 80,000 × 0.6 = 36,000 lx, and

Hp = 0.77096 × 36,000 + 0.75903 × 14,000 = 38,381 lx. s On the other hand, if the window has a diffusing transmittance τ = τ d = 0.7, then Ho2 = 0 and, from equation (6.35), J1 = τ(qsky + qsun cos 36.87◦ ) = 0.7 × (20,000 + 80,000 × 0.8) = 58,800 lx. This results in

Hp = 0.75903 × 58,800 = 44,631 lx. For a diffusing window the light is more evenly distributed throughout the hallway, resulting in higher illumination at point p.

6.5 Radiative Exchange Between Nongray Surfaces We noted in Chapter 3 that surface properties may vary considerably across the spectrum (cf., for example, Figs. 3.12 through 3.15). Indeed, some surfaces are specifically engineered to obtain certain spectral behavior (selective surfaces, e.g., Fig. 3.34). In addition, irradiation upon a surface may come from sources with varying spectral ranges (see, for example, Fig. 5.5). Clearly, there are important applications where the nongray behavior

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 215

of surfaces must be accounted for. In this section we shall consider radiative exchange between nongray surfaces that are directionally ideal: Their absorptances and emittances are independent of direction, while their reflectance is idealized to consist of purely diffuse and/or specular components. We will first consider the case of purely diffuse reflectors. For such a situation equation (5.37) becomes, on a spectral basis, N   δi j j=1

  N    1 δi j − Fi−j Ebλ j − Hoλi , − − 1 Fi− j qλ j =

λ j

λ j 

i = 1, 2, . . . , N.

(6.37)

j=1

Diffuse view factors are purely geometric quantities and, therefore, never depend on wavelength. In principle, equation (6.37) may be solved for all the unknown qλ j and/or Ebλ j . This operation is followed by integrating the results over the entire spectrum, leading to  ∞  ∞ qj = qλ j dλ, Ebj = Ebλ j dλ. (6.38) 0

0

In matrix form this may be written, similar to equation (5.38), as Cλ · qλ = Aλ · ebλ − hoλ ,

(6.39)

where Aλ , ebλ , Cλ , qλ , and hoλ are defined as in Chapter 5, but on a spectral basis. Assuming that all the q j are unknown (and all temperatures are known), equation (6.39) may be solved and integrated as  ∞  ∞ q= qλ dλ = C−1 (6.40) λ · [Aλ · ebλ − hoλ ] dλ. 0

0

A similar expression may be found if the heat flux is specified over some of the surfaces (with temperatures unknown). Branstetter [21] carried out integration of equation (6.40) for two infinite, parallel plates with platinum surfaces. In practice, accurate numerical evaluation of equation (6.40) is considered too complicated for most applications: For every wavelength used in the numerical integration (or quadrature) the matrix C needs to be inverted, which—for large numbers of nodes—is generally done by iteration. Therefore, nongray effects are usually addressed by simplified models such as the semigray approximation or the band approximation.

Semigray Approximation In some applications there is a natural division of the radiative energy within an enclosure into two or more distinct spectral regions. For example, in a solar collector the incoming energy comes from a high-temperature source with most of its energy below 3 μm, while radiation losses for typical collector temperatures are at wavelengths above 3 μm. In the case of laser heating and processing the incoming energy is monochromatic (at the laser wavelength), while reradiation takes place over the entire near- to midinfrared (depending on the workpiece temperature), etc. In such a situation equation (6.37) may be split into two sets of N equations each, one set for each spectral range, and with different radiative properties for each set. For example, consider an enclosure subject to external irradiation, which is confined to a certain spectral range “(1)”. The surfaces in the enclosure, owing to their temperature, emit over spectral range “(2)”.5 Then from equation (6.37), ⎡ ⎤ ⎛ ⎞ N ⎢  ⎥⎥ ⎟⎟ ⎢⎢ δi j ⎜⎜⎜ 1 ⎢⎢ (1) − ⎜⎜ (1) − 1⎟⎟⎟ Fi−j ⎥⎥⎥ q(1) (6.41a) ⎣

⎦ j = −Hoi , ⎝

⎠ j=1

j

j

⎡ ⎤ ⎛ ⎞ N ⎢ N     ⎥⎥ ⎟⎟ ⎢⎢ δi j ⎜⎜⎜ 1 ⎢⎢ (2) − ⎜⎜ (2) − 1⎟⎟⎟ Fi−j ⎥⎥⎥ q(2) δ Ebj , = − F i j i−j ⎣

⎦ j ⎝

⎠ j j j=1 j=1 qi = q(1) + q(2) , i i

i = 1, 2, . . . , N,

where (1) is the average emittance for surface j over spectral interval (1), and so on. j 5. Note that spectral ranges “(1)” and “(2)” do not need to cover the entire spectrum and, indeed, they may overlap.

(6.41b) (6.41c)

216 Radiative Heat Transfer

Example 6.7. Repeat Example 5.3 for a groove coated with a diffusely reflecting, selective absorber coating, whose emittance may be idealized as

λ1 = λ2

⎧ ⎪ ⎪ ⎨1.0, λ < λc = 4 μm, =⎪ ⎪ ⎩0.1, λ > λc .

Solution In order to solve the problem with the semigray approximation, we let (1) = (1) = (1) = 1 for the solar radiation, since 2 1 (2) (2) (2) f (4 μm × 5777 K) = 99% of it is below 4 μm. Similarly, we choose 1 = 2 = = 0.1, since most of the surface emission will be at wavelengths larger than 4 μm. Thus, from equation (6.41), for range (1) q(1) = −Ho1 = −qsol cos α, 1

q(1) = −Ho2 = −qsol sin α, 2

which is particularly easy to determine since (1) = 1. For range (2) we get  1 − 1 F1−2 q(2) = Eb1 − F1−2 Eb2 = (1 − F1−2 )Eb , 2

(2)

(2)   q(2) 1 + 2(2) = −F2−1 Eb1 + Eb2 = (1 − F2−1 )Eb . − (2) − 1 F2−1 q(2) 1



q(2) 1

−



√ We observe that, since both surfaces have the same temperature and emittance, and with F1−2 = F2−1 = F = 1 − 1/2 2 = 0.2929, q(2) = q(2) = 2 1

(1 − F)Eb . 1/ (2) − (1/ (2) − 1) F

Adding both ranges together, we obtain  Q = a(q1 + q2 ) = a

 2(1 − F)σT 4 − q (cos α + sin α) . sol 1/ (2) − (1/ (2) − 1) F

Sticking values for F and (2) it follows √ Q = a(q1 + q2 ) = a

 2σT 4 − qsol (cos α + sin α) . 7.36

Therefore, the (admittedly highly ideal) selective surface ensures full collection of solar energy, while re-emission losses are reduced by more than a factor of 7 compared to the black case of Example 5.3.

Selective surfaces can have enormous impact on radiative heat fluxes in configurations with irradiation from high-temperature sources. Subroutine semigraydf is provided in Appendix F for the solution of the simultaneous equations (6.41), requiring surface information and a partial view factor matrix as input (i.e., the code is limited to two spectral ranges, separating external irradiation from surface emission). The solution to Example 6.7 is also given in the form of program semigrxchdf, which may be used as a starting point for the solution to other problems. Fortran90, C++ as well as Matlab versions are provided. The semigray approximation is not limited to two distinct spectral regions. Each surface of the enclosure may be given a set of absorptances and reflectances, one value for each different surface temperature (with its different emission spectra). Armaly and Tien [22] have indicated how such absorptances may be determined. However, while simple and straightforward, the method can never become “exact,” no matter how many different values of absorptance and reflectance are chosen for each surface. Bobco and coworkers [23] have given a general discussion of the semigray approximation. The method has been applied to solar irradiation falling into a V-groove cavity with a spectrally selective, diffusely reflecting surface by Plamondon and Landram [24]. Comparison with exact (i.e., spectrally integrated) results proved the method to be very accurate. Shimoji [25] used the semigray approximation to model solar irradiation onto conical and V-groove cavities whose reflectances had purely diffuse and specular components.

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 217

Band Approximation Another commonly used method of solving equation (6.37) is the band approximation. In this method the spectrum is broken up into M bands, over which the radiative properties of all surfaces in the enclosure are approximated as constant. Therefore, ⎛ ⎞ ⎡ ⎤ N ⎢ N    ⎜⎜ 1 ⎟⎟ ⎥⎥  (m) ⎢⎢ δi j ⎥ (m) ⎜ ⎟ (m) δi j − Fi−j Ebj − Hoi , i = 1, 2, . . . , N, m = 1, 2, . . . , M; (6.42a) ⎢⎢⎢ (m) − ⎜⎜⎜ (m) − 1⎟⎟⎟ Fi− j ⎥⎥⎥ q j = ⎣

⎦ ⎝

⎠ j=1

j

j=1

j

Ebj =

M 

(m) Ebj ,

qj =

m=1

M 

(m) qj ,

Hoi =

m=1

M 

(m)

Hoi .

(6.42b)

m=1

Equation (6.42) is, of course, nothing but a simple numerical integration of equation (6.37), using the trapezoidal rule with varying steps. This method has the advantage that the widths of the bands can be tailored to the spectral variation of properties, resulting in good accuracy with relatively few bands. For very few bands the accuracy of this method is similar to that of the semigray approximation, but is a little more cumbersome to apply, and requires an iterative approach if some surfaces have prescribed radiative flux rather than temperature. On the other hand, the band approximation can achieve any desired accuracy by using many bands even for surfaces displaying extremely nongray characteristics. Example 6.8. Repeat Example 6.7 using the band approximation. Solution Since the emittances in this example have been idealized to have constant values across the spectrum with the exception of a step at λ = 4 μm, a two-band approximation (λ < λc = 4 μm and λ > 4 μm) will produce the “exact” solution (within the framework of the net radiation method). From equation (6.42) q(m) 1

(m) −



−

1

(m)



1

(m)

 (m) − 1 F1−2 q(m) = E(m) − F1−2 E(m) − Ho1 , 2 b b

 q(m) (m) 2 − 1 F2−1 q(m) + = −F2−1 E(m) +E(m) − Ho2 , 1 b b

(m)

m = 1, 2,

These are two sets of two equations for the two sets of unknowns q(m) and q(m) , which are readily solved (setting 2 1 F1−2 = F2−1 = F) as         (m) (m) 1/ (m) + 1/ (m) − 1 F − 1/ (m) H01 + 1/ (m) − 1 FH02 (1 − F)E(m) b (m) q1 = (1/ (m) )2 + (1/ (m) − 1)2 F2         (m) (m) 1/ (m) + 1/ (m) − 1 F − 1/ (m) H02 + 1/ (m) − 1 FH01 (1 − F)E(m) b (m) q2 = (1/ (m) )2 + (1/ (m) − 1)2 F2  λc   where E(1) = 0 Ebλ dλ = f (λc T)Eb , E(2) = 1 − f (λc T) Eb , b b (1) Ho1 = f (λc Tsun )qsol cos α = 0.99qsol cos α,

(2) Ho1 = 0.01qsol cos α,

(1) = 0.99qsol sin α, Ho2

(2) Ho2 = 0.01qsol sin α.

For range (1), with (1) = 1, this simplifies greatly to (1) = (1 − F)E(1) − H01 , q(1) 1 b

(1) q(1) = (1 − F)E(1) − H02 . 2 b

To compare results from the two analyses we will look at the special case of T = 400 K, qsol = 1000W/m2 , and α = 45◦ (leading to q1 = q2 ). Sticking in the numbers leads to W W − 707.1 2 m2 m W 1026.4 W − 707.1 2 = 2 7.36 m m

q = q1 = q2 = 1026.4

W m2 W = −567.7 2 m W = −544.3 2 m = +319.3

black groove semigray band appr.

218 Radiative Heat Transfer

(where we left out the details for the band approximation). As expected, for the present example the band approximation offers little improvement while complicating the analysis. The semigray approximation is off by some 23 W/m2 , primarily because emission below 4 μm is neglected (just under 2% or 29 W/m2 ). However, the band approximation is the method of choice if no distinct spectral regions are obvious and/or the spectral behavior of properties is more involved.

Subroutine bandappdf is provided in Appendix F for the solution of the simultaneous equations (6.42), requiring surface information and a partial view factor matrix as input. The solution to Example 6.8 is also given in the form of a program bandmxchdf, which may be used as a starting point for the solution to other problems. Fortran90, C++ as well as Matlab versions are provided. Dunkle and Bevans [26] applied the band approximation to the same problem as Branstetter [21] (infinite, parallel, tungsten plates) as well as to some other configurations, showing that the band approximation generally achieves accuracies of 2% and better with very few bands, while a gray analysis may result in errors of 30% or more. One further advantage of the band approximation is that it is easily paired with a number of part-spectrum models for participating media, as described in Chapter 19.

Inclusion of Specular Reflectors Both the semigray and band approximations are also readily applied in the presence of partly specular surfaces. Starting with equation (6.20) this leads to N  

δi j −

s (1−ρλs j )Fλ,i− j



Ebλ j

j=1

⎛ ⎞ d N ⎜  ⎜⎜ δi j ρλ j s ⎟⎟⎟ s ⎜⎜ ⎟ = ⎝ λ j − λ j Fλ,i− j ⎟⎠ qλ j + Hoλi ,

i = 1, 2, . . . , N.

(6.43)

j=1

This equation now contains specular view factors, which may depend on wavelength through the spectral dependence of specular reflectances ρλs . For the semigray approximation equation (6.41) is then replaced by Semigray Approximation ⎡ ⎤ N ⎢  ⎥ ρ dj (1) ⎢⎢⎢ δi j ⎥ s(1) ⎥ = −Hois , ⎢⎢ (1) − (1) Fi−j ⎥⎥⎥ q(1) ⎣

⎦ j

j j j=1 ⎡ ⎤ d (2) N ⎢ N    ⎥⎥  ⎢⎢ δi j ρ j ⎥ s (2) s(2) = δi j − (1−ρ sj (2) )Fi−j Ebj , ⎢⎢⎢ (2) − (2) Fi−j ⎥⎥⎥ q(2) j ⎣

⎦

j=1

j

j

(6.44a)

(6.44b)

j=1

qi = q(1) + q(2) , i i

i = 1, 2, . . . , N,

(6.44c)

while equation (6.42) for the band approximation is extended to Band Approximation N  

s(m)

δi j − (1 − ρ j

 s(m)

)Fi− j

j=1

(m)

Ebj

⎡ ⎤ d(m) N ⎢  ⎥ ρj ⎢⎢⎢ δi j ⎥ (m) s(m) ⎥ s(m) = ⎢⎢ (m) − (m) Fi− j ⎥⎥⎥ q j + Hoi , ⎣

⎦

j=1 j j i = 1, 2, . . . , N,

Ebj =

M  m=1

(m)

Ebj ,

qj =

M  m=1

(m)

qj ,

Hois =

m = 1, 2, . . . , M; M 

s(m)

Hoi .

(6.45a) (6.45b)

m=1

Example 6.9. Repeat Example 6.7, but let the material be specularly reflecting for λ > 4 μm (i.e., ρ d(2) = 0, ρ s(2) = 1 − s(2) = 0.9). Solution The solution proceeds as for the diffuse case and, since range (1) is black, the heat fluxes for that range are unaffected, i.e., = −qsol cos α, q(1) 1

q(1) = −qsol sin α. 2

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 219

Even though over range (2) the material is a specular reflector, the view factors are unaffected (specularly reflected emission s(2) = F1−2 = F2−1 = F. Also, since the material has no diffusely reflecting component, the bounces out of the groove), F1−2 reflection terms in equations (6.44) disappear, and we obtain = q(2) = (1 − F) (2) Eb . q(2) 2 1 It follows then that

  Q = a (2) 2(1 − F)σT 4 − qsol (cos α + sin α)  √  = a 0.1 2σT 4 − qsol (cos α + sin α) .

For T = 400 K, qsol = 1000 W/m2 , and α = 45◦ this results in q1 = q2 = 0.1 × 1026.4 − 707.1 = −604.5 W/m2 . The collected solar flux is seen to increase a little because none of the surface emission returns to itself after hitting the other surface; all is specularly reflected out of the cavity.

Semigray and band approximation computer codes for the case of specular reflectors are also provided in Appendix F. Subroutine semigray is provided for the solution of the simultaneous equations (6.44), and the solution to Example 6.9 may be found with program semigrxch. Similarly, subroutine bandapp and program bandmxch, all in Fortran90, C++ as well as Matlab versions, are provided.

6.6 Directionally Nonideal Surfaces In the vast majority of applications the assumption of “directionally ideal” surfaces gives results of sufficient accuracy, i.e., surfaces may be assumed to be diffusely emitting and absorbing and to be diffusely and/or specularly reflecting (with the magnitude of reflectance independent of incoming direction). However, that these results are not always accurate and that heat fluxes are not necessarily bracketed by the diffuse- and specular-reflection cases have been shown in Fig. 6.8 for V-grooves. There will be situations where (i) the directional properties, (ii) the geometrical considerations, and/or (iii) the accuracy requirements are such that the directional behavior of radiation properties must be addressed. If radiative properties with arbitrary directional behavior are to be accounted for, it is no longer possible to reduce the governing equation to an integral equation in a single quantity (the radiosity) that is a function of surface location only (but not of direction). Rather, applying conservation of energy to this problem produces an equation governing the directional intensity leaving a surface that is a function of both location on the enclosure surface and direction.

The Governing Equation for Intensity Consider the arbitrary enclosure shown in Fig. 6.11. The spectral radiative heat flux leaving an infinitesimal surface element dA into the direction of sˆ  and arriving at surface element dA is Iλ (r , λ, sˆ  ) dAp dΩ = Iλ (r , λ, sˆ  )(dA cos θ )

dA cos θi , S2

(6.46)

where S = |r − r| is the distance between dA and dA, cos θ = sˆ  · nˆ  is the cosine of the angle between the unit direction vector sˆ  = (r − r )/S and the outward surface normal nˆ  at dA and, similarly, cos θi = (−ˆs ) · nˆ at dA. This irradiation at dA coming from dA may also be expressed, from equation (3.32), as Hλ (r, λ, sˆ  ) dA dΩ i = Iλ (r, λ, sˆ  ) dA cos θi

dA cos θ . S2

(6.47)

Equating these two expressions, we find Iλ (r, λ, sˆ  ) = Iλ (r , λ, sˆ  ), that is, the radiative intensity remains unchanged as it travels from dA to dA.

(6.48)

220 Radiative Heat Transfer

FIGURE 6.11 Radiative exchange in an enclosure with arbitrary surface properties.

The outgoing intensity at dA into the direction of sˆ consists of two contributions: locally emitted intensity and reflected intensity. The locally emitted intensity is, from equation (3.1),

λ (r, λ, sˆ )Ibλ (r, λ). The amount of irradiation at dA coming from dA [equation (6.47)] that is reflected into a solid angle dΩ o around the direction sˆ is, from the definition of the bidirectional reflection function, equation (3.33),   ˆ  , sˆ ) Hλ (r, λ, sˆ  ) dΩ i dΩ o , dIλ (r, λ, sˆ ) dΩ o = ρ λ (r, λ, s or ˆ  , sˆ )Iλ (r, λ, sˆ  ) cos θi dΩ i dIλ (r, λ, sˆ ) = ρ λ (r, λ, s cos θi cos θ ˆ  , sˆ )Iλ (r, λ, sˆ  ) = ρ dA . λ (r, λ, s S2 Integrating the reflected intensity over all incoming directions (or over the entire enclosure surface), and adding the locally emitted intensity, we find an expression for the outgoing intensity at dA as  ˆ  , sˆ )Iλ (r , λ, sˆ  ) cos θi dΩ i ρ Iλ (r, λ, sˆ ) = λ (r, λ, sˆ )Ibλ (r, λ) + λ (r, λ, s 2π cos θi cos θ  ˆ  , sˆ )Iλ (r , λ, sˆ  ) = λ (r, λ, sˆ )Ibλ (r, λ) + ρ dA . (6.49) λ (r, λ, s 2 S A Equation (6.49) is an integral equation for outgoing intensity (nˆ · sˆ > 0) anywhere on the surface enclosure. Once a solution to equation (6.49) has been obtained (analytically, numerically, or statistically; approximately or “exactly”), the net radiative heat flux is determined from qλ (r, λ) = qout − qin   = Iλ (r, λ, sˆ ) cos θ dΩ − Iλ (r, λ, sˆ  ) cos θi dΩ i ˆ s>0 ˆ s0 n·ˆ A

(6.50)

or, equivalently, from qλ (r, λ) = qemission − qabsorption = λ Ebλ − αλ Hλ   cos θi cos θ =

λ (r, λ, sˆ ) cos θ dΩ Ibλ (r, λ) − αλ (r, λ, sˆ  )Iλ (r , λ, sˆ  ) dA . S2 ˆ s>0 n·ˆ A

(6.51)

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 221

FIGURE 6.12 Isothermal V-groove with specularly reflecting, directionally dependent reflectance (Example 6.10).

Both forms of equation (6.49) (solid angle and area integration) may be employed, depending on the problem (r, λ, sˆ  , sˆ ) = ρλ (r, λ)/π at hand. For example, if dA is a diffuse emitter and reflector then, from equation (3.38), ρ λ  and, from equation (5.19), Iλ (r, λ, sˆ ) = Jλ (r, λ)/π. If dA is also diffuse, we obtain from the second form of equation (6.49):  Jλ (r, λ) = λ (r, λ)Ebλ (r, λ) + ρλ (r, λ) Jλ (r , λ) dFdA−dA , (6.52) A

which is nothing but the spectral form of equation (5.24) without external irradiation.6 Similarly, equation (6.50) reduces to  qλ (r, λ) = Jλ (r, λ) − Jλ (r , λ) dFdA−dA , (6.53) A

the spectral form of equation (5.25). On the other hand, if dA is a specular reflector the first form of equation (6.49) becomes more convenient: For a specular surface we have ρ = 0 for all sˆ  except for sˆ  = sˆ s , where sˆ s is the “specular direction” from which λ a beam must originate in order to travel on into the direction of sˆ after specular reflection. For that direction ρ → ∞, and it is clear that the integrand of the integral in equation (6.49) will be nonzero only in the immediate λ vicinity of sˆ  = sˆ s . In that vicinity Iλ (r , λ, sˆ  ) varies very little and we may remove it from the integral. From the definition of the spectral, directional–hemispherical reflectance, equation (3.37), and the law of reciprocity for the bidirectional reflectance function, equation (3.35), we obtain       ˆ  , sˆ ) cos θi dΩ i ρλ (r, λ, sˆ , sˆ )Iλ (r, λ, sˆ ) cos θi dΩ i = Iλ (r , λ, sˆ s ) ρ λ (r, λ, s 2π 2π   = Iλ (r , λ, sˆ s ) ρ s, −ˆs ) cos θi dΩ i λ (r, λ, −ˆ 2π

= Iλ (r , λ, sˆ s )ρλ (r, λ, −ˆs), where −ˆs denotes an incoming direction, pointing toward dA, and ρλ (r, λ, −ˆs) is the directional–hemispherical reflectance. From the same Kirchhoff’s law used to establish equation (3.35), it follows that ρλ (r, λ, −ˆs) = ρλ (r, λ, sˆ s ) and Iλ (r, λ, sˆ ) = λ (r, λ, sˆ )Ibλ (r, λ) + ρλ (r, λ, sˆ s )Iλ (r , λ, sˆ s ).

6. External irradiation is readily included in equations (6.49) and (6.50) by replacing Iλ with Iλ + Ioλ inside the integrals.

(6.54)

222 Radiative Heat Transfer

Example 6.10. Consider a very long V-groove with an opening angle of 2γ = 90◦ and with optically smooth metallic surfaces with index of refraction m = n − ik = 23.452(1 − i), i.e., the surfaces are specularly reflecting and their directional dependence obeys Fresnel’s equations. The groove is isothermal at temperature T and no external irradiation is entering the configuration. Calculate the local net radiative heat loss as a function of the distance from the vertex of the groove. Solution This is one of the problems studied by Toor [20], using the Monte Carlo method (the solid line in Fig. 6.8). The directional emittance may be calculated from Fresnel’s equations for a metal, equations (3.75) and (3.76), as

 (θ) = 1 − ρ (θ) =

2n cos θ 2n cos θ + , (n + cos θ)2 + k2 (n cos θ + 1)2 + (k cos θ)2

while the hemispherical emittance follows from equation (3.77) or Fig. 3.10 as = 0.1. The present problem is particularly simple since the surfaces are specular reflectors and since the opening angle of the groove is 90◦ (cf. Fig. 6.12). Any radiation leaving surface A1 traveling toward A2 will be absorbed by A2 or reflected out of the groove; none can be reflected back to A1 . This fact implies that all radiation arriving at A1 is due to emission from A2 , which is a known quantity. Therefore, for those azimuthal angles ψ2 pointing toward A1 we have −

π π < ψ2 < : 2 2

I2 (θ2 ) =  (θ2 )Ib ,

and the local heat flux follows from equation (6.51) as 

 (θ1 )I2 (θ2 ) cos θ1 dΩ 1

q(x) = Eb − 2π

 = Eb − 2 or q(x) 2 =1−

Eb π

π/2



ψ1 =0



π/2

π/2

θ1 =θ1min (ψ1 )



ψ1 =0

π/2

θ1 =θ1min (ψ1 )

 (θ1 )  (θ2 )Ib cos θ1 sin θ1 dθ1 dψ1 ,

 (θ1 )  (θ2 ) cos θ1 sin θ1 dθ1 dψ1 .

Here the limits on the integral express the fact that the solid angle, with which A2 is seen from A1 , is limited. It remains to express θ1min as well as θ2 in terms of θ1 and ψ1 . From Fig. 6.12 it follows that cos θ1 =

y , S

cos θ2 =

x , S

S sin θ1 =

x . cos ψ1

From these three relations and the fact that the minimum value of θ1 occurs when y = L, we find cos θ2 = sin θ1 cos ψ1

and

θ1min (ψ1 ) = tan−1

x . L cos ψ1

Using Fresnel’s equation for the directional emittance, the nondimensional local heat flux q(x)/ Eb may now be calculated using numerical integration. The resulting heat flux is shown as the solid line in Fig. 6.8. This result should be compared with the simpler case of diffuse emission, or  (θ) = = 0.1 = const. For that case the integral above is readily integrated analytically, resulting in the dash-dotted line of Fig. 6.8. The two results are very close, with a maximum error of 2% near the vertex of the groove.

While the evaluation of the “exact” heat flux, using Fresnel’s equations, was quite straightforward in this very simple problem, these calculations are normally much, much more involved than the diffuse-emission approximation. Before embarking on such extensive calculations it is important to ask oneself whether employing Fresnel’s equations will lead to substantially different results for the problem at hand. Few numerical solutions of the exact integral equations have appeared in the literature. For example, Hering and Smith [27] considered the same problem as Example 6.10, but for varying opening angles and for rough surface materials (with the bidirectional reflection function as given in an earlier paper [28]). Lack of detailed knowledge of bidirectional reflection distributions, as well as the enormous complexity involved in the solution of the integral equation (6.49), makes it necessary in practice to make additional simplifying assumptions or to employ a different approach, such as the Monte Carlo method (to be discussed in Chapter 7).

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 223

Net Radiation Method It is possible to apply the net radiation method to surfaces with directionally nonideal properties, although its application is considerably more difficult and restrictive. Breaking up the enclosure into N subsurfaces we may write equation (6.49), for r pointing to a location on subsurface Ai , as 

I(r, λ, sˆ ) = (r, λ, sˆ )Ib (r, λ) + π

N 

ρj (r, λ, sˆ )I j (r, λ)Fdi− j (r),

(6.55)

j=1

where we have dropped the subscript λ for simplicity of notation, and where ρj and I j are “suitable” average values between point r and surface A j . Averaging equation (6.55) over Ai leads to Ii (λ, sˆ ) = i (λ, sˆ )Ibi (λ) + π

N 

ρji (λ, sˆ )I ji (λ)Fi−j ,

i = 1, 2, . . . , N.

(6.56)

j=1

Here I ji is an average value of the intensity leaving surface A j traveling toward Ai , and ρji is a corresponding value for the bidirectional reflection function. If we assume that the enclosure temperature and surface properties are known everywhere, then equation (6.56) has N unknown intensities I ji (j = 1, 2, . . . , N) for each subsurface Ai . Thus, if equation (6.56) is averaged over all the solid angles with which subsurface Ak is seen from Ai , it becomes a set of N × N equations in the N2 unknown Iik : Iik (λ) = ik (λ)Ibi (λ) + π

N 

ρ jik (λ)I ji (λ)Fi−j ,

i, k = 1, 2, . . . , N.

(6.57)

j=1

Here ρ jik is an average value of the bidirectional reflection function for radiation traveling from A j to Ak via reflection at Ai . For a diffusely emitting, absorbing, and reflecting enclosure we have ik = i , πρ jik = ρi , and equation (6.57) becomes, with I ji = I j = Jj /π, Ji = i Ebi + ρi

N 

Jj Fi−j ,

i = 1, 2, . . . , N,

(6.58)

j=1

which is identical to equations (5.30) and (5.31) (without external irradiation). If the N subsurfaces are relatively small (as compared with the distance-squared between them), average properties ik and ρ jik may be obtained simply by evaluating  and ρ at the directions given by connecting the centerpoints of surface Ai with A j and Ak . For larger subsurfaces a more elaborate averaging may be desirable. A discussion on that subject has been given by Bevans and Edwards [29]. Once the N2 unknown Iik have been determined, the average heat flux on Ai may be calculated from equations (6.57) and (6.50) or (6.51) as qi (λ) = π

N 

Iik (λ)Fi−k − π

I ji (λ)Fi−j = π

j=1

k=1

= i (λ)Ebi (λ) − π

N 

N 

αi j (λ)I ji (λ)Fi−j ,

N 

(Ii j −I ji )Fi−j

(6.59a)

j=1

i = 1, 2, . . . , N,

(6.59b)

j=1

where i is the hemispherical emittance of Ai and αi j is the average absorptance of subsurface Ai for radiation coming from A j . It is apparent from equations (6.49) and (6.57) that the net radiation method for directionally nonideal surfaces is valid (i) if each Ibi varies little over each subsurface Ai , (ii) if each Iik varies little between any two positions on Ai and Ak , and (iii) if similar restrictions apply to ik , αi j , and ρ jik . Restrictions (ii) and (iii) are likely to be easily violated unless the surfaces are near-diffuse reflectors or are very small (as compared with the distance between them).

224 Radiative Heat Transfer

FIGURE 6.13 (a) Geometry for Example 6.11, (b) bidirectional reflection function in plane of incidence for θi = 0◦ and θi = 45◦ , for the material of Example 6.11.

Equations (6.49) and (6.57) are valid for an enclosure with gray surface properties, or on a spectral basis. For nongray surface properties the governing equations are readily integrated over the spectrum using the methods outlined in the previous section. To illustrate the difficulties associated with directionally nonideal surfaces, we shall consider one particularly simple example. Example 6.11. Consider the isothermal corner of finite length as depicted in Fig. 6.13a. The surface material is similar to the one of the infinitely long corner of the previous example, i.e., the absorptance and emittance obey Fresnel’s equations with m = n − ik = 23.452(1 − i), and a hemispherical emittance of = 0.1. However, in the present example we assume that the material is reflecting in a nonspecular fashion with a bidirectional reflection function of ρ (ˆs i , sˆ r ) =

ρ (ˆs i ) (1 + sˆ s · sˆ r )n , πCn (ˆs i )

where sˆ i is the direction of incoming radiation, sˆ s is the specular reflection direction (i.e., θs = θi , ψs = ψi + π), and sˆ r is the actual direction of reflection. This form of the bidirectional reflection function describes a surface that has a reflectance maximum in the specular direction, and whose reflectance drops off equally in all directions away from the specular direction (i.e., with changing polar angle and/or azimuthal angle). Since the directional–hemispherical reflectance must obey ρ (ˆs i ) = 1 −  (ˆs i ), the function Cn (ˆs i ) follows from equation (3.37) as  1 (1 + sˆ s · sˆ r )n cos θr dΩ r . Cn (ˆs i ) = π 2π Determine the local radiative heat loss rates from the plates for the case that both plates are isothermal at the same temperature. Solution The direction vectors sˆ may be expressed in terms of polar angle θ and azimuthal angle ψ, or sˆ = sin θ(cos ψˆt1 + sin ψˆt2 ) + ˆ where nˆ is the unit surface normal and ˆt1 and ˆt2 are two perpendicular unit vectors tangential to the surface. cos θn, Therefore, the bidirectional reflection function may be written as ρ (θi , ψi )  n 1 + cos θi cos θr − sin θi sin θr cos(ψi − ψr ) , πCn (θi )  2π  π/2 n  1 Cn (θi ) = 1 + cos θi cos θ + sin θi sin θ cos ψ cos θ sin θ dθ dψ. π 0 0

ρ (θi , ψi , θr , ψr ) =

(6.60a) (6.60b)

The bidirectional reflection function within the plane of incidence (ψr = ψi or ψi + π) is shown in Fig. 6.13b for two different incidence directions and three different values of n. Obviously, for n = 0 the surface reflects diffusely (but the amount of reflection, as well as absorption and emission, depends on direction through Fresnel’s equation). As n grows,

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 225

the surface becomes more specular, and purely specular reflection would be reached with n → ∞. For this configuration and surface material we would like to determine the heat lost from the plates using the net radiation method. As indicated in Fig. 6.13a we shall apply the net radiation method, equations (6.57) and (6.59), by breaking up each surface into M × N subsurfaces (M divisions in the x- and y-directions, N in the z-direction). Considering the intensity at node (i, k) on the bottom surface directed toward node ( jo , ko ) on the vertical wall, we find that equation (6.57) becomes, after division by Ib , Φi,k→jo ,ko =

M  N  Ii,k→jo ,ko = i,k→jo ,ko + πρ ji ,ki →i,k→ jo ,ko Fi,k→ji ,ki Φ ji ,ki →i,k . Ib j =1 i

(6.61)

ki =1

In this relation we have made use of the fact that a node on the bottom surface can only see nodes on the side wall and vice versa. Also, by symmetry we have Φi,k→jo ,ko = Φ j,k→io ,ko

if

j=i

and io = jo ,

and Φi,k→jo ,ko = Φi,N+1−k→ jo ,N+1−ko , that is, the intensity must be symmetric to the two planes x = y and z = L/2. We, therefore, have a total of M × (N/2) unknowns (assuming N to be even) and need to apply equation (6.61) for i = 1, 2, . . . , M and k = 1, 2, . . . , N/2. To calculate the necessary  and ρ values, one must establish a number of polar and azimuthal angles. From Fig. 6.13a it follows that y ji , (cos θi ) i,k→ji ,ki = ( 2 2 xi + y j + (zk − zki )2 i

y jo (cos θr ) i,k→jo ,ko = ( . 2 2 xi + y jo + (zk − zko )2 Using the values for (cos θr ) i,k→jo ,ko one can readily calculate the directional emittances i,k→ jo ,ko = 1 − ρ (cos θr ) from Fresnel’s equation as given in Example 6.10. Similarly, ρ (cos θi ) and Cn (cos θi ) are determined from Fresnel’s equation and equation (6.60),7 respectively; and all values of ρ ji ,ki →i,k→jo ,ko follow from equation (6.60). All necessary view factors may be calculated from equation (4.41), for arbitrarily oriented perpendicular plates. For all view factors the opposing surfaces are of identical and constant size with x2 − x1 = y2 − y1 = w/M and z1 = z3 − z2 = L/N. Offsets x1 and y1 may vary between 0 and (M − 1)w/M and z2 between 0 and (N − 1)L/N. Thus, using symmetry and reciprocity, one must evaluate a total of (M/2) × M × N view factors. In many of today’s workstations and computers all different values of directional emittance, the factor ρ /Cn in the bidirectional reflection function, and all view factors may be calculated— once and for all—and stored (requiring memory allocation for often millions of numbers). The bidirectional reflection function itself depends on surface locations and on all possible incoming as well as all possible outgoing directions. Even after employing symmetry and reciprocity (for the bidirectional reflection function), this would require storing [M × (N/2)] × [M × N]2 /2 = (MN)3 /4 numbers. Unless relatively few subdivisions are used (say M, N < 10), it will be impossible to precalculate and store values of the bidirectional reflection function; rather, part of it must be recalculated every time it is required. The nondimensional intensities are now easily found from equation (6.61) by successive approximation: A first guess for the intensity field is made by setting Φi,k→jo ,ko = i,k→jo ,ko . Improved values for Φi,k→jo ,ko are found by evaluating equation (6.61) again and again until the intensities have converged to within specified error bounds. The local net radiative heat flux may then be determined from equation (6.59b) as Ψi,k =

M N qi,k 1  =1−

i,k→ji ,ki Fi,k→ji ,ki Φ ji ,ki →i,k .

Eb

j =1 i

ki =1

Some representative results for the local radiative heat flux near z = L/2 (i.e., for k = N/2) are shown in Fig. 6.14 for the case of w = L (square plates). Clearly, taking into consideration substantially different reflective properties has rather small effects on the local heat transfer rates. Obviously, as the surface becomes more specular (increasing n) the heat loss rates increase (since less radiation will be reflected back to the emitting surface), but the increases are very minor except for the region close to the vertex (and even there, they are less than 4%). 7. For integer values of n the integration may be carried out analytically, either by hand or on a computer using a symbolic mathematics analyzer (the latter having been used here).

226 Radiative Heat Transfer

FIGURE 6.14 Nondimensional, local heat fluxes for the corner geometry of Example 6.11, for w/L = 1. Solid symbols: Surfaces are broken up into 2 × 2 subsurfaces; open symbols: 4 × 4 subsurfaces; lines: 20 × 20 subsurfaces.

The directional distribution of the emittance is just as important as that of the bidirectional reflection function: The curve labeled “diffuse” shows the case of diffuse emission and reflection, i.e.,  (ˆs) = α (ˆs) = = 0.1 and πρ (ˆs i , sˆ r ) = ρ = 1 − = 0.9. In contrast, the curve labeled “Fresnel, n = 0” corresponds to the case of  (ˆs) = α (ˆs) = 1 − ρ (ˆs) evaluated from Fresnel’s equation and πρ(ˆsi , sˆ r ) = ρ (ˆs i ). All lines in Fig. 6.14 have been calculated by breaking up each surface into 20 × 20 subsurfaces. Also included are the data points for results obtained by breaking up each surface into only 2 × 2 (solid symbols) and 4 × 4 surfaces (open symbols). Local heat fluxes are predicted accurately with few subsurfaces, even for strongly nondiffuse reflection. Total heat loss is predicted even more accurately, with maximum errors of < 0.6% (2 × 2 subsurfaces) and < 0.3% (4 × 4 subsurfaces), respectively. The results should be compared with those of Toor [20] for w/L → 0, as shown in Fig. 6.8: The “diffuse” case of Fig. 6.14 virtually coincides with the corresponding case in Fig. 6.8, while the n = 8 case falls very close to the specular case with Fresnel-varying reflectance of Toor (solid line in Fig. 6.8).

For the present example at least, taking into account the directional behavior of emittance and reflectance is rarely justifiable in view of the additional complexity and computational effort required. Only if the radiative properties are known with great accuracy, and if heat fluxes need to be determined with similar accuracy, should this type of analysis be attempted. Similar statements may be made for most other configurations. For example, if Example 6.11 is recalculated for directly opposed parallel quadratic plates, the effects of Fresnel’s equation and the bidirectional reflection function are even less: Heat fluxes for diffuse reflection—whether Fresnel’s equation is used or not—differ by less than 0.6%, while differences due to the value of n in the bidirectional reflection function never exceed 0.2%. Only in configurations with collimated irradiation and/or strong beam-channeling possibilities should one expect substantial impact as a result of the directional variations of surface properties.

6.7 Analysis for Arbitrary Surface Characteristics The discussion in the previous two sections has demonstrated that the evaluation of radiative transfer rates in enclosures with nonideal surface properties, while relatively straightforward to formulate, is considerably more complex and time consuming. If one considers nongray surface properties, the computational effort increases roughly by a factor of M if M spectral bands (band approximation) or M sets of property values (semigray approximation) are employed. In an analysis with directional properties for an enclosure with N subsurfaces, the computational effort is increased roughly by a factor of N (an enormous increase if a substantial number of subdivisions are made). If the radiative properties are both nongray and directionally varying, the problem becomes even more difficult. While it is relatively simple to combine the methods of the previous two sections for the analysis of an enclosure with such surface properties, to the authors’ knowledge, this has not yet been done in any reported work. Few analytical solutions for such problems can be found (for the very simplest of geometries), and even standard numerical techniques may fail for nontrivial geometries; because of the

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 227

four-dimensional character, huge matrices would have to be inverted. Therefore, such calculations are normally carried out with statistical methods such as the Monte Carlo method (to be discussed in detail in Chapter 7). For example, Toor [20] has studied the radiative interchange between simply arranged flat surfaces having theoretically determined directional surface properties; Modest and Poon [30] and Modest [31] evaluated the heat rejection and solar absorption rates of the U.S. Space Shuttle’s heat rejector panels, using nongray and directional properties determined from experimental data. The validity and accuracy of several directional models have been tested and verified experimentally by Toor and Viskanta [32,33]. They studied radiative transfer among three simply arranged parallel rectangles, comparing experimental results with a simple analysis employing (i) the semigray model, (ii) Fresnel’s equation for the evaluation of directional properties, and (iii) reflectances consisting of purely diffuse and specular parts. They found good agreement with experiment and concluded that, for the gold surfaces studied, (i) directional effects are more pronounced than nongray effects and (ii) in the presence of one or more diffusely reflecting surfaces the effects of specularity of other surfaces become unimportant. Employing a combination of band approximation and the net radiation method has the disadvantage that (i) either large amounts of directional properties and/or view factors must be calculated repeatedly in the iterative solution process (making the method numerically inefficient) or (ii) large amounts of precalculated properties and/or view factors must be stored (requiring enormous amounts of computer storage). In addition, the number of view factors and property calculations scale nonlinearly with number of faces. Thus, from a computational time standpoint, it becomes increasingly difficult to use this method as the problem size gets larger. On the other hand, it avoids the statistical scatter that is always present in Monte Carlo solutions. In light of today’s rapid development in the computer field, with many small workstations and personal computers boasting internal storage capacities of several gigabytes, as well as rapidly increasing multi-processor speeds, it appears that the methods discussed in this chapter may become attractive alternatives to the Monte Carlo method.

Problems 6.1 An infinitely long, diffusely reflecting cylinder is opposite a large, infinitely long plate of semiinfinite width (in plane of paper) as shown in the adjacent sketch. The plate is specularly reflecting with ρs2 = 0.5. As the center of the cylinder moves from x = +∞ to x = −∞ plot Fs1−1 vs. position h (your plot should include at least three precise values).

6.2 Two infinitely long black plates of width D are separated by a long, narrow channel, as indicated in the adjacent sketch. One plate is isothermal at T1 and the other is isothermal at T2 . The emittance of the insulated channel wall is

. Determine the radiative heat flux between the plates if the channel wall is (a) specular and (b) diffuse. For simplicity you may treat the channel wall as a single node. The diffuse case approximates the behavior of a light guide, a device used to pipe daylight into interior, windowless spaces. 6.3 Two circular black plates of diameter D are separated by a long, narrow tubular channel, as indicated in the sketch next to Problem 6.2. One disk is isothermal at T1 and the other is isothermal at T2 . The channel wall is a perfect reflector, i.e., = 0. Determine the radiative heat flux between the disks if the channel wall is (a) specular and (b) diffuse. For simplicity, you may treat the channel wall as a single node. If the channel is made of a transparent material, the specular arrangement approximates the behavior of an optical fiber; if the channel is filled with air, the diffuse case approximates the behavior of a light guide, a device used to pipe daylight into interior, windowless spaces. 6.4 Two infinitely long parallel plates of width w are spaced h = 2w apart. Surface 1 has 1 = 0.2 and T1 = 1000 K and Surface 2 has 2 = 0.5 and T2 = 2000 K. Calculate the heat transfer on these plates if (a) the surfaces are diffuse reflectors and (b) the surfaces are specular. 6.5 Consider the rectangular enclosure shown in Fig. 5.7. Surfaces A1 and A2 are purely specular, and Surfaces A3 and A4 are purely diffuse reflectors. Top and bottom walls are at T1 = T3 = 1000 K, with 1 = 1 − ρs1 = 3 = 1 − ρ3d = 0.3; the side walls are at T2 = T4 = 600 K with emittances 2 = 1 − ρs2 = 4 = 1 − ρ4d = 0.8. Determine the net radiative heat flux for each surface. Compare the results against those of Example 5.4.

228 Radiative Heat Transfer

6.6 A long duct has the cross section of an equilateral triangle with side lengths L = 1 m. Surface 1 is a diffuse reflector to which an external heat flux at the rate of Q1 = 1 kW/m length of duct is supplied. Surfaces 2 and 3 are isothermal at T2 = 1000 K and T3 = 500 K, respectively, and are purely specular reflectors with 1 = 2 = 3 = 0.5. (a) Determine the average temperature of Surface 1, and the heat fluxes for Surfaces 2 and 3. (b) How would the results change if Surfaces 2 and 3 were also diffusely reflecting?

6.7 Consider the infinite groove cavity shown. The entire surface of the groove is isothermal at T and coated with a gray, diffusely emitting material with emittance

. (a) Assuming the coating is a diffuse reflector, what is the total heat loss (per unit length) of the cavity? (b) If the coating is a specular reflector, what is the total heat loss for the cavity?

6.8 To calculate the net heat loss from a part of a spacecraft, this part may be approximated by an infinitely long black plate at temperature T2 = 600 K, as shown. Parallel to this plate is an (infinitely long) thin shield that is gray and reflects specularly with the same emittance 1 on both sides. You may assume the surroundings to be black at 0 K. Calculate the net heat loss from the black plate.

6.9 A long isothermal plate (at T1 ) is a gray, diffuse emitter ( 1 ) and purely specular reflector, and is used to reject heat into space. To regulate the heat flux the plate is shielded by another (black) plate, which is perfectly insulated as illustrated in the adjacent sketch. Give an expression for heat loss as a function of shield opening angle (neglect variations along plates). At what opening angle 0 ≤ φ ≤ 180◦ does maximum heat loss occur?

6.10 Reconsider Problem 6.9, but assume the entire configuration to be isothermal at temperature T, and covered with a partially diffuse, partially specular material, = 1 − ρs − ρ d . Determine an expression for the heat lost from the cavity. 6.11 An infinitely long cylinder with a gray, diffuse surface ( 1 = 0.8) at T1 = 2000 K is situated with its axis parallel to an infinite plane with 2 = 0.2 at T2 = 1000 K in a vacuum environment with a background temperature of 0 K. The axis of the cylinder is two diameters from the plane. Specify the heat loss from the cylinder when the plate surface is (a) gray and diffuse or (b) gray and specular.

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 229

6.12 A pipe carrying hot combustion gases is in radiative contact with a thin plate as shown. Assuming (a) the pipe to be isothermal at 2000 K and black and (b) the thin plate to be coated on both sides with a gray, diffusely emitting/specularly reflecting material ( = 0.1), determine the radiative heat loss from the pipe. The surroundings are at 0 K and convection may be neglected.

6.13 Repeat Problem 5.9 for the case that the flat part of the rod (A1 ) is a purely specular reflector. 6.14 A long furnace may, in a simplified scenario, be considered to consist of a strip plate (the material to be heated, A1 : 1 = 0.2, T1 = 500 K, specular reflector), unheated refractory brick (flat sides and bottom, A2 : 2 = 0.1, diffuse reflector), and a cylindrical dome of heated refractory brick (A3 : 3 = 1, T3 = 1000 K). Heat release inside the heated brick is qh (W/m2 ). The total heat release is radiated into the furnace cavity and is removed by convection, such that the convective heat loss is uniform everywhere (at qc W/m2 on all three surfaces). (a) Express the net radiative fluxes on all three surfaces in terms of qh . (b) Determine the qh necessary to maintain the indicated temperatures. 6.15 Repeat Example 5.9 for purely specularly reflecting shields. The wall material (steel) may be diffusely or specularly reflecting. 6.16 A typical space radiator may have a shape as shown in the adjacent sketch, i.e., a small tube to which are attached a number of flat plate fins, spaced at equal angle intervals. Assume that the central tube is negligibly small, and that a fixed amount of specularlyreflecting fin material is available ( = ρs = 0.5), to give (per unit length of tube) a total, one-sided fin area of A = N × L. Also assume the whole structure to be isothermal. Develop an expression for the total heat loss from the radiator as a function of the number of fins (each fin having length L = A /N). Does an optimum exist? Qualitatively discuss the more realistic case of supplying a fixed amount of heat to the bases of the fins (rather than assuming isothermal fins). 6.17 Repeat Problem 5.18 for the case that the stainless steel, while being a gray and diffuse emitter, is a purely specular reflector (all four surfaces). 6.18 Repeat Problem 5.19 for the case that both the platinum sphere as well as the aluminum shield, while being gray and diffuse emitters, are purely specular reflectors. 6.19 Repeat Problem 5.32, but assume steel and silver to be specular reflectors. 6.20 A long, thin heating wire, radiating energy in the amount of S = 300 W/cm (per cm length of wire), is located between two long, parallel plates as shown in the adjacent sketch. The bottom plate is insulated and specularly reflecting with

2 = 1 − ρs2 = 0.2, while the top plate is isothermal at T1 = 300 K and diffusely reflecting with 1 = 1 − ρ1d = 0.5. Determine the net radiative heat flux on the top plate.

6.21 An infinitely long corner of characteristic length w = 1 m is a gray, diffuse emitter and purely specular reflector with = ρs = 12 . The entire corner is kept at a constant temperature T = 500 K, and is irradiated externally by a line source of strength S = 20 kW/m, located a distance w away from both sides of the corner, as shown in the sketch. What is the total heat flux Q (per m length) to be supplied or extracted from the corner to keep the temperature at 500 K?

230 Radiative Heat Transfer

6.22 A long greenhouse has the cross section of an equilateral triangle as shown. The side exposed to the sun consists of a thin sheet of glass (A1 ) with reflectance ρ1 = 0.1. The glass may be assumed perfectly transparent to solar radiation, and totally opaque to radiation emitted inside the greenhouse. The other side wall (A2 ) is opaque with emittance 2 = 0.2, while the floor (A3 ) has 3 = 0.8. Both walls (A1 and A2 ) are specular reflectors, while the floor reflects diffusely. For simplicity, you may assume surfaces A1 and A2 to be perfectly insulated, while the floor loses heat to the ground according to q3,conduction = U(T3 − T∞ ) where T∞ = 280 K is the temperature of the ground and U = 19.5 W/m2 K is an overall heat transfer coefficient. Determine the temperatures of all three surfaces for the case that the sun shines onto the greenhouse with strength qsun = 1000 W/m2 in a direction parallel to surface A2 . 6.23 Two long plates, parallel to each other and of width w, are spaced a distance √ L = 3w/2 apart, and are facing each other as shown. The bottom plate is a gray, diffuse emitter and specularly reflecting with emittance 1 and temperature T1 . The top plate is a gray, diffuse emitter and diffusely reflecting with emittance

2 and temperature T2 . The bottom plate is irradiated by the sun as shown (strength qsol [W/m2 ], angle θ). Determine the net heat fluxes on the two plates. How accurate do you expect your answer to be? What would be a first step to achieve better accuracy?

6.24 Consider the solar collector shown. The collector plate is gray and diffuse, while the insulated guard plates are gray and specularly reflecting. Sun strikes the cavity at an angle α (α < 45◦ ). How much heat is collected? Compare with a collector without guard plates. For what values of α is your theory valid?

6.25 Reconsider the spacecraft of Problem 6.8. To decrease the heat loss from Surface 2 the specularly reflecting shield 1 is replaced by an array of N shields (parallel to each other and very closely spaced), of the same dimensions as the black surface and made of the original, specularly reflecting shield material with emittance = 0.1. Determine the net heat loss from the black plate as a function of shield number N.

6.26 Repeat Problem 6.21 using subroutine graydifspec of Appendix F (or modifying the sample program grspecxch). Break up each surface into N subsurfaces of equal width (n = 1, 2, 4, 8). 6.27 Repeat Problem 6.20 using subroutine graydifspec of Appendix F (or modifying the sample program grspecxch). Break up each surface into N subsurfaces of equal width (n = 1, 2, 4, 8). 6.28 Repeat Example 5.8 for an absorber plate made of black chrome (Fig. 3.34) and a glass cover made of soda–lime glass (Fig. 3.28). Use the semigray or the band approximation. 6.29 Repeat Problem 5.36 for the case that the top of the copper shield is coated with white epoxy paint (Fig. 3.34).

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 231

6.30 Two identical circular disks of diameter D = 1 m are connected at one point of their periphery by a hinge. The configuration is then opened by an angle φ. Disk 1 is a diffuse reflector, but emits and absorbs according to ⎧ ⎪ ⎪ ⎨0.95 cos θ, λ ≤ 3 μm, 

λ = ⎪ ⎪ ⎩0.5, λ > 3 μm. Disk 2 is black. Both disks are insulated. Assuming the opening angle to be φ = 60◦ , calculate the average equilibrium temperature for each of the two disks, with solar radiation entering the configuration parallel to Disk 2 with a strength of qsun = 1000 W/m2 . 6.31 Reconsider Problem 6.30 for the case that surfaces A1 and A2 are long, rectangular plates. 6.32 A cubical enclosure has five of its surfaces maintained at 300 K, while the sixth is isothermal at 1200 K. The entire enclosure is coated with a material that emits and reflects diffusely with ⎧ ⎪ ⎪ ⎨0.2, 0 ≤ λ < 4 μm,

λ = ⎪ ⎪ ⎩0.8, 4 μm < λ < ∞. Determine the net radiative heat fluxes on the surfaces. 6.33 Consider the configuration shown, consisting of a conical cavity A1 and an opposing circular disk with a hole at the center, as shown (d = 1 cm). Defocused laser radiation at 10.6 μm enters the configuration through the hole in the disk as shown, the beam having a strength of qL = 103 W/cm2 . The down-facing disk A2 is a gray, diffuse material with 2 = 0.1, and is perfectly insulated (toward top). Surface A1 is kept at a constant temperature of 500 K. No other external surfaces or sources affect the heat transfer. (a) Assuming surface A1 to be gray and diffuse with 1 = 0.3 determine the amount of heat that needs to be removed from A1 (Q1 ). (b) If A1 were coated with the material of Problem 3.12, how would you determine Q1 ? Set up any necessary equations and indicate how you would solve them (no actual solution necessary). Would you expect Q1 to increase/decrease/stay the same (and why)? (c) What other simple measures can you suggest to improve the accuracy of the solution (to either (a) or (b))?

6.34 Repeat Problem 6.11 for the case that Surface 1 is coated with the material described in Problem 6.32. 6.35 Repeat Problem 6.21 for the case that the corner is coated with a diffusely emitting, specularly reflecting layer whose spectral behavior may be approximated by ⎧ ⎪ ⎪ ⎨0.8, 0 ≤ λ < 3 μm,

λ = ⎪ ⎪ ⎩0.2, 3 μm < λ < ∞. The line source consists of a long filament at 2500 K inside a quartz tube, i.e., the source behaves like a gray body for λ < 2.5 μm but has no emission beyond 2.5 μm. 6.36 Repeat Problem 6.22 for the case that the side wall A2 is coated with a diffusely emitting, specularly reflecting layer whose spectral behavior may be approximated by ⎧ ⎪ ⎪ ⎨0.1, 0 ≤ λ < 3 μm,

λ = ⎪ ⎪ ⎩0.8, 3 μm < λ < ∞. 6.37 Repeat Problem 5.29 for the case that A1 is coated with a material that has a spectral, directional emittance of /

λ =

0.9 cos θ, 0.3,

π λ < 4μm, 0≤θ< . 2 λ > 4μm.

232 Radiative Heat Transfer

6.38 In the solar energy laboratory at UC Merced parabolic concentrators are employed to enhance the absorption of tubular solar collectors as shown in the sketch. Assume that solar energy enters the cavity normal to the opening, with a strength of qsun = 1000 W/m2 (per unit area normal to the rays). The parabolic receiver is coated with a highly reflective gray, diffuse material with 1 = 0.05, and is kept cold by convection (i.e., emission from it is negligible). Calculate the collected solar energy as a function of tube outer temperature (say, for 300 K, 400 K, 500 K), (a) assuming the tube to be gray with emittance 2 = 0.90, (b) assuming the tube to be covered with black nickel, using the 2-band approach. It is sufficient to treat tube and concentrator each as single zones.

6.39 A small spherical heat source outputting Qs = 10 kW power, spreading equally into all directions, is encased in a reflector as shown, consisting of a hemisphere of radius R = 40 cm, plus a ring of radius R and height h = 30 cm. The arrangement is used to heat a disk of radius = 25 cm a distance of L = 30 cm below the reflector. Reflector A2 is gray and diffuse with emittance of 2 = 0.1 and is insulated. Disk A1 is diffuse and coated with a selective absorber, i.e., ⎧ ⎪ ⎪ ⎨0.8, 0 ≤ λ < 3 μm,

1λ = ⎪ ⎪ ⎩0.2, 3 μm < λ < ∞. The source is of the tungsten–halogen type, i.e., the spectral variation of its emissive power follows that of a blackbody at 4000 K. (a) Determine (per unit area of receiving surface) the irradiation from heat source to reflector and to disk. (b) Determine all relevant view factors. (c) Outline how you would obtain the temperature of the disk, if 0.4 kW of power is extracted from it. (“Outline” implies setting up all the necessary equations, plus a sentence on how you would solve them.) 6.40 Repeat Problem 6.35 using subroutine bandapp of Appendix F (or modifying the sample program bandmxch). Break up each surface into N subsurfaces of equal width (n = 1, 2, 4, 8). 6.41 Repeat Problem 5.28 for the case that the insulated cylinder is coated with a material that has

2λ

⎧ ⎪ ⎪ ⎨0.2, =⎪ ⎪ ⎩0.8,

0 ≤ λ < 4 μm, 4 μm < λ < ∞

(the flat surface remains gray with 3 = 0.5). Note that the wire heater is gray and diffuse and at a temperature of T1 = 3000 K. (a) Find the solution using the semigray method; also set up the same problem and find the solution by using program semigrayxchdf. (b) Set up the solution using the band approximation, i.e., to the point of having a set of simultaneous equations and an outline of how to solve them. Also find the solution using program bandmxchdf. 6.42 Repeat Problem 5.2 assuming that the furnace walls are made of alumina ceramic (aluminum oxide, Fig. 1.14). Use subroutine bandapp of Appendix F (or modifying the sample program bandmxch). Break up the spectrum into several parts, and compare your results for N = 1, 2, 3, and 5. 6.43 Repeat Problem 5.22 assuming that the furnace walls are made of alumina ceramic (aluminum oxide, Fig. 1.14). Use subroutine semigray of Appendix F (or modifying the sample program semigrxch). Break up the groove surface into N subsurfaces of equal size (N = 2 and 4), but only consider incidence angles of θ = 0◦ and 60◦ .

Radiative Exchange Between Nondiffuse and Nongray Surfaces Chapter | 6 233

6.44 Repeat Problem 6.21 for the case that the corner is cold (i.e., has negligible emission), and that the surface is gray and specularly reflecting with = ρ s = 0.5, but has a directional emittance/absorptance of

 (θ) = n cos θ. Determine local and total absorbed radiative heat fluxes. 6.45 Consider two infinitely long, parallel plates of width w = 1 m, spaced a distance h = 0.5 m apart (see Configuration 32 in Appendix D). Both plates are isothermal at 1000 K and are coated with a gray material with a directional emittance of

 (θi ) = α (θi ) = 1 − ρ (θi ) = n cos θi and a hemispherical emittance of = 0.5. Reflection is neither diffuse nor specular, but the bidirectional reflection function of the material is ρ (θi , θr ) =

3  ρ (θi ) cos θr . 2π

Write a small computer program to determine the total heat lost (per unit length) from each plate. Compare with the case for a diffusely emitting/reflecting surface.

References [1] A.F. Sarofim, H.C. Hottel, Radiation exchange among non-Lambert surfaces, ASME Journal of Heat Transfer 88 (1966) 37–44. [2] R.C. Birkebak, E.M. Sparrow, E.R.G. Eckert, J.W. Ramsey, Effect of surface roughness on the total and specular reflectance of metallic surfaces, ASME Journal of Heat Transfer 86 (1964) 193–199. [3] C.R. Wylie, Advanced Engineering Mathematics, 5th ed., McGraw-Hill, New York, 1982. [4] E.R.G. Eckert, E.M. Sparrow, Radiative heat exchange between surfaces with specular reflection, International Journal of Heat and Mass Transfer 3 (1961) 42–54. [5] E.M. Sparrow, L.U. Albers, E.R.G. Eckert, Thermal radiation characteristics of cylindrical enclosures, ASME Journal of Heat Transfer 84 (1962) 73–81. [6] S.H. Lin, E.M. Sparrow, Radiant interchange among curved specularly reflecting surfaces, application to cylindrical and conical cavities, ASME Journal of Heat Transfer 87 (1965) 299–307. [7] E.M. Sparrow, S.L. Lin, Radiation heat transfer at a surface having both specular and diffuse reflectance components, International Journal of Heat and Mass Transfer 8 (1965) 769–779. [8] M. Perlmutter, R. Siegel, Effect of specularly reflecting gray surface on thermal radiation through a tube and from its heated wall, ASME Journal of Heat Transfer 85 (1963) 55–62. [9] E.M. Sparrow, V.K. Jonsson, Radiant emission characteristics of diffuse conical cavities, Journal of the Optical Society of America 53 (1963) 816–821. [10] L.G. Polgar, J.R. Howell, Directional thermal-radiative properties of conical cavities, NASA TN D-2904, 1965. [11] D.S. Tsai, F.G. Ho, W. Strieder, Specular reflection in radiant heat transport across a spherical void, Chemical Engineering Science–Genie Chimique 39 (1984) 775–779. [12] D.S. Tsai, W. Strieder, Radiation across a spherical cavity having both specular and diffuse reflectance components, Chemical Engineering Science 40 (1) (1985) 170. [13] E.M. Sparrow, V.K. Jonsson, Absorption and emission characteristics of diffuse spherical enclosures, NASA TN D-1289, 1962. [14] E.M. Sparrow, V.K. Jonsson, Absorption and emission characteristics of diffuse spherical enclosures, ASME Journal of Heat Transfer 84 (1962) 188–189. [15] J.A. Plamondon, T.E. Horton, On the determination of the view function to the images of a surface in a nonplanar specular reflector, International Journal of Heat and Mass Transfer 10 (5) (1967) 665–679. [16] D.G. Burkhard, D.L. Shealy, R.U. Sexl, Specular reflection of heat radiation from an arbitrary reflector surface to an arbitrary receiver surface, International Journal of Heat and Mass Transfer 16 (1973) 271–280. [17] E.M. Sparrow, R.D. Cess, Radiation Heat Transfer, Hemisphere, New York, 1978. [18] M.B. Ziering, A.F. Sarofim, The electrical network analog to radiative transfer: allowance for specular reflection, ASME Journal of Heat Transfer 88 (1966) 341–342. [19] P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Macmillan, New York, 1963. [20] J.S. Toor, Radiant heat transfer analysis among surfaces having direction dependent properties by the Monte Carlo method, M.S. thesis, Purdue University, Lafayette, IN, 1967. [21] J.R. Branstetter, Radiant heat transfer between nongray parallel plates of tungsten, NASA TN D-1088, 1961. [22] B.F. Armaly, C.L. Tien, A note on the radiative interchange among nongray surfaces, ASME Journal of Heat Transfer 92 (1970) 178–179.

234 Radiative Heat Transfer

[23] R.P. Bobco, G.E. Allen, P.W. Othmer, Local radiation equilibrium temperatures in semigray enclosures, Journal of Spacecraft and Rockets 4 (8) (1967) 1076–1082. [24] J.A. Plamondon, C.S. Landram, Radiant heat transfer from nongray surfaces with external radiation. Thermophysics and temperature control of spacecraft and entry vehicles, Progress in Astronautics and Aeronautics 18 (1966) 173–197. [25] S. Shimoji, Local temperatures in semigray nondiffuse cones and v-grooves, AIAA Journal 15 (3) (1977) 289–290. [26] R.V. Dunkle, J.T. Bevans, Part 3, a method for solving multinode networks and a comparison of the band energy and gray radiation approximations, ASME Journal of Heat Transfer 82 (1) (1960) 14–19. [27] R.G. Hering, T.F. Smith, Surface roughness effects on radiant energy interchange, ASME Journal of Heat Transfer 93 (1) (1971) 88–96. [28] R.G. Hering, T.F. Smith, Apparent radiation properties of a rough surface, AIAA paper no. 69-622, 1969. [29] J.T. Bevans, D.K. Edwards, Radiation exchange in an enclosure with directional wall properties, ASME Journal of Heat Transfer 87 (3) (1965) 388–396. [30] M.F. Modest, S.C. Poon, Determination of three-dimensional radiative exchange factors for the space shuttle by Monte Carlo, ASME paper no. 77-HT-49, 1977. [31] M.F. Modest, Determination of radiative exchange factors for three dimensional geometries with nonideal surface properties, Numerical Heat Transfer 1 (1978) 403–416. [32] J.S. Toor, R. Viskanta, A critical examination of the validity of simplified models for radiant heat transfer analysis, International Journal of Heat and Mass Transfer 15 (1972) 1553–1567. [33] J.S. Toor, R. Viskanta, Experiment and analysis of directional effects on radiant heat transfer, ASME Journal of Heat Transfer 94 (November 1972) 459–466.

Chapter 7

The Monte Carlo Method for Surface Exchange 7.1 Introduction Very few exact, closed-form solutions to thermal radiation problems exist, even in the absence of a participating medium. Under most circumstances the solution has to be found by numerical means. For most engineers, who are used to dealing with partial differential equations, this implies use of deterministic methods, such as finite difference, finite volume, and finite element techniques. These methods are, of course, applicable to thermal radiation problems whenever a solution method is chosen that transforms the governing equations into sets of partial differential equations. For surface exchange, however, radiative transfer is governed by integral equations, which may be solved numerically by employing numerical quadrature for the evaluation of integrals, or more approximately using the “net radiation method” of the previous two chapters. With these techniques the solutions to relatively simple problems are readily found. However, if the geometry is complex, and/or if radiative properties vary with direction and/or wavelength, then a solution by conventional numerical techniques may quickly become extremely involved if not impossible. Many mathematical problems may also be solved by statistical methods, through sampling techniques, to any degree of accuracy. For example, consider predicting the outcome of the next presidential elections. Establishing a mathematical model that would predict voter turnout and voting behavior is, of course, impossible, let alone finding the analytical solution to such a model. However, if an appropriate sampling technique is chosen, the outcome can be predicted by conducting a poll. The accuracy of its prediction depends primarily on the sample size, i.e., how many people have been polled. Solving mathematical problems statistically always involves the use of random numbers, which may be picked, e.g., by placing a ball into a spinning roulette wheel. For this reason these sampling methods are called Monte Carlo methods (named after the principality of Monte Carlo in the south of France, famous for its casino). There is no single scheme to which the name Monte Carlo applies. Rather, any method of solving a mathematical problem with an appropriate statistical sampling technique is commonly referred to as a Monte Carlo method. Problems in thermal radiation are particularly well suited to solution by a Monte Carlo technique, since energy travels in discrete parcels (photons) over (usually) relatively long distances along a (usually) straight path before interaction with matter. Thus, solving a thermal radiation problem by Monte Carlo implies tracing the history of a statistically meaningful random sample of photons from their points of emission to their points of absorption. The advantage of the Monte Carlo method is that even the most complicated problem may be solved with relative ease, as schematically indicated in Fig. 7.1. For a trivial problem, setting up the appropriate photon sampling technique alone may require more effort than finding the analytical (or even a small numerical) solution. As the complexity of the problem increases, however, the complexity of formulation and the solution effort increase much more rapidly for conventional (i.e., deterministic) techniques. For problems beyond a certain complexity, the Monte Carlo solution will be preferable. In engineering problems, “complexity” can arise from many different sources, such as geometry, number of independent and/or dependent variables, nonlinearities, and so on. Unfortunately, there is no way to determine a priori precisely where this crossover point in complexity lies. The disadvantage of Monte Carlo methods is that, as statistical methods, they are subject to statistical error (very similar to the unavoidable error associated with experimental measurements). The name and the systematic development of Monte Carlo methods dates from about 1944 [1], although some crude mathematical sampling techniques were used off and on during previous centuries. Their first use as a research tool stems from the attempt to model neutron diffusion in fission material, for the development of the atomic bomb during the World War II. The method was first applied to thermal radiation problems in the early 1960s by Fleck [2,3] and Howell and Perlmutter [4–6]. Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00015-8 Copyright © 2022 Elsevier Inc. All rights reserved.

235

236 Radiative Heat Transfer

FIGURE 7.1 Comparison of Monte Carlo and conventional solution techniques.

For a thorough understanding of Monte Carlo methods, a good background in statistical methods is necessary, which goes beyond the scope of this book. In this chapter the method as applied to thermal radiation is outlined, and statistical considerations are presented in an intuitive way rather than in a rigorous mathematical fashion. For a more detailed description, the reader may want to consult the books by Hammersley and Handscomb [1], Cashwell and Everett [7], and Schreider [8], or the monographs by Kahn [9], Brown [10], Halton [11], and HajjiSheikh [12]. A first monograph dealing specifically with Monte Carlo methods as applied to thermal radiation has been given by Howell [13]. Another more recent one by Walters and Buckius [14] emphasizes the treatment of scattering. An exhaustive review of the literature up until 1997, that uses some form of radiative Monte Carlo analysis, has been given also by Howell [15]. Since then, a large number of researchers have applied Monte Carlo simulations to a vast array of problems, ranging from nanoscale radiation properties to large-scale tomography, surface radiation, participating media, transient radiation, combined modes heat transfer, etc., too numerous to review in this book.

Probability Distributions When a political poll is conducted, people are not selected at random from a telephone directory. Rather, people are randomly selected from different groups according to probability distributions, to ensure that representative numbers of barbers, housewives, doctors, smokers, gun owners, bald people, heat transfer engineers, etc. are included in the poll. Similarly, in order to follow the history of radiative energy bundles in a statistically meaningful way, the points, directions and wavelengths of emission, reflective behavior, etc. must be chosen according to probability distributions. As an example, consider the total radiative heat flux being emitted from a surface, i.e., the total emissive power, 

∞

E= 0

 Eλ dλ =

∞

λ Ebλ dλ.

(7.1)

0

Between the wavelengths of λ and λ + dλ the emitted heat flux is Eλ dλ = λ Ebλ dλ, and the fraction of energy emitted over this wavelength range is Eλ dλ Eλ P(λ) dλ =  ∞ dλ. = E Eλ dλ 0

(7.2)

We may think of all the photons leaving the surface as belonging to a set of N energy bundles (or “photon bundles”) of equal energy (each consisting of many photons of a single wavelength). Then each bundle carries the amount of energy (E/N) with it, and the probability that any particular bundle has a wavelength between λ and λ + dλ is given by the probability density function P(λ). The fraction of energy emitted over all wavelengths

The Monte Carlo Method for Surface Exchange Chapter | 7 237

between 0 and λ is then 

λ

R(λ) = 0

λ P(λ) dλ =  0∞ 0

Eλ dλ Eλ dλ

.

(7.3)

It is immediately obvious that R(λ) is also the probability that any given energy bundle has a wavelength between 0 and λ, and it is known as the cumulative distribution function. The probability that a bundle has a wavelength between 0 and ∞ is, of course, R(λ → ∞) = 1, a certainty. Equation (7.3) implies that if we want to simulate emission from a surface with N energy bundles of equal energy, then the fraction R(λ) of these bundles must have wavelengths smaller than λ. Now consider a pool of random numbers equally distributed between the values 0 and 1. Since they are equally distributed, this implies that a fraction R of these random numbers have values less than R itself. Let us now pick a single random number, say R0 . Inverting equation (7.3), we find λ(R0 ), i.e., the wavelength corresponding to a cumulative distribution function of value R0 , and we assign this wavelength to one energy bundle. If we repeat this process many times, then the fraction R0 of all energy bundles will have wavelengths below λ(R0 ), since the fraction R0 of all our random numbers will be below this value. Thus, in order to model correctly the spectral variation of surface emission, using N bundles of equal energy, their wavelengths may be determined by picking N random numbers between 0 and 1, and inverting equation (7.3).

Random Numbers If we throw a ball onto a spinning roulette wheel, the ball will eventually settle on any one of the wheel’s numbers (between 0 and 36). If we let the roulette wheel decide on another number again and again, we will obtain a set of random numbers between 0 and 36 (or between 0 and 1, if we divide each number by 36). Unless the croupier throws in the ball and spins the wheel in a regular (nonrandom) fashion,1 any number may be chosen each time with equal probability, regardless of what numbers have been picked previously. However, if sufficiently many numbers are picked, we may expect that roughly half (i.e., 18/37) of all the picked numbers will be between 0 and 17, for example. During the course of a Monte Carlo simulation, generally somewhere between 105 and 107 random numbers need to be drawn (and even more, as computing power continues to increase), and they need to be drawn very rapidly. Obviously, spinning a roulette wheel would be impractical. One solution to this problem is to store an (externally determined) set of random numbers. However, such a table would require a prohibitive amount of computer storage, unless it were a relatively small table, that would be used repeatedly (thus destroying the true randomness of the set). The only practical answer is to generate the random numbers within the computer itself. This appears to be a contradiction, since a digital computer is the incarnation of logic (nonrandomness). Substantial research has been carried out on how to generate sets of sufficiently random numbers using what are called pseudorandom number generators. A number of such generators exist that, after making the choice of a starting point (or seed), generate a new pseudorandom number from the previous one. The randomness of such a set of numbers depends on the quality of the generator as well as the choice of the starting point and should be tested by different “randomness tests.” For a more detailed discussion of pseudorandom number generators, the reader is referred to Hammersley and Handscomb [1], Schreider [8], or Taussky and Todd [16].

Accuracy Considerations Since Monte Carlo methods are statistical methods, the results, when plotted against number of samples, will generally fluctuate randomly around the correct answer. If a set of truly random numbers is used for the sampling, then these fluctuations will decrease as the number of samples increases. Let the answer obtained from the Monte Carlo method after tracing N energy bundles be S(N), and the exact solution obtained after sampling infinitely many energy bundles S(∞). For some simple problems it is possible to calculate directly the probability that the obtained answer, S(N), differs by less than a certain amount from the correct answer, S(∞). Even if it were possible to directly calculate the confidence level for more complicated situations, this would not take into account the pseudorandomness of the computer-generated random number set. This can have 1. This is, of course, the reason casinos tend to employ a number of croupiers, each of whom works only for a very short period each day.

238 Radiative Heat Transfer

FIGURE 7.2 Convergence of Monte Carlo method: (a) convergence of subsamples with different random number seeds; (b) convergence rates as functions of photon bundles and number of subsamples.

rather substantial effects if a random number generator is not thoroughly tested for randomness, or if it is used improperly. Thankfully, most random number generators today generate excellent, large sets of pseudo-random numbers that are relatively foolproof. Nevertheless, care must always be taken to avoid periodic repetition of random number sequences: see, for example, the potential problem with the popular RAN1 from Numerical Recipes [17], as observed by Baker [18]. Figure 7.2a shows typical results for the Monte Carlo evaluation of the view factor between two parallel black plates employing the RAN1 random number generator. Results are shown for two subsamples, each using between 100 and 5000 rays obtained with different seeds for RAN1, and also the mean of altogether 10 sets. The results oscillate rather wildly if very few bundles are traced (right and left triangles), but in the mean (squares) the answer clearly approaches the exact result. Visual inspection shows that the results become “acceptable” if at least 10 × 3000 bundles are traced, but a formula is needed to quantify accuracy. For radiative heat transfer calculations the most straightforward way of estimating the error associated with the sampling result S(N) is to break up the result into a number of I subsamples S(Ni ), such as was shown in Fig. 7.2a (although there the purpose was to demonstrate the difference of results obtained from different random number seeds). Then N = N1 + N2 + . . . + N I =

I 

Ni ,

(7.4)

i=1

S(N) =

I   1 1  N1 S(N1 ) + . . . + NI S(NI ) = Ni S(Ni ). N N

(7.5)

i=1

Normally, each subsample would include identical amounts of bundles, leading to Ni = N/I; S(N) =

1 I

i = 1, 2, . . . , I, I 

S(Ni ).

(7.6) (7.7)

i=1

The I subsamples may be treated as if they were independent experimental measurements of the same quantity. We may then calculate the standard deviation (or standard error) σ and its best estimate (or adjusted standard error)

The Monte Carlo Method for Surface Exchange Chapter | 7 239

σm or its square, the variance σ2m [19]:  1 σ2 = [S(Ni ) − S(N)]2 . I − 1 I(I − 1) I

σ2m =

(7.8)

i=1

The central limit theorem states that the mean S(N) of I measurements S(Ni ) follows a Gaussian distribution, whatever the distribution of the individual measurements. This implies that we can say with 68.3% confidence that the correct answer S(∞) lies within the limits of S(N) ± σm , with 95.5% confidence within S(N) ± 2σm , or with 99% confidence within S(N) ± 2.58σm . Details on statistical analysis of errors may be found in any standard book on experimentation, for example, the one by Barford [19]. It has been the authors’ experience that practical Monte Carlo results never deviate from the correct answer by more than one σm . The progress of adjusted standard error σm for the view factor calculation with number of samples, as calculated with equation (7.8), is depicted in Fig. 7.2b. Clearly, the adjusted standard error decreases as N−1/2 , as would be expected for actual laboratory experiments [19]. This implies that, in order to decrease uncertainty of Monte Carlo simulations by half, we need to trace four times as many photon bundles. It is also of interest to investigate the impact of number of subsamples on the result (akin of making few accurate vs. many relatively inaccurate measurements in actual experiments). Figure 7.2b demonstrates that, regardless of the number of subsamples, the standard error decays as 1/N1/2 . Clearly, predictions with larger numbers of subsamples do so more smoothly. However, data storage for Monte Carlo simulations is proportional to number of subsamples I and, thus, small numbers, such as I = 10, are generally preferred.

7.2 Numerical Quadrature by Monte Carlo Before discussing how statistical methods can be used to solve complicated radiative transfer problems, we will quickly demonstrate that the Monte Carlo method can also be employed to evaluate integrals numerib cally (known as numerical quadrature). Consider the integral a f (x) dx. The most primitive form of numerical quadrature is the midpoint rule, in which f (x) is assumed constant over a small interval Δx, i.e. [20,21], 

b a

N    f (x) dx f xi = (i − 12 )Δx Δx;

Δx =

i=1

b−a . N

(7.9)

For large enough values of N equation (7.9) converges to the correct result. Note that the values of xi are equally distributed across the interval between a and b. If we were to draw N random locations equally distributed between a and b, we would achieve the same result in a statistical sense. Therefore, we can evaluate any integral via the Monte Carlo method as 

b

f (x) dx a

N 

f [xi = a + (b − a)Ri ] Δx; Δx =

i=1

b−a , N

(7.10)

where Ri is a set of random numbers equally distributed between 0 and 1. Equation (7.10) is an efficient means of integration if the integrand f (x) is poorly behaved as, e.g., in the evaluation of k-distributions in Chapter 10 (integration over spectral variations of the absorption coefficient of molecular gases). However, if f (x) varies by orders of magnitude (but in a predictable manner) across a ≤ x ≤ b, picking equally distributed xi results in putting equal emphasis on important as well as unimportant regions. The stochastic integration can be made more efficient by determining the xi from a probability density function (PDF) p(x). We may write 



b

b

f (x) dx = a

a

f (x) p(x) dx = p(x)

 0

1

f (x(ξ)) dξ p(x(ξ))

(7.11)

where  ξ(x) =



x

a

b

p(x) dx ≡ 1.

p(x) dx, a

(7.12)

240 Radiative Heat Transfer

The PDF is chosen in such a way that f /p remains relatively constant across a ≤ x ≤ b, assuring that each stochastic sample makes roughly the same contribution to the result. The integral may then be evaluated as 

f (x) dx a

b − a  f (xi ) , N p(xi ) N

b

xi = ξ−1 (Ri ).

(7.13)

i=1

This is known as importance sampling. Equations (7.10) and (7.13) are also useful if integration is an integral part of a Monte Carlo simulation, such as the Backward Monte Carlo scheme described in Chapter 20. Finally, extension to two- and higher-dimensional integrals is obvious and trivial.

7.3 Heat Transfer Relations for Radiative Exchange Between Surfaces In the absence of a participating medium and assuming a refractive index of unity, the radiative heat flux leaving or going into a certain surface, using the Monte Carlo technique, is governed by the following basic equation:  dFdA →dA dA ,

(r ) σT 4 (r ) (7.14) q(r) = (r)σT 4 (r) − dA A where q(r) T(r)

(r) A dFdA →dA

= = = = =

local surface heat flux at location r, surface temperature at location r, total hemispherical emittance of the surface at r, surface area of the enclosure, and generalized radiation exchange factor between surface elements dA and dA.

In equation (7.14) the first term on the right-hand side describes the emission from the surface, and the integrand of the second term is the fraction of energy, originally emitted from the surface at r , which eventually gets absorbed at location r. Therefore, the definition for the generalized exchange factor must be: dFdA →dA ≡ fraction of the total energy emitted by dA that is absorbed by dA, either directly or after any number and type of reflections.

(7.15)

This definition appears to be the most compatible one for solution by ray-tracing techniques and is therefore usually employed for calculations by the Monte Carlo method. Figure 7.3 shows a schematic of an arbitrary enclosure with energy bundles emitted at dA and absorbed at dA. If the enclosure is not closed, i.e., has openings into space, some artificial closing surfaces must be introduced. For example, an opening directed into outer space without irradiation from the sun or Earth can be replaced by a black surface at a temperature of 0 K. If the opening is irradiated by the sun, it is replaced by a nonreflecting surface with zero emittance for all angles but the solar angle, etc. The enclosure surface is now divided into J subsurfaces, and equation (7.14) reduces to  Qi = Ai

qi dAi = i σTi4 Ai −

J 

j σT4j A j Fj→i − q ext As Fs→i ,

1 ≤ i ≤ J,

(7.16)

j=1

where q ext As

= =

external energy entering through any opening in the enclosure, area of the opening irradiated from external sources,

and the j and T j are suitable average values for each subsurface, i.e.,  1 4

σT 4 dA.

j σT j = A j Aj

(7.17)

The Monte Carlo Method for Surface Exchange Chapter | 7 241

FIGURE 7.3 Possible energy bundle paths in an arbitrary enclosure.

Although heat flow rates Qi can be calculated directly by the Monte Carlo method, it is of advantage to instead determine the exchange factors: although the Qi ’s depend on all surface temperatures in the enclosure, the Fi→ j ’s either do not (gray surfaces) or depend only on the temperature of the emitting surface (nongray surfaces), provided that surface reflectances (and absorptances) are independent of temperature (as they are to a very good degree of accuracy). Since all emitted energy must go somewhere, and, by the Second Law of Thermodynamics the net exchange between two equal temperature surfaces must be zero, the summation rule and reciprocity also hold for exchange factors, i.e., J 

Fi→ j = 1,

(7.18)

j=1

i Ai Fi→ j = j A j F j→i ,

(7.19)

(the former, of course, only for enclosures without openings). A large statistical sample of energy bundles Ni is emitted from surface Ai , each of them carrying the amount of radiative energy ΔEi = i σTi4 Ai /Ni .

(7.20)

If Ni j of these bundles become absorbed by surface A j either after direct travel or after any number of reflections, the exchange factor may be calculated from     Ni j Ni j . (7.21) Fi→ j = lim Ni →∞ Ni Ni Ni 1 MONT3D is a publicly available Fortran code [22–25], given in Appendix F, that calculates general exchange factors for complicated three-dimensional geometries. Monte Carlo calculations of exchange factors, by their nature, automatically obey the summation rule, equation (7.18), but—due to the inherent statistical scatter—reciprocity, equation (7.19), is not fulfilled. Several smoothing schemes have been given in the literature that assures that both equations (7.18) and (7.19) are satisfied [26–30]. Many Monte Carlo simulations today are carried out bypassing the exchange factor formulation, and calculate the Qi directly for each surface. Then, Qi = i σTi4 Ai −

J  j=1

ΔE j N ji − ΔEext Next,i ,

1 ≤ i ≤ J.

(7.22)

242 Radiative Heat Transfer

FIGURE 7.4 Surface description in terms of a position vector; (a) planar element, (b) general curved surface.

For enclosures without external irradiation it is also common to base the amount of energy carried by individual bundles on the emission from the entire enclosure, i.e., ΔE =

J 

j σT4j A j /N,

(7.23)

j=1

where N is the total number of photon bundles to be traced (from all surfaces combined). Then ⎞ ⎛ J  ⎜⎜ ⎟⎟ ⎟ ⎜⎜ Qi = ΔE ⎜⎜Ni − N ji ⎟⎟⎟ , 1 ≤ i ≤ J. ⎠ ⎝

(7.24)

j=1

7.4 Surface Description When Monte Carlo simulations are applied to very simple configurations such as rectangular or circular flat plates, e.g., Toor and Viskanta [31], the determination of bundle emission location, intersection points, intersection angles, reflection angles, etc., are relatively obvious and straightforward. If more complicated surfaces are considered, such as discrete, arbitrarily-oriented planar surface elements produced by a mesh generator, or a second-order polynomial description, e.g., by Weiner and coworkers [32], or the arbitrary-order polynomial description by Modest and Poon [33] and Modest [34], a systematic way to describe surfaces is preferable. It appears most logical to describe surfaces in vectorial form, as indicated in Fig. 7.4, r=

3 

xi (v1 , v2 ) êi ,

v1min ≤ v1 ≤ v1max ,

v2min (v1 ) ≤ v2max (v1 ),

(7.25)

i=1

that is, r is the vector pointing from the origin to a point on the surface, v1 and v2 are two surface parameters, the ˆ into the x, y, z directions, xi are the (x, y, z) coordinates of a point on the surface, and the êi are unit vectors (î, jˆ, k) respectively. In most engineering calculations, the computational domain is discretized by a mesh. Since the radiation calculation is generally part of a larger heat transfer calculation that employs this mesh, the same mesh is typically used for radiation calculations, as well. Discretization of complex geometries often requires mesh elements— both volume and surface—that are nonregular. For surface elements, this implies triangles or nonrectangular quadrilaterals, among other types. In many practical engineering calculations curved surfaces are tesselated into planar faces and the only information available is the coordinates of a set of points on each face. Such planar surface elements are readily defined by the coordinates of 3 points A, B, and C in space, as shown in Fig. 7.4a.

The Monte Carlo Method for Surface Exchange Chapter | 7 243

Two unit tangents on the surface follow as ˆt1 = a/|a| and ˆt2 = b/|b|, where a = rA − rC and b = rB − rC . Any point on its surface is then described by r = rC + v1 ˆt1 + v2 ˆt2 ,

(7.26)

where the two surface parameters, v1 and v2 , represent the distances traveled along the unit vectors from point C. The span of the planar surface is defined by the values of these two parameters. For example, it is easy to see that if 0 ≤ v1 ≤ |a|, and 0 ≤ v2 ≤ |b|, the resulting surface is the parallelogram CADB. By applying the general limits for v1 and v2 as given by equation (7.25), planar surfaces of any size and shape can be described. For the much more common case of simple triangles or parallelograms, it is more convenient to use the vectors a and b, since determining the unit tangent vectors requires the unnecessary calculation of |a| and |b| followed by an extra division. Then equation (7.26) can be rewritten as r = rC + μ1 a + μ2 b,

(7.27)

where 0 ≤ μ1 ≤ 1 and 0 ≤ μ2 ≤ 1 now define the surface of the parallelogram CADB. It is also easy to establish that, if the additional constraint μ1 +μ2 ≤ 1 is placed, then the equation for the triangular surface CAB is obtained, and the special case of μ1 + μ2 = 1 defines the line AB. If the surface is not planar, the description must be generalized. The vector r pointing to arbitrary locations on the surface, as shown in Fig. 7.2b, must be known to properly describe it. We may then define two unit tangents to the surface at any point, as shown in Fig. 7.4b, by ∂r ˆt1 = ∂v1

.  ∂r   , ∂v1 

∂r ˆt2 = ∂v2

.  ∂r   . ∂v2 

(7.28)

While it is usually a good idea to choose the surface parameters v1 and v2 perpendicular to one another (making ˆt1 and ˆt2 perpendicular to each other), this is not necessary. In either case, one can evaluate the unit surface normal as ˆt1 × ˆt2 , |tˆ1 × ˆt2 |

nˆ =

(7.29)

where it has been assumed that v1 and v2 have been ordered such that nˆ is the outward surface normal.

7.5 Random Number Relations for Surface Exchange In order to calculate the exchange factor by tracing the history of a large number of energy bundles, we need to know how to pick statistically meaningful energy bundles as explained at the end of Section 7.1: for each emitted bundle we need to determine a point of emission, a direction of emission, and a wavelength of emission. Upon impact of the bundle onto another point of the enclosure surface, we need to decide whether the bundle is reflected and, if so, into what direction.

Points of Emission Similar to equation (7.1) we may write for the total emission from a surface A j :  Ej =

σT 4 dA.

(7.30)

Aj

Since integration over an area is a double integral, we may rewrite this equation, e.g., for simple rectangular surfaces, as 

X





Y

Ej =

X

σT dy dx = 4

x=0

y=0

0

Ej dx,

(7.31)

244 Radiative Heat Transfer

where  Ej (x) =

Y

σT 4 dy.

(7.32)

0

Thus, we may apply equation (7.3) and find 1 Rx = Ej



x

0

Ej dx.

(7.33)

This relationship may be inverted to find the x-location of the emission point as a function of a random number Rx : x = x(Rx ).

(7.34)

Once the x-location has been determined, equation (7.3) may also be applied to equation (7.32), leading to an expression for the y-location of emission: 1 Ry =  E j (x)



y

σT 4 dy,

(7.35)

0

and y = y(R y , x).

(7.36)

Note that the choice for the y-location depends not only on the random number R y , but also on the location of x. If the emissive power may be separated in x and y, i.e., if E = σT 4 = Ex (x)E y (y),

(7.37)

then equation (7.33) reduces to 

.

x

Rx =

X

Ex (x) dx 0

Ex (x) dx,

(7.38)

E y (y) dy,

(7.39)

0

and equation (7.35) simplifies to  Ry =

.

y

Y

E y (y) dy 0

0

that is, choices for x- and y-locations become independent of one another. In the simplest case of an isothermal surface with constant emittance, these relations reduce to x = Rx X,

y = R y Y.

(7.40)

Example 7.1. Given a ring surface element on the bottom of a black isothermal cylinder with inner radius ri = 10 cm and outer radius ro = 20 cm, as indicated in Fig. 7.5, calculate the location of emission for a pair of random numbers Rr = 0.5 and Rφ = 0.25. Solution We find 



2π



ro

Eb dA = Eb

E= A

r dr dφ. 0

ri

The Monte Carlo Method for Surface Exchange Chapter | 7 245

FIGURE 7.5 Geometry for Example 7.1.

Since this expression is separable in r and φ, this leads to 

.

φ

Rφ =

2π

dφ =

dφ 0

0

φ , 2π

or

φ = 2πRφ ,

and 

.

r

Rr =

ro

r dr ri

ri

r dr =

r2 − r2i r2o − r2i

,

or r=

(

r2i + (r2o − r2i )Rr .

% Therefore, φ = 2π × 0.25 = π/2 and r = 100 + (400 − 100)0.5 = 15.8 cm. While, as expected for a random number of 0.25, the emission point angle is 90◦ away from the φ = 0 axis, the r-location does not fall onto the midpoint. This is because the cylindrical ring has more surface area at larger radii, resulting in larger total emission. This implies that more energy bundles must be emitted from the outer part of the ring.

As mentioned earlier, in many engineering calculations the computational domain is discretized into a mesh of triangles or nonrectangular quadrilaterals. Furthermore, it is generally assumed that every surface element or face is isothermal and has a uniform emittance. Since any planar shape can be split into a set of nonoverlapping triangles, here we describe the procedure for determining the point of emission inside a planar triangular face with constant emittance. Following equation (7.27), the position vector of any emission location within the triangle shown in Fig. 7.4a can be calculated using re = rC + R1 a + R2 b,

if R1 + R2 ≤ 1,

(7.41)

where R1 and R2 are now two random numbers between 0 and 1. Since the coordinates of the vertices of the triangle are known from the mesh generator, the vectors rC , a, and b are readily calculated. Procedurally, a pair of random numbers is drawn, and it is checked to see if R1 + R2 ≤ 1. If the criterion is satisfied, then equation (7.41) is used to calculate the emission location. Else, the two random numbers are discarded, and the procedure is repeated (alternatively, one may replace R1 and R2 by a new and equivalent set R1 = 1 − R1 and R2 = 1 − R2 to find the emission point). It is easy to see that, if R1 and R2 are both varied uniformly between 0 and 1, uniformly spaced points will be distributed over a parallelogram. By placing the constraint R1 + R2 ≤ 1, the parallelogram area is simply halved (to a triangle), while still retaining the original uniform distribution of points on the triangle. Therefore, the constraint of equation (7.30) is satisfied. If points of emission from a curved surface must be found, one needs to apply equation (7.30) for the general vectorial surface description given by equation (7.25). Then an infinitesimal area element on the curved surface

246 Radiative Heat Transfer

FIGURE 7.6 Rocket nozzle diffuser geometry for Example 7.2.

may be described by      ∂r  ∂r   ∂r    ∂r  dv1 dv2 = |ˆt1 × ˆt2 |    dv1 dv2 . × dA =  ∂v1 ∂v2  ∂v1   ∂v2 

(7.42)

Thus, if we replace x by v1 and y by v2 , emission points (v1 , v2 ) are readily found from nesting relations similar to equations (7.33) and (7.35). Example 7.2. Consider the axisymmetric rocket nozzle diffuser shown in Fig. 7.6. Assuming that the diffuser is gray and isothermal, establish the appropriate random number relationships for the determination of emission points. Solution The diffuser surface is described by the formula z = a(r2 − r20 ),

0 ≤ z ≤ L,

r0 ≤ r ≤ r L ,

a=

1 , 2r0

where L is the length of the diffuser and r0 and rL are its radius at z = 0 and L, respectively. In vectorial form, we may write ˆ r = xî + yˆj + zkˆ = r cos φî + r sin φˆj + a(r2 − r20 )k, where φ is the azimuthal angle in the x-y-plane, measured from the x-axis. This suggests the choice v1 = r and v2 = φ. The two surface tangents are now calculated from equation (7.28) as cos φî + sin φˆj + 2arkˆ , √ 1 + 4a2 r2 ˆt2 = − sin φî + cos φˆj.

ˆt1 =

It is seen that tˆ1 · ˆt2 = 0, i.e., the tangents are perpendicular to one another. The surface normal is then found from equation (7.29) as ⎛ ⎜⎜ î ⎜⎜ 1 ⎜⎜ nˆ = ˆt1 × tˆ2 = √ ⎜⎜ cos φ 1 + 4a2 r2 ⎜⎝ − sin φ

ˆj sin φ cos φ

⎞ kˆ ⎟⎟⎟ −2ar(cos φî + sin φˆj) + kˆ ⎟⎟ , √ 2ar⎟⎟⎟ = ⎟⎠ 1 + 4a2 r2 0

and, finally, an infinitesimal surface area is determined from equation (7.42) as √ ˆ − r sin φî + r cos φˆj| dr dφ = 1 + 4a2 r2 r dr dφ. dA = | cos φî + sin φˆj + 2ark|| Since there is no dependence on azimuthal angle φ in either dA or the emissive power, we find immediately

The Monte Carlo Method for Surface Exchange Chapter | 7 247

Rφ =

φ , 2π

or φ = 2πRφ ,

and for the radial position parameter r r √ (1 + 4a2 r2 )3/2 r 1 + 4a2 r2 r dr (1 + 4a2 r2 )3/2 − (1 + 4a2 r20 )3/2 r0 0 = = . Rr =  rL √ r L (1 + 4a2 r2L )3/2 − (1 + 4a2 r20 )3/2 (1 + 4a2 r2 )3/2 |r0 1 + 4a2 r2 r dr r 0

The above expression is readily solved to give an explicit expression for r = r(Rr ).

Wavelengths of Emission Once an emission location has been chosen, the wavelength of the emitted bundle needs to be determined (unless all surfaces in the enclosure are gray; in that case the wavelength of the bundle does not enter the calculations, and its determination may be omitted). The process of finding the wavelength has already been outlined in Section 7.1, leading to equation (7.3), i.e., 1 Rλ =

σT 4



λ

λ Ebλ dλ,

(7.43)

0

and, after inversion, λ = λ(Rλ , x, y).

(7.44)

We note that the choice of wavelength, in general, depends on the choice for the emission location (x, y), unless the surface is isothermal with constant emittance. If the surface is black or gray, equation (7.43) reduces to the simple case of 1 Rλ = σT 4



λ

Ebλ dλ = f (λT).

(7.45)

0

Directions of Emission The spectral emissive power (for a given position and wavelength) is  Eλ = 2π

λ Ibλ cos θ dΩ =

1 Ebλ π



2π



0

π/2

0

λ cos θ sin θ dθ dψ.

(7.46)

As we did for choosing the (two-dimensional) point of emission, we write Rψ =

Ebλ πEλ

 0

ψ

 0

π/2

λ cos θ sin θ dθ dψ =

1 π

 0

ψ



π/2 0

λ

λ

cos θ sin θ dθ dψ,

(7.47)

or ψ = ψ(Rψ , x, y, λ).

(7.48)

We note from equation (7.47) that ψ does not usually depend on emission location, unless the emittance changes across the surface. However, ψ does depend on the chosen wavelength, unless spectral and directional dependence of the emittance are separable. Once the azimuthal angle ψ is found, the polar angle θ is determined from .  π/2  θ 

λ cos θ sin θ dθ

λ cos θ sin θ dθ, (7.49) Rθ = 0

0

or θ = θ(Rθ , x, y, λ, ψ).

(7.50)

248 Radiative Heat Transfer

Most surfaces tend to be isotropic so that the directional emittance does not depend on azimuthal angle ψ. In  π/2 that case λ = 2 0 λ cos θ sin θ dθ, and equation (7.47) reduces to Rψ =

ψ , 2π

or

ψ = 2πRψ ,

(7.51)

and the choice of polar angle becomes independent of azimuthal angle. For a diffuse emitter, equation (7.49) simplifies to % Rθ = sin2 θ, or θ = sin−1 Rθ . (7.52)

Order of Evaluation In the foregoing we have chosen to first determine an emission location, followed by an emission wavelength and, finally, the direction of emission, as is most customary. However, the only constraint that we need to satisfy in a statistical manner is the total emitted energy from a surface, given by   ∞  4

σT dA =

λ Ibλ cos θ dΩ dλ dA. (7.53) E= A

A

0

2π

While we have obtained the random number relationships by peeling the integrals in equation (7.53) in the order shown, integration may be carried out in arbitrary order (e.g., first evaluating emission wavelength, etc.).

Absorption and Reflection When radiative energy impinges on a surface, the fraction αλ will be absorbed, which may depend on the wavelength of irradiation, the direction of the incoming rays, and, perhaps, the local temperature. Of many incoming bundles the fraction αλ will therefore be absorbed while the rest, 1 − αλ , will be reflected. This can clearly be simulated by picking a random number, Rα , and comparing it with αλ : If Rα ≤ αλ , the bundle is absorbed, while if Rα > αλ , it is reflected. The direction of reflection depends on the bidirectional reflection function of the material. The fraction of energy reflected into all possible directions is equal to the directional–hemispherical spectral reflectance, or   ρ ρλ (λ, θi , ψi ) = λ (λ, θi , ψi , θr , ψr ) cos θr dΩ r 

2π 2π



= 0

π/2

0

ρ λ (λ, θi , ψi , θr , ψr ) cos θr sin θr dθr dψr .

(7.54)

As before, the direction of reflection may then be determined from Rψr

1 =  ρλ





ψr

π/2

0

0

ρ λ (λ, θi , ψi , θr , ψr ) cos θr sin θr dθr dψr ,

(7.55)

and  Rθr = 

θr

ρ λ (λ, θi , ψi , θr , ψr ) cos θr sin θr dθr

0 π/2

0

ρ λ (λ, θi , ψi , θr , ψr ) cos θr

.

(7.56)

sin θr dθr

 (λ, θi , ψi , θr , ψr ) = ρ (λ) = ρλ (λ)/π, then equations (7.55) and (7.56) If the surface is a diffuse reflector, i.e., ρ λ λ reduce to Rψ r =

ψr , 2π

or ψr = 2πRψr ,

(7.57)

The Monte Carlo Method for Surface Exchange Chapter | 7 249

FIGURE 7.7 Vector description of emission direction and point of impact.

and Rθr = sin2 θr ,

or

θr = sin−1

% Rθr ,

(7.58)

which are the same as for diffuse emission. For a purely specular reflector, the reflection direction follows from the law of optics as ψr = ψi + π,

θr = θi ,

(7.59)

that is, no random numbers are needed.2

7.6 Ray Tracing Once a point of emission has been found, a wavelength and a direction are calculated from equations (7.43), (7.47), and (7.49). As shown in Fig. 7.7, the direction may be specified as a unit direction vector with polar angle θ measured from the surface normal, and azimuthal angle ψ measured from ˆt1 , leading to sˆ =

 sin θ  ˆ sin(α − ψ)ˆt1 + sin ψˆt2 + cos θn, sin α

(7.60)

and   sin α = ˆt1 × ˆt2  ,

(7.61)

where α is the angle between ˆt1 and ˆt2 . If ˆt1 and ˆt2 are perpendicular (α = π/2), equation (7.60) reduces to   ˆ sˆ = sin θ cos ψˆt1 + sin ψˆt2 + cos θn.

(7.62)

2. Mathematically, equation (7.59) may also be obtained from equations (7.55) and (7.56) by replacing ρ by an appropriate Dirac-delta λ function.

250 Radiative Heat Transfer

As also indicated in Fig. 7.7, the intersection point of an energy bundle emitted at location re , traveling into the direction sˆ , with a surface described in vectorial form may be determined as re + Dˆs = r,

(7.63)

where r is the vector describing the intersection point and D is the distance traveled by the energy bundle. In general, intersections with all possible surfaces in the enclosure must be sought, in order to find the correct one. Equation (7.63) may be written in terms of its x, y, z components and solved for D by forming the dot products ˆ with unit vectors î, jˆ, and k: D=

y(v1 , v2 ) − ye z(v1 , v2 ) − ze x(v1 , v2 ) − xe = = . sˆ · î sˆ · ˆj sˆ · kˆ

(7.64)

Equation (7.64) is a set of three equations in the three unknowns v1 , v2 , and D: First v1 and v2 are calculated, and it is determined whether the intersection occurs within the confines of the surface under scrutiny. If so, and if more than one intersection is a possibility (in the presence of convex surfaces, protruding corners, etc.), then the path length D is also determined; if more than one intersection is found, the correct one is the one after the shortest positive path. If the bundle is reflected, and if reflection is nonspecular, a reflection direction is chosen from equations (7.55) and (7.56). This direction is then expressed in vector form using equation (7.60). If the surface is a specular reflector (or a plane of symmetry), the direction of reflection is determined from equation (7.59), or in vector form as ˆ n. ˆ sˆ r = sˆ i + 2|ˆs i · n|

(7.65)

Once the intersection point and the direction of reflection have been determined, a new intersection may be found from equation (7.64), etc., until the bundle is absorbed. Example 7.3. Consider again the geometry of Example 7.2. An energy bundle is emitted from the origin (x = y = z = 0) ˆ Determine the intersection point on the diffuser and the direction of reflection, assuming into the direction sˆ = 0.8î + 0.6k. the diffuser to be a specular reflector. Solution With re = 0 and equations (7.60) and (7.64), we find D=

r cos φ r sin φ a(r2 − r20 ) = = . 0.8 0 0.6

Obviously, φ = 0,3 and solving the quadratic equation for r, r2 − r20 =

3r 1 = 3 rr0 , or r = 2r0 and z = (4r2 − r20 ) = 32 r0 . 4a 2 2r0 0

At that location we form the unit vectors as given in Example 7.2, 1 ˆ ˆt2 = ˆj, and nˆ = √1 (−2î + k). ˆ ˆt1 = √ (î + 2k), 5 5 Therefore, the direction of reflection is determined from equation (7.65) as    −2 × 0.8 + 0.6  −2î + kˆ ˆ ˆ sˆ r = 0.8î + 0.6k + 2  = k, √  √  5 5 as is easily verified from Fig. 7.6.

For tesselated planar surfaces, it is generally more convenient to formulate the intersection problem in terms of the only known quantities, namely the coordinates of the vertices of a face and the direction vector of the emitted ray. Since all planar surfaces can be split into nonoverlapping triangles, it is sufficient to find the 3. In computer calculations care must be taken here and elsewhere to avoid division by zero.

The Monte Carlo Method for Surface Exchange Chapter | 7 251

intersection of a ray with a triangular face. Therefore, the vector describing the intersection point, following Fig. 7.4a and equation (7.27), follows as r = re + D sˆ = rC + μ1 a + μ2 b.

(7.66)

Equation (7.66) represents a set of three linear equations with unknowns D, μ1 , and μ2 . The criteria that the intersection point lies within the triangle (including its edges and vertices) are D > 0, μ1 ≥ 0, μ2 ≥ 0, and μ1 + μ2 ≤ 1. During ray tracing, each candidate boundary face is visited and the system of equations is solved whereby D is first calculated. If D ≤ 0, the tracing proceeds to the next candidate face. Else, μ1 and μ2 are also determined, and the remaining criteria are checked. If all criteria are satisfied, the intersection point is found by using equation (7.66). If multiple intersections are possible, the remaining surfaces must still be checked. If more than one intersection is found, the correct one is the one with shortest positive path D. Example 7.4. Consider a triangle with vertices C(0,0,0), A(0,0,1), and B(0,1, 12 ), and a ray emitted from a point E( 14 , 14 ,0) ˆ Does this ray intersect the triangle? If so, determine the intersection point. with direction vector sˆ = − 13 î + 23 ˆj + 23 k. Solution ˆ The vectors a and b are readily determined from the coordinates of the vertices as a = rA − rC = kˆ and b = rB − rC = ˆj + 12 k. Substituting the necessary vectors into the second part of equation (7.66), we get ˆ = 0 + μ1 kˆ + μ2 [ˆj + 1 k]. ˆ [ 14 î + 14 ˆj] + D [− 13 î + 23 jˆ + 23 k] 2 Separating the three components of the above vector equation, we obtain 1 4

− 13 D = 0,

1 4

+ 23 D = μ2 ,

and

2 3

D = μ1 + 12 μ2 ,

resulting in D = 34 , μ1 = 18 , μ2 = 34 , and μ1 + μ2 = 78 . Hence, all necessary criteria for intersection are satisfied, and the position vector of the intersection point is calculated using equation (7.66) as ˆ = 3 jˆ + 1 k. ˆ rP = re + D sˆ = [ 14 î + 14 jˆ] + 34 [− 13 î + 23 jˆ + 23 k] 4 2 Thus, the point of intersection of the ray with the triangle is P(0, 34 , 12 ).

7.7 Efficiency Considerations The efficiency of a surface-to-surface Monte Carlo calculation is dictated by two issues: the time taken to sample and trace each photon bundle or ray, and the number of rays that need to be traced to arrive at a preset statistical error, typically measured by the variance [equation (7.8)]. The time taken to sample a ray is generally governed by the complexity of the nonlinear random number relations that need to be inverted, while the time taken to trace a ray is governed by how many ray–surface intersection calculations (checks) need to be performed. Finally, the statistical error (variance) depends on how much energy each photon bundle is carrying, and may be reduced either by infusing deterministic ideas into an otherwise statistical calculation, or by using wellestablished variance-reduction techniques, such as importance sampling [cf. equation (7.13)]. These issues are discussed next.

Inversion of Random Number Relations Many of the random number relationships governing emission location, wavelength, direction, etc., cannot be inverted explicitly. For example, to determine the wavelength of emission, even for a simple black surface, for a given random number Rλ requires the solution of the transcendental equation (7.43),  λ 1 Rλ = Ebλ dλ = f (λT). (7.67) σT 4 0 In principle, this requires guessing a λ, calculating Rλ , etc. until the correct wavelength is found; this would then be repeated for each emitted photon bundle. It would be much more efficient to invert equation (7.67) once and for all before the first energy bundle is traced as λT = f −1 (Rλ ).

(7.68)

252 Radiative Heat Transfer

This is done by first calculating Rλ, j corresponding to a (λT) j for a sufficient number of points j = 0, 1, . . . , J. These data points may then be used to obtain a polynomial description λT = A + BRλ + CR2λ + · · · ,

(7.69)

as proposed by Howell [13]. With the math libraries available today on most digital computers it would, however, be preferable to invert equation (7.68) using a (cubic) spline. Even more efficient is the method employed by Modest and Poon [33] and Modest [34], who used a cubic spline to determine values of (λT) j for (J + 1) equally spaced random numbers   j −1 (λT) j = f Rλ = , j = 0, 1, 2, . . . , J. (7.70) J If, for example, a random number Rλ = 0.6789 is picked, it is immediately known that (λT) lies between (λT)m and (λT)m+1 , where m is the largest integer less than J × Rλ (= 67 if J = 100). The actual value for (λT) may then be found by (linear) interpolation. The quantity to be determined may depend on more than a single random number. For example, to fix an emission wavelength on a surface with nonseparable emissive power (say a surface in the x-y-plane with locally varying, nongray emittance) requires the determination of x = x(Rx ),

y = y(R y , x),

λ = λ(Rλ , x, y).

(7.71)

That is, first the x-location is chosen, requiring the interpolation between and storage of J data points x j (R j ); next the y-location is determined, requiring a double interpolation and storage of a J × K array for y jk (Rk , x j ); and finally λ is found from a triple interpolation from a J × K × L array for λ jkl (Rl , x j , yk ). This may lead to excessive computer storage requirements if J, K, L are chosen too large: If J = K = L = 100, an array with one million numbers needs to be stored for the determination of emission wavelengths alone! The problem may be alleviated by choosing a better interpolation scheme together with smaller values for J, K, L (for example, a choice of J = K = L = 40 reduces storage requirements to 64,000 numbers).

Energy Partitioning In the general Monte Carlo method, a ray of fixed energy content is traced until it is absorbed. In the absence of a participating medium, the decision whether the bundle is absorbed or reflected is made after every impact on a surface. Thus, on the average it will take 1/α tracings until the bundle is absorbed. Therefore, it takes 1/α tracings to add one statistical sample to the calculation of one of the Fi→ j ’s. If the configuration has openings, a number of bundles may be reflected a few times before they escape into space without adding a statistical sample to any of the Fi→ j ’s. Thus, the ordinary Monte Carlo method becomes extremely inefficient for open configurations and/or highly reflective surfaces. The former problem may be alleviated by partitioning the energy of emitted bundles. This was first applied by Sparrow and coworkers [35,36], who, before determining a direction of emission, split the energy of the bundle into two parts: the part leaving the enclosure through the opening (equal to the view factor from the emission point to the opening) and the rest (which will strike a surface). A direction is then determined, limited to those that make the bundle hit an enclosure surface. The procedure is repeated after every reflection. This method guarantees that each bundle will contribute to the statistical sample for exchange factor evaluation. A somewhat more general and more easily implemented energy partitioning scheme was applied by Modest and Poon [33,34]: Rather than drawing a random number Rα to decide whether a bundle is (fully) absorbed or not, they partition the energy of a bundle at each reflection into the fraction α, which is absorbed, and the fraction ρ = 1 − α, which is reflected. The bundle is then traced until it either leaves the enclosure or until its energy is depleted (below a certain fraction of original energy content). This method adds to the statistical sample of an Fi→ j with every traced ray and thus leads to vastly faster statistical convergence for highly reflective surfaces.

Acceleration of Photon Bundle (Ray) Tracing In surface-to-surface Monte Carlo calculations, the vast majority of the computational time is spent on tracing the photon bundles or rays. In such calculations, the central computational issue is the determination of the

The Monte Carlo Method for Surface Exchange Chapter | 7 253

intersection point or points between an infinite ray and the boundary, which often consists of a large set of discrete surface elements or faces. If N rays are to be traced, and the computational domain boundary is comprised of M faces, the number of intersection calculations (or checks) that need to be performed is N × M. In general, all M surface elements or faces must be checked since a propagating ray may potentially intersect more than one face. This then has to be followed by a shortest-positive-distance check to determine the legitimate intersection point. In 2D, each intersection calculation (or check) requires 4 long floating point operations (multiplication or division), while in 3D, it requires 12 long operations, in addition to several logical checks, rendering intersection calculations the most time-consuming aspect. In the computer graphics literature, many advanced algorithms are available to accelerate ray tracing. Rather than conduct a “brute force” intersection check of all the boundary faces—henceforth referred to as the direct method—these algorithms first narrow down the search space (potential faces to be checked for intersection) by using efficient search algorithms. Broadly, algorithms used for acceleration of ray tracing in the context of surface-to-surface Monte Carlo calculations can be categorized into the following types: (1) the Bounding Box (BB) algorithm [37–41], (2) the Binary Spatial Partitioning (BSP) algorithm [42–45], (3) the Uniform Spatial Division (USD) algorithm [39,40,46], and (4) the Volume-by-Volume Advancement (VVA) algorithm [47]. Of the four, the first three algorithms share the common philosophy of narrowing the search using coarser geometric entities, while the fourth algorithm makes use of a volumetric mesh. Bounding Box (BB) Algorithm. In the BB algorithm [37–41], contiguous sets of boundary faces are first enclosed within larger bounding boxes. For example, if ray tracing is conducted inside a cubical enclosure, each of the six surfaces of the cube may be enclosed by a bounding box. Generally, large boundary patches with uniform boundary condition are enclosed in separate boxes. The bounding boxes are constructed by scanning all the vertices of each face that constitutes the boundary patch and using the minimum and maximum x, y, and z values. The result is a box whose faces are aligned to the Cartesian planes. Next, ray–box intersection checks are performed. An extremely efficient algorithm proposed by Kay and Kajiya [48] is used for this purpose. This algorithm requires only 12 floating point operations to determine if a ray strikes a box (i.e., the same number of floating point operations needed to search just one face in 3D). Once it has been determined which box (or boxes in the case of geometry with obstructions) is the target box, detailed search (ray-face intersection calculation) of only those faces that are enclosed within the target box is conducted. The BB algorithm has been used for surfaceto-surface radiation Monte Carlo calculations by Mazumder and Kersch [41], and has been shown to result in substantial computational savings over the direct method for both canonical problems (such as box-in-box) as well as for full-scale rapid thermal processing reactors, although rarely exceeding one order of magnitude. In the BB algorithm, if NBB bounding boxes are used, the number of actual intersection calculations, on an average, will be N × (M/NBB + NBB ). Hence, the computational time requirement will be 1/NBB times that required for direct calculations, assuming that the overheads, such as time spent in calculating ray–box intersections is negligible. As NBB becomes large, this will clearly not be the case. Also, in a domain with obstructions, since a given ray may intersect several boxes as it passes through, more than one box will have to be searched to guarantee that the nearest intersection point is found. All these factors slow down the BB algorithm. Its efficiency, although substantially superior to the direct method, rarely scales as the number of bounding boxes used. Its advantage over other methods is that it is extremely easy to implement. Binary Spatial Partitioning (BSP) Algorithm. The BSP algorithm is probably the most widely used algorithm for ray tracing in computer graphics. In the BSP algorithm [42–45,47], all faces within the computational domain are first placed into a large box (or voxel) whose planes are aligned with Cartesian planes. The box is then recursively bisected along the Cartesian directions, resulting in a set of hierarchical sub-boxes. The relationship between each parent box and its two “children” sub-boxes are stored in the form of a binary tree, along with information about their bounding planes and the enclosed faces. The bisection stops when a box has only a few faces, typically less than 10. During ray tracing, ray–box intersections are first identified using the aforementioned Kay and Kajiya algorithm [48]. As opposed to the BB algorithm, not all boxes are checked for intersection. Starting from the largest box, only pertinent sub-boxes are checked by traversing the binary tree. In theory, the BSP algorithm requires only O(N × log2 M) intersection calculations. In practice, perfect logarithmic scaling is difficult to attain because of the possibility of multiple intersection points, and lack of a perfectly balanced tree, i.e., the two sub-boxes of a box may not have equal numbers of faces. In general, as the number of faces, M, is increased, (log2 M)/M decreases strongly, implying that the computational benefits of the BSP algorithm increase as the problem size increases. Thus, the BSP algorithm is particularly well-suited for practical applications involving large number of mesh faces. Details pertaining to the implementation of this algorithm for surface-to-surface Monte Carlo calculations may be found in Mazumder [47], who performed ray tracing

254 Radiative Heat Transfer

in an open unit cube and a 0.6 × 0.6 × 0.6 cube inscribed inside the unit cube. Calculations were conducted for meshes with M up to ∼50,000. Computational time was found to be reduced by factors of about 37 and 52 for the open cube and cube-in-cube cases, respectively. In a more recent study, Naeimi and Kowsary [49] observed a factor of 39 improvement in computational time when using the BSP algorithm for two different 3D enclosures with obstructions and with M ∼ 20, 000. Uniform Spatial Division (USD) Algorithm. While the BSP algorithm is efficient, its implementation is not trivial: it requires advanced data structures, and recursive procedures, among other complexities. In the Uniform Spatial Division (USD) algorithm [39,40,46], a uniform coarse Cartesian mesh is superposed on the computational domain solely for the purpose of ray tracing. The mesh essentially breaks up the computational domain into a set of equal sized Cartesian boxes. Rays are then traced from box to box, and larger intersected boxes are subsequently searched in detail to locate intersection points. The algorithm has been used extensively by Burns and coworkers [25,46] in their code MONT3D. In refined versions of the method, particular emphasis has been placed on accelerating the process of identifying the “next” box as a ray exits a certain box. The so-called “mailbox technique” has also been suggested to accelerate ray tracing by storing information about previously traced rays and using this information later. With all of these improvements in place, computational gain of a factor of 80 has been reported for complex 3D geometries by Zeeb et al. [46]. Volume-by-Volume Advancement (VVA) Algorithm. First proposed by Mazumder [47] for surface-tosurface Monte Carlo calculations, this algorithm recognizes the fact that such radiation calculations are rarely performed without accompanying heat transfer calculations that typically require a volumetric mesh. Consequently, this same mesh may be utilized for ray tracing. In this algorithm, an emitted ray enters the cell adjacent to it. As it exits, its intersection with one of the bounding faces is found. This intersection point then becomes the next emission location and the process is continued until the ray finally strikes a boundary face. Not only is this boundary face a potential intersection point, but also it is the nearest intersection point. In other words, multiple intersection point determination, followed by shortest distance check, is not required in this method. This is an important advantage of the VVA algorithm over the BB, the BSP, and the USD algorithms. Another crucial advantage of the VVA concept is that the number of volumes (or cells) that a ray passes through, scales with the number of grid points only in one direction even for a 3D geometry, while the total number of boundary faces to be searched in other algorithms scales as the square of the number of grid points in one direction. Thus, for completely unobstructed geometries, the VVA algorithm is an O(N × M1/2 ) algorithm. For geometries with obstructions, as is the case in most practical applications, its performance is expected to be much better. For the aforementioned open cube and cube-in-cube test cases, Mazumder [47] found the VVA algorithm to produce computational improvements by factors of 131 and 335, respectively. Clearly, this algorithm is most suited for geometries with obstructions. Naeimi and Kowsary [49] compared the BSP, the USD, and the VVA algorithms for two different 3D furnace-like enclosures, and also found that the VVA algorithm outperformed the other two algorithms by a significant margin (factor of 99–201 gain), USD being the second best (factor of 79–137 gain). The VVA algorithm requires no pre-processing and is easy to implement for any mesh topology—structured or unstructured. In participating media, photon bundles must be traced through the volumetric mesh (see Chapter 20 for details), implying that the VVA algorithm is a special case of the general ray tracing algorithm used for Monte Carlo calculations in participating media.

Deviational Monte Carlo In many practical situations, even tracing a large number of rays fails to produce meaningful results. For example, consider a radiant furnace in which a cold surface faces away from the heater and directly sees only other cold surfaces at the same temperature. If the emittance of all surfaces in the furnace is close to that of a blackbody, many rays will have to be traced and many reflection events will have to occur before some rays from the heater find their way to the target cold surface. Theoretically, the heat flux on this cold surface must be negative (pointing into the surface) since it is surrounded by other cold surfaces at the same temperature and there is at least one surface in the enclosure that is hotter than itself. In a Monte Carlo calculation, whether the net heat flux on this cold surface will be positive or negative will be determined by a delicate balance of how many rays are emitted from it vs. how many rays strike it. Surplus or deficit of even a single ray can result in the heat flux to have a wrong sign and/or a large error in its computed value. Deviational Monte Carlo is a technique to reduce the statistical error or variance by using a so-called control function, and is sometimes also referred to as control variate Monte Carlo. To apply this technique to surface-to-

The Monte Carlo Method for Surface Exchange Chapter | 7 255

surface Monte Carlo calculations, we first rewrite equation (7.16) for M subsurfaces and without any external radiation: Qi = i Ebi Ai −

M 

j Ebj A j Fj→i ,

1 ≤ i ≤ M.

(7.72)

j=1

If all surfaces of the enclosure are at the same temperature TC , then, by definition, the heat flux on all surfaces must be zero regardless of what surface properties and exchange factors are used. Therefore, 0 = i EbC Ai −

M 

j EbC A j Fj→i ,

1 ≤ i ≤ M,

(7.73)

j=1

where EbC = Eb (TC ). Henceforth, we will refer to TC as the control temperature. Subtracting equation (7.73) from equation (7.72) results in Qi = i [Ebi − EbC ]Ai −

M 

j [Ebj − EbC ]A j Fj→i ,

1 ≤ i ≤ M.

(7.74)

j=1

Equation (7.74) is identical to equation (7.72), except that the energy emitted by each surface, i Ebi Ai , has been replaced by i [Ebi − EbC ]Ai , or the deviational energy. Equation (7.74) can be solved by using the exact same Monte Carlo procedure as for equation (7.72) wherein the actual energy emitted by each subsurface is replaced by its deviational energy. As far as tracing of the rays is concerned, no change is needed. Prior to discussing results produced by the deviational Monte Carlo method, it is instructive to examine why it may produce results that are more accurate. In the deviational Monte Carlo method, each ray carries deviational energy, which, by definition, is smaller than the actual energy. In any Monte Carlo calculation, the ideal scenario, i.e., when variance is zero, is realized when an infinite number of rays are traced and each ray carries an infinitesimal amount of energy. Since the deviational scheme always reduces the energy carried by each ray, it is expected to always produce smaller variance than standard Monte Carlo. If the temperatures of the vast majority of surfaces in the enclosure are close to the control temperature, then the deviational energy carried by each ray will be significantly smaller than the actual energy, and the benefits of the deviation scheme are expected to be largest. Otherwise, the deviational scheme may produce only marginal benefits. Since the ray tracing procedure is unaltered by this scheme, no additional computational time is needed to execute the deviational scheme over the standard scheme. Hence, in the worst-case scenario, deviational Monte Carlo will default to the standard method without any computational penalty. One critical question in deviational Monte Carlo is the choice of the control temperature. While no particular temperature is disallowed, choosing a temperature that keeps the deviational energy positive is helpful in interpreting the computed results. Mazumder [50] employed the minimum temperature in the computational domain, i.e., TC = min(Ti ). This choice was prompted by previous studies in phonon transport [51], and has also been used in deviational Monte Carlo calculations in participating media by Soucasse et al. [52] (see Chapter 20 for details). With this choice, the two schemes were compared for a square enclosure with a heated patch in the middle of one of the walls and various boundary conditions, including coupled radiation-convection cases. Figure 7.8 shows sample results from [50] for a case when TH = 600 K and TC = 300 K. Only 103 photon bundles were traced to generate these results so that the statistical error bars, corresponding to one standard deviation, are clearly visible. It is observed that the statistical errors (variance) are significantly reduced by the deviational Monte Carlo method.

Data Smoothing Virtually all Monte Carlo implementations to date have been of 0th order, i.e., all properties within a given cell are considered constant throughout the cell, without connectivity to surrounding cells. This makes the estimation of local gradients difficult, if not impossible. Several smoothing schemes have been proposed for the exchange factors of equation (7.21), the simpler ones without restrictions on the size of corrections [27,28], and others that find the smallest corrections that make the exchange factors satisfy, both, the summation and reciprocity relationships [29,30].

256 Radiative Heat Transfer

FIGURE 7.8 Comparison of standard and deviational Monte Carlo for a surface-to-surface radiation exchange problem; reproduced from [50]: (a) geometry and boundary conditions, (b) heat flux on bottom wall, and (c) heat flux on right wall.

Problems Because of the nature of the Monte Carlo technique, most of the following problems require the development of a small computer code. However, all problem solutions can be outlined by giving relevant relations, equations, and a detailed flow chart. 7.1 Prepare a little Monte Carlo code that integrates I(z) = 

π/2

si(z) = −

b a

f (z, x) dx. Apply your code to a few simple integrals, plus

e−z cos x cos(z sin x) dx = Si(z) −

0

π . 2

Note: Si(1) = 0.94608. 7.2 In a Monte Carlo simulation involving the plate of Problem 3.9 but of finite width w, a photon bundle is to be emitted from the plate with a wavelength of λ = 2 μm. Find the emission point and direction of this photon bundle in terms of random numbers. 7.3 A triangular, isothermal surface as shown has the following spectral emittance: ⎧ ⎪ 0.1, λ < 2μm; θ ≤ 60◦ ⎪ ⎪ ⎪ ⎨

λ = ⎪ 0.6, λ > 2μm; θ ≤ 60◦ ⎪ ⎪ ⎪ ⎩0.0, all λ; θ > 60◦ For a Monte Carlo simulation (a) find a point of bundle emission in terms of random numbers, (b) find a wavelength of bundle emission in terms of random numbers, and (c) find a direction of bundle emission in terms of random numbers. 7.4 A semicircular disk as shown has a temperature distribution given by T(r) =   T0 / 1 + (r/R)2 , and its emittance is gray and nondiffuse with /

 = λ (λ, θ, ψ) =

0.6, 0,

0 ≤ θ ≤ 30◦ , θ > 30◦ .

For a Monte Carlo simulation (a) find a point of emission in terms of random numbers, (b) find a direction of emission in terms of random numbers. You may leave your answer in simple implicit form. 7.5 A light pipe with direct solar irradiation is to be investigated via a Monte Carlo method. Such a device consists of a straight or curved tube covered with a highly reflective material to pipe light into a room. At visible wavelengths the reflectance from the pipe wall is ρλ (θout ) = 1.5ρλ cos θout , with reflection angle θout measured from the local surface

The Monte Carlo Method for Surface Exchange Chapter | 7 257

normal, and visible light intensity due to direct sunshine may be approximated by Lλ = Kλ Iλ,sun = C exp[−A2 (λ − λ0 )2 ], λ0 = 0.56μm, A = 20/μm. (a) Find the pertinent relationship to determine wavelengths of emission as a function of random number. (b) Find an expression for reflection angle vs. random number. 7.6 Determine the intersection point of a ray with direction vector sˆ = 13 î + 23 ˆj + 23 kˆ and emitted from the origin, with the inside surface of a unit cube whose centroid is located at the origin. If the ray gets reflected specularly upon striking the walls of the cube, determine the next intersection point. 7.7 At the Aaronsburg (Pennsylvania) Apple Fest you have won a large piece of elderberry pie (yumh!) as shown. The wheels in the oven must have been spinning, because it appears that the number of elderberries per unit area increases linearly proportional with radius! If there are 1000 elderberries otherwise randomly distributed on the slice, make a scatter plot of elderberries on the pie slice.

2

7.8 Consider a black disk 0 ≤ r ≤ R with temperature distribution T4 (r) = T04 e−C(r/R) . Develop the random number relations for points of emission; draw random numbers for 1000 emission points and draw them in a scattergram for the cases of C = 0 and C = 5. Use R = 10 cm. 7.9 A disk of radius R is opposed by a square plate (sides of length R) parallel to it, and a distance R away. Find the view factor from disk to square plate. Use 100,000 bundles, plotting updated results after every 5,000 bundles. 7.10 Consider two infinitely long parallel plates of width w spaced a distance h apart (see Configuration 32 in Appendix D). (a) Calculate F1−2 via Monte Carlo for the case that the top plate is horizontally displaced by a distance L. Use L = h = w. s via Monte Carlo for the case that both plates are specular (with identical reflectances ρs1 = ρs2 = 0.5), (b) Calculate F1−2 but not horizontally displaced. Use L = 0, h = w. Prepare a figure similar to Fig. 7.2, also including analytical results for comparison. 7.11 Two directly opposed quadratic plates of width w = 10 cm are spaced a distance L = 10 cm apart, with a third centered quadratic plate of dimension b × b (b = 5 cm) in between at a distance l = 5 cm from the bottom. Determine the view factor F1−2 via Monte Carlo. In order to verify your code (and to have a more flexible tool) it may be best to allow for arbitrary and different top and bottom w as well as b.

7.12 Consider two concentric parallel disks of radius R, spaced a distance H apart. Both plates are isothermal (at T1 and T2 , respectively), are gray diffuse emitters with emittance , and are gray reflectors with diffuse reflectance component ρd and purely specular component ρs . Write a computer code that calculates the generalized exchange factor F1→2 and, taking advantage of the fact that F1→2 = F2→1 , calculate the total heat loss from each plate. Compare with the analytical solution treating each surface as a single node. 7.13 Repeat Problem 7.12, but calculate heat fluxes directly, i.e., without first calculating exchange factors. 7.14 Determine the view factor for Configuration 39 of Appendix D, for h = w = l. Compare with exact results. 7.15 Consider the conical geometry of Problem 5.11: breaking up the sidewall into strips (say 4), calculate all relevant view factors (base-to-rings, ring-to-rings) via Monte Carlo.

258 Radiative Heat Transfer

7.16 Reconsider Problem 5.7: (a) find the solution by writing a small Monte Carlo program and (b) augment this program to allow for nongray, temperature-dependent emittances. 7.17 Repeat Problem 5.41 for T1 = T2 = 1000 K, 1 = 2 = 0.5. Use the Monte Carlo method, employing the energy partitioning of Sparrow and coworkers [35,36]. 7.18 Repeat Problem 5.42. Compare with the exact solutions for several values of . 7.19 Repeat Problem 6.2, using the Monte Carlo method. Compare with the solution from Chapter 6 for a few values of D/L and , and T1 = 1000 K, T2 = 2000 K. How can the problem be done by emitting bundles from only one surface? 7.20 Repeat Problem 6.8 using the Monte Carlo method. 7.21 Repeat Example 6.10 using the Monte Carlo method. 7.22 Repeat Example 6.11 using the Monte Carlo method. 7.23 Repeat Problem 6.45 using the Monte Carlo method. 7.24 Write a computer program to calculate the view factor for Example 4.15 using the Monte Carlo method. 7.25 Consider two 1 m × 1 m diffuse parallel plates at T1 = 1500 K, 1 = 0.5 and T2 = 1000 K, 2 = 0.2. The plates are spaced 1 m apart. (a) Write a computer program to calculate q1−2 = Q1−2 /A1 using the Monte Carlo method. (b) Repeat Part (a) using the deviational Monte Carlo method. Hint: For best statistics, all photon bundles should carry (approximately) the same amount of energy. 7.26 Consider a gray-diffuse spherical enclosure. The upper half of the enclosure is at constant temperature T1 = 1500 K with emittance 1 = 0.5, and the lower half is constant temperature T2 = 1000 K with emittance 2 = 0.2. The diameter of the sphere is 1 m. (a) Write a computer program to calculate q1−2 = Q1−2 /A1 using the Monte Carlo method; compare your answer to results from the net radiation method (cf. Problem 5.43). (b) Repeat Part (a) using the deviational Monte Carlo method.

References [1] J.M. Hammersley, D.C. Handscomb, Monte Carlo Methods, John Wiley & Sons, New York, 1964. [2] J.A. Fleck, The calculation of nonlinear radiation transport by a Monte Carlo method, Technical Report UCRL-7838, Lawrence Radiation Laboratory, 1961. [3] J.A. Fleck, The calculation of nonlinear radiation transport by a Monte Carlo method: Statistical physics, Methods in Computational Physics 1 (1961) 43–65. [4] J.R. Howell, M. Perlmutter, Monte Carlo solution of thermal transfer through radiant media between gray walls, ASME Journal of Heat Transfer 86 (1) (1964) 116–122. [5] J.R. Howell, M. Perlmutter, Monte Carlo solution of thermal transfer in a nongrey nonisothermal gas with temperature dependent properties, AIChE Journal 10 (4) (1964) 562–567. [6] M. Perlmutter, J.R. Howell, Radiant transfer through a gray gas between concentric cylinders using Monte Carlo, ASME Journal of Heat Transfer 86 (2) (1964) 169–179. [7] E.D. Cashwell, C.J. Everett, A Practical Manual on the Monte Carlo Method for Random Walk Problems, Pergamon Press, New York, 1959. [8] Y.A. Schreider, Method of Statistical Testing – Monte Carlo Method, Elsevier, New York, 1964. [9] H. Kahn, Applications of Monte Carlo, Report for Rand Corp., vol. Rept. No. RM-1237-AEC (AEC No. AECU-3259), 1956. [10] G.W. Brown, Monte Carlo methods, in: Modern Mathematics for the Engineer, McGraw-Hill, New York, 1956, pp. 279–307. [11] J.H. Halton, A retrospective and prospective survey of the Monte Carlo method, SIAM Rev. 12 (1) (1970) 1–63. [12] A. Haji-Sheikh, Monte Carlo methods, in: Handbook of Numerical Heat Transfer, John Wiley & Sons, New York, 1988, pp. 673–722. [13] J.R. Howell, Application of Monte Carlo to heat transfer problems, in: J.P. Hartnett, T.F. Irvine (Eds.), Advances in Heat Transfer, vol. 5, Academic Press, New York, 1968. [14] D.V. Walters, R.O. Buckius, Monte Carlo methods for radiative heat transfer in scattering media, Annual Review of Heat Transfer, vol. 5, Hemisphere, New York, 1992, pp. 131–176. [15] J.R. Howell, The Monte Carlo method in radiative heat transfer, ASME Journal of Heat Transfer 120 (3) (1998) 547–560. [16] O. Taussky, J. Todd, Generating and testing of pseudo-random numbers, in: Symposium on Monte Carlo Methods, John Wiley & Sons, New York, 1956, pp. 15–28. [17] W.H. Press, B.P. Flannery, S.A. Tenkolsky, W.T. Vetterling, Numerical Recipes – The Art of Scientific Computing, 1st ed., 1989, Cambridge, New York. [18] B. Baker, A note on the proper use of the numerical recipes RAN1 random number generator, Computational Statistics & Data Analysis 25 (2) (1997) 237–239.

The Monte Carlo Method for Surface Exchange Chapter | 7 259

[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52]

N.C. Barford, Experimental Measurements: Precision, Error and Truth, Addison-Wesley, London, 1967. C.E. Fröberg, Introduction to Numerical Analysis, Addison-Wesley, Reading, MA, 1969. M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover Publications, New York, 1965. J.D. Maltby, Three-dimensional simulation of radiative heat transfer by the Monte Carlo method, M.S. thesis, Colorado State University, Fort Collins, CO, 1987. P.J. Burns, J.D. Maltby, Large-scale surface to surface transport for photons and electrons via Monte Carlo, Computing Systems in Engineering 1 (1) (1990) 75–99. J.D. Maltby, P.J. Burns, Performance, accuracy and convergence in a three-dimensional Monte Carlo radiative heat transfer simulation, Numerical Heat Transfer – Part B: Fundamentals 16 (1991) 191–209. C.N. Zeeb, P.J. Burns, K. Branner, J.S. Dolaghan, User’s manual for Mont3d – Version 2.4, Colorado State University, Fort Collins, CO, 1999. M.E. Larsen, J.R. Howell, Least-squares smoothing of direct-exchange areas in zonal analysis, ASME Journal of Heat Transfer 108 (1) (1986) 239–242. J. van Leersum, A method for determining a consistent set of radiation view factors from a set generated by a nonexact method, International Journal of Heat and Fluid Flow 10 (1) (1989) 83. D.A. Lawson, An improved method for smoothing approximate exchange areas, International Journal of Heat and Mass Transfer 38 (16) (1995) 3109–3110. R.I. Loehrke, J.S. Dolaghan, P.J. Burns, Smoothing Monte Carlo exchange factors, ASME Journal of Heat Transfer 117 (2) (1995) 524–526. K.J. Daun, D.P. Morton, J.R. Howell, Smoothing Monte Carlo exchange factors through constrained maximum likelihood estimation, ASME Journal of Heat Transfer 127 (10) (2005) 1124–1128. J.S. Toor, R. Viskanta, A numerical experiment of radiant heat exchange by the Monte Carlo method, International Journal of Heat and Mass Transfer 11 (5) (1968) 883–887. M.M. Weiner, J.W. Tindall, L.M. Candell, Radiative interchange factors by Monte Carlo, ASME paper no. 65-WA/HT-51, 1965. M.F. Modest, S.C. Poon, Determination of three-dimensional radiative exchange factors for the space shuttle by Monte Carlo, ASME paper no. 77-HT-49, 1977. M.F. Modest, Determination of radiative exchange factors for three dimensional geometries with nonideal surface properties, Numerical Heat Transfer 1 (1978) 403–416. R.P. Heinisch, E.M. Sparrow, N. Shamsundar, Radiant emission from baffled conical cavities, Journal of the Optical Society of America 63 (2) (1973) 152–158. N. Shamsundar, E.M. Sparrow, R.P. Heinisch, Monte Carlo solutions — effect of energy partitioning and number of rays, International Journal of Heat and Mass Transfer 16 (1973) 690–694. J. Foley, A. Van Dam, S. Feiner, J. Hughes, Computer Graphics Principles and Practice, Addison-Wesley Publishing Company, 1990. J. Arvo, D. Kirk, A Survey of Ray Tracing Acceleration Techniques, Academic Press, 1989. A.S. Glassner, Space subdivision for fast ray tracing, IEEE Computer Graphics Applications 4 (1984) 15–22. A.S. Glassner, An Introduction to Ray Tracing, Academic Press, 1989. S. Mazumder, A. Kersch, A fast Monte Carlo scheme for thermal radiation in semiconductor processing applications, Numerical Heat Transfer – Part B: Fundamentals 37 (2) (2000) 185–199. K. Sung, P. Shirley, Ray Tracing with a BSP Tree, AP Professional, 1992. N. Chin, A Walk Through BSP Trees, AP Professional, 1995. J.D. MacDonald, K.S. Booth, Heuristics for ray tracing using space subdivision, The Visual Computer: International Journal of Computer Graphics 6 (1990) 153–166. V. Havran, T. Kopal, J. Bittner, J. Zara, Fast robust BSP tree traversal algorithm for ray tracing, Journal of Graphics Tools 2 (1998) 15–23. C.N. Zeeb, J.S. Dolaghan, P.J. Burns, An efficient Monte Carlo particle tracing algorithm for large, arbitrary geometries, Numerical Heat Transfer – Part B: Fundamentals 39 (4) (2001) 325–344. S. Mazumder, Methods to accelerate ray tracing in the Monte Carlo method for surface-to-surface radiation transport, ASME Journal of Heat Transfer 128 (9) (2006) 945–952. T.L. Kay, J.T. Kajiya, Ray tracing complex scenes, Computer Graphics 20 (4) (1986) 269–278. H. Naeimi, F. Kowsary, An optimized and accurate Monte Carlo method to simulate 3D complex radiative enclosures, International Communications in Heat and Mass Transfer 84 (2017) 150–157. S. Mazumder, Application of a variance reduction technique to surface-to-surface Monte Carlo radiation exchange calculations, International Journal of Heat and Mass Transfer 131 (2019) 424–431. J.P. Peraud, N.G. Hadjiconstantinou, Efficient simulation of multidimensional phonon transport using energy-based variance-reduced Monte Carlo formulations, Physical Review B 84 (2011) 205331. L. Soucasse, P. Rivière, A. Soufiani, Monte Carlo methods for radiative transfer in quasi-isothermal participating media, Journal of Quantitative Spectroscopy and Radiative Transfer 128 (2013) 34–42.

Chapter 8

Surface Radiative Exchange in the Presence of Conduction and Convection 8.1 Introduction In the previous few chapters we have considered only the analysis of radiative exchange in enclosures with specified wall temperatures or fluxes, i.e., we have neglected interaction with other modes of heat transfer. In practical systems, it is nearly always the case that radiation from a boundary is affected by conduction into the solid and/or by convection from the surface. Consequently, two or three modes of heat transfer must be accounted for simultaneously. The interaction may be quite simple, or it may be rather involved. For example, heat loss from an isothermal surface of known temperature, adjacent to a radiatively nonparticipating medium, may occur by convection as well as radiation; however, convective and radiative heat fluxes are independent of one another, can be calculated independently, and may simply be added. If boundary conditions are more complex (i.e., surface temperatures are not specified), then radiation and the other modes of heat transfer are coupled through the boundary condition, and the heat fluxes due to the various modes cannot be computed independently. In a number of important applications, a conduction analysis needs to be performed on an opaque medium, which loses (or gains) heat from its surfaces by radiation (and, possibly, convection). In such conjugate heat transfer problems, radiation enters the conduction problem as a nonlinear boundary condition; however, the radiative flux in this boundary condition may depend on the radiative exchange in the surrounding enclosure. In other applications, conduction and/or convection in a transparent gas or liquid needs to be evaluated, bounded by opaque, radiating walls. Again, radiation enters only as a boundary condition, with the transparent medium itself occupying the enclosure governing the radiative transfer. In both types of applications, radiation and conduction/convection are interdependent, i.e., a change in radiative heat flux disturbs the overall energy balance at the surface, causing a change in temperature as well as conductive/convective fluxes, and vice versa. We begin with a section that highlights the difficulties introduced by the aforementioned nonlinear boundary condition in the solution of combined mode heat transfer problems. This is followed by a section that outlines numerical procedures for robust treatment of the nonlinear boundary condition that dictates the coupling of the surface-to-surface radiation exchange equations with the overall energy conservation equation. Both conductive and convective coupling are discussed in this section, and several examples, representative of practical engineering applications, are presented. The chapter concludes with two sections that present specific applications of such coupling. First, radiative fins are discussed as an example of conduction–radiation coupling, while flow of nonparticipating gases within a tube, which is prevalent in many process industry and automotive applications, is discussed in the final section as an example of convection–radiation coupling.

8.2 Challenges in Coupling Surface-to-Surface Radiation with Conduction/Convection To demonstrate the challenges in coupling surface-to-surface radiation with other modes of heat transfer, let us first consider an enclosure bounded by N gray diffuse surfaces in which the temperatures of all surfaces are known, except one, as shown in Fig. 8.1, referred to as the target surface. The back (or external) side of this surface is subjected to convection (Newton cooling boundary condition), as also illustrated in Fig. 8.1. The objective is to determine the temperature of the target surface at steady state. The scenario just described is representative of a radiant furnace in which a target surface (or object) exchanges radiation with the heating elements mounted Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00016-X Copyright © 2022 Elsevier Inc. All rights reserved.

261

262 Radiative Heat Transfer

FIGURE 8.1 Schematic illustration of an enclosure with N surfaces, with one surface, S, having a convective boundary condition on the back side.

on the walls of the furnace. Applying equation (5.32) to the target surface, S, yields '

 N  N )     1 1 1 4 − − 1 FS−S qS − − 1 FS− j q j = (1 − FS−S ) σTS − FS− j σT4j ,

S

S

j j=1 jS

(8.1)

j=1 jS

where the temperature of the surface, TS , and the heat flux, qS , are both unknown. Substituting the energy balance at the surface, i.e., qS = ho (T∞ − TS ), into equation (8.1) results in '

 N  N )     1 1 1 4 − − 1 FS−S ho (T∞ − TS ) − − 1 FS− j q j = (1 − FS−S ) σTS − FS− j σT4j .

S

S

j j=1 jS

(8.2)

j=1 jS

For the other surfaces, i.e., i  S, again using qS = ho (T∞ − TS ) in equation (5.32), yields  N  N    qi  1 1 4 4 − − 1 Fi−S ho (T∞ − TS ) − − 1 Fi−j q j = σTi − Fi−S σTS − Fi−j σT4j .

i

S

j j=1 jS

(8.3)

j=1 jS

Equations (8.2) and (8.3) represent a set of N nonlinear equations with unknowns TS and qi (for i  S). The nonlinearity is a result of the fact that in the presence of convection, both linear and fourth power dependence on temperature appears in the equations and, consequently, it is not possible to solve for the emissive power, as was done in Chapter 5 for problems involving pure radiation. The solution of equations (8.2) and (8.3) requires a nonlinear simultaneous equation solver, such as the Newton’s method. While, in principle, the solution of these equations using a nonlinear solver is straightforward, difficulties in convergence arise when the number of equations (number of surfaces in the enclosure) is large, as would often be the case in practical engineering computations. These difficulties have been discussed further in a recent paper [1]. Many applications require determination of the temporal behavior of objects when they are subjected to radiative heating or cooling. Rapid thermal processing (RTP), used for semiconductor manufacturing, is one such application. In RTP, a semiconductor wafer is placed inside a low-pressure reactor and is subjected to radiation from tungsten filament lamps. The spatiotemporal evolution of the wafer temperature dictates the thermal stresses on the wafer as well as the quality of the thin semiconductor films that are eventually grown on the wafer by chemical vapor deposition. As a representative problem for mimicking RTP, we consider the problem discussed in the preceding two paragraphs, but with a slight variation. The target surface is now the top surface of a thin slab (representative of a wafer), which is subjected to convection on the underside, as shown in Fig. 8.2. The slab has a finite thickness LS , but for the purposes of this analysis, we assume that it is thin enough

Surface Radiative Exchange in the Presence of Conduction and Convection Chapter | 8 263

FIGURE 8.2 Schematic illustration of an enclosure with N surfaces, with one surface, S, being the top surface of a slab of finite thermal mass that is subjected to a convective boundary condition on the back side.

to be considered a lumped mass. Since the temporal thermal behavior of the slab is now of interest, we consider the transient energy conservation equation for the slab employing the lumped mass approximation: ρS cpS LS

dTS = ho (T∞ − TS ) − qS , dt

(8.4)

where ρS and cpS are the density and specific heat capacity of the slab, respectively. Rearrangement of equation (8.4) yields qS = ho (T∞ − TS ) − ρS cpS LS

dTS . dt

(8.5)

Substitution of equation (8.5) into equation (8.1) results in the following equation for surface S: '

   N  N )    1 1 dTS 1 − − 1 FS−S ho (T∞ − TS ) − ρS cpS LS − 1 FS− j q j = (1 − FS−S ) σTS4 − FS− j σT4j . −

S

S dt

j j=1 jS

(8.6)

j=1 jS

For the other surfaces of the enclosure (i  S), substitution of equation (8.5) into the corresponding radiation exchange equations yields     N  N   qi  1 1 dTS − − 1 Fi−S ho (T∞ − TS ) − ρS cpS LS − 1 Fi−j q j = σTi4 − Fi−S σTS4 − Fi− j σT4j . −

i

S dt

j j=1 jS

(8.7)

j=1 jS

Equations (8.6) and (8.7) represent a set of N nonlinear ordinary differential equations (ODEs) with unknowns TS and qi (i  S). These equations may be solved using well-known methods such as the Runge-Kutta method or other advanced methods, such as Gear’s method [2], that account for stiffness of the equations. While nonlinear sets of ODEs are routinely solved in other disciplines such as nonlinear control and chemical kinetics, once again, difficulties arise when the number of equations (number of surfaces in the enclosure) is large. Furthermore, for stability reasons, most ODE solvers impose restrictions on the time-step size that may be used. In the preceding two examples, representative of two important applications of surface-to-surface radiation transport, the multi-mode nature of heat transfer manifested itself only at the boundaries. As a result, it was possible to directly substitute the energy balance equation into the radiation exchange equations. At least in principle, this led to an implicitly coupled system of equations, although their solution, as discussed, can be quite difficult. To mitigate these difficulties, radiation is often decoupled from the other modes of heat transfer, and treated using iterative coupling procedures. These procedures are discussed in the next section.

264 Radiative Heat Transfer

FIGURE 8.3 Geometry, boundary conditions, and numbering scheme for various surfaces considered in Example 8.1.

8.3 Coupling Procedures A review of the literature reveals that the most common procedure for coupling surface-to-surface radiation with other modes of heat transfer is iterative coupling. In such a coupling procedure, the radiation exchange equations and the energy equation are not solved simultaneously (or fully implicitly) but, rather, in a segregated manner within an iteration loop. This iteration loop—often referred to as an outer iteration loop—serves the dual purpose of not only addressing the nonlinearity of the system, but also coupling of the two equations. The most common iterative coupling procedure is the so-called explicit procedure, which may be summarized as follows: 1. The temperature at boundaries, where it is not prescribed, is guessed, and denoted by TB∗ . 2. The radiation exchange equations are solved using the guessed boundary temperatures, and the radiative heat fluxes at boundaries are computed. Since this quantity is computed based on the guessed (or old) boundary temperatures, it is denoted by q∗R,B . The solution of the radiation exchange equations may be obtained using any solver of choice outlined in the preceding chapters. 3. The radiative heat flux at the boundaries, q∗R,B , is substituted into the boundary conditions of the overall energy equation. This equation is then solved to obtain the new temperature field, Tnew , including temperatures at boundaries where they are not prescribed, denoted by TBnew . 4. Steps 2 and 3 are repeated after replacing the old boundary temperature, TB∗ , by the new boundary temperature, TBnew . The iteration process is continued until convergence is reached, i.e., both the radiation exchange equations and the overall energy equation are simultaneously satisfied. While, in principle, the iterative explicit coupling procedure, just outlined, is straightforward to implement, convergence problems are often encountered. Since the radiative heat flux has a fourth power dependence on temperature, a small change in boundary temperature can cause a large change in the radiative heat flux, thereby making the iterations unstable. To mitigate such instability, under-relaxation is often used, whereby the new boundary temperature is not substituted back directly into the radiative exchange equations in Step 4. Rather, an under-relaxed boundary temperature given by TB = ωu TBnew + (1 − ωu )TB∗ is used in the next step, where the under-relaxation factor ωu is a number between 0 and 1. Using ωu = 1 implies use of the new boundary temperature in the next iteration. The following example highlights the convergence difficulties, and the use of under-relaxation. Example 8.1. A long ceramic slab is heat treated in a radiant furnace, as shown in Fig. 8.3. All dimensions shown in the figure are in mm. The heating element has an emittance of 0.18, and is at 2000 K. The surfaces of the slab have an emittance of 0.7. The (highly reflective) walls of the furnace are maintained at a constant temperature of 350 K, and have an emittance of 0.1. The slab is 1 mm thick, and has a thermal conductivity of k = 30 W/mK. The back side of the slab loses

Surface Radiative Exchange in the Presence of Conduction and Convection Chapter | 8 265

heat to the ambient, which is at a temperature of 300 K, and the convective heat transfer coefficient is ho = 10 W/m2 K. Assume that all surfaces are gray and diffuse, and surface-to-surface radiation is the only mode of heat transfer within the furnace. Determine the temperature distribution on the slab at steady state. Solution First, since the slab is very thin compared to its lateral size, we will assume that the temperature variation across the thickness of the slab is negligible. Second, since the lateral temperature distribution in the slab is sought, we will break it up into 20 control volumes of equal size. The walls of the furnace are isothermal, and therefore, need not be discretized. Consequently, the discretized model will have only 25 surfaces, as shown in Fig. 8.3. The geometry under consideration is two-dimensional. Thus, the view factors can be easily computed using the crossed-strings method described in Section 4.7. Once the view factors have been computed, equation (5.32) can be solved directly with the assumption that all Ti (or Ebi ), including those of the slab, are known. This yields qR,i for all surfaces. The radiative heat fluxes on surfaces A4 through A23 are then related to conductive and convective fluxes in energy balances on the slab control volumes. For a generic control volume, i, shown in Fig. 8.3, this energy balance yields     dT dT LS = −k LS + q∗R,i Li + ho (Ti − T∞ )Li , −k dx w,i dx e,i where, LS is the thickness of the slab, and Li is the area (length) of the top or bottom surface of the i-th control volume and the subscripts e and w refer to eastern and western faces of the control volume, respectively. The radiative heat flux is written with a superscript “*” to clearly denote that it is computed from the old (previous iteration) temperature field. Using the central difference approximation [3] for the derivatives, followed by some rearrangement, we obtain   kLS kLS 2kLS + ho Li Ti − Ti+1 − Ti−1 = ho Li T∞ − q∗R,i Li . i = 5, 6, ..., 22 : Li Li Li For the first (i = 4) and last (i = 23) control volumes, the conductive flux through the external faces must be replaced by the convective flux, resulting in   kLS kLS + ho (Li + LS ) Ti − Ti+1 = ho (Li + LS )T∞ − q∗R,i Li , i=4: Li Li   kLS kLS + ho (Li + LS ) Ti − Ti−1 = ho (Li + LS )T∞ − q∗R,i Li . i = 23 : Li Li The above equations represent a tridiagonal system of 20 linear algebraic equations. Their solution is straightforward, and yields a new set of temperatures of the slab control volumes, Tinew . These new temperatures are next substituted back into the right-hand side of the radiation exchange equations. As discussed earlier, under-relaxation may be used prior to substitution. The procedure is repeated until convergence. In this particular case, convergence is monitored using a residual of the temperature change of the slab control volumes between successive iterations, computed as ⎤1/2 ⎡ 23 ⎥ ⎢⎢ new ∗ 2⎥ ⎢ ⎢ RT = ⎢⎣ (Ti − Ti ) ⎥⎥⎥⎦ . i=4

A tolerance of 10−6 is used to terminate iterations. Figure 8.4a shows the convergence behavior for various values of the under-relaxation factor, ωu . For a value of ωu = 0.4 (or larger), convergence could not be attained, and ωu = 0.35 resulted in the slowest convergence. The optimum value of the under-relaxation factor appears to be in the vicinity of 0.3. The final converged steady-state temperature distribution in the slab is shown in Fig. 8.4b. A Fortran90 program for solution to this example problem, ExStoSEn1D, is provided in Appendix F. In summary, this example demonstrates that, although the commonly used explicit coupling procedure is straightforward to implement, convergence can often be elusive. While moderate to strong under-relaxation of the temperature field obtained by solving the energy equation can be used to mitigate the problem, the exact value of the under-relaxation factor that will lead to convergence is strongly problem dependent and requires numerical experiments to determine. The same under-relaxation factor may not result in a converged solution if the boundary temperatures are changed.

One way to alleviate the fickle convergence behavior associated with the explicit coupling approach is to treat the radiative flux in a more implicit manner. In the explicit coupling approach (described earlier), the radiative flux at a surface was written as q∗R,i = i E∗bi − αi Hi∗ ,

(8.8)

266 Radiative Heat Transfer

FIGURE 8.4 Results for the problem considered in Examples 8.1 and 8.2: (a) convergence behavior for the explicit coupling procedure with various under-relaxation factors, and the semi-implicit method, (b) steady state temperature distribution in the slab.

where all quantities on the right-hand side of equation (8.8) are computed using the temperature field from the previous iteration. Specifically, E∗bi = Eb (Ti∗ ). In reality, since the radiation exchange equations and the energy equation need to be considered simultaneously (implicitly), the heat flux should be computed using the temperature field at the current iteration, i.e., qR,i = i Ebi − αi Hi∗ .

(8.9)

In equation (8.9), it is assumed that the emission term is treated implicitly (the reason why the superscript “*” is absent from the emission term as well as the heat flux), while the incident radiation term is still treated explicitly. In other words, equation (8.9) is a semi-implicit equation. Expanding the first term on the right-hand side of equation (8.9) using a Taylor series, and discarding all higher order terms beyond the linear term, we obtain  ∗   dEbi qR,i = i E∗bi + (Ti − Ti∗ ) − αi Hi∗ . dT

(8.10)

Substituting equation (8.8) into equation (8.10) yields 

dEbi qR,i = qR,i + i dT ∗

∗

(Ti − Ti∗ ).

(8.11)

Equation (8.11) is essentially a higher order expression for the heat flux in which, rather than expressing the heat flux in terms of the temperature field at the previous iteration (first term on the right-hand side), an additional term that expresses the temperature dependence of the heat flux is introduced. The idea of using such a linearization procedure to enhance implicitness in the treatment of coupled conduction-radiation boundary conditions was apparently first proposed in 1977 by Williams and Curry [4], and later adopted and extended by other researchers [1,5–7]. In this new formulation, instead of substituting only the first term on the right-hand side of equation (8.11) into the energy equation, the entire right-hand side is substituted. The benefit of this semi-implicit coupling procedure is demonstrated in the example below. Example 8.2. Repeat Example 8.1, but using the new semi-implicit formulation just described. Solution As in Example 8.1, the first step is to guess a temperature field (T∗ ) and to compute the radiative heat flux (q∗R,i ) using equation (5.32). Noting that [dEbi /dT]∗ = 4σTi∗3 , and substituting equation (8.11) into the discrete equations derived from

Surface Radiative Exchange in the Presence of Conduction and Convection Chapter | 8 267

the energy balance in Example 8.1, we obtain     kLS kLS 2kLS + ho Li Ti − Ti+1 − Ti−1 = ho Li T∞ − q∗R,i + 4 i σTi∗3 (Ti − Ti∗ ) Li . i = 5, 6, ..., 22 : Li Li Li Rearranging the above equation yields     2kLS kLS kLS ∗3 i = 5, 6, ..., 22 : + ho Li + 4 i σTi Li Ti − Ti+1 − Ti−1 = ho Li T∞ − q∗R,i − 4 i σTi∗4 Li . Li Li Li Likewise, substitution of equation (8.11) into the discrete energy equations for the two end control volumes, followed by rearrangement, yields     kLS kLS i=4: + ho (Li + LS ) + 4 i σTi∗3 Li Ti − Ti+1 = ho (Li + LS )T∞ − q∗R,i − 4 i σTi∗4 Li , Li Li     kL kLS S + ho (Li + LS ) + 4 i σTi∗3 Li Ti − Ti−1 = ho (Li + LS )T∞ − q∗R,i − 4 i σTi∗4 Li . i = 23 : Li Li Once again, as in Example 8.1, the above three equations represent a tridiagonal system of linear algebraic equations. The solution to this system yields the new temperature of the slab control volumes, which can then be substituted back into the radiation exchange equations, and the procedure is repeated until convergence. Here, convergence is defined in exactly the same manner as in Example 8.1. Figure 8.4a also includes the convergence using this semi-implicit procedure. In this case, no under-relaxation is necessary, although its use is not prohibited. While the convergence is not superior to the convergence obtained using the explicit coupling procedure discussed in Example 8.1 when using the optimum under-relaxation factor, the fact that no under-relaxation is necessary is an important benefit, since one does not have to conduct any numerical experiments to attain convergence. Although not shown here, if the heater temperature is increased to 3000 K, the explicit procedure fails to converge even with an under-relaxation factor as low as 0.1, while the semi-implicit procedure converges in 103 iterations. The converged results, shown in Fig. 8.4b, are identical for both methods. A Fortran90 program for solution to this example problem, ImStoSEn1D, is provided in Appendix F.

The semi-implicit coupling procedure, just described, may be used not only with deterministic formulations for surface-to-surface radiation exchange (Chapter 5), but the heat flux, q∗R,i , may also be determined using the Monte Carlo method (Chapter 7), which is then substituted into the energy equation using the exact same procedure outlined above. In the preceding two examples, surface-to-surface radiation was coupled to the one-dimensional heat conduction equation. Consequently, the discrete form of the energy equation resulted in a tridiagonal system of equations that could be solved using a direct solver (Gaussian Elimination). In the general case of the multidimensional energy equation, the discrete form will no longer be tridiagonal, and will necessitate use of an iterative linear algebraic equation solver. Iterative solvers are known to produce rapid convergence if the diagonal element (term pre-multiplying Ti ) of the matrix is much larger, magnitude wise, compared to the off-diagonal elements [3] (terms pre-multiplying Ti+1 and Ti−1 ). Inspection of the diagonal elements obtained using the semiimplicit coupling procedure against the same elements obtained using the explicit coupling procedure show that the diagonal has been enhanced (its magnitude is larger) using the semi-implicit procedure, since 4 i σTi∗3 Li > 0 is added to a term that is always positive. In general, the semi-implicit coupling procedure is likely to be more beneficial in the case when surface-to-surface radiation is coupled with the multidimensional energy equation, as is demonstrated in the next example that couples surface-to-surface radiation with convection. Example 8.3. Air flows through a duct, as shown in Fig. 8.5. For simplicity, assume all thermophysical properties of air to be constant (ρ = 1.2 kg/m3 , cp = 1000 J/kg K, k = 0.023 W/mK) and the flow to be hydrodynamically fully-developed (also known as Poiseuille flow). The air enters at 300 K, and at a velocity, um , such that the product of the Reynolds and Prandtl number (also known as the Péclet number), um H/α, is equal to 200, where H is the height of the channel and α = k/ρcp is the thermal diffusivity. The bottom wall of the duct is subjected to an incoming heat flux of qB = 1000 W/m2 , while the top wall is perfectly insulated. All surfaces are assumed to be gray and diffuse, and their emittances are shown in Fig. 8.5. Air is assumed to be radiatively nonparticipating. Determine the temperature distribution in the channel with and without radiative heat transfer. Perform calculations using both the explicit and the semi-implicit coupling procedures and compare the convergence behavior. Solution First, the duct is discretized using a uniform 200 (= K) × 40 (= M) mesh (8000 control volumes), such that Δx = L/K and Δy = H/M. Since the flow is laminar and fully-developed (Poiseuille flow), the x- and y-velocity components are

268 Radiative Heat Transfer

FIGURE 8.5 Geometry and boundary conditions for the problem considered in Example 8.3.

u(y) = 6 um [(y/H) − (y/H)2 ] and v = 0, respectively. At steady state, energy conservation on a generic control volume (i, j) [see Fig. 8.6(a)] yields Qw + Qe + Qn + Qs = 0, where Qw , Qe , etc. denote outgoing net heat transfer rates on various faces of the control volume, and may be split into three components: conductive, convective, and radiative. Thus, the heat transfer rate at the western face of the control volume, Qw , for example, may be written as 

dT Qw = −k dx

 Aw + m˙ w cp Tw + qR,w Aw , w

where Fourier’s law has been used for the conductive flux. Aw denotes the area of the western face, which, for a 2D planar geometry, is equal to Δy. The quantity [dT/dx]w may be expressed using a finite-difference approximation as [dT/dx]w ≈ (Ti,j − Ti−1,j )/Δx. m˙ w denotes the mass flow rate normal to the face w, and may be written as m˙ w = ρuw Aw = ρuw Δy. Since u is invariant in the x-direction in a fully-developed flow and only a function of y, the mass flow rate further reduces to m˙ w = ρu j Δy. Furthermore, since the mass flow in this problem is from the left to the right, the energy transport by the flow—commonly referred to as advection—will also be from left to right. Thus, the temperature of the western face may be approximated by its upstream value, i.e., Tw ≈ Ti−1,j . Substituting these expressions into the above equation for the heat transfer rate across the face w, we obtain     Ti,j − Ti−1,j + ρu j cp Ti−1,j + qR,w Δy. Qw = −k Δx Likewise, for the heat transfer rate on the southern face, it follows that     Ti,j − Ti,j−1 Qs = −k + qR,s Δx, Δy where the advective term has been dropped since v = 0 everywhere. Developing similar expressions for the eastern and northern faces, substituting all four expressions into the overall energy balance equation, and dividing through by ρcp , we find     Δy Δy Δy Δx Δx Δx u j Δy + 2α + 2α Ti,j − α Ti+1,j − u j Δy + α Ti−1,j − α Ti,j+1 − α Ti,j−1 Δx Δy Δx Δx Δy Δy  1  =− (qR,w + qR,e )Δy + (qR,s + qR,n )Δx . ρcp The quantity within square brackets on right-hand side of the above equation is the net imbalance of energy due to radiation transfer in or out of the control volume. For a nonparticipating medium, as is the case here, this term vanishes, since any photon entering the control volume must also leave. Thus, the energy conservation equation for an interior control volume (i, j) may be written in final form as     Δy Δy Δy Δx Δx Δx u j Δy + 2α + 2α Ti,j − α Ti+1,j − u j Δy + α Ti−1,j − α Ti,j+1 − α Ti,j−1 = 0. Δx Δy Δx Δx Δy Δy

Surface Radiative Exchange in the Presence of Conduction and Convection Chapter | 8 269

FIGURE 8.6 Schematic representation of control volumes used for solving the problem considered in Examples 8.3: (a) interior control volume (cell) setup and notations used, (b) cell adjacent to the bottom wall showing energy balance.

The energy balances in control volumes adjacent to the boundaries are affected by boundary conditions, which may be written as Inlet: Outlet: Bottom Wall: Top Wall:

T(0, y) = Tin ,   ∂T −k = 0, ∂x (L,y)   ∂T + qR (x, 0) = qB −k ∂y (x,0)   ∂T −k − qR (x, H) = 0. ∂y (x,H)

For a control volume adjacent to the bottom wall [see Fig. 8.6b, for example], the heat transfer rate is given by the bottom wall boundary condition as Qs = Q(x,0) = qB Δx, which may be substituted into the overall energy balance equation along with the expressions for the northern, eastern and western faces (which remain unchanged since they are interior faces). Following some simplifications, we obtain an algebraic form of the energy conservation equation for the control volumes next to the bottom boundary ( j = 1):     q − q  Δy Δy Δy Δx Δx B R,i +α Ti+1,1 − u1 Δy + α Ti,2 = αΔx . Ti,1 − α Ti−1,1 − α u1 Δy + 2α Δx Δy Δx Δx Δy k In a similar manner, the energy balance principle may be applied to derive the algebraic form of the energy conservation equation for all other cells in the computational domain. The full set of algebraic equations is not presented here for the sake of brevity. Next, we discuss the solution algorithm in which the explicit coupling procedure is used. The steps are as follows: ∗ . Noting that the temperatures at the top and bottom wall 1. Guess temperatures for all cells. These are denoted by Ti,j ∗ ∗ faces are also unknown, these need to be guessed as well, and are denoted by TB,i and TT,i , respectively. 2. Start of outer iteration loop: solve the radiation exchange equations [equation (5.32)]. This yields the radiative heat flux on all boundary faces, denoted by q∗R,i . 3. Replace qR,i by q∗R,i in the algebraic equations derived using conservation of energy for the cells adjacent to the boundaries, and solve the resulting set of equations (for all cells) using an iterative solver of choice. Only partial convergence is warranted (typically a few inner iterations or a few orders of magnitude reduction in the residual) at this stage since the solution to this system of equations will have to be repeated in the next outer iteration. Here, the Alternating Direction Implicit (ADI) method [3] was used with 1 row-wise sweep and 1 column-wise sweep. The new . solution to this linear system of equations yields new cell temperatures, denoted by Ti,j new ∗ 4. Under-relax the temperature field if necessary: Ti,j = ωu Ti,j + (1 − ωu )Ti,j . 5. Recompute wall temperatures using the appropriate boundary condition. For example, for the bottom wall, the boundary condition, [∂T/∂y]B = (qB − q∗R,i )/k, is first written in discrete form using a second-order accurate expression

270 Radiative Heat Transfer

for the derivative [3] to yield (qB − q∗R,i ) 9Ti,1 − Ti,2 − 8TB,i = , 3Δy k This equation can be rearranged to solve for the boundary temperature: TB,i =

3Δy(qB − q∗R,i ) 9 1 Ti,1 − Ti,2 + . 8 8 8k

6. End of outer iteration loop: repeat Steps 2 through 5 until convergence. The residual is defined as the net energy imbalance summed over all cells. As discussed earlier, the semi-implicit method differs from the explicit method in the way the radiative heat flux is treated. Instead of replacing qR,i by q∗R,i in Step 3 of the above algorithm, equation (8.11) is used. Substituting equation (8.11) into the discrete form of the bottom wall boundary condition shown above, we obtain TB,i =

∗3 ∗ (TB,i − TB,i )] 3Δy [qB − q∗R,i − 4 i σTB,i 9 1 Ti,1 − Ti,2 + , 8 8 8k

which, upon rearrangement, may be written as TB,i = ai Ti,1 + bi , where the coefficients, ai and bi are as follows:  . 3Δy 9 ∗3 ai = , 1+

i σTB,i 8 2k .     3Δy(qB − q∗R,i ) 3Δy 3Δy 1 ∗4 ∗3 8 1+ . bi = − Ti,2 + +

i σTB,i

i σTB,i 8 8k 2k 2k In the semi-implicit coupling procedure, the equation TB,i = ai Ti,1 + bi is used to calculate the boundary temperature rather than the equation provided in Step 5 of the above algorithm for the explicit method. Furthermore, this equation and equation (8.11) can be substituted into the discrete form of the energy balance equation for j = 1 to yield     Δy Δy Δy Δx Δx +α Ti,1 − α Ti+1,1 − u1 Δy + α Ti−1,1 − α Ti,2 u1 Δy + 2α Δx Δy Δx Δx Δy   ∗3 ∗ (ai Ti,1 + bi − TB,i ) qB − q∗R,i − 4 i σTB,i . = αΔx k Rearrangement of the above equation shows that the diagonal element of the resulting matrix (term pre-multiplying Ti,1 ) and the right-hand side (source) gets modified as follows: ∗3 ai /k Diagonal (semi-implicit) = Diagonal (explicit) + 4α Δx i σTB,i ∗3 ∗ (TB,i − bi )/k. Source (semi-implicit) = Source (explicit) + 4α Δx i σTB,i ∗3 ai /k to an already positive diagonal, diagonal Once again, it is clear that by adding the positive term 4α Δx i σTB,i dominance has been enhanced, which, as stated earlier, is expected to produce faster convergence. Figure 8.7 shows the convergence behavior using the explicit and semi-implicit coupling procedures. For the explicit coupling procedure, as in the case of Example 8.1, strong under-relaxation is necessary. For an under-relaxation factor of 0.2, convergence could not be attained. In contrast, for the semi-implicit coupling procedure, no under-relaxation is necessary, and convergence is attained in only 70 iterations as opposed to the 655 iterations required for the explicit coupling procedure. The benefits of the semi-implicit coupling procedure are amplified for this multidimensional problem compared to the one-dimensional problem considered in Examples 8.1 and 8.2. The temperature distributions with and without radiation are shown in Fig. 8.8. For the case without radiation, the bottom wall heats up to approximately 1422 K, and a well-defined thermal boundary layer is observed at the bottom wall. The bottom wall is hottest at the downstream end, as is evident in Fig. 8.8a. When surface-to-surface radiation is included, the bottom wall loses large amounts of energy by radiation. This escaping energy is captured by the top wall, which heats up significantly since it is externally insulated. Hence, the middle of the channel stays cold, while the two walls are hot, as shown in Fig. 8.8b. This type of “cup-shaped” temperature distribution is unique to problems involving radiation, where long-range radiation influences entities that are far away. In contrast, in the case of pure convection, the top wall is always the coldest point at a given cross-sectional plane. Furthermore, when radiation is included, the highest temperature is not at the downstream end but somewhere in the middle, since the ends lose significant amount of heat by radiation to the inlet and outlet.

Surface Radiative Exchange in the Presence of Conduction and Convection Chapter | 8 271

FIGURE 8.7 Convergence behavior using the explicit and semi-implicit coupling procedures for the problem considered in Example 8.3.

FIGURE 8.8 Temperature distributions for the problem considered in Example 8.3: (a) top and bottom walls (with and without radiation), (b) various axial locations (with radiation); x1 = 0.04875 m, x2 = 0.24875 m, and x3 = 0.49875 m.

In this section, procedures for coupling surface-to-surface radiation with conduction and convection were discussed. Starting from the 1960s, a number of researchers [8–30] have investigated scenarios that involve coupling between conduction and surface-to-surface radiation. While most of these investigations were conducted for one-dimensional media, some works have dealt with more complex geometries, surface properties (including nongray effects), irradiation conditions, etc. More recently, some researchers have considered combined conduction–surface radiation in media with cavities, such as porous media [31,32], packed beds of spheres [33], mirror furnaces [34], honeycomb panels [35–37], and catalytic reactors [38]. Important applications that involve coupling of surface radiation with both conduction and convection include thermal management in the under-hood of an automobile [39–43] wherein radiation from the surface of the engine interacts with other components, design of the passenger compartment of vehicles [44,45] wherein solar radiation interacts with components within the cabin, chemical vapor deposition reactors [46–48] in which high-power tungsten filament lamps are used for thermal processing of wafers, and cooking ovens [49] wherein radiant heaters are used.

272 Radiative Heat Transfer

8.4 Radiative Heat Transfer Coefficient Consider a gray body at a fixed temperature T, and surface emittance , being surrounded on all sides by the ambient at another fixed temperature T∞ . Following the arguments used for the development of equation (5.36), the heat flux on the surface of that body may be written as 4 q = σ(T 4 − T∞ ).

(8.12)

For thermal analysis, it is often convenient to express the fourth power temperature difference in the radiative flux expression in terms of a linear difference in temperature, analogous to a convective heat flux [equation (1.2)]. 4 This is easily achieved by factorizing T 4 − T∞ , leading to 4 2 ) = [ σ(T + T∞ )(T 2 + T∞ )] (T − T∞ ) = hR (T − T∞ ), q = σ(T 4 − T∞

(8.13)

2 ) hR = σ(T + T∞ )(T 2 + T∞

(8.14)

where

is the radiative heat transfer coefficient. Unlike the convective heat transfer coefficient h, which is generally weakly dependent on temperature (through temperature dependence of the thermophysical properties), the radiative heat transfer coefficient is a strong function of T even if temperature dependence of the emittance was to be neglected, as evident from equation (8.14). This is not surprising in light of the fact that radiation is fundamentally nonlinear in T and equation (8.13) is simply an attempt at linearizing it. Nonetheless, the concept of a radiative heat transfer coefficient has value in practical thermal analysis. For example, it is often necessary to make the decision whether the transient thermal response of a body, such as a silicon wafer placed inside a rapid thermal processing reactor used in semiconductor manufacturing, can be modeled by treating the body as a lumped mass. A lumped mass approximation, which considerably simplifies the analysis, is generally assumed to be valid if the Biot number, Bi = hL/k, is less than approximately 0.1 [50], where L is the characteristic size of the body, and k its thermal conductivity. In the presence of radiation, the definition of the Biot number can be conveniently changed to Bi = (h + hR )L/k, and the same criterion may still be used, wherein an estimated upper limit of T may be employed to estimate hR and the corresponding worst-case (highest possible) Biot number. Another utility of the radiative heat transfer coefficient is in expressing boundary conditions in combined mode heat transfer problems in a compact and unified form. Consider a one-dimensional slab of thickness L, such as the wall of a furnace, with its external surface being exposed to the ambient at T∞ . Accounting for both convective and radiative heat loss from the furnace wall, the boundary condition on the external surface is written as  dT  4  = h(TS − T∞ ) + S σ(TS4 − T∞ −k ), (8.15) dx  S

where the subscript “S” denotes quantities evaluated at the external surface. Using equation (8.14), we get  dT   = (h + hR )(TS − T∞ ), −k (8.16) dx S where T in equation (8.14) has been replaced by TS . This form of the boundary condition is convenient because it is straightforward to either swap convection for radiation or include both in the analysis. This is particularly useful in the development of numerical methods. Of course, one must always remember that even though equation (8.16) appears linear in temperature, in reality, it is a linearized form of the strongly nonlinear boundary condition given by equation (8.15). An approximate expression for the radiative heat transfer coefficient may be derived by replacing the (unknown) temperature T in equation (8.14) with T∞ , resulting in: 3 . hR ≈ 4 σT∞

(8.17)

This approximate expression for the radiative heat transfer coefficient has the advantage it is not a function of temperature (disregarding the temperature dependence of the emittance). However, the difference between the actual and approximate hR can be significant if the difference between T and T∞ is large. In such cases it is preferable to replace T∞ in equation (8.17) with a value between T and T∞ to yield more accurate results.

Surface Radiative Exchange in the Presence of Conduction and Convection Chapter | 8 273

FIGURE 8.9 Schematic representation of a space radiator tube with longitudinal fins.

8.5 Conduction and Surface Radiation—Fins An important application of combined conduction–surface radiation involves heat transfer through vacuum, namely, heat loss from space vehicles or vacuum insulations. As a single example, we will discuss here the performance of a simple rectangular-fin radiator used to reject heat from a spacecraft. Consider a tube with a set of radial fins, as schematically shown in Fig. 8.9. In order to facilitate the analysis, we will make the following assumptions: 1. The thickness of each fin, 2t, is much less than its length in the radial direction, L, which in turn is much less than the fin extent in the direction of the tube axis. This implies that heat conduction within the fin may be calculated by assuming that the fin temperature is a function of radial distance, x, only. 2. End losses from the fin tips (by convection and radiation) are negligible, i.e., ∂Ti /∂xi (L) 0. 3. The thermal conductivity of the fin material, k, is constant. 4. The base temperatures of all fins are the same, i.e., T1 (0) = T2 (0) = Tb , and the fin arrangement is symmetrical, i.e., T1 (x1 ) = T2 (x2 = x1 ), etc. 5. The surfaces are coated with an opaque, gray, diffusely emitting and reflecting material of uniform emittance

. 6. There is no external irradiation falling into the fin cavities (Ho = 0, T∞ = 0). The first three assumptions are standard simplifications made for the analysis of thin fins (see, e.g., Holman [51]), and the other three have been made to make the radiation part of the problem more tractable. Performing an energy balance on an infinitesimal volume element (of unit length in the axial direction) dV = 2t dx, one finds: conduction going in at x across cross-sectional area (2t) = conduction going out at x+dx + net radiative loss from top and bottom surfaces (2 dx) or

  dT  dT   = −2tk  + 2qR dx. −2tk dx x dx x+dx

Expanding the outgoing conduction term into a truncated Taylor series,    dT  dT  d2 T    + dx 2  + · · · , = dx x+dx dx x dx x then leads to

274 Radiative Heat Transfer

d2 T 1 = qR . tk dx2

(8.18)

Here qR (x) is the net radiative heat flux leaving a surface element of the fin, which may be determined in terms of surface radiosity, J, from equations (5.24) and (5.25) as1  qR (x1 ) = J(x1 ) −

L

x2 =0

J(x2 ) dFd1−d2 , 

J(x1 ) = σT (x1 ) + (1 − )

(8.19)

L

4

x2 =0

J(x2 ) dFd1−d2 .

(8.20)

The expression for radiative heat flux may be simplified by eliminating the integral, equation (5.26), qR (x1 ) =



 4 σT1 (x1 ) − J1 (x1 ) . 1−

(8.21)

The view factor between two infinitely long strips may be found from Appendix D, Configuration 5, or from Example 4.1 as Fd1−d2 =

x1 sin2 α x2 dx2 sin2 α x1 x2 dx2 = . 2(x21 − 2x1 x2 cos α + x22 )3/2 2S30

(8.22)

Equation (8.18) requires two boundary conditions, namely, T(x = 0) = Tb ,

dT (x = L) = 0. dx

(8.23)

Before we attempt a numerical solution, it is a good idea to summarize the mathematical problem in terms of nondimensional variables and parameters, θ(ξ) =

T(x) , Tb

J(ξ) =

J(x) , σTb4

Nc =

kt , σTb3 L2

ξ=

x , L

(8.24)

where θ and J are nondimensional temperature and radiosity, and Nc is usually called the conduction-to-radiation parameter, sometimes also known as the Planck number. With these definitions,  d2 θ 1  4 = (ξ) − J(ξ) , θ Nc 1 −

dξ2  1 J(ξ ) K(ξ, ξ ) dξ , J(ξ) = θ 4 (ξ) + (1 − ) ξ =0

K(ξ, ξ ) =

1 ξ ξ sin2 α 2 , 2 (ξ − 2ξξ cos α + ξ 2 )3/2

(8.25a) (8.25b) (8.25c)

subject to θ(ξ = 0) = 1,

dθ (ξ = 1) = 0. dξ

(8.25d)

As for convection-cooled fins, a fin efficiency, η f , is defined, comparing the heat loss from the actual fin to that of an ideal fin (a black fin, which is isothermal at Tb ). The total heat loss from an ideal fin ( = 1, J = σTb4 ) is readily determined from equation (8.19) and Appendix D, Configuration 34, as Qideal = 2L qR,ideal = 2L σTb4 (1 − F1−2 ) = 2L sin

α 4 σT , 2 b

(8.26)

1. For the radiative exchange it is advantageous to attach subscripts 1 and 2 to the x-coordinates to distinguish contributions from different plates, even though T(x), J(x), qR (x), etc., are the same along each of the fins.

Surface Radiative Exchange in the Presence of Conduction and Convection Chapter | 8 275

FIGURE 8.10 Radiative fin efficiency for longitudinal plate fins [52].

while the actual heat loss follows from Fourier’s law applied to the base, or by integrating over the length of the fin, as  L  dT   =2 Qactual = −2tk qR (x) dx. (8.27) dx x=0 0 Thus,  1  Qactual

Nc dθ  1  =− = (θ 4 − J ) dξ, (8.28) ηf = Qideal sin α dξ  sin α 1 −

2

0

2

0

where the last expression is obtained by integrating equation (8.25a) along the length L of the fin. The set of equations (8.25) is readily solved by a host of different methods, including the net radiation method [finite-differencing equation (8.25b) into finite-width isothermal strips, to which equation (5.34) can be applied] or any of the solution methods for Fredholm equations discussed in Section 5.6. Because of the nonlinear nature of the equations it is always advisable to employ the method of successive approximations, i.e., a temperature field is guessed, a radiosity distribution is calculated, an updated temperature field is determined by solving the differential equation (for a known right-hand side), etc., usually requiring under-relaxation. For enhanced numerical stability and convergence, the semi-implicit coupling procedure may be used. In that case, the θ4 term in equation (8.25a) must be linearized and the resulting equation rearranged and solved, as described in Section 8.3. Sample results for the efficiency, as obtained by Sparrow and coworkers [52], are shown in Fig. 8.10. The variation of the fin efficiency is similar to that for a convectively cooled fin [with the heat transfer coefficient replaced by the radiative heat transfer coefficient given by hR = 4 σTb3 , as given by equation (8.17)]. Maximum efficiency is obtained for Nc → ∞, i.e., when conduction dominates and the fin is essentially isothermal. For

< 1 the efficiency is limited to values η f < 1 since a black configuration will always lose more heat. It is also observed that the fin efficiency (but not the actual heat lost) increases as the opening angle α decreases: For small opening angles irradiation from adjacent fins reduces the net radiative heat loss by a large fraction, but not as much as for the “ideal” fin (with irradiation from adjacent fins, which are black and at Tb ). Many studies on radiative fins may be found in the literature. For example, Hering [53] and Tien [54] considered the fins of Fig. 8.9 with specularly reflecting surfaces, and Sparrow and coworkers [52] investigated the influence of external irradiation. Fins connecting parallel tubes were studied by Bartas and Sellers [55], Sparrow and coworkers [56,57], and Lieblein [58]. Single annular fins (i.e., annular disks attached to the outside of tubes)

276 Radiative Heat Transfer

FIGURE 8.11 Forced convection and radiation of a transparent medium flowing through a circular tube, subject to constant wall heat flux.

were studied by Chambers and Sommers [59] (rectangular cross-section), Keller and Holdredge [60] (variable cross-section), and Mackay [61] (with external irradiation), while Sparrow and colleagues [62] investigated the interaction between adjacent fins.

8.6 Convection and Surface Radiation—Tube Flow As in the case of pure convection heat transfer, it is common to distinguish between external flow and internal flow applications. If the flowing medium is air or some other relatively inert gas, the assumption of a transparent, or radiatively nonparticipating, medium is often justified. As an example we will consider here the case of a transparent gas flowing through a cylindrical tube of diameter D = 2R and length L, which is heated uniformly at a rate of qw (per unit surface area). As schematically shown in Fig. 8.11, the fluid enters the tube at x = 0 with a mean, or bulk, temperature Tm1 . Over the length of the tube the supplied heat flux qw is dissipated from the inner surface by convection (to the fluid) and radiation (to the openings and to other parts of the tube wall), while the outer surface of the tube is insulated. The two open ends of the tube are exposed to radiation environments at temperatures T1 and T2 , respectively. The inner surface of the tube is assumed to be gray, diffusely emitting and diffusely reflecting, with a uniform emittance . Finally, for a simplified analysis, we will assume that the convective heat transfer coefficient, h, between tube wall and fluid is constant, independent of the radiative heat transfer, and known. With these simplifications an energy balance on a control volume dV = πR2 × dx yields: enthalpy flux in at x + convective flux in over dx = enthalpy flux out at x+dx, or

  dTm ˙ p Tm (x+dx) = mc ˙ p Tm (x) + ˙ p Tm (x) + h [Tw (x) − Tm (x)] 2πR dx = mc (x) dx , mc dx

(8.29)

or dTm 2h = [Tw (x)−Tm (x)] , dx ρcp um R

(8.30)

where axial conduction has been neglected, and the mass flow rate has been expressed in terms of mean velocity as m˙ = ρum πR2 . Equation (8.30) is a single equation for the unknown wall and bulk temperatures Tw (x) and Tm (x) and is subject to the inlet condition Tm (x = 0) = Tm1 .

(8.31)

An energy balance for the tube surface states that the prescribed heat flux qw is dissipated by convection and radiation or, applying equation (5.26) for the radiative heat flux, 

 4 qw = h [Tw (x) − Tm (x)] + (8.32) σTw (x) − J(x) . 1−

Surface Radiative Exchange in the Presence of Conduction and Convection Chapter | 8 277

The radiosity J(x) is found from equation (5.24) as /  J(x) = σTw4 (x) + (1− ) σT14 Fdx−1 + σT24 Fdx−2 +

L

0 J(x ) dFdx−dx ,

(8.33)

0

where Fdx−1 is the view factor from the circular strip of width dx at x to the opening at x = 0, Fdx−2 is the one to the opening at x = L, and dFdx−dx is the view factor between two circular strips located at x and x , as indicated in Fig. 8.11. All view factors are readily determined from Appendix D, Configurations 9 and 31, and will not be repeated here. Equations (8.30), (8.32), and (8.33) are a set of three simultaneous equations in the unknown Tw (x), Tm (x), and J(x), which must be solved numerically. Before we attempt such a solution, it is best to recast the equations in nondimensional form, by defining the following variables and parameters: x ξ= , D St =



σT 4 θ(ξ) = qw

h , ρcp um

H=

1/4 ,

J , qw

J(ξ) =

  h qw 1/4 , qw σ

(8.34a) (8.34b)

among which is the Stanton number, St, a commonly used parameter in convective heat transfer, comparing heat transfer to thermal heat capacity of the flow [63]. This transforms equations (8.30) through (8.33) to dθm = 4 St [θw (ξ) − θm (ξ)] , dξ

θm (ξ = 0) = θm1 , 

 4 1 = H [θw (ξ) − θm (ξ)] + θw (ξ) − J(ξ) , 1−

/ 0  L/D 4 4 4  J(ξ) = θw (ξ) + (1 − ) θ1 Fdξ−1 + θ2 Fdξ−2 + J(ξ ) dFdξ−dξ .

(8.35) (8.36) (8.37)

0

Equation (8.36) becomes indeterminate for = 1. For the case of a black tube J = θw4 , and equations (8.36) and (8.37) may be combined as  1 = H [θw (ξ) − θm (ξ)] + θw4 (ξ) − θ14 Fdξ−1 − θ24 Fdξ−2 −

0

L/D

θw4 (ξ ) dFdξ−dξ .

(8.38)

Example 8.4. A transparent gas flows through a black tube subject to a constant heat flux. The convective heat transfer coefficient is known to be constant such that Stanton numbers and the nondimensional heat transfer coefficient are evaluated as St = 2.5 × 10−3 and H = 0.8. The environmental temperatures at both ends are equal to the local gas temperatures, i.e., θ1 = θm1 and θ2 = θm2 = θm (ξ = L/D), and the nondimensional inlet temperature is given as θm1 = 1.5. Determine the (nondimensional) wall temperature variation as a function of relative tube length, L/D, using the numerical quadrature approach of Example 5.11. Solution Since the tube wall is black we have only two simultaneous equations, (8.35) and (8.38), in the two unknowns θm and θw . However, the equations are nonlinear; therefore, an iterative procedure is necessary. For simplicity, we will adopt a simple backward finite-difference approach for the solution of equation (8.35), and the numerical quadrature scheme of equation (5.51) for the integral in equation (8.38). Evaluating temperatures at N + 1 nodal points ξi = iΔξ (i = 0, 1, . . . , N) where Δξ = L/(ND), this implies 

dθm dξ  L/D 0

 ξi

θm (ξi ) − θm (ξi−1 ) , Δξ

θw4 (ξ )

i = 1, 2, . . . , N,

N dFdξ−dξ  L  dξ c j θw4 (ξ j ) K(ξi , ξ j ), dξ D j=0

i = 0, 1, . . . , N,

278 Radiative Heat Transfer

where the c j are quadrature weights and, from Configuration 9 in Appendix D,2 K(ξi , ξ j ) = 1 −

Xij (2Xij2 +3) 2(Xij2 +1)

;

Xij = |ξi − ξ j |.

Similarly, the two view factors to the openings are evaluated from Configuration 31 in Appendix D as Xij2 + 12 − Xij , Fdξi −k = ( Xij2 + 1 where j=0

if

k = 1 (opening at ξ = ξ0 = 0),

j=N

if

k = 2 (opening at ξ = ξN = L/D).

To solve for the unknown θm (ξi ) and θw (ξi ), we adopt the following iterative procedure: 1. A wall temperature is guessed for all wall nodes, say, θw (ξi ) = θ1 ,

i = 0, 1, . . . , N.

2. A temperature difference is calculated from equation (8.38), i.e., φi = H [θw (ξi ) − θm (ξi )] = 1 − θw4 (ξi ) + θ14 Fdξi −1 + θ24 Fdξi −2 +

N L  c j θw4 (ξ j ) K(ξi , ξ j ). D j=0

3. The gas bulk temperature is calculated from equation (8.35) as θm (ξi ) = θm (ξi−1 ) +

4 St Δξ φi ; H

θm (ξ0 ) = θ1 .

4. An updated value for the wall temperatures is then determined from the definition for φi , that is, ) ' 1 θwnew (ξi ) = ω θm (ξi ) + φi + (1 − ω) θwold (ξi ), H where ω is known as the relaxation factor. The iteration scheme is called under-relaxed if ω < 1, and over-relaxed if ω > 1 [65,66]. Since problems involving radiation are nonlinear, it is customary to use under-relaxation, denoted by ωu , as described in Section 8.3. Some representative results are shown in Fig. 8.12 for several values of L/D. A large numbers of nodes are necessary to achieve good accuracy (N 40L/D). Because of the strong nonlinearity of the problem, explicit coupling procedure employed here, strong under-relaxation (ωu < 0.02) is necessary. For the case of pure convection ( = 0, or φi ≡ 1) the tube wall temperature rises linearly with axial distance, since constant wall heat flux implies a linear increase in bulk temperature and, therefore (assuming a constant heat transfer coefficient) in surface temperature. This is not the case if radiation is present, in particular for short tubes (small L/D). Near both ends of the tube, much of the radiative energy leaves through the openings, causing a distinct drop in surface temperature. For long tubes (L/D > 50) the surface temperature rises almost linearly over the central parts of the tube, although the temperature stays below the convection-only case: Due to the higher temperatures downstream, some net radiative heat flux travels upstream, making overall heat transfer a little more efficient. It should be noted here that the assumption of a constant heat transfer coefficient is not particularly realistic, since it implies a fully developed thermal profile. It is well known that for pure convection h → ∞ at the inlet and, thus, θw (ξ = 0) = 1 [51]. Near the inlet of a tube the actual temperature distribution for pure convection is very similar to the one depicted in Fig. 8.11, which is driven by radiation losses. Although for pure convection a fully developed thermal profile and constant h are eventually reached (at L/D > 20 for turbulent flow), in the presence of radiation a constant heat transfer coefficient is never reached (because the radiation term makes the governing equations nonlinear).

A number of researchers have investigated combined convection and surface-to-surface radiation for a transparent flowing medium. Early research focused primarily on flow through circular tubes and ducts, and include studies conducted by Siegel and coworkers [67–69] for a number of situations, but always assuming a constant and known heat transfer coefficient. Dussan and Irvine [70] and Chen [71] calculated the local 2. Note that K(ξ, ξ ) has a sharp peak at ξ = ξ. Therefore, and also in light of the truncation error in the finite-differencing of dθm /dξ, it is best to limit the quadrature scheme to Simpson’s rule [64].

Surface Radiative Exchange in the Presence of Conduction and Convection Chapter | 8 279

FIGURE 8.12 Axial surface temperature development for combined convection and surface radiation in a black tube subjected to constant wall heat flux.

convection rate by solving the two-dimensional energy equation for the flowing medium, but they made severe simplifications in the evaluation of radiative heat fluxes. The most general tube flow analysis has been carried out by Thorsen and Kanchanagom [72,73]. Similar problems for parallel-plate channel flow were investigated by Keshock and Siegel [74] (for a constant heat transfer coefficient) and Lin and Thorsen [75] (for two-dimensional convection calculations). Combined radiation and forced convection of external flow across a flat plate has been addressed by Cess [76,77], Sparrow and Lin [78], and Sohal and Howell [79]. Finally, the interaction between surface radiation and free convection has been studied, both numerically and experimentally [80–89].

Problems 8.1 A satellite shaped like a sphere (R = 1 m) has a gray-diffuse surface coating with s = 0.3 and is fitted with a long, thin, cylindrical antenna, as shown in the adjacent sketch. The antenna is a specular reflector with a = 0.1, ka = 100 W/m K, and d = 1 cm. Satellite and antenna are exposed to solar radiation of strength qsol = 1300 W/m2 from a direction normal to the antenna. Assuming that the satellite produces heat at a rate of 4 kW and— due to a high-conductivity shell—is essentially isothermal, determine the equilibrium temperature distribution along the antenna. (Hint: Use the fact that d R not only for conduction calculations, but also for the calculation of view factors.) 8.2 A long, thin, cylindrical needle (L  D) is attached perpendicularly to a large, isothermal base plate at T = Tb = const. The base plate is gray and diffuse ( b = αb ), while the needle is nongray and diffuse (  α). The needle exchanges heat by convection and radiation with a large, isothermal environment at T∞ . (a) Neglecting heat losses from the free tip of the needle, formulate the problem for the calculation of needle temperature distribution, total heat loss, and fin efficiency. (b) Implement the solution numerically for L = 1 m, D = 1 cm, k = 10 W/m K, h = 40 W/m2 K, = 0.8, α = 0.4,

b = 0.8, Tb = 1000 K, T∞ = 300 K. 8.3 A thermocouple with a 0.5 mm diameter bead is used to measure the local temperature of a hot, radiatively nonparticipating gas flowing through an isothermal, gray-diffuse tube (Tw = 300 K, w = 0.8). The thermocouple is a diffuse emitter/specular reflector with b = 0.5, and the heat transfer coefficient between bead and gas is 30 W/m2 K. (a) Determine the thermocouple error as a function of gas temperature (i.e., |Tb − T g | vs. T g ). (b) In order to reduce the error, a radiation shield in the form of a thin, stainless-steel cylinder ( = 0.1, R = 2 mm, L = 20 mm) is placed over the thermocouple. This also reduces the heat transfer coefficient between bead and gas to 15 W/m2 K, which is equal to the heat transfer coefficient on the inside of the shield. On the outside of the cylinder the heat transfer coefficient is 30 W/m2 K. Determine error vs. gas temperature for this case.

280 Radiative Heat Transfer

To simplify the problem, you may make the following assumptions: (i) the leads of the thermocouple may be neglected, (ii) the shield is very long as far as the radiation analysis is concerned, and (iii) the shield reflects diffusely. 8.4 In the emissometer of Vader and coworkers [90] and Sikka [91], the sample is kept inside a long silicon carbide tube that in turn, is inside a furnace, as shown in the sketch. The furnace is heated with a number of SiC heating elements, providing a uniform flux over a 45 cm length as shown. Assume that there is no heat loss through the refractory brick or the bottom of the furnace, that the inside heat transfer coefficient for free convection (with air at 600◦ C) is 10 W/m2 K, that the silicon carbide tube is gray diffuse ( = 0.9, k = 100 W/m K), and that the sample temperature is equal to the SiC tube temperature at the same height. What must be the steady-state power load on the furnace to maintain a sample temperature of 1000◦ C? In this configuration a detector receiving radiation from a small center spot of the sample is supposedly getting the same amount as from a blackbody at 1000◦ C (cf. Table 5.1). What is the actual emittance sensed by the detector, i.e., what systematic error is caused by this near-blackbody, if the sample is gray and diffuse with s = 0.5? 8.5 Repeat Problem 5.36 for the case in which a radiatively nonparticipating, stationary gas (k = 0.04 W/m K) is filling the 1 cm thick gap between surface and shield. 8.6 Consider an oven that is cylindrical in shape with a radius of 0.1 m and a height of 0.2 m. Imagine that the bottom surface of the cylinder represents a target surface that is being heated. There is some heat loss from the underside of this surface, which can be modeled using a constant heat transfer coefficient ho = 5 W/m2 K and an ambient temperature of To = 300 K. The target surface has on emittance of 0.7. A heating element covers the center of the top surface up to a radius of 0.04 m, and may be assumed to be black and at 1000 K. The remaining surfaces of the cylinder may be considered the walls of the furnace. These may be assumed to be isothermal at 300 K with an emittance of 0.5. (a) Assuming that the gas in the furnace is completely transparent and that radiation is the only mode of heat transfer inside the furnace, determine the steady state temperature of the target surface. (b) Write down all necessary governing equations and boundary conditions if there is also conduction within the furnace, and explain how you will determine the steady state temperature of the target surface in this case. 8.7 Consider a parallel-plate configuration in which the left plate represents the surface of an engine, and the right plate represents the surface of an object adjacent to the engine. The plates are separated by 0.1 m. Both plates may be assumed black. The left plate (engine surface) has a temperature of 1000 K, while the back side of the right plate has convective cooling with a heat transfer coefficient of ho = 50 W/m2 K and an ambient temperature of To = 300 K. Neglect external radiation. (a) Assuming that the medium between the plates is transparent static air with a thermal conductivity of k = 0.023 W/m K, determine the steady state temperature distribution in the air gap and the temperature of the right plate. What is the heat flux between the two plates? (b) To protect the right plate, a radiation shield of negligible thickness and with an emittance of 0.2 is placed halfway between the two plates. Determine the temperature distribution and heat flux for this new setup. Comment on your results. Will the location of the shield have any impact on your results? 8.8 Consider steady hydrodynamically fully-developed laminar flow (Poiseuille flow) of hot combustion gas entering a circular pipe of radius ro and length L at a uniform (across the cross-section of the pipe) temperature Tin . The mean gas velocity is uo , and the gas may be assumed to be radiatively nonparticipating, and have constant thermophysical properties. The outer wall of the pipe is exposed to the ambient at temperature, To , with a constant convective heat transfer coefficient, ho , and emittance, o . Assume that the thickness of the pipe wall is negligibly small. The inner wall of the pipe has an emittance i . All walls are gray and diffuse. The inlet and outlet of the pipe may be assumed to be black. (a) Write down the governing equations and boundary conditions in the cylindrical coordinate system (2D axisymmetric) needed to determine the temperature distribution within the pipe and heat flux on the pipe wall. Also outline how you will determine the axial temperature distribution on the wall of the pipe. Is radiation expected to play any role in the solution to the problem? Explain from a physical perspective. (b) Discretize the computational domain (pipe) into N (axial) × M (radial) control volumes, and develop discrete energy balance and radiation exchange equations using Example 8.3 as a guide. (c) Write a computer program to determine the temperature distribution within the pipe, and the heat flux (both radiative and total) on the pipe wall. Compare the solutions with and without radiation. The following values may be used: ro = 1 cm, L = 10 cm, Tin = 2000 K, uo = 1 cm/s, To = 300 K, ho = 100 W/m2 K, o = 0.1, and i = 0.8. The thermophysical properties of the combustion gas are as follows: density ρ = 1.2 kg/m3 , kinematic viscosity ν = 10−5 m2 /s, specific heat capacity cp = 1000 J/kg K, and thermal conductivity k = 0.023 W/mK.

Surface Radiative Exchange in the Presence of Conduction and Convection Chapter | 8 281

References [1] R.E. Hogan, D.K. Gartling, Solution strategies for coupled conduction/radiation problems, Communications in Numerical Methods in Engineering 24 (2008) 523–542. [2] C.W. Gear, Numerical Initial-Value Problems in Ordinary Differential Equations, Prentice Hall, 1971. [3] S. Mazumder, Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods, Academic Press, 2016. [4] S.D. Williams, D.M. Curry, An implicit-iterative solution of the heat conduction equation with a radiation boundary condition, International Journal for Numerical Methods in Engineering 11 (1977) 1605–1619. [5] R. Bialecki, A.J. Nowak, Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions, Applied Mathematical Modelling 5 (1981) 417–421. [6] Z.F. Shen, T.F. Smith, P. Hix, Linearization of the radiation terms for improved convergence by use of the zone method, Numerical Heat Transfer 6 (1983) 377–382. [7] S. Mazumder, A new numerical procedure for coupling radiation in participating media with other modes of heat transfer, ASME Journal of Heat Transfer 127 (9) (2005) 1037–1045. [8] S.S. Abarbanel, Time dependent temperature distribution in radiating solids, Journal of Mathematical Physics 39 (4) (1960) 246–257. [9] E.R.G. Eckert, T.F. Irvine, E.M. Sparrow, Analytical formulation for radiating fins with mutual irradiation, American Rocket Society Journal 30 (1960) 644–646. [10] E.N. Nilson, R. Curry, The minimum weight straight fin of triangular profile radiating to space, Journal of the Aerospace Sciences 27 (1960) 146. [11] R.S. Hickman, Transient response and steady-state temperature distribution in a heated, radiating, circular plate, Technical Report 32-169, California Institute of Technology, Jet Propulsion Laboratory, 1961. [12] M.A. Heaslet, H. Lomax, Numerical predictions of radiative interchange between conducting fins with mutual irradiations, NASA TR R-116, 1961. [13] L.D. Nichols, Surface-temperature distribution on thin-walled bodies subjected to solar radiation in interplanetary space, NASA TN D-584, 1961. [14] L.H. Schreiber, R.P. Mitchell, G.D. Gillespie, T.M. Olcott, Techniques for optimization of a finned-tube radiator, ASME Paper No. 61-SA-44, June 1961. [15] W.E. Olmstead, S. Raynor, Solar heating of a rotating spherical space vehicle, International Journal of Heat and Mass Transfer 5 (1962) 1165–1177. [16] J.E. Wilkins, Minimum-mass thin fins and constant temperature gradients, Journal of the Society for Industrial and Applied Mathematics 10 (1) (1962) 62–73. [17] P. Hrycak, Influence of conduction on spacecraft skin temperatures, AIAA Journal 1 (1963) 2619–2621. [18] B.V. Karlekar, B.T. Chao, Mass minimization of radiating trapezoidal fins with negligible base cylinder interaction, International Journal of Heat and Mass Transfer 6 (1963) 33–48. [19] N.O. Stockman, J.L. Kramer, Effect of variable thermal properties on one-dimensional heat transfer in radiating fins, NASA TN D-1878, 1963. [20] K. Kotan, O.A. Arnas, On the optimization of the design parameters of parabolic radiating fins, ASME Paper No. 65-HT-42, August 1965. [21] H.F. Mueller, N.D. Malmuth, Temperature distribution in radiating heat shields by the method of singular perturbations, International Journal of Heat and Mass Transfer 8 (1965) 915–920. [22] L.D. Russell, A.J. Chapman, Analytical solution of the ‘known-heat-load’ space radiator problem, Journal of Spacecraft and Rockets 4 (3) (1967) 311–315. [23] W. Frost, A.H. Eraslan, An iterative method for determining the heat transfer from a fin with radiative interaction between the base and adjacent fin surfaces, AIAA Paper No. 68-772, June 1968. [24] R.C. Donovan, W.M. Rohrer, Radiative conducting fins on a plane wall, including mutual irradiation, ASME Paper No. 69-WA/HT-22, November 1969. [25] N.M. Schnurr, A.B. Shapiro, M.A. Townsend, Optimization of radiating fin arrays with respect to weight, ASME Journal of Heat Transfer 98 (4) (1976) 643–648. [26] R.G. Eslinger, B.T.F. Chung, Periodic heat transfer in radiating and convecting fins or fin arrays, AIAA Journal 17 (10) (1979) 1134–1140. [27] D.S. Gerencser, A. Razani, Optimization of radiative–convective arrays of pins fins including mutual irradiation between fins, International Journal of Heat and Mass Transfer 38 (1995) 899–907. [28] B.T.F. Chung, B.X. Zhang, E.T. Lee, A multi-objective optimization of radiative fin array systems in a fuzzy environment, ASME Journal of Heat Transfer 118 (3) (1996) 642–649. [29] C.K. Krishnaprakas, Optimum design of radiating rectangular plate fin array extending from a plane wall, ASME Journal of Heat Transfer 118 (1996) 490–493. [30] C.K. Krishnaprakas, Optimum design of radiating longitudinal fin array extending from a cylindrical surface, ASME Journal of Heat Transfer 119 (1997) 857–860. [31] S.P. Fitzgerald, W. Strieder, Radiation heat transfer down an elongated spheroidal cavity, AIChE Journal 43 (1997) 2–12.

282 Radiative Heat Transfer

[32] X.G. Liang, W. Qu, Effective thermal conductivity of gas–solid composite materials and the temperature difference effect at high temperature, International Journal of Heat and Mass Transfer 42 (10) (1999) 1885–1893. [33] B.P. Singh, M. Kaviany, Effect of solid conductivity on radiative heat transfer in packed beds, International Journal of Heat and Mass Transfer 37 (16) (1994) 2579–2583. [34] R. Haya, D. Rivas, J. Sanz, Radiative exchange between a cylindrical crystal and a monoellipsoidal mirror furnace, International Journal of Heat and Mass Transfer 40 (1997) 323–332. [35] K.G.T. Hollands, K. Iynkaran, Analytical model for the thermal conductance of compound honeycomb transparent insulation, with experimental validation, Solar Energy 51 (1993) 223–227. [36] P.D. Jones, Correlation of combined radiation and conduction in evacuated honeycomb-cored panels, Journal of Solar Energy Engineering 118 (1996) 97–100. [37] H. Schweiger, A. Oliva, M. Costa, C.D. Segarra, Monte Carlo method for the simulation of transient radiation heat transfer: application to compound honeycomb transparent insulation, Numerical Heat Transfer. Part B, Fundamentals 35 (1999) 113–136. [38] M. Grimm, S. Mazumder, Numerical investigation of radiation effects in monolithic catalytic combustion reactors, International Journal of Chemical Reactor Engineering 9 (2011) A44. [39] C. Rauch, T. Moshammer, A. Raimund, Thermal radiation benchmarks for the automotive industry, in: VTMS 8 - Vehicle Thermal Management Systems Conference and Exhibition, vol. C640, 2007, pp. 695–704. [40] C. Rauch, Thermal radiation modeling for automotive applications, Ph.D. thesis, Graz University of Technology, Graz, Austria, 2012. [41] S. Sahoo, A. Kuchhal, S. Vineeth, V. Chauhan, Thermal radiation heat transfer model and its application for automobile exhaust components, SAE Technical Paper, SAE International, 2016, pp. 2016–28–0051. [42] N.N. Verma, A. Iacob, S. Mazumder, A. Selamet, Experimental and computational study of flow and heat transfer around a surrogate engine mount, in: Proceedings of International Mechanical Energy Congress and Exposition, Paper IMECE2017-70580, 2017. [43] N.N. Verma, A. Iacob, S. Mazumder, A. Selamet, Combined computational and experimental analysis of cooldown of a surrogate engine mount assembly, in: Proceedings of International Mechanical Energy Congress and Exposition, Paper IMECE2018-87139, 2018. [44] J.W. Lee, E.Y. Jang, S.H. Lee, H.S. Ryou, S. Choi, Y. Kim, Influence of the spectral solar radiation on the air flow and temperature distributions in a passenger compartment, International Journal of Thermal Sciences 75 (2014) 36–44. [45] J.H. Moon, J.W. Lee, C.H. Jeong, H.L. Seong, Thermal comfort analysis in a passenger compartment considering the solar radiation effect, International Journal of Thermal Sciences 107 (2016) 77–88. [46] S. Mazumder, A. Kersch, Effect of thin films on radiative transport in chemical vapor deposition systems, in: Proceedings of International Mechanical Engineering Congress and Exposition, vol. 364-3, 1999, pp. 9–13. [47] S. Mazumder, A. Kersch, A fast Monte Carlo scheme for thermal radiation in semiconductor processing applications, Numerical Heat Transfer. Part B, Fundamentals 37 (2) (2000) 185–199. [48] A. Kersch, T. Schafbauer, Thermal modeling of RTP and RTCVD processes, Thin Solid Films 365 (2) (2000) 307–321. [49] V. Nicholas, P. Glouannec, J.P. Ploteau, P. Salagnac, V. Jury, Experiment and multiphysic simulation of dough baking by convection, infrared radiation and direct conduction, International Journal of Thermal Sciences 115 (2017) 65–78. [50] T.L. Bergman, A.S. Lavine, F.P. Incropera, D.P. DeWitt, Introduction to Heat Transfer, sixth ed., John Wiley & Sons, New York, 2011. [51] J.P. Holman, Heat Transfer, 7th ed., McGraw-Hill Book Company, New York, 1990. [52] E.M. Sparrow, E.R.G. Eckert, T.F. Irvine, The effectiveness of radiating fins with mutual irradiation, Journal of the Aerospace Sciences 28 (1961) 763–772. [53] R.G. Hering, Radiative heat exchange between conducting plates with specular reflection, ASME Journal of Heat Transfer C 88 (1966) 29–36. [54] C.L. Tien, Approximate solutions of radiative exchange between conducting plates with specular reflection, ASME Journal of Heat Transfer 89C (1967) 119–120. [55] J.G. Bartas, W.H. Sellers, Radiation fin effectiveness, ASME Journal of Heat Transfer 82C (1960) 73–75. [56] E.M. Sparrow, E.R.G. Eckert, Radiant interaction between fins and base surfaces, ASME Journal of Heat Transfer C 84 (1962) 12–18. [57] E.M. Sparrow, V.K. Jonsson, W.J. Minkowycz, Heat transfer from fin-tube radiators including longitudinal heat conduction and radiant interchange between longitudinally non-isothermal finite surfaces, NASA TN D-2077, 1963. [58] S. Lieblein, Analysis of temperature distribution and radiant heat transfer along a rectangular fin, NASA TN D-196, 1959. [59] R.L. Chambers, E.V. Sommers, Radiation fin efficiency for one-dimensional heat flow in a circular fin, ASME Journal of Heat Transfer 81C (4) (1959) 327–329. [60] H.H. Keller, E.S. Holdredge, Radiation heat transfer for annular fins of trapezoid profile, ASME Journal of Heat Transfer 92 (6) (1970) 113–116. [61] D.B. Mackay, Design of Space Powerplants, Prentice-Hall, Englewood Cliffs, NJ, 1963. [62] E.M. Sparrow, G.B. Miller, V.K. Jonsson, Radiating effectiveness of annular-finned space radiators including mutual irradiation between radiator elements, Journal of the Aerospace Sciences 29 (1962) 1291–1299. [63] W.M. Kays, M.E. Crawford, Convective Heat and Mass Transfer, 3rd ed., McGraw-Hill, 1993. [64] C.E. Fröberg, Introduction to Numerical Analysis, Addison-Wesley, Reading, MA, 1969. [65] R.W. Hornbeck, Numerical Methods, Quantum Publishers, Inc., New York, 1975. [66] J.H. Ferziger, Numerical Methods for Engineering Application, John Wiley & Sons, New York, 1981.

Surface Radiative Exchange in the Presence of Conduction and Convection Chapter | 8 283

[67] R. Siegel, M. Perlmutter, Convective and radiant heat transfer for flow of a transparent gas in a tube with gray wall, International Journal of Heat and Mass Transfer 5 (1962) 639–660. [68] M. Perlmutter, R. Siegel, Heat transfer by combined forced convection and thermal radiation in a heated tube, ASME Journal of Heat Transfer C 84 (1962) 301–311. [69] R. Siegel, E.G. Keshock, Wall temperature in a tube with forced convection, internal radiation exchange and axial wall conduction, NASA TN D-2116, 1964. [70] B.I. Dussan, T.F. Irvine, Laminar heat transfer in a round tube with radiating flux at the outer wall, in: Proceedings of the Third International Heat Transfer Conference, vol. 5, Hemisphere, Washington, D.C., 1966, pp. 184–189. [71] J.C. Chen, Laminar heat transfer in a tube with nonlinear radiant heat-flux boundary conditions, International Journal of Heat and Mass Transfer 9 (1966) 433–440. [72] R.S. Thorsen, Heat transfer in a tube with forced convection, internal radiation exchange, axial wall heat conduction and arbitrary wall heat generation, International Journal of Heat and Mass Transfer 12 (1969) 1182–1187. [73] R.S. Thorsen, D. Kanchanagom, The influence of internal radiation exchange, arbitrary wall heat generation and wall heat conduction on heat transfer in laminar and turbulent flows, in: Proceedings of the Fourth International Heat Transfer Conference, vol. 3, Elsevier, New York, 1970, pp. 1–10. [74] E.G. Keshock, R. Siegel, Combined radiation and convection in asymmetrically heated parallel plate flow channel, ASME Journal of Heat Transfer 86C (1964) 341–350. [75] S.T. Lin, R.S. Thorsen, Combined forced convection and radiation heat transfer in asymmetrically heated parallel plates, in: Proceedings of the Heat Transfer and Fluid Mechanics Institute, Stanford University Press, 1970, pp. 32–44. [76] R.D. Cess, The effect of radiation upon forced-convection heat transfer, Applied Scientific Research Part A 10 (1962) 430–438. [77] R.D. Cess, The interaction of thermal radiation with conduction and convection heat transfer, in: Advances in Heat Transfer, vol. 1, Academic Press, New York, 1964, pp. 1–50. [78] E.M. Sparrow, S.H. Lin, Boundary layers with prescribed heat flux–application to simultaneous convection and radiation, International Journal of Heat and Mass Transfer 8 (1965) 437–448. [79] M. Sohal, J.R. Howell, Determination of plate temperature in case of combined conduction, convection and radiation heat exchange, International Journal of Heat and Mass Transfer 16 (1973) 2055–2066. [80] S. Gianoulakis, D.E. Klein, Combined natural convection and surface radiation in the annular region between volumetrically heated inner tube and a finite conducting outer tube, Nuclear Technology 104 (1993) 241–251. [81] C. Balaji, S.P. Venkateshan, Natural convection in L-corners with surface radiation and conduction, ASME Journal of Heat Transfer 118 (1996) 222–225. [82] V.R. Rao, S.P. Venkateshan, Experimental study of free convection and radiation in horizontal fin arrays, International Journal of Heat and Mass Transfer 39 (1996) 779–789. [83] V.R. Rao, C. Balaji, S.P. Venkateshan, Interferometric study of interaction of free convection with surface radiation in an l corner, International Journal of Heat and Mass Transfer 40 (1997) 2941–2947. [84] K.S. Jayaram, C. Balaji, S.P. Venkateshan, Interaction of surface radiation and free convection in an enclosure with a vertical partition, ASME Journal of Heat Transfer 119 (1997) 641–645. [85] X. Cheng, U. Müller, Turbulent natural convection coupled with thermal radiation in large vertical channels with asymmetric heating, International Journal of Heat and Mass Transfer 41 (12) (1998) 1681–1692. [86] N. Ramesh, S.P. Venkateshan, Effect of surface radiation on natural convection in a square enclosure, Journal of Thermophysics and Heat Transfer 13 (3) (1999) 299–301. [87] E. Yu, Y.K. Joshi, Heat transfer in discretely heated side-vented compact enclosures by combined conduction, natural convection, and radiation, ASME Journal of Heat Transfer 121 (4) (1999) 1002–1010. [88] V.H. Adams, Y.K. Joshi, D.L. Blackburn, Three-dimensional study of combined conduction, radiation, and natural convection from discrete heat sources in a horizontal narrow-aspect-ratio enclosure, ASME Journal of Heat Transfer 121 (4) (1999) 992–1001. [89] K. Velusamy, T. Sundararajan, K.N. Seetharamu, Interaction effects between surface radiation and turbulent natural convection in square and rectangular enclosures, ASME Journal of Heat Transfer 123 (6) (2001) 1062–1070. [90] D.T. Vader, R. Viskanta, F.P. Incropera, Design and testing of a high-temperature emissometer for porous and particulate dielectrics, Review of Scientific Instruments 57 (1) (1986) 87–93. [91] K.K. Sikka, High temperature normal spectral emittance of silicon carbide based materials, M.S. thesis, The Pennsylvania State University, University Park, PA, 1991.

Chapter 9

The Radiative Transfer Equation in Participating Media (RTE) 9.1 Introduction In previous chapters we have looked at radiative transfer between surfaces that were separated by vacuum or by a transparent (“radiatively nonparticipating”) medium. However, in many engineering applications the interaction of thermal radiation with an absorbing, emitting, and scattering (“radiatively participating”) medium must be accounted for. Examples in the heat transfer area are the burning of any fuel (be it gaseous, liquid, or solid; be it for power production, within fires, within explosions, etc.), rocket propulsion, hypersonic shock layers, ablation systems on reentry vehicles, nuclear explosions, plasmas in fusion reactors, and many more. In the present chapter we shall develop the general relationships that govern the behavior of radiative heat transfer in the presence of an absorbing, emitting, and/or scattering medium. We shall begin by making a radiative energy balance, known as the radiative transfer equation, or RTE, which describes the radiative intensity field within the enclosure as a function of location (fixed by location vector r), direction (fixed by unit direction vector sˆ ), and spectral variable (wavenumber η).1 To obtain the net radiative heat flux crossing a surface element, we must sum the contributions of radiative energy irradiating the surface from all possible directions and for all possible wavenumbers. Therefore, integrating the radiative transfer equation over all directions and wavenumbers leads to a conservation of radiative energy statement applied to an infinitesimal volume. Finally, this will be combined with a balance for all types of energy (including conduction and convection), leading to the Overall Conservation of Energy equation. In the following three chapters we shall deal with the radiation properties of participating media, i.e., with how a substance can absorb, emit, and scatter thermal radiation. In Chapter 10 we discuss how a molecular gas can absorb and emit photons by changing its energy states, how to predict the radiation properties, and how to measure them experimentally. Chapter 11 is concerned with how small particles interact with electromagnetic waves—how they absorb, emit, and scatter radiative energy. Again, theoretical as well as experimental methods are covered. Finally, in Chapter 12 a very brief account is given of the radiation properties of solids and liquids that allow electromagnetic waves of certain wavelengths to penetrate into them for appreciable distances, known as semitransparent media.

9.2 Attenuation by Absorption and Scattering If the medium through which radiative energy travels is “participating,” then any incident beam will be attenuated by absorption and scattering while it travels through the medium, as schematically shown in Fig. 9.1. In the following we shall develop expressions for this attenuation for a light beam which travels within a pencil of rays into the direction sˆ . The present discussion will be limited to media with constant refractive index, i.e., media through which electromagnetic waves travel along straight lines [while a varying refractive index will bend the ray, as shown by Snell’s law, equation (2.72), for an abrupt change]. It is further assumed that the medium is stationary (as compared to the speed of light), that it is nonpolarizing, and that it is (for most of the discussion) at local thermodynamic equilibrium (LTE). 1. In our discussion of surface radiative transport we have used wavelength λ as the spectral variable throughout, largely to conform with the majority of other publications. However, for gases, frequency ν or wavenumber η are considerably more convenient to use. Again, to conform with the majority of the literature, we shall use wavenumber throughout this part. Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00017-1 Copyright © 2022 Elsevier Inc. All rights reserved.

285

286 Radiative Heat Transfer

FIGURE 9.1 Attenuation of radiative intensity by absorption and scattering.

Absorption The absolute amount of absorption has been observed to be directly proportional to the magnitude of the incident energy as well as the distance the beam travels through the medium. Thus, we may write, (dIη ) abs = −κη Iη ds,

(9.1)

where the proportionality constant κη is known as the (linear) absorption coefficient, and the negative sign has been introduced since the intensity decreases. As will be discussed in the following chapter, the absorption of radiation in molecular gases depends also on the number of receptive molecules per unit volume, so that some researchers use a density-based absorption coefficient or a pressure-based absorption coefficient, defined by (dIη ) abs = −κρη Iη ρ ds = −κpη Iη p ds.

(9.2)

The subscripts ρ and p are used here only to demonstrate the differences between the coefficients. The reader of scientific literature often must rely on the physical units to determine the coefficient used. Integration of equation (9.1) over a geometric path s results in ⎛ s ⎞ ⎜⎜  ⎟⎟ ⎜ (9.3) Iη (s) = Iη (0) exp ⎝⎜− κη ds⎟⎠⎟ = Iη (0) e−τη , 0

where

 τη =

s

κη ds

(9.4)

0

is called the optical thickness (for absorption) through which the beam has traveled and Iη (0) is the intensity entering the medium at s = 0. Equation (9.3) is commonly known as Beer’s law.2 Note that the (linear) absorption coefficient is the inverse of the mean free path for a photon until it undergoes absorption. One may also define an absorptivity for the participating medium (for a given path within the medium) as αη ≡

Iη (0) − Iη (s) Iη (0)

= 1 − e−τη .

(9.5)

2. Historically, the law was apparently first discovered by Pierre Bouguer in 1729, noting that the absorbance (optical thickness in this text) = − ln[Iη (s)/Iη (0)] of a solution remains the same as long as product of concentration (which is proportional to absorption coefficient) and layer thickness is constant. This was reported by Johann Heinrich Lambert in 1760. Much later, August Beer discovered a similar attenuation relation in 1852. Consequently, the law is also known by several variations of the term Bouguer–Lambert–Beer Law.

The Radiative Transfer Equation in Participating Media (RTE) Chapter | 9 287

Scattering Attenuation by scattering, or “out-scattering” (away from the direction under consideration), is very similar to absorption, i.e., a part of the incoming intensity is removed from the direction of propagation, sˆ . The only difference between the two phenomena is that absorbed energy is converted into internal energy, while scattered energy is simply redirected and appears as augmentation along another direction (discussed in the next section), also known as “in-scattering.” Thus, we may write (dIη ) sca = −σsη Iη ds,

(9.6)

where the proportionality constant σsη is the (linear) scattering coefficient for scattering from the pencil of rays under consideration into all other directions. Again, scattering coefficients based on density or pressure may be defined. It is also possible to define an optical thickness for scattering, where the scattering coefficient is the inverse of the mean free path for scattering.

Total Attenuation The total attenuation of the intensity in a pencil of rays by both absorption and scattering is known as extinction. Thus, an extinction coefficient is defined3 as βη = κη + σsη .

(9.7)

The optical distance based on extinction is defined as 

s

τη =

βη ds.

(9.8)

0

As for absorption and scattering, the extinction coefficient is sometimes based on density or pressure.

9.3 Augmentation by Emission and Scattering A light beam traveling through a participating medium in the direction of sˆ loses energy by absorption and by scattering away from the direction of travel. But at the same time it also gains energy by emission, as well as by scattering from other directions into the direction of travel sˆ .

Emission The rate of emission from a volume element will be proportional to the magnitude of the volume. Therefore, the emitted intensity (which is the rate of emitted energy per unit area) along any path again must be proportional to the length of the path, and it must be proportional to the local energy content in the medium. Thus, (dIη ) em = jη ds,

(9.9)

where jη is termed the emission coefficient. Since, at local thermodynamic equilibrium (LTE), the intensity everywhere must be equal to the blackbody intensity, it will be shown in Chapter 10, equation (10.22), that jη = κη Ibη

and (dIη ) em = κη Ibη ds,

(9.10)

that is, at LTE the proportionality constant for emission is the same as for absorption. Similar to absorptivity, one may also define an emissivity of an isothermal medium as the amount of energy emitted over a certain path s that escapes into a given direction (without having been absorbed between point of emission and point of exit), as compared to the maximum possible. Combining equations (9.1) and (9.10) gives the complete radiative transfer equation for an absorbing–emitting (but not scattering) medium as dIη ds

= κη (Ibη − Iη ),

(9.11)

3. Care must be taken to distinguish the dimensional extinction coefficient βη from the absorptive index, i.e., the imaginary part of the index of refraction complex k (sometimes referred to in the literature as the “extinction coefficient”).

288 Radiative Heat Transfer

FIGURE 9.2 Redirection of radiative intensity by scattering.

where the first term of the right-hand side is augmentation due to emission and the second term is attenuation due to absorption. The solution to the radiative transfer equation for an isothermal gas layer of thickness s is   (9.12) Iη (s) = Iη (0) e−τη + Ibη 1 − e−τη , where the optical distance has been defined in equation (9.4). If only emission is considered, Iη (0) = 0, and the emissivity is defined as

η = Iη (s)/Ibη = 1 − e−τη ,

(9.13)

which, as is the case with surface radiation, is identical to the expression for absorptivity.

Scattering Augmentation due to scattering, or “in-scattering,” has contributions from all directions and, therefore, must be calculated by integration over all solid angles. Consider the radiative heat flux impinging on a volume element dV = dA ds, from an infinitesimal pencil of rays in the direction sˆ i as depicted in Fig. 9.2. Recalling the definition for radiative intensity as energy flux per unit area normal to the rays, per unit solid angle, and per unit wavenumber interval, one may calculate the spectral radiative heat flux impinging on dA from within the solid angle dΩ i as Iη (ˆs i )(dA sˆ i · sˆ ) dΩ i dη. This flux travels through dV for a distance ds/ˆs i · sˆ . Therefore, the total amount of energy scattered away from sˆ i is, according to equation (9.6),    ds  (9.14) σsη Iη (ˆs i )(dA sˆ i · sˆ ) dΩi dη = σsη Iη (ˆs i ) dA dΩ i dη ds. sˆ i · sˆ Of this amount, the fraction Φη (ˆs i , sˆ ) dΩ/4π is scattered into the cone dΩ around the direction sˆ . The function Φη is called the scattering phase function and describes the probability that a ray from one direction, sˆ i , will be scattered into a certain other direction, sˆ . The constant 4π is arbitrary and is included for convenience [see equation (9.17)]. The amount of energy flux from the cone dΩ i scattered into the cone dΩ is then σsη Iη (ˆs i ) dA dΩ i dη ds

Φη (ˆs i , sˆ ) 4π

dΩ.

(9.15)

We can now calculate the energy flux scattered into the direction sˆ from all incoming directions sˆ i by integrating:    dΩ , σsη Iη (ˆs i ) dA dΩ i dη ds Φη (ˆs i , sˆ ) dIη (ˆs) dA dΩ dη = sca 4π 4π or

The Radiative Transfer Equation in Participating Media (RTE) Chapter | 9 289

FIGURE 9.3 Pencil of rays for radiative energy balance.



dIη

 sca

(ˆs) = ds

σsη 4π

 Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i .

(9.16)

4π

Returning to equation (9.15), we find that the amount of energy flux scattered from dΩ i into all directions is  1 σsη Iη (ˆs i ) dA dΩ i dη ds Φη (ˆs i , sˆ ) dΩ, 4π 4π which must be equal to the amount in equation (9.14). We conclude that  1 Φη (ˆs i , sˆ ) dΩ ≡ 1. 4π 4π

(9.17)

Therefore, if Φη = const, i.e., if equal amounts of energy are scattered into all directions (called isotropic scattering), then Φη ≡ 1. This is the reason for the inclusion of the factor 4π.

9.4 The Radiative Transfer Equation We can now make an energy balance on the radiative energy traveling in the direction of sˆ within a small pencil of rays as shown in Fig. 9.3. The change in intensity is found by summing the contributions from emission, absorption, scattering away from the direction sˆ , and scattering into the direction of sˆ , from equations (9.1), (9.6), (9.9), and (9.16) as  σsη Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i ds. (9.18) Iη (s+ds, sˆ , t+dt) − Iη (s, sˆ , t) = jη (s, t) ds − κη Iη (s, sˆ , t) ds − σsη Iη (s, sˆ , t) ds + 4π 4π This equation is Lagrangian in nature, i.e., we are following a ray from s to s+ds; since the ray travels at the speed of light c, ds and dt are related through ds = c dt. The outgoing intensity may be developed into a truncated Taylor series, or Iη (s+ds, sˆ , t+dt) = Iη (s, sˆ , t) + dt

∂Iη ∂t

+ ds

∂Iη ∂s

,

(9.19)

so that equation (9.18) may be simplified to σsη 1 ∂Iη ∂Iη + = jη − κη Iη − σsη Iη + c ∂t 4π ∂s

 Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i .

(9.20)

4π

In this radiative transfer equation (commonly abbreviated as RTE), or equation of transfer, all quantities may vary with location in space, time, and wavenumber, while the intensity and the phase function also depend on direction sˆ (and sˆ i ). Only the directional dependence, and only whenever necessary, has been explicitly indicated in this and the following equations, to simplify notation. As indicated earlier, the development of this equation is subject to a number of simplifying assumptions, viz., the medium is homogeneous and at rest (as compared to the speed of light), the medium is nonpolarizing and the state of polarization is neglected, and the medium has a constant index of refraction. An elaborate discussion of these limitations has been given by Viskanta and Mengüç [1]. The RTE for a medium with varying refractive index is discussed in Section 9.7. Equation (9.20) is valid anywhere inside an arbitrary enclosure. Its solution requires knowledge of the intensity for each direction at some location s, usually the intensity entering the medium through or from

290 Radiative Heat Transfer

FIGURE 9.4 Enclosure for derivation of radiative transfer equation.

the enclosure boundary into the direction of sˆ , as indicated in Fig. 9.4. We have not yet brought the radiative transfer equation into its most compact form so that the four different contributions to the change of intensity may be clearly identified. Equation (9.20) is the transient form of the radiative transfer equation, valid at local thermodynamic equilibrium as well as nonequilibrium. Over the last few years, primarily due to the development of short-pulsed lasers, with pulse durations in the ps or fs range, transient radiation phenomena have been becoming of increasing importance [2]. However, for the vast majority of engineering applications, the speed of light is so large compared to local time and length scales that the first term in equation (9.20) may be neglected. There are also several important applications that take place at thermodynamic nonequilibrium, such as the strong nonequilibrium radiation hitting a hypersonic spacecraft entering Earth’s atmosphere [3] (creating a high-temperature plasma ahead of it; cf. Fig. 10.7). Nevertheless, most engineering applications are at local thermodynamic equilibrium. We have presented here the full equation for completeness, but will omit the transient and nonequilibrium terms during the remainder of this book (with the exception of a very brief discussion of nonequilibrium properties in Chapter 10, and a somewhat more detailed consideration of transient radiation in Chapter 18). After introducing the extinction coefficient defined in equation (9.7), one may restate equation (9.20) in its equilibrium, quasi-steady form as dIη ds

= sˆ · ∇Iη = κη Ibη − βη Iη +



σsη

Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i ,

4π

(9.21)

4π

where the intensity gradient has been converted into a total derivative since we assume the process to be quasisteady. The radiative transfer equation is often rewritten in terms of nondimensional optical coordinates (see Fig. 9.4),  s  s τη = (κη + σsη ) ds = βη ds, (9.22) 0

0

and the single scattering albedo, first defined in equation (1.61) as ωη ≡

σsη κη + σsη

=

σsη βη

,

(9.23)

leading to dIη dτη

= −Iη + (1 − ωη )Ibη +

ωη 4π

 Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i . 4π

(9.24)

The Radiative Transfer Equation in Participating Media (RTE) Chapter | 9 291

The last two terms in equation (9.24) are often combined and are then known as the source function for radiative intensity,  ωη Sη (τη , sˆ ) = (1 − ωη )Ibη + Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩi . (9.25) 4π 4π Equation (9.24) then assumes the deceptively simple form of dIη dτη

+ Iη = Sη (τη , sˆ ),

(9.26)

which is, of course, an integro-differential equation (in space, and in two directional coordinates with local origin). Furthermore, the Planck function Ibη is generally not known and must be found by considering the overall energy equation (adding derivatives in the three space coordinates and integrations over two more directional coordinates and the wavenumber spectrum).

9.5 Formal Solution to the Radiative Transfer Equation If the source function is known (or assumed known), equation (9.26) can be formally integrated by the use of an integrating factor. Thus, multiplying through by eτη results in d  τη  Iη e = Sη (τη , sˆ ) eτη , dτη

(9.27)

which may be integrated from a point s = 0 at the wall to a point s = s inside the medium (see Fig. 9.4), so that  τη  −τη Iη (τη ) = Iη (0) e + Sη (τη , sˆ ) e−(τη −τη ) dτη , (9.28) 0

τη



where is the optical coordinate at s = s . Physically, one can readily appreciate that the first term on the right-hand side of equation (9.28) is the contribution to the local intensity by the intensity entering the enclosure at s = 0, which decays exponentially due to extinction over the optical distance τη . The integrand of the second term, Sη (τη ) dτη , on the other hand, is the contribution from the local emission at τη , attenuated exponentially by self-extinction over the optical distance between the emission point and the point under consideration, τη − τη . The integral, finally, sums all the contributions over the entire emission path. Equation (9.28) is a third-order integral equation in intensity Iη . The integral over the source function must be carried out over the optical coordinate (for all directions), while the source function itself is also an integral over a set of direction coordinates (with varying local origin) containing the unknown intensity. Furthermore, usually the temperature and, therefore, the blackbody intensity are not known and must be found in conjunction with overall conservation of energy. There are, however, a few cases for which the radiative transfer equation becomes considerably simplified.

Nonscattering Medium If the medium only absorbs and emits, the source function reduces to the local blackbody intensity, and  τη  −τη Iη (τη ) = Iη (0) e + Ibη (τη ) e−(τη −τη ) dτη . (9.29) 0

This equation is an explicit expression for the radiation intensity if the temperature field is known. However, generally the temperature is not known and must be found in conjunction with overall conservation of energy. Example 9.1. What is the spectral intensity emanating from an isothermal sphere bounded by vacuum or a cold black wall? Solution Because of the symmetry in this problem, the intensity emanating from the sphere surface is only a function of the exit

292 Radiative Heat Transfer

FIGURE 9.5 Isothermal sphere for Example 9.1.

angle. Examining Fig. 9.5, we see that equation (9.29) reduces to 

τs

Iη (τR , θ) = 0



Ibη (τs ) e−(τs −τs ) dτs .

But for a sphere τs = 2τR cos θ, regardless of the azimuthal angle. Therefore, with Ibη (τs ) = Ibη = const, the desired intensity turns out to be     2τR cos θ = Ibη 1 − e−2τR cos θ . Iη (τR , θ) = Ibη e−(2τR cos θ−τs )  0

Thus, for τR  1 the isothermal sphere emits equally into all directions, like a black surface at the same temperature.

The Cold Medium If the temperature of the medium is so low that the blackbody intensity at that temperature is small as compared with incident intensity, then the radiative transfer equation is decoupled from other modes of heat transfer. However, the governing equation remains a third-order integral equation, namely,  −τη

Iη (τη , sˆ ) = Iη (0) e

τη

+

ωη



4π

0



4π

Iη (τη , sˆ i ) Φη (ˆs i , sˆ ) dΩ i e−(τη −τη ) dτη .

(9.30)

If the scattering is isotropic, or Φ ≡ 1, the directional integration in equation (9.30) may be carried out, so that −τη

Iη (τη , sˆ ) = Iη (0) e

1 + 4π

where

 0

τη



ωη Gη (τη ) e−(τη −τη ) dτη ,

(9.31)

 Gη (τ) ≡ 4π

Iη (τη , sˆ i ) dΩ i

(9.32)

is known as the incident radiation function (since it is the total intensity impinging on a point from all sides). The problem is then much simplified since it is only necessary to find a solution for G [by direction-integrating equation (9.31)] rather than determining the direction-dependent intensity.

The Radiative Transfer Equation in Participating Media (RTE) Chapter | 9 293

FIGURE 9.6 Geometry for Example 9.2.

Purely Scattering Medium If the medium scatters radiation, but does not absorb or emit, then the radiative transfer is again decoupled from other heat transfer modes. In this case ωη ≡ 1, and the radiative transfer equation reduces to a form essentially identical to equation (9.30), i.e.,  τη  1  Iη (τη , sˆ i ) Φη (ˆs i , sˆ ) dΩ i e−(τη −τη ) dτη . (9.33) Iη (τη , sˆ ) = Iη (0) e−τη + 4π 0 4π Again, for isotropic scattering, this equation may be simplified by introducing the incident radiation, so that  τη 1  Gη (τη , sˆ ) e−(τη −τη ) dτη . (9.34) Iη (τη , sˆ ) = Iη (0) e−τη + 4π 0 Example 9.2. A large isothermal black plate is covered with a thin layer of isotropically scattering, nonabsorbing (and, therefore, nonemitting) material with unity index of refraction. Assuming that the layer is so thin that any ray emitted from the plate is scattered at most once before leaving the scattering layer, estimate the radiative intensity above the layer in the direction normal to the plate. Solution The exiting intensity in the normal direction (see Fig. 9.6) may be calculated from equation (9.34) by retaining only terms of order τη or higher (since τη 1). This process leads to e−τη = 1 − τη + O(τ2η ), G(τη ) = G(τη ) + O(τη ) (radiation to be  scattered arrives unattenuated at a point), and e−(τη −τη ) = 1 − O(τη ) (scattered radiation will leave the medium without further attenuation), so that Inη = Ibη (1 − τη ) +

1 Gη τη + O(τ2η ), 4π

where the intensity emanating from the plate is known since the plate is black. The incident radiation at any point is due to unattenuated emission from the bottom plate arriving from the lower 2π solid angles, and nothing coming from the top 2π solid angles, i.e., Gη ≈ 2πIbη and  τη  1 + O(τ2η ). Inη = Ibη (1 − τη ) + Ibη τη + O(τ2η ) = Ibη 1 − 2 2 Physically this result tells us that the emission into the normal direction is attenuated by the fraction τη (scattered away from the normal direction), and augmented by the fraction τη /2 (scattered into the normal direction): Since scattering is isotropic, exactly half of the attenuation is scattered upward and half downward; the latter is then absorbed by the emitting plate. Thus, the scattering layer acts as a heat shield for the hot plate.

9.6 Boundary Conditions for the Radiative Transfer Equation The radiative transfer equation in its quasi-steady form, equation (9.21), is a first-order differential equation in intensity (for a fixed direction sˆ ). As such, the equation requires knowledge of the radiative intensity at a single point in space, into the direction of sˆ . Generally, the point where the intensity can be specified independently lies on the surface of an enclosure surrounding the participating medium, as indicated by the formal solution in

294 Radiative Heat Transfer

FIGURE 9.7 Radiative intensity reflected from a surface.

equation (9.28). This intensity, leaving a wall into a specified direction, may be determined by the methods given in Chapter 5 (diffusely emitting and reflecting surfaces), and Chapter 6 (surfaces with nonideal characteristics).

Diffusely Emitting and Reflecting Opaque Surfaces For a surface that emits and reflects diffusely, the exiting intensity is independent of direction. Therefore, at a point rw on the surface, from equations (5.18) and (5.19), I(rw , sˆ ) = I(rw ) = J(rw )/π = (rw ) Ib (rw ) + ρ(rw ) H(rw )/π,

(9.35)

where H(rw ) is the hemispherical irradiation (i.e., incoming radiative heat flux) defined by equation (3.41), leading to  ρ(rw ) I(rw , sˆ ) = (rw ) Ib (rw ) + I(rw , sˆ  ) |nˆ · sˆ  | dΩ , (9.36) π ˆ s n2 ).

where dA is an infinitesimal area element on the interface, and we have chosen frequency ν as the spectral variable, because only frequency remains unchanged as light passes through media with different refractive indices. Eliminating solid angle dΩ = sin θdθdψ (and azimuthal angle ψ, which is unaffected by passing from one medium to the next), this simplifies to Iν1 (θ1 )(1 − ρ12 ) sin θ1 cos θ1 dθ1 = Iν1 (θ2 ) sin θ2 cos θ2 dθ2 .

(9.42)

From Snell’s law, equations (2.72) and (3.59), we have n1 sin θ1 = n2 sin θ2 , and, after differentiation, n1 cos θ1 dθ1 = n2 cos θ2 dθ2 .

(9.43)

Finally, sticking these two relations into equation (9.42), we obtain Iν1 (θ1 )(1 − ρ12 ) n21

=

Iν2 (θ2 ) . n22

(9.44)

Note that, since n1 > n2 , refraction in Medium 2 is away from the surface normal, i.e., θ2 > θ1 , and there is a critical angle θ1 = θc , as given by equation (2.100), at which θ2 = 90◦ and for larger θ1 there will be total internal reflection, and nothing is transmitted into Medium 2: θ1 > θc = sin−1



 n2 : n1

ρ12 = 1; Iν2 (θ2 ) = 0.

(9.45)

This is indicated in Fig. 9.9 by showing several additional incident directions (with thin dashed lines and open arrows), together with their transmitted (for θ1 < θc only) and reflected directions. Employing equations (9.44) and (9.45), we can now make a full energy balance for the interface, comprising intensity coming in from inside Medium 1, Iν1i (θ1 ), the fraction of it that is reflected, Iν1r (θ1 ) (with specular reflection angle θr = θ1 ), and the fraction transmitted into Medium 2, Iν1t (θ2 ), along with similar contributions from intensity striking the interface from inside Medium 2, as depicted in Fig. 9.9: 

 n2 2 Iν2 (θ2 ) = ρ21 Iν2i (θ2 ) + Iν1t (θ2 ) = ρ21 Iν2i (θ2 ) + (1 − ρ12 ) Iν1i (θ1 ), n1  2 n1 Iν1 (θ1 ) = ρ12 Iν1i (θ1 ) + Iν2t (θ1 ) = ρ12 Iν1i (θ1 ) + (1 − ρ21 ) Iν2i (θ2 ), n2

(9.46a) (9.46b)

The Radiative Transfer Equation in Participating Media (RTE) Chapter | 9 297

where, from equation (2.96),

ρ12 = ρ21

⎧     ⎪ ⎪ 1 n1 cos θ2 − n2 cos θ1 2 n1 cos θ1 − n2 cos θ2 2 ⎪ ⎪ + , ⎪ ⎨ 2 n cos θ + n cos θ n1 cos1 +n2 cos θ2 1 2 2 1 =⎪ ⎪ ⎪ ⎪ ⎪ ⎩1,

θ1 < θc ,

(9.47)

θ1 ≥ θc .

The intensity entering the optically less dense Medium 2 from the interface, Iν2 (θ2 ), will have a transmitted contribution from Medium 1 for all values of θ2 (but coming from within a cone with opening angle θc ). Intensity entering Medium 1, Iν1 (θ1 ), on the other hand, will have a transmitted component from Medium 2 only if θ1 < θc .

9.7 RTE for a Medium with Graded Refractive Index In recent decades, due to emerging applications in optical and optoelectronic devices, there has been a growing interest in studying radiation transport in media with a graded refractive index, henceforth referred to as graded media. In this section, we provide a brief outline of the modifications needed to the RTE to address radiative transfer in such media. For additional details, the reader is referred to the text by Pomraning [4] and journal articles on this topic [5–8]. An energy balance across an interface between two different materials of different refractive indices was formulated in equation (9.41), finally resulted in a relationship between the intensities on the two sides of the interface, as given by equation (9.44). The theory of electromagnetic wave propagation stipulates that reflection at an interface can only occur if there is a discontinuity or step in the refractive index, as evidenced in equations (2.89) through (2.94). While the refractive index in a graded medium changes from point to point, it is still continuous. Therefore, at an interface between two control volumes in a graded medium, ρ12 = 0, and equation (9.44) reduces to Iν1 (θ1 ) Iν2 (θ2 ) = , (9.48) n21 n22 where the difference between n1 and n2 may be thought of as infinitesimally small. In other words, the quantity I/n2 is conserved in the absence of emission, absorption, or scattering. Consequently, the RTE in a non-graded medium [such as equation (9.21)] may be extended to a graded medium by simply replacing I by I/n2 to write [7,8]            Ibη σsη Iη Iη Iη (ˆs i ) d Iη = sˆ · ∇ 2 = κη 2 − βη 2 + Φη (ˆs i , sˆ ) dΩ i . (9.49) ds n2 4π 4π n2 n n n Noting that n2 is not a function of direction, equation (9.49) may be rearranged to      Iη σsη Iη 2 d 2 Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i . n = n sˆ · ∇ 2 = κη Ibη − βη Iη + ds n2 4π 4π n

(9.50)

Likewise, if I is replaced by I/n2 in the boundary condition given by equation (9.36), it is straightforward to see that the boundary condition remains unchanged. A rigorous derivation of the RTE for a graded medium and polarized radiation (so-called vector RTE) has been presented by Zhao et al. [8]. One of the telltale signs of radiation transport through a graded medium is that a ray propagating through it continuously changes direction, i.e., it propagates along curvilinear paths rather than straight paths. This is explained by Snell’s law, n sin θ = const, and differentiating it to yield sin θ dn + n cos θ dθ = 0. Upon rearrangement, we get dθ = −(tan θ/n) dn. This means that a small change in n will cause a small change in θ, resulting in bending of the rays in a graded medium. The bending of rays often has serious implications in practical applications. For example, laser beams used in optical diagnostics may go out of alignment because of change of the refractive index of the medium due to local heating or cooling. In nature, the mirage effect witnessed in deserts is also caused by spatial variations in the refractive index. Solutions to the RTE for a graded one-dimensional plane-parallel medium are presented in Chapter 13, at which point, mathematical formulations governing the ray trajectory and additional physical concepts pertaining to radiation transport in graded media will be discussed.

298 Radiative Heat Transfer

9.8 Radiation Energy Density A volume element inside an enclosure is irradiated from all directions and, at any instant in time t, contains a certain amount of radiative energy in the form of photons. Consider, for example, an element dV = dA ds irradiated perpendicularly to dA with intensity Iη (ˆs) as shown in Fig. 9.3. Therefore, per unit time radiative energy in the amount of Iη (ˆs) dΩ dA enters dV. From the development in Chapter 1, equation (1.51), we see that this energy remains inside dV for a duration of dt = ds/c, before exiting at the other side. Thus, due to irradiation from a single direction, the volume contains the amount of radiative energy Iη (ˆs) dΩ dA ds/c = Iη (ˆs) dΩ dV/c at any instant in time. Adding the contributions from all possible directions, we find the total radiative energy stored within dV is uη dV, where uη is the spectral radiation energy density uη ≡



1 c

Iη (ˆs) dΩ.

(9.51)

4π

Integration over the spectrum gives the total radiation energy density,  u= 0

∞

1 uη dη = c

 4π



∞

0

1 Iη (ˆs) dη dΩ = c

 I(ˆs) dΩ.

(9.52)

4π

Although the radiation energy density is a very basic quantity akin to internal energy for energy stored within matter, it is not widely used by heat transfer engineers. Instead, it is common practice to employ the incident radiation Gη , which is related to the energy density through  Gη ≡

Iη (ˆs) dΩ = cuη ;

G = cu.

(9.53)

4π

9.9 Radiative Heat Flux The spectral radiative heat flux onto a surface element has been expressed in terms of incident and outgoing intensity in equation (1.41) as  (9.54) qη · nˆ = Iη nˆ · sˆ dΩ. 4π

This relationship also holds, of course, for a hypothetical (i.e., totally transmissive) surface element placed arbitrarily inside an enclosure. Removing the surface normal from equation (1.41), we obtain the definition for the spectral, radiative heat flux vector inside a participating medium. To obtain the total radiative heat flux, equation (9.54) needs to be integrated over the spectrum, and  q= 0

∞



∞

qη dη =

 Iη (ˆs) sˆ dΩ dη.

0

(9.55)

4π

Depending on the coordinate system used, or the surface being described, the radiative heat flux vector may be separated into its coordinate components, for example, qx , q y , and qz (for a Cartesian coordinate system), or into components normal and tangential to a surface, and so on. Example 9.3. Evaluate the total heat loss from an isothermal spherical medium bounded by vacuum, assuming that κη = const (i.e., does not vary with location, temperature, or wavenumber). Solution Here we are dealing with a spherical coordinate system, and we are interested in the radial component of the radiative heat flux (the other two being equal to zero by symmetry). We saw in Example 9.1 that the intensity emanating from the sphere is   Iη (τR , θ) = Ibη 1 − e−2τR cos θ ,

0≤θ≤

π , 2

The Radiative Transfer Equation in Participating Media (RTE) Chapter | 9 299

FIGURE 9.10 Control volume for derivation of divergence of radiative heat flux.

where θ is measured from the surface normal pointing away from the sphere (Fig. 9.5). Since the sphere is bounded by vacuum, there is no incoming radiation and Iη (τR , θ) = 0,

π ≤ θ ≤ π. 2

Therefore, from equation (9.55), 









π/2   Iη (τR , θ) cos θ sin θ dθ dψ dη = 2π Ibη 1 − e−2τR cos θ cos θ sin θ dθ dη 0 0 0 0 0 0 0 / /   1  1  −2τR 2 4 = πIb 1 − 2 1 − (1 + 2τR ) e = n σT 1 − 2 1 − (1 + 2τR ) e−2τR , 2τR 2τR

q(τR ) =

∞

2π

π

∞

where n is the refractive index of the medium (usually n ≈ 1 for gases, but n > 1 for semitransparent liquids and solids). As discussed in the previous example, if τR → ∞ the heat flux approaches the same value as the one from a black surface.

If the sphere in the last example is optically thin τR 1 (i.e., the medium emits radiative energy, but does not absorb any of the emitted energy), then the total heat loss (total emission) from the sphere is Q = 4πR2 q = 4πR2 × 43 τR n2 σT 4 = 4κn2 σT 4 V.

(9.56)

This result may be generalized to govern emission from any isothermal volume V without self-absorption, or Qemission = 4κn2 σT 4 V.

(9.57)

9.10 Divergence of the Radiative Heat Flux While the heat transfer engineer is interested in the radiative heat flux, this interest usually holds true only for fluxes at physical boundaries. Inside the medium, on the other hand, we need to know how much net radiative energy is deposited into (or withdrawn from) each volume element. Thus, making a radiative energy balance on an infinitesimal volume dV = dx dy dz as shown in Fig. 9.10, we have ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ ⎜⎜radiative energy⎟⎟ ⎜⎜rad. energy generated⎟⎟ ⎜⎜rad. energy destroyed⎟⎟ ⎜⎜ flux in at x − flux out at x + dx ⎟⎟ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜⎜ ⎜⎜ stored in dV ⎟⎟⎟ − ⎜⎜⎜ (emitted) by dV ⎟⎟⎟ + ⎜⎜⎜ (absorbed) by dV ⎟⎟⎟ = ⎜⎜⎜+ flux in at y − flux out at y + dy⎟⎟⎟ . ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ per unit time per unit time per unit time + flux in at z − flux out at z + dz The right-hand side may be written in mathematical form as

300 Radiative Heat Transfer

q(x) dy dz − q(x + dx) dy dz + q(y) dx dz − q(y + dy) dx dz + q(z) dx dy − q(z + dz) dx dy

⎫ ⎪   ⎪ ⎪ ⎪ ∂q ∂q ∂q ⎬ =− + + dx dy dz = −∇ · q dV. ⎪ ⎪ ⎪ ∂x ∂y ∂z ⎪ ⎭

Thus, within the overall energy equation, it is the divergence of the radiative heat flux that is of interest inside the participating medium.5 We can derive this energy balance also directly from the radiative transfer equation [for example, equation (9.21), if we limit ourselves to quasi-steady, equilibrium problems],  dIη σsη = sˆ · ∇Iη = κη Ibη − βη Iη (ˆs) + Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i , (9.58) ds 4π 4π which is a radiation balance for an infinitesimal pencil of rays. Thus, in order to get a volume balance, we integrate this equation over all solid angles, or      σsη sˆ · ∇Iη dΩ = κη Ibη dΩ − βη Iη (ˆs) dΩ + Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i dΩ, (9.59) 4π 4π 4π 4π 4π 4π and



 Iη sˆ dΩ = 4πκη Ibη −

∇· 4π

βη Iη (ˆs) dΩ + 4π

σsη 4π





 Φη (ˆs i , sˆ ) dΩ dΩ i .

Iη (ˆs i ) 4π

(9.60)

4π

On the left side of equation (9.60) the integral and the direction vector were taken into the gradient since direction and space coordinates are all independent from one another.6 The expression inside the operator is now, of course, the spectral radiative heat flux. On the right side of equation (9.60) the order of integration has been changed, applying the Ω-integration to the only part depending on it, the scattering phase function Φη . This last integration can be carried out using equation (9.17), leading to   ∇ · qη = 4πκη Ibη − βη Iη (ˆs) dΩ + σsη Iη (ˆsi ) dΩ i . (9.61) 4π

4π

Since Ω and Ω i are dummy arguments for integration over all solid angles, the last two terms can be pulled together, using κη = βη − σsη :      Iη dΩ = κη 4πIbη − Gη . ∇ · qη = κη 4πIbη − (9.62) 4π

Equation (9.62) states that physically the net loss of radiative energy from a control volume is equal to emitted energy minus absorbed irradiation. This direction-integrated form of the radiative transfer equation no longer contains the scattering coefficient. This fact is not surprising since scattering only redirects the stream of photons; it does not affect the energy content of any given unit volume. Equation (9.62) is a spectral relationship, i.e., it gives the heat flux per unit wavenumber at a certain spectral position. If the divergence of the total heat flux is desired, the integration over the spectrum is carried out to give   ∞  ∞    ∞   ∇·q=∇· qη dη = κη 4πIbη − Iη dΩ dη = κη 4πIbη − Gη dη. (9.63) 0

0

4π

0

Equation (9.63) is a statement of the conservation of radiative energy. For the special case of a gray medium (κη = κ = constant) this may be simplified to      4 ∇ · q = κ 4σT − I dΩ = κ 4σT 4 − G . (9.64) 4π

5. For simplicity, this equation was derived for a Cartesian coordinate system but the result holds, of course, for any arbitrary coordinate system. 6. While this statement is always true, care must be taken in non-Cartesian coordinate systems: Although the direction vector is independent from space coordinates, the three components may be tied to locally defined unit vectors. For example, in a cylindrical coordinate system the direction vector is usually defined in terms of êr and êθ , which vary with r and θ.

The Radiative Transfer Equation in Participating Media (RTE) Chapter | 9 301

FIGURE 9.11 Divergence of radiative heat flux at center of isothermal sphere as function of optical thickness.

Example 9.4. Calculate the divergence of the total radiative heat flux at the center and at the surface of the gray, isothermal spherical medium in the previous example. Solution We already know the intensity at the surface of the sphere and, therefore,  π/2   1 − e−2τR cos θ sin θ dθ sin θIη dθ = 2πIbη 0 0 ⎛  ⎞ −2τR cos θ π/2 ⎟  πIbη  ⎜⎜ e ⎟   ⎟⎟⎠ = = 2πIbη ⎜⎜⎝ 1 − 2τR − 1 + e−2τR , 2τR 0 τR 

Gη (τR ) = 2π

π

and, after integration over all wavenumbers, ∇ · q(τR ) = κ (4πIb − G) =

 σT 4  2τR + 1 − e−2τR . R

(9.65)

At the center of the sphere the intensity is easily evaluated as Iη (0) = Ibη (1 − e−τR ) , and Gη (0) = 4πIbη (1 − e−τR ) , so that ∇ · q(0) = 4κσT 4 e−τR .

(9.66)

The right-hand sides of equations (9.65) and (9.66) are radiative heat losses per unit time and volume, which must be made up for by a volumetric heat source if the sphere is to stay isothermal.

Equation (9.66) also demonstrates important typical behavior for ∇·q (= emission – absorption) at a point deep inside a (near-)isothermal medium, as shown in Fig. 9.11 for a nondimensional divergence R ∇·q(0)/σT 4 : If τR 1 (“optically thin” medium) ∇·q increases linearly with κ (emission only). As optical thickness increases absorption becomes also important; and for τR  1 (“optically thick”) all locally emitted radiation is absorbed again in the immediate vicinity (known as self-absorption), and ∇·q → 0. A maximum ∇·q is always reached at intermediate optical thickness (τR = 1 in the present example).

302 Radiative Heat Transfer

FIGURE 9.12 Enclosure for the derivation of the integral form of the radiative transfer equation.

9.11 Integral Formulation of the Radiative Transfer Equation In order to obtain incident radiation, radiative heat flux, or its divergence, it is sometimes desirable to use an integral formulation of the radiative transfer equation. We start with the formal solution, equation (9.28), but rewritten in terms of the vectors shown in Fig. 9.12, ⎤  s ⎤ ⎡ s ⎡ s ⎥ ⎥ ⎢⎢  ⎢⎢   ⎥   ⎥ ⎥ ⎢ ⎢ Iη (r, sˆ ) = Iwη (rw , sˆ ) exp ⎢⎣− βη ds ⎥⎦ + Sη (r , sˆ ) exp ⎢⎣− βη ds ⎥⎥⎦ βη ds , 0

0

(9.67)

0

where s = |r − rw | and the direction of integration has been switched to go along s (from point r toward the wall). From the definition of the incident radiation, equation (9.32), we have  Gη (r) = 4π

⎤ ⎡ s  ⎥⎥ ⎢⎢   Iwη (rw , sˆ ) exp ⎢⎢⎣− βη ds ⎥⎥⎦ dΩ +

with, from equation (1.28),



4π

0

⎧ ⎪ dA ⎪ ⎪ ⎪ , ⎪ ⎪ 2 ⎪ ⎪ ⎨ |r − r | dΩ = ⎪ ⎪ ⎪ ⎪ ⎪ nˆ · sˆ dAw ⎪ ⎪ , ⎪ ⎩ |r − rw |2

0

s

⎤ ⎡ s ⎥⎥ ⎢⎢   Sη (r , sˆ ) exp ⎢⎢⎣− βη ds ⎥⎥⎦ βη ds dΩ, 

(9.68)

0

inside volume, (9.69) at the wall,

where dA is an infinitesimal area perpendicular to the integration path (and ds ), such that dV = ds dA is an infinitesimal volume. Therefore, equation (9.68) may be rewritten as ⎤ ⎤ ⎡ s ⎡ s  ⎥⎥ nˆ · sˆ dAw ⎥⎥ βη dV ⎢⎢  ⎢⎢     Gη (r) = Iwη (rw , sˆ ) exp ⎢⎢⎣− βη ds ⎥⎥⎦ + Sη (r , sˆ ) exp ⎢⎢⎣− βη ds ⎥⎥⎦ , 2 |r − rw | |r − r |2 Aw V 0 0 

(9.70)

with the local unit direction vector found from sˆ =

r − r . |r − r |

(9.71)

The radiative flux (and any higher moment) can be determined similarly, after first multiplying equation (9.67) by sˆ , as ⎤ ⎤ ⎡ s ⎡ s  ⎥ (nˆ · sˆ )ˆs dAw ⎥ βη sˆ dV ⎢⎢  ⎢⎢   ⎥   ⎥ ⎥ ⎢ ⎢ qη (r) = Iwη (rw , sˆ ) exp ⎢⎣− βη ds ⎥⎦ + Sη (r , sˆ ) exp ⎢⎣− βη ds ⎥⎥⎦ . 2 |r − rw | |r − r |2 Aw V 0 0 

(9.72)

The Radiative Transfer Equation in Participating Media (RTE) Chapter | 9 303

For a nonscattering medium Sη = Ibη , and equation (9.70) is the explicit solution for incident radiation Gη , provided the temperature field is known, and if the walls are black. For isotropic scattering the source function depends only on Ibη (or temperature) and incident radiation. For such a case (and if the walls are black) equation (9.70) is a single, independent integral equation for the incident radiation; once Gη has been determined qη is found from equation (9.72). For reflecting walls and anisotropic scattering, equations (9.70) and (9.72) (and, perhaps, higher-order moments) must be solved simultaneously. Also, for a nonparticipating medium (βη = 0) with diffusely reflecting surfaces (Iw = J/π), equation (5.25) is readily recovered from equation (9.72); this is left as an exercise (Problem 9.15). Example 9.5. Repeat Example 9.3 using the integral formulation of the RTE. Solution In this simple problem with a cold, black (i.e., nonreflecting) wall with Iwη = 0, and in the absence of scattering with Sη = Ibη = const we can determine qη directly from equation (9.72) as  ˆ · sˆ dV  n qη (R) = −qη (rw ) · nˆ = −Ibη κη e−κη s , (s )2 V where s is the distance between any point inside the medium (at r ) and the chosen point on the wall, r = rw . It is tempting at this point to introduce a spherical coordinate system at the center of the sphere to evaluate the volume integral for qη ; however, this would lead to a very difficult integral. Instead, we introduce a spherical coordinate system at the chosen point at the wall, i.e., rw = 0 (point τs in Fig. 9.5). An arbitrary location inside the sphere can then be specified as ˆ r = −ˆss = s (cos ψ sin θ î + sin ψ sin θ ˆj + cos θ k), where kˆ = nˆ is pointing toward the center of the sphere and î and ˆj are arbitrary (as long as they form a right-handed coordinate system). Then, with a maximum value for smax = 2R cos θ, as given in Example 9.1,  qη (R) = −Ibη κη  = 2πIbη

2π ψ=0

π/2 θ=0



π/2

θ=0



2R cos θ

s =0



e−κη s

(− cos θ) sin θ dθ dψ(s )2 ds (s )2

(1 − e−2τR cos θ ) cos θ sin θ dθ,

exactly as in Example 9.3.

9.12 Overall Energy Conservation Thermal radiation is only one mode of transferring heat which, in general, must compete with conductive and convective heat transfer. Therefore, the temperature field must be determined through an energy conservation equation that incorporates all three modes of heat transfer. The radiation intensity, through emission and temperature-dependent properties, depends on the temperature field and, therefore, cannot be decoupled from the overall energy equation. The general form of the energy conservation equation for a moving compressible fluid may be stated as   ∂u Du =ρ + v · ∇u = −∇ · q − p∇ · v + μΦ + Q˙  , (9.73) ρ Dt ∂t where u is internal energy, v is the velocity vector, q is the total heat flux vector, Φ is the dissipation function, and Q˙  is heat generated within the medium (such as energy release due to chemical reactions). For a detailed derivation of equation (9.73), the reader is referred to standard textbooks, such as [9,10]. If the medium is radiatively participating through emission, absorption, and scattering, then the conservation equations for momentum and energy are altered by three effects [11]: 1. The heat flux term in equation (9.73), which without radiation is in most applications due only to molecular diffusion (heat conduction), now has a second component, the radiative heat flux, due to radiative energy interacting with the medium within the control volume. 2. The internal energy now contains a radiative contribution [the radiation energy density uR (as defined in equation (9.52)), due to the first term in equation (9.20) after integration over all directions].

304 Radiative Heat Transfer

3. The radiation pressure tensor, as briefly discussed in Section 1.8, must be added to the traditional fluid dynamics pressure tensor. We have already seen that the second effect is almost always negligible, and the same is true for the augmentation of the pressure tensor. Under these conditions the energy conservation equation can be simplified. If we assume that du = cv dT, and that Fourier’s law for heat conduction holds, q = qC + qR = −k∇T + qR ,

(9.74)

  ∂T DT = ρcv + v · ∇T = ∇ · (k∇T) − p∇ · v + μΦ + Q˙  − ∇ · qR . ρcv Dt ∂t

(9.75)

equation (9.73) becomes

Note that, while the conductive flux depends only on the local temperature gradient, the radiative flux generally depends on the temperature of the entire computational domain. Therefore, qR remains unresolved in the overall energy equation. Equation (9.75) is a partial differential equation for the calculation of the temperature field, which must be solved in conjunction with the RTE, e.g., equation (9.21) to determine the divergence of the radiative flux from (9.62). This coupling makes combined mode heat transfer problems involving radiation very challenging. It is apparent that the negative divergence of the radiative flux acts as, and is called, a radiative heat source7 , Q˙  = −∇ · qR = (absorption – emission)/volume R

(9.76)

The set of the overall energy equation, Equation (9.75), together with the RTE has up to seven dimensions (time, 3 space coordinates, 2 direction coordinates, and spectral variable). Obviously, a complete solution of this equation, even with the recent advent of supercomputers, is a truly formidable task. Example 9.6. State the radiative transfer equation and its boundary conditions for the case of combined steady-state conduction and radiation within a one-dimensional, planar, gray, and nonscattering medium, bounded by isothermal black walls. Solution Since the problem is steady state and there is no movement in the medium, the left side of equation (9.75) vanishes, and only the first (conduction) and last (radiation) terms on the right side remain. For a one-dimensional planar medium this reduces to8   d dT − qR = 0, (9.77) k dz dz and the divergence of radiative heat flux is related to temperature and incident radiation through equation (9.62), dqR = κ(4σT 4 − G), dz where the spectral integration for the gray medium has been carried out by simply dropping the subscript η. Finally, the incident radiation is found from direction-integrating equation (9.29) (not a trivial task). The necessary boundary conditions are T = Ti , i = 1, 2 at the two walls (for conduction) and I(0, sˆ ) = σTi4 /π (for radiation) needed in equation (9.29). Solution of this seemingly simple problem is by no means trivial, and can only be achieved through relatively involved numerical analysis.

Radiative Equilibrium Much attention in the following chapters will be given to the situation in which radiation is the dominant mode of heat transfer, meaning that when conduction and convection are negligible. This situation is referred to as radiative equilibrium, meaning that thermodynamic equilibrium within the medium is achieved by virtue of 7. This should not be confused with the “source function” defined in equation (9.25). 8. While in the science of conduction the variable x is usually employed for one-dimensional planar problems, for thermal radiation problems the variable z is more convenient. The reason for this is that, by convention, the polar angle for the direction vector is measured from the z-axis.

The Radiative Transfer Equation in Participating Media (RTE) Chapter | 9 305

thermal radiation alone. As is commonly done in the discussion of “pure” conduction or convection, we allow volumetric heat sources throughout the medium. Thus, we may write ρcv

∂T + ∇ · qR = Q˙  , ∂t

(9.78)

which is identical in form to the basic transient heat conduction equation (before substitution of Fourier’s law). In the vast majority of cases radiative transfer occurs so fast that radiative equilibrium is achieved before a noticeable change in temperature occurs [i.e., when the unsteady term in equation (9.20) can be dropped]. Then the statement of radiative equilibrium reduces to its steady-state form ∇ · qR = Q˙  .

(9.79)

Radiative equilibrium is often a good assumption in applications with extremely high temperatures, such as plasmas, nuclear explosions, and such. The inclusion of a volumetric heat source allows the treatment of conduction and convection “through the back door:” A guess is made for the temperature field and the nonradiation terms in equation (9.75) are calculated to give Q˙  for the radiation calculations. This process is then repeated until a convergence criterion is met.

9.13 Solution Methods for the Radiative Transfer Equation Exact analytical solutions to the radiative transfer equation [equation (9.21)] are exceedingly difficult, and explicit solutions are impossible for all but the very simplest situations. Therefore, research on radiative heat transfer in participating media has generally proceeded in two directions: (i) exact (analytical and numerical) solutions of highly idealized situations, and (ii) approximate solution methods for more involved scenarios. Phenomena that make a radiative heat transfer problem difficult may be placed into four different categories: Geometry: The problem may be one-dimensional, two-dimensional, or three-dimensional. Most attempts to find exact analytical solutions to date have dealt with one-dimensional geometries, and the vast majority of these have dealt with the simplest case of a one-dimensional plane-parallel slab. Temperature Field: The least difficult situation arises if the temperature profile within the medium is known, making equation (9.21) a relatively “simple” integral equation. Consequently, the most basic case of an isothermal medium has been studied extensively. Alternatively, if radiative equilibrium prevails, the temperature field is unknown but uncoupled from conduction and convection, and must be found from directional and spectral integration of the radiative transfer equation. In the most complicated scenario, radiative heat transfer is combined with conduction and/or convection, resulting in a highly nonlinear integro-differential equation. Scattering: The solution to a radiation problem is greatly simplified if the medium does not scatter. In that case the radiative transfer equation reduces to a simple first-order differential equation if the temperature field is known, and a relatively simpler integral equation if radiative equilibrium prevails. If scattering must be considered, isotropic scattering is often assumed. Relatively few investigations have dealt with the case of anisotropic scattering, and most of those are limited to the case of linear-anisotropic scattering (see Section 11.9). Properties: Although most participating media display strong nongray character, as discussed in the following three chapters, the majority of investigations to date have centered on the study of gray media. In addition, while radiative properties also generally depend strongly on temperature, concentration, etc., most calculations were limited to situations with constant properties. Only over the past 30 years have nongray solutions gradually become more common, in particular for combustion product gases. Most “exact” solutions are limited to gray media with constant properties in one-dimensional, mainly planeparallel geometries. The media are isothermal or at radiative equilibrium, and if they scatter, the scattering is usually isotropic. Since the usefulness of such one-dimensional solutions in heat transfer applications is limited, they are only briefly discussed in Chapter 13. Several chapters are devoted to the various approximate methods that have been devised for the solution of the radiative transfer equation. Still, these seven chapters by no means cover all the different methods that have been and still are used by investigators in the field. A number of approximate methods for one-dimensional

306 Radiative Heat Transfer

problems are discussed in Chapter 14. The optically thin and diffusion (or optically thick) approximations have historically been developed for a one-dimensional plane-parallel medium, but can readily be applied to more complicated geometries. Similarly, the Schuster–Schwarzschild or two-flux approximation [12,13] is a forerunner to the multidimensional discrete ordinates method. In this method the intensity is assumed to be constant over discrete parts of the total solid angle of 4π. Several other flux methods exist, but they are usually tailored toward special geometries, and cannot easily be applied to other scenarios, for example, the six-flux methods of Chu and Churchill [14] and Shih and coworkers [15,16]. Another early one-dimensional model was the moment method or Eddington approximation [17]. In this model the directional dependence is expressed by a truncated series representation (rather than discretized). In general geometries this expansion is usually achieved through the use of spherical harmonics, leading to the spherical harmonics method. Several variations to the moment method that are tailored toward specific geometries have been proposed [18,19], but these are of limited general utility. Finally, the exponential kernel approximation, already discussed in Chapter 5 for surface radiation problems, may be used as a tool for many one-dimensional problems. However, its extension to multidimensional geometries is problematic. A survey of the literature over the past 50 years demonstrates that some solution methods have been used frequently, while others that appeared promising at one time are no longer employed on a regular basis. Apparently, some methods have been found to be more readily adapted to more difficult situations than others (such as multidimensionality, variable properties, anisotropic scattering, and/or nongray effects). The majority of radiative heat transfer analyses today appear to use one of four methods: (i) the spherical harmonics method or a variation of it, (ii) the discrete ordinates method or its more modern form, the finite angle method (commonly known as finite volume method), and other derivatives, (iii) the zonal method, and (iv) the Monte Carlo method. The first two of these have already been discussed briefly above with the one-dimensional approximations. The zonal method was developed by Hottel [20] in his pioneering work on furnace heat transfer. Unlike the spherical harmonics and discrete ordinates methods, the zonal method approximates spatial, rather than directional, behavior by breaking up an enclosure into finite, isothermal subvolumes. On the other hand, the Monte Carlo method [21] is a statistical method, in which the history of bundles of photons is traced as they travel through the enclosure. While the statistical nature of the Monte Carlo method makes it difficult to match it with other calculations, it is the only method that can satisfactorily deal with effects of irregular radiative properties (nonideal directional and/or nongray behavior). Because of their importance, an entire chapter is devoted to each of these four solution methods. Several other methods that can be found in the literature are not covered in this book (except for brief descriptions in appropriate places). For example, the discrete transfer method, proposed by Shah [22] and Lockwood and Shah [23], combines features of the discrete ordinates, zonal, and Monte Carlo methods. Another hybrid proposed by Edwards [24] combines elements of the Monte Carlo and zonal methods. Over the years it has also become clear that one method may do well for one type of problem, while another excels for different conditions. For example, diffusion and low-order spherical harmonics methods shine in optically thick media, in which the discrete ordinates methods perform poorly, while the opposite is true in optically thin conditions. This naturally leads to hybrid methods, employing separate RTE solvers for separate subdomains (spatially or spectrally). Research in the development of such hybrid methods is still in its infancy [25–27], but is expected to become an important topic during the coming years. It is important to recognize that wavenumber (or wavelength) is, in addition to space and time, an independent variable in the RTE. Since wavenumber is independent of space and time and does not appear in operator form, it may be treated as a parameter rather than a variable. However, it is generally not possible to a priori integrate the RTE over the spectrum—the RTE must be solved first (for many spectral values) before integration over the spectrum is carried out, either directly or with a simplifying spectral model. Accordingly, nongray RTE solutions are discussed in a separate chapter after all RTE solution methods have been presented.

Problems 9.1 A semi-infinite medium 0 ≤ z < ∞ consists of a gray, absorbing–emitting gas that does not scatter, bounded by vacuum at the interface z = 0. The gas is isothermal at 1000 K, and the absorption coefficient is κ = 1 m−1 . The interface is nonreflecting; conduction and convection may be neglected.

The Radiative Transfer Equation in Participating Media (RTE) Chapter | 9 307

(a) What is the local heat generation that is necessary to keep the gas at 1000 K? (b) What is the intensity distribution at the interface, that is, I(z = 0, θ, ψ), for all θ and ψ? (c) What is the total heat flux leaving the semi-infinite medium? 9.2 Reconsider the semi-infinite medium of Problem 9.1 for a temperature distribution of T = T0 e−z/L , T0 = 1000 K, L = 1 m. What are the exiting intensity and heat flux for this case? Discuss how the answer would change if κ varied between 0 and ∞. 9.3 Repeat Problem 9.1 for a medium of thickness L = 1 m. Discuss how the answer would change if κ varied between 0 and ∞. 9.4 A semi-infinite, gray, nonscattering medium (n = 2, κ = 1 m−1 ) is irradiated by the sun normal to its surface at a rate of qsun = 1000 W/m2 . Neglecting emission from the relatively cold medium, determine the local heat generation rate due to absorption of solar energy. Hint: The solar radiation may be thought of as being due to a radiative intensity which has a large value Io over a very small cone of solid angles δΩ, and is zero elsewhere, i.e., ⎧ ⎪ ⎪ ⎨Io I(ˆs) = ⎪ ⎪ ⎩0 and

ˆ over δΩ along n, elsewhere,

 qsun =

I(ˆs)nˆ · sˆ dΩ = Io δΩ. 4π

9.5 A 1 m thick slab of an absorbing–emitting gas has an approximately linear temperature distribution as shown in the sketch. On both sides the medium is bounded by vacuum with nonreflecting boundaries. (a) If the medium has a constant and gray absorption coefficient of κ = 1 m−1 , what is the intensity (as a function of direction) leaving the hot side of the slab? (b) Give an expression for the radiative heat flux leaving the hot side.

9.6 A semitransparent sphere of radius R = 10 cm has a parabolic temperature profile T = Tc (1 − r2 /R2 ), Tc = 2000 K. The sphere is gray with κ = 0.1 cm−1 , n = 1.0, does not scatter, and has nonreflective boundaries. Outline how to calculate the total heat loss from the sphere (i.e., there is no need actually to carry out cumbersome integrations). 9.7 Repeat Problem 9.6, but assume that the temperature is uniform at 2000 K. What must the local production of heat be if the sphere is to remain at 2000 K everywhere? Note: The answer may be left in integral form (which must be solved numerically). Carry out the integration for r = 0 and r = R. 9.8 Repeat Problem 9.6, but assume that the temperature is uniform at 2000 K. Also, there is no heat production, meaning that the sphere cools down. How long will it take for the sphere to cool down to 500 K (the heat capacity of the medium is ρc = 1000 kJ/m3 K and the conductivity is very large, i.e., the sphere is isothermal at all times)? 9.9 A relatively cold sphere with a radius of Ro = 1 m consists of a nonscattering gray medium that absorbs with an absorption coefficient of κ = 0.1 cm−1 and has a refractive index n = 2. At the center of the sphere is a small black sphere with radius Ri = 1 cm at a temperature of 1000 K. On the outside, the sphere is bounded by vacuum. What is the total heat flux leaving the sphere? Explain what happens as κ is increased from zero to a large value. 9.10 A laser beam is directed onto the atmosphere of a (hypothetical) planet. The planet’s atmosphere contains 0.01% by volume of an absorbing gas. The absorbing gas has a molecular weight of 20 and, at the laser wavelength, an absorption coefficient κη = 10−4 cm−1 /(g/m3 ). It is known that the pressure and temperature distributions of the atmosphere can be approximated by p = p0 e−2z/L and T = T0 e−z/L , where p0 = 0.75 atm, T0 = 400 K are values at the planet surface z = 0, and L = 2 km is a characteristic length. What fraction of the laser energy arrives at the planet’s surface?

308 Radiative Heat Transfer

9.11 A CO2 laser with a total power output of Q = 10 W is directed (at right angle) onto a 10 cm thick, isothermal, absorbing/emitting (but not scattering) medium at 1000 K. It is known that the laser beam is essentially monochromatic at a wavelength of 10.6 μm with a Gaussian power distribution. Thus, the intensity falling onto the medium is 2

I(0) ∝  e−(r/R) /(δΩ δη), Q=

0 ≤ r ≤ ∞;

I(0) dA δΩ δη, A

where r is distance from beam center, R = 100 μm is the “effective radius” of the laser beam, δΩ = 5 × 10−3 sr is the range of solid angles over which the laser beam outputs intensity (assumed uniform over δΩ), and δη is the range of wavenumbers over which the intensity is distributed (also assumed uniform). At 10.6 μm the medium is known to have an absorption coefficient κη = 0.15 cm−1 . Assuming that the medium has nonreflecting boundaries, determine the exiting total intensity in the normal direction (transmitted laser radiation plus emission, assuming the medium to be gray). Is the emission contribution important? How thick would the medium have to be to make transmission and emission equally important? 9.12 Repeat Problem 9.11 for a medium with refractive index n = 2, bounded by vacuum (i.e., a slab with reflecting surfaces). Hint: (1) Part of the laser beam will be reflected when first hitting the slab, part will penetrate into the slab. Part of this energy will be absorbed by the layer, part will hit the rear face, where a fraction will be reflected back into the slab, and the rest will emerge from the slab, etc. Similar multiple internal reflections will take place with the emitted energy before emerging from the slab. (2) To calculate the slab–surroundings reflectance, show that the value of the absorptive index is negligible. 9.13 A thin column of gas of cross-section δA and length L contains a uniform suspension of small particles that absorb and scatter radiation. The scattering is according to the phase function (a) Φ = 1 (isotropic scattering), (b) Φ = 1 + A1 cos Θ (linear anisotropic scattering, A1 is a constant), and (c) Φ = 34 (1 + cos2 Θ) (Rayleigh scattering), where Θ is the angle between incoming and scattered directions. A laser beam hits the column normal to δA. What is the transmitted fraction of the laser power? What fraction of the laser flux goes through an infinite plane at L normal to the gas column? What fraction goes back through a plane at 0? What happens to the rest? 9.14 Repeat Example 9.2 for (a) Φ = 1 + A1 cos Θ (linear anisotropic scattering, A1 = const), and (b) Φ = (Rayleigh scattering), and Θ is the angle between incoming and scattered directions.

3 (1 4

+ cos2 Θ)

9.15 Show that, by setting βη = 0 and Iw = J/π, the radiosity integral equation (5.25) can be recovered from equation (9.72) for a nonparticipating medium surrounded by diffusely reflecting walls. Hint: Break up the heat flux in equation (9.72) into two parts, incoming radiation H and exiting radiation J. For the latter assume r to be an infinitesimal distance above the surface and evaluate the integral in equation (9.72).

References [1] R. Viskanta, M.P. Mengüç, Radiation heat transfer in combustion systems, Progress in Energy and Combustion Science 13 (1987) 97–160. [2] S. Kumar, K. Mitra, Microscale aspects of thermal radiation transport and laser applications, in: Advances in Heat Transfer, vol. 33, Academic Press, New York, 1999, pp. 187–294. [3] L. Hartung, R. Mitcheltree, P. Gnoffo, Stagnation point nonequilibrium radiative heating and influence of energy exchange models, Journal of Thermophysics and Heat Transfer 6 (3) (1992) 412–418. [4] G.C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press, New York, 1973. [5] P. Ben-Abdallah, V. Le Dez, Thermal emission of a semi-transparent slab with variable spatial refractive index, Journal of Quantitative Spectroscopy and Radiative Transfer 67 (2000) 185–198. [6] P. Ben-Abdallah, V. Le Dez, Temperature field inside an absorbing-emitting semi-transparent slab at radiative equilibrium with variable spatial refractive index, Journal of Quantitative Spectroscopy and Radiative Transfer 65 (2000) 595–608. [7] C.-Y. Wu, M.-F. Hou, Integral equation solutions based on exact ray paths for radiative transfer in a participating medium with formulated refractive index, International Journal of Heat and Mass Transfer 55 (2012) 6600–6608. [8] J.M. Zhao, J.Y. Tan, L.H. Liu, On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media, Journal of Quantitative Spectroscopy and Radiative Transfer 113 (2012) 239–250. [9] W.M. Rohsenow, H.Y. Choi, Heat, Mass and Momentum Transfer, Prentice Hall, Englewood Cliffs, NJ, 1961. [10] W.M. Kays, M.E. Crawford, Convective Heat and Mass Transfer, 3rd ed., McGraw-Hill, 1993. [11] E.M. Sparrow, R.D. Cess, Radiation Heat Transfer, Hemisphere, New York, 1978. [12] A. Schuster, Radiation through a foggy atmosphere, The Astrophysical Journal 21 (1905) 1–22. [13] K. Schwarzschild, Über das Gleichgewicht der Sonnenatmosphären (Equilibrium of the sun’s atmosphere), Nachrichten der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse 195 (1906) 41–53.

The Radiative Transfer Equation in Participating Media (RTE) Chapter | 9 309

[14] C.M. Chu, S.W. Churchill, Numerical solution of problems in multiple scattering of electromagnetic radiation, Journal of Physical Chemistry 59 (1960) 855–863. [15] T.M. Shih, Y.N. Chen, A discretized-intensity method proposed for two-dimensional systems enclosing radiative and conductive media, Numerical Heat Transfer 6 (1983) 117–134. [16] T.M. Shih, A.L. Ren, Combined radiative and convective recirculating flows in enclosures, Numerical Heat Transfer 8 (2) (1985) 149–167. [17] A.S. Eddington, The Internal Constitution of the Stars, Dover Publications, New York, 1959. [18] Y.S. Chou, C.L. Tien, A modified moment method for radiative transfer in non-planar systems, Journal of Quantitative Spectroscopy and Radiative Transfer 8 (1968) 719–733. [19] G.E. Hunt, The transport equation of radiative transfer with axial symmetry, SIAM Journal on Applied Mathematics 16 (1) (1968) 228–237. [20] H.C. Hottel, E.S. Cohen, Radiant heat exchange in a gas-filled enclosure: allowance for nonuniformity of gas temperature, AIChE Journal 4 (1958) 3–14. [21] J.R. Howell, Application of Monte Carlo to heat transfer problems, in: J.P. Hartnett, T.F. Irvine (Eds.), Advances in Heat Transfer, vol. 5, Academic Press, New York, 1968. [22] N.G. Shah, New method of computation of radiation heat transfer in combustion chambers, Ph.D. thesis, Imperial College of Science and Technology, London, England, 1979. [23] F.C. Lockwood, N.G. Shah, A new radiation solution method for incorporation in general combustion prediction procedures, in: Eighteenth Symposium (International) on Combustion, The Combustion Institute, 1981, pp. 1405–1409. [24] D.K. Edwards, Hybrid Monte-Carlo matrix-inversion formulation of radiation heat transfer with volume scattering, in: Heat Transfer in Fire and Combustion Systems, vol. HTD-45, ASME, 1985, pp. 273–278. [25] R. Yadav, A. Kushari, A.K. Verma, V. Eswaran, Weighted sum of gray gas modeling for nongray radiation in combusting environment using the hybrid solution methodology, Numerical Heat Transfer – Part B: Fundamentals 64 (2013) 174–197. [26] P.J. Coelho, N. Crouseilles, P. Pereira, M. Roger, Multi-scale methods for the solution of the radiative transfer equation, Journal of Quantitative Spectroscopy and Radiative Transfer 172 (2016) 36–49. [27] Y. Sun, X. Zhang, Contributions of gray gases in SLW for non-gray radiation heat transfer and corresponding accuracies of FVM and P1 method, International Journal of Heat and Mass Transfer 121 (2018) 819–831.

Chapter 10

Radiative Properties of Molecular Gases 10.1 Fundamental Principles Radiative transfer characteristics of an opaque wall can often be described with good accuracy by the very simple model of gray and diffuse emission, absorption, and reflection. The radiative properties of a molecular gas, on the other hand, vary so strongly and rapidly across the spectrum that the assumption of a “gray” gas is almost never a good one [1]. In the present chapter a short development of the radiative properties of molecular gases is given. Other elaborate discussions can be found, for example, in the book by Goody and Yung [2], in the monograph by Tien [3], and in the somewhat more recent treatise of Taine and Soufiani [4]. Most of the earlier work was not in the area of heat transfer but rather was carried out by astronomers, who had to deal with light absorption within Earth’s atmosphere, and by astrophysicists, who studied the spectra of stars. The study of atmospheric radiation was apparently initiated by Lord Rayleigh [5] and Langley [6] in the late nineteenth century. The radiation spectra of stars started to receive attention in the early twentieth century, for example, by Eddington [7] and Chandrasekhar [8,9]. The earliest measurements of radiation from hot gases were reported by Paschen, a physicist, in 1894 [10], but his work was apparently ignored by heat transfer engineers for many years [11]. The last few decades have seen much progress in the understanding of molecular gas radiation, in particular the radiation from water vapor and carbon dioxide, which is of great importance in the combustion of hydrocarbon fuels, and which also dominates atmospheric radiation with its thermodynamic implications on Earth’s atmosphere. The combination of the two, i.e., the man-made strong increases in the atmosphere’s CO2 content, giving rise to “global warming,” is perhaps the most pressing problem facing mankind today. Much of the pioneering work since the late 1920s was done by Hottel and coworkers [12–19] (measurements and practical calculations) and by Penner [20] and Plass [21,22] (theoretical basis). When a photon (or an electromagnetic wave) interacts with a gas molecule, it may be either absorbed, raising the molecule’s energy level, or scattered, changing the direction of travel of the photon. Conversely, a gas molecule may spontaneously lower its energy level by the emission of an appropriate photon. As will be seen in the next chapter on particle properties (since every molecule is, of course, a very small particle), the scattering of photons by molecules is always negligible for heat transfer applications. There are three different types of radiative transitions that lead to a change of molecular energy level by emission or absorption of a photon: (i) transitions between nondissociated (“bound”) atomic or molecular states, called bound–bound transitions, (ii) transitions from a “bound” state to a “free” (dissociated) one (absorption) or from “free” to “bound” (emission), called bound–free transitions, and (iii) transitions between two different “free” states, called free–free transitions. The internal energy of every atom and molecule depends on a number of factors, primarily on the energies associated with electrons spinning at varying distances around the nucleus, atoms within a molecule spinning around one another, and atoms within a molecule vibrating against each other. Quantum mechanics postulates that the energy levels for atomic or molecular electron orbit as well as the energy levels for molecular rotation and vibration are quantized; i.e., electron orbits and rotational and vibrational frequencies can only change by certain discrete amounts. Since the energy contained in a photon or electromagnetic wave is directly proportional to frequency, quantization means that, in bound–bound transitions, photons must have a certain frequency (or wavelength) in order to be captured or released, resulting in discrete spectral lines for absorption and emission. Since, according to Heisenberg’s uncertainty principle, the energy level of an atom or molecule cannot be fixed precisely, this phenomenon (and, as we shall see, some others as well) results in a slight broadening of these spectral lines. Changing the orbit of an electron requires a relatively large amount of energy, or a high-frequency photon, resulting in absorption–emission lines at short wavelengths between the ultraviolet and the near-infrared (between 10−2 μm and 1.5 μm). Vibrational energy level changes require somewhat less energy, so that their spectral lines Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00018-3 Copyright © 2022 Elsevier Inc. All rights reserved.

311

312 Radiative Heat Transfer

are found in the infrared (between 1.5 μm and 10 μm), while changes in rotational energy levels call for the least amount of energy and, thus, rotational lines are found in the far infrared (beyond 10 μm). Changes in vibrational energy levels may (and often must) be accompanied by rotational transitions, leading to closely spaced groups of spectral lines that, as a result of line broadening, may partly overlap and lead to so-called vibration–rotation bands in the infrared. Similarly, electronic transitions in molecules (as opposed to atoms) are always accompanied by vibrational and rotational energy changes, generally in the ultraviolet to the near-infrared. If the initial energy level of a molecule is very high (e.g., in very high-temperature gases), then the absorption of a photon may cause the breaking-away of an electron or the breakup of the entire molecule because of too strong vibration, i.e., a bound–free transition. The post-absorption energy level of the molecule depends on the kinetic energy of the separated part, which is essentially not quantized. Therefore, bound–free transitions result in a continuous absorption spectrum over all wavelengths or frequencies for which the photon energy exceeds the required ionization or dissociation energy. The same is true for the reverse process, emission of a photon in a free–bound transition (often called radiative combination). In an ionized gas free electrons can interact with the electric field of ions resulting in a free–free transition (also known as Bremsstrahlung, which is German for brake radiation); i.e., the release of a photon lowers the kinetic energy of the electron (decelerates it), or the capture of a photon accelerates it (inverse Bremsstrahlung). Since kinetic energy levels of electrons are essentially not quantized, these photons may have any frequency or wavelength. Bound–free and free–free transitions generally occur at very high temperatures (when dissociation and ionization become substantial). The continuum radiation associated with them is usually found at short wavelengths (ultraviolet to visible). Therefore, these effects are of importance only in extremely high-temperature situations. Most engineering applications occur at moderate temperature levels, with little ionization and dissociation, making bound–bound transitions most important. At combustion temperatures the emissive power has its maximum in the infrared (between 1 μm and 6 μm), giving special importance to vibration–rotation bands. In this book we will focus our discussion on the most important case of bound–bound transitions.

10.2 Emission and Absorption Probabilities There are three different processes leading to the release or capture of a photon, namely, spontaneous emission, induced or stimulated emission (also called negative absorption), and absorption. The absorption and emission coefficients associated with these transitions may, at least theoretically, be calculated from quantum mechanics. Complete descriptions of the microscopic phenomena may be found in books on statistical mechanics [23,24] or spectroscopy [25,26]. An informative (rather than precise) synopsis has been given by Tien [3] that we shall essentially follow here. Let there be nu atoms or molecules (per unit volume) at a nondegenerate higher energy state u and nl at a lower energy state l. “Nondegenerate” means that, if there are several states with identical energies (degeneracy), each state is counted separately. The difference of energy between the two states is hν. The number of transitions from state u to state l by release of a photon with energy hν (spontaneous emission) must be proportional to the number of atoms or molecules at that level. Thus   dnu = −Aul nu , (10.1) dt u→l where the proportionality constant Aul is known as the Einstein coefficient for spontaneous emission. Spontaneous emission is isotropic, meaning that the direction of the emitted photon is random, resulting in equal emission intensity in all directions. Quantum mechanics postulates that, in addition to spontaneous emission, incoming radiative intensity (or photon streams) with the appropriate frequency may induce the molecule to emit photons into the same direction as the incoming intensity (stimulated emission). Therefore, the total number of transitions from state u to state l may be written as      dnu = −nu Aul + Bul Iν dΩ , (10.2) dt u→l 4π where Iν is the incoming intensity, which must be integrated over all directions to account for all possible transitions, and Bul is the Einstein coefficient for stimulated emission. Finally, part of the incoming radiative intensity

Radiative Properties of Molecular Gases Chapter | 10 313

may be absorbed by molecules at energy state l. Obviously, the absorption rate will be proportional to the strength of incoming radiation as well as the number of molecules that are at energy state l, leading to    dnl = nl Blu Iν dΩ, (10.3) dt l→u 4π where Blu is the Einstein coefficient for absorption. The three Einstein coefficients may be related to one another by considering the special case of equilibrium radiation. Equilibrium radiation occurs in an isothermal black enclosure, where the radiative intensity is everywhere equal to the blackbody intensity Ibν and where the average number of molecules at any given energy level is constant at any given time, i.e., the number of transitions from all upper energy levels u to all lower states l is equal to the ones from l to u, or         dnu dnl + gl = −gu nu Aul + Bul Ibν dΩ + gl nl Blu Ibν dΩ = 0, (10.4) gu dt u→l dt l→u 4π 4π where gu and gl are the degeneracies of the upper and lower energy state, respectively, i.e., the number of different arrangements with which a molecule can obtain this energy level. At local thermodynamic equilibrium the number of particles at any energy level is governed by Boltzmann’s distribution law [23], leading to + (10.5) nl /nu = e−El /kT e−Eu /kT = ehν/kT , where Eu and El are the energy levels associated with states u and l, respectively. Thus, the blackbody intensity may be evaluated from equation (10.4) as Ibν =

Aul /Bul 1 .   4π gl Blu /gu Bul ehν/kT − 1

(10.6)

Comparison with Planck’s law, equation (1.9), shows that all three Einstein coefficients are dependent upon another, namely, Aul =

8πhν3 Bul , c20

gu Bul = gl Blu .

(10.7)

The Einstein coefficients are universal functions for a given transition and, therefore, the relationships between them hold also if local thermodynamic equilibrium does not prevail [i.e., the energy level populations do not obey Boltzmann’s distribution, equation (10.5)]. The one remaining independent Einstein coefficient is clearly an indicator of how strongly a gas is able to emit and absorb radiation. This is most easily seen by examining the number of induced transitions (by absorption and emission) in a single direction (or within a thin pencil of rays). If   dn d = (gl nl Blu − gu nu Bul )Iν (10.8) g dΩ dt l↔u is the net number of photons removed from the pencil of rays per unit time and per unit volume, then—since each photon carries the energy hν—the change of radiative energy per unit time, per unit area and distance, and per unit solid angle is   d dn = −(gl nl Blu − gu nu Bul )hνIν . (10.9) − hν g dΩ dt l↔u This relation is equivalent to equation (9.1), except that in reality the spectral line associated with a transition between an upper energy state u and a lower energy state l is “broadened,” i.e., transitions occur across a (very small) range of frequencies, and equation (10.9) captures all of these transitions. Accounting for this slight spread in frequencies (and recalling the definition of intensity, Section 1.6), we have   d Iν dν = −(gl nl Blu − gu nu Bul )hνIν = − (gl nl Blu − gu nu Bul )hνIν dν, (10.10) ds Δν Δν

314 Radiative Heat Transfer

i.e., the Einstein probabilities are not defined for a single transition frequency, but rather are spread over a small but finite frequency range Δν due to broadening, with [27] Aul = Aul φν ,

Bul = Bul φν ,

Blu = Blu φν ,

and φν (ν) is a normalized line shape function (assumed here to be equal for all three probabilities),  φν (ν) dν = 1. Δν

(10.11)

(10.12)

The exact shape of line broadening will be discussed in detail in Section 10.4. Using equation (10.11) we can rewrite equation (10.10) as   d Iν dν = −(gl nl Blu − gu nu Bul ) hνφν Iν dν. (10.13) ds Δν Δν This relation gives the absorption of an entire line, and we define the line strength or line intensity as  Sν = (gl nl Blu − gu nu Bul ) hνφν dν = (gl nl Blu − gu nu Bul )hν. Δν

(10.14)

In the last expression of equation (10.14) the (line-center) frequency has been taken out of the integral, since ν varies very little across a narrow spectral line. By the definition of the absorption coefficient, the line strength is the (linear) absorption coefficient integrated across a line. On a spectral basis across Δν, this becomes  Sν = κν dν, and κν = Sν φν , (10.15) Δν

so that dIν = −κν Iν , ds

(10.16)

which is, of course, identical to equation (9.1). The absorption coefficient as defined here is often termed the effective absorption coefficient since it incorporates stimulated emission (or negative absorption). Sometimes a true absorption coefficient is defined from  κν dν = gl nl Blu hν. (10.17) Δν

Since stimulated emission and absorption always occur together and cannot be separated, it is general practice to incorporate stimulated emission into the absorption coefficient, so that only the effective absorption coefficient needs to be considered.1 Examination of equation (10.14) shows that the absorption coefficient is proportional to molecular number density. Therefore, as mentioned earlier, a number of researchers take the number density out of the definition for κν either in the form of density or pressure, by defining a density-based absorption coefficient or a pressure-based absorption coefficient, respectively, as κρν ≡

κν , ρ

κpν ≡

κν , p

(10.18)

and similarly for Sν . If a mass or pressure absorption coefficient is used, then a ρ or p must, of course, be added to equation (10.16).2 The negative of equation (10.1) gives the rate at which molecules emit photons of strength hν randomly into all directions (into a solid angle of 4π) and per unit volume. Thus, multiplying this equation by −hν and dividing 1. Since it is experimentally impossible to distinguish stimulated emission from absorption, its existence had initially been questioned. Equation (10.6) is generally accepted as proof that stimulated emission does indeed exist: Without it Bul → 0 and the blackbody intensity would be governed by Wien’s distribution, equation (1.19), which is known to be incorrect. 2. Thus, depending on what spectral variable is employed (wavelength λ, wavenumber η, or frequency ν), a spectrally integrated absorption coefficient may appear in nine different variations. Often the only way to determine which definition has been used is to carefully check the units given.

Radiative Properties of Molecular Gases Chapter | 10 315

by 4π gives isotropic energy emitted per unit time, per unit solid angle, per unit area and distance along a pencil of rays or, in short, the change of intensity per unit distance due to spontaneous emission:    d d dn Iν dν = −hν = gu nu Aul hν/4π. (10.19) ds Δν dΩ dt u→l This is the emission of an entire line and, on a spectral basis across Δν this becomes dIν = gu nu Aul hν/4π = jν , ds

(10.20)

and jν is called the emission coefficient, which is related to the absorption coefficient through equations (10.7), (10.14), and (10.15), leading to jν = κ ν

2hν3 nu , c20 nl − nu

(10.21)

At local thermodynamic equilibrium energy levels are populated according to Boltzmann’s distribution, equation (10.5), and the emission coefficient and equation (10.20) reduce to dIν = jν = κν Ibν , ds

(10.22)

which represents the augmentation of directional intensity due to spontaneous emission, as given by equation (9.10).

10.3 Atomic and Molecular Spectra We have already seen that the emission or absorption of a photon goes hand in hand with the change of rotational and/or vibrational energy levels in molecules, or with the change of electron orbits (in atoms and molecules). This change, in turn, causes a change in radiative intensity resulting in spectral lines. In this section we discuss briefly how the position of spectral lines within a vibration–rotation band can be calculated, since it is these bands that are of great importance to the heat transfer engineer. More detailed information as well as discussion of electronic spectra, and bound–free and free–free transitions may be found in more specialized books on quantum mechanics [24,25,28] or spectroscopy [26,29–31], in the book on atmospheric radiation by Goody and Yung [2], or in the monographs on gas radiation properties by Tien [3] and Taine and Soufiani [4]. Since every particle moves in three-dimensional space, it has three degrees of freedom: It can move in the forward–backward, left–right, and/or upward–downward directions. If two or more particles are connected with each other (diatomic and polyatomic molecules), then each of the atoms making up the molecule has three degrees of freedom. However, it is more convenient to say that a molecule consisting of N atoms has three degrees of freedom for translation, and 3N − 3 degrees of freedom for relative motion between atoms. These 3N − 3 degrees of internal freedom may be further separated into rotational and vibrational degrees of freedom. This fact is illustrated in Fig. 10.1 for a diatomic molecule and for linear and nonlinear triatomic molecules. The diatomic molecule has three internal degrees of freedom. Obviously, it can rotate around its center of gravity within the plane of the paper or, similarly, perpendicularly to the paper (with the rotation axis lying in the paper). It could also rotate around its own axis; however, neither one of the atoms would move (except for rotating around itself). Thus, the last degree of freedom must be used for vibrational motion between the two atoms as indicated in the figure. The situation gets rapidly more complicated for molecules with increasing number of atoms. For linear triatomic molecules (e.g., CO2 , N2 O, HCN) there are, again, only two rotational modes. Since there are six internal degrees of freedom, there are four vibrational modes, as indicated in Fig. 10.1. However, two of these vibrational modes are identical, or degenerate (except for taking place in perpendicular planes). In contrast, a nonlinear triatomic molecule has three rotational modes: In this case rotation around the horizontal axis in the plane of the paper is legitimate, so there are only three vibrational degrees of freedom. Depending on the axis of rotation, a polyatomic molecule may have different moments of inertia for each of the three rotational modes. If symmetry is such that all three moments of inertia are the same, the molecule is classified as a spherical top (e.g., CH4 ). It is called a symmetric top, if two are the same (e.g., NH3 , CH3 Cl, C2 H6 , SF6 ), and an asymmetric top, if all three are different (e.g., H2 O, O3 , SO2 , NO2 , H2 S, H2 O2 ).

316 Radiative Heat Transfer

FIGURE 10.1 Rotational and vibrational degrees of freedom for (a) diatomic, (b) linear triatomic, and (c) nonlinear triatomic molecules.

Rotational Transitions To calculate the allowed rotational energy level from quantum mechanics using Schrödinger’s wave equation (see, for example, [23,24]), we generally assume that the molecule consists of point masses connected by rigid massless rods, the so-called rigid rotator model. The solution to this wave equation dictates that possible energy levels for a linear molecule are limited to Ej =

2 j(j + 1) = hc 0 Bj(j + 1), 2I

j = 0, 1, 2, . . . (j integer),

(10.23)

where  = h/2π is the modified Planck’s constant, I is the moment of inertia of the molecule, j is the rotational quantum number, and the abbreviation B has been introduced for later convenience. Allowed transitions are Δj = ±1 and 0 (the latter being of importance for a simultaneous vibrational transition); this expression is known as the selection rule. In the case of the absorption of a photon ( j → j + 1 transition) the wavenumbers of the resulting spectral lines can then be determined3 as η = (E j+1 − E j )/hc 0 = B(j + 1)(j + 2) − Bj(j + 1) = 2B(j + 1),

j = 0, 1, 2, . . . .

(10.24)

The results of this equation produce a number of equidistant spectral lines (in units of wavenumber or frequency), as shown in the sketch of Fig. 10.2. The rigid rotator model turns out to be surprisingly accurate, although for high rotation rates (j  0) a small correction factor due to the centrifugal contribution (stretching of the “rod”) may be considered. Not all linear molecules exhibit rotational lines, since an electric dipole moment is required for a transition to occur. Thus, diatomic molecules such as O2 and N2 never undergo rotational transitions, while symmetric molecules such as CO2 show a rotational spectrum only if accompanied by a vibrational transition [3]. Evaluation of the spectral lines of nonlinear polyatomic molecules is always rather complicated and the reader is referred to specialized treatises such as the one by Herzberg [30]. 3. In our discussion of surface radiative transport we have used wavelength λ as the spectral variable throughout, largely to conform with the majority of other publications. However, for gases frequency ν or wavenumber η are considerably more convenient to use [see, for example, equation (10.24)]. Again, to conform with the majority of the literature, we shall use wavenumber throughout this part.

Radiative Properties of Molecular Gases Chapter | 10 317

FIGURE 10.2 Spectral position and energy levels for a rigid rotator.

Vibrational Transitions The simplest model of a vibrating diatomic molecule assumes two point masses connected by a perfectly elastic massless spring. Such a model leads to a harmonic oscillation and is, therefore, called the harmonic oscillator. For this case the solution to Schrödinger’s wave equation for the determination of possible vibrational energy levels is readily found to be Ev = hνe (v + 12 ),

v = 0, 1, 2, . . . (v integer),

(10.25)

where νe is the equilibrium frequency of harmonic oscillation or eigenfrequency and v is the vibrational quantum number. The selection rule for a harmonic oscillator is Δv = ±1 and, thus, one would expect a single spectral line at the same frequency as the harmonic oscillation, or at a wavenumber η = (Ev+1 − Ev )/hc 0 = (νe /c 0 )(v + 1 − v) = νe /c 0 ,

(10.26)

as indicated in Fig. 10.3. Unfortunately, the assumption of a harmonic oscillator leads to considerably less accurate results than the one of a rigid rotator. This fact is easily appreciated by looking at Fig. 10.4, which depicts the molecular energy level of a diatomic molecule vs. interatomic distance: When atoms move toward each other repulsive forces grow more and more rapidly, while the opposite is true when the atoms move apart. The heavy line in Fig. 10.4 shows the minimum and maximum distances between atoms for any given vibrational

FIGURE 10.3 Spectral position and energy levels for a harmonic oscillator.

318 Radiative Heat Transfer

FIGURE 10.4 Energy level vs. interatomic distance.

energy state (showing also that the molecule may dissociate if the energy level becomes too high). In a perfectly elastic spring, force increases linearly with displacement, leading to a symmetric quadratic polynomial for the displacement limits as also indicated in the figure. If a more complicated spring constant is included in the analysis, this results in additional terms in equation (10.25); and the selection rule changes to Δv = ±1, ±2, ±3, . . ., producing several approximately equally spaced spectral lines. The transition corresponding to Δv = ±1 is called the fundamental, or the first harmonic, and usually is by far the strongest one. The transition corresponding to Δv = ±2 is called the first overtone or second harmonic, and so on. For example, CO has a strong fundamental band at η0 = 2143 cm−1 and a much weaker first overtone band at η0 = 4260 cm−1 (see the data in Table 10.4 in Section 10.10). In the literature the vibrational state of a molecule is identified by the values of the vibrational quantum numbers. For example, the vibrational state of a nonlinear, triatomic molecule, such as H2 O, with its three different vibrational modes, is identified as (v1 v2 v3 ). The case is a little bit more complicated for molecules with degeneracies. For example, the linear CO2 molecule has three different modes, the second one being  vibrational  l2 doubly degenerate (see Fig. 10.1); its vibrational state is defined by v1 v2 v3 or (v1 v2 l2 v3 ), where 0 ≤ l2 ≤ v2 is an angular momentum quantum number, describing the rotation of the molecule caused by different vibrations in perpendicular planes. More details on these issues are given by Taine and Soufiani [4] and by Herzberg [30].

Combined Vibrational–Rotational Transitions Since the energy required to change the vibrational state is so much larger than that needed for rotational changes, and since both transitions can (and indeed often must) occur simultaneously, this requirement leads to many closely spaced lines, also called a vibration–rotation band, centered around the wavenumber η = νe /c 0 , which is known as the band origin or band center. For the simplest model of a rigid rotator combined with a harmonic oscillator, assuming both modes to be independent, the combined energy level at quantum numbers j, v is given by Ev j = hνe (v + 12 ) + Bv j(j + 1),

v, j = 0, 1, 2, . . . .

(10.27)

Since the small error due to the assumption of a totally rigid rotator can result in appreciable total error when a large collection of simultaneous vibration–rotation transition is considered, allowance has been made in the above expression for the fact that Bv (or the molecular moment of inertia) may depend on the vibrational energy level. The allowed transitions (Δv = ±1 combined with Δj = ±1, 0) lead to three separate branches of the band, namely, P (Δj = −1), Q (Δj = 0), and R (Δj = +1) branches, with spectral lines at wavenumbers ηP = η0 − (Bv+1 + Bv )j + (Bv+1 − Bv )j2 ,

j = 1, 2, 3, . . .

(10.28a)

ηQ = η0 + (Bv+1 − Bv )j + (Bv+1 − Bv )j ,

j = 1, 2, 3, . . .

(10.28b)

2

Radiative Properties of Molecular Gases Chapter | 10 319

FIGURE 10.5 Typical spectrum of vibration–rotation bands.

ηR = η0 + 2Bv+1 + (3Bv+1 − Bv )j + (Bv+1 − Bv )j2 ,

j = 0, 1, 2, . . .

(10.28c)

where j is the rotational state before the transition. It is seen that there is no line at the band origin. If Bv+1 = Bv = const, then the Q-branch vanishes and the two remaining branches yield equally spaced lines on both sides of the band center. If Bv+1 < Bv (larger moment of inertia I at higher vibrational level), then the R-branch will, for sufficiently large j, fold back toward and beyond the band origin. In that case all lines within the band are on one side of a limiting wavenumber. Those bands, where this occurs close to the band center (i.e., for small j where the line strength is strong), are known as bands with a head. A sketch of a typical vibration–rotation band spectrum is shown in Fig. 10.5. Note that in linear molecules the Q-branch often does not occur as a result of forbidden transitions [3]. Many more complicated combined transitions are possible, since every molecule has a number of rotational and vibrational energy modes, any number of which could undergo a transition simultaneously. An example is given in Fig. 10.6, which shows a calculated spectrum of the pressure-based absorption coefficient across the 4.3 μm CO2 band (the strongest vibrational transition together with its rotational lines) for small amounts of CO2 in nitrogen (pCO2 p), generated from the HITRAN database [32]. It is apparent that this band has no Q-branch. A short list of the strongest bands of important radiating gases in combustion and fires can be found in Table 10.1. Band strength or band intensity, α, is generally defined as  ∞ κρη dη, (10.29) α= 0

where the density-based absorption coefficient is employed since it renders α independent of pressure. For fundamental bands (which all bands in Table 10.1 are except for small contributions to the 2.7 μm band of water vapor), the band strength is also independent of temperature. Note that water vapor has a rotational “band” (no vibrational transition) with strong rotational lines for λ  10 μm.

Electronic Transitions Electronic energy transitions, i.e., changing the orbital radius of an electron, requires a substantially larger amount of energy than vibrational and rotational transitions, with resulting photons in the ultraviolet and visible parts of the spectrum. Transitions of interest in heat transfer applications (i.e., at wavelengths above 0.25 μm) generally occur only at very high temperatures (above several thousand degrees Kelvin) and/or in the presence of large numbers of free electrons (such as fluorescent lights). At extreme temperatures atoms and molecules may also become ionized through a bound–free absorption event, or an ion and electron can recombine

320 Radiative Heat Transfer

FIGURE 10.6 Pressure-based spectral absorption coefficient for small amounts of CO2 in nitrogen; 4.3 μm band at p = 1.0 bar, T = 296 K.

TABLE 10.1 Strong vibration–rotation bands of gases found in combustion systems and the atmosphere. Band Location λc [μm]

ηc [cm−1 ]

Vibr. Quantum Step

Band strength

(δk )

α [cm−1 /(g/m2 )]

Band Location λc [μm]

ηc [cm−1 ]

Vibr. Quantum Step

Band strength

(δk )

α [cm−1 /(g/m2 )]

CO2

H2 O  10 μm

rotational

(0, 0, 0)

5.455

15 μm

667 cm−1

(0, 1, 0)

19.0

6.3 μm

1600 cm−1

(0, 1, 0)

41.2

4.3 μm

2410 cm−1

(0, 0, 1)

110.0

2.7 μm CO

3760 cm−1

(1, 0, 0)a

25.9a

2.7 μm CH4

3660 cm−1

(1, 0, 1)

4.0

4.7 μm

2143 cm−1

(1)

20.9

7.7 μm

1310 cm−1

(0, 0, 0, 1)

28.0

3.3 μm

3020 cm−1

(0, 0, 1, 0)

46.0

a Includes

small contributions from (0,2,0) and (1,0,0) transitions.

(free–bound emission). In addition, a free electron colliding with a molecule may absorb or emit a photon (free– free transition). If the gas is monatomic, radiation can alter only electronic energy states. Still, this results in some 914 lines for monatomic nitrogen and 682 for monatomic oxygen [33], contributing to heat transfer in hightemperature applications, such as the air plasma in front of a hypersonic spacecraft entering Earth’s atmosphere. As an example Fig. 10.7 shows the absorption coefficient of atomic nitrogen at T = 10,860 K, as encountered in the shock layer of the Stardust spacecraft [34]. Many of the monatomic lines are extremely strong (with absorption coefficients near 106 m−1 ), and continuum radiation (bound–free and free–free transitions) is substantial. In this part of the spectrum otherwise radiatively inert molecules, e.g., diatomic nitrogen, also emit and absorb photons, leading to simultaneous electronic–vibration–rotation bands. For comparison, the absorption coefficient for N2 is also included in Fig. 10.7, consisting of 5 electronic bands, each containing many vibration–rotation subbands. At temperatures above 10,000 K N2 is nearly completely dissociated, making its absorption coefficient small in comparison to that of monatomic N. At lower temperatures, nearly all molecules are at the lowest electronic energy level, and only the bands with η > 50,000 cm−1 , or λ < 0.2 μm remain (of no importance in most engineering applications).

Radiative Properties of Molecular Gases Chapter | 10 321

FIGURE 10.7 Linear spectral absorption coefficient of monatomic and diatomic nitrogen in a hypersonic boundary air plasma.

Strength of Spectral Lines within a Band In equation (10.14) we related the spectral absorption coefficient to the Einstein coefficients Blu and Bul before knowing how such a transition takes place. We now want to develop equation (10.14) a little further to learn how the strength of individual lines (and, through it, the absorption coefficient) varies across vibration–rotation bands, and how they are affected by variations in temperature and pressure. For a combined vibrational (from vibrational quantum number v to v ± 1) and rotational (from rotational quantum number j to j or j ± 1) transition, the line intensity or line strength may be rewritten in terms of wavenumber (i.e., after division by c 0 ) as Sη = (nl gl Blu − nu gu Bul )hη,

(10.30)

where η is the associated transition wavenumber from equations (10.28). Using equations (10.5) and (10.7) this becomes  nl gu Aul  1 − e−hc 0 η/kT . (10.31) Sη = 2 8πc 0 η The number of molecules at the lower energy state, nl , may be related to the total number of particles per unit volume, n, through [23] nl e−El /kT = , n Q(T)

n=

p , kT

(10.32)

where Q(T) is the rovibrational partition function (a summation over all the possible rotational and vibrational energy levels of the molecule). Substituting this into equation (10.31) and relating the Einstein coefficient to matrix elements of the molecule’s electric dipole moment [20], ul , leads to Sη =

 p  8π3 η |ul |2 1 − e−hc 0 η/kT e−El /kT . 3hc 0 k Q(T)T

(10.33)

The rovibrational partition function Q(T) and dipole elements |ul |2 can, at least in principle, be calculated from quantum mechanics through very lengthy and complex calculations. For example, much of Penner’s book [20] is devoted to this subject. To gain some insight into the relative strengths of lines within a vibration–rotation band, we will look at the case of a rigid rotator–harmonic oscillator, with the additional assumptions that the bandwidth is small

322 Radiative Heat Transfer

compared with the wavenumber at the band center and that only the P and R branches are important. For such a case the evaluation of the |ul |2 is relatively straightforward [20], and equation (10.33) may be restated as SP j = Cj e−hc 0 Bv j( j+1)/kT , −hc 0 Bv j( j+1)/kT

SRj = C(j + 1) e

,

j = 1, 2, 3, . . .

(10.34a)

j = 0, 1, 2, . . .

(10.34b)

where Er j = hc 0 Bv j(j + 1) is the rotational contribution to the lower energy state from equation (10.23) (i.e., before transition for absorption of a photon; after transition for emission) and C collects the coefficients in equation (10.33), as well as the vibrational contribution to the lower energy state. Examination of equations (10.34) shows that line strength first increases linearly with increasing j (as long as hc 0 Bv j(j + 1)/kT 1), levels off √ around j kT/hc 0 Bv , then drops off exponentially with large values of j. It is apparent that the band widens with temperature, and lines farther away from the band center become most important. An example is given in Fig. 10.6 for the calculated spectrum of the 4.3 μm CO2 band, generated from the HITRAN database [32]. At room temperature the 4.3 μm band is dominated by the 000 0 → 000 1 vibrational transition, centered at 2349 cm−1 . It is clear that this band has no Q-branch, and that the line strengths of the P- and R-branches closely follow equation (10.34). Temperature and pressure dependence As seen from equation (10.33) the linear line strength Sη is directly proportional to the pressure of the absorbing/emitting gas; therefore, pressure-based line strength Spη and densitybased line strength Sρη are functions of temperature only. The temperature dependence comes from three contributions: (i) from the partition function Q(T), (ii) from the stimulated emission term, exp(−hc 0 η/kT), and (iii) from the lower energy state El . Evaluation of the partition function is extremely difficult, and approximations need to be made. To a good degree of accuracy rotational and vibrational contributions can be separated, i.e., Q(T) Qv (T)Qr (T). The vibrational partition function can then be determined, assuming a harmonic oscillator, as [30] : −gk , (10.35) Qv (T) = 1 − e−hc 0 ηk /kT k

where the product is over all the different vibrational modes with their harmonic oscillation wavenumbers ηk [= νe /c 0 in equation (10.25)], and gk is the degeneracy of the vibrational mode. The rotational partition function depends on the symmetry of the molecule and on the moments of inertia for rotation around two (linear molecule) or three (nonlinear molecule) axes. For moderate to high temperatures, i.e., when 2IkT/2  1 [23,30], 1 2IkT ∝ T, σ 2  1/2 1 : 2Ii kT ∝ T3/2 , Nonlinear molecules: Qr (T) = 2 σ  i=x,y,z

Linear molecules (Ix = I y = I):

Qr (T) =

(10.36a) (10.36b)

where σ is a symmetry number, or the number of distinguishable rotational modes. Examining the separate contributions to the temperature dependence we note that, at moderate temperatures, the rotational partition function causes the line strength to decrease with temperature as 1/T or 1/T3/2 , while the influences of the vibrational partition function and of stimulated emission are very minor (but may become important for T > 1000 K). The influence of the lower energy state El can be negligible or dramatic, depending on the size of El : for small values of El (low vibrational levels) exp(−El /kT) 1 and further raising the temperature will not change this value. On the other hand, large values of El (associated with high vibrational levels) make line strengths very small at low temperatures, but produce sharply increasing line strengths at elevated temperatures (when more molecules populate the higher vibrational levels), giving rise to so-called “hot lines” and “hot bands.” An example of the temperature dependence of the spectral absorption coefficient (including effects of line broadening and spacing) will be given in the next section, in Fig. 10.13.

10.4 Line Radiation In the previous two sections we have seen that quantum mechanics postulates that a molecular gas can emit or absorb photons at an infinite set of distinct wavenumbers or frequencies. We already observed that no

Radiative Properties of Molecular Gases Chapter | 10 323

spectral line can be truly monochromatic; rather, absorption or emission occurs over a tiny but finite range of wavenumbers. The results are broadened spectral lines that have their maxima at the wavenumber predicted by quantum mechanics. In this section we will briefly look at line strengths, the causes of line broadening, and at line shapes, i.e., the variation of line strength with wavenumber for an isolated line. More detailed accounts may be found in more specialized works [2,3,20,26]. The effects of line overlap, which usually occurs in vibration–rotation bands in the infrared, will be discussed in Section 10.8, “Narrow Band Models.” Numerous phenomena cause broadening of spectral lines. The four most important ones are natural line broadening, collision broadening, Stark broadening, and Doppler broadening, with collision and, to a lesser extent, Doppler broadening dominating in most engineering applications. These models have been developed for isolated lines, i.e., interaction between overlapping lines is not considered, and was found to be accurate for lowto-moderate pressures. However, at elevated pressures (roughly 10 bar) collisional interference (or line mixing) effects should be accounted for [35,36].

Natural Line Broadening Every excited molecule will have its energy levels decay spontaneously to a lower state by emitting a photon, even if the molecule is completely undisturbed. According to Heisenberg’s uncertainty principle no energy transition can occur with precisely the same amount of energy, thus causing the energy of emitted photons to vary slightly and the spectral lines to be broadened. The mechanism of decay for that of spontaneous emission is the same as that for collision broadening as discussed in the next section, resulting in identical line shapes. However, the average time for spontaneous decay is much larger than the average time between molecular collisions. Therefore, natural line broadening is generally not important from an engineering point of view, and its effect is invariably small compared to collision broadening. Its small effect may be accounted for by adding a line half-width γN to the collision line half-width γC discussed below.

Collision Broadening As the name indicates, collision broadening of spectral lines is attributable to the frequency of collisions between gas molecules. The shape of such a line can be calculated from the electron theory of Lorentz∗ or from quantum mechanics [2,37] as  γC S κη = = SφLη (γC , η − η0 ), S ≡ κη dη, (10.37) π (η − η0 )2 + γ2C Δη where S is the line-integrated absorption coefficient or line strength, γC is the so-called line half-width in units of wavenumber (half the line width at half the maximum absorption coefficient), and η0 is the wavenumber at the line center. The line shape function is a normalized Lorentz profile, such that  φLη (η) dη = 1. (10.38) Δη

The line shape function is not dimensionless, but has the units of reciprocal spectral variable. In equation (10.38) this is reciprocal wavenumber (or cm), since κη is expressed in terms of wavenumber. The shape of a collisionbroadened line is identical to that of natural line broadening, and the combined effect is generally termed Lorentz broadening with a line half-width γL . The spectral distribution of a Lorentz line is shown in Fig. 10.8 (together with the shape of Doppler- and Voigt-broadened lines). Since molecular collisions are proportional to the number √ density of molecules (n ∝ ρ ∝ p/T) and to the average molecular speed (vav ∝ T), it is not surprising that the half-width for a pure gas can be calculated from kinetic theory [2] as   n p T0 D2 p 2 γC = √ , (10.39) = γC0 √ p T π c 0 mkT 0 where D is the effective diameter of the molecule, m is its mass, p is total gas pressure, T is absolute temperature, and the subscript “0” denotes a reference state. The collisional diameter depends on the temperature of the gas ∗

A biographical footnote for Hendrik A. Lorentz may be found in Section 2.6.

324 Radiative Heat Transfer

FIGURE 10.8 Spectral line shape for Lorentz (collision), Doppler, and Voigt broadening (for equal line strength and half-width).

and the value for the exponent n must, in general, be found from experiment. If the absorbing–emitting gas is part of a mixture, the fact that collisions involving only nonradiating gases do not cause broadening, and that the nonradiating gases have different molecular diameters, must be accounted for, and equation (10.39) must be generalized to &   ni   2 pi T0 2  σi pi 1 1 1/2  γC = + = γC0,i , (10.40) √ π p0 T c 0 kT m mi i

i

where pi and mi are partial pressure and molecular mass of the various broadening gases (including the radiating gas), respectively, and σi is the effective collisional diameter with species i. The HITRAN and HITEMP databases list these parameters for individual lines in a mixture with air as  nair, j   T0 γC, j (p, T) = γair, j (p0 , T0 )(p − pself ) + γself, j (p0 , T0 )pself , for j−th spectral line. (10.41) T Temperature-dependent broadening coefficients for some absorbing gases have also been tabulated by Rosenmann et al. [38] (CO2 ), Delaye et al. [39] (H2 O), and Hartmann et al. [40], all for mixtures containing N2 , O2 , CO2 , and H2 O. Soufiani and Taine [41] conducted a bibliographical survey and suggested the following correlations for mean line widths to be used with band models: 0 /  0.5 p   T0 T0 (10.42a) 0.462 xH2 O + γH2 O = 0.0792(1 − xCO2 − xO2 ) + 0.106xCO2 + 0.036xO2 , p0 T T p  T0 0.7   γCO2 = 0.07xCO2 + 0.058(1 − xCO2 − xH2 O ) + 0.1xH2 O , (10.42b) p0 T 0 /  0.82  0.7  0.6 p T0 T0 T0 γCO = xCO2 + 0.12 xH2 O + 0.06 (1 − xCO2 − xH2 O ) . (10.42c) 0.075 p0 T T T

Stark Broadening Stark broadening occurs if the radiative transition occurs in the presence of a strong electric field. The electrical field may be externally applied, but it is most often due to an internal field, such as the presence of ions and

Radiative Properties of Molecular Gases Chapter | 10 325

free electrons in a high-temperature plasma. At low-enough pressures Stark broadened lines are symmetric and have Lorentzian shape, equation (10.37). Line widths depend strongly on free electron number density, ne , and free electron temperature, Te , and may be calculated as [26,42] γS = γS0



Te T0

n 

 ne , n0

(10.43)

where again the subscript “0” denotes a reference state. The Stark effect can also result in a shift in the line’s spectral position.

Doppler Broadening According to the Doppler effect a wave traveling toward an observer appears slightly compressed (shorter wavelength or higher frequency) if the emitter is also moving toward the observer, and slightly expanded (longer wavelength or lower frequency) if the emitter is moving away. This is true whether the wave is a sound wave (for example, the pitch of a whistle of a train passing an observer) or an electromagnetic wave. Thus,   v · sˆ , ηobs = ηem 1 + c

(10.44)

where v is the velocity of the emitter and sˆ is a unit vector pointing from the emitter to the observer. Assuming local thermodynamic equilibrium, so that Maxwell’s velocity distribution applies, the probability for a relative velocity v = v · sˆ between an emitting/absorbing molecule and an observer is     m 1/2 mv2 p (v) = exp − , (10.45) 2πkT 2kT where m is the mass of the radiating molecule. For small v this leads to a Doppler shift in observed wavenumber of v (10.46) η − η0 = η0 . c Substituting equation (10.46) into (10.45) one can calculate the line profile as [20] ⎡  2 ⎤ η − η0 ⎥⎥ ln 2 ⎢⎢⎢ ⎥⎥ , κη = S φDη (γD , η − η0 ) = S √ exp ⎢⎣−(ln 2) ⎦ γD γD π √

(10.47)

where γD is the Doppler line half-width, given by η0 γD = c0

&

2kT ln 2. m

(10.48)

Note that, unlike during collision and natural line broadening, the Doppler line width depends on its spectral position. The different line shapes are compared in Fig. 10.8. For equal overall strength, the Doppler line is much more concentrated near the line center.

Combined Effects

√ In most engineering applications collision broadening, which is proportional to p/ T, is by far the most important broadening mechanism. Only at very high temperatures (when, owing to the distribution of the Planck function, transitions at large η are most important; and/or through the opposing temperature dependencies of γL and γD ) √ and/or low pressures may Doppler broadening, with its proportionality to η T, become dominant. Figure 10.9 shows typical line half-widths for CO2 and water vapor in their 2.7 μm bands as a function of temperature. It is seen that at low pressures (p = 0.1 bar) Doppler broadening always dominates. At higher pressures (p ≥ 1 bar) collision broadening dominates, unless extremely high temperatures (T > 2000 K) are encountered. Even then the lines retain their Lorentz shape in the all-important line wings (since in gas columns line centers tend to be

326 Radiative Heat Transfer

FIGURE 10.9 Lorentz and Doppler line half-widths for the 2.7 μm bands of CO2 and H2 O.

opaque, regardless of line shape, radiative behavior is usually governed by the strengths of the line wings). A study by Wang and Modest [43] quantifies the conditions under which combined pressure–Doppler broadening must be considered. Combined broadening behavior is also encountered in low-pressure plasmas, where both Doppler and Stark broadening can be substantial, especially for monatomic gases. If combined effects need to be considered, it is customary to assume collision and Doppler broadening to be independent of one another (which is not strictly correct). In that case a collision-broadened line would be displaced by the Doppler shift, equation (10.46), and averaged over its probability, equation (10.45). This leads to the Voigt profile [2], &  2 SγL +∞ e−x dx m . (10.49) , x=v κη = 3/2   2 π 2kT −∞ xγD 2 + γL η − η0 − √ ln 2 No closed-form solution exists for the Voigt profile. It has been tabulated in the meteorological literature in terms of the parameter 2γL /γD . How the shape of the Voigt profile changes from pure Doppler broadening (γL /γD = 0) to pure collision broadening (γL /γD → ∞) is also shown in Fig. 10.8 (for constant line half-widths). Several fast algorithms for the calculation of the Voigt profile have also been reported [44–47]. A Fortran subroutine voigt is given in Appendix F, which calculates the Voigt κη as a function of S, γL , γD , and |η − η0 | based on the Humlí˘cek algorithm [47]. Example 10.1. The half-width of a certain spectral line of a certain gas has been measured to be 0.05 cm−1 at room temperature (300 K) and 1 atm. When the line half-width is measured at 1 atm and 3000 K, it turns out that the width has remained unchanged. Estimate the contributions of Doppler and collision broadening in both cases. Solution As a first approximation we assume that the widths of both contributions may be added to give the total line half-width (this is a fairly good approximation if one makes a substantially larger contribution than the other). Therefore, we may estimate γC1 + γD1 ≈ γ1 = γ2 ≈ γC2 + γD2 and, from equations (10.39) and (10.48), γC2 = γC1

&

T1 1 = √ , T2 10

γD2 = γD1

&

T2 √ = 10. T1

Radiative Properties of Molecular Gases Chapter | 10 327

Eliminating the Doppler widths from these equations we obtain √ √ γC1 γC1 γ2 = √ + 10γD1 = √ + 10(γ1 − γC1 ), 10 10 √   γC1 γ2 10 √ = 0.76, = 10 − γ1 9 γ1 and

  γC2 1 √ γ1 = 10 − 1 = 0.24. γ2 9 γ2

We see that at room temperature, collision broadening is about three times stronger than Doppler broadening, while exactly the reverse is true at 3000 K.

Line Mixing Effects The ideal impact collision theories that result in the Lorentz line shape are generally accurate in the line core regions, but become inaccurate in high-density gases (high pressure, but also low temperature), when the ratio of collision duration to time between collisions becomes appreciable, so that broadening of multiple lines may interfere with one another. In such cases the far wings of strong lines decay much faster than predicted by the Lorentz profile. These effects are commonly referred to as line mixing and have been recognized by spectroscopists for many years. For the heat transfer engineer they become important in high-pressure applications, such as engines, rocket plumes, gas turbines, etc. To deal with faster-than-Lorentz decrease of the absorption coefficient, spectroscopists introduced a semi-empirical χ-factor to augment the line shape function to [48,49] ⎧ ⎪ ⎪ ⎨ φLη (γc , η − η0 ), φη = ⎪   ⎪ ⎩ φLη (γc , η − η0 )χ(p, T, η − η0 ),

  η − η0  < Δηb ,   η − η0  > Δηb ,

(10.50)

where Δη0 is a wavenumber distance from the line center beyond which the absorption coefficient decreases faster than predicted by φLη . Many χ-factors for CO2 band wings and H2 O have been reported. The only ones dealing with higher temperatures seem to be the ones by Perrin and Hartmann [50] for the far wing of the CO2 4.3 μm band (2400 cm−1 –2600 cm−1 ), and by Hartmann et al. [51] for H2 O for the range 4100 cm−1 –4600 cm−1 . The simplest possible χ-factor is to truncate line wings, i.e., setting χ ≡ 0 beyond Δηb . That was done by Alberti et al. [52–54] while calculating total emissivities for CO2 and H2 O (see Section 10.11), and band absorptances for CO (see Section 10.10). Using published experimental data for CO2 [55,56], H2 O [51], and CO [55,57] they developed a simple cut-off criterion  n Δηb T p0 = Ab ; γc T0 p

T0 = 296 K, p0 = 1 bar;

(10.51)

where Ab = 430, n = 0.822 for CO2 ;

Ab = 687, n = 0.833 for H2 O and CO.

(10.52)

Aside from the inconvenience that χ-factors have to be determined individually for each vibration–rotation band wing, they, as well as Alberti et al.’s cut-off criterion, suffer from the fact that they remove energy from the collision. In reality, at elevated pressure collisional line-mixing effects transfer intensity from regions of weak absorption (far line wings) to those of strong absorption, i.e., they lead to an increase in absorption coefficients near strong line centers [50]. This is demonstrated in Fig. 10.10, which shows the absorptivity, as defined in equation (9.5), across the strong CO2 4.3 μm band at high pressures with a very small concentration of CO2 (88 ppm). In the calculations line parameters (position, strength, and width) were taken from the high-resolution databases HITRAN 2016 [58] and HITEMP 2010 [59], which will be described in Section 10.6. It is seen that both χ-factor and line wing cut-off methods, unlike the standard Lorentz profile, follow experimental values very well in the band wings, but strongly underpredict absorption across the core of the band.

328 Radiative Heat Transfer

FIGURE 10.10 Absorptivity across the CO2 4.3 μm band of an L = 4.4 cm layer at T = 296 K and high pressures and low concentration (88ppm CO2 in N2 ).

Many variations to the Lorentz (and Voigt) profile that conserve energy have been proposed. The simplest of these is the pseudo-Lorentz profile given by [60] a n , κη = Sφη =  η − η0  + γn

a=

  π n−1 S n sin , γ 2 n π

(10.53)

with n > 1. For n = 2 this reduces to the Lorentz profile. For n > 2 absorption in the wings falls off faster than with the Lorentz profile, and this is known as sub-Lorentzian. The opposite holds for n < 2, resulting in a super-Lorentzian wing. Benedict et al. [60] considered a composite line profile, in which the basic Lorentz shape is used for a core region |η − η0 | < Δηb (> γc ), and a sub-Lorentzian profile beyond (normalized to preserve energy). This type of composite profile can be easily extended to a nonsymmetric form. The pseudo-Lorentz profile was successfully applied to the CO2 4.3 μm band by Westlye et al. [61], who determined the n for two elevated temperatures as   0.1 n = 2 + e−1 − e−(p/p0 ) b(T); b(296 K) = 25.0, b(627 K) = 8.01, b(1000 K) = 7.19 (10.54) (the third 296 K value was found by fitting the experimental data in Fig. 10.10). Absorptivities calculated with the pseudo-Lorentz are also included in that figure, and are seen to follow the experimental data very closely even in the core. However, general values for b(T) for other gases, other vibration–rotation bands, and other temperatures remain unknown at this time. A more advanced description of line mixing is given in Chapter 4 of [48]. Based on that information Lamouroux et al. [62,63] constructed a Fortran routine using the “Energy Corrected Sudden” approximation and determined necessary relaxation parameters from fits of CO2 line broadening data measured at various temperatures between 200 and 300 K. The program can be downloaded from the HITRAN website at https:// hitran.org/suppl/LM/. Absorptivities calculated with the Lamouroux package are also included in Fig. 10.10 and are seen to give the best fit with the experimental data. The conditions in Fig. 10.10 were chosen to elucidate the shortcomings of χ-factors as well as the cut-off criterion. In practical applications, concentrations of CO2 and H2 O will be significant, and many of the vibration– rotation bands will be saturated (zero transmissivity across the core of the band), in particular the strong CO2 4.3 μm band. Figure 10.11 shows the transmissivities of the same band, but for 5% CO2 in N2 , for a path of L = 3.3028 cm at 1000 K and a pressure of 80 bar, as measured by Christiansen and coworkers [64], employing all the different line mixing schemes described here. It is obvious that ignoring line mixing grossly underpredicts

Radiative Properties of Molecular Gases Chapter | 10 329

FIGURE 10.11 Transmissivity across the CO2 4.3 μm band of an L = 3.3 cm layer at T = 1000 K and p = 80.12 bar at a concentration of 5% CO2 in N2 .

transmission in the band wings (HITEMP predicting a slightly lower transmissivity than HITRAN, apparently due to missing hot lines in the latter). The Perrin and Hartmann χ-factors and Westlye et al.’s pseudo Lorentz profile follow the experiment extremely well, while Alberti et al.’s cut-off criterion underestimates line mixing in the region beyond 2400 cm−1 . Even though the Lamouroux package has only been fitted for low temperatures the model performs quite well.

Radiation from Isolated Lines Combining equations (10.16) and (10.22) gives the complete equation of transfer for an absorbing–emitting (but not scattering) medium, dIη ds

= κη (Ibη − Iη ),

(10.55)

where the first term of the right-hand side represents augmentation due to emission and the second term is attenuation due to absorption. Let us assume we have a layer of an isothermal and homogeneous gas of thickness L. Then neither Ibη nor κη is a function of location and the solution to the equation of transfer is   Iη (X) = Iη (0) e−κη X + Ibη 1 − e−κη X ,

(10.56)

where the optical path length X is equal to L if a linear absorption coefficient is used (geometric path length), or equal to L multiplied by partial density (density path length) or pressure (pressure path length) of the radiating gas if either mass or pressure absorption coefficient is used. Thus, the difference between entering and exiting intensity, integrated over the entire spectral line, is     I(X) − I(0) = [Iη (X) − Iη (0)] dη ≈ [Ibη − Iη (0)] 1 − e−κη X dη, (10.57) Δη

Δη

where the assumption has been used that neither incoming nor blackbody intensity can vary appreciably over the width of a single spectral line. The integrand of the factor    W= 1 − e−κη X dη (10.58) Δη

330 Radiative Heat Transfer

is the fraction of incoming radiation absorbed by the gas layer at any given wavenumber, and it is also the fraction of the total emitted radiation that escapes from the layer (not undergoing self-absorption). W is commonly called the equivalent line width since a line of width W with infinite absorption coefficient would have the identical effect on absorption and emission; the dependence of the increase of W with increasing optical path X is sometimes called the curve of growth. The equivalent line width for a Lorentz line may be evaluated by substituting equation (10.37) into equation (10.58) to yield W = 2πγL x e−x [I0 (x) + I1 (x)] = 2πγL L (x),

(10.59)

where γL ≡ γC + γN ,

x ≡ SX/2πγL ,

(10.60)

the I0 and I1 are modified Bessel functions and L(x) is called the Ladenburg–Reiche function, after the authors who originally developed it [65]. For simpler evaluation, equation (10.59) may be approximated as reported in [2] as   5/4 −2/5 πx L(x) x 1 + , (10.61) 2 with a maximum error of approximately 1% near x = 1. Asymptotic values for W are easily obtained as W = SX, % W = 2 SXγL ,

x 1,

(10.62a)

x  1.

(10.62b)

Comparing equation (10.60) with equation (10.37), evaluated at half-height (|η − η0 | = γL ), shows that x is the nondimensional optical thickness of the gas layer, κη X, at that location. Therefore, the parameter x gives an indication of the strength of the line. For a weak line (x 1) little absorption takes place so that every position in the gas layer receives the full irradiation, resulting in a linear absorption rate (with distance). In the case of a strong line (x  1) the radiation intensity has been appreciably weakened before exiting the gas layer, resulting in locally lesser absorption and causing the square-root dependence of equation (10.62b).

10.5 Nonequilibrium Radiation There are many radiation applications, in which local thermal equilibrium cannot be assumed, such as in the plasma generated during atmospheric entry of spacecraft, ballistic ranges, high-speed shock tubes, arc jets, etc. When a gas is not in thermal equilibrium, its state cannot be described by a single temperature [66], and the populations of internal energy states do not follow Boltzmann distributions, equation (10.5). The thermodynamic state may then be described using a multitemperature approach (i.e., a Boltzmann distribution is assumed for each internal mode with a specific temperature) [67]. Alternatively, level population distributions may be calculated directly, taking into account collisional and radiative processes. This is known as the Collisional– Radiative (CR) model [68,69] or, if infinitely fast reaction rates are assumed, the Quasi-Steady State (QSS) approximation [67]. Most often the more closely spaced energy levels for translation, rotation, vibration, and free electrons are assumed to have individual equilibrium distributions with up to four different temperatures (Tt , Tr , Tv , Te ), while the widely spaced electronic energy levels are modeled using the QSS/CR approach. Once all energy state distributions have been determined, the emission is given by equation (10.20). Relating it to the absorption coefficient one may define a nonequilibrium Planck function, from equation (10.21), as (in terms of wavenumbers) ne = Ibη

jη κη

= 2hc20 η3

nu . nl − nu

(10.63)

An example is given in Fig. 10.12, showing the nonequilibrium Planck function for diatomic CN (a strongly radiating ablation product from thermal protection systems) [70]. In this graph a two-temperature model was adopted with Tt = Tr = 15,000 K and Tv = Tel = Te = 10,000 K (with electronic energy levels in equilibrium at Tel ), and only Doppler broadening was considered. The ultraviolet CN band (1 ↔ 3 electronic transition) is shown, including many vibration–rotation subbands. For example, the lines labeled Δv = vu − vl = −2 imply that the

Radiative Properties of Molecular Gases Chapter | 10 331

FIGURE 10.12 Nonequlibrium Planck function for CN for a two-temperature model (electronic, vibrational, and electron states at equilibrium with Tv , rotational and translational states with Tr ).

vibrational energy of the upper (electronic) level is two levels lower than that of the lower (electronic) energy state, and so on. The nonequilibrium Planck function displays line structure similar to that of the absorption coefficient. This can be better understood by looking at the special case of negligible stimulated emission and no line overlap (both good approximations for the present case). Then [cf. equation (10.32)] ne (Tv , Tr ) Ibη

Ibη (Tv )

=

'  ) nu hc 0 η/ kTv [Qvr,l /Qvr,u ]ne (Tv , Tr ) Eru − Erl 1 1 exp , e = − nl [Qvr,l /Qvr,u ](Tv ) k Tv Tr

(10.64)

where Qvr is the rovibrational partition function (depending on temperature only) and Er is the rotational energy level. Note that u and l refer to the upper and lower states of the total transition, always determined by the electronic level, i.e., Eru − Erl is the rotational energy change for a given transition (spectral line), which can be negative (lines below the equilibrium Planck function in Fig. 10.12). As can be appreciated from the discussion in this section, and on electronic transitions in Section 10.3, radiation in high-temperature nonequilibrium plasmas is considerably more complicated than usually encountered in engineering, and is beyond the scope of the present text. The reader is referred to the literature dedicated to such problems [67,71].

10.6 High-Resolution Spectroscopic Databases During the past 40 years or so, due to the advent of high-resolution spectroscopy (mostly FTIR spectrometers), it has become possible to measure strengths and positions of individual spectral lines. A first collection of spectral data was assembled in the late 1960s by the Air Force Cambridge Research Laboratories for atmospheric scientists, including low-temperature data for the major constituents of the Earth’s atmosphere, and was published in 1973 as an Air Force report [72]. With contributions from many researchers across the world this grew into the HITRAN database (an acronym for HIgh resolution TRANsmission molecular absorption), first published in 1987 [73]. The database is maintained by the Harvard–Smithsonian Center for Astrophysics, with periodic updates [32,58,74–77]. The latest version (as of 2020) is HITRAN 2016 [58], which includes detailed information on 49 species with a total of about 9.2 million lines (however, little of importance to the heat transfer engineer has been added or changed since the 2008 edition, which contained 2.7 million lines). Among hydrocarbons only methane (CH4 ) was represented in the earlier versions, while in the latest version 4 more species have been added. As the popularity of HITRAN grew, the need for a database valid at elevated temperatures became obvious. A first attempt was made by the group around Taine in France, who augmented HITRAN 1986 data for water vapor

332 Radiative Heat Transfer

and carbon dioxide through theoretical calculations [78,79]. A development by the HITRAN group resulted in a first version of HITEMP (1995) [80] for H2 O, CO2 , CO, and OH, using theoretical models. Comparison with experiment [81–84] indicated that HITEMP 1995 greatly overpredicted CO2 emissivities above 1000 K, while agreement for H2 O was acceptable. More accurate and extensive calculations for CO2 were carried out in Russia, resulting in several versions of the CDSD-1000 database [85,86] (with the 2008 version containing 4 million lines), which were shown to agree well with experiment. The latest version of CDSD, called CDSD-4000 [87], aims to be accurate up to 4000 K, and has 628 million lines, requiring 23 GB of storage. More recently, the same group also published a most exhaustive CO2 database for atmospheric application (i.e., low temperature), called CDSD296 [88], which includes about 530,000 lines. Several extensive high-temperature collections were developed for H2 O: the Ames database [89] includes 300 million lines, SCAN [90] contains 3 billion, and the BT2 collection [91] has 500 million lines; building up on the Ames database, Perez et al. [92] rejected lines from the Ames collection that remain weak below 3000 K, and combined it with well-established lines from HITRAN 2001 and HITEMP 1995, culminating in a manageable collection with 1.3 million lines. Finally, in 2010 a new version of HITEMP was released [59], designed for temperatures up to 3000 K. Citing best agreement against experimental data, HITEMP 2010 incorporates and extends CDSD 2008 for CO2 (11 million lines) and a slimmed-down version of BT2 for H2 O (111 million lines). HITEMP 2010 also includes data for three diatomic gases (CN, CO, and OH) with their relatively few lines. Several researchers have compared data for CO2 and H2 O obtained from HITEMP 2010 and CDSD-1000 against their experiments up to 1773 K noting very good agreement [93–95]. Comparisons against measurements of hydrocarbon gases have exposed some shortcomings: Lecoustre and coworkers [96] found that HITRAN 2012 agrees very well with room temperature data for ethylene, although one band was entirely missing from the database, and at elevated temperature (500 K) line strengths in the database are clearly too low. For ethane agreement was less satisfactory even at room temperature. For methane approximate high-temperature data (up to 2000 K) extrapolated from the HITRAN database can be found in [97]. An example calculation is given in Fig. 10.13, showing a small part of the artificial spectrum of the 4.3 μm CO2 band, generated from the HITRAN 2012 database [98], employing the Voigt line shape, and containing more than 1,500 spectral lines. The top frame of Fig. 10.13 shows the pressure-based absorption coefficient of CO2 at low partial pressure in air at a total pressure of 10 mbar. Because of the relatively low total pressure, the lines are fairly narrow, resulting in little overlap, and are dominated by Doppler broadening. If the total pressure is raised to 1 bar, shown in the center frame, lines become strongly broadened by collisions, leading to substantial line overlap, and a smoother variation in the absorption coefficient (with considerably lower maxima and higher minima). At the high temperatures usually encountered during combustion the spectral lines narrow considerably [see equation (10.39)], decreasing line overlap; at the same time the strengths of the lines that were most important at low temperature decrease according to equation (10.36) and finally, at high temperatures “hot lines,” that were negligible at room temperature, become more and more important. To be valid up to 3000 K, HITEMP 2010 [59] lists more than 22,000 spectral lines for this small wavenumber range. The result is a fairly erratic looking absorption coefficient as depicted in the bottom frame of Fig. 10.13. If high temperatures are combined with low total pressures (not shown), the spectral behavior of the absorption coefficient resembles high-frequency electronic noise. Fortunately, heat transfer calculations in media at low total pressure are rare (they are important, though, in meteorological applications dealing with the low-pressure upper atmosphere). Similar efforts have been made by the plasma radiation community. RAD/EQUIL is perhaps the earliest attempt, including contributions from atomic lines and continua, and approximate models for molecules, but only for thermodynamic equilibrium conditions [99]. The NonEQuilibrium AIr Radiation (NEQAIR) model [100] was originally developed for the study of radiative properties of nonequilibrium, low density air plasmas. The updated NEQAIR96 model [101] includes spectral line data for spontaneous emission, stimulated emission, and absorption for 14 monatomic and diatomic species, as well as bound–free and free–free transition data for atoms. Nonequilibrium electronic level populations are determined using the QSS approximation (cf. Section 10.5). Since the creation of NEQAIR various improvements have been made by Laux [102] and others, leading to the SPECAIR database [103]. In Japan the SPRADIAN database was assembled [104], which was later updated in cooperation with KAIST [105]. A new High-temperature Aerothermodynamic RAdiation model (HARA) developed by Johnston [68,106] utilizes comprehensive and updated atomic line data obtained from the National Institute of Standards and Technology (NIST) online database [107] and the Opacity Project [108], as well as atomic bound-

Radiative Properties of Molecular Gases Chapter | 10 333

FIGURE 10.13 Spectral pressure-based absorption coefficient for small amounts of CO2 in nitrogen, across a small portion of the CO2 4.3 μm band; top frame: p = 10 mbar, T = 300 K; center frame: p = 1 bar, T = 300 K; bottom frame: p = 1 bar, T = 1000 K.

free cross-sections from the TOPbase [109]. Since the above databases are generally stand-alone programs, incorporating several other tools, such as primitive RTE solvers, Sohn et al. [110] extracted the relevant data from NEQAIR96 to form an efficient radiative property module. This database has been updated for high-speed retrieval rates and to incorporate the state-of-the-art data in HARA [111].

10.7 Spectral Models for Radiative Transfer Calculations A single spectral line at a certain spectral position is fully characterized by its strength (the intensity, or integrated absorption coefficient) and its line half-width (plus knowledge of the broadening mechanism, i.e., collision and/or Doppler broadening). However, a vibration–rotation band has many closely spaced spectral lines that may overlap considerably. While the absorption coefficients for individual lines may simply be added to give

334 Radiative Heat Transfer

the absorption coefficient of an entire band at any spectral position,  κη = κη j ,

(10.65)

j

the resulting function tends to gyrate violently across the band (as seen in Figs. 10.6 and 10.13), unless the lines overlap very strongly. This difficulty, together with the fact that there may be literally millions of spectral lines, makes radiative transfer calculations a truly formidable task, if the exact relationship is to be used in the spectral integration for total intensity [equation (9.28)], total radiative heat flux [equation (9.55)], or the divergence of the heat flux [equation (9.62)]. Exact evaluation of these quantities would require roughly one million solutions to the spectral RTE, known as line-by-line calculations. For many years, starting long before detailed spectroscopic databases became available, spectroscopists and engineers developed a number of approximate spectral models to make the problem more manageable. Exact and approximate methods may be loosely put into four groups (in order of decreasing complexity and accuracy): (1) line-by-line calculations, (2) narrow band calculations, (3) wide band calculations, and (4) global models. Line-By-Line Calculations With the advent of powerful computers and the necessary high-resolution spectroscopic databases, a number of spectrally resolved or “line-by-line (LBL) calculations” have been performed, a few for actual heat transfer calculations, e.g., [112–114], some to prepare narrow band model correlations, e.g., [115,116], and others to validate global spectral models, e.g., [117–119]. Such calculations rely on very detailed knowledge of every single spectral line, taken from one of the high-resolution spectroscopic databases described in Section 10.6. As indicated earlier, the spectral radiative transfer problem must be solved for up to one million wavenumbers, followed by integration over the spectrum, if standard (deterministic) RTE solvers are employed. While such calculations may be the most accurate to date, they require vast amounts of computer resources, even while radiative calculations usually are only a small part of a sophisticated, overall fire/combustion code. In addition, high-resolution gas property data (resolution of better than 0.01 cm−1 ), as required for accurate line-by-line calculations, are generally found from theoretical calculations and mostly still remain to be validated against experimental data. In particular, temperature and pressure dependence of spectral line broadening is very complicated and simply not well enough understood to extrapolate room temperature data to the high temperatures important in combustion environments. For these reasons it is fair to assume that, for the foreseeable future, line-by-line calculations together with standard RTE solution methods will only be used as benchmarks for the validation of more approximate spectral models. On the other hand, it has been demonstrated recently, e.g., [120–122], that LBL-accurate calculations can be made very efficiently with Monte Carlo solutions, as will be outlined in Chapter 20. Narrow Band Models When calculating spectral radiative fluxes from a molecular gas one finds that the gas absorption coefficient (and with it, the radiative intensity) varies much more rapidly across the spectrum than other quantities, such as blackbody intensity, etc. It is, therefore, in principle possible to replace the actual absorption coefficient (and intensity) by smoothed values appropriately averaged over a narrow spectral range. A number of such “narrow band models” were developed some 40–50 years ago, and will be examined in the following section. In principle, narrow band calculations can be as accurate as line-by-line calculations, provided an “exact” narrow band average can be found. The primary disadvantages of such narrow band models are that they are difficult to apply to nonhomogeneous gases and the fact that heat transfer calculations, based on narrow band data and using general solution methods, are limited to nonscattering media within a black-walled enclosure. An alternative to the “traditional” narrow band models is the so-called “correlated k-distribution.” In this method it is observed that, over a narrow spectral range, the rapidly oscillating absorption coefficient κη attains the same value many times (at slightly different wavenumbers η), each time resulting in identical intensity Iη and radiative flux (provided the medium is homogeneous, i.e., has an absorption coefficient independent of position). Since the actual wavenumbers are irrelevant (across the small spectral range), in the correlated k-distribution method the absorption coefficient is reordered, resulting in a smooth dependence of absorption coefficient vs. artificial wavenumber (varying across the given narrow range). This, in turn, makes spectral integration very straightforward. k-distributions are relatively new, and are still undergoing development. While attractive, they also are difficult to apply to nonhomogeneous media. Wide Band Models Wide band models make use of the fact that, even across an entire vibration–rotation band, blackbody intensity does not vary substantially. In principle, wide band correlations are found by integrating

Radiative Properties of Molecular Gases Chapter | 10 335

narrow band results across an entire band, resulting in only slightly lesser accuracy. Wide band model calculations have been very popular in the past, due to the facts that the necessary calculations are relatively simple and that much better spectral data were not available. However, it is well recognized that wide band correlations have a typical correlational accuracy of ±30%, and in some cases may be in error by as much as 70%; substantial additional but unquantified errors may be expected due to experimental inaccuracies. One of the attractions of the correlated k-distributions is that they can be readily adapted to wide band calculations. Global Models In heat transfer calculations it is generally only the (spectrally integrated) total radiative heat flux or its divergence that are of interest. Global models attempt to calculate these total fluxes directly, using spectrally integrated radiative properties. Most early global methods employ the total emissivities and absorptivities of gas columns, but more recently full-spectrum correlated k-distributions have also been developed. During the remainder of this chapter we will discuss the smoothing of spectral radiative properties of molecular gases over narrow bands and wide bands, as well as the evaluation of total properties. Actual heat transfer calculations using these data will be deferred until Chapter 19 (i.e., until after the discussion of particulate properties and of solution methods for the radiative transfer equation). Global models require manipulation of the RTE and, thus, will also be deferred to Chapter 19.

10.8 Narrow Band Models Examination of the formal solution to the equation of radiative transfer, equation (9.28), shows that all spectral integrations may be reduced to four cases, namely, 

∞

 κη I(b)η dη

∞

and

I(b)η

0

0

⎛ X ⎡ ⎞⎤ ⎜  ⎢⎢ ⎟⎥ ⎢⎢1 − exp ⎜⎜⎜− κη dX⎟⎟⎟⎥⎥⎥ dη, ⎝ ⎣ ⎠⎦

(10.66)

0

where I(b)η denotes that either Ibη or Iη can occur, and X is the optical path length introduced in equation (10.56). It is clear from inspection of Fig. 1.5 that the Planck function will never vary appreciably over the spectral range of a few lines, considering that adjacent lines are very closely spaced (measured in fractions of cm−1 ). Local radiation intensity Iη , on the other hand, may vary just as strongly as the absorption coefficient, since emission within the gas takes place at those wavenumbers where κη is large [see equation (9.10)]. However, if we limit our consideration to nonscattering media bounded by black (or no) walls, the formal solution of the radiative equation of transfer, equation (9.29), shows that all spectral integrations involve only the Planck function, and not the local intensity. For such a restricted scenario4 we may simplify expressions (10.66), with extremely good accuracy, to 

/

∞

Ibη 0

and

 0

∞



1 Δη

η+Δη/2

η−Δη/2

0 κη dη dη

(10.67a)

⎧ ⎫ ⎛ X ⎞⎤  η+Δη/2 ⎡ ⎪ ⎪ ⎪ ⎜⎜  ⎢⎢ ⎟⎟⎥⎥  ⎪ ⎨ 1 ⎢⎢1 − exp ⎜⎜− κη dX⎟⎟⎥⎥ dη ⎬ dη. Ibη ⎪ ⎪ ⎝ ⎣ ⎦ ⎠ ⎪ ⎪ ⎩ Δη η−Δη/2 ⎭ 0

(10.67b)

The expressions within the large braces are local averages of the spectral absorption coefficient and of the spectral emissivity, respectively, indicated by an overbar:5 κη (η) =

1 Δη

1

η (η) = Δη

 

η+Δη/2

η−Δη/2 η+Δη/2

η−Δη/2

κη dη ,

(10.68)

⎛ X ⎡ ⎞⎤ ⎜⎜  ⎢⎢ ⎟⎥ ⎢⎢1 − exp ⎜⎜− κη dX⎟⎟⎟⎥⎥⎥ dη . ⎝ ⎣ ⎠⎦

(10.69)

0

4. If the Monte Carlo method is employed as the solution method, this restriction is not necessary, since integration over local intensity is avoided even for reflecting walls/scattering media; see Section 20.3. 5. It should be understood that the definition of κ in equation (10.68) is not sufficient since  1 − exp(−κs). This fact will be demonstrated in Example 10.2.

336 Radiative Heat Transfer

FIGURE 10.14 Typical spectral line arrangement for (a) Elsasser and (b) statistical model.

One can expect the spectral variation of κ and to be relatively smooth over the band, making spectral integration of radiative heat fluxes feasible. Furthermore, due to the limited sensitivity of detectors, experimental determination of spectral gas absorption coefficients has relied on the measurement of such smoothed emissivities or, more often, transmissivities67 ⎛ X ⎞  η+Δη/2 ⎜⎜  ⎟⎟ 1 ⎜ τη (η) = 1 − η (η) = exp ⎝⎜− κη dX⎟⎠⎟dη . (10.70) Δη η−Δη/2 0 To find spectrally averaged or “narrow band” values of the absorption coefficient and the emissivity, some information must be available on the spacing of individual lines within the group and on their relative strengths. A number of models have been proposed to this purpose, of which the two extreme ones are the Elsasser model, in which equally spaced lines of equal intensity are considered, and the statistical models, in which the spectral lines are assumed to have random spacing and/or intensity. A typical spectral line arrangement for these two extreme models is shown in Fig. 10.14. The main distinction between the two models is the difference in line overlap. Both models will predict the same narrow band parameters for optically thin situations or nonoverlap conditions (since overlap has no effect), as well as for optically very strong situations (since no beam can penetrate through the gas, regardless of the overlapping characteristics). Under intermediate conditions the Elsasser model will always predict a higher emissivity/absorptivity than the statistical models, since regular spacing always results in less overlap (for the same average absorption coefficient) [3]. The deviation between the models is never more than 20%. In the following we will limit our discussion to lines of Lorentz shape, since collision broadening generally dominates at the relatively high pressures encountered in heat transfer applications. Discussion on models for Doppler and Voigt line shapes can be found in the meteorological literature, e.g., [2]. A very exhaustive discussion of various line shapes and dozens of band models has been given by Young [60].

The Elsasser Model We saw earlier in this chapter that diatomic molecules and linear polyatomic molecules have only two, identical rotational modes, resulting in a single set of lines (consisting of two or three branches, as shown in Fig. 10.2 and Fig. 10.5). For these gases one may expect spectral lines with nearly constant spacing and slowly varying intensity, in particular if the Q-branch is unimportant (or “forbidden”) and if the folding back of the R-branch gives also only a small contribution. Summing up the contributions from infinitely many Lorentz lines on both sides of an arbitrary line with center at η0 , we get κη =

∞  γL S , π (η − η0 − iδ)2 + γ2L

(10.71)

i=−∞

6. While it is true that modern FTIRs can measure transmissivities at the resolutions required for LBL calculations (see Section 10.13), the scope of such measurements would be unachievable. 7. Unfortunately, in the radiative transfer literature the letter τ is used for, both, transmissivity as well as for optical thickness, which can lead to confusion. For this reason we will use = 1 − τ whenever feasible in this chapter.

Radiative Properties of Molecular Gases Chapter | 10 337

where δ is the (constant) spacing between spectral lines.8 This series may be evaluated in closed form, as was first done by Elsasser, resulting in [123] κη =

sinh 2β S , δ cosh 2β − cos(z − z0 )

(10.72)

where β ≡ πγL /δ,

z ≡ 2πη/δ.

(10.73)

From equation (10.68), the average absorption coefficient is simply κη =

S . δ

(10.74)

This also follows without integration from the fact that S is each line’s contribution to the integrated absorption coefficient [see equation (10.37)], and that the lines are spaced δ wavenumbers apart, i.e., for every δ wavenumbers S is added to the integrated absorption coefficient. The spectrally averaged emissivity may be evaluated from equation (10.69) as    π 2βx sinh 2β 1

η = 1 − exp − dz, (10.75) 2π −π cosh 2β − cos z where, since the absorption coefficient is a periodic function, one full period was chosen for the averaging wavenumber range and, thus, the arbitrary location z0 could be eliminated. As one may see from its definition, equation (10.73), β is the line overlap parameter: β gives an indication of how much the individual lines overlap each other, and x, already defined in equation (10.60), is the line strength parameter. At this point we may also define another nondimensional parameter, the narrow band optical thickness τ = κX, so that we now have three characterizing parameters, namely, x=

SX , 2πγL

β=π

γL , δ

τ=

S X = 2βx. δ

(10.76)

Equation (10.75) cannot be solved in closed form, but an accurate approximate expression, known as the Godson approximation, has been given [2]:

η ≈ erf

√   √ √  πW π S −x X e [I0 (x) + I1 (x)] = erf πβL(x) = erf 2 δ 2 δ

(10.77)

where erf is the error function and is tabulated in standard mathematical texts [124]. The Godson approximation is reasonably accurate for small-to-moderate line overlap (β < 1). For larger values of β, and for hand calculations it is desirable to have simpler expressions. We can distinguish among three different limiting regimes: weak lines (x 1) : strong overlap (β > 1) : strong lines (x  1) : no overlap (β 1) :

  S

η = 1 − exp − X = 1 − e−τ , δ ⎞ ⎛& %  ⎜⎜ S γL ⎟⎟⎟ ⎜ X⎟⎠ = erf τβ ,

η = erf ⎜⎝ π δ δ

η =

W = 2βL(x), δ

(10.78a) (10.78b) (10.78c)

where the W/δ in equation (10.78c) can possibly be further simplified using equations (10.62a) and (10.62b). These relations are summarized in Table 10.2. 8. Since we are using wavenumber here, the value for δ is measured in units of wavenumbers, cm−1 . If we were to use frequency or wavelength, the definition and units of δ would correspondingly change.

338 Radiative Heat Transfer

The Statistical Models In the statistical models it is assumed that the spectral lines are not equally spaced and of equal strength but, rather, are of random strength and are randomly distributed across the narrow band. This assumption can be expected to be an accurate representation for complex molecules for which lines from different rotational modes overlap in an irregular fashion. In several early studies Goody [125] and Godson [126] showed that any narrow band model with randomly placed spectral lines, with arbitrary strengths and line shape (i.e., Lorentzian or other), leads to the same expression for the spectrally averaged emissivity 

η = 1 − exp −

W δ

 ,

(10.79)

where W is an average over the N lines contained in the spectral interval, W=

N 1  Wi , N

(10.80)

i=1

and δ is the average line spacing, defined as Δη . N

δ=

(10.81)

A number of statistical models have been developed, in which lines are placed at random across Δη with random strengths picked from different probability distributions. We will limit our brief discussion to three different models, which excel due to their simplicity and/or their success to model actual spectral distributions. The simplest statistical model is the uniform statistical model, in which all lines have equal strengths, or S = S = const.

Uniform statistical model:

(10.82)

A more realistic representation must allow for varying lines strengths, given by a probability density function p(S). The properties of the narrow band are then found by averaging line properties with the probability density function. A frequently used such probability distribution is the exponential form proposed by Goody [125], Goody model:

  S p(S) = , exp − S S 1

0 ≤ S < ∞,

(10.83)

which is popular due to its simplicity. However, Malkmus [127] recognized that in many cases this exponential intensity distribution severely underpredicts the number of low-strength lines. He modified the physically plausible 1/S distribution proposed by Godson [126] to obtain an exponential-tailed 1/S distribution, now known as the Malkmus model: Malkmus model:

  1 S p(S) = exp − , S S

0 ≤ S < ∞.

(10.84)

All three distribution functions, equations (10.82), (10.83), and (10.84), have identical average line strengths S. Finding the average equivalent line width W for the uniform statistical model is trivial, since every equivalent line width from equation (10.82) is identical, and W = W (single line). For the Goody and Malkmus model the sum in equation (10.80) can, for a large statistical sample, be replaced by an integral:  W −→

N→∞

0

∞

 p(S)W(S) dS =



∞

p(S) 0

+∞ −∞

  1 − e−κη (S)X dη dS.

(10.85)

Radiative Properties of Molecular Gases Chapter | 10 339

Substituting equations (10.83) and (10.84) and carrying out the integrations leads to, for Lorentz lines, W

Uniform statistical model:

δ W

Goody model:

δ W

Malkmus model:

δ

 SX = 2βL(x) = 2βL(τ/2β), = 2π L 2πγL δ . 1/2 + S SX = τ (1 + τ/β)1/2 , = X 1+ πγL δ ⎡ 1/2 ⎤  ⎥⎥ β  πγL ⎢⎢⎢ 4SX ⎢⎢ 1 + = − 1⎥⎥⎥⎦ = (1 + 4τ/β)1/2 − 1 , ⎣ 2 πγL 2δ γL



(10.86) (10.87) (10.88)

where L(x) is the Ladenburg–Reiche function given by equation (10.59). In these models the narrow band parameters γL /δ and S/δ are either found by fitting experimental data, or from high-resolution spectral data such as the HITRAN database [32]. In the latter case, it is desirable to have the models yield exact results in the limits of weak lines (x 1) as well as strong lines (x  1). In the weak line limit we have, for all three models, weak lines (x 1) :

W δ

→

S δ

X = 2βx = τ,

(10.89)

while the models lead to slightly different strong line limits, i.e., strong lines (x  1) : W

Uniform statistical:

Goody/Malkmus:

δ W δ

( 2 γL SX

→ (

→

δ πγL SX δ

= 2β(2x/π)1/2 = 2(τβ/π)1/2 ,

= β(2x)1/2 = (τβ)1/2 .

(10.90a)

(10.90b)

Satisfying these two conditions requires [2] N S 1  Si , = Δη δ i=1

γL δ

=

Cγ Δη

5

2 N 1/2 i=1 (Si γLi ) 5N

i=1

Si

,

(10.91)

with Cγ = 1 for the uniform statistical model and Cγ = 4/π for the Goody and Malkmus models; the latter two models will always have some weak lines, resulting in a smaller value for W/δ, even in the strong line limit (based on average line strength). The results from the statistical models have also been summarized in Table 10.2. The narrow band emissivities from all four models are compared in Fig. 10.15 as a function of the optical path of an average spectral line (i.e., average absorption coefficient S/2πγL multiplied by distance X). Note that all predictions are relatively close to each other, although the statistical models may predict up to 20% lower emissivities for optically thick situations. The Goody and Malkmus models more or less coincide for small values of β, giving somewhat lower emissivities than the uniform statistical model because of their different strong line behavior. For optically thin situations (x < 1) the uniform statistical and Goody’s model move toward the Elsasser model, with lower emissivities predicted by the Malkmus model. Note that the Elsasser lines were drawn from numerical evaluations of equation (10.75), not from equation (10.77), which would show serious error for the β = 1 line. Example 10.2. The following data are known at a certain spectral location for a pure gas at 300 K and 0.75 atm: The mean line spacing is 0.6 cm−1 , the mean line half-width is 0.03 cm−1 , and the mean line strength (or integrated absorption coefficient) is 0.08 cm−2 atm−1 . What is the mean spectral emissivity for geometric path lengths of 1 cm and 1 m, if the gas is diatomic (such as CO), or if the gas is polyatomic (such as water vapor)? Solution Since the units of the given line strength tell us that a pressure absorption coefficient has been used, we need to employ a pressure path length X = ps. For a path length of 1 cm we get X = 0.75 atm × 1 cm = 0.75 cm atm and

340 Radiative Heat Transfer

TABLE 10.2 Summary of effective line widths and narrow band emissivities for Lorentz lines.

Single line, W W δ Elsasser model W δ

Weak line

Strong line

No overlap

x1

x1

β1

τ

% 2 SXγL % 2 τβ/π

τ

% 2 τβ/π

SX

1 − e−τ

η

erf

All regimes 2πγL L(τ/2β) 2βL(τ/2β)

%  τβ

2βL(τ/2β)  √π W  erf 2 δ

W δ

Statistical models W δ W δ W δ

(S = const)

τ

% 2 τβ/π

(Goody)

τ

% τβ

(Malkmus)

τ

% τβ

1 − e−τ

  1 − exp −W/δ

η Definitions: x=

SX ; 2πγL

β=π

γL δ

; τ=

S δ

X = 2βx;

2βL(τ/2β) τ β 2

W δ

+%

1 + τ/β

%  1 + 4τ/β − 1

  1 − exp −W/δ

  5/4 −2/5 πx L(x) x 1 + 2

FIGURE 10.15 Mean spectral emissivities for Lorentz lines as a function of average optical depth (S/δ)X.

x = SX/2πγ = 0.08 cm−2 atm−1 × 0.75 cm atm/(2π 0.03 cm−1 ) = 1/π, while the overlapping parameter turns out to be β = πγ/δ = π × 0.03 cm−1 /0.6 cm−1 = π/20, and τ = 2βx = 2(π/20)(1/π) = 0.1. For a diatomic gas for which the Elsasser model should be more accurate, we can use either equation (10.77) or (since β 1) equation (10.78c). Evaluating the Ladenburg–Reiche function from (10.61) gives L and

η = erf

  −2/5 1 1 1 + 0.55/4 = 0.2766, π π

 √ π π π 0.2766 = erf (0.0770) = 0.0867 2 0.2766 = 0.0869 = 8.7%. 20 20

Radiative Properties of Molecular Gases Chapter | 10 341

If the gas is polyatomic we may want to use one of the statistical models. Choosing the Malkmus model, equation (10.88), we obtain ⎧ ⎡ ⎤⎫ 1/2 ⎪ ⎪ ⎪ ⎥⎥⎪ 4 × 0.1 ⎬ ⎨ 1 π ⎢⎢⎢ = 0.0670. −

η = 1 − exp ⎪ − 1⎥⎥⎦⎪ ⎢⎣ 1 + ⎪ ⎪ ⎭ ⎩ 2 20 (π/20) If the path length is a full meter, we have X = 75 cm atm and x = 100/π while β is still β = π/20 and now τ = 10. Thus we % are in the strong-line region. For the diatomic gas, from equation (10.78b) η = erf[ 10(π/20)] = erf(1.2533) = 0.924. For the polyatomic gas, again using equation (10.88), we get η = 0.692. In the first two cases, using the simple relation = 1 − exp(−κs) actually would have given fairly good results (0.095) because the gas is optically thin resulting in essentially linear absorption at every wavenumber. For the larger path we would have gotten 1 − e−10 ≈ 1. Thus, using an average value for the absorption coefficient makes the gas opaque at all wavenumbers rather than only near the line centers. Example 10.3. For a certain polyatomic gas the line-width-to-spacing ratio and the average absorption coefficient for a vibration–rotation band in the infrared are known as       S S −2|η−η0 |/ω S ≈ e , = 10 cm−1 , (10.92) δ η δ 0 δ 0 γ ≈ 0.1 ≈ const. ω = 50 cm−1 , δ Find an expression for the averaged spectral emissivity and for the total band absorptance defined by   ∞  A≡ 1 − e−κη X dη,

η dη = band

0

for a path length of 20 cm. Solution Calculating the optical thickness τ0 = (S/δ) 0 X = 10 × 20 = 200, the overlap parameter β = π/10, and the line strength x0 = τ0 /2β = 1000/π, we find that this band falls into the “strong-line” regime everywhere except in the (unimportant) far band wings. Since we have a polyatomic molecule with exponential decay of intensity, one of the statistical models should provide the best answer. As seen from Fig. 10.15, all three statistical models give very similar results, and the (more appropriate) Goody and Malkmus models go to the same strong line limit, equation (10.90b), or  % 

η = 1 − e −W/δ ≈ 1 − exp − τβ , since τ/β  1. Employing equation (10.92) to evaluate τ in this expression yields the spectral emissivity,  % 

η = 1 − exp − τ0 β e−|η−η0 |/ω . Integrating this equation over the entire band gives the total band absorptance,  ∞  %  1 − exp − τ0 β e−|η−η0 |/ω dη. A= 0

Realizing that this integral has two symmetric parts and setting ln z = −(η − η0 )/ω, we have  A = 2ω 0

1



 %  dz 1 − exp − τ0 βz . z

This integral may be solved in terms of exponential integrals9 as given, for example, in Abramowitz and Stegun [124]. This leads to   % % A = 2ω E1 ( τ0 β) + ln( τ0 β) + γE = 264.7 cm−1 , where γE = 0.57721 . . . is Euler’s constant. 9. Exponential integrals are discussed in some detail in Appendix E.

342 Radiative Heat Transfer

Most available narrow band property data, such as the RADCAL database [128,129], have been correlated with the Goody model. The correlation by Malkmus is a relative latecomer, but is today recognized as the best model for polyatomic molecules. While commonly used in the atmospheric sciences this correlation was widely ignored by the heat transfer community for many years. Taine and coworkers [115,116,130] have generated artificial narrow band properties from HITRAN 1992 line-by-line data. Employing the Malkmus model with a resolution of 25 cm−1 they observed a maximum 10% error between line-by-line and narrow band absorptivities. Using two narrow spectral ranges of H2 O and CO2 Lacis and Oinas [131] showed that (for a resolution of 10 cm−1 , and for total gas pressures above 0.1 atm) the correlational accuracy of the Malkmus model can be improved to better than 1% if the model parameters are found through least square fits of the HITRAN 1992 line-by-line data. Soufiani and Taine [41] have assembled a Malkmus-correlated EM2C narrow band database (25 cm−1 resolution) for various gases for atmospheric pressure and a temperature range 300 K ≤ T ≤ 2500 K, using the HITRAN 1992 database together with some proprietary French high-temperature extensions. This was updated in 2012 by Rivière and Soufiani [132] based on CDSD-4000 [87] and HITEMP 2010 [59], adding data for CO and CH4 , and extending the temperature range for H2 O and CO2 to 4,000 K. A similar database, specifically devoted to nine common fuels, was generated by Consalvi and Liu [133]. To date very few experimental narrow band data have been correlated with the Malkmus model: Phillips has measured and correlated the 2.7 μm H2 O band [134] and the 4.3 μm CO2 band [135], both between room temperature and 1000 K. Recently, Lecoustre and coworkers [96] took medium resolution FTIR measurements of ethylene, propylene, and ethane, and provided Malkmus narrow band parameters (S/δ) and β in graphical form, also for bands of 25 cm−1 widths. Both the RADCAL and the EM2C databases are included in Appendix F. More recently, André and Vaillon [119,136] found that, if the natural logarithm of the transmissivity, ln(1 − ), is expanded into a series of k-moments μi , where μi =

1 Δη

 Δη

κiη dη,

(10.93)

and if this expansion is truncated after just two moments, the resulting transmissivity is identical to the Malkmus distribution, with β evaluated from β=

2μ21 μ2 − μ21

,

(10.94)

i.e., no data fit (to experiment or spectroscopic database) is necessary. Comparison against LBL calculations showed it to be very accurate. Two generalizations of the Malkmus model have also been developed, a multiscale model for nonhomogeneous gases [137] (see also below) and a generalized model more appropriate for Doppler-dominated regimes [138].

Gas Mixtures Experimental data for narrow band properties, such as line overlap (γ/δ) and average absorption coefficient (S/δ), are usually given from correlations of measurements performed on a homogeneous column involving a single absorbing gas species. In practical applications, on the other hand, radiative properties of mixtures that contain several absorbing gas species, such as CO2 , H2 O, CO, etc., are generally required. Over large portions of the spectrum spectral lines from different species do not overlap each other, and the expressions given in Table 10.2 remain valid. However, there are regions of the spectrum where spectral line overlap is substantial and must be accounted for. For example, the two most important combustion gases, water vapor and CO2 , both have strong bands in the vicinity of 2.7 μm. Mixture values for (γ/δ) and (S/δ) are found from their definitions, equation (10.91), by setting  i

Si =

 n

i

Sni ;

% i

Si γi =

% n

i

Sni γni ,

(10.95)

Radiative Properties of Molecular Gases Chapter | 10 343

where the subscript n identifies the gas species. Comparing equation (10.91) for the mixture and its individual components readily leads to 

S



δ

= mix

 S  n

δ



γ

;



δ

n





S δ

mix

mix

 ⎤2 ⎡ ⎢⎢  γ   S  ⎥⎥ ⎥⎥ ⎢⎢ = ⎢⎢ ⎥⎥ . ⎣ δ n δ n⎦

(10.96)

n

Expressions in Table 10.2 together with equation (10.96) can then be used to evaluate the transmissivity of a gas mixture. Other expressions for mixture values of (γ/δ) and (S/δ) have been discussed by Liu and coworkers [139]. Taine and Soufiani [4] pointed out that there is no physical reason why there should be any significant correlation between the spectral variation of absorption coefficients of different gas species. If one treats the absorption coefficients of the M species as statistically independent random variables, the transmissivity of a mixture can be evaluated as the product of the individual species’ transmissivities, τη,mix = 1 − η,mix =

M :

τη,m .

(10.97)

m=1

This was first discovered by Burch et al. [140]. For example, comparing the mixture transmissivity of a room temperature water vapor–carbon dioxide mixture for the overlapping 2.7 μm region, calculated directly from the HITRAN database and from equation (10.97), Taine and Soufiani found them to be virtually indistinguishable.

Nonhomogeneous Gases Up to this point in calculating narrow band emissivities we have tacitly assumed that the gas is isothermal, and has constant total and partial pressure of the absorbing gas everywhere, i.e., we replaced the integral X κ dX in equation (10.69) by κX. We now want to expand our results to include nonhomogeneous gases. For 0 the Elsasser model the solution to equation (10.75) is possible, but too cumbersome to allow a straightforward solution if properties are path-dependent. For the more important statistical models the same is true, especially if not only line strength, S, but also the line overlap parameter, β, varies along the path. Instead, one resorts to approximations. The best known and most widely used approximation is known as the Curtis–Godson two parameter scaling approximation [2,141], which has been fairly successful. Other scaling approximations have been developed, e.g., the one by Lindquist and Simmons [142]. In the Curtis–Godson approximation the values of τ and β used in equation (10.75) or (10.77) (Elsasser model) and equations (10.79) plus (10.86) through (10.88) (statistical models) are replaced by path-averaged values * τ and * β. The proper values (scaling) for * τ and * β are found by satisfying both the optically thin and optically thick limits. Thus, we find from equations (10.62a) and (10.62b), for a single line “i”,  x 1:

X

Wi =

Si (X) dX,

(10.98)

0

  x1:

X

Wi = 2

Si (X) γLi (X) dX.

(10.99)

0

For many lines, from equation (10.80), x 1:

 X N  1  X W= Si (X) dX = S(X) dX. N 0 0

(10.100)

i=1

Now, from equation (10.78a) or (10.89), * τ=

W δ



X

= 0



S δ

 dX.

(10.101)

344 Radiative Heat Transfer

For strong lines we obtain N 2  W= N

x1:

 

X

Si (X) γLi (X) dX.

(10.102)

0

i=1

If one assumes Si and γLi to be separable, i.e., they can be written as, e.g., Si (X) = Si0 fs (X), where Si0 is a different constant for each line, and fs (X) is a function of the path (but the same for each line), one can—after some manipulation—rewrite equation (10.102) as [4] 

2 W = N 2

x1:

2 

X

0

⎡ N ⎤2 ⎢⎢ % ⎥⎥ ⎢⎢ Si (X) γLi (X) ⎥⎥⎥⎦ dX. ⎢⎣

(10.103)

i=1

Comparing with equation (10.62b) [or (10.90)], and utilizing equation (10.91) we obtain 

W

2

δ

4/π * 4 * τβ = 2 = Cγ δ

or 1 * β= * τ



X



S(X) γL (X) dX

(10.104)

0

S δ

0

X

β dX.

(10.105)

Equations (10.77) and (10.86) through (10.88) may now be used with* τ and * β to calculate narrow band emissivities for nonhomogeneous paths. The accuracy of various early scaling approximations was tested by Hartmann and coworkers [116,130] for various nonhomogeneous conditions in CO2 –N2 and H2 O–N2 mixtures. It was found that the Malkmus model together with the Curtis–Godson scaling approximation generally gave the most accurate results, except in the presence of strong (total) pressure gradients. More recently, André and Vaillon [143] used the k-moments expansion of second order, equation (10.93), to find the scaled * β as 1 1 = * τ β *



X 0

S 1 dX. δ β

(10.106)

Testing against LBL calculations of transmission through a nonhomogeneous layer of CO2 they found both methods equally accurate. Finally, a multiscale Malkmus model was developed by Bharadwaj and Modest [137] to improve its accuracy for nonhomogeneous paths. In this scheme it is assumed that high-temperature spectral lines (coming from elevated vibrational energy levels, i.e., with larger lower level energy El ) are uncorrelated from lower temperature lines. This implies that transmissivities of the individual “scales” are multiplicative [equation (10.97)]. Separating the gas accordingly into scales and applying equation (10.84) to each scale m as well as the Curtis–Godson approximation leads to ⎡ ⎤ 1/2  β˜m ⎢ ⎥⎥ 4˜τm ⎢⎢ − 1⎥⎥⎦ , = ⎢⎣ 1 + ˜ 2 βm δ m

W

(10.107)

with τ˜ m and β˜m from equations (10.101) and (10.105). Bharadwaj and Modest also outlined how scales are to be defined, whether using experimental data or data from spectroscopic databases. Testing the method with various nonhomogeneous CO2 –H2 O–N2 mixtures, they found the 2-scale Malkmus model to be a factor of 2 to 5 more accurate than the standard Curtis–Godson approach. Several other methods to treat nonhomogeneous paths can be found in [60,144,145].

10.9 Narrow Band k-Distributions As in the case of “traditional” narrow band models (i.e., Elsasser and statistical models), we will start by looking at a homogeneous medium (constant temperature, pressure, and concentrations), i.e., a medium whose

Radiative Properties of Molecular Gases Chapter | 10 345

FIGURE 10.16 Extraction of k-distributions from spectral absorption coefficient data: (a) simplified absorption coefficient across a small portion of the CO2 15 μm band (p = 1.0 bar, T = 296 K); (b) corresponding k-distribution f (k) and cumulative k-distribution k(g).

absorption coefficient is a function of wavenumber alone. In such a medium the spectral intensity depends on geometry, the Planck function, Ibη , emittance of bounding surfaces, η , the absorption and scattering coefficients of suspended particles, κpη and σsη , and finally the absorption coefficient of any absorbing gas. Over a small spectral interval, such as a few tens of wavenumbers, the Planck function and nongaseous radiation properties remain essentially constant. Thus, across such a small spectral interval the intensity varies with gas absorption coefficient alone. On the other hand, Fig. 10.13 shows that the gas absorption coefficient varies wildly even across a very narrow spectrum, attaining the same value for κη many times, each time producing the identical intensity field within the medium. Thus, carrying out line-by-line calculations across such a spectrum would be rather wasteful, repeating the same calculation again and again. It would, therefore, be advantageous to reorder the absorption coefficient field into a smooth, monotonically increasing function, assuring that each intensity field calculation is carried out only once. This reordering idea was first reported in the Western literature by Arking and Grossman [146], but they give credit to Kondratyev [147], who in turn credits a 1939 Russian paper. Other early publications on k-distributions are by Goody and coworkers [148], Lacis and Oinas [131], and Fu and Liou [149], all in the field of meteorology (atmospheric radiation). In the heat transfer area most of the work on k-distributions again is due to the group around Taine and Soufiani in France [41,150–152]. The narrow band average of any spectral quantity that depends only on the gaseous absorption coefficient, such as intensity Iη , transmissivity τη , etc., can be rewritten in terms of a k-distribution f (k) as follows (here expressed for transmissivity τη ):   ∞ 1 τη (X) = e−κη X dη = e−kX f (k) dk. (10.108) Δη Δη 0 The nature of k-distributions and how to evaluate them is best illustrated by looking at a very small part of the spectrum with very few lines. Figure 10.16a shows a fraction of the CO2 15 μm band at 1 bar and 296 K and, to minimize irregularity, with only the strongest 10 lines considered (two of them having their centers slightly outside the depicted spectral range). It is seen that the absorption coefficient goes through a number of minima and maxima; between any two of these the integral may be rewritten as     κη,max −κη X −κη X  dη  e dη = e   dκη . dκη  κη,min The absolute value sign comes from the fact that, where dκη /dη < 0, we have changed the direction of integration (always from κη,min to κη,max ). Therefore, integration over the entire range Δη gives f (k) as a weighted sum of the

346 Radiative Heat Transfer

number of points where κη = k,

  1   dη  f (k) =   . Δη dκη i

(10.109)

i

Mathematically, this can be put into a more elegant form as  1 δ(k − κη ) dη, f (k) = Δη Δη

(10.110)

where δ(k − κη ) is the Dirac-delta function defined by ⎧ ⎪ 0, |x| > δ , ⎪ ⎪ ⎨ δ(x) = lim ⎪ 1 ⎪ δ →0 ⎪ ⎩ , |x| < δ , 2δ

 ∞ δ(x) dx = 1.

or

(10.111a)

(10.111b)

−∞

The k-distribution of the absorption coefficient in Fig. 10.16a is shown as the thin solid line in Fig. 10.16b. Even for this minuscule fraction of the spectrum with only three dominant lines, f (k) shows very erratic behavior: wherever the absorption coefficient has a maximum or minimum f (k) → ∞ since |dκη /dη| = 0 at those points (6 in the present case); and wherever a semistrong line produces a wiggle in the absorption coefficient, f (k) has a strong maximum. Thankfully, the k-distribution itself is not needed during actual calculations. Introducing the cumulative k-distribution function g(k) as  k g(k) = f (k) dk, (10.112) 0

we may rewrite the transmissivity (or any other narrow band-averaged quantity) as 

∞

τη (X) =



1

e−kX f (k) dk =

0

e−k(g)X dg,

(10.113)

0

with k(g) being the inverse function of g(k), which is shown in Fig. 10.16b as the thick solid line. Sticking equation (10.112) into (10.110) leads to  g(k) = 0

k

1 f (k) dk = Δη

 Δη

 0

k



1 δ(k − κη ) dk dη = Δη

Δη

H(k − κη ) dη,

(10.114)

where H(k) is Heaviside’s unit step function, ⎧ ⎪ ⎪ ⎨0, H(x) = ⎪ ⎪ ⎩1,

x < 0, x > 0.

(10.115)

Thus, g(k) represents the fraction of the spectrum whose absorption coefficient lies below the value of k and, therefore, 0 ≤ g ≤ 1 [this can also be seen by setting X = 0 in equations (10.108) or (10.113), leading to τη = 1]. g acts as a nondimensional wavenumber (normalized by Δη), and the reordered absorption coefficient k(g) is a smooth, monotonically increasing function, with minimum and maximum values identical to those of κη (η). In actual reordering schemes values of k are grouped over small ranges k j ≤ k < k j + δk j = k j+1 , as depicted in Fig. 10.16, so that   1   δη  1  dg(k j ) = f (k j )δk j δηi (k j ), (10.116)   δk j = Δη δκη i Δη i

i

Radiative Properties of Molecular Gases Chapter | 10 347

FIGURE 10.17 CO2 k-distributions for the three cases depicted in Fig. 10.13.

where the summation over i collects all the occurrences where k j < κη < k j+1 , as also indicated in the figure. If the absorption coefficient is known from line-by-line data, the k-distribution is readily calculated from equation (10.116). The k-distributions for the three cases in Fig. 10.13 are shown in Fig. 10.17. Because of the many maxima and minima in the absorption coefficient these functions show very erratic behavior, as expected. Numerically, one can never obtain the singularities f (k) → ∞, and they appear as sharp peaks [strongly dependent on the spacing used for η and δk in equation (10.116)]. Inaccurate evaluation of f (k) (such as its peaks) has little influence on k(g), which is much easier to determine accurately. This, and the fact that g(k) represents the fraction of wavenumbers with kη ≤ k, suggests a very simple method to evaluate f (k)δk and g(k): the wavenumber range Δη is broken up into N intervals δη of equal width. The absorption coefficient at the center of each interval is evaluated and, if k j ≤ κη < k j+1 , the value of f (k j )δk j is incremented by 1/N. After all intervals have been tallied f (k j )δk j contains the fraction of wavenumbers with k j ≤ κη < k j+1 , and g(k j+1 ) =

j 

f (k j )δk j = g(k j ) + f (k j )δk j .

(10.117)

j =1

The k(g) for the three cases in Fig. 10.13 are shown in Fig. 10.18. Program nbkdistdb in Appendix F is a Fortran code that calculates such a g(k) distribution directly from a spectroscopic database, while nbkdistsg determines a single k-distribution from a given array of wavenumber–absorption coefficient pairs. As an example for the determination of k-distributions, the instructions to nbkdistdb show how to obtain the distributions of Figs. 10.17 and 10.18. An additional way of assembling cumulative k-distribution was recently described by André et al. [153], using a series expansion of moments of the log of the absorption coefficient. The k-distribution can be found more easily if accurate narrow band transmissivity data are available: inspection of equation (10.108) shows that τη is the Laplace transform of f (k), i.e., f (k) = L −1 {τη (X)},

(10.118)

where L −1 indicates inverse Laplace transform. This was first recognized by Domoto [154], who also found an analytical expression for the k-distribution based on the Malkmus model, equation (10.88): 1 f (k) = 2

&

   β κβ κ k exp , 2− − 4 k κ πk3

κ=

S δ

.

(10.119)

348 Radiative Heat Transfer

FIGURE 10.18 k-values as a function of cumulative k-distribution g for the three CO2 cases depicted in Fig. 10.13.

The cumulative k-distribution can also be determined analytically as & ⎞⎤ & ⎞⎤ ⎡ % ⎛& ⎡ % ⎛& ⎢⎢ β ⎜⎜ κ k ⎟⎟⎟⎥⎥⎥ 1 β k ⎟⎟⎟⎥⎥⎥ 1 ⎢⎢⎢ β ⎜⎜⎜ κ ⎜⎜ − + g(k) = erfc ⎢⎣ ⎟⎥ + e erfc ⎢⎢⎣ ⎟⎥ , ⎜ 2 2 ⎝ k 2 ⎝ k κ ⎠⎦ 2 κ ⎠⎦

(10.120)

where erfc is the complementary error function [124] and, by convention, erfc(−∞) = 2. Example 10.4. A certain diatomic gas is found to have an absorption coefficient that obeys Elsasser’s model across a narrow band of width Δη = 10 cm−1 . The gas conditions are such that mean absorption coefficient (S/δ) and overlap parameter β are known for the N = Δη/δ lines across the narrow band. Determine the narrow band k-distribution of the gas. Solution From equation (10.73) the absorption coefficient may be written as κη =

S δ

sinh 2β ,  η − ηc cosh 2β − cos 2β γ

ηl < η < ηl + Δη,

(10.121)

where ηl is the minimum wavenumber of the narrow band and ηc is the line center position of any one line in the band. Because of the periodic nature of an Elsasser band (see Fig. 10.14a), there will be exactly 2N wavelengths where kmin =

S sinh 2β S sinh 2β < k = κη < kmax = δ cosh 2β + 1 δ cosh 2β − 1

with identical |dκη /dη| each time. Therefore, from equation (10.109) or (10.110)       dη  2N  dη  1 f (k) = dκη = . δ(k−κη )     dκη k=κ Δη Δη  dκη k=κ η

But

η

      κ2η 2β 2β sinh 2β η − ηc η − ηc  dκη  S sin 2β = sin 2β ,   =    2 γ S dη δ γ γ γ η − ηc sinh 2β cosh 2β − cos 2β δ γ

and      η − ηc S sinh 2β sin 2β = = sin cos−1 cosh 2β − γ δ κη



 2 S sinh 2β 1 − cosh 2β − . δ κη

Radiative Properties of Molecular Gases Chapter | 10 349

Therefore,  S 2  sinh 2β . S sinh 2β 2 γ δ 1 − cosh 2β − f (k) = δ 2β k2 δ k S sinh 2β 1 δ = . & 2  π S 2 k k − k cosh 2β − sinh 2β δ Integrating f (k) according to equation (10.113) we obtain (using integration tables),   S sinh 2β 1 g(k) = 1 − cos−1 cosh 2β − π δ k or, after inversion, k=

sinh 2β S . δ cosh 2β − cos π(1 − g)

(10.122)

This is, of course, just equation (10.121) with 2β(η − ηc )/γ replaced by π(1 − g): the k-distribution recognizes that, in the Elsasser scheme, the same structure is repeated 2N times (of that N times as a mirror image), and a single half-period is stretched across the entire reordered range 0 ≤ g ≤ 1. The present k-distribution can also be obtained by precalculating an array of absorption coefficients across Δη from equation (10.121) and using subroutine nbkdistsg in Appendix F.

Comparing equation (10.113) with the first expression in equation (10.108), we note that the integration in equation (10.113) is equivalent in difficulty to the integration over half of a single line. Given that a narrow spectral range can contain thousands of little overlapping lines, we conclude that the CPU time savings over line-by-line calculations can be enormous! However, the generation of the necessary k-distributions from the large number of spectral lines contained in the various spectroscopic databases is tedious and time consuming. A first database of narrow band k-distributions for CO2 and H2 O was offered by Soufiani and Taine [41] as part of their EM2C narrow band database. It contains k-distribution data for fairly wide spectral intervals (larger than 100 cm−1 ; 17 bands for CO2 and 44 for H2 O), and are valid for atmospheric pressure and temperatures up to 2500 K. Each k-distribution is defined by 7 k-values, to be used with a 7-point Gaussian quadrature for spectral integration. Like their Malkmus parameter counterparts they are generated from the HITRAN 1992 database plus proprietary extensions (cf. p. 342). A more accurate, highly compact database, also for CO2 and H2 O, was generated by Wang and Modest [155] based on CDSD-1000 [85] (for CO2 ) and HITEMP 1995 [80] (for H2 O), valid for total pressures between 0.1 bar and 30 bar, and temperatures between 300 K and 2500 K. The spectrum is divided into 248 narrow bands for all gases (allowing the determination of mixture k-distributions from those of individual species). Nested Gauss–Chebychev quadrature with up to 128 quadrature points is used to guarantee 0.5% accuracy for all absorption coefficient and emissivity calculations, and to allow for variable order spectral quadrature. Both of these databases have since been updated. Rivière and Soufiani [132] employed HITEMP 2010 [59] (for H2 O) (51 bands) and CDSD-4000 [87] for CO2 (40 bands) assembled from their narrow band database described in Section 10.8. The Wang and Modest Narrow Band K-Distribution for InfraRed (NBKDIR) database has been augmented by Cai and Modest [156] to include additional species (CO, CH4 , and C2 H4 ), and is continuously updated to incorporate the newest spectroscopic data; at the time of print all k-distributions have been obtained from HITEMP 2010 [59] (H2 O, CO2 , and CO) and HITRAN 2008 [32], (CH4 and C2 H4 ). Again, a database similar to the one of Rivière and Soufiani, but specifically devoted to nine common fuels, was generated by Consalvi and Liu [133]. Both EM2C and NBKDIR are included in Appendix F.

Gas Mixtures The k-distributions for mixtures can, in principle, be calculated directly, simply by adding the linear, spectral absorption coefficients of all components in the mixture before applying the reordering process, equation (10.116). Since assembling k-distributions is a tedious, time-consuming affair, it is desirable to obtain them from databases. However, determining an exact k-distribution for a mixture from those of individual species is in general impossible, because k-distributions never retain any information pertaining to the spectral location of individual

350 Radiative Heat Transfer

absorption lines. Only in two simple situations is exact manipulation of k-distributions feasible: (1) a gas “mixing” with itself, i.e., changing the concentration of the absorbing gas species and (2) adding a gray (across the given narrow band) material to the nongray absorbing gas. Variable Mole Fraction of a Single Absorbing Gas Consider a gas whose absorption coefficient is linearly dependent on its partial pressure, i.e., a gas whose line broadening is unaffected by its own partial pressure. This is always true for molecules that have the same size as the surrounding broadening gas (such as CO2 in air), and for all gases whenever Doppler broadening dominates. Then κxη (T, p, x; η) = xκη (T, p; η),

(10.123)

where κη is the absorption coefficient of the pure gas and x is its mole fraction in a mixture. Comparing the two k-distributions  1 f (T, p; k) = δ(k − κη ) dη, (10.124) Δη Δη  1 fx (T, p, x; kx = xk) = δ(kx − κxη ) dη, (10.125) Δη Δη we see that they both are populated by exactly the same spectral locations (i.e., kx = κxη wherever k = κη ), so that fx (T, p, x; kx ) d(xk) = f (T, p; k) dk or fx (T, p, x; kx ) =

1 f (T, p; kx /x). x

(10.126)

fx (T, p, x; kx ) dkx = gx (T, p, x; kx ),

(10.127)

Integrating equation (10.126) leads to 



k

f (T, p; k) dk =

g(T, p; k) =

kx

0

0

i.e., the k vs. g behavior is independent of mole fraction. In a k vs. g plot the lines are simply vertically displaced by a multiplicative factor of x, or kx (g) = xk(g),

(10.128)

as demonstrated in Fig. 10.19 for a k-distribution based on the Malkmus model, equation (10.120) (using an unrealistically large overlap parameter of β = 10 for better visibility). Single Absorbing Gas Mixed with Gray Medium Consider a gas that is mixed with a gray medium (say, particles), with constant absorption coefficient κp . Then κpη (T, p, κp ; η) = κη (T, p, η) + κp .

(10.129)

Proceeding as in the previous paragraph we obtain   1 1 fp (T, p, κp ; kp ) = δ(kp − [κxη + κp ]) dη = δ([kp − κp ] − κxη ) dη Δη Δη Δη Δη = f (T, p; k = kp − κp )

(10.130)

and g(T, p; k) = gp (T, p, κp ; kp = k + κp ),

(10.131)

i.e., the k vs. g behavior is also independent of any gray additions. In a k vs. g plot the lines are simply vertically displaced by a constant amount of κp , kp (g) = κp + k(g), as also shown in Fig. 10.19.

(10.132)

Radiative Properties of Molecular Gases Chapter | 10 351

FIGURE 10.19 Scalability of narrow band k-distributions: k: pure gas; kx : gas with mole fraction x = 0.5; kp : gas mixed with gray medium of κp = 2 cm−1 .

Multispecies Mixtures Several approximate mixing models for k-distributions have been proposed that rely on assumptions about the statistical relationships between the absorption lines of the individual species, mostly by Solovjov and Webb [157] (full spectrum models only), such as their convolution, superposition, multiplication, and hybrid approaches, and by Modest and Riazzi [158], exploiting the uncorrelatedness between species. All of these approaches produce a single mixture k-distribution, but rely on different assumptions and methodologies to achieve their goal. It was found that the approach of Modest and Riazzi results in negligible errors for all conditions tested (low to moderate pressures). Pal and Modest [159] found that their methodology works equally well at very high pressures (up to 30 bar), even though broadened spectral lines overlap much more strongly. Consequently, we will present here only the Modest and Riazzi mixing scheme. Earlier it was shown how the idea of uncorrelated absorption coefficients can be used to obtain the transmissivity of a mixture, as given by equation (10.97). Through simple mathematical manipulation, it is possible to extend this logic to the mixing of cumulative k-distributions. We begin by recalling that the definition of the transmissivity, in terms of the k-distribution for a single absorbing species, is also the definition of the Laplace transform of f (k) [154], equation (10.118). Using this and the product of transmissivities model, the transmissivity of a mixture of M species may be expressed as the product of the Laplace transforms of the component k-distributions, or τη,mix = L [ fmix (k)] =

M :

M :

τη,m =

m=1

L [ fm (k)].

(10.133)

i=m

In terms of the cumulative k-distributions, the transmissivity of an individual component is given by  τm =

1

e−km L dgm ,

(10.134)

0

and for a binary mixture this becomes 

1

τmix = L [ fmix (k)] =

 −k1 L

e 0

dg1

1

 −k2 L

e

dg2 =

0

1



g1 =0

1 g2 =0

e−[k1 (g1 )+k2 (g2 )]L dg2 dg1 .

(10.135)

Using the integral property of the Laplace transform we obtain  L 0

k

  fmix (k) = L [gmix (k)] =

1 g1 =0



1 g2 =0

 −[k1 (g1 )+k2 (g2 )]L

e

dg2 dg1

1 = L



1 g1 =0



1 g2 =0

e−[k1 (g1 )+k2 (g2 )]L dg2 dg1 , (10.136) L

352 Radiative Heat Transfer

FIGURE 10.20 Narrow band transmissivity of a CO2 –H2 O mixture from individual species k-distributions, equation (10.137).

or, when the inverse transform is taken, with H being the Heaviside step function,  gmix (kmix ) =



1 g1 =0



1 g2 =0

H[kmix − (k1 + k2 )]dg2 dg1 =

1 g1 =0

g2 (kmix − k1 ) dg1 .

(10.137)

In the second, once integrated expression, it is assumed that gm (k < km,min ) = 0 (i.e., all absorption coefficients are above km,min ) and gm (k > km,max ) = 1 (i.e., all absorption coefficients are below km,max ). This relation may also be readily extended to a mixture of M species,  gmix (kmix ) =



1 g1 =0

....

1 gM =0

H[kmix − (k1 + .... + kM )]dgM ....dg1 .

(10.138)

This integral may be evaluated by multiple Gaussian quadrature, leading to a single mixture k-distribution at specific k-values while using the component k-distributions stored at quadrature points with their associated weights. The k-values for this new mixture distribution must be predetermined and chosen such that they cover the entire range of values of all component species. Carrying out mixing with this model consistently outperforms the models of Solovjov and Webb [157] (by a factor of 10 or more). Its accuracy is demonstrated in Fig. 10.20 for a mixture of water vapor and carbon dioxide in the 2.7 μm region (where both gases heavily overlap), with absolute errors mostly below 0.005 (roughly the same as obtained by direct multiplication of transmissivities). The mixing scheme described here is incorporated into the NBKDIR database in Appendix F, i.e., NBKDIR allows for the retrieval of mixture k-distributions. Example 10.5. Consider a mixture of two diatomic gases, both having absorption coefficients, κ1η and κ2η , that obey Elsasser’s model across a narrow band of width Δη = 10 cm−1 . The following is known for the two gases: γ1 S1 π = 0.314; = 1 cm−1 ; β1 = π = δ1 δ1 10 γ2 S2 7π = 0.110. = Gas 2 : δ2 = 0.1429 cm−1 , γ2 = 0.0050 cm−1 , = 2 cm−1 ; β2 = π δ2 δ2 200

Gas 1 : δ1 = 0.2500 cm−1 , γ1 = 0.0250 cm−1 ,

Determine the narrow band k-distribution for this mixture. Solution The individual k-distributions for the two component gases are given from the previous example as ki =

  sinh 2βi S ; δ i cosh 2βi − cos π(1 − gi )

i = 1, 2.

(10.139)

Radiative Properties of Molecular Gases Chapter | 10 353

FIGURE 10.21 Narrow band k-distribution for a two-component mixture (Example 10.5): (a) absorption coefficients, (b) k-distributions.

The k-distribution of the mixture is immediately found from the rightmost expression in equation (10.137) as  gmix (kmix ) =

1 g1 =0

g2 (kmix − k1 ) dg1 ,

kmin = k1 min + k2 min ≤ kmix ≤ kmax = k1 max + k2 max ,

(10.140)

where k1 is obtained from equation (10.139), while g2 is found from its inverse, or ⎧ ⎪ ⎪ 0, k < k2 min , ⎪ ⎪ ⎪  ⎪   sinh 2β  ⎪ ⎨ S 1 2 g2 (k) = ⎪ 1 − cos−1 cosh 2β2 − , k2 min < k < k2 max , ⎪ ⎪ π δ 2 k ⎪ ⎪ ⎪ ⎪ ⎩1, k > k2 max . The integration in equation (10.140) is best carried out numerically. Here care must be taken that the argument of cos−1 does not fall outside its allowable range (between −1 and +1). The same holds true for the mixing of any two k-distributions, i.e., g2 ≡ 0 for kmix − k1 ≤ k2 min , and g2 ≡ 1 for kmix − k1 ≥ k2 max . The result of a simple trapezoidal rule integration is shown in Fig. 10.21. Frame (a) shows the absorption coefficients for the mixture and the two component gases, and Frame (b) the corresponding k-distributions. The mixture k-distribution is calculated in two ways: “exactly,” using the absorption coefficient in Fig. 10.21a, or equation (10.121) (with random and different η1 for each gas), and with the Modest&Riazzi scheme, equation (10.140). Semilog plots are employed to better separate the various absorption coefficients and k-distributions. It is apparent that both mixture k-distributions virtually coincide (in fact, transmissivities calculated with both k-distributions coincide to within 5 digits).

Nonhomogeneous Gases Correlated-k Like the statistical models the k-distribution is not straightforward to apply to nonhomogeneous paths. However, it was found that for many important situations the k-distributions are essentially “correlated,” i.e., if k-distributions k(g) are known at two locations in a nonhomogeneous medium, then the absorption coefficient can essentially be mapped from one location to the other (documented to some extent by Lacis and Oinas [131]). This implies that all the values of η that correspond to one value of κ and g at one location, more or less map to the same value of g (but a different κ) at another location [131,149]: pressure changes affect all lines equally (causing more or less broadening by higher/lower total pressure p, increasing line strengths uniformly by changes in partial pressure of the absorbing gas, pa ). We may then write, with good accuracy, 1 τη (0 → X) = Δη

 Δη

⎛ X ⎞  ⎜⎜  ⎟⎟ ⎜ ⎟ exp ⎜⎝− κη dX⎟⎠ dη 0

1 0

⎛ X ⎞ ⎜⎜  ⎟⎟ ⎜ exp ⎜⎝− k(X, g) dX⎟⎟⎠ dg. 0

(10.141)

354 Radiative Heat Transfer

This assumption of a correlated k-distribution has proven very successful in the atmospheric sciences, where temperatures change only from about 200 K to 320 K, but pressure changes can be very substantial [131,148,149]. Scaled-k A more restrictive, but mathematically precise condition for correlation of k-distributions is to assume the dependence on wavenumber and location in the absorption coefficient to be separable, i.e., κη (η, T, p, pa ) = kη (η)u(T, p, pa ),

(10.142)

where kη (η) is the absorption coefficient at some reference condition and u(T, p, pa ) is a nondimensional function depending on local conditions of the gas, but not on wavenumber. This is commonly known as the scaling approximation. Substituting this into equation (10.141) gives ⎛ ⎞     X ⎜⎜ ⎟⎟ 1 1 ⎜ ⎟ τη (0 → X) = exp ⎜⎝−kη (η) u dX⎟⎠ dη = exp −kη X dη , (10.143) Δη Δη Δη Δη 0 where X is now a path-integrated value for X. Comparing with equation (10.108), we find that in this case there is only a single k-distribution, based on the reference absorption coefficient kη , and  1  X −k(g)X τη (0 → X) = e dg; X= u dX. (10.144) 0

0

As for homogeneous media equations (10.141) and (10.144) provide reordered absorption coefficients, which can be used in arbitrary radiation solvers without restrictions. At first glance, equation (10.141) looks superior to equation (10.144), since the assumption of a scaled absorption coefficient is more restrictive. However, in practice one needs to approximate an actual absorption coefficient, which is neither scaled nor correlated: if the scaling method is employed, the scaling function u(T, p, pa ) and its reference state for kη can be freely chosen and, thus, optimized for a problem at hand. On the other hand, if the correlated-k method is used, the absorption coefficient is simply assumed to be correlated (even though it is not), and the inherent error cannot be minimized. Following Modest and Zhang [160] and assuming constant total pressure, reference state temperature T0 and partial pressure pa0 may be chosen from  1 pa dV, (10.145) pa0 = V V  1 κη (T0 , x0 )Ibη (T0 ) = κη (T, x)Ibη (T) dV, (10.146) V V  where κη = Δη κη dη/Δη is the average absorption coefficient, i.e., volume-averaged partial pressure and a mean temperature based on average emission from the volume. For the scaling function Modest and Zhang suggest equating exact and approximate radiation leaving from a homogeneous slab of the length under consideration, or  1  1     exp −k(T, pa , g)L dg = exp −k(T0 , pa0 , g)u(T, p, pa )L dg. (10.147) 0

0

Correlated-k and scaled-k are about equally efficient numerically: both require evaluation of the local kdistribution k(T, pa , g) everywhere along the path. As an illustration a simple (yet severe) example is shown in Fig. 10.22, showing transmissivity through, and emissivity from, a slab of hot gas at 1000 K adjacent to a cold slab at 300 K. Both layers are at the same total and partial pressures, and are of equal width [161]. The transmissivity for a blackbody beam Ibη (Th = 1000 K), through such a double layer is, from Chapter 9,  Iη (L) tr 1 = τη = exp[−κη (Th , x)Lh − κη (Tc , x)Lc ] dη, (10.148) Ibη (Th ) Δη Δη while the emissivity is defined here as the intensity of emitted radiation exiting the cold layer, as compared to the Planck function of the hot layer. Employing equation (9.29) this is readily evaluated as     Ibη (Tc )  Iη (L) em 1 −κη (Tc ,x)Lc −κη (Tc ,x)Lc −κη (Th ,x)Lh −κη (Tc ,x)Lc =

η = −e + dη. (10.149) e 1−e Ibη (Th ) Δη Δη Ibη (Th )

Radiative Properties of Molecular Gases Chapter | 10 355

FIGURE 10.22 Narrow band transmissivities and emissivities for two-temperature slab, as calculated by the LBL, scaled-k, and correlated-k methods: (a) 2.7 μm band of CO2 with pCO2 = 0.1 bar, (b) 6.3 μm band of H2 O with pH2 O = 0.2 bar.

Note that, while transmissivities are more regularly shown in the narrow band literature, the emissivity is generally more descriptive of heat transfer problems. Figure 10.22a shows these narrow band transmissivities and emissivities for the 2.7 μm band of CO2 for a partial pressure of pCO2 = 0.1 bar, as calculated by the LBL, scaled-k, and correlated-k methods, using the original HITEMP 1995 database [80], and all for a resolution of Δη = 5 cm−1 (lines) and 25 cm−1 (symbols). Both correlated and scaled k-distributions predict transmissivity very accurately with the exception of small discrepancies near the minima at 3600 cm−1 and 3700 cm−1 . Similar errors also show up in the emissivity, but are somewhat amplified. This amplification was observed for all bands studied (i.e., the effect is not limited to regions of small emissivities, as in this figure). For both, transmissivity and emissivity, results from the two k-distributions are virtually identical, although correlated-k performs slightly better for the 2.7 μm band (in the case of the 4.3 μm band, not shown, roles are reversed and scaled-k slightly outperforms correlated-k). Figure 10.22b shows transmissivities and emissivities for the wide 6.3 μm water vapor band. Conditions are the same as for Fig. 10.22a, except that pH2 O = 0.2 bar and only a Δη = 25 cm−1 resolution is shown (a resolution of 5 cm−1 results in a very irregular shape which, while the k-distributions follow this behavior accurately, makes them difficult to compare). Again, both k-distributions predict transmissivities rather accurately, and the slight errors are somewhat amplified in the emissivities. And, again, both k-distributions give virtually the same results, with scaled-k being a little more accurate for this band. In summary, one may say that both models perform about equally well; this implies that—for narrow bands and for temperatures not exceeding 1000 K—the absorption coefficients for water vapor and carbon dioxide are relatively well correlated. Note also that the present case, with a sharp step in temperature, is rather extreme; accuracy can be expected to be significantly better in more realistic combustion systems. Unfortunately, for nonhomogeneous media with even more extreme temperature gradients the correlation between k-distributions at different temperatures breaks down. The reason for this is that different lines can have vastly different temperature dependence through the exponential term in equations (10.33): at low temperatures lines near the band center are strongest (with largest κη ), while at high temperatures lines away from the band center exhibit the largest κη . Since the correlated k-distribution pairs values of equal absorption coefficients, this results in pairing wrong spectral values in hot and cold regions. This is not only true for wide spectral ranges, but also on a narrow band level, since a vibration–rotation band consists of many slightly displaced subbands, generated by different levels of vibrational energies (different Bv ), some of which undergo transitions only at elevated temperatures [large values for El in equation (10.33)], known as “hot lines.” For more detail the reader may want to consult the monograph by Taine and Soufiani [4]. The lack of correlation in nonisothermal media was first recognized by Rivière and coworkers [150–152], who devised the so-called “fictitious gas technique”: starting with a high-resolution database, they grouped lines according to the values of their lower energy levels, E j = hcBv j(j + 1) (i.e., according to their temperature dependence), found the k-distribution for each of the fictitious gases and, in a further approximation, estimated the gas transmissivity as the product of the

356 Radiative Heat Transfer

transmissivities of the fictitious gases, τη =

ng  : i=1

0

1

⎡ X ⎤ ⎢⎢  ⎥⎥ exp ⎢⎣⎢− ki (g, X) dX⎥⎥⎦ dg,

(10.150)

0

where n g is the number of fictitious gases. A very similar approach was taken by Bharadwaj and Modest [137], employing the fictitious gas approach applied to k-distributions obtained from the Malkmus model. Unfortunately, these methods can only supply the mean transmissivity for a gas layer, i.e., they lose all the advantages of the k-distributions, and are limited in their application in the same way as the statistical narrow band models.

Comparison of k-Distributions and Statistical Models The k-distribution method has a number of important advantages over the statistical narrow band models, although the statistical models, in particular the Malkmus model combined with the Curtis–Godson scaling approximation, outperform k-distributions in a couple of respects: 1. Perhaps the greatest advantage that k-distributions have is that they formulate radiative properties in terms of a (reordered) absorption coefficient. This implies that radiative heat transfer rates may be calculated using any desired solution method for the radiative transfer equation. If based on exact line-by-line property data, the method is essentially exact (for a homogeneous medium). Statistical narrow band models, on the other hand, calculate gas column transmissivity, and heat transfer rates can only be determined in terms of these transmissivities. 2. Statistical narrow band models are, due to the transmissivity approach, limited to application in black enclosures without scattering (unless a ray tracing approach is used to solve the RTE, such as the Monte Carlo method described in Chapter 20). No such restriction is necessary for k-distributions (as long as wall reflectance and scattering properties remain constant across the narrow band). 3. The k-distribution method is valid for spectral lines of any shape; statistical narrow band models, on the other hand, are generally limited to Lorentz lines (although some formulations for Doppler and Voigt profiles exist). This is not unimportant, since in combustion applications the lines often have Voigt profiles as seen from Fig. 10.9. 4. Statistical narrow band models return an explicit expression for averaged transmissivity, while the kdistribution requires integration (quadrature) over the (reordered) narrow spectrum. On the other hand, the narrow band is limited to several tens of wavenumbers for statistical models (to avoid significant changes in statistical parameters, such as S and δ), but can span several hundreds of wavenumbers for k-distributions (only limited by changes in Planck function and, if present, spectral variations of wall emittances and scattering properties). 5. Neither method treats nonhomogeneous paths to complete satisfaction. In fields with moderate temperature gradients and moderate-to-strong pressure variations the correlated-k approach performs extremely well, while the Curtis–Godson approximation loses accuracy in the presence of strong pressure variations. On the other hand, in fields with extreme temperature fields all methods have some problems; under such conditions only the correlated-k, fictitious-gas approach performs well. However, the fictitious-gas approach calculates gas layer transmissivities only, i.e., it is under the same limitations as the statistical methods.

10.10 Wide Band Models The heat transfer engineer is usually only interested in obtaining heat fluxes or divergences of heat fluxes integrated over the entire spectrum. Therefore, it is desirable to have models that can more readily predict the total absorption or emission from an entire band as was done in Example 10.3. These models are known as wide band models since they treat the spectral range of the entire band. It is theoretically possible to use quantum mechanical relations, such as equations (10.34), to accurately predict the radiative behavior of entire bands. This has been attempted by Greif and coworkers [162,163] in a series of

Radiative Properties of Molecular Gases Chapter | 10 357

FIGURE 10.23 The box model for the approximation of total band absorptance.

papers. While such calculations are more accurate, they tend to be too involved, so simpler methods are sought for practical applications. In the following we will give an account of the two most important wide band models, namely, the box model and the exponential wide band model, which enjoyed considerable popularity for a time, especially the latter. However, with the advent of k-distributions and spectroscopic databases their use has plummeted, with very little modern research making use of them.

The Box Model In this very simple model the band is approximated by a rectangular box of width Δηe (the effective band width) and height κ as shown in Fig. 10.23. With these assumptions we can calculate the total band absorptance for a homogeneous gas layer as   ∞    A≡

η dη = 1 − e−κη X dη = Δηe 1 − e−κX , (10.151) band

0

where both Δηe and κ may be functions of temperature and pressure. The box model was developed by Penner [20] and successfully applied to diatomic gases. However, the determination of the effective band width is something of a “black art.” Once Δηe has been found (by using the somewhat arbitrary criterion given by Penner [20] or some other means), κ may be related to the band intensity α, defined as  α≡

∞

 κη dη =

0

0

∞



S δ

 dη,

(10.152)

η

leading to κ = α/Δηe .

(10.153)

If the molecular gas layer forms a radiation barrier between two surfaces of unequal temperature, then a suitable choice for the effective band width can give quite reasonable results. However, if emission from a hot gas is considered, then the results become very sensitive to the correct choice of Δηe . Nevertheless, the box model— because of its great simplicity—enjoys considerable popularity for use in heat transfer models (see Chapter 19). Example 10.6. Calculate the effective band width Δηe for which the box model predicts the correct total band absorptance for Example 10.3. Solution Integrating equation (10.92) over the entire band gives α = (S/δ) 0 × ω = 500 cm−2 and κX = αX/Δηe = 10,000 cm−1 /Δηe . Equation (10.151) then, with A = 264.7 cm−1 , results in Δηe = 264.7 cm−1 by trial and error. Δηe is seen to be substantially larger than ω and essentially equal to A, because the band in this example is optically very thick. Even in the band wings far away from the band center the band is optically opaque (τ  1). This result must be accounted for in the choice of Δηe . For optically thick gases finding the correct Δηe is equivalent to finding A itself. Drawing a box seemingly best approximating the actual band shape can lead to large errors!

358 Radiative Heat Transfer

FIGURE 10.24 Band shapes for exponential wide band model.

The Exponential Wide Band Model The exponential wide band model, first developed by Edwards and Menard [164], is by far the most successful of the wide band models. The original model has been further developed in a series of papers by Edwards and coworkers [165–168]. The word “successful” here implies that the model is able to correlate experimental data for band absorptances with an average error of approximately ±20% (but with maximum errors as high as 50% to 80%). We present here the latest version of Edwards, together with its terminology (based on Goody’s narrow band model), followed by a short discussion of newer models by Felske and Tien [169] (Goody’s model) and Wang [170] (Malkmus’ model). For a more exhaustive discussion on Edwards’ model the reader may want to consult Edwards’ monograph on gas radiation [1]. Since it is known from quantum mechanics that the line strength decreases exponentially in the band wings far away from the band center,10 Edwards assumed that the smoothed absorption coefficient S/δ has one of the following three shapes, as shown in Fig. 10.24: with upper limit head symmetric band with lower limit head

S δ S δ S δ

=

α −(ηu −η)/ω e , ω

(10.154a)

=

α −2|ηc −η|/ω e , ω

(10.154b)

=

α −(η−ηl )/ω e , ω

(10.154c)

where α is the integrated absorption coefficient or the band strength parameter (or area under the curves in Fig. 10.24), which was defined in equation (10.152), and ω is the band width parameter,11 giving the width of the band at 1/e of maximum intensity. The band can be expected to be fairly symmetric if, during rotational energy changes, the B does not change too much [recall equations (10.28a) through (10.28c)]. ηc is then the wavenumber connected with the vibrational transition. On the other hand, if the change in B is substantial, then either the Ror the P-branch may fold back, leading to bands with upper or lower head. Thus, the wavenumbers ηu and ηl are the wavenumbers where this folding back occurs, and not the band center. The sharp exponential apex is, of course, not very realistic. The rationale is that, if the band center is optically thick, then it is opaque no matter what the shape, while if it is thin, then only the total α is of importance. Edwards and Menard [164] proceeded to evaluate the band absorptance using the general statistical model by substituting expressions (10.154) into equation (10.87) and carrying out the integration in an approximate fashion. Since equation (10.87) contains the line overlap parameter β and the optical thickness τ, the authors were able to describe the total band absorptance 10. This fact is easily seen by letting j  1 in equations (10.28a) and (10.34a) for the P-branch, and in equations (10.28c) and (10.34b) for the R-branch. 11. The band width parameter ω, as used here, applies only to the wide band correlation. If equations (10.154) are used for spectral (i.e., narrow band) calculations, Edwards [1] suggests increasing the value of ω by 20% for better agreement between wide band model and band-integrated narrow band model calculations.

Radiative Properties of Molecular Gases Chapter | 10 359

TABLE 10.3 Exponential wide band correlation for an isothermal gas. β≤1

β≥1

β ≤ τ0 ≤ 1/β

A∗ = τ0 % A∗ = 2 τ0 β − β

Square root regime

1/β ≤ τ0 < ∞

A∗ = ln(τ0 β) + 2 − β

Logarithmic regime

0 ≤ τ0 ≤ 1

A∗

1 ≤ τ0 < ∞

A∗ = ln τ0 + 1

0 ≤ τ0 ≤ β

Linear regime

= τ0

Linear regime Logarithmic regime

α, β, and ω from Table 10.4 and equations (10.156) through (10.159), τ0 = αX/ω.

as a function of three parameters, namely, A∗ = A/ω = A∗ (α, β, τ0 ),

(10.155)

where τ0 is the optical thickness at the band center (symmetric band) or the band head. Their results are summarized in Table 10.3.12 Example 10.7. Determine the total band absorptance of the previous two examples by the exponential wide band model. Solution From Example 10.3 we have τ0 = 200 and β = π/10. Thus, since τ0 > 1/β, we find from Table 10.3 A∗ = ln(τ0 β) + 2 − β = ln(200 × π/10) + 2 − π/10 = 5.826 and A = A∗ ω = 5.826 × 50 = 291.3 cm−1 . The difference between the two results is primarily due to the fact that in Example 10.3 we treated the optically thin band wings as optically thick.

The parameters α, β, and ω are functions of temperature and must be determined experimentally. Values for the most important combustion gases—H2 O, CO2 , CO, CH4 , NO, and SO2 —for a reference temperature of T0 = 100 K are given in Table 10.4. Most of these correlation data are based on work by Edwards and coworkers and are summarized in [1]. Data for the purely rotational band of H2 O have been taken from the more modern work of Modak [171]. Values for other bands and other gases may be found in the literature, e.g., for H2 O, CO2 , and CH4 [1,165,168,172–175], for CO [1,165,168,176–178], for SO2 [1,168,179], for NH3 [180], for NO [181], for N2 O [182], and for C2 H2 [183] (in the older of these references the parameters for the slightly different original model are given; in a number of papers a pressure path length has been used instead of a density path length). The temperature dependence of the band correlation parameters for vibration–rotation bands is given by Edwards [1] as α(T) = α0

Ψ(T) , Ψ(T0 ) &

β(T) = β∗ Pe = β∗0 & T ω(T) = ω0 , T0 and

(10.156) T0 Φ(T) Pe , T Φ(T0 )

(10.157) (10.158)



 n p pa Pe = , 1 + (b − 1) p0 p

(p0 = 1 atm, T0 = 100 K),

(10.159)

where m ∞ :  (vk + gk + |δk | − 1)! −u (T)v e k k ⎧ ⎛ m ⎞⎫ (gk − 1)! vk ! ⎪ ⎪  ⎜ ⎟ ⎪ ⎪ v =v k=1 ⎜ ⎟ k 0,k ⎬ ⎨ 1 − exp ⎜⎜⎜⎝− Ψ(T) = ⎪ uk (T)δk ⎟⎟⎟⎠⎪ , ⎪ ⎪ m  ∞ : ⎭ ⎩ (vk + gk − 1)! −u (T)v k=1 e k k (gk − 1)! vk ! v =0 k=1

(10.160)

k

% 12. In the original version the parameters C1 = α, C3 = ω, and C2 = 4C1 C3 β∗ were used, where β∗ is the value of β for a gas mixture at a total pressure of 1 atm with zero partial pressure of the absorbing gas. Also, limits between regimes were slightly different, using A itself rather than τ0 .

360 Radiative Heat Transfer

TABLE 10.4 Wide band model correlation parameters for various gases. Band Location λ [μm]

Vibr. Quantum Step

ηc [cm−1 ]

Pressure Parameters n

(δk )

m = 3, η1 = 3652 cm−1 , η2 = 1595 cm−1 , η3 = 3756 cm−1 , gk = (1, 1, 1)

H2 O

b

71 μma

ηc = 140 cm−1

(0, 0, 0)

1

6.3 μm

ηc = 1600 cm−1

(0, 1, 0)

1

( T 8.6 T0 + 0.5 ( T 8.6 T0 + 0.5

2.7 μm

ηc =

3760 cm−1

1

( T 8.6 T0 + 0.5

1.87 μm

ηc =

5350 cm−1

(0, 1, 1)

1

1.38 μm

ηc = 7250 cm−1

(1, 0, 1)

1

(0, 2, 0) (1, 0, 0) (0, 0, 1)

Correlation Parameters α0 [cm−1 /(g/m2 )]

β∗0

ω0

5.455

0.143

69.3

41.2

0.094

56.4

0.132b,c

60.0b

3.0

0.082

43.1

2.5

0.116

32.0

0.2 2.3 22.4

( T 8.6 T0 + 0.5 ( T 8.6 T0 + 0.5

m = 3, η1 = 1351 cm−1 , η2 = 666 cm−1 , η3 = 2396 cm−1 , gk = (1, 2, 1)

CO2 15 μm

ηc = 667 cm−1

(0, 1, 0)

0.7

1.3

19.0

0.062

12.7

10.4 μmd

ηc = 960 cm−1

(−1, 0, 1)

0.8

1.3

2.47×10−9

0.040

13.4

9.4 μmd

ηc = 1060 cm−1

(0, −2, 1)

0.8

1.3

2.48×10−9

0.119

10.1

4.3 μm

ηu = 2410 cm−1

(0, 0, 1)

0.8

1.3

110.0

0.247

11.2

2.7 μm

ηc = 3660 cm−1

(1, 0, 1)

0.65

1.3

4.0

0.133

23.5

2.0 μm

ηc = 5200 cm−1

(2, 0, 1)

0.65

1.3

0.060

0.393

34.5

m = 1, η1 = 2143 cm−1 , g1 = 1

CO

4.7 μm

ηc = 2143 cm−1

(1)

0.8

1.1

20.9

0.075

25.5

2.35 μm

ηc = 4260 cm−1

(2)

0.8

1.0

0.14

0.168

20.0

m = 4, η1 = 2914 cm−1 , η2 = 1526 cm−1 , η3 = 3020 cm−1 , gk = (1, 2, 3, 3)

CH4 7.7 μm

ηc = 1310 cm−1

(0, 0, 0, 1)

0.8

1.3

28.0

0.087

21.0

3.3 μm

ηc =

3020 cm−1

(0, 0, 1, 0)

0.8

1.3

46.0

0.070

56.0

2.4 μm

ηc = 4220 cm−1

(1, 0, 0, 1)

0.8

1.3

2.9

0.354

60.0

1.7 μm

ηc = 5861 cm−1

(1, 1, 0, 1)

0.8

1.3

0.42

0.686

45.0

(1)

0.65

1.0

9.0

0.181

20.0

m = 1, η1 = 1876 cm−1 , g1 = 1

NO

ηc = 1876 cm−1

5.3 μm

m = 3, η1 = 1151 cm−1 , η2 = 519 cm−1 , η3 = 1361 cm−1 , gk = (1, 1, 1)

SO2 19.3 μm

ηc = 519 cm−1

(0, 1, 0)

0.7

1.28

4.22

0.053

33.1

8.7 μm

ηc = 1151 cm−1

(1, 0, 0)

0.7

1.28

3.67

0.060

24.8

7.3 μm

ηc =

1361 cm−1

(0, 0, 1)

0.65

1.28

29.97

0.493

8.8

4.3 μm

ηc = 2350 cm−1

(2, 0, 0)

0.6

1.28

0.423

0.475

16.5

4.0 μm

ηc = 2512 cm−1

0.6

1.28

0.346

0.589

10.9

a For

(1, 0, 1)  √ √ the rotational band α = α0 exp −9( T0 /T − 1) , β∗ = β∗0 T0 /T.

b Combination c Line



of three bands, all but weak (0, 2, 0) band are fundamental bands, α0 = 25.9 cm−1 /(g/m2 ).

overlap for overlapping bands from equation (10.166).

d “Hot

bands,” very weak at room temperature, exponential growth in strength at high temperatures. ( ( n p  p T α = α0 ΨΨ , ω = ω0 TT , β = β∗ Pe = β∗0 T0 ΦΦ Pe , Pe = p 1 + (b − 1) pa . 0

0

0

0

Ψ from equations (10.156) and (10.160), Φ from equation (10.161), T0 = 100 K, p0 = 1 atm.

Radiative Properties of Molecular Gases Chapter | 10 361

Φ(T) =

 ⎫2 ⎧ m ∞ ⎪ :  ⎪ ⎪ (vk + gk + |δk | − 1)! −u (T)v ⎪ ⎬ ⎨ e k k⎪ ⎪ ⎪ ⎪ ⎭ ⎩ (g − 1)! v ! k k v =v k=1

0,k

k

m ∞ :  (vk + gk + |δk | − 1)! −u (T)v e k k (g − 1)! v ! k k v =v k=1

k

(10.161)

0,k

and

/ uk (T) = hcηk /kT,

,

v0,k =

δk ≥ 0, δk ≤ 0.

0 for |δk | for

(10.162)

In these rather complicated expressions the vk are vibrational quantum numbers, δk is the change in vibrational quantum number during transition (±1 for a fundamental band, etc.), and the gk are statistical weights for the transition (degeneracy = number of ways the transition can take place). Values for the ηk , δk , and gk are given in Table 10.4. The effective pressure Pe gives the pressure dependence of line broadening due to collisions of absorbing molecules with other absorbing molecules and with nonabsorbing molecules that may be present (for example, nitrogen and other inert gases contained in a mixture). Note that the definition for Pe is slightly different here from equation (10.40) (this was done for empirical reasons, to achieve better agreement with experimental data). For the case of nonnegative δk or v0,k = 0 (the majority of gas bands listed in Table 10.4), the series in the expression for Ψ and the denominator of Φ may be simplified [124] to ∞  (vk + gk + δk − 1)! −uk vk (gk + δk − 1)!  −g −δ e = 1 − e−uk k k . (g − 1)!v ! (g − 1)! k k k v =0

(10.163)

k

If v0,k  0, then v0,k terms need to be subtracted from the above result. Because of the low reference temperature of T0 = 100 K, the values for u0,k are relatively large, so both Φ0 and Ψ0 are very simple to evaluate and, for v0,k = 0, Ψ0 ≈

m : (gk + δk − 1)! k=1

(gk − 1)!

,

Φ0 ≈ 1.

(10.164)

If only one of the vibrational modes undergoes a transition (only one δk  0), then all other modes cancel out of the expression for Ψ; and if the transition results in a fundamental band (single transition with δk = 1), then Ψ ≡ 1. This implies that, for a fundamental band, α(T) = α0 = const. Unfortunately, the temperature dependence of the broadening mechanism is always more complicated, and Φ must generally be evaluated from equation (10.161). If several bands overlap each other (e.g., the three H2 O bands situated around 2.7 μm), then also the individual lines overlap lines from other bands, resulting in an effective overlap parameter β that is larger than for any of the individual bands. The band strength and overlap parameter for overlapping bands are calculated [1] from α=

J 

α j,

(10.165)

j=1

⎤2 ⎡ J ( ⎥⎥ 1 ⎢⎢⎢⎢ ⎥ β = ⎢⎢ α j β j ⎥⎥⎥ , ⎦ α⎣

(10.166)

j=1

where J is the number of overlapping bands. When the exponential wide band model was first presented by Edwards and Menard, the temperature dependence for the broadening parameter was not calculated by quantum statistics but was rather correlated from experimental data that, because of their scatter, generally resulted in fairly simple formulae; but extrapolation to higher temperatures tended to be very inaccurate. Most of the bands listed in Table 10.4 are fundamental bands, not because calculations for these bands are simpler, but because fundamental bands tend to be much stronger than overtones or combined-mode bands, often making them the only important ones for heat transfer calculations.

362 Radiative Heat Transfer

FIGURE 10.25 Temperature dependence of the line overlap parameter, β∗ , and band strength parameter, α, for water vapor.

FIGURE 10.26 Temperature dependence of line overlap parameter, β∗ , and band strength parameter, α, for carbon dioxide.

To facilitate hand calculations, the temperature dependence of band strength parameters α (for nonfundamental bands) and overlap parameters β∗ are shown in graphical form in Fig. 10.25 for water vapor. A similar plot is given in Fig. 10.26 for the important bands of carbon dioxide, and Fig. 10.27 shows the temperature dependence of the line overlap parameter for the fundamental bands of methane and carbon monoxide (with α = α0 = const). For more accurate computer calculations the subroutines wbmh2o, wbmco2, wbmch4, wbmco, wbmno, and wbmso2 are given in Appendix F. Alternatively, very accurate polynomial fits for these functions have been given by Lallemant and Weber [184]. Example 10.8. Consider a water vapor–air mixture at 3 atm and 600 K, with 5% water vapor by volume. What is the most important H2 O band and what is its total band absorptance for a path of 10 cm? Solution At 600 K the Planck function has its maximum around 5 μm. Since total emission will depend on the blackbody intensity [see equation (10.66)], we seek a band with large α in the vicinity of 5 μm. Inspection of Table 10.4 shows that the strongest vibration–rotation band for water vapor √ in. From the table we √ lies at 6.3 μm and is, therefore, the band we √ are interested find α = α0 = 41.2 cm−1 /(g/m2 ), β = β∗0 T0 /T(Φ/Φ0 )Pe with β∗0 = 0.094, and ω = ω0 T/T0 = 56.4 600/100 = 138.15 cm−1 . √ To evaluate the effective broadening pressure we find n = 1 and b = 8.6 100/600 + 0.5 = 4.01 and with a volume fraction

Radiative Properties of Molecular Gases Chapter | 10 363

FIGURE 10.27 Temperature dependence of the line overlap parameter, β∗ , for the fundamental bands of methane and carbon monoxide.

x = pa /p the effective pressure becomes Pe = {(p/1 atm)[1+(b−1)x]}n = 3[1+3.01×0.05] = 3.452. Estimating the temperature dependence of the line overlap parameter from Fig. 10.25 leads to β∗ /β∗0 0.65 and β = 0.094 × 0.65 × 3.452 = 0.211. Since all values for α in Table 10.4 are based on a mass absorption coefficient, we must calculate X as X = ρa s, where ρa is the partial density of the absorbing gas (not the density of the gas mixture). For our water vapor with a partial pressure of pa = 0.05 × 3 = 0.15 atm and a molecular weight of M = 18 g/mol, we get from the ideal gas law ρa =

Mpa Ru T

=

18 g/mol × 0.15 atm 1.0132 × 105 J/m3 = 54.84 g/m3 8.3145 J/mol K × 600 K 1 atm

and X = 54.84 × 0.1 = 5.48 g/m2 . Finally, from τ0 = αX/ω we get τ0 = 41.2 × 5.48/138.15 = 1.634. Since the value of τ0 lies between the values of β and 1/β we are in the square-root regime and √ % A∗ = 2 τ0 β − β = 2 1.634 × 0.211 − 0.211 = 0.964 or A = 0.964 × 138.15 = 133 cm−1 .

The calculation of exact values for Φ and Ψ for nonfundamental bands is rather tedious and is best left to computer calculations with the subroutines given in Appendix F.

Continuous Wide Band Correlations While the correlation in Table 10.3 is simple and straightforward (aside from the temperature dependence of α and β), it is often preferable to have a single continuous correlation formula. A simple analytical expression can be obtained for the high-pressure limit, i.e., when the lines become very wide from broadening resulting in very strong overlap, or β → ∞, leading to κη = (S/d)η and A∗ = E1 (τ0 ) + ln τ0 + γE = Ein(τ0 ),

β → ∞,

(10.167)

where E1 (τ) is known as an exponential integral function, which is discussed in some detail in Appendix E. Felske and Tien [169] have given a formula for all ranges of β, based on results from the numerical quadrature of equation (10.87):  ⎛ ⎞ ⎛  ⎞ ⎛ ⎞   ⎜⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎟ ⎟ ⎜ ⎟ ⎜ 1 + 2β τ0 β τ β τ /β τ /β 1 0 0 0 ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ∗ A = 2E1 ⎜⎜ + 2γE , ⎟ + E1 ⎜⎜ ⎟ − E1 ⎜⎜ ⎟ + ln ⎝ 1 + β/τ0 ⎟⎠ ⎝ 2 1 + β/τ0 ⎟⎠ ⎝ 2 1 + β/τ0 ⎟⎠ (1 + β/τ0 )(1 + 2β)

(10.168)

364 Radiative Heat Transfer

FIGURE 10.28 Comparison of various band absorptance correlations.

or, more compactly,     1 w +1 w , A = 2 Ein(w) + Ein − Ein 2β 2β 

∗



W w= δ

 =% 0,Goody

τ0 1 + τ0 /β

.

(10.169)

A previous, somewhat simpler expression by Tien and Lowder [185] is known today to be seriously in error for small values of β [169,186], and is not recommended. Edwards’ wide band model, given in Table 10.3, as well as the continuous correlation by Felske and Tien are based on equation (10.79) together with Goody’s statistical model, equation (10.87), the best narrow band model available at the time of Edwards and Menard’s [164] original paper. Since then it has been found that the Malkmus model introduced in 1967, equation (10.88), describes the radiative behavior of most gases better than Goody’s model [130]. It was shown by Wang [170] that an exact closed-form solution for the band absorptance can be found if equation (10.154) is combined with Malkmus’ narrow band model, leading to    β %  W ∗ β A = e E1 (β + w) − E1 (β) + ln(1 + w/β) + Ein(w), w= = 1 + 4τ0 /β − 1 . (10.170) δ 0,Malkmus 2 Results from Wang’s model, equation (10.170), are compared in Fig. 10.28 with those of Edwards and Menard’s, Table 10.3, as well as against the Felske and Tien’s model, equation (10.168). The agreement between all three models is good. However, the band absorptance based on Malkmus’ model, equation (10.170), is always slightly below that predicted by Goody’s model, equation (10.168). Both the Felske and Tien and the Wang models go to the correct strong-overlap limit (β → ∞), equation (10.167), while the older Edwards and Menard model shows its more approximate character, substantially overpredicting band absorptances for large β, particularly for intermediate values of τ0 . A considerable number of other band correlations are available in the literature, based on numerous variations of the Elsasser and statistical models. An exhaustive discussion of the older (up to 1978) correlations and their accuracies (as compared with numerical quadrature results based on the plain Elsasser and the general statistical models) has been given by Tiwari [187]. Example 10.9. Repeat Example 10.8, using the Felske and Tien and the Wang models. Solution All relations developed for Edwards and Menard’s model, equations (10.156) through (10.159), are equally valid for these two models, as are the data in Table 10.4. Thus, we have again τ0 = 1.634 and β = 0.211. Sticking these numbers into equations (10.168) and (10.170) (or, rather, using the Fortran functions ftwbm and wangwbm, or the stand-alone program wbmodels, all supplied in Appendix F) gives A∗FT = 0.6916,

A∗Wang = 0.6427.

Radiative Properties of Molecular Gases Chapter | 10 365

As expected, the results are fairly close to each other, with the Malkmus-based Wang correlation predicting an about 7% lower band absorptance. Both values are significantly lower than those predicted by Edwards and Menard’s model, which—as inspection of Fig. 10.28 shows—considerably overpredicts band absorptances for strong line overlap (large β) at intermediate optical thicknesses τ0 .

Wide Band Model for Nonhomogeneous Gases As indicated in the previous section on narrow band models, the spectral emissivity for a nonhomogeneous path (with varying temperature and/or gas pressures) [cf. equation (10.69)] is ⎛ X ⎞ ⎜⎜  ⎟⎟ (10.171)

η = 1 − exp ⎜⎜⎝− κη dX⎟⎟⎠, 0

from which we may calculate the total band absorptance as ⎛ X ⎞⎤  ∞⎡  ∞ ⎜⎜  ⎢⎢ ⎟⎟⎥⎥ A=

η dη = ⎢⎢⎣1 − exp ⎜⎜⎝− κη dX⎟⎟⎠⎥⎥⎦ dη. 0

0

(10.172)

0

Here we have replaced the geometric path s by X in case a linear absorption coefficient is not used, but rather one based on density (as was done for the correlation parameters in Table 10.4) or pressure. Since we would still like to use the simple wide band model, appropriate path-averaged values for the correlation parameters α, β, and ω must be found. Attempts at such scaling were made by Chan and Tien [188], Cess and Wang [189], and Edwards and Morizumi [190], and are summarized by Edwards [1]. The average value for α follows readily from the weak line limit (linear regime in Table 10.3) as 1 * α≡ X



X



0

∞

0

1 κη dη dX = X



X



0

0

∞



S δ

 dη dX = η

1 X



X

α dX.

(10.173)

0

The definition of an average value for ω is *≡ ω

1 * αX



X

ωα dX,

(10.174)

0

while the averaged value for β is found by comparison with the square root regime in Table 10.3 as * β≡

1 ** ω αX



X

βωα dX.

(10.175)

0

* and * There is little theoretical justification for the choice of ω β,13 but comparison with spectral calculations using equations (10.101), (10.105), and (10.87) showed that they give excellent results [190]. Example 10.10. Reconsider Example 10.8, but assume that the water vapor–air mixture temperature varies linearly between 400 K and 800 K over its path of 10 cm. How does this affect the total band absorptance for the 6.3 μm band? Solution We may express the temperature variation as T = 400 K(1 + s /s), where s is distance along path s, and the density variation as ρ600 600 K T0 = 6ρ600 = 2  . T T 1 + s /s 3

ρa = ρ600 Thus,  X= 0

s

ρa ds = 6ρ600

 s 0

 1  dξ T0 3 ds = ρ600 s = 3 X600 ln 2 = 1.040 X600 = 5.702 g/m2 . T 2 1 +ξ 2 0

13. Note that there are two different definitions for * β, one for narrow band calculations and the present one for the wide band model.

366 Radiative Heat Transfer

The path-averaged band strength becomes  s 1 1 * αρa ds = α0 X = α0 = 41.2 cm−1 /(g/m2 ), α= X 0 X * we get, from since the 6.3 μm band is a fundamental band and α is independent of temperature. For the averaged ω √ √ √ ω = ω0 T/T0 = ω0 4 1 + s /s,  s  &  6ω0 ρ600 s T T0  3ω0 ρ600 s 1 dξ 1 *= ds = ωαρa ds = ω √ X T0 T X * αX 0 1+ξ 0 0  √  √  X600 6 2−1 1 % 3ω0 X600 = × 2 1 + ξ = 6 2 − 1 ω0 = × 56.4 cm−1 3 X X ln 2 0 2 = 134.8 cm−1 . And, finally, the overlap parameter is obtained from * β=

1 ** ω αX



= 6 β∗0 Pe

s

⎛ & ⎞   s β∗ ⎜⎜ 6ρ600 T ⎟⎟⎟ T0 β∗0 Pe ∗ ⎜⎜⎝ω0 ds ⎟ β0 T0 ⎠ T *X 0 ω & β∗ T0  dξ . β∗0 T

βωαρa ds =

0

ω0 X600 * X ω



1 0

√ √ Inspection of Fig. 10.25 reveals that the integrand varies from 0.59/ 4 0.30 (at 400 K) to 0.66/ 6 0.27 (at 600 K), back √ to 0.80/ 8 0.29 (at 800 K); i.e., the integrand is relatively constant. Keeping in mind the inherent inaccuracies of the wide band model, the integral may be approximated by using an average value of 0.28. Then 0.28×6β∗0 Pe 0.28×0.09427×3.4515 ω0 X600 *  = = √ = 0.220. β 0.28 × 6β∗0 Pe √ X * ω 2−1 6 2−1 The effective optical thickness at the band center is now τ0 = * αX/* ω = 41.2 × 5.702/134.8 = 1.743. Again we are in the square root regime and ( √ β−* β = 2 1.743×0.220 − 0.220 = 1.018 and A = 137 cm−1 . A∗ = 2 τ0 * Thus, although the temperature varied considerably over the path (by a factor of two) values for α, β, and ω changed only slightly, and the final value for the band absorptance changed by less than 3%. In view of the accuracy of the wide band correlation, the assumption of an isothermal gas can often lead to satisfactory results. This has been corroborated by Felske and Tien [191], who suggested a linear average for temperature, and a second independent linear average for density (as opposed to density evaluated at average temperature). They found negligible discrepancy for a large number of nonisothermal examples.

Wide Band k-Distributions Wide band models allow us to determine the radiative emission (or the absorption of incoming radiation) from a volume of gas over an entire vibration–rotation band with a single calculation; but they are inherently less accurate than narrow band models, and they have the same limitations, i.e., they are difficult to apply to nonhomogeneous gases, and they cannot be used at all in enclosures that have nonblack walls and/or in the presence of scattering particles. The k-distribution method, on the other hand, smoothes the spectrum by simply reordering it, rather than supplying an effective transmissivity, and, therefore, it can readily be applied to nonblack walls as well as to scattering media. For a homogeneous medium the method is essentially exact, even for an entire vibration– rotation band, except for the assumption that the Planck function, Ibη , is invariable across the band. This has prompted a number of researchers to generate wide band k-distributions based on exponential wide band correlation data. The first such k-distribution was generated by Wang and Shi [192], using the Malkmus narrow band

Radiative Properties of Molecular Gases Chapter | 10 367

model together with exponentially decaying average line strength. In order to obtain a finite-range reordered wavenumber, 0 ≤ g ≤ 1, as was done for narrow band k-distributions, they truncated the exponentially decaying band wings [see Fig. 10.24 and equation (10.154b)]. This resulted in an analytical expression for the wide band k-distribution, F(k). However, evaluation of the reordered wavenumber, g(k) = F dk, and its inversion to k(g) required numerical integration. Marin and Buckius [193] took a very similar approach but used the exponential wide band model together with the Malkmus model and also the Goody model; they also provided approximate, explicit expressions for water vapor and carbon dioxide [194–196]. Lee et al. [197,198] were able to find the k-distribution directly from wide band correlations, using a rather obscure version of Edwards’ model. This approach was further refined by Parthasarathy et al. [199], using Wang’s wide band model [170]. Denison and Fiveland [200] also provided closed-form approximations for the cumulative k-distribution, based on Edwards’ original wide band model given in Table 10.3. Comparison with narrow band calculations has shown that results from this model have very respectable accuracy [201]. Employing wide band correlation constants limit the accuracy of the resulting k-distributions to the accuracy of the underlying wide band model (perhaps ±20%); however, k-distributions can be incorporated into the RTE, i.e., they are not limited to line-of-sight calculations. More accurate wide band k-distributions can be determined directly from HITEMP 2010 [59] or quickly assembled from the narrow band database generated by Wang and Modest [155]. They, however, do not offer the simple analytical form for F(κ) as described below. The band absorptance for a vibration–rotation band is given by equation (10.151). Assuming a symmetric band, such as given by equation (10.154b), and reordering according to Section 10.9 leads to  ∞  ∞  ∞    −κη X −κX A=2 1−e d|η − ηc | = 2 1−e F(κ) dκ = 2 1 − e−κ(g)X dg, (10.176) 0

0

where the k-distribution

0



∞

F(κ) =

δ(κ − κη ) dη,

(10.177)

0

is defined over an unbounded (wide band) spectral range Δη → ∞ and, thus, g is also unbounded [cf. equations (10.110) and (10.113)] and equivalent to |η − ηc |. The reordered band can also be regarded as symmetrical, if desired (with g going into both directions away from ηc ). Nondimensionalizing equation (10.176) gives  ∞  ∞   A ∗ ∗ ∗ =2 A∗ = 1 − e−κ τ0 F∗ (κ∗ ) dκ∗ = 1 − e−κ (g )τ0 dg∗ , ω 0 0 g α κω αF , F∗ = 2 , g∗ = 2 . (10.178) τ0 = X, κ∗ = ω α ω ω Differentiating equation (10.178) with respect to τ0 , and using Wang’s expression for band absorptance, equation (10.170), yields  ⎧ ⎡ ⎛ ⎞⎤ ⎫  ∞ ⎢⎢ β ⎜⎜ ⎟⎟⎥⎥⎪ ⎪ ⎪ dA∗ 1 ⎪ 4τ ∗ ⎬ ⎨ 0 ⎥ ⎢ ⎜ ⎟ ⎥ ⎢ ⎜ ⎟ =2 = e−κ τ0 κ∗ F∗ (κ∗ ) dκ∗ . (10.179) ⎪1 − exp ⎢⎣ ⎜⎝1 − 1 + ⎟⎥ ⎪ ⎭ ⎩ dτ0 τ0 ⎪ 2 β ⎠⎦ ⎪ 0 Comparing both sides of this equation it is apparent that F∗ (κ∗ ) is related to the inverse Laplace transform of dA∗ /dτ0 ,  ∗ −1 dA ∗ ∗ ∗ 2κ F (κ ) = L . (10.180) dτ0 Using Wang’s model an analytical expression can be obtained for the inverse [199]: ⎧ ⎡%  ⎡%  ⎤ ⎤⎫ ⎪ ⎪ ⎢⎢ β √ ⎥⎥ ⎥⎥⎪ ⎢⎢ β √ 1 ⎪ 1 1 ⎬ ⎨ ∗ ∗ β F (κ ) = ∗ ⎪ κ∗ − √ ⎥⎥⎦ − e erfc ⎢⎢⎣ κ∗ + √ ⎥⎥⎦⎪ ⎪erfc ⎢⎢⎣ ⎪. ∗ ∗ 4κ ⎩ 2 2 κ κ ⎭

(10.181)

The cumulative k-distribution g∗ , or reordered wavenumber, must be found and inverted numerically from  ∞ g 1 (10.182) F∗ (κ∗ ) dκ∗ = g∗ = . 2 ω κ∗

368 Radiative Heat Transfer

FIGURE 10.29 Nondimensional reordered absorption coefficient κ∗ for an exponential wide band vs. nondimensional cumulative kdistribution g∗ .

Figure 10.29 shows the resulting reordered, nondimensional absorption coefficient κ∗ vs. artificial, normalized wavenumber g∗ . For large values of β there is strong line overlap and κη (S/δ) η , and essentially no reordering is ∗ necessary. For that case F∗ approaches F∗ → 1/2κ∗ for κ∗ < 1 and F∗ → 0 for κ∗ > 1, leading to κ∗ → e−g , g∗  0.1.14 For smaller values of β, or less line overlap, but with identical average absorption coefficient the maximum value of the spectral absorption coefficient increases, and fewer spectral positions will have intermediate values, making the distribution more and more compressed toward small g∗ , with larger values near g∗ = 0. Example 10.11. The water vapor–air mixture of Example 10.8 is contained in a nonblack furnace of varied dimensions mixed with soot and scattering particles. In order to make accurate predictions of the radiative heat flux possible across the 6.3 μm water vapor band, determine a reordered correlated k-distribution for this mixture. Solution For the water vapor–air mixture of Example 10.8 we have for the 6.3 μm band α = 41.2 cm−1 /(g/m2 ), ω = 138.15 cm−1 , β = 0.211, and ρa = 54.84 g/m3 . Obtaining a reordered, nondimensional absorption coefficient κ∗ = κ∗ (g∗ ) from equation (10.182) [by utilizing the Fortran subroutine wbmkvsg given in Appendix F], we get from equation (10.178)     ρa α ∗ 2g ρa α ∗ 2|η − ηc | κ κ κ(g) = κ(|η − ηc |) = = , ω ω ω ω where we have replaced the α in equation (10.178) by ρa α in order to obtain a linear, rather than density-based, absorption coefficient [see equation (10.18)], which is generally preferred for spectral calculations. This equivalent spectral absorption coefficient for the 6.3 μm water vapor band, centered at ηc = 1600 cm−1 , is shown in Fig. 10.30, and is compared with the spectral narrow band average absorption coefficient, (S/δ) η for the same conditions. Since, for β = 0.211, there is relatively little line overlap, average values (S/δ) η must come from strongly varying κη with values much larger and much smaller than the average; thus the abundance of large κ (near η = ηc ) with a quick drop-off away from the band center.

Figure 10.30 makes the band appear less wide than indicated by the band width parameter ω. This was done for mathematical convenience: as Fig. 10.13 shows, a band with small β contains many strong lines separated by small κ; we have simply chosen to collect all the large values of κ near the band center. The wide band k-distribution presented here requires numerical integration of equation (10.182) and its inversion to obtain the reordered absorption coefficient k(g); the reordered absorption coefficient recovers the total band absorptance as defined by exponential wide band model parameters. On the other hand, in the work of Marin and Buckius [194–196] explicit (albeit cumbersome) expressions are given for k(g), which approximate the wide band k-distributions obtained from the HITEMP 1995 database [80]. While probably more accurate 14. By convention erfc(x) = 0 for x → +∞ and erfc(x) = 2 for x → −∞ [124].

Radiative Properties of Molecular Gases Chapter | 10 369

FIGURE 10.30 Reordered absorption coefficient for Example 10.11.

below 1000 K (the limit of applicability of HITEMP), the Marin and Buckius formulation depends strongly on the arbitrary and nonphysical choice for the cutoff wavenumber (chosen to find a best fit with HITEMP-generated k-distributions).

10.11 Total Emissivity and Mean Absorption Coefficient Total Emissivity In less sophisticated, more practical engineering treatment it is usually sufficient to evaluate the emission from a hot gas (usually considered isothermal) that reaches a wall. The total emissivity is defined as the portion of total emitted radiation over a path X that is not attenuated by self-absorption, divided by the maximum possible emission. Before the advent of high-resolution spectral databases total emissivities were measured experimentally (see following section) either directly or by measuring emissivities for the various vibration– rotation bands across the spectrum. Then, from equation (10.56) and considering only emission within the gas,  ∞  ∞    N  N     Ibη 1 − e−κη X dη  Ibη η dη πIbη0 πIbη0 0 0 −κη X ∞ Ai , (10.183) = = 1 − e dη =

≡ ∞ σT 4 i Δηband σT 4 i I dη I dη bη bη i=1 i=1 0 0 where two simplifying assumptions have been made: (i) the spectral width of each of the N bands is so narrow that the Planck function varies only negligibly over this range and (ii) the bands do not overlap. While the first assumption is generally very good (with the exception of pure rotational bands such as the one for water vapor listed in Table 10.4), bands of different species in a mixture do sometimes overlap (for example, water vapor and carbon dioxide both have bands in the vicinity of 2.7 μm). If two or more bands of the species contained in a gas mixture overlap, the emission from the mixture will be smaller than the sum of the individual contributions (because of increased self-absorption). This problem has been dealt with, in an approximate fashion, by Hottel and Sarofim [11]. They argued that the transmissivities of species a and b over the overlapping region Δη are independent from one another, that is,    1 1 1 −κηa X −κηb X −κηa X τa+b = e e dη ≈ e dη e−κηb X dη = τa τb . (10.184) Δη Δη Δη Δη Δη Δη Since = 1 − τ this expression leads to the total emissivity of two overlapping bands as

a+b = a + b − a b .

(10.185)

This equation is only accurate if both bands fully overlap. If the overlap is only partial, then the correction term,

a b , should be calculated based on the fractions of band emissivity that pertain to the overlap region (i.e., a quantity that is not available from wide band correlations). An approximate way of dealing with this problem has been suggested by Felske and Tien [191].

370 Radiative Heat Transfer

A total absorptivity for the gas may be defined in the same way as equation (10.183). However, as for surfaces, in the absorptivity the absorption coefficient must be evaluated at the temperature of the gas, while the Planck function is based on the blackbody temperature of the radiation source. It is clear from equation (10.183) that the total emissivity is equal to the sum of band absorptances multiplied by the weight factor (πIbη0 /σT 4 ). Since the band absorptance is roughly proportional to the band strength parameter α (exactly proportional for small values of optical path X), comparison of the factors [α(πIbη0 /σT 4 )]i gives an idea of which bands need to be considered for the calculation of the total emissivity. Example 10.12. What is the total emissivity of a 20 cm thick layer of pure CO at 800 K and 1 atm? Solution For these conditions CO has a single important absorption band in the infrared. Comparing αIbη0 for the 4.7 μm and 2.35 μm bands (see Table 10.4) we find with (η0 /T) 4.3 = 2143 cm−1 /800 K = 2.679 cm−1 /K and (η0 /T) 2.35 = 4260 cm−1 /800 K = 5.325 cm−1 /K,   .  αEbη0 αEbη0 20.9 × 1.5563 = 874. = T3 4.7 T3 2.35 0.14 × 0.2659  4.7 μm band is much stronger (α4.7 /α2.35 150) and located in a more important part of the spectrum Therefore, since the Ebη4.7 /Ebη2.35 6 , the influence of the 2.35 μm band can be neglected. We first need to calculate the band absorptance for the 4.7 μm band. Since values in Table 10.4 are based on the mass absorption coefficient, we need to calculate the density of the CO from the ideal gas law, as we did in Example 10.8: ρa =

Mpa Ru T

=

28 g/mol × 1 atm 1.0132 × 105 J/m3 = 426.6 g/m3 8.3145 J/mol K × 800 K 1 atm

and X = ρa s = 85.32 g/m . We also find from Table 10.4 that n = 0.8 and b = 1.1, so Pe = 1.10.8 = 1.079 and β∗0 Pe = √ 0.075 × 1.079 = 0.081. Further we find α = 20.9 cm−1 /(g/m2 ), ω = 25.5 800/100 = 72.125 cm−1 , and τ0 = αX/ω = 20.9 × 85.32/72.125 = 24.72. From Fig. 10.27 or subroutine wbmco we obtain β∗ /β∗0 = 0.529 and β = (β∗ /β∗0 )β∗0 Pe = 0.529 × 0.081 = 0.043. Thus, τ0 > 1/β and we are in the logarithmic regime, and 2

A∗ = ln(τ0 β) + 2 − β = 2.018 and A = 145.6 cm−1 . Sticking this into equation (10.183), 

CO (800 K, 1 atm) =

πIbη0



σT 4



η0 =2143 cm−1

= 1.5563×10−8

m2

×A=

Ebη0 T3

 η0 =2143 cm−1

×

A σT

W 145.6 cm−1 × −1 3 cm K 5.670×10−8 ×800 W/m2 K3

= 0.0500.

If only total emissivities are desired, it would be very convenient to have correlations, tables, or charts from which the total emissivity can be read directly, rather than having to go through the algebra of the wide band correlations plus equation (10.183). A number of investigators have included total emissivity charts with their wide band correlation data; for example, Brosmer and Tien [174,183] compiled data on CH4 and C2 H2 , and Tien and coworkers [182] did the same for N2 O. However, by far the most monumental work has been collected by Hottel [18] and Hottel and Sarofim [11]. They considered primarily combustion gases, but they also presented charts for a number of other gases. Their data for total emissivity and absorptivity are presented in the form

= (pa L, p, T g ),



   T g 1/2 Ts

pa L , p, Ts , α = α(pa L, p, T g , Ts ) ≈ Ts Tg

(10.186) (10.187)

where T g is the gas temperature and Ts is the temperature of an external blackbody (or gray) source such as a hot surface. Originally, the power for T g /Ts recommended by Hottel was 0.65 for CO2 and 0.45 for water vapor, but with greater theoretical understanding the single value of 0.5 has become accepted [11]. In equation (10.187) pa is the partial pressure of the absorbing gas and p is the total pressure. (Hottel and Sarofim preferred a pressure path

Radiative Properties of Molecular Gases Chapter | 10 371

FIGURE 10.31 Total emissivity of water vapor at a total gas pressure of 1 bar and zero partial pressure, from Hottel [18] (solid lines) and Leckner [202] (dashed lines).

length over the density path length used by Edwards.) The emissivities were given in chart form vs. temperature, with pressure path length as parameter, and for an overall pressure of 1 atm. Later work by Leckner [202], Ludwig and coworkers [203,204], Sarofim and coworkers [205], and others has shown that the original charts by Hottel [11,18], while accurate for many conditions (in particular, over the ranges covered by experimental data of the times), are seriously in error for some conditions (primarily those based on extrapolation of experimental data). New charts, based on the integration of spectral data, have been prepared by Leckner [202] and Ludwig and coworkers [203,204], and show good agreement among each other. Emissivity charts, comparing the newly calculated data by Leckner [202] with Hottel’s [18], are shown in Fig. 10.31 for water vapor and in Fig. 10.32 for carbon dioxide. These charts give the emissivities for the limiting case of vanishing partial pressure of the absorbing gas (pa → 0). The original charts by Hottel also included pressure correction charts for the evaluation of cases with pa  0 and p  1 bar, as well as charts for the overlap parameter Δ . Again, these factors were found to be somewhat inaccurate under extreme conditions and have been improved upon in later work. Particularly useful for calculations are the correlations given by Leckner [202], which (for temperatures above 400 K) have a maximum error of 5% for water vapor and 10% for CO2 , respectively, compared to his spectrally integrated emissivities (i.e., the dashed lines in Figs. 10.31 and 10.32). In his correlation the zero-partial-pressure (or “standard”) emissivity is given by ⎡   j i ⎤⎥ M  N ⎢⎢ T ⎥⎥ p L g a ⎢ ⎥⎥ , T0 = 1000 K, (pa L) 0 = 1 bar cm,

0 (pa L, p=1 bar, T g ) = exp ⎢⎢⎢ c ji (10.188) log10 ⎣ T0 (pa L) 0 ⎥⎦ i=0 j=0

and the c ji are correlation constants given in Table 10.5 for water vapor and carbon dioxide. The emissivity for different pressure conditions is then found from ⎛  2 ⎞ ⎜⎜ (pa L) m ⎟⎟ (a−1)(1−PE ) ⎟⎟ , ⎜ =1− exp ⎜⎝−c log10 ⎠

0 (pa L, 1 bar, T g ) a+b−1+PE pa L

(pa L, p, T g )

(10.189)

where PE is an effective pressure, and a, b, c, and (pa L) m are correlation parameters, also given in Table 10.5.

372 Radiative Heat Transfer

FIGURE 10.32 Total emissivity of carbon dioxide at a total gas pressure of 1 bar and zero partial pressure, from Hottel [18] (solid lines) and Leckner [202] (dashed lines).

TABLE 10.5 Correlation constants for the determination of the total emissivity for water vapor and carbon dioxide [202].

c00 .. . c0M

Gas

Water Vapor

Carbon Dioxide

M, N

2, 2

2, 3

...

cN0

−2.2118 0.85667 −0.10838

−1.1987 0.035596 0.93048 −0.14391 −0.17156 0.045915 √ (p + 2.56pa / t)/p0

.. . . . . . cNM ..

PE (pa L) m /(pa L) 0

−3.9893 1.2710 −0.23678

2.7669 −1.1090 0.19731

2.144, 1.888 − 2.053 log10 t,

0.39163 −0.21897 0.044644

(p + 0.28pa )/p0

13.2t2

a

−2.1081 1.0195 −0.19544

0.054/t2 ,

t < 0.7

0.225t2 ,

t > 0.7

t < 0.75 t > 0.75

1 + 0.1/t1.45

b

1.10/t1.4

0.23

c

0.5

1.47

T0 = 1000 K, p0 = 1 bar, t = T/T0 , (pa L) 0 = 1 bar cm

As noted before, in a mixture that contains both carbon dioxide and water vapor, the bands partially overlap and another correction factor must be introduced, which is found from

Δ = with ζ=

'

ζ − 0.0089ζ10.4 10.7 + 101ζ

pH2 O . pH2 O + pCO2

) log10

(pH2 O + pCO2 )L (pa L) 0

2.76 ,

(10.190)

(10.191)

Radiative Properties of Molecular Gases Chapter | 10 373

This factor is directly applicable to emissivity and absorptivity. Modak [206] observed that Leckner’s Δ is overpredicted for lower temperatures and proposed a correction factor F(t) = −1.0204t2 + 2.2449t − 0.23469,

(10.192)

with t defined in Table 10.5, such that Δ = Δ Leckner F(t)

if

(pH2 O + pCO2 )L > 0.1. (pa L) 0

(10.193)

F(t) has a maximum at 1100 K of F(t = 1.1) = 1 and becomes negative for T > 2085 K; therefore, its use for temperatures above 1100 K is questionable. To summarize, the total emissivity and absorptivity of gases containing CO2 , water vapor, or both, may be calculated from:  

i (pi L, p, T g ) = 0i (pi L, 1 bar, T g ) (pi L, p, T g ), i = CO2 or H2 O, (10.194a)

0 i   1/2  Tg Ts αi (pi L, p, T g , Ts ) =

i pi L , p, Ts , (10.194b) i = CO2 or H2 O, Ts Tg  

CO2 +H2 O = CO2 + H2 O − Δ pH2 O L, pCO2 L , (10.194c)   Ts Ts . (10.194d) αCO2 +H2 O = αCO2 + αH2 O − Δ pH2 O L , pCO2 L Tg Tg For the convenience of the reader Appendix F contains the Fortran routines totemiss and totabsor, which calculate the total emissivity or absorptivity of a CO2 –water vapor mixture from Leckner’s correlation, and which can also be called from the stand-alone program Leckner through user prompts. Example 10.13. Consider a 1 m thick layer of a gas mixture at 1000 K and 5 bar that consists of 10% carbon dioxide, 20% water vapor, and 70% nitrogen. What is the total normal intensity escaping from this layer? Solution From equations (10.56) and (10.183) we see that the exiting total intensity is 

∞

I=

   Ibη 1 − e−κη X dη =

0

0

∞

Ibη η dη =

σT 4 , π

where is the total emissivity of the water vapor–carbon dioxide mixture. First we calculate the emissivity of CO2 at a total pressure of 1 bar from Table 10.5: With pCO2 L = 0.1 × 5 m bar = 50 bar cm and T g = 1000 K we find CO2 ,0 (1 bar) = 0.157 (which may also be estimated from Fig. 10.32); for a total pressure of 5 bar we find from Table 10.5 the effective pressure is PE = 5.14, a = 1.1, b = 0.23, c = 1.47, and (pa L) m = 0.225 bar cm. Thus, from equation (10.189) 



0



=1−

CO2

   0.1 × (−4.14) 0.225 2 ≈ 1.00, exp −1.47 × log10 0.33 + 5.14 50

and

CO2 ≈ 0.157. Similarly, for water vapor with pH2 O L = 0.2 × 5 m bar = 100 bar cm we find H2 O,0 (1 bar) ≈ 0.359 and the pressure correction factor becomes, with PE = 7.56, a = 1.88, b = 1.1, c = 0.5, and (pa L) m = 13.2 bar cm, 



0

 H2 O

=1−

   0.888 × (−6.56) 13.2 2 = 1.414, exp −0.5 × log10 1.988 + 7.56 100

and

H2 O ≈ 0.359 × 1.414 = 0.508. Finally, since we have a mixture of carbon dioxide and water vapor, we need to deduct for the band overlaps: From equation (10.190), with ζ = 23 , Δ = 0.072 (or 0.071 if Modak’s correction is applied). Thus, the total emissivity is

374 Radiative Heat Transfer

= 0.157 + 0.508 − 0.072 = 0.593. Alternatively, and more easily, using subroutine totemiss with ph2o = 1., pco2 = .5, ptot = 5, L = 100, and Tg = 1000 returns the same numbers. The total normal intensity is then I = 0.593 × 5.670×10−8 W/(m2 K4 ) × (1000 K)4 /π sr = 10.70 kW/m2 sr.

It is apparent from this example that the calculation of total emissivities is far from an exact science and carries a good deal of uncertainty. Carrying along three digits in the above calculations is optimistic at best. The reader should understand that accurate emissivity values are difficult to measure, and that too many parameters are involved to make simple and accurate correlations possible. Total emissivities can, of course, also be calculated directly from modern high-resolution databases, although such rather tedious calculations would defeat the purpose of using such values for simple calculations, unless they are collected in easy-to-use databases. This has recently been done by Alberti et al. for CO2 [52] and H2 O [53]. For CO2 they found that Hottel’s values at atmospheric pressure may be off by as much as 25%, while at 40 bar the errors reach 80%. Leckner’s correlation was shown to be fairly accurate, especially at higher temperatures with a maximum error of 10%, but the error may reach 35% at lower temperatures. For H2 O Hottel’s charts may be in error by as much as 300%, while Leckner’s correlations were found to be quite accurate, with about ±5% errors for standard emissivities, and somewhat worse when the pressure correction is applied. Values from the databases can be extracted from Excel spreadsheets, which are also included in Appendix F. Alberti et al. also investigated the total emissivity of CO as well as overlap effects for binary and tertiary mixtures [207,208]. They noted that (10.194c) holds for any binary CO2 –H2 O–CO mixture, and is a good approximation for a tertiary mixture (all 3 species present), since the overlap between all 3 species is small and can generally be neglected, i.e.,

CO2 +H2 O+CO CO2 + H2 O + CO − Δ (pH2 O L, pCO2 L) − Δ (pH2 O L, pCO L) − Δ (pCO L, pCO2 L).

(10.195)

Comparing many conditions against LBL calculations they noted that for low temperatures Hottel’s charts are fairly accurate while Leckner’s formula overpredicts Δ (thus, Modak’s correction should be applied); for temperatures above 800 K the reverse is true (Modak’s correction should not be applied). They present numerous graphs for standard emissivities, pressure correction charts, as well as overlap factors too numerous to include here. Ref. [208] provides an Excel spreadsheet to calculate the total emissivity of an arbitrary CO2 –H2 O–CO mixture for a5temperature range of 300 K < T < 3000 K, a pressure range of 0.1 bar < p < 100 bar and pressure path lengths i pi L < 6000 bar cm. The accuracy of the predicted total emissivities is generally within 1% of LBL values. This file is also included in Appendix F. For example, using Alberti et al.’s Excel sheets for the conditions given in Example 10.13 yields CO2 = 0.174 and H2 O = 0.502, as well as a mixture emissivity of 0.578 (which differs by 2.5% from the answer in Example 10.13). Very recently, Alberti et al. [209] also correlated total absorptivities for CO2 , H2 O, and CO (but not for mixtures), based on HITEMP 2010 and their line wing truncation. Results are presented in terms of a modified version of equation (10.187), i.e., 

1/2   s  Ts α=

pa L , p, Ts , Ts Tg Tg

(10.196)

with the exponent s depending on species and gas conditions, as tabulated in [209]. Since their work has not yet included mixture absorptivities, for that purpose equation (10.194d) must be used. The group’s Excel sheets to calculate total emissivities of CO2 (CO2Emissivity [52]), H2 O (H2OEmissivity [53]), and mixtures (MixEmissivity [208]) are all included in Appendix F.

Mean Absorption Coefficients We noted in the previous chapter that the emission term in the equation of transfer, equation (9.21), and in the divergence of the radiative heat flux, equation (9.62), is proportional to κη Ibη . Thus, for the evaluation of total intensity or heat flux divergence it is convenient to define the following total absorption coefficient, known as the Planck-mean absorption coefficient: ∞  ∞ Ibη κη dη π κP ≡ 0 ∞ Ibη κη dη. (10.197) = σT 4 0 Ibη dη 0

Radiative Properties of Molecular Gases Chapter | 10 375

FIGURE 10.33 Planck-mean absorption coefficients for carbon dioxide, and water vapor.

Using narrow band averaged values for the absorption coefficient, and making again the assumption that the Planck function varies little across each vibration–rotation band, equation (10.197) may be restated as κP =

  N   πIbη0 i=1

σT 4

i

Δηband



S δ

 dη =

 N   πIbη0 i=1

σT 4

αi ,

(10.198)

i

where the sum is over all N bands, and the Ibη0 are evaluated at the center of each band. It is interesting to note that the Planck-mean absorption coefficient depends only on the band strength parameter α and, therefore, on temperature (but not on pressure). Values for α have been measured and tabulated by a number of investigators for various gases and, using them, Planck-mean absorption coefficients have been presented by Tien [3], but these values are today known to be seriously in error. Alternatively, the Planck-mean absorption coefficient can be calculated directly from high-resolution databases such as HITRAN [32] and HITEMP [59] as [210]  ∞    πIbη0   ∞   πIbη0  π κP = I κ dη = κ dη = S j, (10.199) ηj ηj bη σT 4 0 σT 4 j 0 σT 4 j j j j where the summation is now over all the spectral lines of the gas, and the Ibη0 are evaluated at the center of each line. Figures 10.33 through 10.35 show Planck-mean absorption coefficients calculated from the HITEMP 2010 (CO2 , H2 O, and CO) and HITRAN 2008 databases (all gases). For some gases, which saw major updates in the most recent HITRAN 2008 version, the values obtained from HITRAN 1996 [75] are also shown for comparison. At higher temperatures the Planck-mean absorption coefficients from HITRAN 2008 are generally larger than those from HITRAN 1996, due to the inclusion of many more lines from higher vibrational energy levels. Accordingly, today’s HITRAN 2008 can be used with confidence up to about 1000 K. The latest version of HITEMP [59] includes many more “hot lines,” and strives to be accurate for temperatures up to 3000 K. Sometimes the Planck-mean absorption coefficient is required for absorption (rather than emission), for example, when gas and radiation source are at different temperatures. This expression is known as the modified Planck-mean absorption coefficient and is defined as ∞ Ibη (Ts )κη (T) dη κm (T, Ts ) ≡ 0  ∞ . (10.200) Ibη (Ts ) dη 0 An approximate expression relating κm to κP has been given by Cess and Mighdoll [211] as κm (T, Ts ) = κP (Ts )



 Ts . T

(10.201)

376 Radiative Heat Transfer

FIGURE 10.34 Planck-mean absorption coefficients for ammonia, nitrous oxide, and sulfur dioxide.

FIGURE 10.35 Planck-mean absorption coefficients for carbon monoxide, nitric oxide, and methane.

In later chapters we shall see that in optically thick situations the radiative heat flux becomes proportional to 1 1 dIbη ∇T. ∇Ibη = κη κη dT

(10.202)

This has led to the definition of an optically thick or Rosseland-mean absorption coefficient as 1 ≡ κR

 0

∞

1 dIbη dη κη dT

. 0

∞

dIbη dT

dη =

π 4σT3

 0

∞

1 dIbη dη. κη dT

(10.203)

Even though they noted the difficulty of integrating equation (10.203) over the entire spectrum (with zero absorption coefficient between bands), Abu-Romia and Tien [178] and Tien [3] attempted to evaluate the Rosseland-mean absorption coefficient for pure gases. Since the results are, at least by this author, regarded as very dubious they will not be reproduced here. We shall return to the Rosseland absorption coefficient when its use is warranted, i.e., when a medium is optically thick over the entire spectrum (for example, an optically thick particle background with or without molecular gases).

Radiative Properties of Molecular Gases Chapter | 10 377

10.12 Gas Properties of Earth’s Atmosphere and Climate Change In Section 3.10, the role of the surface properties of the Earth on climate change was highlighted. In the analyses presented therein, the role of the atmosphere in the Earth’s energy balance was neglected. In this section, the critical role that the atmosphere—in particular, the greenhouse gases in the atmosphere—plays in altering the climate of the Earth is discussed. First, a brief outline of the composition (temperature, pressure, species concentrations) of the various layers of the atmosphere is given, followed by a discussion of the absorption coefficient of the two most important greenhouse gases, namely carbon dioxide and water vapor. While previous sections have discussed such properties extensively, here the focus is on conditions found in the Earth’s atmosphere, where low pressures and temperatures prevail. These radiative properties are then utilized to calculate the transmissivity (both spectral and total) of the atmosphere and a simple model is presented to predict the Earth’s equilibrium surface temperature, with the primary goal being to elucidate the relationship between rising carbon dioxide levels in the atmosphere and the Earth’s temperature.

The Atmosphere The atmosphere is conventionally split into six layers: the troposphere, which extends to roughly 12 km from the surface of the Earth; the stratosphere, which extends from about 12 km to 50 km; the mesosphere, the thermosphere, the exosphere, and the ionosphere. About 75–80% of the mass of the atmosphere, including essentially all of its water is contained within the troposphere [212]. Almost all of the remaining mass is contained within the stratosphere. Most notably, the so-called ozone layer is contained within the stratosphere at an approximate height of 25 km from the surface of the Earth. Beyond about 20 km, the partial pressure of the gases is very low and, therefore, the atmosphere may be considered transparent beyond this altitude. Although the concentrations of the various gases constituting the Earth’s atmosphere vary significantly with altitude, seasons, and geographic location, the following average values (by volume, as of 2016) may be used as a guide for determining which of these gases may play a pivotal role in dictating the Earth’s radiation exchange: nitrogen (78.084%); oxygen (20.946%); argon (0.934%); water vapor (0.25%); carbon dioxide (0.04%); neon (0.0018%); helium (0.00052%); methane (0.00018%); ozone, carbon monoxide, hydrogen, krypton, sulfur dioxide, nitrous oxide, nitric oxide, and hydrogen sulfide (all less than 0.0001%). Of these gases, water vapor and carbon dioxide are the two most abundant greenhouse gases. In the atmospheric sciences literature, the term greenhouse gas is used for a gas that emits and absorbs radiation within the infrared part of the spectrum. Although methane, ozone, and carbon monoxide are other strong greenhouse gases, their concentrations are so small that their effect on atmospheric transmissivity is negligible. Therefore, for the purposes of the present analysis and discussion, only water vapor and carbon dioxide are considered. A summary of the composition of the atmosphere (average values) as a function of altitude is provided in Table 10.6 (broken up into 9 layers, across each of which composition may be assumed constant). The temperature decreases almost linearly across the troposphere starting from 15 ◦ C on the Earth’s surface, and thereafter remains constant across the lower part of the stratosphere. The pressure, on the other hand, decreases near-exponentially through the entire atmosphere. Literature [213,214] suggests that the carbon dioxide concentration is fairly uniform across the lower parts of the atmosphere, with an average value of 400 ppm (as of 2016). In contrast, water vapor resides primarily in the lowest parts of the atmosphere, and almost disappears above 20 km.

Absorption Coefficient and Transmissivity The HITRAN2012 database [98] may be used to calculate the absorption coefficients of carbon dioxide and water vapor for the conditions tabulated in Table 10.6, and are shown in Fig. 10.36 for three different layers of the atmosphere—one close to the Earth’s surface, one midway into the troposphere, and one in the stratosphere. The 4.3 μm and 15 μm bands are evidently the two most dominant bands of carbon dioxide. For water vapor the two most dominant bands are at 2.7 μm and 6.3 μm; in addition, there are several weaker bands in the near infrared, and the purely rotational band beyond about 10 μm showing continuous absorption. For both gases, peak values of the absorption coefficient increase substantially with decrease in pressure, while mean and total values increase slightly with decreasing temperature (cf. discussion of line strength and broadening in Sections 10.3 and 10.4).

378 Radiative Heat Transfer

TABLE 10.6 Variation of the composition (average values) of the atmosphere with altitude, compiled from [212–216]. Layer

Altitude (km)

Pressure (bar)

Temperature (K)

CO2 (ppm)

H2 O (ppm)

1

0–1

0.98

285

400

14200

2

1–3

0.87

275

400

11000

3

3–5

0.73

262

400

4560

4

5–7

0.58

249

400

812

5

7–9

0.44

236

400

198

6

9–11

0.30

224

400

93.5

7

11–13

0.21

217

400

52.8

8

13–17

0.15

217

400

7.3

9

17–20

0.08

217

400

3.1

FIGURE 10.36 Pressure-based absorption coefficients under various atmospheric conditions: (a) carbon dioxide and (b) water vapor. The numbers 1, 5, and 9 indicate the layers of the atmosphere, which are shown in Table 10.6.

With the absorption coefficients of the two gases known for the various layers, along with their partial pressures and thicknesses (Table 10.6), the spectral transmissivity of the atmosphere may be calculated as ⎡ N ⎤ layer ⎢⎢  ⎥⎥ ⎢⎢ ⎥ τλ = exp ⎢⎢− (κpλ,CO2 ,i pCO2 ,i + κpλ,H2 O,i pH2 O,i )Li ⎥⎥⎥ , ⎣ ⎦

(10.204)

i=1

where Li is the thickness of the i-th layer of the atmosphere and Nlayer is the total number of layers used to represent the atmosphere—9 according to Table 10.6; κpλ denotes the spectral pressure-based absorption coefficient and pi is the partial pressure in layer i. Figure 10.37 shows the resulting spectral transmissivity of the atmosphere over the spectrum of interest. The low transmissivity locations correspond to the absorption band centers of the two gases. Also shown in the same figure are the blackbody emissive powers computed using the temperature of the sun (TS = 5777 K) and the Earth (TE = 288 K). It is clear from Fig. 10.37 that there is strong overlap between all of the spectral energy distribution of the Earth’s emission with the low transmission bands of the two gases. On the other hand, about 50% of the energy emitted by the sun (at wavelengths below 0.7 μm) is unaffected by the atmosphere since its transmissivity in this range is unity. The disparity between how the atmosphere filters incoming solar irradiation vs. how it filters outgoing energy emitted by the Earth’s surface results in the socalled Atmospheric Greenhouse Effect. In general, the greenhouse effect is exhibited by any medium that transmits short wavelength radiation effectively, but transmits long wavelength radiation poorly. Perhaps the best way to quantify the atmospheric greenhouse effect is to calculate the total (spectrally integrated) transmissivity of the atmosphere for solar irradiation and radiation emitted by the Earth separately. Thus, assuming that sun and

Radiative Properties of Molecular Gases Chapter | 10 379

FIGURE 10.37 Spectral transmissivity of the atmosphere (only due to CO2 and H2 O) and blackbody emissive power of the sun and the Earth.

Earth both behave as blackbodies, we obtain  ∞ 1 τin = τλ Ebλ (TS ) dλ; σ TS4 0

τout =

1 σ TE4



∞

τλ Ebλ (TE ) dλ.

(10.205)

0

Using the data presented in Fig. 10.37 with equation (10.205), one obtains τin = 0.850719 and τout = 0.226829 for the model atmosphere (with just carbon dioxide and water vapor and without scattering) considered here, i.e., about 85% of solar irradiation reaches the Earth’s surface but only about 23% of the energy emitted by it leaves. In reality, the fraction of solar irradiation that reaches the Earth’s surface is believed to be significantly less than 85% [212]. A significant fraction of the solar irradiation is scattered back to space by the outer layers of the atmosphere and by clouds, and another significant fraction is absorbed by clouds and dust particles—both of which are critical for the Earth’s energy balance, as elaborated further in the next subsection. Nonetheless, the calculated difference in the incoming and outgoing total transmissivities demonstrates the radiation blocking mechanism of the two dominant greenhouse gases in the atmosphere. For centuries prior to the industrial revolution, carbon dioxide levels in the atmosphere hovered around 250 ppm [217,218]. Since the late 1970s, there has been a dramatic upsurge in the amount of carbon dioxide in the atmosphere, with the current (as of 2019) reported value being slightly above 400 ppm [217,218]. The latest report by the Intergovernmental Panel on Climate Change (IPCC) [217] predicts the concentration of carbon dioxide to reach 440 ppm by 2050. One important question is what effect this increase from 250 ppm to 400 ppm may have had on the transmissivity of the atmosphere for incoming and outgoing radiation and, consequently, on the temperature of the Earth’s surface. To answer this question, the results shown in Figs. 10.36a and 10.37 are recomputed with 250 ppm CO2 . Figure 10.38 shows the change in spectral transmissivity of the atmosphere, defined as its value computed using 400 ppm CO2 subtracted from its value computed using 250 ppm CO2 (with all other conditions, namely, temperature, total pressure, and water vapor concentrations kept unchanged). A significant decrease (about 20%) in the spectral transmissivity is observed. As expected, the changes are pronounced around the two dominant bands of CO2 centered at 4.3 μm and 15 μm. However, since only about 0.4% of solar irradiation is absorbed by the atmosphere’s CO2 (as opposed to roughly 6% by water vapor), a 20% decrease has only a very minor effect: the total transmissivity for incoming radiation, τin , changed over time from 0.851859 (for 250 ppm CO2 ) to 0.850719 (for 400 ppm CO2 ). For outgoing radiation, on the other hand, the effect is much stronger: τout changed from 0.233146 (for 250 ppm CO2 ) to 0.226829 (for 400 ppm CO2 ). In other words the increase in the CO2 level left the incoming solar irradiation essentially unchanged, but increased the radiation blocking capacity of the atmosphere by 2.7%.

Earth’s Surface Temperature In the presence of the atmosphere, only the transmitted fraction of the solar irradiation, qsol , reaches the Earth’s surface. In reality, about 6% of the solar radiation approaching the Earth is scattered back into outer space

380 Radiative Heat Transfer

FIGURE 10.38 Change in the transmissivity of the atmosphere caused by altering CO2 level from 400 ppm to 250 ppm.

by the outer layers of the atmosphere [219,220], primarily due to Rayleigh scattering by the gases. Thus, only qsol,in = 0.94 × qsol is actually transmitted through the final 20 km closest to the Earth’s surface. In addition, due to absorption and scattering by clouds and dust particles within the final 20 km, the effective transmissivity of the atmosphere is smaller than the values computed in the preceding section, which only considers the effects of carbon dioxide and water vapor. Various research groups, based on observations, have now reached a unanimous conclusion that, overall, clouds help to cool the Earth by virtue of them preventing more of the incoming short wavelength solar radiation from reaching the Earth’s surface than trapping the outgoing long wavelength radiation emitted by the Earth. It is estimated that due to clouds, approximately qred = 0 to 110 W/m2 (based on the total surface area of the Earth) less radiation is available for heating of the Earth’s surface [219,220], where large values of qred are typically observed in the northern hemisphere, which has significantly more cloud cover than the southern hemisphere. Following the development of equations (3.107) and (3.108), the amount of radiative energy absorbed and emitted by the Earth’s surface may then be altered to account for the atmosphere:   Qabs = − αE τin qsol,in (πR2E ) − qred (4πR2E ) , (10.206a) Qem = E τout (σTE4 ) (4πR2E ).

(10.206b)

Equation (10.206b) essentially represents the radiation energy leaving the atmosphere. For simplicity (to avoid a multilayer energy balance), we will use it here for energy balance at the Earth’s surface, which carries the implicit assumption that any radiation emitted by the Earth and absorbed by the atmosphere is returned back to the Earth’s surface. The energy absorbed by the Earth’s surface is also partially lost to the atmospheric layers closest to it via other modes of heat transfer. In the present simplified analysis, this loss is treated using a simple convection-like heat transfer term, Qloss = U(TE − T∞ )(4πR2E ), where U is an overall heat transfer coefficient that encapsulates all mechanisms of heat loss from the Earth’s surface other than by radiation, and T∞ is the average temperature of a layer close to the Earth’s surface, which may be chosen to be the temperature of layer 1 shown in Table 10.6. An energy balance on the Earth’s surface then yields Qem + Qabs + Qloss = 0, which may be expanded and simplified to write 4 E τout σTE4 − αE τin qsol,in + 4qred + 4U(TE − T∞ ) = 0,

(10.207)

wherein data for the Earth’s surface properties presented in Section 3.10 and the absorption coefficient data of the present section can be used. Equation (10.207) is a nonlinear equation that may be solved to calculate TE . However, this requires an iterative procedure. First, the Earth’s surface temperature, TE , is guessed. A good guess is the recorded value of 288 K. Next, the effective total absorptance of the Earth’s surface, αE , and the effective total emittance, E , of the Earth’s surface are calculated using equation (3.113). The incoming and outgoing transmissivities are also calculated using equation (10.205). These values are then substituted

Radiative Properties of Molecular Gases Chapter | 10 381

into equation (10.207), and a new surface temperature is computed. The procedure is repeated until convergence. During the iterative process, the incoming transmissivity remains unchanged. Generally, it takes 2–3 iterations to obtain a converged temperature that is accurate up to the second decimal place. Since the resolution sought in these calculations is within a few tenths of a degree, performing iterations is necessary. In the present analysis, the values of U and qred were adjusted to yield a value of TE within ±1 K of the measured value of 288 K. The computed surface temperature using this procedure for an atmosphere with 400 ppm CO2 is 288.08 K. While the present model is simplistic, and the values of U and qred were adjusted to match the measured temperature of the Earth, it may still be explored to understand trends. For example, if the CO2 concentration is changed to the historic value of 250 ppm, the surface temperature is computed to be 286.96 K. In other words, the aforementioned increased opacity of the atmosphere due to increasing CO2 levels is predicted to have resulted in a temperature rise of 1.12 K, which agrees well with the recorded data [217,218] of the Earth’s surface temperature that shows a rise of about 1.2 K from the average temperature between 1951 and 1980. Further increase of the CO2 level to 440 ppm, as predicted for 2050 by the IPCC, is expected to increase the temperature by a further 0.26 K from 400 ppm conditions according to our simple model. The contribution to the increase in the Earth’s temperature due to increased radiation trapping by a particular greenhouse gas is often referred to in the literature as radiative forcing by that gas. In summary, the present section elucidated the role of the spectral absorption coefficient of the two main greenhouse gases, namely, carbon dioxide and water vapor, on the Earth’s climate (temperature of the atmosphere) change. While the energy balance model presented here is simplistic in nature, it does account for the detailed spectral nature of the two most abundant greenhouse gases. The predictions clearly delineate the causeand-effect relationship between rising CO2 levels and rising temperature. Methane, another strong greenhouse gas, whose concentration is rapidly rising due to dairy farming and natural gas production, will have to be accounted for in future analysis. Far more sophisticated models of the atmosphere pertaining to global climate change are available, and the reader is referred to Refs. [221–224] and the references cited therein for further reading.

10.13 Experimental Methods Before going on to employ the above concepts of radiation properties of molecular gases in the solution of the radiative equation of transfer and the calculation of radiative heat fluxes, we want to briefly look at some of the more common experimental methods of determining these properties. While light sources, monochromators, detectors, and optical components are similar to the ones used for surface property measurements, as discussed in Section 3.11, gas property measurements result in transmission studies (as opposed to reflection measurements for surfaces). All transmission measurements resemble one another to a certain extent: They consist of a light source, a monochromator or FTIR spectrometer (unless, for measurements over a narrow spectral range, a tunable laser is used as source), a chopper, a test cell with the (approximately isothermal) gas whose properties are to be measured, a detector, associated optics, and an amplifier–recorder device. The chopper often serves two purposes: (i) a pyroelectric detector cannot measure radiative intensity, rather, it measures changes in intensity and (ii) if the beam is chopped before going through the sample gas then, by measuring the difference in intensity between chopper open and closed conditions, indeed only transmission of the incident light beam is measured. That is, any emission from the (possibly very hot) test gas and/or stray radiation will not be part of the signal. A typical setup is shown in Fig. 10.39, depicting an apparatus used by Tien and Giedt [225]. A chopper is not required if an FTIR spectrometer is used, since the light is modulated inside the unit. However, for high test gas temperatures care must be taken to eliminate sample emission from the signal [226,227]. Usually, gas temperatures are measured independently, and knowledge of gas absorption coefficients is acquired. But it is also possible to radiatively determine the gas temperature, if accurate knowledge of the absorption coefficient is given, such as detailed line structure of diatomic molecules together with FTIR spectrometry [228–230]. Measurements of radiative properties of gases may be characterized by the nature of the test gas containment and by the spectral width of the measurements. As indicated by Edwards [1], we distinguish among (1) hot window cell, (2) cold window cell, (3) nozzle seal cell, and (4) free jet devices; these may be used to make (a) narrow band measurements, (b) total band absorptance measurements, or (c) total emissivity/absorptivity measurements.

382 Radiative Heat Transfer

FIGURE 10.39 General setup of gas radiation measurement apparatus [225].

FIGURE 10.40 Top: 3D drawing of HTPGC. Bottom: cross-section sketch of HPTGC including dimensions [64].

The hot window cell uses an isothermal gas within a container that is closed off at both ends by windows that are kept at the same temperature as the gas. While this setup is the most nearly ideal situation for measurements, it is generally very difficult to find window material that (i) can withstand the high temperatures at which gas properties are often measured, (ii) are transparent in the spectral regions where measurements are desired (usually near-infrared to infrared) and do not experience “thermal runaway” (strong increase in absorptivity at a certain temperature level), and (iii) do not succumb to chemical attack from the test gas and other gases. Such cells have been used, for example, by Penner [20], Goldstein [231], and Oppenheim and Goldman [232]. The best material for such hot windows is generally sapphire (Al2 O3 ), which allows temperatures of close to 2000 K but has a limited wavelength range, turning opaque for wavelengths above 6 μm (cf. Fig. 12.1 for room temperature). Recently, such a device was constructed in Denmark by Christiansen et al. [64] to measure gas transmissivities at elevated pressures up to 200 bar and 1300 K. A schematic of their High-Temperature and high-Pressure Gas Cell (HTPGC) is shown in Fig. 10.40. Intended for high-pressure measurements the device has a fixed optical path

Radiative Properties of Molecular Gases Chapter | 10 383

FIGURE 10.41 Schematic of the high-temperature gas furnace used by Tien and Giedt [225].

of 30.28 mm. Three separate heating coils are wound around the outer ceramic tube (not shown in the figure). Their optical setup includes a high-resolution FTIR spectrometer with highest nominal spectral resolution of 0.125 cm−1 . A sample of their experimental results was shown in Fig. 10.11 demonstrating the importance of line-mixing at elevated pressures. The cold window cell, as the name implies, lets the probing beam enter and exit the test cell through watercooled windows. This method has the advantage that the problems in a hot window cell are nearly nonexistent. However, if the geometric path of the gas is relatively short, this method introduces serious temperature and density variations along the path. Tien and Giedt [225] designed a high-temperature furnace, consisting of a zirconia tube surrounded by a graphite heater, that allowed temperatures up to 2000 K. The furnace was fitted with water-cooled, movable zinc selenide windows, which are transmissive between 0.5 μm and 20 μm and stay inert to reactions with water vapor and carbon oxides for temperatures below 550 K. A schematic of their furnace is shown in Fig. 10.41. Their design allows for a variable optical path, but is limited to atmospheric pressure. While able to make measurements at high temperatures, it is impossible to obtain truly isothermal gas columns with such a device. For example, for a nominal cell at 1750 K of 30 cm length, they found that the temperature gradually varied by a rather substantial 350 K over the central 2/3 of the cell, and then rapidly dropped to 330 K over the outer 1/3. This apparatus was used by Tien and coworkers to measure the properties of various gases [177,179–182,233]. In the early 2000s Modest constructed a modern high-temperature gas transmissometer, shown schematically in Fig. 10.42 and used by Bharadwaj et al. [81,83,84], to measure transmissivities of carbon dioxide and water vapor. The device is based on the infrared emissometer [234–236] shown in Fig. 3.47 and combines the advantages of hot-window and cold-window absorption cells. In essence, the apparatus consists of a hermetically sealed high-temperature furnace, a motorized tube fitted with an optical window, a sealed optical path, and an FTIR spectrometer with internal infrared light source and an external detector, which can only detect the modulated light from the FTIR. Light from the FTIR is imaged onto a platinum mirror inside the furnace; the reflected light, in turn, is imaged onto the external detector. The cold drop-tube with an optical window is placed into position and retracted by a high-speed motor. The gas column between platinum mirror and optical window forms an isothermal absorption cell and, since the optical window resides within the furnace’s hot zone for only a few seconds at a time, this device is able to measure transmissivities of truly isothermal high-temperature gas columns. Nozzle seal cells are open flow cells in which the absorbing gas is contained within the cell by layers on each end of inert gases such as argon or nitrogen. This system eliminates some of the problems with windows, but may also cause density and temperature gradients near the seal; in addition, some scattering may be introduced by the turbulent eddies of the mixing flows [237]. This type of apparatus has been used by Hottel and Mangelsdorf [13]

384 Radiative Heat Transfer

FIGURE 10.42 Schematic of a drop-tube transmissometer [81].

and Eckert [238] for total emissivity measurements of water vapor and carbon dioxide. Most of the measurements made by Edwards and coworkers also used nozzle seal cells [165,166,168,176,237,239,240]. Another FTIR-based transmissometer built at the Technical University of Denmark is an atmospheric pressure flow cell with a so-called “laminar window” nozzle seal, of which a schematic is shown in Fig. 10.43 [93]. Special care was taken to obtain a uniform gas temperature profile (296—1873 K) and a well-defined path length, fitted with KBr windows, which allow spectral measurements in the region 0.25—25μm. Comparison of their high resolution (0.125 cm−1 ) transmission data for CO2 with HITEMP 2010 and CDSD-1000 attests to, both, the accuracy of the databases and the experiment (with smaller uncertainties than those of [84]). Using a burner and jet for gas radiation measurements eliminates the window problems, and is in many ways similar to the nozzle seal cell. Free jet devices can be used for extremely high temperatures, but they also introduce considerable uncertainty with respect to gas temperature and density distribution and to path length. Ferriso and Ludwig [241] used such a device for spectral measurements of the 2.7 μm water vapor band. Recently, Depraz and coworkers [242] designed an experiment to measure high-resolution (0.01–0.1 cm−1 ) emissivities from a microwave discharge producing a very well defined CO2 plasma with temperatures up to 5000 K. While

FIGURE 10.43 Schematic of nozzle seal gas containment system by Evseev and coworkers [93]. A representative IR ray emitted by the blackbody source passing through the HGC is shown by a line with arrows.

Radiative Properties of Molecular Gases Chapter | 10 385

comparison against CDSD-4000 [87,243] showed the general accuracy of that database, quantitative comparison was difficult because of experimental uncertainties. All multispectral diagnostic techniques discussed so far have employed single-detector monochromator or FTIR spectroscopy. Such devices can provide spectral scans in a wide range of resolutions and of great accuracy, but to obtain a spectrally resolved measurement with good signal-to-noise ratio takes tens of seconds for lowresolution narrow-band scans to hours for high-resolution full-spectrum measurements. Very few attempts have been made to date to obtain time-resolved multispectral signals from turbulent systems, because—to obtain snapshots of a turbulent flow field—exposure times must be of order of 0.1 ms or less. Richardson et al. [244,245] were perhaps the first to attempt such measurements, using a 32-element InSb linear array detector fitted with a grating monochromator. The apparatus was quite similar to the one shown in Fig. 10.39, except that there is no need to rotate the monochromator’s prism or grating, with the spectrally separated light hitting different elements of the array detector simultaneously. Their device was able to collect a 32-spectrum signal over 160 μs, storing 250 samples for each detector element. This resulted in an equivalent FTIR resolution of 32 cm−1 when collecting a spectrum of 250 cm−1 , with a signal-to-noise ratio of about 50. Their improved second device was able to hold 2048 full spectra collected every 16 μs. A similar apparatus was built by Keltner et al. [246], using a 256 × 256 MCT array detector. They argued that the use of (dual) prisms is preferable to grating monochromators in connection with array detectors. This dual prism arrangement was also used by Ji et al. [247], together with a 160-element PbSe linear array detector. The resulting high-speed spectrometer, is capable of taking near-instantaneous snapshots at a rate of 390 Hz. The device was calibrated against a blackbody, and spectra from a laminar premixed flame were compared with measurements using a grating spectrometer–InSb detector combination. Later measurements have been carried out with this high-speed infrared array spectrometer, to provide radiation data for the otherwise well-documented Sandia Workshop flames [248–250], and for a sooty ethylene air diffusion flame [251].

Data Correlation The half-width of a typical spectral line in the infrared is on the order of 0.1 cm−1 . To get a strong enough signal with a monochromator, any spectral measurement is by experimental necessity an average over several wavenumbers and, therefore, dozens or even hundreds of lines, unless an extremely monochromatic laser beam is employed. Thus, the measured transmissivity or (after subtracting from unity) absorptivity/emissivity is of the narrow band average type. Most FTIR measurements also fall into this category, although they generally have much better resolution than monochromators; resolutions better than 0.1 cm−1 are possible with high-end spectrometers. A correlation for the average absorption coefficient may be found by inverting equation (10.77) or equation (10.79), depending on whether the Elsasser or one of the statistical models is to be used, in either case yielding S δ

=

S δ

( η , X, γ/δ),

(10.208)

where the η and X (density or pressure path length) are measured quantities, and the width-to-spacing ratio must be determined independently. Most early measurements have assumed a constant γ/δ for the entire band, in which case the width-to-spacing ratio can be obtained in a number of ways: (i) direct prediction of γ and δ, (ii) using an independently determined band intensity, α, as the closing parameter, or (iii) finding a best fit for β (which is directly related to γ/δ) in the exponential wide band model. With the advent of high-resolution databases it has been recognized that line spacing can vary dramatically across a band. The first narrow band correlation with variable β was done by Brosmer and Tien [252] for propylene, using Goody’s model and least-mean-square-error fits. In medium-resolution measurements of CO2 Modest and Bharadwaj [81] correlated their experimental transmissivities to the Malkmus model through a least-mean-square-error fit. As an example the 2.7 μm bands of CO2 at 300 and 1000 K are shown in Fig. 10.44 and compared with data obtained from the most accurate databases of the time, HITRAN 1996 [75] and EM2C [41]. CO2 is seen to have two bands around 2.7 μm, one centered at 3615 cm−1 and the other at 3715 cm−1 . Agreement between experiment-based correlation and HITRAN 1996 is seen to be excellent except near the four S/δ peaks, where the absorption coefficient is dominated by a few widely spaced strong lines (about 1.8 cm−1 apart). This leads to a jagged appearance if the statistical definition

386 Radiative Heat Transfer

FIGURE 10.44 Narrow band correlation for the 2.7 μm band of carbon dioxide; experimental data from [81].

for S/δ is used, equation (10.91), and even if straight averaging over 4 cm−1 (equal to the experimental resolution) is carried out. The line labeled “HITRAN/FTIR avg.” was obtained by averaging the absorption coefficient with the FTIR’s instrument response function [253] as weight factor, which comes close to simulating the actual experiment. Results from the EM2C database are also shown for comparison. Because of its relatively low resolution of 25 cm−1 this database cannot capture the dual peaks, but agreement with experiment is excellent if the lower resolution is accounted for. Measured spectral absorptivities may be integrated to determine total band absorptances. Plotting those band absorptances that fall into the logarithmic regime vs. XPe on semilog paper gives a straight line whose √ slope is the band width parameter (cf. Table 10.3). Preparing a linear plot of A/Pe vs. X/Pe for data in the % square root regime gives again a straight line, this time with αωβ∗ as the slope (where β∗ = β/Pe = πγ/δ is the width-to-spacing ratio for a dilute mixture, cf. Tables 10.3 and 10.4). Finally, total emissivity values may be calculated by substituting the measured total band absorptances into equation (10.183).

Experimental Errors Most of the earlier gas property measurements were subject to considerable experimental errors, as listed by Edwards [237]: (1) inhomogeneity and uncertainty in the values of temperature, pressure, and composition, (2) scattering by mixing zones in nozzle seals and free jets, (3) reflection and scattering by optical windows, and/or (4) deterioration of the window material due to adsorption or “thermal runaway.” In addition, essentially all data until the 1980s were poorly correlated, using fixed values for γ/δ (across an entire vibration–rotation band), with a resulting correlational accuracy of ±20% at best. Only the more modern measurements by Phillips [134,135], Bharadwaj et al. [81,83,84], and Fateev and Clausen [64,93] apparently have experimental accuracies better than 5% and have been accurately correlated, with the two newest Danish devices shown in Figs. 10.40 and 10.43 apparently the most accurate.

Problems 10.1 Estimate the eigenfrequency for vibration, νe , for a CO molecule. 10.2 A certain gas at 1 bar pressure has a molecular mass of m = 10−22 g and a diameter of D = 5 × 10−8 cm. At what temperature would Doppler and collision broadening result in identical broadening widths for a line at a wavenumber of 4000 cm−1 ? 10.3 Water vapor is known to have spectral lines in the vicinity of λ = 1.38 μm. Consider a single, broadened spectral line centered at λ0 = 1.33 μm. If the water vapor is at a pressure of 0.1 atm and a temperature of 1000 K, what would

Radiative Properties of Molecular Gases Chapter | 10 387

you expect to be the main cause for broadening? Over what range of wavenumbers would you expect the line to be appreciable, i.e., over what range is the absorption coefficient at least 1% of its value at the line center? 10.4 Compute the half-width for a spectral line of CO2 at 2.8 μm for both Doppler and collision broadening as a function of pressure and temperature. Find the temperature as a function of pressure for which both broadening phenomena result in the same half-width. (Note: The effective diameter of the CO2 molecule is 4.0×10−8 cm.) 10.5 Methane is known to have a vibration-rotation band around 1.7 μm. It is desired to measure the Doppler half-width of a spectral line in that band at room temperature (T = 300 K). In order to make sure that collision broadening is negligible, the pressure of the CH4 is adjusted so that the expected collision half-width is only 1/10 of the Doppler half-width. What is this pressure? (For methane: D = 0.381 nm.) 10.6 Repeat Problem 10.4 for CO at a spectral location of 4.8 μm (Note: The effective diameter of the CO molecule is 3.4×10−8 cm.) 10.7 A certain gas has two important vibration–rotation bands centered at 4 μm and 10 μm. Measurements of spectral lines in the 4 μm band (taken at 300 K and 1 bar = 105 N/m2 ) indicate a half-width of γη = 0.5 cm−1 . Predict the half-width in the 10 μm band for the gas at 500 K, 3 bar. (The diameter of the gas molecules is known to be between 5 Å < D < 40 Å.) 10.8 It is desired to measure the volume fraction of CO in a hot gas by measuring the transmissivity of a 10 cm long column, using a blackbody source and a detector responsive around 4.7 μm. The conditions in the column are 1000 K, 1 atm, −1 and properties for CO around 4.7 μm are known to be S = 0.8 cm−2 atm , γ = 0.02 cm−1 , and δ = 0.05 cm−1 . Give an expression relating measured transmissivity to CO volume fraction. 10.9 A polyatomic gas has an absorption band in the infrared. For a certain small wavelength range the following is known: Average line half-width: 0.04 cm−1 , Average integrated absorption coefficient: 2.0 × 10−4 cm−1 /(g/m2 ), Average line spacing: 0.25 cm−1 , The density of the gas at STP is 3 × 10−3 g/cm3 . For a 50 cm thick gas layer at 500 K and 1 atm calculate the mean spectral emissivity for this wavelength range using (a) the Elsasser model, (b) the statistical model. Which result can be expected to be more accurate? 10.10 Consider a gas for which the semistatistical model is applicable, i.e., η = 1 − exp(−W η /δ). To predict η for arbitrary situations, a band-averaged (or constant) value for γη /δ must be known. Experimentally available are values for  α = Δη (Sη /δ) dη and η = η (η) (for optically thick situations) for given pe and T. It is also known that ⎛ ⎞  1/2 ⎜⎜ γη ⎟⎟ T0 ⎜⎝⎜ ⎟⎠⎟ pe . T δ δ 0

γη

Outline how an average value for (γη /δ) 0 can be found. 10.11 The following is known for a gas mixture at 600 K and 2 atm total pressure and in the vicinity of a certain spectral position: The gas consists of 80% (by volume) N2 and 20% of a diatomic absorbing gas with a molecular weight of 20 g/mol, a mean line half-width γ = 0.01 cm−1 , a mean line spacing of δ = 0.1 cm−1 , and a mean line strength of S = 8 × 10−5 cm−2 /(g/m3 ). (a) For a gas column 10 cm thick determine the mean spectral emissivity of the gas. (b) What happens if the pressure is increased to 20 atm? (Since no broadening parameters are known you may assume the effective broadening pressure to be equal to the total pressure.) 10.12 Repeat Problem 10.11 for a four-atomic gas. 10.13 1 kg of a gas mixture at 2000 K and 1 atm occupies a container of 1 m height. The gas consists of 70% nitrogen (by volume) and 30% of an absorbing species. It is known that, at a certain spectral location, the line half-width is γ = 300 MHz, the mean line spacing is δ = 2000 MHz, and the line strength is S = 100 cm−1 MHz. (a) Calculate the mean spectral emissivity under these conditions. (b) What will happen to the emissivity if the sealed container is cooled to 300 K? 10.14 A 50 cm thick layer of a pure gas is maintained at 1000 K and 1 atm. It is known that, at a certain spectral location, the mean line half-width is γ = 0.1 nm, the mean line spacing is δ = 2 nm, and the mean line strength is S = 0.002 cm−1 nm atm−1 = 2 × 10−10 atm−1 . What is the mean spectral emissivity under these conditions? (1 nm = 10−9 m)

388 Radiative Heat Transfer

10.15 The following data for a diatomic gas at 300 K and 1 atm are known: the mean line spacing is 0.6 cm−1 and the mean line half-width is 0.03 cm−1 ; the mean line strength (= integrated absorption coefficient) is 0.8 cm−2 atm−1 (based on a pressure absorption coefficient). Calculate the mean spectral emissivity for a path length of 1 cm. In what band approximation is the optical condition? 10.16 The average narrow band transmissivity of a homogeneous gas mixture has, at a certain wavenumber η, been measured as 0.70 for a length of 10 cm, and as 0.58 for a length of 20 cm. What is the expected transmissivity for a gas column of 30 cm length, assuming the Malkmus model to hold? 10.17 1 kg of a gas mixture at 2000 K and 1 atm occupies a container of 1 m height. The gas consists of 70% nitrogen (by volume) and 30% of an absorbing species. It is known that, at a certain spectral location, the nitrogen-broadening line half-width at STP (1 atm and 300 K) is γn0 = 0.05 cm−1 , the self-broadening line half-width is γa0 = 0.02 cm−1 , the mean line spacing is δ = 0.4 cm−1 , and the density and mean line strength (for the given mixture conditions) are ρ = 0.800 kg/m3 and S = 4 × 10−3 cm−1 /(g/m2 ), respectively. Under these conditions collision broadening is expected to dominate. (a) Calculate the mean spectral emissivity based on the height of the container. (b) What will happen to the emissivity if the sealed container is cooled to 300 K at constant pressure (with fixed container cross-section and sinking top end)? Note: The mean line intensity is directly proportional to the number of molecules of the absorbing gas and otherwise constant. The line half-width is given by & γ = [γn0 pn + γa0 pa ]

T0 T

(p in atm, T0 = 300 K),

where pn and pa are partial pressures of nitrogen and absorbing species. 10.18 A certain gas is known to behave almost according to the rigid-rotor/harmonic-oscillator model, resulting in gradually changing line strengths (with wavenumber) and somewhat irregular line spacing. Calculate the mean emissivity for a 1 m thick layer of the gas at 0.1 atm pressure. In the wavelength range of interest, it is known that the integrated absorption coefficient is equal to 0.80 cm−2 atm−1 , the line half-width is 0.04 cm−1 , and the average line spacing is 0.40 cm−1 . 10.19 Estimate the transmissivities of the Earth’s atmosphere for incoming (solar irradiation) and outgoing (Earth’s emission) radiation. Assume the atmosphere to consist of 9 homogeneous layers as provided in Table 10.6 (and ignore all other species). Write a computer program employing the statistical narrow band parameters of Rivière and Soufiani [132] given in App. F, together with the Curtis-Godson approximation, equation (10.105), as well as the k-moment method, equation (10.106), to account for the inhomogeneity of the atmosphere. Compare the values calculated here against the transmissivities listed in Section 10.12. 10.20 A narrow band of a certain absorbing gas contains a single spectral line of Lorentz shape at its center. For a narrow band width of Δη = 10γ, determine the corresponding reordered k vs. g distribution. Hint: This can be achieved without a lot of math. 10.21 Consider the spectral absorption coefficient for a narrow band range of Δη as given by the sketch. Carefully sketch the corresponding k-distribution. Determine the mean narrow band emissivity of a layer of thickness L from this k-distribution.

10.22 Consider the spectral absorption coefficient for a narrow band range of Δη as given by the sketch. Carefully sketch the corresponding k-distribution. Verify your sketch through calculations.

Radiative Properties of Molecular Gases Chapter | 10 389

10.23 Consider the (highly artificial) absorption coefficient shown. Mathematically, this may be expressed as κη = A[η + 3h(η)]

0 1000◦ C. 10.37 Estimate the total band absorptance of the 2.7 μm CO2 band at 833 K, a total pressure of 10 atm, a partial pressure of 1 atm, and a mass-path length of ρCO2 L = 2440 g/m2 , from Fig. 1.16. Compare with the result from the exponential wide band model. 10.38 A mixture of nitrogen and sulfur dioxide (with 5% SO2 by volume) is at 1 atm total pressure. To measure the temperature of the mixture in a furnace environment (T > 1000 K), an instrument is used that measures total band absorptance for the strong SO2 band √ centered at ηc = 1361 cm−1 . For that band it is known that α = 2340 (T0 /T) cm−2 atm−1 , √ β = 0.357 T/T0 Pe , ω = 8.8 T/T0 cm−1 , b = 1.28 and n = 0.65. What is the temperature of the mixture if the total band absorptance has been measured as 142 cm−1 for a 1 m thick gas layer?

Radiative Properties of Molecular Gases Chapter | 10 391

10.39 The Earth’s pollution with sulfur dioxide (SO2 )is determined by measuring the transmission of a light beam from a satellite. Assuming that the band absorptance of the 7.3 μm band has been measured as 10.0 cm−1 , and that the atmosphere may be approximated as a 10 km thick isothermal layer of nitrogen (with a trace of SO2 ) at 0.5 atm and the volume fraction of SO2 . Use wbmso2 from Appendix F to calculate the overlap parameter β or −10◦ C, determine √ use β 0.357 T/T0 Pe . 10.40 To determine the average atmospheric temperature on a distant planet, the total band absorptance for the 3.3 μm CH4 band has been measured as A3.3 = 100 cm−1 . It is known from other measurements that methane is a trace element in the atmosphere (which contains mostly nitrogen and whose total pressure is 2 atm), and that the absorption path length for methane on that planet, for which A3.3 was measured, is 4.14 g/m2 . What is the temperature? 10.41 Using the exponential wide band model, evaluate the total emissivity of a 1 m thick layer of a nitrogen–water vapor mixture at 2 atm and 400 K if the water vapor content by volume is (a) 0.01%, (b) 1%, or (c) 100%. Compare with Leckner’s model using subroutine totemiss. 10.42 Using the exponential wide band model, evaluate the total emissivity of a 1 m thick layer of a nitrogen–CO2 mixture at 0.75 atm and 600 K if the CO2 content by volume is (a) 0.01%, (b) 1%, or (c) 100%. Compare with Leckner’s model using subroutine totemiss. 10.43 Evaluate the Planck-mean absorption coefficients for the two gases in Problems 10.41 and 10.42, based on the data given in Table 10.4. Compare the results with Fig. 10.33. 10.44 Write a small computer program that calculates the total emissivity of a CO2 –inert gas mixture, based on wide band property data from Table 10.4, as a function of temperature, pressure, CO2 volume fraction, and path length. For a given set of pressure, volume fraction, and length, compare with values obtained from Leckner’s model using subroutine totemiss and plot the emissivity as a function of temperature. 10.45 Repeat Problem 10.41 for a path with a temperature profile given by T = 300 K[1 + 4s(L − s)/L2 ], where s is distance across the gas layer. ∞ 10.46 Develop a simple box model for the evaluation of the effective band width, i.e., A = 0 η dη = η Δη, based on an average emissivity (rather than absorption coefficient). You may assume that the line spacing and line intensity are constant across the band. Calculate the total band absorptance of water vapor at 0.1 atm and 400 K for path lengths of 1 mm and 1 m, assuming that Δη ≈ ω, where ω is the band width parameter from the exponential wide band model. Compare with results from that model. 10.47 Consider a mixture of nitrogen with 10% CO2 and/or 20% H2 O at 1 bar pressure. For a gas column of 100 cm length, prepare a plot of total emissivity vs. temperature (300 K ≤ T ≤ 1500 K), to evaluate the accuracy of the Leckner model, by comparing with calculations using the Alberti et al. [208] MixEmissivity Excel code. 10.48 Consider a mixture of nitrogen with 10% CO2 and/or 20% H2 O at 1 bar pressure. For a gas column of 100 cm length, prepare a plot of total emissivity vs. temperature (300 K ≤ T ≤ 1500 K), employing the EM2C database. (a) Use statistical narrow band parameters from the database to calculate narrow band transmissivities [together with multiplicative mixing, equation (10.97), when appropriate]; then determine the total emissivities by integrating over all narrow bands. (b) Evaluate the accuracy of the results, by comparing with calculations using the Alberti et al. [208] MixEmissivity Excel code.

References [1] [2] [3] [4] [5] [6] [7] [8]

D.K. Edwards, Molecular gas band radiation, in: Advances in Heat Transfer, vol. 12, Academic Press, New York, 1976, pp. 115–193. R.M. Goody, Y.L. Yung, Atmospheric Radiation – Theoretical Basis, 2nd ed., Oxford University Press, New York, 1989. C.L. Tien, Thermal radiation properties of gases, in: Advances in Heat Transfer, vol. 5, Academic Press, New York, 1968, pp. 253–324. J. Taine, A. Soufiani, Gas IR radiative properties: from spectroscopic data to approximate models, in: Advances in Heat Transfer, vol. 33, Academic Press, New York, 1999, pp. 295–414. L. Rayleigh, On the light from the sky, its polarization and colour, Philosophical Magazine 41 (1871) 107–120, 274–279, reprinted in in: Scientific Papers by Lord Rayleigh, vol. I: 1869–1881, No. 8, Dover, New York, 1964. S.P. Langley, Experimental determination of wave-lengths in the invisible prismatic spectrum, Memoirs of the National Academy of Sciences 2 (1883) 147–162. A.S. Eddington, The Internal Constitution of the Stars, Cambridge University Press, England, 1926, also Dover Publications, New York, 1959. S. Chandrasekhar, An Introduction to the Study of Stellar Structure, University of Chicago Press, 1939.

392 Radiative Heat Transfer

[9] S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960, originally published by Oxford University Press, London, 1950. [10] F. Paschen, Annalen der Physik und Chemie 53 (1894) 334. [11] H.C. Hottel, A.F. Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967. [12] H.C. Hottel, Heat transmission by radiation from non-luminous gases, Transactions of AIChE 19 (1927) 173–205. [13] H.C. Hottel, H.G. Mangelsdorf, Heat transmission by radiation from non-luminous gases II. Experimental study of carbon dioxide and water vapor, Transactions of AIChE 31 (1935) 517–549. [14] H.C. Hottel, V.C. Smith, Radiation from non-lumunious flames, Transactions of ASME, Journal of Heat Transfer 57 (1935) 463–470. [15] H.C. Hottel, I.M. Stewart, Space requirement for the combustion of pulverized coal, Industrial and Engineering Chemistry 32 (1940) 719–730. [16] H.C. Hottel, R.B. Egbert, The radiation of furnace gases, Transactions of ASME, Journal of Heat Transfer 63 (1941) 297–307. [17] H.C. Hottel, R.B. Egbert, Radiant heat transmission from water vapor, Transactions of AIChE 38 (1942) 531–565. [18] H.C. Hottel, Radiant heat transmission, in: W.H. McAdams (Ed.), Heat Transmission, 3rd ed., McGraw-Hill, New York, 1954, ch. 4. [19] H.C. Hottel, E.S. Cohen, Radiant heat exchange in a gas-filled enclosure: allowance for nonuniformity of gas temperature, AIChE Journal 4 (1958) 3–14. [20] S.S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities, Addison Wesley, Reading, MA, 1960. [21] G.N. Plass, Models for spectral band absorption, Journal of the Optical Society of America 48 (10) (1958) 690–703. [22] G.N. Plass, Spectral emissivity of carbon dioxide from 1800–2500 cm−1 , Journal of the Optical Society of America 49 (1959) 821–828. [23] C.L. Tien, J.H. Lienhard, Statistical Thermodynamics, rev. ed., McGraw–Hill Inc., New York, 1978. [24] N. Davidson, Statistical Mechanics, McGraw-Hill, New York, 1962, also Dover Publications, 2003. [25] W. Heitler, The Quantum Theory of Radiation, 3rd ed., Dover Publications, New York, 2010. [26] H.R. Griem, Spectral Line Broadening by Plasmas (Pure & Applied Physics), Academic Press, New York, 1974. [27] D. Salzmann, Atomic Physics in Hot Plasmas, Oxford University Press, New York, 1998. [28] H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Systems, Academic Press, New York, 1957, also Dover Publications, 2008. [29] G. Herzberg, Atomic Spectra and Atomic Structure, 2nd ed., Van Nostrand, New York, 1944, also Dover Publications, 2010. [30] G. Herzberg, Molecular Spectra and Molecular Structure, Vol. II: Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand, Princeton, NJ, 1945. [31] G. Herzberg, Molecular Spectra and Molecular Structure, Vol. I: Spectra of Diatomic Molecules, 2nd ed., Van Nostrand, Princeton, NJ, 1950, also Reitell Press, 2007. [32] L.S. Rothman, I.E. Gordon, A. Barbe, D.C. Benner, P.F. Bernath, M. Birk, V. Boudon, L.R. Brown, A. Campargue, J.-P. Champion, K. Chance, L.H. Coudert, V. Dana, V.M. Devi, S. Fally, J.-M. Flaud, R.R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W.J. Lafferty, J.-Y. Mandin, S.T. Massie, S.N. Mikhailenko, C.E. Miller, N. Moazzen-Ahmadi, O.V. Naumenko, A.V. Nikitin, J. Orphal, V.I. Perevalov, A. Perrin, A. Predoi-Cross, C.P. Rinsland, M. Rotger, M. Simeckova, M.A.H. Smith, K. Sung, S.A. Tashkun, J. Tennyson, R.A. Toth, A.C. Vandaele, J.V. Auwera, The HITRAN 2008 molecular spectroscopic database, Journal of Quantitative Spectroscopy and Radiative Transfer 110 (2009) 533–572. [33] C. Park, Stagnation-point radiation for apollo 4, Journal of Thermophysics and Heat Transfer 18 (1) (2004) 349–357. [34] A.M. Feldick, M.F. Modest, D.A. Levin, Closely coupled flowfield–radiation interactions during hypersonic reentry, Journal of Thermophysics and Heat Transfer 25 (4) (2011) 481–492. [35] F. Niro, N.F. Boulet, J.-M. Hartmann, Spectra calculations in central and wing regions of CO2 IR bands between 10 and 20 μm. I: model and laboratory measurements, Journal of Quantitative Spectroscopy and Radiative Transfer 88 (2004) 483–498. [36] A. Farooq, J.B. Jeffries, R.K. Hanson, High-pressure measurements of CO2 absorption near 2.7μm: line mixing and finite duration collision effects, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (7–8) (2010) 949–960. [37] R.G. Breene, Theories of Spectral Line Shape, John Wiley & Sons, New York, 1981. [38] L. Rosenmann, J.-M. Hartmann, M.-Y. Perrin, J. Taine, Accurate tabulation of IR and Raman CO2 line broadening by CO2 , H2 O, N2 , O2 in the 300–2400 K temperature range, Applied Optics 27 (1988) 3902–3907. [39] C. Delaye, J.-M. Hartmann, J. Taine, Calculated tabulations of H2 O line broadening by H2 O, N2 , O2 and CO2 at high temperature, Applied Optics 28 (1989) 5080–5087. [40] J.-M. Hartmann, L. Rosenmann, M.-Y. Perrin, J. Taine, Accurate calculated tabulations of CO line broadening by H2 O, N2 , O2 and CO2 in the 200–3000 K temperature range, Applied Optics 27 (1988) 3063–3065. [41] A. Soufiani, J. Taine, High temperature gas radiative property parameters of statistical narrow-band model for H2 O, CO2 and CO, and correlated-k model for H2 O and CO2 , International Journal of Heat and Mass Transfer 40 (4) (1997) 987–991. [42] L. Hartung-Chambers, Predicting radiative heat transfer in thermochemical nonequilibrium flow fields, NASA Technical Memorandum 4564, 1994. [43] A. Wang, M.F. Modest, Importance of combined Lorentz–Doppler broadening in high-temperature radiative heat transfer applications, ASME Journal of Heat Transfer 126 (5) (2004) 858–861. [44] S.R. Drayson, Rapid computation of the Voigt profile, Journal of Quantitative Spectroscopy and Radiative Transfer 16 (1976) 611–614. [45] J.H. Pierluissi, P.C. Vanderwood, R.B. Gomez, Fast calculational algorithm for the Voigt profile, Journal of Quantitative Spectroscopy and Radiative Transfer 18 (1977) 555–558.

Radiative Properties of Molecular Gases Chapter | 10 393

[46] A.K. Hui, B.H. Armstrong, A.A. Wray, Rapid computation of the Voigt and complex error functions, Journal of Quantitative Spectroscopy and Radiative Transfer 19 (1978) 509. [47] J. Humlí˘cek, Optimized computation of the Voigt and complex probability functions, Journal of Quantitative Spectroscopy and Radiative Transfer 27 (1982) 437. [48] J.-M. Hartmann, C. Boulet, D. Robert, Collisional Effects on Molecular Spectra, Elsevier, 2008. [49] J.-M. Hartmann, H. Tran, R. Armante, C. Boulet, A. Campargue, F. Forget, L. Gianfrani, I. Gordon, S. Guerlet, M. Gustafsson, J.T. Hodges, S. Kassi, D. Lisak, F. Thibault, G.C. Toon, Recent advances in collisional effects on spectra of molecular gases and their practical consequences, Journal of Quantitative Spectroscopy and Radiative Transfer 213 (2018) 178–227. [50] M.-Y. Perrin, J.-M. Hartmann, Temperature-dependent measurements and modeling of absorption by CO2 -N2 mixtures in the far line-wings of the 4.3 μm CO2 band, Journal of Quantitative Spectroscopy and Radiative Transfer 42 (4) (1989) 311–317. [51] J.-M. Hartmann, M.-Y. Perrin, Q. Ma, R.H. Tipping, The infrared continuum of pure water vapor: calculations and high-temperature measurements, Journal of Quantitative Spectroscopy and Radiative Transfer 49 (1993) 675–691. [52] M. Alberti, R. Weber, M. Mancini, Re-creating Hottel’s emissivity charts for carbon dioxide and extending them to 40 bar pressure using HITEMP-2010 database, Combustion and Flame 162 (2015) 597–612. [53] M. Alberti, R. Weber, M. Mancini, Re-creating Hottel’s emissivity charts for water vapor and extending them to 40 bar pressure using HITEMP-2010 database, Combustion and Flame 169 (2016) 141–153. [54] M. Alberti, R. Weber, M. Mancini, Absorption of infrared radiation by carbon monoxide at elevated temperatures and pressures: Part A. Advancing the line-by-line procedure based on HITEMP-2010, Journal of Quantitative Spectroscopy and Radiative Transfer 200 (2017) 258–271. [55] M. Fukabori, T. Nakazawa, M. Tanaka, Absorption properties of infrared active gases at high pressures—i CO2 , Journal of Quantitative Spectroscopy and Radiative Transfer 36 (1986) 265–270. [56] D. Scutaru, L. Rosenmann, J. Taine, R.B. Wattson, L.S. Rothman, Measurements and calculations of CO2 absorption at high temperature in the 4.3 and 2.7 μm regions, Journal of Quantitative Spectroscopy and Radiative Transfer 50 (1993) 179–191. [57] C. Brodbeck, J.P. Bouanich, V.T. Nguyen, J.-M. Hartmann, B. Khalil, R.L. Doucen, Absorption of radiation by gases from low to high pressures. II. Measurements and calculations of CO infrared spectras, Journal of Physics II France 4 (12) (1994) 2101–2118. [58] I.E. Gordon, L.S. Rothman, C. Hill, R.V. Kochanov, Y. Tan, P.F. Bernath, M. Birk, V. Boudon, A. Campargue, K.V. Chance, B.J. Drouin, J.-M. Flaud, R.R. Gamache, J.T. Hodges, D. Jacquemart, V.I. Perevalov, A. Perrin, K.P. Shine, M.-A.H. Smith, J. Tennyson, G.C. Toon, H. Tran, V.G. Tyuterev, A. Barbe, A.G. Császár, V.M. Devi, T. Furtenbacher, J.J. Harrison, J.-M. Hartmann, A. Jolly, T.J. Johnson, T. Karman, I. Kleiner, A.A. Kyuberis, J. Loos, O.M. Lyulin, S.T. Massie, S.N. Mikhailenko, N. Moazzen-Ahmadi, H.S.P. Müller, O.V. Naumenko, A.V. Nikitin, O.L. Polyansky, M. Rey, M. Rotger, S.W. Sharpe, K. Sung, E. Starikova, S.A. Tashkun, J. Vander Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, E.J. Zak, The HITRAN 2016 molecular spectroscopic database, Journal of Quantitative Spectroscopy and Radiative Transfer 203 (2017) 3–69. [59] L.S. Rothman, I.E. Gordon, R.J. Barber, H. Dothe, R.R. Gamache, A. Goldman, V.I. Perevalov, S.A. Tashkun, J. Tennyson, HITEMP, the high-temperature molecular spectroscopic database, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (15) (2010) 2139–2150. [60] S.J. Young, Band Model Theory of Radiation Transport, Aerospace Press, El Segundo, CA, 2013. [61] F.R. Westlye, B.A.K. Hartz, A. Ivarsson, A. Fateev, S. Clausen, Experimental and theoretical evaluation of spectral radiative transfer in high-pressure flames, Journal of Quantitative Spectroscopy and Radiative Transfer (2021), to appear. [62] J. Lamouroux, H. Tran, A.L. Laraia, R.R. Gamache, L.S. Rothman, I.E. Gordon, J.-M. Hartmann, Updated database plus software for line-mixing in CO2 infrared spectra and their test using laboratory spectra in the 1.5–2.3μm region, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (2011) 2321–2331. [63] J. Lamouroux, L. Régalia, X. Thomas, J.V. der Auwera, R.R. Gamache, J.-M. Hartmann, CO2 line-mixing database and software update and its tests in the 2.1μm and 4.3μm regions, Journal of Quantitative Spectroscopy and Radiative Transfer 151 (2015) 88–96. [64] C. Christiansen, T. Stolberg-Rohr, A. Fateev, S. Clausen, High temperature and high pressure gas cell for quantitative spectroscopic measurements, Journal of Quantitative Spectroscopy and Radiative Transfer 169 (2016) 96–103. [65] R. Ladenburg, F. Reiche, Über selektive Absorption, Annalen der Physik 42 (1913) 181. [66] J.-M. Lamet, P. Rivière, M.-Y. Perrin, A. Soufiani, Narrow-band model for nonequilibrium air plasma radiation, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (1) (2010) 87–104. [67] C. Park, Nonequilibrium Hypersonic Aerothermodynamics, Wiley, New York, 1990. [68] C.O. Johnston, B.R. Hollis, K.A. Sutton, Non-Boltzmann modeling for air shock-layer radiation at lunar-return conditions, Journal of Spacecraft and Rockets 45 (5) (2008) 879–890. [69] M. Panesi, T. Magin, A. Bourdon, A. Biltel, O. Chazot, Fire II flight experiment analysis by means of a collisional-radiative model, Journal of Thermophysics and Heat Transfer 23 (2) (2009) 236–248. [70] A. Bansal, M.F. Modest, Spectral modeling of radiative heat transfer in carbonaceous atmospheres using new k-distribution models, in: 49th AIAA Aerospace Sciences Conference, 2011, AIAA Paper 2011-0247. [71] M. Capitelli, Nonequilibrium Vibrational Kinetics, Topics in Current Physics, Springer Verlag, New York, 1986. [72] R.A. McClatchey, W.S. Benedict, S.A. Clough, D.E. Burch, K. Fox, L.S. Rothman, J.S. Garing, AFCRL atmospheric absorption line parameters compilation, Technical Report AFCRL-TR-0096, 1973.

394 Radiative Heat Transfer

[73] L.S. Rothman, R.R. Gamache, A. Goldman, L.R. Brown, R.A. Toth, H.M. Pickett, R.L. Poynter, J.-M. Flaud, C. Camy-Peyret, A. Barbe, N. Husson, C.P. Rinsland, M.A.H. Smith, The HITRAN database: 1986 edition, Applied Optics 26 (19) (1987) 4058–4097. [74] L.S. Rothman, R.R. Gamache, R.H. Tipping, C.P. Rinsland, M.A.H. Smith, D.C. Benner, V.M. Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S.T. Massie, L.R. Brown, R.A. Toth, The HITRAN molecular database: editions of 1991 and 1992, Journal of Quantitative Spectroscopy and Radiative Transfer 48 (5/6) (1992) 469–507. [75] L.S. Rothman, C.P. Rinsland, A. Goldman, S.T. Massie, D.P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-Y. Mandin, J. Schroeder, A. McCann, R.R. Gamache, R.B. Wattson, K. Yoshino, K.V. Chance, K.W. Jucks, L.R. Brown, V. Nemtchinov, P. Varanasi, The HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition, Journal of Quantitative Spectroscopy and Radiative Transfer 60 (1998) 665–710. [76] L.S. Rothman, A. Barbe, D.C. Benner, L.R. Brown, C. Camy-Peyret, M.R. Carleer, K. Chance, C. Clerbaux, V. Dana, V.M. Devi, A. Fayt, J.-M. Flaud, R.R. Gamache, A. Goldman, D. Jacquemart, K.W. Jucks, W.J. Lafferty, J.-Y. Mandin, S.T. Massie, V. Nemtchinov, D.A. Newnham, A. Perrin, C.P. Rinsland, J. Schroeder, K.M. Smith, M.A.H. Smith, K. Tang, R.A. Toth, J.V. Auwera, P. Varanasi, K. Yoshino, The HITRAN spectroscopic molecular database: edition of 2000 including updates through 2001, Journal of Quantitative Spectroscopy and Radiative Transfer 82 (1–4) (2003) 5–44. [77] L.S. Rothman, D. Jacquemart, A. Barbe, D.C. Benner, M. Birk, L.R. Brown, M.R. Carleer, C. Chackerian Jr., K. Chance, L.H. Coudert, V. Dana, V.M. Devi, J.-M. Flaud, R.R. Gamache, A. Goldman, J.-M. Hartmann, K.W. Jucks, A.G. Maki, J.-Y. Mandin, S.T. Massie, J. Orphal, A. Perrin, C.P. Rinsland, M.A.H. Smith, J. Tennyson, R.N. Tolchenov, R.A. Toth, J.V. Auwera, P. Varanasi, G. Wagner, The HITRAN 2004 molecular spectroscopic database, Journal of Quantitative Spectroscopy and Radiative Transfer 96 (2005) 139–204. [78] D. Scutaru, L. Rosenmann, J. Taine, Approximate band intensities of CO2 hot bands at 2.7, 4.3 and 12 μm for high temperature and medium resolution applications, Journal of Quantitative Spectroscopy and Radiative Transfer 52 (1994) 765–781. [79] P. Rivière, S. Langlois, A. Soufiani, J. Taine, An approximate data base of H2 O infrared lines for high temperature applications at low resolution. Statistical narrow-band model parameters, Journal of Quantitative Spectroscopy and Radiative Transfer 53 (1995) 221–234. [80] L.S. Rothman, R.B. Wattson, R.R. Gamache, J. Schroeder, A. McCann, HITRAN, HAWKS and HITEMP high temperature databases, Proceedings of SPIE 2471 (1995) 105–111. [81] M.F. Modest, S.P. Bharadwaj, High-resolution, high-temperature transmissivity measurements and correlations for carbon dioxide– nitrogen mixtures, Journal of Quantitative Spectroscopy and Radiative Transfer 73 (2–5) (2002) 329–338. [82] T. Fleckl, H. Jäger, I. Obernberger, Experimental verification of gas spectra calculated for high temperatures using the HITRAN/HITEMP database, Journal of Physics D: Applied Physics 35 (23) (2002) 3138–3144. [83] S.P. Bharadwaj, M.F. Modest, R.J. Riazzi, Medium resolution transmission measurements of water vapor at high temperature, ASME Journal of Heat Transfer 128 (2006) 374–381. [84] S.P. Bharadwaj, M.F. Modest, Medium resolution transmission measurements of CO2 at high temperature – an update, Journal of Quantitative Spectroscopy and Radiative Transfer 103 (2007) 146–155. [85] S.A. Tashkun, V.I. Perevalov, A.D. Bykov, N.N. Lavrentieva, J.-L. Teffo, Carbon Dioxide Spectroscopic databank (CDSD), available from ftp://ftp.iao.ru/pub/CDSD-1000, 2002. [86] S.A. Tashkun, V.I. Perevalov, Carbon dioxide spectroscopic databank (CDSD): updated and enlarged version for atmospheric applications, in: Tenth HITRAN Conference, Cambridge, MA, 2008, Paper T2.3, available from ftp://ftp.iao.ru/pub/CDSD-2008. [87] S.A. Tashkun, V.I. Perevalov, CDSD-4000: high-resolution, high-temperature carbon dioxide spectroscopic databank, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (9) (2011) 1403–1410, available from ftp://ftp.iao.ru/pub/CDSD-4000. [88] S.A. Tashkun, V.I. Perevalov, R.R. Gamache, J. Lamouroux, CDSD-296, high-resolution carbon dioxide spectroscopic databank: an update, Journal of Quantitative Spectroscopy and Radiative Transfer 228 (2019) 124–131. [89] H. Partridge, D.W. Schwenke, The determination of an accurate isotope dependent potential energy surface for water from extensive ab initio calculations and experimental data, Journal of Chemical Physics 106 (11) (1997) 4618–4639. [90] U.G. Jørgensen, P. Jensen, G.O. Sørensen, B. Aringer, H2 O in stellar atmospheres, Astronomy and Astrophysics 372 (2001) 249–259. [91] R.J. Barber, J. Tennyson, G.J. Harris, R.N. Tolchenov, A high-accuracy computed water line list, Monthly Notices of the Royal Astronomical Society 368 (2006) 1087–1094. [92] P. Perez, A. Boischot, L. Ibgui, A. Roblin, A spectroscopic database for water vapor adapted to spectral properties at high temperature, and moderate resolution, Journal of Quantitative Spectroscopy and Radiative Transfer 103 (2) (2007) 231–244. [93] V. Evseev, A. Fateev, S. Clausen, High-resolution transmission measurements of CO2 at high temperatures for industrial applications, Journal of Quantitative Spectroscopy and Radiative Transfer 113 (2012) 2222–2233. [94] M. Alberti, R. Weber, M. Mancini, M.F. Modest, Comparison of models for predicting band transmissivity of carbon dioxide and water vapour at high temperatures, International Journal of Heat and Mass Transfer 64 (2013) 910–925. [95] M. Alberti, R. Weber, M. Mancini, A. Fateev, S. Clausen, Validation of HITEMP-2010 for carbon dioxide and water vapour at high temperatures and atmospheric pressures in 450–7600cm−1 spectral range, Journal of Quantitative Spectroscopy and Radiative Transfer 157 (2015) 14–33. [96] V.R. Lecoustre, K. Wakatsuki, G.S. Jackson, Fitting narrow-band models to temperature-dependent, spectral absorption coefficients of fuel vapors, Journal of Quantitative Spectroscopy and Radiative Transfer 147 (2014) 24–37. [97] M.-Y. Perrin, A. Soufiani, Approximate radiative properties of methane at high temperature, Journal of Quantitative Spectroscopy and Radiative Transfer 103 (2007) 3–13.

Radiative Properties of Molecular Gases Chapter | 10 395

[98] L.S. Rothman, I.E. Gordon, Y. Babikov, A. Barbe, D.C. Benner, P.F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L.R. Brown, et al., The HITRAN 2012 molecular spectroscopic database, Journal of Quantitative Spectroscopy and Radiative Transfer 130 (2013) 4–50. [99] W.E. Nicolet, User’s manual for RAD/EQUIL/1973, a general purpose radiation transport program, NASA CR-132470, 1973. [100] E. Whiting, C. Park, Radiative heating at the stagnation point of the AFE vehicle, NASA Technical Memorandum 102829 1990, pp. 395–418. [101] E. Whiting, C. Park, Y. Liu, J. Arnold, J. Paterson, NEQAIR96, nonequilibrium and equilibrium radiative transport and spectra program: User’s manual, NASA reference publication 1389, NASA/Ames Research Center, Moffett Field, CA 94035-1000, December 1996. [102] C.O. Laux, Optical diagnostics and radiative emission of air plasmas, Ph.D. thesis, Stanford University, Stanford, CA, 1993. [103] C.O. Laux, Radiation and nonequilibrium collisional-radiative models, in: D. Fletcher, J.-M. Charbonnier, G.S.R. Sarma, T. Magin (Eds.), Physico-Chemical Modeling of High Enthalpy and Plasma Flows, in: von Karman Institute Lecture Series 2002-07, 2002, Rhode-Saint-Genèse, Belgium, available from http://specair-radiation.net/. [104] K. Fujita, T. Abe, SPRADIAN, structured package for radiation analysis: theory and application, Technical Report No. 669, Japanese Aerospace Exploration Agency (JAXA), 1997. [105] S.Y. Hyun, C. Park, K.S. Chang, H. Katsurayama, K. Fujita, T. Abe, User’s manual: Program SPRADIAN07 (structured package for radiation analysis 2007), Technical report, 2007. [106] C.O. Johnston, B.R. Hollis, K.A. Sutton, Spectrum modeling for air shock-layer radiation at lunar-return conditions, Journal of Spacecraft and Rockets 45 (5) (2008) 865–878. [107] Y. Ralchenko, A.E. Kramida, J. Reader, NIST atomic spectra database, version 4, National Institute of Standards and Technology (NIST), Physics Lab, 2010, available from http://www.nist.gov/pml/data/asd.cfm. [108] The Opacity Project Team, The opacity project, vol. 1, Technical report, Institute of Physics Publishing, Bristol and Philadelphia, 1995. [109] W. Cunto, C. Mendoza, F. Ochsenbein, C.J. Zeippen, TOPbase at the CDS, Astronomy and Astrophysics 275 (1993) L5–L8. [110] I. Sohn, A. Bansal, D.A. Levin, M.F. Modest, Advanced radiation calculations of hypersonic reentry flows using efficient databasing schemes, Journal of Thermophysics and Heat Transfer 24 (3) (2010) 623–637. [111] A. Bansal, A.M. Feldick, M.F. Modest, Simulation of hypersonic flow and radiation over a Mars reentry vehicle using OpenFOAM, in: 50th AIAA Aerospace Sciences Conference, 2012, Paper No. AIAA-2012-0650. [112] G.N. Schenker, B. Keller, Line-by-line calculations of the absorption of infrared radiation by water vapor in a box-shaped enclosure filled with humid air, International Journal of Heat and Mass Transfer 38 (1995) 3127–3134. [113] S. Chauveau, C. Deron, M.-Y. Perrin, P. Rivière, A. Soufiani, Radiative transfer in LTE air plasmas for temperatures up to 15,000 K, Journal of Quantitative Spectroscopy and Radiative Transfer 77 (2003) 113–130. [114] K. Beier, E. Lindermeir, Comparison of line-by-line and molecular band IR modeling of high altitude missile plume, Journal of Quantitative Spectroscopy and Radiative Transfer 105 (2007) 111–127. [115] J. Taine, A line-by-line calculation of low-resolution radiative properties of CO2 –CO–transparent nonisothermal gases mixtures up to 3000 K, Journal of Quantitative Spectroscopy and Radiative Transfer 30 (4) (1983) 371–379. [116] J.-M. Hartmann, R. Levi Di Leon, J. Taine, Line-by-line and narrow-band statistical model calculations for H2 O, Journal of Quantitative Spectroscopy and Radiative Transfer 32 (2) (1984) 119–127. [117] H. Zhang, M.F. Modest, Multi-group full-spectrum k-distribution database for water vapor mixtures in radiative transfer calculations, International Journal of Heat and Mass Transfer 46 (19) (2003) 3593–3603. [118] V.P. Solovjov, B.W. Webb, The cumulative wavenumber method for modeling radiative transfer in gas mixtures with soot, Journal of Quantitative Spectroscopy and Radiative Transfer 93 (1–3) (2005) 273–287. [119] F. André, R. Vaillon, The k-moment method for modeling the blackbody weighted transmission function for narrow and wide band radiative properties of gases, Journal of Quantitative Spectroscopy and Radiative Transfer 108 (1) (2007) 1–16. [120] T. Ozawa, M.F. Modest, D.A. Levin, Spectral module for photon Monte Carlo calculations in hypersonic nonequilibrium radiation, ASME Journal of Heat Transfer 132 (2010) 023406. [121] A.M. Feldick, M.F. Modest, A spectrally accurate tightly-coupled 2-D axisymmetric photon Monte-Carlo RTE solver for hypersonic entry flows, ASME Journal of Heat Transfer 134 (12) (2012) 122701. [122] T. Ren, M.F. Modest, Hybrid wavenumber selection scheme for line-by-line photon Monte Carlo simulations in high-temperature gases, ASME Journal of Heat Transfer 135 (8) (2013) 084501. [123] W.M. Elsasser, Heat Transfer by Infrared Radiation in the Atmosphere, Harvard University Press, Cambridge, MA, 1943. [124] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover Publications, New York, 1965. [125] R.M. Goody, A statistical model for water-vapour absorption, Quarterly Journal of the Royal Meteorological Society 78 (1952) 165. [126] W.L. Godson, The computation of infrared transmission by atmospheric water vapour: I and II, Journal of Meteorology 12 (1955) 272 and 533. [127] W. Malkmus, Random Lorentz band model with exponential-tailed S−1 line-intensity distribution function, Journal of the Optical Society of America 57 (3) (1967) 323–329. [128] W.L. Grosshandler, Radiative transfer in nonhomogeneous gases: a simplified approach, International Journal of Heat and Mass Transfer 23 (1980) 1447–1457. [129] W.L. Grosshandler, RADCAL: a narrow-band model for radiation calculations in a combustion environment, Technical Report NIST Technical Note 1402, National Institute of Standards and Technology, 1993.

396 Radiative Heat Transfer

[130] A. Soufiani, J.-M. Hartmann, J. Taine, Validity of band-model calculations for CO2 and H2 O applied to radiative properties and conductive–radiative transfer, Journal of Quantitative Spectroscopy and Radiative Transfer 33 (1985) 243–257. [131] A.A. Lacis, V. Oinas, A description of the correlated-k distribution method for modeling nongray gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres, Journal of Geophysical Research 96 (D5) (1991) 9027–9063. [132] P. Rivière, A. Soufiani, Updated band model parameters for H2 O, CO2 , CH4 and CO radiation at high temperature, International Journal of Heat and Mass Transfer 55 (2012) 3349–3358. [133] J.-L. Consalvi, F. Liu, A database of narrow-band parameters for fuels commonly encountered in fire applications, Fire Safety Journal 78 (2015) 202–218. [134] W.J. Phillips, Band model parameters of the 2.7 μm band of H2 O, Journal of Quantitative Spectroscopy and Radiative Transfer 43 (1990). [135] W.J. Phillips, Band model parameters of the 4.3 μm CO2 band in the 300–1000 K temperature range, Journal of Quantitative Spectroscopy and Radiative Transfer 48 (1992). [136] F. André, R. Vaillon, A simple and accurate method to derive SNB parameters for the radiative properties of gases, in: Proceedings of Eurotherm Seminar 78, Elsevier, Poitiers, France, April 2006. [137] S.P. Bharadwaj, M.F. Modest, A multiscale Malkmus model for treatment of inhomogeneous gas paths, International Journal of Thermal Sciences 46 (2007) 479–490. [138] P. Rivière, A. Soufiani, Generalized Malkmus line intensity distribution for CO2 infrared radiation in Doppler broadening regime, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (3) (2011) 475–485. [139] F. Liu, G.J. Smallwood, O.L. Gülder, Application of the statistical narrow-band correlated-k method to non-grey gas radiation in CO2 mixtures: approximate treatments of overlapping bands, Journal of Quantitative Spectroscopy and Radiative Transfer 68 (2001) 401–417. [140] D.E. Burch, J.N. Howard, D. Williams, Infrared transmission of synthetic atmospheres. V. Absorption laws for overlapping bands, Journal of the Optical Society of America 46 (1956) 452–455. [141] S.J. Young, Nonisothermal band model theory, Journal of Quantitative Spectroscopy and Radiative Transfer 18 (1977) 1–28. [142] G.H. Lindquist, F.S. Simmons, A band model formulation for very non uniform paths, Journal of Quantitative Spectroscopy and Radiative Transfer 12 (1972) 807–820. [143] F. André, R. Vaillon, The k-moment method for the narrow band modeling of radiative properties of nonuniform gaseous media, Journal of Quantitative Spectroscopy and Radiative Transfer 109 (2) (2008) 258–268. [144] F. André, L. Hou, M. Roger, R. Vaillon, The multispectral gas radiation modeling: a new theoretical framework based on a multidimensional approach to k-distribution methods, Journal of Quantitative Spectroscopy and Radiative Transfer 147 (2014) 178–195. [145] F. André, Effective scaling factors in non-uniform gas radiation modeling, Journal of Quantitative Spectroscopy and Radiative Transfer 206 (2018) 105–116. [146] A. Arking, K. Grossman, The influence of line shape and band structure on temperatures in planetary atmospheres, Journal of the Atmospheric Sciences 29 (1972) 937–949. [147] K.Y. Kondratyev, Radiation in the Atmosphere, Academic Press, New York, 1969. [148] R.M. Goody, R. West, L. Chen, D. Crisp, The correlated k method for radiation calculations in nonhomogeneous atmospheres, Journal of Quantitative Spectroscopy and Radiative Transfer 42 (1989) 539–550. [149] Q. Fu, K.N. Liou, On the correlated k-distribution method for radiative transfer in nonhomogeneous atmospheres, Journal of the Atmospheric Sciences 49 (22) (1992) 2139–2156. [150] P. Rivière, A. Soufiani, J. Taine, Correlated-k and fictitious gas methods for H2 O near 2.7 μm, Journal of Quantitative Spectroscopy and Radiative Transfer 48 (1992) 187–203. [151] P. Rivière, D. Scutaru, A. Soufiani, J. Taine, A new ck data base suitable from 300 to 2500 K for spectrally correlated radiative transfer in CO2 –H2 O transparent gas mixtures, in: Tenth International Heat Transfer Conference, Taylor & Francis, 1994, pp. 129–134. [152] P. Rivière, A. Soufiani, J. Taine, Correlated-k and fictitious gas model for H2 O infrared radiation in the Voigt regime, Journal of Quantitative Spectroscopy and Radiative Transfer 53 (1995) 335–346. [153] F. André, V.P. Solovjov, L. Hou, R. Vaillon, D. Lemonnier, The generalized k-moment method for the modeling of cumulative kdistributions of H2 O at high temperature, Journal of Quantitative Spectroscopy and Radiative Transfer 143 (2014) 92–99. [154] G.A. Domoto, Frequency integration for radiative transfer problems involving homogeneous non-gray gases: the inverse transmission function, Journal of Quantitative Spectroscopy and Radiative Transfer 14 (1974) 935–942. [155] A. Wang, M.F. Modest, High-accuracy, compact database of narrow-band k-distributions for water vapor and carbon dioxide, Journal of Quantitative Spectroscopy and Radiative Transfer 93 (2005) 245–261. [156] J. Cai, M.F. Modest, Improved full-spectrum k-distribution implementation for inhomogeneous media using a narrow-band database, Journal of Quantitative Spectroscopy and Radiative Transfer 141 (2013) 65–72. [157] V.P. Solovjov, B.W. Webb, SLW modeling of radiative transfer in multicomponent gas mixtures, Journal of Quantitative Spectroscopy and Radiative Transfer 65 (2000) 655–672. [158] M.F. Modest, R.J. Riazzi, Assembly of full-spectrum k-distributions from a narrow-band database; effects of mixing gases, gases and nongray absorbing particles, and mixtures with nongray scatterers in nongray enclosures, Journal of Quantitative Spectroscopy and Radiative Transfer 90 (2) (2005) 169–189.

Radiative Properties of Molecular Gases Chapter | 10 397

[159] G. Pal, M.F. Modest, k-distribution methods for radiation calculations in high pressure combustion, in: 50th Aerospace Sciences Meeting, 2012, Paper No. AIAA-2012-0529. [160] M.F. Modest, H. Zhang, The full-spectrum correlated-k distribution for thermal radiation from molecular gas–particulate mixtures, ASME Journal of Heat Transfer 124 (1) (2002) 30–38. [161] M.F. Modest, Narrow-band and full-spectrum k-distributions for radiative heat transfer—correlated-k vs. scaling approximation, Journal of Quantitative Spectroscopy and Radiative Transfer 76 (1) (2003) 69–83. [162] T.C. Hsieh, R. Greif, Theoretical determination of the absorption coefficient and the total band absorptance including a specific application to carbon monoxide, International Journal of Heat and Mass Transfer 15 (1972) 1477–1487. [163] K.H. Chu, R. Greif, Theoretical determination of band absorption for nonrigid rotation with applications to CO, NO, N2 O, and CO2 , ASME Journal of Heat Transfer 100 (1978) 230–234. [164] D.K. Edwards, W.A. Menard, Comparison of models for correlation of total band absorption, Applied Optics 3 (1964) 621–625. [165] D.K. Edwards, L.K. Glassen, W.C. Hauser, J.S. Tuchscher, Radiation heat transfer in nonisothermal nongray gases, ASME Journal of Heat Transfer 89 (1967) 219–229. [166] M.M. Weiner, D.K. Edwards, Non-isothermal gas radiation in superposed vibration–rotation bands, Journal of Quantitative Spectroscopy and Radiative Transfer 8 (1968) 1171–1183. [167] D.K. Edwards, Radiative transfer characteristics of materials, ASME Journal of Heat Transfer 91 (1969) 1–15. [168] D.K. Edwards, A. Balakrishnan, Thermal radiation by combustion gases, International Journal of Heat and Mass Transfer 16 (1973) 25–40. [169] J.D. Felske, C.L. Tien, A theoretical closed form expression for the total band absorptance of infrared-radiating gases, ASME Journal of Heat Transfer 96 (1974) 155–158. [170] W.C. Wang, An analytical expression for the total band absorptance of infrared-radiating gases, Journal of Quantitative Spectroscopy and Radiative Transfer 29 (1983) 279–281. [171] A.T. Modak, Exponential wide band parameters for the pure rotational band of water vapor, Journal of Quantitative Spectroscopy and Radiative Transfer 21 (1979) 131–142. [172] D.K. Edwards, W.A. Menard, Correlations for absorption by methane and carbon dioxide gases, Applied Optics 3 (1964) 847–852. [173] D.K. Edwards, W.A. Menard, Correlation of absorption by water vapor at temperatures from 300 K to 1100 K, Applied Optics 4 (1965) 715–721. [174] M.A. Brosmer, C.L. Tien, Infrared radiation properties of methane at elevated temperatures, Journal of Quantitative Spectroscopy and Radiative Transfer 33 (5) (1985) 521–532. [175] S.P. Fuss, O.A. Ezekoye, M.J. Hall, The absorptance of infrared radiation by methane at elevated temperatures, ASME Journal of Heat Transfer (1996) 918–923. [176] D.K. Edwards, Absorption of radiation by carbon monoxide gas according to the exponential wide-band model, Applied Optics 4 (10) (1965) 1352–1353. [177] M.M. Abu-Romia, C.L. Tien, Measurements and correlations of infrared radiation of carbon monoxide at elevated temperatures, Journal of Quantitative Spectroscopy and Radiative Transfer 6 (1966) 143–167. [178] M.M. Abu-Romia, C.L. Tien, Appropriate mean absorption coefficients for infrared radiation of gases, ASME Journal of Heat Transfer 89C (1967) 321–327. [179] S.H. Chan, C.L. Tien, Infrared radiation properties of sulfur dioxide, ASME Journal of Heat Transfer 93 (1971) 172–177. [180] C.L. Tien, Band and total emissivity of ammonia, International Journal of Heat and Mass Transfer 16 (1973) 856–857. [181] R.M. Green, C.L. Tien, Infrared radiation properties of nitric oxide at elevated temperatures, Journal of Quantitative Spectroscopy and Radiative Transfer 10 (1970) 805–817. [182] C.L. Tien, M.F. Modest, C.R. McCreight, Infrared radiation properties of nitrous oxide, Journal of Quantitative Spectroscopy and Radiative Transfer 12 (1972) 267–277. [183] M.A. Brosmer, C.L. Tien, Thermal radiation properties of acetylene, ASME Journal of Heat Transfer 107 (1985) 943–948. [184] N. Lallemant, R. Weber, A computationally efficient procedure for calculating gas radiative properties using the exponential wide band model, International Journal of Heat and Mass Transfer 39 (1996) 3273–3286. [185] C.L. Tien, J.E. Lowder, A correlation for total band absorption of radiating gases, International Journal of Heat and Mass Transfer 9 (1966) 698–701. [186] R.D. Cess, S.N. Tiwari, Infrared radiative energy transfer in gases, in: Advances in Heat Transfer, vol. 8, Academic Press, New York, 1972, pp. 229–283. [187] S.N. Tiwari, Models for infrared atmospheric radiation, in: Advances in Geophysics, vol. 20, Academic Press, New York, 1978. [188] S.H. Chan, C.L. Tien, Total band absorptance of non-isothermal infrared-radiating gases, Journal of Quantitative Spectroscopy and Radiative Transfer 9 (1969) 1261–1271. [189] R.D. Cess, L.S. Wang, A band absorptance formulation for non-isothermal gaseous radiation, International Journal of Heat and Mass Transfer 13 (1970) 547–555. [190] D.K. Edwards, S.J. Morizumi, Scaling vibration–rotation band parameters for nonhomogeneous gas radiation, Journal of Quantitative Spectroscopy and Radiative Transfer 10 (1970) 175–188. [191] J.D. Felske, C.L. Tien, Infrared radiation from non-homogeneous gas mixtures having overlapping bands, Journal of Quantitative Spectroscopy and Radiative Transfer 14 (1974) 35–48.

398 Radiative Heat Transfer

[192] W.C. Wang, G.Y. Shi, Total band absorptance and k-distribution function for atmospheric gases, Journal of Quantitative Spectroscopy and Radiative Transfer 39 (1988) 387–398. [193] O. Marin, R.O. Buckius, Wide band correlated-k method applied to absorbing, emitting and scattering media, Journal of Thermophysics and Heat Transfer 10 (1996) 364–371. [194] O. Marin, R.O. Buckius, A model of the cumulative distribution function for wide band radiative properties, Journal of Quantitative Spectroscopy and Radiative Transfer 59 (1998) 671–685. [195] O. Marin, R.O. Buckius, A simplified wide band model of the cumulative distribution function for water vapor, International Journal of Heat and Mass Transfer 41 (1998) 2877–2892. [196] O. Marin, R.O. Buckius, A simplified wide band model of the cumulative distribution function for carbon dioxide, International Journal of Heat and Mass Transfer 41 (1998) 3881–3897. [197] P.Y.C. Lee, G.D. Raithby, K.G.T. Hollands, The “reordering” concept of the absorption coefficient for modelling nongray gases, in: Radiative Heat Transfer: Current Research, vol. HTD-276, ASME, 1994, pp. 21–30. [198] P.Y.C. Lee, K.G.T. Hollands, G.D. Raithby, Reordering the absorption coefficient within the wide band for predicting gaseous radiant exchange, ASME Journal of Heat Transfer 118 (2) (1996) 394–400. [199] G. Parthasarathy, J.C. Chai, S.V. Patankar, A simple approach to nongray gas modeling, Numerical Heat Transfer 29 (1996) 394–400. [200] M.K. Denison, W.A. Fiveland, A correlation for the reordered wave number of the wideband absorptance of radiating gases, ASME Journal of Heat Transfer 119 (1997) 853–856. [201] J. Ströhle, P.J. Coelho, On the application of the exponential wide band model to the calculation of radiative heat transfer in one- and two-dimensional enclosures, International Journal of Heat and Mass Transfer 45 (2002) 2129–2139. [202] B. Leckner, Spectral and total emissivity of water vapor and carbon dioxide, Combustion and Flame 19 (1972) 33–48. [203] F.P. Boynton, C.B. Ludwig, Total emissivity of hot water vapor – II, semi-empirical charts deduced from long-path spectral data, International Journal of Heat and Mass Transfer 14 (1971) 963–973. [204] C.B. Ludwig, W. Malkmus, J.E. Reardon, J.A.L. Thomson, Handbook of infrared radiation from combustion gases, Technical Report SP-3080, NASA, 1973. [205] A.F. Sarofim, I.H. Farag, H.C. Hottel, Radiative heat transmission from nonluminous gases. Computational study of the emissivities of carbon dioxide, ASME paper no. 78-HT-55, 1978. [206] A.T. Modak, Radiation from products of combustion, Fire Research 1 (1979) 339–361. [207] M. Alberti, R. Weber, M. Mancini, Overlap corrections for emissivity calculations of H2 O–CO2 –CO–N2 mixtures, Journal of Quantitative Spectroscopy and Radiative Transfer 217 (2017) 1–11. [208] M. Alberti, R. Weber, M. Mancini, Gray gas emissivities for H2 O–CO2 –CO–N2 mixtures, Journal of Quantitative Spectroscopy and Radiative Transfer 219 (2018) 274–291. [209] M. Alberti, R. Weber, M. Mancini, New formulae for gray gas absorptivities of H2 O, CO2 and CO, Journal of Quantitative Spectroscopy and Radiative Transfer 255 (2020) 107227. [210] H. Zhang, M.F. Modest, Evaluation of the Planck-mean absorption coefficients from HITRAN and HITEMP databases, Journal of Quantitative Spectroscopy and Radiative Transfer 73 (6) (2002) 649–653. [211] R.D. Cess, P. Mighdoll, Modified Planck mean coefficients for optically thin gaseous radiation, International Journal of Heat and Mass Transfer 10 (1967) 1291–1292. [212] D.L. Hartmann, Global Physical Climatology, second ed., Elsevier, 2015. [213] T. Machida, K. Kita, Y. Kondo, D. Blake, S. Kawakami, G. Inoue, T. Ogawa, Vertical and meridional distributions of the atmospheric CO2 mixing ratio between northern midlatitudes and southern subtropics, Journal of Geophysical Research 108 (2003) 8401–8410. [214] Y. Shibata, C. Nagasawa, M. Abo, M. Inoue, I. Morino, O. Uchino, Comparison of CO2 vertical profiles in the lower troposphere between 1.6 μm differential absorption Lidar and aircraft measurements over Tsukuba, Sensors 18 (2018) 4064–4073. [215] L. Palchetti, G. Bianchini, B. Carli, U. Cortesi, S. Del Bianco, Measurement of the water vapour vertical profile and of the Earth’s outgoing far infrared flux, Atmospheric Chemistry and Physics 8 (2008) 2885–2894. [216] A. Buis, Earth’s atmosphere: a multi-layered cake, https://climate.nasa.gov/news/2919/earths-atmosphere-a-multi-layered-cake/, 2019. [217] Intergovernmental Panel on Climate Change (IPCC) Report, https://www.ipcc.ch/2019/, 2019. [218] National Centers for Environmental Information (NCEI), https://www.ncdc.noaa.gov/, 2020. [219] A. Henderson-Sellers (Ed.), World Survey of Climatology, Elsevier, 1995, pp. 245–279, ch. 7. [220] G. North, J. Pyle, F. Zhang (Eds.), Encyclopedia of Atmospheric Sciences, 2d ed., Academic Press, New York, 2015, pp. 18–25. [221] M. Etminan, G. Myhre, E.J. Highwood, K.P. Shine, Radiative forcing of carbon dioxide, methane, and nitrous oxide: a significant revision of the methane radiative forcing, Geophysical Research Letters 43 (2016) 12614–12623. [222] D.R. Feldman, W.D. Collins, P.J. Gero, M.S. Torn, E.J. Mlawer, T.R. Shippert, Observational determination of surface radiative forcing by CO2 from 2000 to 2010, Nature 519 (2015) 339–341. [223] M.G. Mlynczak, T.S. Daniels, D.P. Kratz, D.R. Feldman, W.D. Collins, E.J. Mlawer, M.J. Alvarado, J.E. Lawler, L.W. Anderson, D.W. Fahey, L.A. Hunt, J.C. Mast, The spectroscopic foundation of radiative forcing of climate by carbon dioxide, Geophysical Research Letters 43 (2016) 5318–5325. [224] E. Specht, T. Redemann, N. Lorenz, Simplified mathematical model for calculating global warming through anthropogenic CO2 , International Journal of Thermal Sciences 102 (2016) 1–8.

Radiative Properties of Molecular Gases Chapter | 10 399

[225] C.L. Tien, W.H. Giedt, Experimental determination of infrared absorption of high-temperature gases, in: Advances in Thermophysical Properties at Extreme Temperatures and Pressures, ASME, 1965, pp. 167–173. [226] D.B. Tanner, R.P. McCall, Source of a problem with Fourier transform spectroscopy, Applied Optics 23 (14) (1994) 2363–2368. [227] C.P. Tripp, R.A. McFarlane, Discussion of the stray light rejection efficiency of FT-IR spectrometers: the effects of sample emission on FT-IR spectra, Applied Spectroscopy 48 (9) (1994) 1138–1142. [228] R.J. Anderson, P.R. Griffiths, Determination of rotational temperatures of diatomic molecules from absorption spectra measured at moderate resolution, Journal of Quantitative Spectroscopy and Radiative Transfer 17 (1977) 393–401. [229] L.A. Gross, P.R. Griffiths, Temperature estimation of carbon dioxide by infrared absorption spectrometry at medium resolution, Journal of Quantitative Spectroscopy and Radiative Transfer 39 (2) (1988) 131–138. [230] P.J. Medvecz, K.M. Nichols, D.T. Clay, R. Atalla, Determination of gas temperatures at 295–1273 K using CO vibrational–rotational absorption spectra recorded with an FT-IR spectrometer, Applied Spectroscopy 45 (8) (1991) 1350–1359. [231] R.J. Goldstein, Measurements of infrared absorption by water vapor at temperatures to 1000 K, Journal of Quantitative Spectroscopy and Radiative Transfer 4 (1964) 343–352. [232] U.P. Oppenheim, A. Goldman, Spectral emissivity of water vapor at 1200 K, in: Tenth Symposium (International) on Combustion, The Combustion Institute, 1965, pp. 185–188. [233] M.M. Abu-Romia, C.L. Tien, Spectral and integrated intensity of CO fundamental band at elevated temperatures, International Journal of Heat and Mass Transfer 10 (1967) 1779–1784. [234] M.A. Postlethwait, K.K. Sikka, M.F. Modest, J.R. Hellmann, High temperature normal spectral emittance of silicon carbide based materials, Journal of Thermophysics and Heat Transfer 8 (3) (1994) 412–418. [235] M.A. Postlethwait, M.F. Modest, M.A. Botch, J.R. Hellmann, Normal spectral emittance of alumina based materials, in: Radiative Heat Transfer: Current Research; 6th AIAA/ASME Thermophysics and Heat Transfer Conference, vol. HTD-276, ASME, 1994, pp. 73–77. [236] M.J. Challingsworth, J.R. Hellmann, M.F. Modest, Tailoring the spectral emittance of rare Earth oxides via doping, in: Proceedings of the 97th Annual Meeting and Exposition of the American Ceramic Society, 1995. [237] D.K. Edwards, Thermal radiation measurements, in: E.R.G. Eckert, R.J. Goldstein (Eds.), Measurements in Heat Transfer, Hemisphere, Washington, DC, 1976, ch. 10. [238] E.R.G. Eckert, Messung der Gesamtstrahlung von Wasserdampf und Kohlensäure in Mischung mit nichtstrahlenden Gasen bei Temperaturen bis 1300◦ C, VDI Forschungshefte 387 (1937) 1–20. [239] J.T. Bevans, R.V. Dunkle, D.K. Edwards, J.T. Gier, L.L. Levenson, A.K. Oppenheim, Apparatus for the determination of the band absorption of gases at elevated pressures and temperatures, Journal of the Optical Society of America 50 (1960) 130–136. [240] D.K. Edwards, Absorption by infrared bands of carbon dioxide gas at elevated pressures and temperatures, Journal of the Optical Society of America 50 (1960) 617–626. [241] C.C. Ferriso, C.B. Ludwig, Spectral emissivities and integrated intensities of the 2.7 μm H2 O band between 530 and 2200 K, Journal of Quantitative Spectroscopy and Radiative Transfer 4 (1964) 215–227. [242] S. Depraz, M.-Y. Perrin, A. Soufiani, Infrared emission spectroscopy of CO2 at high temperature. Part I: Experimental setup and source characterization, Journal of Quantitative Spectroscopy and Radiative Transfer 113 (2012) 1–13. [243] S. Depraz, M.-Y. Perrin, P. Rivière, A. Soufiani, Infrared emission spectroscopy of CO2 at high temperature. Part II: Experimental results and comparisons with spectroscopic databases, Journal of Quantitative Spectroscopy and Radiative Transfer 113 (2012) 14–25. [244] H.H. Richardson, V.W. Pabst, J.A. Butcher, A novel infrared spectrometer using a linear array detector, Applied Spectroscopy 44 (5) (1990) 822–825. [245] S.M. Alawi, T. Krug, H.H. Richardson, Characterization and application of an infrared linear array spectrometer for time-resolved infrared spectroscopy, Applied Spectroscopy 47 (10) (1993) 1626–1630. [246] Z. Keltner, K. Kayima, A. Lanzarotta, L. Lavalle, M. Canepa, A.E. Dowrey, G.M. Story, C. Marcott, A.J. Sommer, Prism-based infrared spectrographs using modern-day detectors, Applied Spectroscopy 61 (9) (2007) 909–915. [247] J. Ji, J.P. Gore, Y.R. Sivathanu, J. Lim, Fast infrared array spectrometer with a thermoelectrically cooled 160-element pbse detector, Review of Scientific Instruments 75 (2) (2004) 333–339. [248] R.S. Barlow, International Workshop on Measurement and Computation of Turbulent Nonpremixed Flames (TNF), website, http:// www.sandia.gov/TNF/abstract.html. [249] Y. Zheng, R.S. Barlow, J.P. Gore, Measurements and calculations of spectral radiation intensities for turbulent non-premixed and partially premixed flames, ASME Journal of Heat Transfer 125 (2003) 678–686. [250] Y. Zheng, R.S. Barlow, J.P. Gore, Spectral radiation properties of partially premixed turbulent flames, ASME Journal of Heat Transfer 125 (2003) 1065–1073. [251] Y. Zheng, J.P. Gore, Measurements and inverse calculations of spectral radiation intensities of a turbulent ethylene/air jet flame, in: Thirtieth Symposium (International) on Combustion, The Combustion Institute, 2005, pp. 727–734. [252] M.A. Brosmer, C.L. Tien, Thermal radiation properties of propylene, Combustion Science and Technology 48 (1986) 163–175. [253] P.R. Griffiths, J.A. de Haseth, Fourier Transform Infrared Spectrometry, Chemical Analysis, vol. 83, John Wiley & Sons, New York, 1986.

Chapter 11

Radiative Properties of Particulate Media 11.1 Introduction When an electromagnetic wave or a photon interacts with a medium containing small particles, the radiative intensity may be changed by absorption and/or scattering. Common examples of this interaction are sunlight being absorbed by a cloud of smoke (which is nothing but a multitude of fine particles suspended in air), scattering of sunshine by the atmosphere (the atmosphere consists of molecules which are, in fact, tiny particles) resulting in blue skies and red sunsets, and the colors of the rainbow. Radiation scattering by particles was first dealt with by astrophysicists, who were interested in the scattering of starlight by interstellar dust. Scientists from many other disciplines are concerned with the scattering of electromagnetic waves: Meteorologists are concerned with scattering within the Earth’s atmosphere (scattering of sunlight as well as scattering of radar waves for observation of precipitation); electrical engineers and physicists deal with the propagation of radio waves through the atmosphere; physicists, chemists, and engineers today use light scattering as diagnostic tools for nonintrusive and nondestructive measurements in gases, liquids, and solids. Reviews of thermal radiation phenomena in particulate media have been given by Tien and Drolen [1], and also by Dombrovsky and Baillis [2]. How much and into which direction a particle scatters an electromagnetic wave passing through its vicinity depends on (i) the shape of the particle, (ii) the material of the particle (i.e., the complex index of refraction, m = n − ik), (iii) its size relative to wavelength, and (iv) the clearance between particles. In radiative analyses the shape of particles is usually assumed to be spherical (for spherical and irregularly shaped objects) or cylindrical (for long fibrous materials). These simplifying assumptions give generally excellent results, since averaging over many millions of irregular shapes tends to smoothen the irregularities [1]. In the following discussion we shall primarily consider absorption and scattering by spherical particles, as shown in Fig. 11.1. An electromagnetic wave or photon passing through the immediate vicinity of spherical particles will be absorbed or scattered. The scattering is due to three separate phenomena, namely, (i) diffraction (waves never come into contact with the particle, but their direction of propagation is altered by the presence of the particle), (ii) reflection by a particle (waves reflected from the surface of the sphere), and (iii) refraction in a particle (waves that penetrate into the sphere and, after partial absorption, reemerge traveling into a different direction). The vast majority of photons are scattered elastically, i.e., their wavelength (and energy) remain unchanged. A tiny fraction undergoes inelastic or Raman scattering (the photons reemerge with a different wavelength). While very important for optical diagnostics, the Raman effect is unimportant for the evaluation of radiative heat transfer rates, and we shall treat only elastic scattering in this book. If scattering by one particle is not affected by the presence of surrounding particles, we speak of independent scattering, otherwise we have dependent scattering. Thus, the radiative properties of a cloud of spherical particles of radius a, interacting with an electromagnetic

FIGURE 11.1 Interaction between electromagnetic waves and spherical particles. Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00019-5 Copyright © 2022 Elsevier Inc. All rights reserved.

401

402 Radiative Heat Transfer

FIGURE 11.2 Scattering regime map for independent and dependent scattering [1].

wave of wavelength λ, are governed by three independent nondimensional parameters: complex index of refraction: m = n − ik, size parameter: x = 2πa/λ, clearance-to-wavelength ratio: c/λ.

(11.1) (11.2) (11.3)

If scattering is independent (c/λ  1), then only the first two parameters are needed. For the classification of dependent scattering, the clearance-to-wavelength ratio is often replaced by a purely geometric parameter, c/a, which in turn may be related to the volume fraction of particles, fv . While in earlier works, for example that by van de Hulst [3], it was assumed that dependent effects were a function of particle separation only, it is now known that wavelength effects also play a role. This was first recognized by Hottel and coworkers [4]. Since then, a number of investigators, notably Tien and coworkers [1,5–9], have established limits for when dependent effects must be considered. Their results, summarized in Fig. 11.2, show that dependent scattering effects may be ignored as long as fv < 0.006 or c/λ > 0.5. Since these values include nearly all heat transfer applications, only independent scattering is discussed in the present chapter. The reader interested in the prediction of dependent scattering properties should consult the monograph by Tien and Drolen [1].

11.2 Absorption and Scattering from a Single Sphere The scattering and absorption of radiation by single spheres was first discussed during the later part of the nineteenth century by Lord Rayleigh [10,11], who obtained a simple solution for spheres whose diameters are much smaller than the wavelength of radiation (small size parameter, x 1). This work was followed in the 1890s by the work of Lorenz∗ [12,13], in 1908 by the classical paper of Gustav Mie† [14], and in 1909 by a similar ∗

Ludvig Lorenz (1829–1891) Danish mathematician and physicist. Lorenz studied at the Technical University of Copenhagen and, starting in 1876, he served as professor of physics at the Military Academy in Copenhagen. He also independently, and around the time as Hendrik Anton Lorentz (see p. 56), discovered the relationship between the refractive index and the density of a medium, generally known as the Lorenz-Lorentz formula.

†

Gustav Mie (1868–1957) German physicist. After studying at the universities of Rostock and Heidelberg, he served as professor of physics at various German universities.

Radiative Properties of Particulate Media Chapter | 11 403

treatment of Debye [15]. Lorenz’s work was based on his own theory of electromagnetism rather than Maxwell’s, while Mie developed an equivalent solution to Maxwell’s equations [cf. equations (2.11) through (2.14)] for an electromagnetic wave train traveling through a medium with an embedded sphere. Although the work of Lorenz predates that of Mie, the general theory describing radiative scattering by absorbing spheres is generally referred to as the “Mie theory.” More recently, in recognition of Lorenz’s contributions, the terminology “Lorenz–Mie theory” has also become popular. An exhaustive review of the history of the development of particle scattering theory has been given by Kerker [16]. The complicated Lorenz–Mie scattering theory must generally be used if the size of the sphere is such that it is too large to apply the Rayleigh theory, but too small to employ geometric optics (which requires x  1 as well as kx  1). We shall give here a very brief discussion of the Lorenz–Mie theory and some representative results. Detailed derivations may be found in the books on the subject by van de Hulst [3], Kerker [16], Deirmendjian [17], and Bohren and Huffman [18]. The amount of scattering and absorption by a particle is usually expressed in terms of the scattering crosssection, Csca , and absorption cross-section, Cabs . The total amount of absorption and scattering, or extinction, is expressed in terms of the extinction cross-section, Cext = Cabs + Csca .

(11.4)

Often efficiency factors Q are used instead of cross-sections; they are nondimensionalized with the projected surface area of the sphere, or absorption efficiency factor: scattering efficiency factor: extinction efficiency factor:

Cabs , πa2 Csca Qsca = , πa2 Cext , Qext = πa2 Qabs =

(11.5) (11.6) (11.7)

and Qext = Qabs + Qsca .

(11.8)

Radiation interacting with a spherical particle may be scattered away from its original direction by an angle Θ, i.e., the propagation vector of the electric and magnetic fields may be redirected by the scattering angle (Fig. 11.1). This deflection from the incident direction is described by the angle Θ alone because, for a spherical particle, there can be no azimuthal variation. The intensity of the wave scattered by the angle Θ [i.e., the magnitude of the Poynting vector, equation (2.42)] is proportional to two complex amplitude functions S1 (Θ) and S2 (Θ), where the subscripts denote two perpendicular polarizations. Once these amplitude functions have been determined, the intensity of radiation Isca , scattered by an angle Θ from the incident unpolarized beam of strength Iin , may be calculated [3,16,17] from Isca (Θ) 1 i1 + i2 = , Iin 2 x2

(11.9)

where i1 and i2 are the nondimensional polarized intensities calculated from i1 (x, m, Θ) = |S1 |2 ,

i2 (x, m, Θ) = |S2 |2 .

(11.10)

From equation (11.9) it follows that the total amount of energy scattered by one sphere into all directions [3] is   π Isca Csca a2 1 Qsca = = 2 dΩ = 2 (i1 + i2 ) sin Θ dΘ. (11.11) πa2 πa 4π Iin x 0 The fraction of this energy that is scattered into any given direction is denoted by the scattering phase function Φ(Θ), which is normalized such that  1 Φ(ˆs i , sˆ ) dΩ ≡ 1. (11.12) 4π 4π

404 Radiative Heat Transfer

Thus, together with equation (11.9), the scattering phase function may be expressed as Φ(Θ) =

1 4π



i 1 + i2

=2

(i1 + i2 ) dΩ

i1 + i 2 . x2 Qsca

(11.13)

4π

Finally, total extinction by a single particle (absorption within the particle, plus scattering into all directions) is related to the real part of the amplitude functions by Qext =

4 {S(0)}, x2

(11.14)

where the amplitude function S is without a subscript because S1 (0) = S2 (0). The major difficulty in the evaluation of scattering properties lies in the calculation of the complex amplitude functions S1 (Θ) and S2 (Θ). For the general case of arbitrary values for the complex index of refraction m and the size parameter x, the full Lorenz–Mie equations as expressed by van de Hulst [3] must be employed S1 (Θ) = S2 (Θ) =

∞  2n + 1 [an πn (cos Θ) + bn τn (cos Θ)] , n(n + 1)

(11.15)

2n + 1 [bn πn (cos Θ) + an τn (cos Θ)] , n(n + 1)

(11.16)

n=1 ∞  n=1

where the direction-dependent functions πn and τn are related to Legendre polynomials Pn (for a description of these polynomials, see, e.g., Wylie [19]) by πn (cos Θ) =

dPn (cos Θ) , d cos Θ

τn (cos Θ) = cos Θ πn (cos Θ) − sin2 Θ

(11.17) dπn (cos Θ) , d cos Θ

(11.18)

and the Mie scattering coefficients an and bn are complex functions of x and y = mx, ψn (y)ψn (x) − mψn (y)ψn (x) , ψn (y)ζn (x) − mψn (y)ζn (x) mψn (y)ψn (x) − ψn (y)ψn (x) . bn = mψn (y)ζn (x) − ψn (y)ζn (x) an =

(11.19) (11.20)

The functions ψn and ζn are known as Riccati–Bessel functions, and are related to Bessel and Hankel functions [19, 20] by  1/2  1/2 πz πz ψn (z) = Jn+1/2 (z), ζn (z) = Hn+1/2 (z). (11.21) 2 2 Equations (11.15) and (11.16) may be substituted into equations (11.11) and (11.14). Using the fact that—like Legendre polynomials—the functions πn and τn constitute sets of orthogonal functions leads to Qsca = Qext =

∞ 2  (2n + 1)(|an |2 + |bn |2 ), x2 n=1 ∞ 2  (2n + 1){an + bn }. x2 n=1

(11.22) (11.23)

Once all Mie scattering coefficients an and bn have been determined, the phase function Φ may also be evaluated from equation (11.13), but this calculation tends to be extremely tedious because of the nature of equation (11.10),

Radiative Properties of Particulate Media Chapter | 11 405

and because the calculations must be carried out anew for every scattering angle Θ. To facilitate the calculations Chu and Churchill [21,22] expressed the scattering phase function as a series in Legendre polynomials, Φ(Θ) = 1 +

∞ 

An Pn (cos Θ),

(11.24)

n=1

where the coefficients An are directly related to the Mie scattering coefficients an and bn through some rather complicated formulae not reproduced here. The great advantage of this formulation is that, once the An have been determined, the value of the phase function Φ is determined quickly for any or all scattering directions. In many applications the use of the complicated scattering phase function described by equation (11.24) is too involved. For a simpler analysis the directional scattering behavior may be described by the average cosine of the scattering angle, known as the asymmetry factor, and related to the phase function by  1 Φ(Θ) cos Θ dΩ. (11.25) g = cos Θ = 4π 4π For the case of isotropic scattering (i.e., equal amounts are scattered into all directions, and Φ ≡ 1) the asymmetry factor vanishes; g also vanishes if scattering is symmetrical about the plane perpendicular to beam propagation. If the particle scatters more radiation into the forward directions (Θ < π/2), g is positive; if more radiation is scattered into the backward direction (Θ > π/2), g is negative. For spherical particles the asymmetry factor is readily calculated [18] as g = cos Θ =

∞ ' ) 4  n(n + 2) 2n + 1 ∗ ∗ ∗ {a {a a + b b } + b } n n n n . n+1 n+1 n(n + 1) x2 Qsca n=1 n + 1

(11.26)

The calculation of the scattering Mie coefficients an and bn is no trivial matter even in these days of supercomputers: The relationships leading to their determination are involved and require the frequent evaluation of complicated functions with complex arguments. For large size parameters x many terms need to be calculated (nmax ≈ 2x). Recursion formulae for the functions πn , τn , ψn , and ζn have been given by Deirmendjian [17] and others, which evaluate these functions for increasing values of n in terms of previously calculated functions. Deirmendjian observed that the accuracy of calculations decreases for increasing n, causing complete failure of the calculations for large values of the size parameter x (for which many terms are required in the series for the amplitude functions), even if double-precision arithmetic is employed. This problem was overcome by Kattawar and Plass [23] who showed that all four functions may be reduced to functions each belonging to one of two sets: One set has stable recursion formulae for increasing n (i.e., round-off error decreases with growing n), and the other set is stable for decreasing values of n (setting the function to zero for a larger n than required in the series results in very accurate values for slightly smaller n). Wiscombe [24] compared the accuracy and stability of several Lorenz–Mie scattering computer solution routines and discussed the efficiency of different calculation methods (whether to use upward or downward recursion, what recursion formulae to use, etc.). Some representative results of Lorenz–Mie calculations are shown in Figs. 11.3 through 11.5. Figure 11.3 shows typical behavior of efficiency factors, demonstrated with the extinction efficiency of a dielectric (k ≡ 0) for a number of different refractive indices n. Observe that there is a primary oscillation in the variation of Qext with size parameter, upon which secondary oscillations are superimposed (stronger for larger refractive indices). Note also that the oscillations become smaller for larger size parameters, and Qext → 2 as x → ∞ (for dielectrics as well as metals). Figure 11.4 shows the qualitative behavior of efficiency factors for absorption, Qabs , and scattering, Qsca , respectively, for a fixed value of the size parameter (x = 1), as a function of absorptive index k. The absorption efficiency factors may vary by many orders of magnitude over the range of absorptive index k, while the scattering efficiency remains constant over great changes of k. Finally, Fig. 11.5 shows some representative scattering phase functions, Φ(Θ). Figure 11.5a shows the scattering behavior of very small particles (known as Rayleigh scattering): The scattering is symmetric to the plane perpendicular to the incident beam and is nearly isotropic with slight forward- and backward-scattering peaks and somewhat lesser scattering to the sides. Figure 11.5b demonstrates the behavior of particles with refractive indices close to unity (known as Rayleigh–Gans scattering): Nearly all of the scattered energy is scattered into forward directions with some scattering into a few preferred other directions. This behavior becomes more extreme as the size parameter increases. Figure 11.5c

406 Radiative Heat Transfer

FIGURE 11.3 Extinction efficiency factors for dielectric spheres for several refractive indices [3].

FIGURE 11.4 Efficiency factors as functions of complex index of refraction, m = n − ik, for a size parameter of x = 1 [23]: (a) absorption efficiency factor, (b) scattering efficiency factor.

shows the phase function of a typical dielectric: The scattering has a strong forward component; otherwise the scattering behavior demonstrates rapid maxima and minima at varying scattering angles, with much stronger amplitudes than for Rayleigh–Gans scattering (note the change in scale). The variations are not quite so extreme as in Fig. 11.5b owing to the large value for n. The behavior of a typical metal (aluminum at 3.1 μm) is shown

Radiative Properties of Particulate Media Chapter | 11 407

FIGURE 11.5 Polar plot of scattering phase functions for single spherical particles: (a) small sphere with x = 0.001; (b) dielectric with x = 5 and m = 1.0001; (c) dielectric sphere with x = 10 and m = 2; and (d) metallic sphere (aluminum) with x = 10 and m = 4.46 − 31.5i.

in Fig. 11.5d: Besides a strong forward-scattering peak these particles display lesser-degree oscillations than dielectrics. These phase functions have been calculated with the author’s own code, mmmie, which is included in Appendix F for the convenience of the reader.

11.3 Radiative Properties of a Particle Cloud In all problems of radiative heat transfer with particulate scattering and absorption, we have to deal with a large collection of particles. If the scattering is independent, as is assumed in this chapter, then the effects of large numbers of particles are simply additive. For simplicity, it is often assumed that particle clouds consist of spheres that are all equally large. More accurate analyses take into account that particles of many different sizes may occur within a single cloud, and that these sizes often vary by orders of magnitude. We shall briefly describe both approaches in the following paragraphs.

Clouds of Uniform Size Particles The fraction of energy scattered by all particles per unit length along the direction of the incoming beam is called the scattering coefficient [as defined by equation (9.6)] and is equal to the scattering cross-section summed over all particles. If NT is the number of particles per unit volume, all of uniform radius a, then σsλ = NT Csca = πa2 NT Qsca =

3 fv Qsca , 4a

(11.27)

where fv = 43 πa3 NT is the total volume of particles per unit volume, or volume fraction. Similarly, for absorption and extinction, κλ = NT Cabs = πa2 NT Qabs =

3 fv Qabs , 4a

βλ = κλ + σsλ = NT Cext = πa2 NT Qext =

(11.28) 3 fv Qext . 4a

(11.29)

408 Radiative Heat Transfer

Since the scattering phase function (or the directional distribution of scattered energy) in a cloud of uniform particles is the same for each particle, it is also the same for the particle cloud, or ΦTλ (Θ) = Φ(Θ),

(11.30)

gTλ = (cos Θ)Tλ = cos Θ.

(11.31)

and similarly for the asymmetry factor,

In both cases we have temporarily added the subscript T (to distinguish the total cloud of particles from a single particle) and λ to emphasize the fact that both quantities are spectral quantities that may vary with wavelength. If total (i.e., spectrally integrated) properties are desired, equations (11.27) through (11.29) may be integrated to obtain Planck-mean or Rosseland-mean coefficients (for absorption, scattering, and/or extinction), as defined by equations (10.197) and (10.203), or  ∞ π yP = Ibλ yλ dλ, y = κ, σs , or β, (11.32) σT 4 0  ∞ 1 1 dIbλ π dλ, y = κ, σs , or β. = (11.33) yR 4σT3 0 yλ dT Similarly, total emissivities and absorptivities may be obtained from equation (10.183). Since the efficiency factors Q may vary rapidly across the spectrum, these integrations generally need to be done numerically.

Clouds of Nonuniform Size Particles For clouds of particles of nonuniform size it is customary to describe the number of particles as a function of radius in the form of a particle distribution function. A number of different forms for the distribution function have been used by various researchers. We introduce here the so-called modified gamma distribution [17], n(a) = Aaγ exp(−Baδ ),

0 ≤ a < ∞,

(11.34)

which vanishes at a = 0 and a → ∞. This distribution function reduces to the gamma distribution if δ = 1. The four constants A, B, γ, and δ are positive and real, and γ and δ are usually chosen to be integers. They must be determined from measurable quantities such as total number of particles (per unit volume),  NT =

∞



∞

n(a) da = A

0

aγ exp(−Baδ ) da =

0

Here Γ is the gamma function,

 Γ(z) =

∞

AΓ

 γ+1  δ

δB(γ+1)/δ

e−t tz−1 dt,

.

(11.35)

(11.36)

0

and has been tabulated, e.g., by Abramowitz and Stegun [20]. Equation (11.35) shows that the constant A is essentially given by NT . For nonuniform size particles the volume fraction follows as  fv = 0

∞

3 4 3 πa n(a) da

=

4πAΓ

 γ+4  δ

3δB(γ+4)/δ

.

(11.37)

Assuming that all particles have the same optical properties, we may again determine the scattering coefficient for a particle cloud by adding the scattering cross-section over all particles but, because of the particle size distribution, this is now an integral rather than a simple sum,  ∞  ∞ σsλ = Csca n(a) da = π Qsca a2 n(a) da, (11.38) 0

0

Radiative Properties of Particulate Media Chapter | 11 409

FIGURE 11.6 Lorenz–Mie scattering phase function for clouds of (a) absorbing, and (b) dielectric particles [25].

and, similarly, for absorption and extinction,  ∞  ∞ Cabs n(a) da = π Qabs a2 n(a) da, κλ = 0 0  ∞  ∞ Cext n(a) da = π Qext a2 n(a) da. βλ = 0

(11.39) (11.40)

0

For nonuniform particles the scattering phase function is not the same for all particles. From the definition of the phase function, it follows that the scattered energies into a given direction must be summed over all particles and then normalized, or  ∞  ∞ (i1 + i2 ) n(a) da Csca (a) Φ(a, Θ) n(a) da 0 0  ∞  ΦTλ (Θ) = =   ∞ 1 Csca (a) n(a) da (i1 + i2 ) n(a) da dΩ 4π 4π 0 0  ∞ 1 Csca (a) Φ(a, Θ) n(a) da, (11.41) = σsλ 0 and, similarly, gTλ = (cos Θ)Tλ =

1 σsλ



∞

Csca (a) g(a) n(a) da.

(11.42)

0

Again, if total properties are needed, equations (11.38) through (11.40) may be integrated over the entire spectrum. Figure 11.6 shows a few typical scattering phase functions for absorbing and nonabsorbing particle clouds, calculated with program mmmie of Appendix F. Two types of particles are considered, one nonabsorbing with an index of refraction m = 2, the other one absorbing with m = 2 − i. The particles are either in clouds of constant radius a = 5 μm, or in clouds with a distribution function n(a) = 27,230a2 exp(−1.7594a),

(11.43)

which has its maximum at a = 5 μm. All the particle clouds have a number density of 104 particles/cm3 , and the Lorenz–Mie calculations have been carried out for a typical wavelength of λ = 3.1416 μm, resulting in a size parameter of x = 2πa/λ = 10 for the constant-radius clouds, and a range of significant size parameters of 0 < x ≤ 20 for clouds with particle size distribution. The radiative properties for the four different particle clouds are summarized in Table 11.1. Absorption and scattering coefficients for constant-radius and particle

410 Radiative Heat Transfer

TABLE 11.1 Radiative properties of typical particle clouds (NT = 104 /cm3 , λ = 3.1416 μm). Cloud#1

Cloud#2

Cloud#3

Cloud#4

Const. Radius

Size Distr.

Const. Radius

Size Distr.

a = 5 μm

n(a)

a = 5 μm

n(a)

m=2−i

m=2−i

m=2

m=2

Absorption coefficient κ [cm−1 ]

8.307 × 10−3

1.524 × 10−3

0

0

Scattering coefficient σs [cm−1 ]

1.073 × 10−2

1.674 × 10−3

6.420 × 10−2

3.363 × 10−3

Extinction coefficient β [cm−1 ]

1.904 × 10−2

3.198 × 10−3

6.420 × 10−2

3.363 × 10−3

Scattering albedo

0.5634

0.5235

1

1

Terms needed for phase function

26

35

27

33

FIGURE 11.7 The effect of size dispersion on the extinction efficiency for water droplets and visible light (σ = standard deviation in Gaussian distribution function) [18].

distribution clouds differ considerably, primarily because the average particle size in equation (11.43) is less than 5 μm, being 2.33 μm for the volume- or mass-averaged radius, and 1.52 μm for the number-averaged radius. Observe that the phase functions for uniform particle size clouds display strong oscillations due to diffraction peaks, because the phase function is identical to the one of single particles (cf. Fig. 11.5). Since the diffraction peaks shift slightly with changing size parameters, these peaks and valleys are smoothed out for clouds with varying particle sizes. For these types of clouds the phase function becomes very smooth with only a strong forward-scattering peak remaining (plus a weaker backward-scattering peak for dielectric particles). Thus, the analysis of scattering phenomena may actually be simpler if there is a particle size distribution! Figure 11.6 also shows linear anisotropic approximations to these phase functions, as discussed in Section 11.9. Bohren and Huffman [18] have shown that this smoothing effect occurs for the efficiency factors as well as for the phase function, requiring only a small deviation from uniform-size particles to be present. Figure 11.7 shows the extinction efficiency for clouds of water droplets, which are assumed to have a Gaussian distribution function centered around a mean particle size with standard deviation σ. Small deviations from uniform size blur out the high-frequency variation (called the ripple structure), while slightly larger deviations also dampen out the low-frequency variations of the extinction efficiency (called the interference structure). Similar smoothing

Radiative Properties of Particulate Media Chapter | 11 411

effects occur in a cloud of uniform-size particles of irregular shape as shown by Hodkinson [26] for aqueous suspensions of irregular quartz particles.

11.4 Radiative Properties of Small Spheres (Rayleigh Scattering) Radiative scattering by spheres that are small compared with wavelength was first described by Lord Rayleigh [10,11] long before the development of Mie’s theory [14]. However, results for small particles are here most easily obtained by taking the appropriate limits in the general solution to Mie’s equations. If the scattering particles are extremely small, then the size parameter x = 2πa/λ becomes very small. Such behavior is primarily observed with gas molecules (which are, in fact, very tiny particles). There are, however, also some multimolecule solid particles that fall into the Rayleigh scattering regime, e.g., soot particles (whose diameters are often smaller than 10 nm and which, in combustion applications, are irradiated by light of approximately 3 μm, resulting in x ≈ 0.01). In the limit of x → 0 it is relatively straightforward to show that only the a1 in equations (11.19) and (11.20) is nonzero, or m2 − 1 3 (11.44) x cos Θ, m2 + 2 that is, the amplitude function for one polarization is independent of scattering angle Θ. Substitution into equations (11.11) and (11.14) then gives the efficiency factors as S2 (Θ) = S1 (Θ) cos Θ = i

 2 8  m2 − 1  4 Qsca =  2 x, 3m + 2 / 2 0 m −1 Qabs = −4 2 x ≈ Qext , m +2

(11.45) (11.46)

where the last equality in equation (11.46) is due to the fact that x4 x, so scattering may be neglected as compared with absorption. We observe the wavelength dependence of the scattering efficiency to be Qsca ∝

1 ∝ ν4 . λ4

(11.47)

We note in passing that this fact explains the colors of the sky: During most of the day, when the sun’s rays travel a relatively short distance through Earth’s atmosphere (cf. Fig. 11.8), only the shortest wavelengths are scattered away in any appreciable amounts from the sun’s direct path; they are scattered again and again by the molecules in the atmosphere, providing us with a blue sky (blue light having the shortest wavelength within

FIGURE 11.8 Distance traveled through Earth’s atmosphere by solar rays.

412 Radiative Heat Transfer

FIGURE 11.9 Polar diagram of Rayleigh phase function: 1, polarized with electric vector in plane perpendicular to paper; 2, polarized in plane of paper; 1+2, unpolarized.

the visible spectrum). Close to sunset, however, the sun’s rays travel at a grazing angle through the atmosphere to the observer, so that all but the very longest wavelengths (of the visible spectrum) have been scattered away from the direct path, giving the sun a red appearance. Without the atmosphere the sky would appear black to us, as witnessed by the astronauts visiting the (atmosphere-less) moon. The wavelength dependence of the absorption efficiency, on the other hand, is Qabs ∝

1 ∝ ν, λ

(11.48)

which describes the spectral behavior of small particles such as soot reasonably well. The phase function for Rayleigh scattering follows from equations (11.44) and (11.13) as Φ(Θ) = 34 (1 + cos2 Θ),

(11.49)

where the two terms are the contributions from the two perpendicular polarizations, as shown in Fig. 11.9. It is observed that the phase function is symmetric as far as forward and backward scattering is concerned and does not deviate too strongly from isotropic scattering. The absorption coefficient for a cloud of nonuniform-size small particles follows from equations (11.39) and (11.46) as / 2 0 ∞  ∞  m −1 2πa 2 κλ = π πa2 n(a) da. Qabs a n(a) da = −4 2 (11.50) λ m +2 0 0 The integral in this equation may be related to the volume fraction fv , 

∞

fv =



0

 4 3 πa n(a) da, 3

(11.51)

so that the absorption coefficient for small particles reduces to 0 m2 − 1 6π fv κλ = − 2 , λ m +2 /

(11.52)

or, expanding the complex index of refraction, m = n − ik, κλ =

fv 36πnk . (n2 − k2 + 2)2 + 4n2 k2 λ

(11.53)

Radiative Properties of Particulate Media Chapter | 11 413

Therefore, for particles small enough that Rayleigh scattering holds, the absorption coefficient does not depend on particle size distribution, but only on the total volume occupied by all particles (per unit system volume). Example 11.1. During the burning of propane it is observed that the products contain a volume fraction of 10−4 % (1 ppm) of soot with complex index of refraction m = 2.21 − 1.23i (measured at a wavelength of 3 μm). Assuming a mean particle diameter of 0.05 μm, determine the absorption and scattering efficiency of this soot cloud as well as its absorption coefficient, all at a wavelength of 3 μm. Solution For the given diameter and wavelength the particle size parameter is x = π × 0.05 μm/3 μm = 0.0524 1 and we assume Rayleigh scattering to hold for all particles. For all three properties we need to evaluate the complex ratio (m2 − 1)/(m2 + 2): m2 − 1 2.212 − 2 × 2.21 × 1.23i − 1.232 − 1 2.3712 − 5.4366i 5.3712 + 5.4366i = = × m2 + 2 2.212 − 2 × 2.21 × 1.23i − 1.232 + 2 5.3712 − 5.4366i 5.3712 + 5.4366i =

42.2928 − 16.3098i = 0.7241 − 0.2792i. 58.4064

Thus, the efficiencies can be evaluated as Qsca =

8 |0.7241 − 0.2792i|2 × (0.0524)4 = 1.21 × 10−5 , 3

and Qabs = −4 × (−0.2792) × 0.0524 = 5.85 × 10−2 , showing that scattering may indeed be neglected compared with absorption. The absorption coefficient follows from equation (11.53) as 6π × 10−4 /100 = 0.01754 cm−1 , 3 × 10−4 cm

κλ = −(−0.2792) ×

that is, any radiation (at 3 μm) penetrating into such a soot cloud would be attenuated to 1/e of its original intensity over a distance of 1/κλ = 57 cm.

11.5 Rayleigh–Gans Scattering A near-dielectric sphere with k ≈ 0 and with a refractive index close to unity, i.e., |m − 1| 1, has negligible reflectivity and, thus, lets light pass into the sphere unattenuated and unrefracted. If also x|m − 1| 1, then the light will exit the sphere again essentially unattenuated. However, since the phase velocity of light is slightly less inside the particle, light traveling through the sphere will display a small phase lag as opposed to the incident light. This phenomenon is known as Rayleigh–Gans scattering. As described by van de Hulst [3], taking the appropriate limits reduces equations (11.15) and (11.16) to S2 (Θ) = S1 (Θ) cos Θ = ix3 (m − 1)G(u) cos Θ,

(11.54)

where G(u) =

2 (sin u − u cos u), u3

u = 2x sin 12 Θ.

(11.55)

The absorption efficiency is identical to the one for Rayleigh scattering, that is, 0 m2 − 1 x, = −4 2 m +2 /

Qabs

(11.56)

while the scattering efficiency turns out to be  Qsca = |m − 1|2 x4 0

π

G2 (u)(1 + cos2 Θ) sin Θ dΘ.

(11.57)

414 Radiative Heat Transfer

Finally, the phase function for Rayleigh–Gans scattering is now easily determined as Φ(Θ) = 

π

2G2 (u)(1 + cos2 Θ)

.

(11.58)

G (u)(1 + cos Θ) sin Θ dΘ 2

2

0

An example of this phase function is included in Fig. 11.5b for x = 5 and m = 1.0001. The phase function displays a strong forward-scattering peak (which increases with increasing size parameter), with very rapid oscillations of varying amplitude into the other directions.

11.6 Anomalous Diffraction Simple relations for near-dielectric spheres, |m − 1| 1, can also be obtained for arbitrary values of x|m − 1|, provided the particles are large, x  1. This allows separation of (approximately straight) transmission and diffraction, and is called anomalous diffraction by van de Hulst [3]. For this limiting case the efficiency factors are found from     6  7 cos q cos q 4 cos(p−2q) − cos 2q , (11.59) Qext = 4 K 2x(m−1)i = 2 − cos q e−p tan q sin(p−q) + p p p Qabs = 2K(2p tan q),

(11.60)

where  1  1 − 1 − (1 + w) e−w , 2 w2 k ; p tan q = 2xk. p = 2x(n − 1), q = tan−1 n−1

K(w) =

(11.61) (11.62)

Physically, p represents the phase lag experienced by a ray that passes through the center of the sphere. Similar to Rayleigh–Gans scattering, many nonmetallic particles present during combustion come reasonably close to satisfying the |m − 1| 1 conditions.

11.7 Radiative Properties of Large Spheres If the spheres are very large (x  1), very many terms are required in the evaluation of equations (11.15) and (11.16). However, in this case it is sufficient to resort to geometric optics, and one may separate diffraction from reflection and refraction. For very large spheres it is always true that Qext = 2.

(11.63)

This relationship is sometimes called the extinction paradox since it states that a large particle removes exactly twice the amount of light from the beam as it can intercept, and has been discussed by van de Hulst [3]. Since, for geometric optics, the projected area of a particle for reflection and absorption is πa2 , this means that half of the extinction efficiency is due to diffraction. How much of the rest is due to absorption, and how much due to reflection, depends on the value of the complex index of refraction m, or the reflectivity of the sphere’s surface. In the following we shall determine the scattering properties of large opaque spheres, i.e., such spheres for which any ray refracted into the particle will be totally absorbed within, without exiting the sphere at another location. This requires the additional assumption that kx  1 (say, 2 or 3). Thus, k may be fairly small as long as x  1. A consequence of this is that, for a metal, “large particle” may mean x > 10, while for a near-dielectric it may mean x > 10,000. While electromagnetic wave theory always assumes optically smooth surfaces, resulting in specular reflection, very large spheres (as compared with wavelength) may have roughness levels at the sphere’s surface that are also large as compared with wavelength, resulting in nonspecular reflection. Treatment of very irregular directional behavior for the reflectance is, of course, extremely difficult (as it was for surface transport, cf. Chapter 6). However, the extreme case of perfectly diffuse reflection lends itself to straightforward analysis (similar to the treatment of surface transport in Chapter 5), and is, therefore, also included in this section.

Radiative Properties of Particulate Media Chapter | 11 415

FIGURE 11.10 Phase function for diffraction over a large sphere.

Diffraction from Large Spheres The diffraction pattern of light passing through the vicinity of a large sphere is, by Babinet’s principle, equal to that of a circular hole with the same diameter [3]. As a consequence the directional behavior of the diffracted light consists of alternating bright and dark rings. The amplitude functions for diffraction have been given by van de Hulst [3] as S1 (Θ) = S2 (Θ) = x

J1 (x sin Θ) , sin Θ

(11.64)

where J1 is a Bessel function [19]. Therefore, the phase function for diffraction over a large sphere follows from equation (11.13) (noting that Qsca = 1 for diffraction) as Φ(Θ) = 2

J12 (x sin Θ) i 1 + i2 = 4 . x2 sin2 Θ

(11.65)

This phase function, depicted in Fig. 11.10, demonstrates that almost all energy is scattered forward within a narrow cone of Θ < (150/x)◦ from the direction of transmission. Thus, in heat transfer applications we may usually neglect diffraction and treat it as transmission. Then, for large particles without diffraction, Qext = 1.

(11.66)

Large Specularly Reflecting Spheres Consider a specularly reflecting opaque sphere irradiated by an intensity Ii distributed over a thin pencil of rays of solid angle dΩ i as shown in Fig. 11.11. Under these conditions the infinitesimal band at an angle β from the incident direction (indicated by shading in the figure) receives radiation from a direction which is off-normal (from its surface) by an angle β. Recalling the definition of intensity as “heat rate per unit area normal to the rays, per unit solid angle, and per unit wavelength,” the energy intercepted by the band over a wavelength range of dλ is d2Qi = Ii dΩ i dλ (dAband cos β) = Ii dΩ i dλ 2πa sin β a dβ cos β.

(11.67)

Of that, the fraction ρs (β) is reflected into the direction 2β as measured from the incoming pencil of rays. The

416 Radiative Heat Transfer

FIGURE 11.11 Scattering of incident radiation by a large specularly reflecting sphere.

total heat rate intercepted by the sphere is 

π/2

dQi =

Ii dΩ i dλ 2πa2 sin β cos β dβ = Ii dΩ i dλ πa2 ,

(11.68)

0

while the total reflected (or scattered) heat rate is 

π/2

dQs =

ρs (β)Ii dΩ i dλ 2πa2 sin β cos β dβ

0



π/2

= Ii dΩ i dλ πa 2 2

ρs (β) sin β cos β dβ = ρs Ii dΩ i dλ πa2 ,

(11.69)

0

where ρs is the hemispherical reflectance, averaged over all incoming directions [cf. equation (3.46)]: 

π/2

ρ =2 s

ρs (β) sin β cos β dβ.

(11.70)

0

Thus, the scattering efficiency for a large, opaque, specularly reflecting particle is simply Qsca =

dQs = ρs , dQi

(11.71)

and the absorption efficiency follows as Qabs = Qext − Qsca = 1 − ρs = α,

(11.72)

that is, the hemispherical absorptivity. To evaluate the scattering phase function we consider the amount of energy scattered into any given direction Θ, where Θ is measured from the transmission direction sˆ , as also indicated in Fig. 11.11. It is clear that, for a homogeneous sphere, the scattered intensity can only vary with the polar angle Θ (and not azimuthally). Furthermore, for a specularly reflecting sphere the outgoing intensity in a certain direction Θ can only come from a single position on the sphere’s surface. For example, radiation scattered into the direction Θ = π − 2β comes from the shaded band in Fig. 11.11. Recalling that the scattering phase function is defined as 4π × scattered

Radiative Properties of Particulate Media Chapter | 11 417

FIGURE 11.12 Scattering phase functions for large spheres of various materials.

intensity/total scattered heat flux [cf. equation (9.15)] we get, for Θ = π − 2β or β = (π − Θ)/2,   2 ρs (β) d2Qi /dΩ r s π−Θ Ii dΩ i dλ 2πa sin β cos β dβ/dΩ r = 4πρ . Φ(Θ) = 4π dQs 2 ρs Ii dΩ i dλ πa2

(11.73)

The solid angle for the reflection is best visualized by letting the reflected intensity fall upon a concentric (and very large) sphere of radius R. The solid angle is then the area of the illuminated band divided by R2 , or dΩ r = 2π sin 2β d(2β), leading to Φ(Θ) = ρs



π−Θ 2

;

ρs .

(11.74)

(11.75)

Alternatively, we could use the fact that the scattering phase function is proportional to intensity into any given direction, and then normalize the resulting expression with equation (9.17). The actual directional scattering behavior (or the behavior of the phase function) depends on the material of which the particles are made. Figure 11.12 shows a comparison of the phase function between a “typical” metal (aluminum at 3.1 μm with an index of refraction of m = 4.46 − 31.5i) and a “typical” dielectric (m = 2).1 These two phase functions should be compared with Fig. 11.5c,d, which are for identical materials but for a smaller size parameter (and are shown in a polar rather than a Cartesian plot). Since the size parameter in Fig. 11.5c,d is fairly large (x = 10), the major difference between Fig. 11.5c,d and Fig. 11.12 lies in the omission of diffraction in Fig. 11.12. For large particles all materials have their maximum scattering into the forward direction, Θ = 0, since ρs (π/2) = 1 always. However, this peak is considerably more pronounced for dielectrics and is hardly noticeable for the metal because of the dip in reflectance at near-grazing angles (compare also Fig. 2.11, which shows the directional variation of the reflectance of aluminum). Because of their relatively high reflectance at all directions, large metallic particles tend to be almost isotropic scatterers. Example 11.2. Consider glass particles with a complex index of refraction m = 1.5 − 0.1i and a density of ρglass = 2 g/cm3 , suspended in an inert gas, with a particle loading ratio of 1 kg of particles per m3 of suspension volume. Particle sizes range between 100 μm and 1000 μm, with an equal distribution over all sizes by weight-%. Determine the absorption coefficient, the scattering coefficient, and the phase function for the infrared (3 μm < λ < 10 μm). Solution First, we need to determine the particle distribution function by number (rather than mass). Since the mass distribution 1. Note that in the case of a dielectric the absorptive index k is assumed negligible as compared with the refractive index n, but k is assumed large enough to make the spheres opaque.

418 Radiative Heat Transfer

function is a constant we get 1 kg/m3 = 4 πa3 ρglass n(a), 100 μm ≤ a ≤ 1000 μm, (1000 − 100) μm 3 3m(a) = 1.3226 × 10−7 μm−1 /a3 , 100 μm ≤ a ≤ 1000 μm. n(a) = 4πa3 ρglass

m(a) =

Next we need to determine the range of the size parameter x to see whether Rayleigh scattering, Lorenz–Mie scattering, or large-particle scattering must be considered. The minimum value for x will occur for the smallest particle at the longest wavelength, or 2πamin 2π100 = 62.83  1, = λmax 10 = 6.283  1,

xmin = (kx) min

that is, the large-particle assumption will be acceptable for all conditions encountered in this example. Thus, the absorption and scattering coefficients may be related to the hemispherical emissivity of the glass. Since for this glass k n, the material behaves essentially like a dielectric, and the hemispherical emissivity may be found from Fig. 3.19 or equation (3.82). Either method leads to = α = 1 − ρ = 0.91. The absorption and scattering coefficients may then be calculated from equations (11.39) and (11.38) as 

∞

κλ = π

 αa2 n(a) da = πα

0 −7

= 1.3226 × 10

1000 μm

100 μm

a2

1.3226 × 10−7 da μm a3

1000 = 9.60 × 10−3 α cm−1 = 8.74 × 10−3 cm−1 , 100 = 0.86 × 10−3 cm−1 .

−1

μm πα ln

σsλ = 9.60 × 10−3 ρ cm−1

The scattering phase function must be evaluated from equation (11.75) and is also included in Fig. 11.12. Because of the small value for k, the directional behavior is very similar to that of the perfect dielectric (m = 2), but the forward-scattering peak is more pronounced because of the smaller refractive index.

Large Diffusely Reflecting Spheres In equations (11.67) through (11.72) the directional characteristics of the sphere reflectance did not enter the development. Thus, for a diffusely reflecting sphere the amount of incident radiation on a surface element, as well as the expression for the heat flux reflected into all directions, is the same as for a specularly reflecting sphere. Therefore, equations (11.67) through (11.72) also hold for the diffusely reflecting sphere, or Qabs = α, Qsca = ρ.

(11.76) (11.77)

However, while for a specularly reflecting sphere the energy scattered into any given direction resulted from reflection from a single location on the sphere’s surface, this is not true for a diffusely reflecting sphere. This complicates the development for the scattering phase function a bit. Consider Fig. 11.13: Incident radiation traveling into the direction of the unit vector sˆ i illuminates one half of the diffusely reflecting sphere. An observer, located far away from the sphere in the direction of sˆ o , sees a different half of the sphere, part of which is illuminated by the incident radiation (shown by shadowing), part of which is in the shade. This illuminated region seen by the observer has the shape of a circular wedge similar to a slice of lemon. To describe the surface in polar coordinates it is most convenient to define the plane formed by the two unit vectors sˆ i and sˆ o to be the x-y-plane with polar angle β measured from the z-axis and the azimuthal angle ψ measured from the negative x-axis as indicated in Fig. 11.13. With this coordinate system the normal to a surface element in the illuminated region may be expressed as ˆ ˆ ψ) = − sin β cos ψ î + sin β sin ψ jˆ + cos β k, n(β,

(11.78)

and also sˆ i = î,

sˆ o = cos Θ î + sin Θ jˆ.

(11.79)

Radiative Properties of Particulate Media Chapter | 11 419

FIGURE 11.13 Scattering of incident radiation by a large diffusely reflecting sphere.

The energy reflected from an infinitesimal surface area is, as developed in equation (11.67), d2Qs = ρIi dΩ i dλ [dA(−nˆ · sˆ i )],

(11.80)

where dA is two-dimensionally infinitesimal as indicated in Fig. 11.13 (i.e., not a ring as in the previous section, Fig. 11.11). Thus, the radiosity at that location, because of diffuse reflection of incident radiation, is dJ = ρIi dΩ i dλ (−nˆ · sˆ i ).

(11.81)

Some of the reflected radiation will travel toward the observer into the direction of sˆ o . If we assume the observer stands on a large sphere with radius R  a, then the heat flux through a surface element dAR on the large sphere due to reflection from the small sphere is  dIs dΩ =

dJ dFdA−dAR dA,

(11.82)

Ashaded

where dFdA−dAR =

nˆ · sˆ o dAR 1 = nˆ · sˆ o dΩ π πR2

(11.83)

is the view factor between dA and dAR , nˆ · sˆ o is the cosine of the angle between the surface normal at dA and the line to dAR , while the surface normal at dAR points directly to the particle. Thus, dIs =

1 π

 dJ nˆ · sˆ o dA,

(11.84)

Ashaded

and, again recalling that the scattering phase function is equal to 4π × scattered intensity/total scattered heat flux, we get  Φ(ˆs i , sˆ o ) = 4π dIs /dQs = 4

+ (ρIi dΩ i dλ)(−nˆ · sˆ i )(nˆ · sˆ o ) dA ρIi dΩ i dλ πa2

Ashaded

420 Radiative Heat Transfer

=

4 πa2

4 = 2 πa

 (−nˆ · sˆ i )(nˆ · sˆ o ) dA Ashaded



π 2 π 2 −Θ



π

sin β cos ψ sin β(sin ψ sin Θ − cos ψ cos Θ) a2 sin β dβ dψ,

0

which may readily be integrated to yield Φ(Θ) =

8 (sin Θ − Θ cos Θ). 3π

(11.85)

The phase function for diffuse spheres, equation (11.85), is also depicted in Fig. 11.12. Unlike for specularly reflecting spheres, the phase function for diffusely reflecting spheres displays a strong backward-scattering peak, and it is independent of the reflectance (or the complex index of refraction) of the materials.

Particle Beds The scattering regime map of Fig. 11.2 suggests that independent scattering may be assumed for packed and fluidized beds with particle volume fractions as large as 0.7 [5]. However, the classical continuum theory for radiative transfer in particulate media is based on the assumption of infinitesimally small particle size and, thus, negligible shading. Particle beds generally contain relatively large particles (x  1) and, combined with large volume fractions, shading can no longer be ignored [27]. Brewster [28] proposed a simple correction to equation (11.29) as βλ =

3 fv πa2 NT Qext , = 1 − fv (1 − fv )a

(11.86)

where uniform particle size has been assumed, and Qext = 1 for large particles. Comparison with stochastic Monte Carlo simulations [27] showed near-perfect agreement. Another, more recent ray tracing algorithm for densely packed spheres also attests to the accuracy of equation (11.86) [29].

11.8 Absorption and Scattering by Long Cylinders Scattering from cylinders has been studied for almost as long as that from spheres, starting with Lord Rayleigh looking at infinitely long cylinders at normal incidence. In the area of radiative heat transfer scattering from cylinders has become of interest only very recently, to predict transfer rates through optical fibers and fibrous insulation. Consider a cylinder of length L and radius a, with its axis pointed into the direction of sˆ f , that is irradiated obliquely by electromagnetic waves propagating into direction sˆ as indicated in Fig. 11.14. For cylinders it is common to define the angle of incidence with respect to the normal to the cylinder axis, i.e., sˆ · sˆ f = sin φ as shown. Similar to waves impinging obliquely on flat surfaces (see Chapter 2), we need to distinguish between two polarization components: the transverse magnetic (TM, or “Case I”; no magnetic vector component in the sˆ f -direction) mode, and the transverse electric (TE, or “Case II”; no electric vector component in the sˆ f -direction) mode. For short cylinders the scattering behavior is very similar to that of spheres, but with increasing L/a-ratio scattering becomes more and more confined to a conical surface (rather than being spread out over all 4π solid angles). For infinitely long cylinders (L/a → ∞) all scattering is confined to the conical surface described by sˆ f and sˆ as indicated in Fig. 11.14. The nondimensional polarized scattering intensities can be calculated for this case as [16]  2 ∞    i11 (m, x, φ, θ) = |T11 |2 = b0I + 2 bnI cos nθ ,   n=1  2 ∞    i12 (m, x, φ, θ) = |T12 |2 = 2 anI sin nθ ,  n=1 

(11.87a)

(11.87b)

Radiative Properties of Particulate Media Chapter | 11 421

FIGURE 11.14 Scattering of incident radiation by a long cylinder.

 2 ∞    i21 (m, x, φ, θ) = |T21 |2 = 2 bnII sin nθ ,  n=1   2 ∞    i22 (m, x, φ, θ) = |T22 |2 = a0II + 2 anII cos nθ .   n=1

(11.87c)

(11.87d)

As for spheres the an and bn can be expressed in terms of Bessel and Hankel functions and are given by Kerker [16]. For unpolarized incident radiation the nondimensional intensity is evaluated as i(m, x, φ, θ) =

1 (i11 + i12 + i21 + i22 ); 2

i12 = i21 ,

and extinction and scattering cross-sections and efficiencies are evaluated from ⎧ ⎫ ∞ ⎪   1 ⎪ ⎪ ⎪ 2  Cext ⎨ ⎬ b0I + a0II + 2 Qext = =  T (θ = 0) =  ⎪ (bnI + anII )⎪ ⎪ ⎪ ⎭ 2a x x ⎩ n=1 ⎫ ⎧  2π  ∞  ⎪   ⎪ ⎪ 1 Csca 1⎪ ⎨ 2 2 2 2 2 2 ⎬ , Qsca = = i m, x, φ, θ dθ = ⎪ |b0I | + |a0II | + |bnI | + |bnII | + |anI | + |anII | ⎪ ⎪ ⎪ ⎭ 2a πx 0 x⎩

(11.88)

(11.89) (11.90)

n=1

where, as for spheres, x = 2πa/λ, but cross-sections are per unit length of cylinder (i.e., have units of length). The phase function for a single, infinite cylinder is given by [30] i(θ, φ)δ(φ − φ ) Φ(Θ, φ) =  2π , i(θ, φ) dθ 0

cos θ = (cos Θ − sin2 φ )/ cos2 φ ,

(11.91)

where δ(φ − φ ) is the Dirac-delta function,2 and Θ is again the scattering angle away from the sˆ -direction, which is related to polar angle φ and azimuthal angle θ as given. The behavior of infinitely long fibers has been investigated by several researchers, notably the group around Tong [31–36] and by Lee [30,37–43] and others [44–47]. Some of these investigations have concentrated on scattering by single fibers [34,35,37], others on effects of dependent scattering [40–42,44], but most deal with the effects of various fiber arrangements. For a random arrangement of infinitely long fibers with size distribution 2. First defined in Section 10.9, equation (10.111).

422 Radiative Heat Transfer

n(a), extinction and scattering properties can be determined from [37]: 

∞

βλ (m) = 

0

π/2

Cext (m, x, φ) cos φ dφ n(a) da,

(11.92)

Csca (m, x, φ) cos φ dφ n(a) da,

(11.93)

0

∞

σsλ (m) =

 

0

π/2

0

1 4λ Φλ (m, Θ) = σsλ π2

 0

∞



π/2

0

i(θ) dφ n(a) da, sin θ cos φ  + cos θ = cos Θ − sin2 φ cos2 φ.

(11.94)

11.9 Approximate Scattering Phase Functions It is clear from Figs. 11.3 and 11.7 that radiative properties of particles may display strong oscillatory behavior with size parameter and, therefore, wavelength, particularly for the case of large, monodisperse, dielectric particles. Even more bothersome is the fact that the scattering phase function may undergo strong angular oscillations at any given single wavelength, again particularly for the case of large, monodisperse, dielectric particles (cf. Figs. 11.5, 11.6). Since radiative calculations for media with spectrally varying properties are generally carried out on a spectral basis with subsequent integration over all relevant wavelengths, this fact means that these spectral oscillations are somewhat inconvenient, but they do not make the analysis intractable. Strong angular oscillations in the scattering phase function, on the other hand, will enormously complicate the analysis for any given wavelength. Indeed, most solution methods described in the following chapters cannot accept highly oscillatory phase functions, or else they must be carried to unacceptably high orders or node numbers. It is, therefore, common practice to approximate oscillatory phase functions by simpler expressions with more regular behavior. It is observed that large particles generally have strong forward-scattering peaks (due to diffraction, cf. Fig. 11.6). Indeed, if x → ∞, half of the total extinction is due to diffraction into near-forward directions, as described in Section 11.7. Since diffraction was neglected (i.e., treated as transmission) in that section, the phase functions for large particles are in fact simplified. If either geometric optics cannot be used or diffraction effects must be retained for other reasons, then the approximate phase function must accommodate the strong forward-scattering peak. To this purpose many investigators have used the Henyey–Greenstein phase function, ΦHG (Θ) =

1 − g2 , [1 + g2 − 2g cos Θ]3/2

(11.95)

where g is the asymmetry factor. Sometimes the Henyey–Greenstein function is written in the form of a Legendre polynomial series, or ΦHG (Θ) = 1 +

∞ 

(2n + 1)gn Pn (cos Θ).

(11.96)

n=1

Thus, this expression is equivalent to equation (11.24) with approximate values for the An being related to the asymmetry factor. A representative comparison between Lorenz–Mie and Henyey–Greenstein phase functions is given in Fig. 11.15 for a dielectric with index of refraction m = n = 1.33 and size parameter x = 300 (water droplets). Both van de Hulst [48] and Hansen [49] have shown that the Henyey–Greenstein formulation gives very accurate results for radiative heat fluxes as long as the particles are nondielectric: Dielectric particles may have a relatively strong backward-scattering peak besides a strong forward-scattering peak. This situation cannot be described by the asymmetry factor alone, and the Henyey–Greenstein formulation must fail. That neglect of backward-scattering peaks can cause considerable error in heat flux calculations has been shown by Modest and Azad [25]. For many calculations the Henyey–Greenstein phase function is still too complicated. As mentioned earlier, in heat transfer applications forward scattering may usually be treated as transmission. This fact has led a number of researchers to the use of so-called Dirac-delta or Delta–Eddington approximations, where the forward-scattering

Radiative Properties of Particulate Media Chapter | 11 423

FIGURE 11.15 Comparison of Lorenz–Mie, Henyey–Greenstein, linear-anisotropic, and isotropic phase functions for water droplets (m = 1.33, x = 100).

peak is separated from the rest of the scattering phase function by Φ(Θ) ≈ 2 f δ(1−cos Θ) + (1 − f ) Φ∗ (Θ),

(11.97)

where Φ∗ is the new approximate phase function, f is a forward scattering fraction to be determined, and δ is the Dirac-delta function. Substitution of equation (11.97) into equation (9.17) shows that the approximate phase function is properly normalized, that is,  1 Φ∗ (Θ) dΩ = 1. (11.98) 4π 4π Different authors have used different approaches to define f and Φ∗ . Potter [50] was one of the first to use the following scheme for his work on atmospheric scattering. He truncated the peak by extrapolating the phase function from directions outside the peak into the forward direction; otherwise he left the phase function unchanged. Not surprisingly, his method produced excellent results, but it still leaves the approximate phase function in a rather complex form. It appears more promising to express the approximate phase function as a truncated Legendre series, Φ∗ (Θ) = 1 +

M 

A∗n Pn (cos Θ),

(11.99)

n=1

where the constant M is the chosen order of approximation, mostly taken as M = 1 (linear-anisotropic scattering) [25,51–53], or M = 0 (isotropic scattering) [52], while higher-order approximations have been carried out by Crosbie and Davidson [52]. There is considerable disagreement among authors about the criteria to be used to determine the forward fraction f as well as the coefficients A∗n . Both Joseph and coworkers [51] and Crosbie and Davidson [52] agreed that at least one of the moments of equation (11.97) should be satisfied: Multiplying equation (11.97) by Pm (Θ) and integrating over all Θ results in 

π 0

 Φ(Θ)Pm (cos Θ) dΘ = 0

π

2 f δ(1−cos Θ)Pm (cos Θ) dΘ + (1 − f )

M   n=1

0

π

A∗n Pn (cos Θ) Pm (cos Θ) dΘ,

(11.100)

424 Radiative Heat Transfer

or, using the fact that Legendre polynomials are orthogonal functions over the interval (0, π) [19], (1 − f )A∗m = Am − (2m + 1) f,

m = 1, 2, . . . .

(11.101)

If the approximate phase function is to be isotropic, equation (11.101) yields, with A∗1 = 0, f =

A1 = g, 3

(11.102)

and Φ(Θ) ≈ 2gδ(1−cos Θ) + (1 − g).

(11.103)

Joseph and colleagues [51] developed an approximate linear-anisotropic phase function. They employed equation (11.101) for the first two moments to find f and A∗1 , using an approximate value of A2 ≈ 5g2 (from the Henyey–Greenstein phase function). However, their approximate phase function may turn out to be negative for some back scattering directions, which is physically impossible. Crosbie and Davidson [52] overcame this difficulty by applying the second moment only conditionally. From the first moment it follows that g− f . 1− f

A∗1 = 3

(11.104)

Requiring the phase function to be positive for all angles is equivalent to |A∗1 | ≤ 1, or 1 (3g − 1) ≤ f ≤ g. 2

(11.105)

Instead of using the second moment directly, i.e., f = A2 /5, they require | f − A2 /5| to be a minimum without violating equation (11.105). This method can readily be extended to arbitrarily high orders. Their linear-anisotropic and order-10 phase function approximations are also included in Fig. 11.15 for water droplets. It should be noted that this method will work only for positive asymmetry factors. In the case of g < 0 the method breaks down and f = 0 should be used. Even then one may find A∗1 < −1, in which case one has to force A∗1 = −1 to avoid negative forward scattering. The method will break down completely for strong backward-scattering peaks. None of the above approximations allows for simultaneous forward- and backward-scattering peaks. Modest and Azad [25] have shown that neglecting the backward-scattering peaks that may appear in dielectrics may cause considerable error in heat flux calculations. Thus, they proposed a double Dirac-delta phase function approximation. However, this model severely complicates the RTE by requiring a I(−ˆs) term (backward intensity). Example 11.3. Calculate approximate phase functions for monodisperse suspensions of large specular dielectric spheres (m = 2) and diffusely reflecting spheres, using the Henyey–Greenstein function, and the Crosbie and Davidson model. Solution The Henyey–Greenstein function requires the calculation of the asymmetry factor g=

1 A1 = 3 2



+1

−1

Φ(μ)μ dμ,

where μ is the cosine of the scattering angle. The Crosbie and Davidson approximation requires the calculation of g as well as the calculation of  A2 1 +1 = Φ(μ)P2 (μ) dμ. 5 2 −1 Numerical integration of the phase function yields for the specular dielectric spheres g = 0.229 and A2 /5 = 0.138. Since A2 /5 < g it follows for the Crosbie and Davidson model that f = A2 /5 = 0.138 and, from equation (11.104), A∗1 = 0.315. Both approximate phase functions are shown in Fig. 11.16 together with the exact expression. The Henyey–Greenstein function does not try to remove the forward-scattering peak, but is unable to follow the sharp peak for large μ. The Crosbie–Davidson model follows the actual function well, except for the forward peak that has been removed. The integration for the diffuse-sphere phase function could be carried out analytically but is rather tedious. Numerical integration of the phase function yields for the diffuse spheres g = −0.444 and A2 /5 = 0.062. Since g < 0, the scattering is

Radiative Properties of Particulate Media Chapter | 11 425

FIGURE 11.16 Scattering phase function approximations for Example 11.3.

predominantly backward and the Crosbie and Davidson model cannot be applied. Thus, for this model, we force f = 0 and, from equation (11.104), A∗1 = −1.333; since this would result in negative values for forward directions we also force A∗1 = −1. It is seen that the Henyey–Greenstein function does not work very well for back scattering, while the Crosbie and Davidson model gives acceptable results.

11.10 Radiative Properties of Irregular Particles and Aggregates In practical applications particles are rarely, if ever, homogeneous spheres or long cylinders. As noted earlier, averaging over millions of irregularly shaped particles tends to give results very close to those found with the uniform sphere assumption [1]. However, if the average shape of irregular particles does not resemble a sphere or a long fiber, more advanced methods must be employed to study their interaction with electromagnetic waves. Over the years numerous exact and approximate methods have appeared in the literature, recently reviewed by Mishchenko [54] and Wriedt [55], the latter also listing freely available computer codes. An extensive description of general disperse systems has been given by Dombrovsky and Baillis [2]. An in-depth discussion of the treatment of irregular particles is beyond the scope of this text, and the reader is directed toward these three exhaustive references. We will give here only a very brief account of the perhaps most popular methods, the cluster T-matrix method, the generalized multisphere Mie solution, the discrete dipole approximation, and the Finite Difference Time Domain method.

The Cluster T-Matrix Method The method goes back to 1965 as proposed by Waterman [56], and a detailed description is provided in Mishchenko and coworkers [57]. While the method can be applied to particles of any shape, it is best suited for rotationally symmetric particles and can be readily applied to multiparticle clusters, such as fractal aggregates. In the T-matrix method the incident, internal, and scattered electromagnetic fields for the individual particles are expanded into vector spherical harmonics. Coefficients of the scattered field are linearly related to those of the incident field by a matrix called the T- (or transition) matrix. Linearity of Maxwell’s equations then allows the determination of the scattered field of an agglomerate through superposition. One of the advantages of the T-matrix method is that, once the matrix has been computed, it can be applied to arbitrary incidence angles, i.e., the method provides not only scattering and extinction coefficients, but also directional scattering information, which can be of great importance in laser scattering diagnostics. The method has been gathering considerable popularity and, in particular, has been applied to multisphere clusters, such as soot aggregates [58–62] and alumina particle clusters [63,64]. Several T-matrix computer codes are freely available from [65,66].

426 Radiative Heat Transfer

The Generalized Multisphere Mie Solution Like the cluster T-matrix method the generalized multisphere Mie solution (GMM) is also an exact method for scattering from clusters of small particles. In fact, the method shares many features with the T-matrix method, but there are also substantial differences, such as different treatment of far-field interference and in translating field expansions between displaced reference systems [67,68]. The method was developed by Xu [69,70], and enjoys increasing popularity for the modeling of scattering from soot aggregates [71–73]. A GMM computer code may be downloaded from [74].

The Discrete Dipole Approximation Another popular method to deal with scattering from aggregates of small particles is the discrete dipole approximation. In the limit of small point masses (i.e., individual atoms) the particles can be thought of as electrical dipoles, which then allows for an exact formulation of the resulting electromagnetic field. To make the problem manageable, a particle or aggregate may be subdivided into a relatively small number of identical elements, each containing many atoms, but small enough to be represented as a dipole oscillator. The vector amplitude of the field scattered by each dipole is determined iteratively, and the total scattered field is obtained as the sum of all the individual dipole fields. The method was first formulated by Purcell and Pennypecker [75]. It appears to be particularly well suited to model aggregates of many identical primary particles, such as soot, and has been employed, for example, by Mulholland et al. [76,77].

Finite Difference Time Domain Method In recent years, with tremendous increase in computational resources and speed, it has now become possible to solve the Maxwell’s equations numerically in time domain to quantify scattering of light by irregular particles. The Finite Difference Time Domain (FDTD) method uses the finite difference method in both space and time to discretize Maxwell’s equations, which are then solved using a time-marching method. The method was first developed by Yee [78] in the electrical engineering field, and first applied to light scattering by Yang and Liou [79]. A review of it has recently been given by Sun [80]. Free computer codes employing this scheme are available, e.g., at scattport.org.

11.11 Radiative Properties of Combustion Particles Undoubtedly, some of the most important engineering applications of thermal radiation are in the areas of the combustion of gaseous, liquid (usually in droplet form), or solid (often pulverized) fuels, be it for power production or for propulsion. During combustion thermal radiation will carry energy directly from the combustion products to the burner walls, often at rates higher than for convection. In the case of liquid and solid fuels thermal radiation also plays an important role in the preheating of the fuel and its ignition. Nearly all flames are visible to the human eye and are, therefore, called luminous (sending out light). Apparently, there is some radiative emission from within the flame at wavelengths where there are no vibration–rotation bands for any combustion gases. This luminous emission is today known to come from tiny char (almost pure carbon) particles, called soot, which are generated during the combustion process. The “dirtier” the flame is (i.e., the higher the soot content), the more luminous it is. A review of the importance of radiative heat transfer in combustion systems has been given by Sarofim and Hottel [81]. All combustion processes are very complicated. Usually there are many intermediate chemical reactions in sequence and/or parallel, intermittent generation of a variety of intermediate species, generation of soot, agglomeration of soot particles, and subsequent partial burning of the soot. Since thermal radiation contributes strongly to the heat transfer mechanism of the combustion, any understanding and modeling of the process must include knowledge of the radiation properties of the combustion gases as well as any particulates that are present. The most important particles are the relatively large coal and fly ash particles formed during the combustion of pulverized coal as well as the very small soot particles. Because of their great importance, these suspensions will be treated in some detail below.

Radiative Properties of Particulate Media Chapter | 11 427

TABLE 11.2 Representative values for the complex index of refraction in the near infrared for different coals and ashes [90]. Particle Type

m = n − ik

carbon

2.20 − 1.12i

anthracite

2.05 − 0.54i

bituminous

1.85 − 0.22i

lignite

1.70 − 0.066i

fly ash

1.50 − 0.020i

Pulverized Coal and Fly Ash Dispersions To calculate the radiative properties of arbitrary size distributions of coal and ash particles, one must have knowledge of their complex index of refraction as a function of wavelength and temperature. Data for carbon and different types of coal indicate that its real part, n, varies little over the infrared and is relatively insensitive to the type of coal (e.g., anthracite, lignite, bituminous), while the absorptive index, k, may vary strongly over the spectrum and from coal to coal [82–84]. The composition of fly ash and, therefore, its optical properties may vary greatly from coal to coal. The few data in the literature [85–89] report consistent values for the refractive index (n ≈ 1.5) and widely varying values for the absorptive index. Wall and coworkers [87] calculated the absorptive index for a number of Australian coals (based on their ash composition), and found that k varied between 0.008 and 0.020. Nothing at all appears to be known about the temperature dependence of these optical properties. A summary of representative values for the optical constants of coals and ashes has been reported by Viskanta and colleagues [90] and is reproduced in Table 11.2. Somewhat more detailed data may be found in Johansson et al. [91], who collected coal and ash refractive index data from the literature and provide polynomial fits for several coals and ashes as function of wavelength. A first attempt to establish formulae for extinction by carbon particles was made by Tien and coworkers [92], who looked at a single index of refraction (m = 1.5 − 0.5i) for a gamma size distribution of particles [cf. equation (11.34)]. They found a relatively simple (but not very accurate) smooth correlation for the extinction coefficient β. Buckius and Hwang [93] carried out a large number of Lorenz–Mie calculations for a variety of complex indices of refraction (simulating different coals) and a variety of different particle distribution functions [gamma distributions and “rectangular” distributions, i.e., n(a) = const over a certain range of radii]. They found that, when normalized with the Rayleigh small-particle limit, the absorption coefficient and extinction coefficient as well as the asymmetry factor are virtually independent of the particle size distribution function, and only depend on a mean particle diameter. Employing the range for m given by Foster and Howarth [82] for different coals, they found a similar insensitivity of the index of refraction, at least in the limits of small and large particles; in the intermediate size range, deviations of up to nearly ±50% were reported as shown in a sample of their calculations, Fig. 11.17. The spectral results were also wavelength-integrated to yield Planck-mean and Rosseland-mean absorption and extinction coefficients. Considering a temperature range of 750 K to 2500 K they found that their data could be correlated to within 30% for the different coals. Based on their numerical data for different types of coals they developed correlations for a number of nondimensional radiation properties. Spectral properties correlated were the absorption and extinction coefficients and the asymmetry factor, with nondimensional κ and β defined by κ∗ (λ, m) = κ(λ, m, NT )/ fA , where

 fA =

∞

πa2 n(a) da

β∗ (λ, m) = β(λ, m, NT )/ fA ,

(11.106) (11.107)

0

is the total projected area of the particles per unit volume. Thus, these nondimensional values are essentially sizeaveraged absorption and extinction efficiencies [cf. equations (11.39) and (11.40)]. For extremely small particles

428 Radiative Heat Transfer

FIGURE 11.17 Extinction and absorption properties of pulverized coal [93,94].

κ∗ ≈ β∗ may be calculated from Rayleigh scattering theory, equation (11.53), as 0 0 / 2 / 2 m − 1 6π fv m −1 ∗ ∗ , κ0 (λ, m) = β0 (λ, m) = − 2 = −4x¯  2 m + 2 λ fA m +2 where x¯ is a mean size parameter based on a mean particle radius defined by ∞ a3 n(a) da 3 fv r32 = = 0∞ . 4 fA a2 n(a) da 0

(11.108)

(11.109)

The corresponding diameter d32 = 2r32 is also known as the Sauter mean diameter (six times the volume divided by the surface area), named after a German scientist of the 1920s. Many studies agree that use of a single (Sauter mean) diameter (as opposed to a size distribution) is sufficient for coal and ash clouds. Since β∗0 is linear in x¯ it may also be regarded as a weighted (by a function of m) size parameter. The asymmetry factor for Rayleigh scattering is zero (because of its symmetric phase function) and g0 for the small particle limit must be found from a higher-order expansion given by [93], which may be simplified to / 2 0   ∞ 8 (m + 2)(m2 + 3) 2π 2 0 a n(a) da 1 ∞ g0 (λ, m) =  . (11.110) 15 λ 2m2 + 3 a6 n(a) da 0

In a similar fashion, they defined nondimensional Planck-mean and Rosseland-mean absorption and extinction coefficients, all normalized by fA . All correlations obey the same basic formula, 1 1 1 = z + z , yz y0 y∞

(11.111)

where y stands for one of the above nondimensional properties, y0 is that property for small average particle sizes, and y∞ the one for large average particle sizes. The correlation parameters y0 , y∞ , and z for the various properties are summarized in Table 11.3, and results of this correlation are included in Fig. 11.17. A somewhat simpler set of formulae to calculate the radiative properties in pulverized-coal reactors has been given by Kim and Lior [95].

Radiative Properties of Particulate Media Chapter | 11 429

TABLE 11.3 Correlation parameters for the prediction of nondimensional coal properties from y−z = y−z + y−z ∞ [93]. 0 y

y0

y∞

β∗ (λ, m)

β∗0 (1 + 6.78β∗0 2 )

3.09/β∗0 0.1

κ∗ (λ, m)

β∗0 (1 + 2.30β∗0 2 )

1.66/β∗0 0.16

1.6

g(λ, m)

g0

0.9

1.0

β∗P

0.0032φ[1 + (φ/355)1.9 ]

10.99/φ0.02

1.2

+ (φ/485)1.75 ]

10.99/φ0.02

1.2

0.0032φ[1 + (φ/725)1.65 ]

13.75/φ0.13

1.5

0.0032φ[1 + (φ/650)2.3 ]

15.65/φ0.143

1.15

β∗R κ∗P κ∗R

0.0032φ[1

z 1.2

φ = r¯T/1 μm K, β and κ nondimensionalized by fA from equation (11.107); β∗0 from equation (11.108), g0 from equation (11.110), r¯ from equation (11.109).

The results of Buckius and Hwang were essentially corroborated by Viskanta and coworkers [90]. They too found that variations with particle distribution functions are relatively minor, and that the different indices of refraction made a difference only for midsized particles. However, they felt that these differences were too large to use a single correlation and presented individual graphs for different coals. Table 11.3 indicates that— according to Buckius and Hwang [93]—Planck-mean and Rosseland-mean coefficients do not depend on the optical properties of the coal and are very close to one another. Again, this observation was corroborated by Viskanta and coworkers [90] for carbon, anthracite and bituminous coal, as well as for lignite at high temperature (above 1000 K). For fly ash and for lower temperature lignite mean absorption coefficients were considerably lower due to the significantly lower absorptive indices of these materials. Thus, Table 11.3 should be regarded as a relatively crude approximation, which should be replaced when more accurate data for different coals and ashes become available (optical properties varying with wavelength and temperature, particle size distributions). Mengüç and Viskanta [94] applied the approximate theory of equations (11.59) to two very different particle size distributions and several different complex indices of refraction (simulating carbon particles, several coals, and fly ash). They found the approximate solutions to agree very well with full Lorenz–Mie calculations, even for carbon particles [which, with m = 2.20 − 1.12i, significantly violate the limitations on equations (11.59)]. Like Buckius and Hwang [93] they noticed that the particle size distribution has only a very small effect on radiative properties. For comparison, results from equation (11.59) for particles of uniform size are also included in Fig. 11.17, showing good agreement with Buckius and Hwang’s correlation for large particle sizes. Equations (11.59) predict the index of refraction effects more accurately but must fail for small size parameters. Liu and Swithenbank [96] used the same simplified theory, together with the comprehensive experimental data of Goodwin [88], to predict radiative properties of fly ash dispersions. They found that wavelength dependence of the complex index of refraction cannot be ignored: while n remains relatively constant, the absorptive index k of fly ash varies by orders of magnitude across the spectrum, causing large changes in radiative properties. Im and Ahluwalia [97], also using Goodwin’s [88] data, have given a correlation of the complex index of refraction for fly ash, as a function of wavelength and mineral composition. Manickavasagam and Mengüç [98] gave direct correlations for the absorption coefficient of two coals (as a function of wavelength), again finding that particle size distributions did not change κ appreciably. A slightly different approach was taken by Caldas and Semião [99], who used four curve fits to the Lorenz–Mie results for Qext and Qsca , covering different ranges of effective particle sizes. They applied this method to several distributions typical of fly ash and carbon particles. In more recent studies Johansson et al. [91] observed that knowledge of (Sauter mean) sizes is more important than accurate values for m; for large particles (10 μm for ash and 40 μm for coal) they found heat transfer results unaffected by which of the (widely varying) m from the literature was chosen. Also, a T-matrix study by Gronarz et al. [100] showed that nonsphericity of coal and ash particles had little impact on results, with much simpler standard (i.e., independent spheres) Mie calculations always accurate to within 10%. On the other hand, Gronarz et al. [101] found the effects of directional scattering to be important. Both ash and coal are strong forward scatterers (in particular, coal). They determined that the two extremes (all scattering isotropic, or no scattering at all) gave poor results. However, the Delta–Eddington approximation of equation (11.97) was found

430 Radiative Heat Transfer

FIGURE 11.18 Complex index of refraction for several liquid fuels based on different studies; n-decane & n-heptane: Anderson [103], ethanol: Sani and Dell’Oro [112], Diesel, yellow: Dombrovsky et al. [106].

to give accurate results with forward scattering factors determined based on Mie theory, for which an analytical expression is included in their paper. Small Fortran routines for the models of Buckius and Hwang and of Mengüç and Viskanta are included in Appendix F.

Radiative Properties of Fuel Sprays Absorption and scattering properties of droplet clouds can be calculated from Lorenz–Mie theory as long as the complex index of refraction, m, of the fuel is known. Tuntomo et al. [102] reported values of m for heptane and decane in the range of 2.6–15 μm, found from transmission data, and similar measurements were also performed by Anderson [103] (also including data for hexane and nonane); decane (as well as butanol) was also considered by Wang [104]. In addition, Dombrovsky and coworkers [105,106] reported values for several types of Diesel fuels across various ranges from 0.2 to 16 μm. Finally, data for ethanol, ethylene glycol and toluene have been measured or recorded by several others [107–112]. Selected results of these data are displayed in Fig. 11.18; reporting all would overwhelm the figure. Dombrovsky et al. [106,113] also noted that Mie calculations can be computationally expensive, since fuel sprays generally include large semi-transparent droplets (x  1, k 1), which require large numbers of terms in the calculations. Assuming typical fuel droplet radii of 2 μm < a < 50 μm, and a spectral range of importance of 0.4 μm < λ < 6 μm, leads to a range of importance of 2 < x < 800 for the size parameter in Diesel fuel sprays. Making many tests against Mie calculations they devised an approximate scheme to evaluate the absorption and scattering efficiencies for fuel sprays:  4n  1 − e−4kx 2 (n + 1) / 3 p/5 −15k = n(n − 1)e 2 (5/p)γ

Qabs = Qsca

(11.112) p≤5 , p>5

p = 2x(n − 1),

γ = 1.4 − e−80k .

(11.113)

Radiative Properties of Particulate Media Chapter | 11 431

FIGURE 11.19 Results of Mie calculations (solid curves) and calculations based on approximate formulae (11.112) and (11.113) (dashed curves) of the efficiency factor for absorption Qabs (lower curves) and efficiency factor of extinction Qext = Qabs + Qsca (upper curves) of individual spherical droplets with (a) a = 10 μm, and (b) a = 50 μm [106].

Equation (11.112) is valid in the geometric optics limit with small absorptive index (x  1, k 1) but has been shown to be accurate even when these conditions are not satisfied [106]; equation (11.113) was obtained by curve fitting Mie calculations. An example is shown in Fig. 11.19 depicting absorption and extinction efficiencies for uniform size Diesel sprays determined, both, from exact Mie calculations and from the approximate model. While equations (11.112) and (11.113) cannot follow all the oscillations in the Lorenz–Mie calculations, agreement is generally quite acceptable, in particular, in light of the fact that the Mie peaks and valleys are smoothed out in the presence of a droplet size distribution. Dombrovsky and coworkers further noted that fuel sprays obeying gamma-distributions are well approximated by single-sized droplets when employing the Sauter mean radius, equation (11.109). In a more recent publication studying the properties of water mist for fire protection, Dombrovsky et al. [114] noted that the approximate formulas (11.112) and (11.113) are also valid for absorption and scattering efficiencies for water sprays (assuming similar droplet diameters as for fuel).

Radiative Properties of Soot Soot particles are produced in fuel-rich flames, or fuel-rich parts of flames, as a result of incomplete combustion of hydrocarbon fuels and biomass burning. As shown by electron microscopy, soot particles are generally small and spherical, ranging in size between approximately 50 Å and 800 Å (5 nm to 80 nm), and up to about 3000 Å in extreme cases [115,116]. While mostly spherical in shape, soot particles may also appear in agglomerated chunks and even as long agglomerated filaments. Consisting primarily of carbon, they are strong absorbers and emitters of radiation and are, therefore, of great importance to the understanding of combustion processes. Particles emitted from various combustion devices and biomass burning entrained in Earth’s atmosphere have also been identified as major contributors to radiative forcing of climate as well as to various health concerns to humans. In the aerosol community studying such effects soot is commonly referred to as “black carbon” (BC). It has been determined experimentally in typical diffusion flames of hydrocarbon fuels that the volume percentage of soot generally lies in the range between 10−4 % and 10−6 % [81,117,118]. Since soot particles are very small, they are generally at the same temperature as the flame and, therefore, strongly emit thermal radiation in a continuous spectrum over the infrared region. Experiments have shown that soot emission often is considerably stronger than the emission from the combustion gases. In order to predict the radiative properties of a soot cloud, it is necessary to determine the amount, shape, and distribution of soot particles, as well as their optical properties, which depend on chemical composition and particle porosity. It is known today that key steps in soot formation and destruction (oxidation) are [119–127]: formation of gas-phase precursors (polycyclic aromatic hydrocarbons—PAHs) in fuel-rich regions; soot particle inception (∼ 1 nm particle size); particle surface growth involving acetylene (C2 H2 ) and/or PAHs; particle coagulation/agglomeration (up to tens of μm particle size);

432 Radiative Heat Transfer

and oxidation in oxygen-rich regions. High-level soot models often employ concentration moments of the soot distribution function [128–130]: this “method-of-moments” has the advantage that it allows the distribution of soot particles to be computed using essentially the same approach that is used for gas-phase chemical species. Early work on soot radiation properties concentrated on predicting the absorption coefficient κλ for a given flame as a function of wavelength. For all but the largest soot particles the size parameter x = 2πa/λ is very small for all but the shortest wavelengths in the infrared, so one may expect that Rayleigh’s theory for small particles will, at least approximately, hold. This condition would, according to equation (11.53), lead to negligible scattering for unagglomerated soot particles and an absorption coefficient of 0 / 2 fv m − 1 6π fv 36πnk κλ = βλ = − 2 = 2 . (11.114) λ m +2 (n − k2 + 2)2 + 4n2 k2 λ In the soot literature it is common to employ a (soot) “refractive index function,” / 2 0 m −1 6nk E(m) = − 2 = 2 2 m +2 (n − k + 2)2 + 4n2 k2

(11.115)

and the absorption coefficient reduces to fv . (11.116) λ In the aerosol community use of the mass absorption cross-section (MAC), or total absorption per unit mass of black carbon, is preferred, and is related to the linear absorption coefficient by κλ = βλ = 6πE(m)

κλ = MAC ρBC fv ,

(11.117)

where ρBC = 1.8 ± 0.1 g/cm3 is the bulk density, which has been found to be relatively insensitive to different types of soot [131]. Experiments have confirmed that scattering may indeed be neglected for unagglomerated soot [132]. The form of equation (11.116) would lead one to expect that the absorption coefficient should vary with wavelength as 1/λ. However, this assumption is only approximately correct, since the complex index of refraction m (and, in particular, the absorptive index k) can vary significantly across the spectrum. It is customary to write C fv (11.118) κλ = a , λ where C and a are empirical constants. Many different values for the dispersion exponent a have been measured by investigators for many different flame conditions, ranging from as low as 0.7 to as high as 2.2. Earlier theories explained this deviation from Rayleigh theory to be a consequence of particle size. While it is true that Lorenz–Mie theory predicts a growing value for a for increasing particle size, it is easy to show that this alone cannot explain the large values for the dispersion exponent in some flames. Rather, this increase in a must be due to spectral variations of the effective complex index of refraction, resulting from the chemical composition and the porosity of the soot particles. Millikan [133,134] investigated the dependence between dispersion exponent and chemical composition. While for many years soot was assumed to be amorphous carbon, he found the particles contained considerable amounts of hydrogen (up to 40 atom-%), and he determined that a was approximately directly proportional to the hydrogen–carbon ratio of the soot material as shown in Fig. 11.20. He further showed that the radiative properties of the soot were the same for in situ flame measurements as for soot collected from the flame, suggesting that the optical properties are fairly independent of temperature. Unfortunately, his experimental setup did not allow for the determination of the constant C in equation (11.118), so that quantitative evaluation of the extinction coefficient is not possible. The optical properties of soot material, i.e., the complex index of refraction m, have received a very considerable amount of attention during the last 50 years, using different forms of carbon and various experimental methods. Foster and Howarth [82] were the first to report experimental measurements for the complex index of refraction of hydrocarbon soot, based on various carbon black powders. This work was followed shortly thereafter with measurements by Dalzell and Sarofim [135] on soot collected on cooled brass plates from laminar diffusion flames burning either acetylene or propane. In both cases pellets with very smooth, quasi-specular

Radiative Properties of Particulate Media Chapter | 11 433

FIGURE 11.20 Dispersion exponent a of soot deposits vs. hydrogen-to-carbon ratio: 1, pure carbon (arc evaporated); 2, acetylene/oxygen flame; 3, ethylene/oxygen flame; 4, 5, 6, ethylene/air flames [133].

surfaces were formed by compressing small soot samples between optically flat surfaces with pressures up to 2760 bar. The index of refraction was then deduced from reflectance measurements employing Fresnel’s relations for specular reflectors. They found the optical properties of the two different types of soot to be fairly similar, with values for acetylene soot somewhat higher than for propane soot, apparently because of the higher H/C ratio in propane soot. Comparing their results with values reported by Stull and Plass [136] (based on amorphous carbon) and by Howarth, Foster, and Thring [137] (based on pyrographite) they note that optical properties of amorphous or graphitic carbon are not equal to those of soot, primarily because of the different H/C ratios. The data of Dalzell and Sarofim [135] have been employed in many subsequent studies (and continue to be used today). For example, Hubbard and Tien [138] used them to evaluate Planck-mean and Rosseland-mean absorption coefficients for soot clouds and soot–gas mixtures. However, the accuracy of Dalzell and Sarofim’s data has been questioned by a number of researchers. All ex situ measurements suffer from the fact that during the analysis the soot is not in the same state as in the flame. The soot particles are at a different temperature, and they may have different morphologies because of agglomeration during the sampling process. The severest criticism concerns the pellet-reflection technique. Medalia and Richards [139], Graham [140], and Janzen [141] have pointed out that the pellets must contain a considerable amount of void (33% even after compression to 2760 bar, according to Medalia and Richards [139]), since the sample is made by compressing a powder. This technique leads to two serious sources for errors: (i) Since the pellets are actually a two-phase dispersion of soot and air, the inferred index of refraction is the one of the dispersion and not the one of the soot particles themselves, and (ii) at least at short wavelengths the pellet cannot be assumed to be optically smooth and Fresnel’s relations become invalid. Nevertheless, Felske and coworkers [142] returned to the pellet-reflection technique, arguing that—for a carefully prepared pellet—the data in the infrared do obey Fresnel’s relations. They also measured the void fraction over the first few layers of particles (where all absorption occurs) and found that the proportion of voids in these layers is significantly lower (18%) than in the bulk of the material (33%) and determined that their surfaces could be considered specular reflectors for wavelengths λ ≥ 2.0 μm. Their data, even after correction for voidage (which raise the value for n by approximately 0.3, and for k by approximately 0.15), differ significantly from those of other investigations and depend only weakly on wavelength. Problems with the pellet-reflection technique prompted Lee and Tien [143] to obtain soot optical properties from in situ flame transmission data together with application of the dispersion theory [18,144] (i.e., the theory that predicts the wavenumber dependence of the optical constants n and k by relating them to bound- and freeelectron densities).Their results for polystyrene and Plexiglas flame soot, based on data by Buckius and Tien [145] and Bard and Pagni [146], are shown in Fig. 11.21 together with the data of Stull and Plass [136], Howarth and coworkers [137], and the propane soot results of Dalzell and Sarofim [135], Chang and Charalampopoulos [147], and Felske and coworkers [142]. Lee and Tien’s data agree fairly well with those of Dalzell and Sarofim, except for the visible where the pellet-reflection technique is particularly suspect. In contrast to Dalzell and Sarofim as well as Millikan [133,134], Tien and Lee noted that the optical properties varied little from flame to flame

434 Radiative Heat Transfer

FIGURE 11.21 Complex index of refraction for soot based on different studies: 1, Lee and Tien [143] (polystyrene and Plexiglas soot); 2, Stull and Plass [136] (amorphous carbon); 3, Dalzell and Sarofim [135] (propane soot); 4, Howarth and coworkers [137] (pyrographite at 300 K); 5, Chang and Charalampopoulos [147] (propane soot); 6, Felske and coworkers [142] (propane soot).

despite their different fuel (not necessarily soot) H/C ratios. Conceivably the soot of their different flames had similar H/C ratios. They also applied the dispersion theory to determine the temperature dependence of the optical properties, observing that m = n − ik is very insensitive to temperature changes at high temperature levels. This would imply negligible effect of spatial temperature variation on soot properties, as is commonly assumed. It should be noted that, like the pellet-reflection technique, the spectral transmission technique has its own set of difficulties: For its data reduction, a scattering theory and a theory describing the spectral variation of the refractive index (the dispersion theory) must be used. Usually the Lorenz–Mie scattering theory based on monodisperse spherical soot particles is employed. Thus, only when the particles are spherical with a single diameter can these results be used with confidence. The more recent data of Chang and Charalampopoulos [147] show similar values for the refractive index, but somewhat lower absorptive indices. Their data have been confirmed in even more recent studies covering diverse flame conditions [148]. Chang and Charalampopoulos also provided a polynomial expression, valid for the wavelength range 0.4 μm ≤ λ ≤ 30 μm : n = 1.811 + .1263 ln λ + .0270 ln2 λ + .0417 ln3 λ, 2

3

k = .5821 + .1213 ln λ + .2309 ln λ − .0100 ln λ,

(11.119a) λ in μm.

(11.119b)

It is well known today that in most flames soot particles agglomerate into large chunks or long chains, making the use of the spherical-particle assumption very questionable. A number of textbooks have considered scattering by nonspherical particles [3,16,18], and a short introduction to methods for nonspherical particles and agglomerates was provided in Section 11.10. Approximating chunks of soot as prolate spheroids, Jones [149] found their absorption behavior to be considerably different from that of spheres of identical volumes. Lee and Tien [150] investigated the extreme case of long chains approximated by infinite cylinders. They found that the extinction coefficient for spheres drops off in the infrared much faster than the one for cylinders of the same radius. However, the wavelength-integrated extinction coefficient is rather insensitive to particle shape at elevated temperatures, say T > 1000 K (i.e., at flame temperatures where soot emission may be important) [150]. Similar results were found by Mackowski and coworkers [151], who looked at infinite soot cylinders also using Lee and Tien’s optical properties. Investigating the behavior of polydisperse cylindrical soot particles, they found the behavior to be similar to that observed by Buckius and Hwang [93] for polydisperse coal particles. While they generated correlations for absorption and extinction coefficients according to equation (11.111), unfortunately

Radiative Properties of Particulate Media Chapter | 11 435

their correlation is rather cumbersome to use since different sets of parameters apply to each of a large number of wavelengths. In more modern measurements Dobbins and Megaridis [152] built a thermophoretic probe that made it possible to sample soot aggregates from flames for electron microscope studies. They showed that near the start of soot formation small aggregates form (on the order of 10 nm), while clusters up to 1 μm in length can be found in turbulent flames, i.e., sizes clearly too large for the Rayleigh theory to hold. Experiments have further shown that soot aggregates resemble mass fractals, and the number of soot particles in an aggregate is given by3 N = k f (R g /a)D f ,

(11.120)

where a is the radius of the individual soot particles, R g is the radius of gyration, and D f and k f are mass fractal dimension and fractal prefactor, respectively [153,154]. Various soot aggregates of different size and shape have been found to have universal morphology with 1.6 < D f < 1.9 and 2.0 < k f < 2.6, almost independent of fuel or position within a flame [155–158]. On the other hand, fractal clusters generated theoretically by diffusion limited cluster aggregation (DLCA) tend to lead to smaller fractal prefactors of around 1.2 [159], a discrepancy that remains unresolved. For monodisperse particles the radius of gyration is found from R2g =

N 1  2 ri , N

(11.121)

i=1

where ri is the distance from the center of each particle to the center of gravity of the soot aggregate. Accordingly, more recent theoretical studies have modeled agglomerated soot as long chains of spherical particles [156,160–178]. Three different fundamental approaches have been pursued, most using Jones’ formulation [179,180], which in turn is based on Saxon’s integral equation [181]. In this method primary soot particles are assumed to obey Rayleigh scattering and the electric field inside them is taken as uniform, while the field outside the particles is determined from the integral representation of Maxwell’s equations [160–165]. In the discrete dipole approximation (DDA) of Purcell and Pennypecker [75] the soot aggregate is modeled as an array of N polarizable elements (“dipoles”) in vacuum, leading to a set of linear algebraic equations [166–169]. In the model of Iskander and coworkers [182], known as the I-C-P model, the aggregate is divided into cubical or spherical cells with uniform electromagnetic fields. A control volume analysis converts the governing equations to a set of linear algebraic ones [170–172]. Results from these aggregate models show that approximating agglomerated soot as infinitely long cylinders [150,151] leads to significant errors. In recent years a number of numerically exact (and computer intensive) simulations of fractal soot aggregates have also been carried out, using the T-matrix and generalized multisphere Mie (GMM) solutions, as briefly described in Section 11.10 [59,61,71–73,183,184]. Good reviews of the different methods to calculate scattering properties of agglomerated soot have been given by Köylü and Faeth [174], Manickavasagam and Mengüç [177], and by Sorensen [185], also assessing a number of more approximate theories, such as regular Rayleigh scattering (treating particles as independent), equivalent-sphere Mie scattering (replacing the agglomerate by a single sphere of equal volume), and Rayleigh– Debye–Gans (R-D-G) scattering (valid for |m − 1| 1) for both assumed shapes (such as straight chains) and fractal aggregates. While Rayleigh scattering always underpredicts scattering, equivalent-sphere Mie scattering can give acceptable results under certain conditions, in particular for small aggregates, but must also be regarded as unreliable, in general. The R-D-G scattering theory tends to give relatively good results, especially if fractal aggregates are considered (i.e., conforming to experimental observations). However, the R-D-G scattering theory assumes a complex index of refraction near unity, which is clearly not very accurate for soot, especially at larger wavelengths (see Fig. 11.21). Köylü and Faeth [186] pointed out and corrected inconsistencies in the fractal R-D-G model of Dobbins and Megaridis [173], as reported in [174], and revised results have been given by Farias and coworkers [175]. Another variation of the R-D-G approach has been reported by Sorensen and Roberts [159], using a slightly different form factor. Zhao and Ma [61] have compared the two R-D-G models and assessed their accuracy, finding both models to perform about equally well. Results from equivalent-sphere Mie calculations and R-D-G calculations are compared with approximate I-C-P calculations of Nelson [170] (assuming a mean scalar field) in Fig. 11.22 [174]. Figure 11.22a shows that the absorption cross-section of a soot aggregate, Caabs , differs by only a few percent from that of N independent 3. In the early work of Dobbins and Megaridis [152], as well as in some later papers, equation (11.120) was based on primary particle diameter, 2a, which increases k f by a factor of 2D f , while most authors today use the form given here.

436 Radiative Heat Transfer

FIGURE 11.22 Absorption and scattering cross-sections of soot aggregates (C p is cross-section for a single particle, C a for an aggregate, and N is the number of soot particles in the aggregate) [174].

soot particles. This result is also obtained by the R-D-G theory (for assumed or fractal aggregates) since it neglects agglomeration effects on absorption, while the equivalent sphere calculations become totally unreliable for N > 10. Mackowski [187] used the electrostatic approximation (ESA) to calculate Caabs for fractal aggregates with various indices of refraction. He found that agglomeration effects on the absorption coefficient are well correlated by ⎛ a ⎞ ⎛ a ⎞ ⎜⎜ Cabs ⎟⎟ 2(N − 1) ⎜⎜⎜ Cabs ⎟⎟⎟ 3 ⎟⎟ = ⎜⎜ + (11.122) ⎟ , ⎜ p ⎠ ⎝ 2N + 1 2N + 1 ⎝ NC p ⎠ NCabs abs ∞ with the limiting value for N → ∞ depending on the refractive index function E(m). For typical soot optical p a constants (Cabs /NCsca )∞ is relatively small (= 1.08 for m = 1.6 − 0.6i), but can be substantial for larger m. These results were qualitatively confirmed by Liu and Smallwood [71,72], who used the exact GMM formulation. p However, they noted that Cabs also depends on primary particle size, and that equation (11.122) becomes inaccurate for large soot particles (x > 0.1). p a Figure 11.22b indicates that the relative scattering cross-section, Csca /NCsca , strongly increases with particle number—first linearly (for small aggregates with negligible multiple scattering and self-interaction), then tending toward a saturation value for large N. Again, the equivalent sphere predictions become unreliable for N > 10 or so, while the R-D-G model gives plausible results, especially if the same fractal distribution as that of Nelson is used. The fractal R-D-G results in Fig. 11.22b reflect the corrections made by Köylü and colleagues [175,186], which leads to a large-agglomerate limit for the scattering cross-section of  a    kf Csca 3 12 = − , (11.123) p NCsca ∞ (2x)D f 2 − D f (6 − D f )(4 − D f ) while for intermediate values of N the data of Farias and coworkers [175] are accurately correlated by a simple power law,  a 1−N−1/4 a Csca Csca . (11.124) p = p NCsca NCsca ∞ Agreement between R-D-G results and Nelson’s I-C-P model is good except for large clusters, for which Nelson’s results have been shown to underpredict the relative scattering cross-section by the more complete I-C-P calculations of Farias and coworkers [175]. Comparison with exact GMM calculations of Liu and Smallwood [72] showed good agreement with equation (11.124) for all N. Recent measurements of Chakrabarty [158] on ethene soot, determining absorption and scattering coefficients via nephelometry and photoacoustic spectroscopy, produced a fractal prefactor of k f = 2.6, and a mass fractal dimension of D f = 1.7. Comparison with R-D-G theory showed the results to be always within 10%. While perhaps not yet established as a completely reliable tool, it

Radiative Properties of Particulate Media Chapter | 11 437

is generally agreed today that the fractal R-D-G theory provides simple and reliable estimates of the radiative properties of agglomerated soot. All of the above models assume soot aggregates to consist of single size and spherical primary particles. Extensive measurements by Dobbins and Megaridis [152] and by Köylü and colleagues [155,188] have shown that soot primary particles generally vary in size between 15 and 50 nm, depending on fuel and on flame location, but that their local size distribution has indeed a very small standard deviation (almost uniform). The number of particles comprising an aggregate, N, on the other hand, shows strong local and global variations, following a log-normal distribution with geometric standard deviations ranging from 2 to 3.5. Regardless of the method applied to account for agglomeration, the relations for absorption and extinction coefficients need to be corrected accordingly, i.e., fv κλ = MAC ρBC fv = 6πE(m) λ

⎛ a ⎞ ⎜⎜ Cabs ⎟⎟ ⎟, ⎜⎜ p ⎟ ⎠ ⎝ NCabs

(11.125)

and the scattering coefficient is calculated directly from equation (11.124). While the correlation of Chang and Charalampopoulos [147], equation (11.119), enjoys great popularity until today, probably mostly because of its availability and ease of use, the reader is cautioned that great uncertainty persists regarding precise values for the complex index of refraction, m. Several recent studies have directly measured the mass absorption cross-section (MAC) (or absorption coefficient) in flames, using extinction minus scattering techniques, e.g. [189], and photoacoustic spectrometry, e.g., [190]. While only available for visible wavelengths around 550 nm, these measurements give consistent values of MAC (8.0 ± 0.7)m2 /g; this would correspond to a refractive index function of E(m) ≥ 0.32 [using a maximum plausible correction factor of 1.3 in equation (11.125)] [131]. Most E(m) obtained from the values for m listed in Fig. 11.21 fall well below this value, e.g., E(m) = 0.237 for the Chang and Charalampopoulos correlation. The reader is referred to the review articles of Bond and Bergstrom [191] and Liu et al. [131] to get a more complete picture of the current state of soot modeling. For a simplified heat transfer analysis it is generally desirable to use suitably defined mean absorption and extinction coefficients such as the Planck-mean and Rosseland-mean. If the soot particles are very small so that the Rayleigh theory applies for all particles and relevant wavelengths, then the extinction coefficient is described by equation (11.116). By choosing appropriate spectral average values for the refractive index n and absorptive index k one may approximate the extinction coefficient by κλ = βλ = C0

fv , λ

C0 = 6πE(mavg ) =

(n2

36πnk , − + 2)2 + 4n2 k2 k2

(11.126)

where C0 is now a constant depending only on the soot index of refraction. With this simple 1/λ wavelength dependence, Planck-mean and Rosseland-mean extinction coefficients are readily calculated as κP = βP = 3.83 fv C0 T/C2 ,

κR = βR = 3.60 fv C0 T/C2 ,

(11.127)

where C2 = 1.4388 cm K is the second Planck function constant. It is interesting to note that the Planck-mean coefficient (appropriate for optically thin situations) differs by only 6% from the Rosseland-mean (appropriate for optically thick situations). Thus, Felske and Tien [192] have suggested using an average value of κm = βm = 3.72 fv C0 T/C2

(11.128)

for all optical regimes. It is important to keep in mind that the above formulae apply only to very small soot particles, and that the extinction coefficient will increase for aggregates, as indicated in Fig. 11.22, or if primary particle sizes are encountered that exceed Rayleigh scattering limits. Example 11.4. Propane is burned with air under fuel-rich conditions, resulting in a volume fraction of soot of fv = 10−5 (10 ppm). Determine the extinction coefficient for very small particles at a wavelength of 3 μm using the refractive index data of (i) Lee and Tien, (ii) Stull and Plass, (iii) Dalzell and Sarofim, (iv) Chang and Charalampopoulos, and (v) Felske and coworkers. If the soot consisted of long fractal aggregates with 100 soot particles each (a = 50 nm), how would the extinction coefficient change?

438 Radiative Heat Transfer

Solution To determine the extinction coefficient for small spherical soot particles we use equation (11.114) together with optical property data from Fig. 11.21: Lee and Tien: Stull and Plass: Dalzell and Sarofim: Chang and Charalampopoulos Felske and coworkers:

n = 2.21, n = 2.63, n = 2.19, n =1.89, n = 2.31,

k = 1.23, k = 1.95, k = 1.30, k = 0.92, k = 0.71,

κλ κλ κλ κλ κλ

= 0.1754 cm−1 ; = 0.1472 cm−1 ; = 0.1835 cm−1 ; = 0.1904 cm−1 ; = 0.1077 cm−1 .

Thus, the values found from the data of Lee and Tien, Dalzell and Sarofim, and Chang and Charalampopoulos are fairly consistent, while the absorptive index based on Stull and Plass’ data is considerably higher, probably because amorphous carbon simply does not represent soot well. The absorptive index based on the data of Felske and coworkers is by far the lowest. The extinction coefficient for soot aggregates can be estimated from equation (11.124). Reworking Example 11.1 (i.e., using Lee and Tien’s data, but with ten times the volume fraction and twice the particle size), we find p p x = 2π × 0.05/3 = 0.1048, Qabs = 1.17 × 10−1 , and Qsca = 1.94 × 10−4 . For an aggregate, the absorption coefficient remains p a relatively unchanged. If we employ equation (11.122) with (Cabs /NCabs )∞ 1.08, we obtain ⎛ a ⎞ ⎜⎜ Cabs ⎟⎟ 3 2 × 99 ⎟ + × 1.08 1.08, ⎜⎜⎝ p ⎟ = 201 NCabs ⎠ 201 i.e., the absorption coefficient increases by about 6%, or  p  p 3 fv   p p a a a πa2 (NT /N) = Cabs /NCabs Qabs πa2 NT = Cabs /NCabs Qabs κλ = Qabs 4a 3 × 10−5 0.1895 cm−1 . = 1.08 × 0.117 × 4 × 5 × 10−6 cm On the other hand, the scattering coefficient becomes (using the fractal parameters of Fig. 11.22)  p 3 fv  p a a πa2 (NT /N) = Csca /NCsca Qsca σsλ = Qsca 4a  a  ) ' Csca 12 3 1.9 − = 433.5 = p 1.8 0.2 4.2 × 2.2 NCsca ∞ (2 × 0.1048) a Csca 1−100−1/4 = 63.5 p = 433.5 NCsca and σsλ 63.5 × 1.94 × 10−4 ×

3 × 10−5 1.85 × 10−2 cm−1 . 4 × 5 × 10−6 cm

Adding together we find βλ = κλ + σsλ = 0.1895 + .0185 = 0.2080 cm−1 , i.e., while scattering from aggregates is 63 times larger than that from individual particles, and may not be negligible (depending on the physical size of the soot cloud), its impact on the extinction coefficient is fairly small.

11.12 Experimental Determination of Radiative Properties of Particles Experimental measurements of radiative properties of particles and clouds of particles are useful to verify the Lorenz–Mie theory, to ascertain the applicability of the Lorenz–Mie theory (for nonspherical particles, nonisotropic particles, closely spaced particles, etc.), or simply to determine the radiative properties of particles for which no theory exists. A comprehensive review of such experiments up to 1991 has been given by Agarwal and Mengüç [193]. Properties that can be measured are extinction coefficient, absorption coefficient, and scattered intensity. The easiest property to measure is the extinction coefficient. In principle, a standard spectrometer can be used for this measurement. The results, however, may be unreliable unless the detector is modified to eliminate forward-scattered light, which may account for the majority of total extinction [18] (in particular for large particle sizes, cf. Fig. 11.6). A schematic of such an apparatus is shown in Fig. 11.23. Light from a point source is collimated by a lens, transmitted through the sample cell (with its suspension of particles), and then focused onto a detector by a second lens. In order to reject forward-scattered light, the detector is covered by a guard plate with a small pinhole located at the focal point of the second lens. The diameter of the pinhole must be carefully optimized: If

Radiative Properties of Particulate Media Chapter | 11 439

FIGURE 11.23 Schematic for measurement of extinction coefficient (optical) and absorption coefficient (photoacoustic) [18].

the hole is too small then the signal from the transmitted light may become too weak, while a hole too large will admit an unacceptable amount of forward-scattered energy to the detector. Normally the light beam is chopped by a rotating blade since most detectors only respond to changes in irradiation. To distinguish between absorption and scattering, either the absorption coefficient or total scattering must be measured independently. To measure scattering over all (forward and backward) directions is very difficult, requiring a spectrometer capable of collecting radiation going into all directions (usually accomplished with an integrating sphere technique described in Chapter 3; cf., for example, Bryant et al. [194]). Absorption can also be detected fairly easily with a method usually referred to as photoacoustic [18]. Particles irradiated by a chopped beam are heated periodically, causing periodic changes in the particle temperature, which in turn cause slight pressure oscillations that may be detected by a sensitive microphone. These signals are then amplified by a lock-in amplifier synchronized with the light chopper. Since only absorbed light causes a temperature change in the particles, the acoustic signal must be proportional to the absorption coefficient of the suspension. Details may be found in the papers by Roessler and Faxvog [195] and Faxvog and Roessler [196], who measured the absorption coefficients of acetylene smoke and diesel emissions using this method. An ingenious way to separate transmitted and scattered radiation in the visible has been developed by Härd and Nilsson [197], who utilized the Doppler effect that occurs when an electromagnetic wave is scattered by a moving particle. Modern photoacoustic spectrometers use monochromatic or tuneable laser sources, e.g., to determine the absorption coefficient of soots [190]. Angular scattering measurements are carried out with a scattering photometer (sometimes called a nephelometer). We distinguish between measurements with single scattering (i.e., the cell contains a dilute particle mixture that is optically thin, σs L 1, so that every light beam is scattered at most once before exiting the particle layer) and multiple scattering, between monodisperse suspensions (i.e., all particles are exactly the same size) and polydisperse suspensions (i.e., the particle sizes obey a certain distribution function), between near-forward scattering (to measure the strong forward-scattering peak, but separating it from transmission) and scattering into all directions. Angular light scattering measurements are sometimes classified as either absolute or relative. In an absolute measurement the ratio between intercepted and scattered radiation, δIs (Θ)/Ii , is measured directly, while in a relative measurement the scattered intensity is related to intensity scattered into a reference direction, δIs (Θ)/δIs (Θref ). Thus neither measurement is truly “absolute”; in both cases a relative (i.e., nondimensional) intensity is recorded [18]. Since relative measurements are considerably easier to make, this method is employed by most experimentalists. Single scattering experiments have been carried out primarily to verify the Lorenz–Mie theory, or to assess the accuracy of a device to be used for other scattering measurements. Hottel and coworkers [198] described such an experiment, in which they measured the nondimensional polarized intensities given by equation (11.10) for monodisperse polystyrene latex spheres (it appears that polystyrene spheres are favored by most experimenters, since it is relatively easy to manufacture spheres of constant diameter and of known index of refraction in the visible, m = 1.60, i.e., the spheres scatter but do not absorb). Their equipment consisted of a mercury arc and optics to produce an unpolarized near-parallel beam, a polarizer, a test cell manufactured from parallel microscope slides, and optics to confine the received beam to a small divergence angle. Their results for single scattering of small (2a = 0.106 μm) spheres are shown in Fig. 11.24. It is seen that the agreement between experiment and Lorenz–Mie theory is excellent. Hottel and coworkers attribute the small discrepancies primarily

440 Radiative Heat Transfer

FIGURE 11.24 Lorenz–Mie scattering intensities i1 and i2 for 0.106 μm diameter spheres; data points experimental, solid lines theoretical [198].

to the unavoidable spread in particle sizes. A more modern device to measure the scattering phase function for (almost) single scattering, as well as extinction and scattering coefficients, has been reported by Menart and colleagues [199]. Their apparatus employed a globar light source and an open gas-particle column, both mounted on a rotatable table, together with collection optics and a highly sensitive dual element (InSb–HgCdTe) detector. Measurements taken for soda–lime glass beads and aluminum oxide particles in the wavelength range between 2.5 and 11 μm showed good agreement with Lorenz–Mie theory. Multiple scattering experiments were reported by Woodward [200,201], also on polystyrene spheres, using dispersions with narrow size distribution. Woodward found good agreement between his data and the multiplescattering theory of Hartel [202], a somewhat dated approximate solution of the equation of transfer for a purely scattering medium. However, as Smart and coworkers [203] pointed out, Woodward did not correct for the reflection of the emergent beam at the water–glass–air interfaces of the test cell, nor did he account for the change in scattering path length for larger angles. To compensate for these errors, Smart and coworkers [203] devised an apparatus whose schematic is shown in Fig. 11.25. Employing a standard Brice–Phoenix spectrometer, they placed the simple parallel-microscope slide test section inside a special cell filled with Nujol (a liquid paraffin). Nujol has the same refractive index as the glass bounding the test section as well as the outer jacket of the cell. Thus, the Nujol serves two purposes: Reflections at the interfaces are almost eliminated and—from Snell’s law—all scattering angles up to 90◦ are contained in experimental angles below 65◦ , making otherwise impossible-to-measure scattering angles measurable. Some representative results of their multiple-scattering measurements for varying optical thicknesses of the particle suspension are shown in Fig. 11.26. Agreement between experiment and theory is excellent except for very small and very large optical thicknesses. For small thicknesses the experiment could not regenerate the maxima and minima, probably as a result of uncertainty in the particle size distribution. Disagreement for large thicknesses stems from the fact that Smart and coworkers also used Hartel’s approximate theory. Orchard [204] pointed out that using Hartel’s approximation leads to a transmissivity of 0.5 for a medium of infinite optical thickness (rather than the correct value of zero). Therefore, Hottel and coworkers [198] used the method of discrete ordinates, which may be made arbitrarily accurate for sufficient numbers of “ordinates,”4 to calculate bidirectional reflectance and transmissivity for a particle layer. Some representative data in Fig. 11.27 show the excellent agreement between theory and experiment for optical thicknesses up to 775. Very similar experiments, also using the method of discrete ordinates for theoretical calculations, were carried out by Brewster and Tien [5] and Yamada, Cartigny, and Tien [6] for large polydivinyl 4. This method for the solution of the radiative transport equation is described in detail in Chapter 16.

Radiative Properties of Particulate Media Chapter | 11 441

FIGURE 11.25 Schematic for angular scattering experiment [203]: A, Hg arc; B, monochromatic filter; C, lens; D, E, H, light stops; F, test section; G, jacket with Nujol; I, analyzer; J, photomultiplier.

FIGURE 11.26 Relative scattered intensity vs. angle of observation (a) for relatively low concentration, (b) for relatively high concentration; data points experimental, solid lines theoretical [203].

spheres in air, resulting in equally good agreement between experiment and theory. A different approach to avoiding reflection and refraction losses, and to measuring scattering intensities at oblique angles, was taken by Daniel and coworkers [205], who measured the phase function for aqueous suspensions of unicellular algae. They used a rotatable fiber-optic detector immersed inside the large dish filled with a dilute algae suspension. Measurements of the radiative properties of soot have become important for a number of reasons. For one, soot is a very strong radiator and more often than not dominates the heat transfer in sooty flames. For another, soot is a pollutant and various nonintrusive optical (i.e., radiative) experiments have been designed for its in situ detection, such as transmission, scattering, and emission (via laser-induced incandescence or LII) schemes. A state-of-the art device to measure soot extinction coefficients is shown in Fig. 11.28 [206]. The facility is known as the Large Agglomerate Optics Facility (LAOF) at the National Institute for Standards and Technology (NIST), and it is used to measure extinction coefficients of soot generated by various fuels burning under laminar or turbulent conditions. Soot from the laminar or turbulent burner is collected with minimal disturbance of the flame. The collected soot is diluted with air prior to entering the transmission cell. The soot and gas mixture enters the cell at location 1 and exits at location 2. Several near-monochromatic laser sources are available for transmission measurements. The laser beams pass through air-purged “light tubes” at both end, in order to prevent soot deposition on optical surfaces, and the signal can be collected with a choice of detectors (such as the InGaAs and photodiode detectors indicated in the figure). When steady state is reached

442 Radiative Heat Transfer

FIGURE 11.27 Bidirectional reflectance and transmissivity of an aqueous solution of 0.530 μm polystyrene spheres, for varying concentrations; data points experimental, solid lines theoretical [198].

FIGURE 11.28 Large Agglomerate Optics Facility at NIST [206].

the mixture is directed across the filter, and the collected soot is carefully weighed after the experiment (with 2–5 μg uncertainty). In addition to extinction measurements the LAOF also collects total scattering data using a reciprocal nephelometer, as also indicated in the figure. The apparatus has been used to measure radiative properties of soot from turbulent acetylene and ethene flames [206], laminar acetylene and ethene flames [189], and turbulent JP-8 combustion [207].

Radiative Properties of Particulate Media Chapter | 11 443

Other measurements of scattering media include properties of titanium dioxide powders, for which Kuhn and coworkers [208] used a loose layer of a powder (20 nm to 3 μm in diameter) supported on a film, while Cabrera and colleagues [209] suspended the TiO2 particles in water; both used an integrating sphere to capture directional–hemispherical values of the layer’s transmissivity and reflectivity. A similar experiment was used by Yaroslavsky et al. [210] to capture the radiative properties of biological turbid media. Brewster and Yamada [211] discussed how properties of turbid media can be deduced from time-resolved measurements (using ps pulses), overcoming some of the difficulties of traditional methods; they applied the new scheme to solutions of latex particles. The measurement of scattered intensity into the near-forward direction poses a unique set of problems, because the signal may vary by several orders of magnitude over a few degrees of scattering angle, and because separating the transmitted radiation from forward scattering is difficult. Although sometimes employed for particle sizing and the determination of the index of refraction, near-forward scattering is of importance primarily in applications with large geometric paths, such as atmospheric scattering, scattering effects on visibility in the seas, astrophysical applications, and so on. In heat transfer applications forward scattering is generally of small importance, since treating it as transmitted radiation usually results in negligible errors. The reader interested in such experiments is referred to the papers by Spinrad and coworkers [212].

Problems 11.1 A mass of m (kg) of coal is ground into particles of equal size a (μm), which may be assumed to be “large” and black. Determine the optical thickness (based on radius R) of the resulting spherical particle cloud, assuming that the particles are uniformly distributed throughout the volume. 11.2 One way to determine the number of particles in a gas is to measure the absorption coefficient for the cloud. For a cloud of large, diffuse particles (x  1, λ = 0.4), the particle distribution function is known to be of the form / n(a) =

C = const, 100 μm < a < 500 μm, 0, elsewhere.

If κλ is measured as 1 cm−1 , determine C and the total number of particles per cm3 . 11.3 Coal particles (gray and diffuse with = 0.9, m = 1.925 − 0.1i) are burnt in a long cylindrical combustion chamber. The combustor is well stirred, resulting in a uniform distribution of particles with a size distribution of ⎧ ⎪ ⎪ ⎨1.5 × 108 m−4 , n(a) = ⎪ ⎪ ⎩0,

amin = 1 mm < a < amax = 3 mm, otherwise.

Determine the absorption and scattering coefficients of this particle cloud. 11.4 Consider a particle cloud of fixed-size particles (radius a) contained between parallel plates 0 ≤ x ≤ L = 1 m. The volume fraction of particles is fv (x) = f0 + Δ f (x/L), and their temperature is T(x) = T0 + ΔT (x/L), where Δ f / f0 = ΔT/T0 = 1, f0 = 1%, T0 = 500 K. Assuming the particle size to be a = 500 μm, and made of a material with a gray hemispherical emittance of λ = 0.7, show that the large-particle approximation may be used for the infrared. Calculate the local, spectral absorption and scattering coefficients. Determine the local Planck-mean extinction coefficient as well as the total optical thickness of the slab (based on the Planck-mean). 11.5 Black spheres of radius a = 10 μm occupy a semi-infinite space with varying number density NT = N0 e−z/L (N0 = 104 /cm3 , L = 1 m). If a HeNe laser (λ = 0.633μm) shines onto this layer (at z = 0 into the z-direction), what fraction of its energy is directly transmitted? 11.6 Pulverized coal is burned in a combustor. In order to achieve maximum radiative heat transfer rates, it is desired to keep the optical thickness of the particle cloud at intermediate levels, say τL = κL = 1, in the intermediate infrared, or λ = 5 μm, where L = 1 m is a characteristic combustor dimension. Determine the necessary volume fraction of coal dust, if its size distribution may be taken as ⎧ ⎪ ⎪ ⎨ a−3 , 100 μm ≤ a ≤ 1000 μm, n(a) ∝ ⎪ ⎪ ⎩ 0, all other a. The spectral, hemispherical emittance of coal at λ = 5 μm is λ = 0.7.

444 Radiative Heat Transfer

11.7 To maximize radiative heat loss from a hot medium it is usually desirable for the medium to have an intermediate optical thickness. A nonparticipating hot gas at 1000 K occupying the 1 m wide space between two parallel plates is to be seeded with platinum particles to make the optical thickness of the slab (based on extinction) unity (τL = 1.0). For this purpose a total of 2 kg/m3 particles are to be used (density of platinum 20 g/cm3 ). (a) To what particle radius must the platinum be ground in order to achieve the desired result? (b) Instead, if platinum spheres of radii 50 μm and 100 μm are available, how much of each (for a total of 2 kg/m3 ) must be used to achieve the same result? 11.8 The distribution function of a particle cloud may be approximated by an exponential function such as n(a) = Ca2 e−ba , where a is particle radius and b and C are constants. It is proposed to determine the distribution function of a set of particles by suspending a measured mass of particles between parallel plates, followed by measuring extinction across the particle layer. Given that m = 0.05 g/cm2 of particles are present between the plates, which are 10 cm apart, and that the optical thickness based on extinction has been measured as τ0 = 2: (a) Determine the distribution function above (i.e., b and C). (b) If a single particle size were to be used to achieve the same extinction with the same mass of particles, what would the particle radius be? You may assume all particles to be “large” and diffuse spheres with an emittance of 0.7 and a density of ρ = 2 g/cm3 . 11.9 Consider a particle cloud with a distribution function of n(a) = Ca2 e−ba , where a is particle radius and b and C are constants. The particles are coal ( = 1), and measurements show the particles occupy a volume fraction of 1%, while the number density has been measured as NT = 106 /cm3 . Calculate the extinction, absorption, and scattering coefficients of the cloud for the wavelength range 1 μm < λ < 4 μm. 11.10 A LIDAR laser beam (operating in the green at λ = 0.6 μm) is shot into the sky. At a height of 1 km the laser encounters a 200 m thick cloud consisting of water droplets of varying size (100 μm ≤ a ≤ 200 μm), but constant particle distribution function everywhere (n = 500/μm m3 ). What fraction of the laser beam will be transmitted through the cloud? How much will be absorbed? Very approximately, how much would you expect to get scattered back to the Earth’s surface? Carefully justify your statements about absorption and scattering, using estimates, graphs, and/or physical arguments for support. Qualitatively, how would your explanation change, if you take into account that k = 10−7 (i.e., droplets are not opaque)? Note: Water at 0.6 μm has an index of refraction of m 1.35 − 10−7 i. For the sake of this problem you may assume the droplets to be opaque (not really true). 11.11 In a coal-burning plant, pulverized coal is used that is known to have a particle size distribution function of 6

n(a) ∝ a2 e−Aa ,

A = 3 × 10−11 μm−6 .

The coal may be approximated as diffuse spheres with a gray emittance of = 0.3. What is the effective minimum size parameter, xmin (i.e., 90% by weight of all particles have a size parameter larger than that)? You may assume a combustion temperature of ≈ 2000 K, i.e., the relevant wavelengths range from about 1 μm to about 10 μm. If the furnace is loaded with 10 kg coal particles per cubic meter, what are the spectral absorption and scattering coefficients? (Density of the coal = 2000 kg/m3 .) 11.12 Consider nitrogen mixed with spherical particles at a rate of 108 particles/m3 . The particles have a radius of 300 μm and are diffuse-gray with = 0.5. (a) Determine the absorption and scattering coefficients, and the scattering phase function. (b) Show how the phase function can be approximated by a Henyey–Greenstein function. (c) Can the Crosbie–Davidson model be used for this mixture? (d) Compare the different versions of the phase function in a Φ vs. cos Θ plot. 11.13 A semi-infinite space is filled with black spheres. At any given distance, z, away from the plate the particle number density is identical, namely NT = 6.3662 × 108 m−3 . However, the radius of the suspended spheres diminishes monotonically away from the surface as a = a0 e−z/L ;

a0 = 10−4 m,

L = 1 m.

(a) Determine the absorption coefficient as a function of z (you may make the large-particle assumption). (b) Determine the optical coordinate as a function of z. What is the total optical thickness of the semi-infinite space?

Radiative Properties of Particulate Media Chapter | 11 445

11.14 A semi-infinite space is filled with black spheres of uniform radius a = 100 μm. The particle number density is maximum adjacent to the surface, and decays exponentially away from the surface according to NT = N0 e−Cz ;

N0 = 108 m−3 ,

C = π m−1 .

(a) Determine the absorption and extinction coefficients as functions of z. (b) Determine the optical coordinate as a function of z. What is the total optical thickness of the semi-infinite space? 11.15 In a combustion chamber radiatively nonparticipating gases are mixed with soot and coal particles. The following is known (per m3 of mixture): Soot: uniform particle size, as = 10 nm, mass = 10−3 kg, m2 − 1 = 0.5λ2 − 0.1λi (λ in μm). complex index of refraction 2 m +2 Coal: uniform particle size, ac = 1 mm, mass = 1 kg, coal is black. The density of both, coal and soot, is 2,000 kg/m3 . Determine the spectral absorption coefficient of the mixture for the near infrared. 11.16 In a sheet flame confined between two large parallel plates −L ≤ z ≤ +L = 1 m soot is generated mainly in the central flame region, leading to a local soot volume fraction of fv (z) = fv0 [1 − (z/L)2 ], with fv0 = 1.07 × 10−6 . The soot is propane soot with a complex index of refraction of m = 2.21 − 1.23i. (a) Determine the relevant radiative properties of the mixture, assuming the combustion gases to be nonparticipating. (b) What is the spectral optical thickness of the 2L thick layer? 11.17 A laser beam at 633 nm wavelength is probing a 1 m thick layer of gold nanoparticles suspended in air (radius a = 10 nm; for gold at 633 nm: m = 0.47 − 2.83i). If the exiting laser beam is attenuated by 10% due to absorption and scattering, determine (a) the number density of gold particles, (b) their volume fraction. 11.18 A LIDAR laser beam (operating in the green at λ = 0.6μm) is shot into the sky. At a height of 1km the laser encounters a 200m thick cloud consisting of tiny water droplets of varying size (1nm ≤ a ≤ 20nm), but constant particle distribution function everywhere (n = 6 × 1015 /nm m3 ). (a) Determine the water droplet volume fraction in the cloud. (b) Determine its spectral absorption coefficient; compare with equation (11.126). (c) What fraction of the laser beam will be transmitted through the cloud? How much will be absorbed? Note: Water at 0.6μm has an index of refraction of m 1.35 − 10−7 i. 11.19 Redo Problem 11.4 for propane soot with a single mean radius of am = 0.1 μm in a flame with f0 = 10−6 and T0 = 1500 K. Show that the small particle limit is appropriate for, say, λ > 3 μm. For hand calculations you may approximate the index of refraction by a single average value (say, at 3 μm), and the emissive power by Wien’s law. 11.20 Redo Problem 11.19 for the case that the soot has agglomerated into mass fractal aggregates of 1000 soot particles each (D f = 1.77 and k f = 8.1). 11.21 Consider a particle cloud with a distribution function of n(a) = Ca2 e−ba , where a is particle radius and b and C are constants. The particles are soot (m 1.5 − 0.5i), and measurements show the soot occupies a volume fraction of 10−5 , while the number density has been measured as NT = 1012 /cm3 . Calculate the extinction, absorption, and scattering coefficients of the cloud for the wavelength range 1 μm < λ < 4 μm.

References [1] C.L. Tien, B.L. Drolen, Thermal radiation in particulate media with dependent and independent scattering, in: Annual Review of Numerical Fluid Mechanics and Heat Transfer, vol. 1, Hemisphere, New York, 1987, pp. 1–32. [2] L.A. Dombrovsky, D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, New York, 2010. [3] H.C. van de Hulst, Light Scattering by Small Particles, John Wiley & Sons, New York, 1957, also Dover Publications, New York, 1981. [4] H.C. Hottel, A.F. Sarofim, W.H. Dalzell, I.A. Vasalos, Optical properties of coatings. Effect of pigment concentration, AIAA Journal 9 (1971) 1895–1898. [5] M.Q. Brewster, C.L. Tien, Radiative transfer in packed/fluidized beds: dependent vs. independent scattering, ASME Journal of Heat Transfer 104 (1982) 573–579.

446 Radiative Heat Transfer

[6] Y. Yamada, J.D. Cartigny, C.L. Tien, Radiative transfer with dependent scattering by particles, part 2: experimental investigation, ASME Journal of Heat Transfer 108 (1986) 614–618. [7] J.D. Cartigny, Y. Yamada, C.L. Tien, Radiative transfer with dependent scattering by particles, part 1: theoretical investigation, ASME Journal of Heat Transfer 108 (1986) 608–613. [8] B.L. Drolen, C.L. Tien, Independent and dependent scattering in packed-sphere systems, Journal of Thermophysics and Heat Transfer 1 (1987) 63–68. [9] B.L. Drolen, K. Kumar, C.L. Tien, Experiments on dependent scattering of radiation, AIAA paper no. TP-87-210, 1987. [10] L. Rayleigh, On the light from the sky, its polarization and colour, Philosophical Magazine 41 (1871) 107–120, 274–279, reprinted in in: Scientific Papers by Lord Rayleigh, vol. I: 1869–1881, No. 8, Dover, New York, 1964. [11] L. Rayleigh, Philosophical Magazine 12 (1881). [12] L. Lorenz, in: Videnskab Selskab Skrifter, vol. 6, Copenhagen, Denmark, 1890. [13] L. Lorenz, in: Oeuvres Scientifiques, vol. I, Copenhagen, Denmark, 1898, p. 405. [14] G.A. Mie, Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen, Annalen der Physik 25 (1908) 377–445. [15] P. Debye, Annalen der Physik 30 (4) (1909) 57. [16] M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, Academic Press, New York, 1969. [17] D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions, Elsevier, New York, 1969. [18] C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley & Sons, New York, 1983. [19] C.R. Wylie, Advanced Engineering Mathematics, 5th ed., McGraw-Hill, New York, 1982. [20] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover Publications, New York, 1965. [21] C.M. Chu, S.W. Churchill, Representation of the angular distribution of radiation scattered by a spherical particle, Journal of the Optical Society of America 45 (11) (1955) 958–962. [22] G.C. Clark, C.M. Chu, S.W. Churchill, Angular distribution coefficients for radiation scattered by a spherical particle, Journal of the Optical Society of America 47 (1957) 81–84. [23] G.W. Kattawar, G.N. Plass, Electromagnetic scattering from absorbing spheres, Applied Optics 6 (8) (1967) 1377–1383. [24] W.J. Wiscombe, Improved Mie scattering algorithms, Applied Optics 19 (1980) 1505–1509. [25] M.F. Modest, F.H. Azad, The influence and treatment of Mie-anisotropic scattering in radiative heat transfer, ASME Journal of Heat Transfer 102 (1980) 92–98. [26] J.R. Hodkinson, Light scattering and extinction by irregular particles larger than the wavelength, in: M. Kerker (Ed.), Electromagnetic Scattering, Macmillan, New York, 1963, pp. 87–100. [27] B.P. Singh, M. Kaviany, Radiative transfer in packed and fluidized beds: dependent versus independent scattering, International Journal of Heat and Mass Transfer 34 (1991) 2869–2882. [28] M.Q. Brewster, Volume scattering of radiation in packed beds of large, opaque spheres, ASME Journal of Heat Transfer 126 (4) (2004) 1048–1050. [29] J.H. Randrianalisoa, D. Baillis, Radiative properties of densely packed spheres in semitransparent media: a new geometric optics approach, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (10) (2010) 1372–1388. [30] S.C. Lee, Radiation heat-transfer model for fibers oriented parallel to diffuse boundaries, Journal of Thermophysics and Heat Transfer 2 (4) (Oct 1988) 303–308. [31] T.W. Tong, C.L. Tien, Analytical models for thermal radiation in fibrous insulation, Journal of Thermal Insulation 4 (1980) 27–44. [32] T.W. Tong, C.L. Tien, Radiative heat transfer in fibrous insulations—part I: analytical study, ASME Journal of Heat Transfer 105 (1983) 70–74. [33] T.W. Tong, P.S. Swathi, G.R. Cunnington, Reduction of radiative heat transfer in thermal insulations by use of dielectric coated fibers, International Communications in Heat and Mass Transfer 16 (1989) 851–860. [34] P.S. Swathi, T.W. Tong, G.R. Cunnington, Scattering of electromagnetic waves by cylinders coated with a radially-inhomogeneous layer, Journal of Quantitative Spectroscopy and Radiative Transfer 46 (4) (1991) 281–292. [35] G.R. Cunnington, T.W. Tong, P.S. Swathi, Angular scattering of radiation from coated cylindrical fibers, Journal of Quantitative Spectroscopy and Radiative Transfer 48 (4) (1992) 353–362. [36] T.W. Tong, W. Li, Enhancement of thermal emission from porous radiant burners, Journal of Quantitative Spectroscopy and Radiative Transfer 53 (2) (1995) 235–248. [37] S.C. Lee, Radiative transfer through a fibrous medium: allowance for fiber orientation, Journal of Quantitative Spectroscopy and Radiative Transfer 36 (3) (1986) 253–263. [38] S.C. Lee, Effect of fiber orientation on thermal radiation in fibrous media, International Journal of Heat and Mass Transfer 32 (2) (1989) 311–320. [39] S.C. Lee, Scattering phase function for fibrous media, International Journal of Heat and Mass Transfer 33 (10) (1990) 2183–2190. [40] S.C. Lee, Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders, Journal of Applied Physics 68 (10) (1990) 4952–4957. [41] S.C. Lee, Dependent scattering by parallel fibers: effects of multiple scattering and wave interference, Journal of Thermophysics and Heat Transfer 6 (4) (1992) 589–595. [42] S.C. Lee, Angle of incidence and size effects on dependent scattering in fibrous media, ASME Journal of Heat Transfer 118 (4) (1996) 931–936.

Radiative Properties of Particulate Media Chapter | 11 447

[43] G.R. Cunnington, S.C. Lee, S.M. White, Radiative properties of fiber-reinforced aerogel: theory versus experiment, Journal of Thermophysics and Heat Transfer 12 (1) (1998) 17–22. [44] S. Kumar, S.M. White, Dependent scattering properties of woven fibrous insulations for normal incidence, ASME Journal of Heat Transfer 117 (1) (1995) 160–166. [45] L.A. Dombrovsky, Quartz–fiber thermal insulation: infrared radiative properties and calculation of radiative–conductive heat transfer, ASME Journal of Heat Transfer 118 (2) (1996) 408–414. [46] J. Marschall, F.S. Milos, The calculation of anisotropic extinction coefficients for radiation diffusion in rigid fibrous ceramic insulations, International Journal of Heat and Mass Transfer 40 (1997) 627–634. [47] J. Yamada, Y. Kurosaki, Radiative characteristics of fibers with a large size parameter, International Journal of Heat and Mass Transfer 43 (6) (2000) 981–991. [48] H.C. van de Hulst, Asymptotic fitting, a method for solving anisotropic transfer problems in thick layers, Journal of Computational Physics 3 (1968) 291–306. [49] J.E. Hansen, Exact and approximate solutions for multiple scattering by cloudy and hazy planetary atmospheres, Journal of the Atmospheric Sciences 26 (1969) 478–487. [50] J.F. Potter, The delta function approximation in radiative transfer theory, Journal of the Atmospheric Sciences 27 (1970) 943–949. [51] J.H. Joseph, W.J. Wiscombe, J.A. Weinman, The delta-Eddington approximation for radiative flux transfer, Journal of the Atmospheric Sciences 33 (1976) 2452–2459. [52] A.L. Crosbie, G.W. Davidson, Dirac-delta function approximations to the scattering phase function, Journal of Quantitative Spectroscopy and Radiative Transfer 33 (4) (1985) 391–409. [53] R. Davies, Fast azimuthally dependent model of the reflection of solar radiation by plane-parallel clouds, Applied Optics 19 (1980) 250–255. [54] M.I. Mishchenko, Electromagnetic scattering by nonspherical particles: a tutorial review, Journal of Quantitative Spectroscopy and Radiative Transfer 110 (11) (2009) 808–832. [55] T. Wriedt, Light scattering theories and computer codes, Journal of Quantitative Spectroscopy and Radiative Transfer 110 (11) (2009) 833–843. [56] P.C. Waterman, Matrix formulation of electromagnetic scattering, Proceedings of IEEE 53 (8) (1965) 805. [57] M.I. Mishchenko, L.D. Travis, D.W. Mackowski, T-matrix computations of light scattering by nonspherical particles: a review, Journal of Quantitative Spectroscopy and Radiative Transfer 55 (5) (1996) 535–575. [58] D.W. Mackowski, Calculation of total cross sections of multi-sphere clusters, Journal of the Optical Society of America 11 (1994) 2851–2861. [59] L. Liu, M.I. Mishchenko, Scattering and radiative properties of complex soot and soot-containing aggregate particles, Journal of Quantitative Spectroscopy and Radiative Transfer 106 (2007) 262–273. [60] M.I. Mishchenko, N.T. Zakharova, G. Videen, N.G. Khlebtsov, T. Wriedt, Comprehensive T-matrix reference database: a 2007–2009 update, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (2010) 650–658. [61] Y. Zhao, L. Ma, Assessment of two fractal scattering models for the prediction of the optical characteristics of soot aggregates, Journal of Quantitative Spectroscopy and Radiative Transfer 110 (4–5) (2009) 315–322. [62] J.Y. Yin, L.H. Liu, Influence of complex component and particle polydispersity on radiative properties of soot aggregate in atmosphere, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (14) (2008) 2115–2126. [63] J. Li, L. Bai, Z. Wu, L. Guo, Y. Gong, Ultraviolet scattering properties of alumina particle clusters at three phase states in aircraft plume, Journal of Quantitative Spectroscopy and Radiative Transfer 191 (2017) 40–45. [64] J. Li, L. Bai, Z. Wu, L. Guo, Y. Gong, Scattering properties of alumina particle clusters with different radius of monomers in aerocraft plume, Journal of Quantitative Spectroscopy and Radiative Transfer 202 (2017) 233–239. [65] M.I. Mishchenko, L.D. Travis, D.W. Mackowski, T-matrix codes for computing electromagnetic scattering by nonspherical and aggregated particles, available from https://www.giss.nasa.gov/staff/mmishchenko/t_matrix.html. [66] T. Wriedt, SCATTPORT, available from http://www.scattport.org/. [67] Y.-L. Xu, Scattering Mueller matrix of an ensemble of variously shaped small particles, Journal of the Optical Society of America 20 (11) (2003) 2093–2105. [68] Y.-L. Xu, N.G. Khlebtsov, Orientational-averaged radiative properties of an arbitrary configuration of scatterers, Journal of Quantitative Spectroscopy and Radiative Transfer 79–80 (2003) 1121–1137. [69] Y.-L. Xu, Electromagnetic scattering by an aggregate of spheres, Applied Optics 34 (1995) 4573–4588. [70] Y.-L. Xu, Electromagnetic scattering by an aggregate of spheres: far field, Applied Optics 36 (1995) 9496–9508. [71] F. Liu, G.J. Smallwood, Effect of aggregation on the absorption cross-section of fractal soot aggregates and its impact on LII modelling, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (2) (2010) 302–308. [72] F. Liu, G.J. Smallwood, Radiative properties of numerically generated fractal soot aggregates: the importance of configuration averaging, ASME Journal of Heat Transfer 132 (1) (2010) 023308. [73] H. Li, C. Liu, L. Bi, P. Yang, G.W. Kattawar, Numerical accuracy of “equivalent” spherical approximations for computing ensembleaveraged scattering properties of fractal soot aggregates, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (14) (2008) 2127–2132. [74] Y.-L. Xu, Generalized multi-sphere Mie solution programs, available from http://www.scattport.org/files/xu/codes.htm.

448 Radiative Heat Transfer

[75] E.M. Purcell, C.R. Pennypecker, Scattering and absorption by non-spherical dielectric grains, The Astrophysical Journal 186 (1973) 705–714. [76] G.W. Mulholland, C.F. Bohren, K.A. Fuller, Light scattering by agglomerates: coupled electric and magnetic dipole method, Langmuir 10 (1994) 2533–2546. [77] G.W. Mulholland, R.D. Mountain, Coupled dipole calculation of extinction coefficient and polarization ratio for smoke agglomerates, Combustion and Flame 119 (1999) 56–68. [78] S.K. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media, IEEE Transactions on Antennas and Propagation AP-14 (1966) 302–307. [79] P. Yang, K.N. Liou, Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space, Journal of the Optical Society of America 13 (1996) 2072–2085. [80] W. Sun, G. Videen, Q. Fu, S.e. a, Tanev (Eds.): Finite-Difference Time-Domain Solution of Light Scattering by Arbitrarily Shaped Particles and Surfaces, Light Scattering Reviews, vol. 6, Springer, 2012. [81] A.F. Sarofim, H.C. Hottel, Radiative transfer in combustion chambers: influence of alternative fuels, in: Proceedings of the Sixth International Heat Transfer Conference, vol. 6, Hemisphere, Washington, D.C., 1978, pp. 199–217. [82] P.J. Foster, C.R. Howarth, Optical constants of carbons and coals in the infrared, Carbon 6 (1968) 719–729. [83] A.G. Blokh, The problem of flame as a disperse system, in: N.F. Afghan, J.M. Beer (Eds.), Heat Transfer in Flames, Scripta Book Co., Washington, 1974, pp. 111–130. [84] A.G. Blokh, L.D. Burak, Primary radiation characteristics of solid fuels, Thermal Engineering 20 (8) (1973) 65–70. [85] A. Lowe, I.M. Stewart, T.F. Wall, The measurement and interpretation of radiation from fly-ash particles in large pulverized coal flames, in: Seventeenth Symposium (International) on Combustion, The Combustion Institute, 1979, pp. 105–114. [86] A.G. Blokh, Sagadeev, V.D. Vyushin, Experimental and theoretical investigation of radiation properties of flame with coal burning in powerful boiler furnaces, in: Heat and Mass Transfer-VI, vol. VIII, Soviet Academy of Sciences, Minsk, 1980, pp. 70–73 (in Russian). [87] T.F. Wall, A. Lowe, L.J. Wibberley, T. Mai-Viet, R.P. Gupta, Fly-ash characteristics and radiative heat transfer in pulverized-coal-fired furnaces, Combustion Science and Technology 26 (1981) 107–121. [88] D.G. Goodwin, Infrared optical constants of coal slags, Technical Report T-255, Stanford University, California, 1986. [89] D.G. Goodwin, M. Mitchner, Flyash radiative properties and effects on radiative heat transfer in coal-fired systems, International Journal of Heat and Mass Transfer 32 (4) (1989) 627–638. [90] R. Viskanta, A. Ugnan, M.P. Mengüç, Predictions of radiative properties of pulverized coal and fly-ash polydispersions, ASME paper no. 81-HT-24, 1981. [91] R. Johansson, T. Gronarz, R. Kneer, Influence of index of refraction and particle size distribution on radiative heat transfer in a pulverized coal combustion furnace, ASME Journal of Heat Transfer 139 (2017) 042702. [92] C.L. Tien, D.G. Doornink, D.A. Rafferty, Attenuation of visible radiation by carbon smokes, Combustion Science and Technology 6 (1972) 55–59. [93] R.O. Buckius, D.C. Hwang, Radiation properties for polydispersions: application to coal, ASME Journal of Heat Transfer 102 (1980) 99–103. [94] M.P. Mengüç, R. Viskanta, On the radiative properties of polydispersions: a simplified approach, Combustion Science and Technology 44 (1985) 143–159. [95] C. Kim, N. Lior, A simplified model for spectral radiative properties in pulverized coal combustors, in: Heat Transfer in Hazardous Waste Processing, vol. HTD-212, ASME, 1992, pp. 113–122. [96] F. Liu, J. Swithenbank, The effects of particle size distribution and refractive index on fly-ash radiative properties using a simplified approach, International Journal of Heat and Mass Transfer 36 (7) (1993) 1905–1912. [97] K.H. Im, R.K. Ahluwalia, Radiation properties of coal combustion products, International Journal of Heat and Mass Transfer 36 (2) (1993) 293–302. [98] S. Manickavasagam, M.P. Mengüç, Effective optical properties of pulverized coal particles determined from FT-IR spectrometer experiments, Energy & Fuels 7 (6) (1993) 860–869. [99] M. Caldas, V. Semião, Modelling of scattering and absorption coefficients for a polydispersion, International Journal of Heat and Mass Transfer 42 (24) (1999) 4535–4548. [100] T. Gronarz, M. Schnell, C. Siewert, L. Schneiders, W. Schröder, R. Kneer, Comparison of scattering behaviour for spherical and non-spherical particles in pulverized coal combustion, International Journal of Thermal Sciences 111 (2017) 116–128. [101] T. Gronarz, R. Johansson, R. Kneer, Modeling of anisotropic scattering of thermal radiation in pulverized coal combustion, ASME Journal of Heat Transfer 140 (2018) 062701. [102] A. Tuntomo, C.L. Tien, S.H. Park, Optical constants of liquid hydrocarbon fuels, Combustion Science and Technology 84 (1992) 133–140. [103] M.R. Anderson, Determination of infrared optical constants for single component hydrocarbon fuels, M.S. thesis, University of Missouri-Rolla, Department of Mechanical Engineering, Rolla, Missouri, 2000. [104] C.C. Wang, J.Y. Tan, C.Y. Jing, L.H. Liu, Temperature-dependent optical constants of liquid isopropanol, n-butanol, and n-decane, Applied Optics 57 (12) (2018) 3003–3011. [105] L.A. Dombrovsky, S.S. Sazhin, E.M. Sazhina, G. Feng, M.R. Heikal, M.E.A. Bardsley, S.V. Mikhalovsky, Heating and evaporation of semi-transparent diesel fuel droplets in the presence of thermal radiation, Fuel 80 (2001) 1535–1544.

Radiative Properties of Particulate Media Chapter | 11 449

[106] L.A. Dombrovsky, S.S. Sazhin, S.V. Mikhalovsky, R. Wood, M.R. Heikal, Spectral properties of diesel fuel droplets, Fuel 82 (2003) 15–22. [107] U. Liddel, C. Kasper, Spectral differentiation of pure hydrocarbons: a near infrared absorption study, BS Journal of Research 10 (1933) 599–618. [108] J.J. Kelly, C.H. Barlow, T.M. Jinguji, J.B. Callis, Prediction of gasoline octane numbers from near-infrared spectral features in the range 660-1215 nm, Analytical Chemistry 61 (1989) 313–320. [109] J.E. Bertie, R.N. Jones, Y. Apelblat, C.D. Keefe, Infrared intensities of liquids xiii: accurate optical constants and molar absorption coefficients between 6500 and 435 cm−1 of toluene at 25◦ c, from spectra recorded in several laboratories, Applied Spectroscopy 48 (1994) 127–143. [110] S. Kedenburg, M. Vieweg, T. Gissibl, H. Giessen, Linear refractive index and absorption measurements of nonlinear optical liquids in the visible and near-infrared spectral region, Optical Materials Express 2 (2012) 1588–1611. [111] E. Sani, A. Dell’Oro, Optical constants of ethylene glycol over an extremely wide spectral range, Optical Materials 37 (2014) 36–41. [112] E. Sani, A. Dell’Oro, Spectral optical constants of ethanol and isopropanol from ultraviolet to far infrared, Optical Materials 60 (2016) 137–141. [113] L.A. Dombrovsky, Spectral model of absorption and scattering of thermal radiation by diesel fuel droplets, High Temperature 40 (2002) 242–248. [114] L.A. Dombrovsky, S. Dembele, J.X. Wen, A simplified model for the shielding of fire thermal radiation by water mists, International Journal of Heat and Mass Transfer 96 (2016) 199–209. [115] J.M. Singer, J. Grumer, Carbon formation in very rich hydrocarbon–air flames—I: studies of chemical content, temperature, ionization and particulate matter, in: Seventh Symposium (International) on Combustion, The Combustion Institute, 1959, pp. 559–572. [116] B.L. Wersborg, J.B. Howard, G.C. Williams, Physical mechanisms in carbon formation in flames, in: Fourteenth Symposium (International) on Combustion, The Combustion Institute, 1972, pp. 929–940. [117] M. Kunugi, H. Jinno, Determination of size and concentration of soot particles in diffusion flames by a light-scattering technique, in: Eleventh Symposium (International) on Combustion, The Combustion Institute, 1966, pp. 257–266. [118] T. Sato, T. Kunitomo, S. Yoshi, T. Hashimoto, On the monochromatic distribution of the radiation from the luminous flame, Bulletin of JSME 12 (1969) 1135–1143. [119] M. Frenklach, H. Wang, M.J. Rabinowitz, Optimization and analysis of large chemical kinetic mechanisms using the solution mapping method—combustion of methane, Progress in Energy and Combustion Science 18 (1992) 47–73. [120] M. Frenklach, H. Wang, Detailed mechanism and modeling of soot particle formation, in: Soot Formation in Combustion, Springer Verlag, New York, 1994, pp. 162–192. [121] H. Bockhorn, Soot Formation in Combustion, Springer Verlag, New York, 1994. [122] M. Frenklach, On surface growth mechanism of soot particles, in: Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, 1996, pp. 2285–2293. [123] I.M. Kennedy, Models of soot formation and oxidation, Progress in Energy and Combustion Science 23 (1997) 95–132. [124] H. Wang, M. Frenklach, A detailed kinetic modeling study of aromatics formation in laminar premixed acetylene and ethylene flames, Combustion and Flame 110 (1997) 173–221. [125] A. Kazakov, M. Frenklach, Dynamic modeling of soot particle coagulation and aggregation: implementation with the method of moments and application to high-pressure laminar premixed flames, Combustion and Flame 114 (3–4) (1998) 484–501. [126] J. Appel, H. Bockhorn, M. Frenklach, Kinetic modeling of soot formation with detailed chemistry and physics: laminar premixed flames of C2 hydrocarbons, Combustion and Flame 121 (2000) 122–136. [127] H. Richter, J.B. Howard, Formation of polycyclic aromatic hydrocarbons and their growth to soot – a review of chemical reaction pathways, Progress in Energy and Combustion Science 26 (2000) 565–608. [128] M. Frenklach, S.J. Harris, Aerosol dynamics modeling using the method of moments, Journal of Colloid and Interface Science 118 (1987) 252–261. [129] M. Frenklach, S.J. Harris, Aerosol dynamics using the method of moments, Journal of Colloid and Interface Science 130 (1987) 252–261. [130] M. Frenklach, Soot formation codes, http://combustion.berkeley.edu/soot/codes/codes.html. [131] F. Liu, J. Yon, A. Fuentes, P. Lobo, G.J. Smallwood, J.C. Corbin, Review of recent literature on the light absorption properties of black carbon: refractive index, mass absorption cross section, and absorption function, Aerosol Science and Technology 54 (1) (2020) 33–51. [132] A. Becker, Über die strahlung und temperatur der hefnerlampe, Annalen der Physik 333 (5) (1909) 1017–1031. [133] R.C. Millikan, Optical properties of soot, Journal of the Optical Society of America 51 (1961) 698–699. [134] R.C. Millikan, Sizes, optical properties and temperatures of soot particles, in: The Fourth Symposium on Temperature, Its Measurement and Control in Science and Industry, vol. 3, 1961, pp. 497–507. [135] W.H. Dalzell, A.F. Sarofim, Optical constants of soot and their application to heat-flux calculations, ASME Journal of Heat Transfer 91 (1) (1969) 100–104. [136] V.R. Stull, G.N. Plass, Emissivity of dispersed carbon particles, Journal of the Optical Society of America 50 (2) (1960) 121–129. [137] C.R. Howarth, P.J. Foster, M.W. Thring, The effect of temperature on the extinction of radiation by soot particles, in: Proceedings of the Third International Heat Transfer Conference, vol. 5, Hemisphere, Washington, D.C., 1966, pp. 122–128. [138] G.L. Hubbard, C.L. Tien, Infrared mean absorption coefficients of luminous flames and smoke, ASME Journal of Heat Transfer 100 (1978) 235–239.

450 Radiative Heat Transfer

[139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158]

[159] [160] [161] [162] [163] [164] [165] [166]

[167] [168] [169] [170] [171]

A.I. Medalia, L.W. Richards, Tinting strength of carbon black, Journal of Colloid and Interface Science 40 (1972) 233–252. S.C. Graham, The refractive indices of isolated and of aggregated soot particles, Combustion Science and Technology 9 (1974) 159–163. J. Janzen, The refractive index of colloidal carbon, Journal of Colloid and Interface Science 69 (1979). J.D. Felske, T.T. Charalampopoulos, H.S. Hura, Determination of refractive indices of soot particles from the reflectivities of compressed soot particles, Combustion Science and Technology 37 (1984) 263–284. S.C. Lee, C.L. Tien, Optical constants of soot in hydrocarbon flames, in: Eighteenth Symposium (International) on Combustion, The Combustion Institute, 1981, pp. 1159–1166. T.S. Moss, G.J. Burrell, B. Ellis, Semiconductor Opto-Electronics, John Wiley & Sons, New York, 1972. R.O. Buckius, C.L. Tien, Infrared flame radiation, International Journal of Heat and Mass Transfer 20 (1977) 93–106. S. Bard, P.J. Pagni, Carbon particulate in small pool fire flames, ASME Journal of Heat Transfer 103 (1981) 357–362. H. Chang, T.T. Charalampopoulos, Determination of the wavelength dependence of refractive indices of flame soot, Proceedings of the Royal Society (London) A 430 (1880) (1990) 577–591. Ü.Ö. Köylü, G.M. Faeth, Spectral extinction coefficients of soot aggregates from turbulent diffusion flames, ASME Journal of Heat Transfer 118 (1996) 415–421. A.R. Jones, An estimate of the possible effects of particle agglomeration on the emissivity of sooty flames, in: Combustion Institute European Symposium, 1973, pp. 376–381. S.C. Lee, C.L. Tien, Effect of soot shape on soot radiation, Journal of Quantitative Spectroscopy and Radiative Transfer 29 (1983) 259–265. D.W. Mackowski, R.A. Altenkirch, M.P. Mengüç, Extinction and absorption coefficients of cylindrically-shaped soot particles, Combustion Science and Technology 40 (1987) 399–410. R.A. Dobbins, C.M. Megaridis, Morphology of flame-generated soot as determined by thermophoretic sampling, Langmuir 3 (1987) 254–259. R.J. Samson, G.W. Mulholland, J.W. Gentry, Structural analysis of soot agglomerates, Langmuir 3 (1987) 272–281. R. Jullien, R. Botet, Aggregation and Fractal Aggregates, World Scientific Publishing Co., Singapore, 1987. Ü.Ö. Köylü, G.M. Faeth, Structure of overfire soot in buoyant turbulent diffusion flames at long residence times, Combustion and Flame 89 (1992) 140–156. T.L. Farias, M.G. Carvalho, Ü.Ö. Köylü, Radiative heat transfer in soot-containing combustion systems with aggregation, International Journal of Heat and Mass Transfer 41 (17) (1998) 2581–2587. B. Hu, B. Yang, Ü.Ö. Köylü, Soot measurements at the axis of an ethylene/air nonpremixed turbulent jet flame, Combustion and Flame 134 (2003) 93–106. R.K. Chakrabarty, H. Moosmüller, W.P. Arnott, M.A. Garro, J.G. Slowik, E.S. Cross, J.-H. Han, P. Davidovits, T.B. Onasch, D.R. Worsnop, Light scattering and absorption by fractal-like carbonaceous chain aggregates: comparison of theories and experiment, Applied Optics 46 (2007) 6990–7006. C.M. Sorensen, G.C. Roberts, The prefactor of fractal aggregates, Journal of Colloid and Interface Science 186 (1997) 447–452. S. Kumar, C.L. Tien, Effective diameter of agglomerates for radiative extinction and scattering, Combustion Science and Technology 66 (1989) 199–216. J.C. Ku, K.H. Shim, Optical diagnostics and radiative properties of simulated soot agglomerates, ASME Journal of Heat Transfer 113 (4) (1991) 953–958. J.C. Ku, K.H. Shim, A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles, Journal of Quantitative Spectroscopy and Radiative Transfer 47 (1992) 201–220. T.T. Charalampopoulos, P.K. Panigrahi, Depolarization characteristics of agglomerated particulates-reciprocity relations, Journal of Physics D: Applied Physics 26 (1993) 2075–2081. W. Lou, T.T. Charalampopoulos, On the electromagnetic scattering and absorption of agglomerated small spherical particles, Journal of Physics D: Applied Physics 27 (1994) 2258–2270. W. Lou, T.T. Charalampopoulos, On the inverse scattering problem for characterization of agglomerated particulates: partial derivative formulation, Journal of Physics D: Applied Physics 28 (1995) 2585–2594. M.P. Mengüç, A. Mahadeviah, K. Saito, S. Manickavasagam, Application of the discrete dipole approximation to determine the radiative properties of soot agglomerates, in: A.M. Kanury, M.Q. Brewster (Eds.), Heat Transfer in Fire and Combustions Systems, vol. HTD-199, ASME, 1992, pp. 9–16. B.M. Vaglieco, O. Monda, F.E. Corcione, M.P. Mengüç, Optical and radiative properties of particulates at Diesel engine exhaust, Combustion Science and Technology 102 (1994) 283–299. ˘ Ivezi´c, M.P. Mengüç, An investigation of dependent/independent scattering regimes using a discrete dipole approximation, InterZ. national Journal of Heat and Mass Transfer 39 (4) (1996) 811–822. ˘ Ivezi´c, M.P. Mengüç, T.G. Knauer, A procedure to determine the onset of soot agglomeration from multi-wavelength experiments, Z. Journal of Quantitative Spectroscopy and Radiative Transfer 57 (6) (1997) 859–865. J. Nelson, Test of a mean field theory for the optics of fractal clusters, Journal of Modern Optics 36 (1989) 1031–1057. H.Y. Chen, M.F. Iskander, J.E. Penner, Light scattering and absorption by fractal agglomerates and coagulations of smoke aerosols, Journal of Modern Optics 2 (1990) 171–181.

Radiative Properties of Particulate Media Chapter | 11 451

[172] H.Y. Chen, M.F. Iskander, J.E. Penner, Empirical formula for optical absorption by fractal aerosol aggregates, Applied Optics 30 (1991) 1547–1551. [173] R.A. Dobbins, C.M. Megaridis, Absorption and scattering light by polydisperse aggregates, Applied Optics 30 (1991) 4747–4754. [174] Ü.Ö. Köylü, G.M. Faeth, Radiative properties of flame-generated soot, ASME Journal of Heat Transfer 115 (2) (1993) 409–417. [175] T.L. Farias, M.G. Carvalho, Ü.Ö. Köylü, G.M. Faeth, Computational evaluation of approximate Rayleigh–Debye–Gans fractal-aggregate theory for the absorption and scattering properties of soot, ASME Journal of Heat Transfer 117 (1) (1995) 152–159. [176] T.L. Farias, M.G. Carvalho, Ü.Ö. Köylü, The range of validity of the Rayleigh–Debye–Gans theory for optics of fractal aggregates, Applied Optics 35 (1996) 6560–6567. [177] S. Manickavasagam, M.P. Mengüç, Scattering matrix elements of fractal-like soot agglomerates, Journal of Applied Physics 36 (6) (1997) 1337–1351. [178] S.S. Krishnan, K.C. Lin, G.M. Faeth, Spectral extinction coefficients of soot aggregates from turbulent diffusion flames, ASME Journal of Heat Transfer 123 (2001) 331–339. [179] A.R. Jones, Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation, Proceedings of the Royal Society (London) A 366 (1979) 111–127. [180] A.R. Jones, Scattering efficiency factors for agglomerates of small spheres, Journal of Physics D: Applied Physics 12 (1979) 1661–1672. [181] D.S. Saxon, Technical Report CRI 40816, NASA, 1973. [182] M.F. Iskander, H.Y. Chen, J.E. Penner, Optical scattering and absorption by branched-chains of aerosols, Applied Optics 28 (1989) 3083–3091. [183] D.W. Mackowski, M.I. Mishchenko, Calculation of the T matrix and the scattering matrix for ensembles of spheres, Journal of the Optical Society of America 13 (11) (1996) 2266–2278. [184] L. Liu, M.I. Mishchenko, W.P. Arnott, A study of radiative properties of fractal soot aggregates using the superposition T-matrix method, Journal of Quantitative Spectroscopy and Radiative Transfer 109 (15) (2008) 2656–2663. [185] C.M. Sorensen, Light scattering by fractal aggregates: a review, Aerosol Science and Technology 35 (2001) 648–687. [186] Ü.Ö. Köylü, G.M. Faeth, Optical properties of overfire soot in buoyant turbulent diffusion flames at long residence times, ASME Journal of Heat Transfer 116 (1) (1994) 152–159. [187] D.W. Mackowski, A simplified model to predict the effects of aggregation on the absorption properties of soot particles, Journal of Quantitative Spectroscopy and Radiative Transfer 100 (2006) 237–249. [188] Ü.Ö. Köylü, C.S. McEnally, D.E. Rosner, L.D. Pfefferle, Simultaneous measurements of soot volume fraction and particle size/microstructure in flames using a thermophoretic sampling technique, Combustion and Flame 110 (1997) 494–507. [189] J.Y. Zhu, M.Y. Choi, G.W. Mulholland, S.L. Manzello, L.A. Gritzo, J. Suo-Anttila, Measurement of visible and near-IR optical properties of soot produced from laminar flames, Proceedings of the Combustion Institute 29 (2003) 2367–2374. [190] C.D. Zangmeister, R. You, E.M. Lunny, A.E. Jacobson, M. Okumura, M.R. Zachariah, J.G. Radney, Measured in-situ mass absorption spectra for nine forms of highly-absorbing carbonaceous aerosol, Carbon 136 (2018) 85–93. [191] T.C. Bond, R.W. Bergstrom, Light absorption by carbonaceous particles: an investigative review, Aerosol Science and Technology 40 (1) (2006) 27–67. [192] J.D. Felske, C.L. Tien, The use of the Milne–Eddington absorption coefficient for radiative heat transfer in combustion systems, ASME Journal of Heat Transfer 99 (3) (1977) 458–465. [193] B.M. Agarwal, M.P. Mengüç, Forward and inverse analysis of single and multiple scattering of collimated radiation in an axisymmetric system, International Journal of Heat and Mass Transfer 34 (3) (1991) 633–647. [194] F.D. Bryant, B.A. Sieber, P. Latimer, Absolute optical cross sections of cells and chloroplasts, Archives of Biochemistry and Biophysics 135 (1969) 97–108. [195] D.M. Roessler, F.R. Faxvog, Optoacoustic measurement of optical absorption in acetylene smoke, Journal of the Optical Society of America 69 (1979) 1699–1704. [196] F.R. Faxvog, D.M. Roessler, Optoacoustic measurement of Diesel particulate emission, Journal of Applied Physics 50 (1979) 7880–7882. [197] S. Härd, O. Nilsson, Laser heterodyne apparatus for measuring small angle scattering from particles, Applied Optics 18 (1979) 3018–3026. [198] H.C. Hottel, A.F. Sarofim, I.A. Vasalos, W.H. Dalzell, Multiple scatter: comparison of theory with experiment, ASME Journal of Heat Transfer 92 (1970) 285–291. [199] J.A. Menart, H.S. Lee, R.O. Buckius, Experimental determination of radiative properties for scattering particulates, Experimental Heat Transfer 2 (4) (1989) 309. [200] D.H. Woodward, He-Ne laser as source for light scattering measurements, Applied Optics 2 (1963) 1205–1207. [201] D.H. Woodward, Multiple light scattering by spherical dielectric particles, Journal of the Optical Society of America 54 (1964) 1325–1331. [202] W. Hartel, Zur Theorie der Lichstreuung durch trübe Schichten, besonders Trübgläser, Licht 10 (1940) 141–143, 232–234. [203] C. Smart, R. Jacobsen, M. Kerker, P. Kratohvil, E. Matijevic, Experimental study of multiple light scattering, Journal of the Optical Society of America 55 (8) (1965) 947–955. [204] S.E. Orchard, Multiple scattering by spherical dielectric particles, Journal of the Optical Society of America 55 (1965) 737–738. [205] K.J. Daniel, N.M. Laurendeau, F.P. Incropera, Optical property measurements for suspensions of unicellular algae, ASME paper no. 78-HT-14, 1978.

452 Radiative Heat Transfer

[206] G.W. Mulholland, M.Y. Choi, Measurement of the mass specific extinction coefficient for acetylene and ethene smoke using the large agglomerate optics facility, Proceedings of the Combustion Institute 27 (1998) 1515–1522. [207] J.Y. Zhu, A. Irrera, M.Y. Choi, G.W. Mulholland, J. Suo-Anttila, L.A. Gritzo, Measurement of light extinction constant of JP-8 soot in the visible and near-infrared spectrum, International Journal of Heat and Mass Transfer 47 (17–18) (2004) 3643–3648. [208] J. Kuhn, S. Korder, M.C. Arduini-Schuster, Infrared-optical transmission and reflection measurements on loose powders, Review of Scientific Instruments 64 (1993) 2523–2530. [209] M.I. Cabrera, O.M. Alfano, A.E. Cassano, Absorption and scattering coefficients of titanium dioxide particulate suspensions in water, Journal of Physical Chemistry 100 (1996) 20043–20050. [210] I.V. Yaroslavsky, A.N. Yaroslavsky, T. Goldbach, Inverse hybrid technique for determining the optical properties of turbid media from integrating-sphere measurements, Applied Optics 35 (1996) 6797–6809. [211] M.Q. Brewster, Y. Yamada, Optical properties of thick, turbid media from picosecond time-resolved light scattering measurements, International Journal of Heat and Mass Transfer 38 (14) (1995) 2569–2581. [212] R.W. Spinrad, J.R.V. Zaneveld, H. Pak, Volume scattering function of suspended particulate matter at near-forward angles: a comparison of experimental and theoretical values, Applied Optics 17 (7) (1978) 1125–1130.

Chapter 12

Radiative Properties of Semitransparent Media 12.1 Introduction Any solid or liquid that allows electromagnetic waves to penetrate an appreciable distance into it is known as a semitransparent medium. What constitutes an “appreciable distance” depends, of course, on the physical system at hand. If a thick film on top of a substrate allows a substantial amount of photons to propagate, say, 100 μm into it, the film material would be considered semitransparent. On the other hand, if heat transfer within a large vat of liquid glass is of interest, the glass cannot be considered semitransparent for those wavelengths that cannot penetrate several centimeters through the glass. Pure solids with perfect crystalline or very regular amorphous structures, as well as pure liquids, gradually absorb radiation as it travels through the medium, but they do not scatter it appreciably within that part of the spectrum that is of interest to the heat transfer engineer. If a solid crystal has defects, or if a solid or liquid contains inclusions (foreign molecules or particles, bubbles, etc.), the material may scatter as well as absorb. In some instances semitransparent media are inhomogeneous and tend to scatter radiation as a result of their inhomogeneities. An example of such material is aerogel [1], a highly transparent, and low heat-loss window material made of tiny hollow glass spheres pressed together. A number of theoretical models exist to predict the absorption and scattering characteristics of semitransparent media. As for opaque surfaces, the applicability of theories is limited, and they must be used in conjunction with experimental data. In this chapter we shall limit ourselves to absorption within semitransparent media. The models describing scattering behavior are the same as the ones presented in the previous chapter and will not be further discussed here. In particular, scattering from turbid media, insulation, foams, etc., has been summarized near the end of Section 11.12.

12.2 Absorption by Semitransparent Solids The absorption behavior of ionic crystals can be rather successfully modeled by the Lorentz model, which was discussed in some detail in Chapters 2 and 3. The Lorentz theory predicts that an ionic crystal has one or more Reststrahlen bands in the midinfrared (λ > 5 μm) (photon excitation of lattice vibrations). The wavelength at which strong absorption commences because of Reststrahlen bands is often called the long-wavelength absorption edge. The spectral absorption coefficients and their long-wavelength absorption edges are shown for a number of ionic crystals in Fig. 12.1. Note that these crystals are essentially transparent over much of the near infrared and become very rapidly opaque at the onset of Reststrahlen bands. The Lorentz model also predicts that the excitation of valence band electrons, across the band gap into the conduction band, results in several absorption bands at short wavelengths (usually around the ultraviolet). Figure 12.2 shows the absorption coefficient and short-wavelength absorption edge for several halides: Materials that are essentially opaque in the ultraviolet become highly transparent in the visible and beyond. Pure solids are generally highly transparent between the two absorption edges. If large amounts of localized lattice defects and/or dopants (foreign-material molecules called color centers) are present, electronic excitations may occur at other wavelengths in between. A number of models predict the absorption characteristics of such defects, some sophisticated, some simple and semiempirical. For example, Bhattacharyya and Streetman [3] and Blomberg and coworkers [4] developed models predicting the effect of dopants on the absorption coefficient of silicon. Figure 12.3 shows a comparison of the model by Blomberg and coworkers with experimental data of Siregar and colleagues [5] and Boyd and coworkers [6] for phosphorus-doped silicon at 10.6 μm (a wavelength of great importance for materials processing with CO2 lasers). The absorption coefficient increases strongly with Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00020-1 Copyright © 2022 Elsevier Inc. All rights reserved.

453

454 Radiative Heat Transfer

FIGURE 12.1 Spectral absorption coefficients of several ionic crystals at room temperature [2].

FIGURE 12.2 Spectral absorption coefficient of several halides at room temperature [2].

dopant concentration and with temperature. According to both models, the rise with temperature is due to increases in the number of free electrons and to their individual contributions. The same trends were observed by Timans [7] for the wavelength range between 1.1 and 1.6 μm. The absorption behavior of amorphous, i.e., noncrystalline solids is much more difficult to predict, although the general trends are quite similar. By far the most important semitransparent amorphous solid is soda–lime glass (ordinary window glass, as opposed to the quartz or silicon dioxide crystals depicted in Fig. 12.1). A number of investigators measured the absorption behavior of window glass, notably Genzel [8], Neuroth [9, 10], Grove and Jellyman [11], and Bagley and coworkers [12]. Figure 1.17 shows the behavior of the spectral absorption coefficient of window glass for a number of different temperatures. As expected from the data for the transmissivity of window panes (Figs. 3.28 and 3.29), glass is fairly transparent for wavelengths λ < 2.5 μm; beyond that it tends to become rather opaque. The temperature dependence for quartz has been observed to be similar to that of silicon by Beder and coworkers [13], who reported a fourfold increase of the absorption coefficient between room temperature and 1500◦ C.

Radiative Properties of Semitransparent Media Chapter | 12 455

FIGURE 12.3 Spectral absorption coefficient of phosphorus-doped Si at 10.6 μm; solid lines: model of Blomberg and coworkers [4]; square symbols (): data of Boyd and coworkers [6] (dopant concentration of 1.1×1015 cm−3 ); circular symbols (•): data of Siregar and coworkers [5] (dopant concentration unknown).

12.3 Absorption by Semitransparent Liquids The absorption properties of semitransparent liquids are quite similar to those of solids, while they also display some behavior similar to molecular gases. Remnants of intermolecular vibrations (Reststrahlen bands) are observed in many liquids, as are remnants of electronic band gap transitions in the ultraviolet. In the wavelengths in between, molecular vibration bands are observed for molecules with permanent dipole moments, similar to the vibration–rotation bands of gases. Because of its abundance in the world around us (and, indeed, inside our own bodies) the absorption properties of water (and its solid form as ice) are by far the most important and, therefore, have been studied extensively, indeed for centuries. The data of many investigators for clear water and clear ice have been collected and interpreted by Irvine and Pollack [14] and by Ray [15]. Another review, limited to pure water, has been given by Hale and Querry [16]. More recent measurements have been reported by Kou and colleagues [17] (water and ice for wavelengths below 2.5 μm) and by Marley and coworkers [18] (water between 3.3 μm and 11 μm). The spectral absorption coefficient of clear water (at room temperature) and of clear ice (at −10◦ C) is shown in Fig. 12.4, based on the tabulations of Irvine and Pollack [14], Kou and colleagues [17], and Marley and coworkers [18]. Note the similarity between solid ice and liquid water. The lowest points of the absorption spectra of water and ice lie in the visible, making them virtually transparent over short distances. The minimum point lies in the blue part of the visible (λ 0.45 μm): Large bodies of water (or clear ice) transmit blue light the most, giving them a bluish hue. In the near- to midinfrared water and ice display several absorption bands (at 1.45, 1.94, 2.95, 4.7, and 6.05 μm in water, somewhat shifted for ice). These bands are very similar to the water vapor bands at 1.38, 1.87, 2.7, and 6.3 μm (see Table 10.4). Agreement between the data of Irvine and Pollack, and that of Kou and colleagues is excellent, while the data of Marley and coworkers in the longer wavelength region are considerably lower than those of Irvine and Pollack: measurement of such large absorption coefficients is extremely difficult, and the modern measurements of Marley and coworkers list an average estimated error of better than 3%. The temperature dependence of the absorption coefficient of water has been investigated by Goldstein and Penner [19] (up to 209◦ C) and by Hale and coworkers [20] (up to 70◦ C) and was found to be fairly weak. As temperature increases, water becomes somewhat more transparent in relatively transparent regions and somewhat more opaque in absorbing regions. A rather detailed discussion of the absorption behavior of clean water and ice has been given by Bohren and Huffman [21]. Natural waters and ice generally contain significant amounts of particulates (small organisms, detritus) and gas bubbles, which tend to increase the

456 Radiative Heat Transfer

FIGURE 12.4 Spectral absorption coefficient of clear water (at room temperature) and clear ice (at −10◦ C [14] and −25◦ C [17]); from [14] (thick lines), [18] (medium line), and [17] (thin lines).

FIGURE 12.5 Spectral absorption coefficient of LiF for various temperatures; A: 300 K; B: 705 K; C: 835 K; D: 975 K; E: 1160 K. The melting point of LiF is 1115 K [22].

absorption rate as well as to scatter radiation. While a number of measurements have been made on varieties of natural waters and ice, the results are difficult to correlate since the composition of natural waters varies greatly. The similarity of absorption behavior between the solid and liquid states of a substance is not limited to water. Barker [22] has measured the absorption coefficient of three alkali halides (KBr, NaCl, and LiF) for several temperatures between 300 K and temperatures above the melting point. Since Reststrahlen bands tend to widen with increasing temperature (see Section 3.5), the long-wavelength absorption edge moves toward shorter wavelengths. No distinct discontinuity in absorption coefficient was observed as the material changed phase from solid to liquid. As an example, the behavior of lithium fluoride (LiF) is depicted in Fig. 12.5. Semiempirical models for the absorption coefficient of alkali halide crystals, resulting in simple formulae, have been given by Skettrup [23] and Woodruff [24], while a similar formula for alkali halide melts has been developed by Senatore and coworkers [25].

Radiative Properties of Semitransparent Media Chapter | 12 457

FIGURE 12.6 Schematic of ideal foam cell, consisting of struts (with lengths a and curved triangular cross-section diameter b), strut junctions, and, in the case of closed-cell foams, thin walls of thickness dw [40].

FIGURE 12.7 Hemispherical reflectance for carbon foam sample 4.3 mm thick for normal incidence; experimental (Re ) and theoretical (Rt ) results [39].

12.4 Radiative Properties of Porous Solids The applicability of the RTE to heterogeneous media was studied by several investigators, e.g., [26–38]. In this section we will assume that heterogeneous media can be modeled as homogeneous with radiative intensity described by a local average value based on appropriate continuum properties. The radiative properties of open cell carbon foam were studied using experimental techniques and a predictive model by Baillis et al. [39]. The model combined elements of geometric optics and diffraction theory applied to the foam geometry determined by microscopic techniques. Extinction, scattering, and absorption coefficients were determined by assuming open cells to consist of struts with varying thickness and strut junctions, as schematically shown in Fig. 12.6, leading to   G¯ 2 ¯ βλ =N G1 + , (12.1) 2 σsλ =ρλ βλ ,   κ λ = 1 − ρ λ βλ ,

(12.2) (12.3)

where N is the number of struts per unit volume, G¯ 1 and G¯ 2 are the average geometric cross sections of struts and strut junctions, respectively, and ρλ is the spectral hemispherical reflectance of the solid.1 Hemispherical reflectances of foam slabs obtained by solving the RTE with the predicted properties agreed well with measured ones, as shown in Fig. 12.7. Larger discrepancies were observed for the very small, and thus difficult to measure, hemispherical transmittance of a 4.3 mm thick sample. The radiative properties of highly-porous open-cell metallic foams with inhomogeneities in the size range of geometric optics were studied using simple predictive models by Loretz et al. [42]. The foam structure 1. The factor of 12 in equation (12.1) is not present in the original paper [39], but was added in more recent work, e.g., [41], perhaps to account for the fact that foam contains fewer strut junctions than struts.

458 Radiative Heat Transfer

FIGURE 12.8 Radiative conductivity for two different extruded polystyrene foams [40]. Φc is the diameter of the foam cell measured as (a) Φc = 76 ± 30 μm and (b) Φc = 108 ± 30 μm, respectively. Both predicted and “measured” conductivities depend on the unknown solid fraction contained in struts, fs .

was determined using microscopic and tomographic techniques. The cells (Fig. 12.6) were assumed to consist of struts and strut junctions. The extinction coefficient of the cells modeled as pentagon dodecahedrons or tetracaedecahedrons was obtained using the Glicksman and Torpey model [43]: & 1−ε , (12.4) β = 4.09 D2 where ε and D are the porosity and average cell diameter, respectively. For pentagon dodecahedrons with neglected strut junctions equation (12.4) becomes β=

3 b 1.305 2 , 4 a

(12.5)

where a and b are the strut length and average thickness, respectively, as indicated in Fig. 12.6. Silicon carbide open-cell foams were investigated by Tseng et al. [44]. The dimensionless strut diameter was deemed an important parameter in this study, and radiative properties were computed using Mie scattering theory. Comparison of the mean extinction coefficient with experimental data at 1000 K showed good agreement. The radiative properties of closed-cell foams were studied for expanded polystyrene foam by Coquard et al. [45,46], and those for extruded polystyrene foams were predicted and verified experimentally by Kaemmerlen and coworkers [40]. The properties were determined using the integration method of [47] applied to the curvedtriangular foam cell wall and strut geometries of Fig. 12.6. The Rosseland-mean extinction coefficient was calculated by independently determining the extinction coefficients of struts and of thin films of polystyrene, from which they also deduced the material’s radiative conductivity kR for use in the diffusion approximation, an approximate RTE solution method for optically dense materials [cf. Section 14.2 and equation (14.19)]. Due to the low density of the foam, independent scattering was assumed to hold, and the bulk extinction coefficient was determined by adding contributions from struts and walls, similar to equation (12.1). Figure 12.8 shows kR with and without a correction factor to the scattering efficiency to account for the concave shape of circular struts, which leads to a noticeable decrease in the variation of kR with the strut fraction as compared to the uncorrected results. However, as the authors noticed, the trends between predicted and measured radiative conductivities are different. A more extensive discussion of the radiative properties of open-cell and closed-cell foams may be found in the book by Dombrovsky and Baillis [41]. Recent review of both modeling and experimental characterization of radiative transport in open-cell regular and irregular foams may be found in the review articles by Cunsolo and coworkers [48,49]. Monte Carlo ray tracing methods of Chapter 7 have been employed in a number of studies for the determination of effective radiative properties of heterogeneous media based on the geometry and properties of individual medium components. Tancrez and Taine [29] presented a methodology for porous media with opaque solid phase, which was extended to media with semitransparent solids [33]. Coquard and Baillis applied ray tracing

Radiative Properties of Semitransparent Media Chapter | 12 459

FIGURE 12.9 3D rendering of Rh-coated reticulated porous ceramics with nominal pore diameter dnom = 2.54 mm obtained using computed tomography techniques [53].

to determine the radiative properties of beds of opaque, diffusely or specularly reflecting particles [50]. The latter study was extended to beds of spheres containing an absorbing and scattering medium [51] and also applied to the actual geometry of polymeric foams obtained by tomography [52]. Also using tomography, the geometry of reticulated porous ceramics (RPC) with an opaque solid phase was obtained by Petrasch et al. [53] and by Haussener et al. [54]; the latter also used this technique for reacting packed beds with an opaque solid phase [55]. Finally, mullite foam with a semitransparent solid phase was studied by Zeghondy and coworkers [33]. In the tomography-based Monte Carlo methods used to study radiative properties of reticulated porous ceramics (Fig. 12.9) [29,33,53,54,56] the media were assumed to be statistically homogeneous and isotropic, and the solid phase was assumed to be opaque. Diffraction effects were neglected and geometric optics was assumed to be valid. A large number Nr of stochastic rays were launched in the void phase of a subvolume V0 of a representative elementary volume V. Rays were traced until they interacted with the solid–void interface or were lost at the faces of V. For each ray colliding with the solid phase the distance to collision was recorded, and rays were either absorbed or reflected, either specularly or diffusely. The distribution function for attenuation path length was then computed as  s 1 Fs = dN(s) = 1 − exp(−βs), (12.6) Nr s∗ =0 where dN(s) is the number of rays attenuated within ds around s; Fs quantifies the probability of a ray hitting the solid–void interface at a location between 0 and s. The scattering and absorption coefficients were then obtained from equations (12.2) and (12.3). Figure 12.10 shows the radiative intensity obtained numerically and experimentally as a function of normalized path length. The relative difference of 10% between experimental (βex ) and Monte Carlo-determined (βMC ) extinction coefficient was attributed to the effect of local material anisotropy for finite and relatively small RPC samples. Monte Carlo results were integrated over all solid angles, while the experimental measurements were carried out only along a single direction. In other Monte Carlo studies, Arambakam et al. [57] investigated radiative transfer through fibrous insulation material comprised of bimodal translucent fibers. 3D computer models of the media with different packing density (porosity), fiber orientation, and fiber diameter were first created. Monte Carlo calculations were then conducted in a 3D system with a temperature gradient imposed in one of the three directions. Snell’s law and Fresnel’s relations were used to compute ray paths through the translucent fibers. The transmittance of the fibrous medium was found to increase with increasing through-plane orientation of the fibers, but was found to be independent of their inplane orientations. It was also found that fiber orientation had a negligible effect on the temperature distribution. Gladen et al. [58] conducted Monte Carlo calculations to determine the solar-weighted optical properties of potential thermotropic composite materials for overheat protection of polymer solar absorbers used in solar thermochemical reactors. Ganesan et al. [59] employed the Monte Carlo method to obtain spectral (between 0.35–2.2 μm) absorption and scattering coefficients of three-dimensionally ordered macroporous packed ceria

460 Radiative Heat Transfer

FIGURE 12.10 Variation of radiative intensity in Rh-coated reticulated porous ceramics obtained numerically (squares) and experimentally (circles) as a function of the normalized path length, along with exponential fits with βMC = 210 m−1 and βex = 230 m−1 [75].

samples, used in solar thermochemical conversion applications, from measured transmittance and reflectance data. Results showed that altering the morphology of the sample altered the scattering coefficient, but did not affect the absorption coefficient. Marti et al. [60] combined experimental measurements with the Monte Carlo method to estimate the spectral extinction coefficient and scattering phase function of silicon carbide suspensions. Wang et al. [61] studied radiative transfer through a bed of spherical particles in which the particles have an inner semitransparent core and an outer semitransparent coating of a different material. They also investigated the effect of including bidirectional reflection and transmission functions on the computed results [62]. The Monte Carlo method, both forward and backward, has also been used to study radiative transport in metal foams [56,63–66]. The Monte Carlo method is based on geometrical optics, and disregards coherence and dependent scattering within the porous medium. A few studies have been dedicated to investigating these issues [67–69]. Amongst other methods, a number of studies have also used the discrete ordinates method (DOM) and its variant, the finite angle method (FAM), the topics of Chapter 16, to investigate radiative transfer in porous media with volume-averaged properties [59,70–72]. A two-phase radiation transport model based on the P1 approximation has been proposed by Ferkl et al. [73] (Chapter 15).

12.5 Experimental Methods The spectral absorption coefficient of a semitransparent solid or liquid can be measured in several ways. The simplest and most common method is to measure the transmissivity of a sample of known thickness, as described in Section 11.12 for particulate clouds. Since solids and liquids reflect energy at the air interfaces, the transmissivity is often determined by forming a ratio between the transmitted signals from two samples of different thickness. However, the transmission method is not capable of measuring very small or very large absorption coefficients: For samples with large transmissivity small errors in the determination of transmissivity, τ, lead to very large errors for the absorption coefficient, κ (since κ is proportional to ln τ). On the other hand, for a material with large κ sufficient energy for transmission measurements can be passed only through extremely thin samples. Such samples are usually prepared as vacuum-deposited thin films, which do not have the same properties as the parent material [74]. The absorption coefficient may also be determined through a number of different reflection techniques. The reflectivity of an optically smooth interface of a semitransparent medium depends, through the complex index of refraction, on the refractive index n as well as the absorptive index k. In turn, k is related to the absorption coefficient through equation (3.79) as κ = 4πηk/n, where η = 1/λ is the wavenumber of the radiation inside the medium. Thus, two data points are necessary to determine n and k. Noting the directional dependence of

Radiative Properties of Semitransparent Media Chapter | 12 461

FIGURE 12.11 Sample and holder, mounted within heating tube, for device to determine the optical properties of small, semitransparent solid samples [79].

reflectivity on m = n − ik, some researchers have measured the specular reflectivity at two different angles. Leupacher and Penzkofer [76] showed that this can lead to very substantial errors. Other researchers have measured the reflectivity at a single angle, using parallel- and perpendicular-polarized light (known as an ellipsometric technique). However, this may also lead to large errors [76]. A new method overcoming these problems has been proposed by Lu and Penzkofer [77]. Using parallel-polarized light they vary the incidence angle until the point of minimum reflectivity at Brewster’s angle is found (cf. Figs. 2.8 and 2.11). Another reflection technique exploits the fact that a causal relationship exists between n and k, i.e., they are not independent of one another. This causal relationship is known as the Kramers–Kronig relation, which may be expressed as  η ∞ ln ρn (η )  δ(η) = dη , (12.7) π 0 η2 − η 2 where ρn (η) is the spectral, normal reflectivity of the sample surface [cf. equation (2.114)], and δ(η) is the phase angle of the complex reflection coefficient, equation (2.111), * rn =

n − ik − 1 √ . ρn eiδ = n − ik + 1

(12.8)

Thus, if ρn is measured for a large part of the spectrum, the phase angle δ may be determined from equation (12.7) for wavenumbers well inside the measured spectrum; n and k are then readily found from equation (12.8). The method is particularly well suited to experiments employing an FTIR (Fourier transform infrared) spectrometer, which can take broad spectrum measurements over very short times, and which often have a built-in Kramers– Kronig analysis capability. More detailed discussions on the various Kramers–Kronig relations may be found, for example, in the books by Wooten [78] and Bohren and Huffman [21]. A description of the numerical evaluation of equation (12.7) has been given by Wooten [78]. Measurement of physical properties at high temperatures is always difficult, but particularly so for semitransparent media since two properties need to be measured (absorption coefficient as well as interface reflectivity, or equivalently, n and k). Myers and coworkers [79] have given a good review of such methods for solid samples. They also developed a new method to determine the optical properties of small, semitransparent, solid samples. Their device is essentially a compact arrangement of that employed by Stierwalt [80], which takes three different radiance measurements in rapid succession. A front and cross-sectional view of their sample heating arrangement is shown in Fig. 12.11. The slab-shaped sample is mounted within an equalizing nickel block, which is coupled radiatively to the electrically heated tube. The nickel block has four cavities and holes serving as radiance targets. A water-cooled graphite block (not shown) is positioned behind the heating tube to provide a room-temperature background for the through-hole as well as a reference for the detector. Three

462 Radiative Heat Transfer

FIGURE 12.12 Measurement of absorption coefficients of high-temperature liquids. (a) Schematic representation of apparatus of Ebert and Self [83], (b) schematic representation of their submerged reflector arrangement.

radiance measurements are made and compared with the reference: (i) the slab sample positioned in front of the blackbody (cavity-hole), (ii) the freely radiating sample (through-hole), and (iii) the blackbody reference. With the relations given in Section 3.8 one can use these measurements to deduce the optical properties (n, k, and κ). The method has the advantages that measurements at high temperatures ( 1000◦ C) can be taken, that only a single sample is necessary, and that no optically smooth surfaces are required. On the other hand, the method suffers from the standard weaknesses of transmission methods (see discussion at the beginning of this section) and is restricted to high temperatures (to produce a strong enough emission signal). Measurements of the optical properties of a high-temperature liquid are even more challenging. It is more difficult to confine a liquid in a sample holder (which must be horizontal) and more difficult to measure the thickness of the liquid layer. In addition, the layer thickness may be nonuniform because of (often unknown) surface tension effects. Furthermore, high-temperature liquids are often highly reactive, making a sealed chamber necessary. If the vapor pressure becomes substantial at high temperatures, the windows of the sealed chamber will be attacked. Shvarev and coworkers [81] have measured the optical properties of liquid silicon in the wavelength range of 0.4–1.0 μm with such a sealed-chamber furnace apparatus, using an ellipsometric technique. Barker [22,82] designed an apparatus to measure the optical properties of semitransparent solid slabs and corrosive melts. To isolate the specimen he relied on a windowless chamber with continuous inert-gas purging. His data evaluation required independent measurements of the interface reflectivity, the reflectivity of a platinum mirror, the sample overall reflectivity, and the thickness of the sample. In addition, the reflectivity of the platinum–liquid interface must be estimated. As such, Barker’s method appears to be very vulnerable to experimental error. A more accurate device, limited to absorption coefficients of liquids, has been reported by Ebert and Self [83]. A schematic representation of their apparatus is shown in Fig. 12.12a. The aperture of a blackbody source at 1700◦ C is imaged (by the spherical mirror M3) onto the platinum mirror located in an alumina crucible inside the furnace. The reflected signal is focused onto the monochromator and detector via another spherical mirror (M5). The beam is chopped to eliminate emission as well as background radiation from the signal. The transmissivity of the liquid is measured by what they called a “submerged reflector method,” illustrated in Fig. 12.12b: A platinum mirror, which may be adjusted via three support rods, is submerged below the surface of the liquid filling the crucible. The platinum mirror is tilted slightly from the horizontal to allow the first surface reflection

Radiative Properties of Semitransparent Media Chapter | 12 463

and multiple internal reflections to be rejected from the collection optics. The thickness of the liquid layer is adjusted by raising and lowering the crucible (leaving the platinum mirror in place). As in the transmission technique, signals for two different layer thicknesses (d1 and d2 ) are ratioed, giving the transmissivity for a layer of thickness 2 (d2 − d1 ). By rejecting the first reflection, and by being able to produce and measure very thin liquid layers, they were able to measure absorption coefficients an order of magnitude higher than Barker, reporting values as high as 70 cm−1 for synthetic molten slags [83]. Similar measurements have been carried out by Gupta and Modest [84] (lithium salts), by Makino and coworkers [85] (alkali metal carbonates), and by Zhang and colleagues [86] (liquid glasses). Foams and Porous Solids. Radiative transfer in porous solids has also been investigated experimentally for a variety of porous media and their applications, such as porous screens for fire protection [87], silica aerogel composites for thermal insulation [70,88], foams used in insulation and chemical processing applications [89], fiber-reinforced cement composites for building construction [90], polystyrene and polyurethane foam insulation [91], fiberglass and carbon foam [92–94], and reticulated porous ceramics [95,96]. In the vast majority of these studies, the transmittance and reflectance—either spectral or total—of a layer of the porous material has been measured, and attempts have been made to either extract the radiative properties (absorption and scattering coefficients; scattering phase function) [56,88,89] or the radiative conductivity [70,90] via fitting to the data. The bidirectional reflectance of mullite foam has been measured by Zeghondy and coworkers [97], which agreed well with model results based on the Monte Carlo tool of Tancrez and Taine [29,33]. Cunnington and coworkers [98] measured the scattering from individual, coated silica fibers, and found qualitative agreement with a theoretical model. Cunnington and Lee measured direct transmissivity and hemispherical reflectivity of randomly packed, high-porosity fibrous material (tiles from the Space Shuttle) [99], and for aerogel-reinforced fibrous material [100]; comparison with Lee’s models [101–104] showed excellent agreement for both materials.

Problems 12.1 The absorption coefficient of a liquid, confined between two parallel and transparent windows, is to be measured by the transmission method. The detector signals from transmission measurements with varying liquid thickness are to be used. (a) Using transmission measurements for two thicknesses, show how the absorption coefficient κ may be deduced. Determine how errors in the transmissivity value and the liquid layer thickness affect the accuracy of κ. (b) If transmission measurements are made for many thicknesses, can you devise a method that measures small absorption coefficients more accurately? 12.2 Show how the optical properties (n, k, and κ) of a semitransparent solid may be deduced from the three measurements taken with the apparatus of Myers and coworkers [79], as depicted in Fig. 12.11.

References [1] R. Caps, J. Fricke, Infrared radiative heat transfer in highly transparent silica aerogel, Solar Energy 36 (4) (1986) 361–364. [2] A. Smakula, Synthetic crystals and polarizing materials, Optica Acta 9 (1962) 205–222. [3] A. Bhattacharyya, B.G. Streetman, Theoretical considerations regarding pulsed CO2 laser annealing of silicon, Solid State Communications 36 (1980) 671–675. [4] M. Blomberg, K. Naukkarinen, T. Tuomi, V.M. Airaksinen, M. Luomajarvi, E. Rauhala, Substrate heating effects in CO2 laser annealing of ion-implanted silicon, Journal of Applied Physics 54 (5) (1983) 2327–2328. [5] M.R.T. Siregar, W. Lüthy, K. Affolter, Dynamics of CO2 laser heating in the processing of silicon, Applied Physics Letters 36 (1980) 787–788. [6] I.W. Boyd, J.I. Binnie, B. Wilson, M.J. Colles, Absorption of infrared radiation in silicon, Journal of Applied Physics 55 (8) (1984) 3061–3063. [7] P.J. Timans, Emissivity of silicon at elevated temperatures, Journal of Applied Physics 74 (10) (1993) 6353–6364. [8] L. Genzel, Messung der Ultrarot-Absorption von Glas zwischen 20◦ C and 1360◦ C (Measurement of infrared absorption of glass between 20◦ C and 1360◦ C), Glastechnische Berichte 24 (3) (1951) 55–63. [9] N. Neuroth, Der Einfluss der Temperatur auf die spektrale Absorption von Gläsern im Ultraroten, I (Effect of temperature on spectral absorption of glasses in the infrared, I), Glastechnische Berichte 25 (1952) 242–249. [10] N. Neuroth, Der Einfluss der Temperatur auf die spektrale Absorption von Gläsern im Ultraroten, II (Effect of temperature on spectral absorption of glasses in the infrared, II), Glastechnische Berichte 26 (1953) 66–69. [11] F.J. Grove, P.E. Jellyman, The infrared transmission of glass in the range from room temperature to 1400◦ C, Journal of the Society of Glass Technology 39 (186) (1955) 3–15.

464 Radiative Heat Transfer

[12] B.G. Bagley, E.M. Vogel, W.G. French, G.A. Pasteur, J.N. Gan, J. Tauc, The optical properties of soda-lime-silica glass in the region from 0.006 to 22 eV, Journal of Non-Crystalline Solids 22 (1976) 423–436. [13] E.C. Beder, C.D. Bass, W.L. Shackleford, Transmissivity and absorption of fused quartz between 0.2 μm and 3.5 μm from room temperature to 1500◦ C, Applied Optics 10 (1971) 2263–2268. [14] W.M. Irvine, J.B. Pollack, Infrared optical properties of water and ice spheres, Icarus 8 (1968) 324–360. [15] P.S. Ray, Broadband complex refractive indices of ice and water, Applied Optics 11 (1972) 1836–1844. [16] G.M. Hale, M.R. Querry, Optical constants of water in the 200 nm to 200 μm wavelength region, Applied Optics 12 (1973) 555–563. [17] L. Kou, D. Labrie, P. Chylek, Refractive indices of water and ice in the 0.65 to 2.5 μm spectral range, Applied Optics 32 (1993) 3531–3540. [18] N.A. Marley, J.S. Gaffney, M.M. Cunningham, Lambert absorption coefficients of water in the frequency range of 3000–934 cm−1 , Applied Optics 33 (1994) 8041–8054. [19] R.J. Goldstein, S.S. Penner, The near-infrared absorption of liquid water at temperatures between 27 and 209◦ C, Journal of Quantitative Spectroscopy and Radiative Transfer 4 (1964) 441–451. [20] G.M. Hale, M.R. Querry, A.N. Rusk, D. Williams, Influence of temperature on the spectrum of water, Journal of the Optical Society of America 62 (1972) 1103–1108. [21] C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley & Sons, New York, 1983. [22] A.J. Barker, The effect of melting on the multiphonon infrared absorption spectra of KBr, NaCl, and LiF, Journal of Physics C: Solid State Physics 5 (1972) 2276–2282. [23] T. Skettrup, Urbach’s rule and phase fluctuations of the transmitted light, Physica Status Solidi B 103 (1981) 613–621. [24] T.O. Woodruff, Empirically derived formula for the energies of the first ultraviolet absorption maximum of 20 alkali-halide crystals, Solid State Communications 46 (1983) 139–142. [25] G. Senatore, M.P. Tosi, T.O. Woodruff, A simple formula for the fundamental optical absorption of alkali halide melts, Solid State Communications 52 (2) (1984) 173–176. [26] B.P. Singh, M. Kaviany, Radiative transfer in packed and fluidized beds: dependent versus independent scattering, International Journal of Heat and Mass Transfer 34 (1991) 2869–2882. [27] B.P. Singh, M. Kaviany, Modelling radiative heat transfer in packed beds, International Journal of Heat and Mass Transfer 35 (1992) 1397–1405. [28] J.-L. Consalvi, B. Porterie, J. Loraud, A formal averaging procedure for radiation heat transfer in particulate media, International Journal of Heat and Mass Transfer 45 (2002) 2755–2768. [29] M. Tancrez, J. Taine, Direct identification of absorption and scattering coefficients and phase function of a porous medium by a Monte Carlo technique, International Journal of Heat and Mass Transfer 47 (2004) 373–383. [30] J. Taine, E. Iacona, Upscaling statistical methodology for radiative transfer in porous media: new trends, ASME Journal of Heat Transfer 134 (2012) 031012. [31] D. Baillis, J.-F. Sacadura, Thermal radiation properties of dispersed media: theoretical prediction and experimental characterization, Journal of Quantitative Spectroscopy and Radiative Transfer 67 (2000) 327–363. [32] R. Coquard, D. Baillis, J.H. Randrianalisoa, Homogeneous phase and multi-phase approaches for modeling radiative transfer in foams, International Journal of Thermal Sciences 50 (2011) 1648–1663. [33] B. Zeghondy, E. Iacona, J. Taine, Determination of the anisotropic radiative properties of a porous material by radiative distribution function identification (RDFI), International Journal of Heat and Mass Transfer 49 (2006) 2810–2819. [34] A.V. Gusarov, Homogenization of radiation transfer in two-phase media with irregular phase boundaries, Physical Review B 77 (2008) 144201. [35] A.V. Gusarov, Model of radiative heat transfer in heterogeneous multiphase media, Physical Review B 81 (2010) 064202. [36] W. Lipinski, ´ J. Petrasch, S. Haussener, Application of the spatial averaging theorem to radiative heat transfer in two-phase media, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (1) (2010) 253–258. [37] W. Lipinski, ´ D. Keene, S. Haussener, J. Petrasch, Continuum radiative heat transfer modeling in media consisting of optically distinct components in the limit of geometrical optics, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (16) (2010) 2474–2480. [38] J. Petrasch, S. Haussener, W. Lipinski, ´ Application of the spatial averaging theorem to radiative heat transfer in two-phase media, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (2011) 1450–1459. [39] D. Baillis, M. Raynaud, J.-F. Sacadura, Determination of spectral radiative properties of open cell foam: model validation, Journal of Thermophysics and Heat Transfer 14 (2) (2000) 137–143. [40] A. Kaemmerlen, C. Vo, F. Asllanaj, G. Jeandel, D. Baillis, Radiative properties of extruded polystyrene foams: predictive model and experimental results, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (2010) 865–877. [41] L.A. Dombrovsky, D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, New York, 2010. [42] M. Loretz, R. Coquard, D. Baillis, E. Maire, Metallic foams: radiative properties/comparison between different models, Journal of Quantitative Spectroscopy and Radiative Transfer 109 (1) (2008) 16–27. [43] L.R. Glicksman, M. Torpey, A study of radiative heat transfer through foam insulation, Technical report, Massachusetts Institute of Technology, Cambridge, MA, 1988. [44] C.C. Tseng, A.D. Swanson, R. Viskanta, R.L. Sikorski, M.Y. Chen, Effect of foam properties on radiative properties of open-cell silicon carbide foams, Journal of Quantitative Spectroscopy and Radiative Transfer 113 (2012) 1503–1507. [45] R. Coquard, D. Baillis, Modeling of heat transfer in low-density EPS foams, ASME Journal of Heat Transfer 128 (2006) 538–549.

Radiative Properties of Semitransparent Media Chapter | 12 465

[46] R. Coquard, D. Baillis, D. Quenard, Radiative properties of expanded polystyrene foams, ASME Journal of Heat Transfer 131 (2009) 012702. [47] E. Placido, M.C. Arduini-Schuster, J. Kuhn, Thermal properties predictive model for insulating foams, Infrared Physics & Technology 46 (2005) 219–231. [48] S. Cunsolo, R. Coquard, D. Baillis, N. Bianco, Radiative properties modeling of open cell solid foam: review and new analytical law, International Journal of Thermal Sciences 104 (2016) 122–134. [49] S. Cunsolo, R. Coquard, D. Baillis, W.K.S. Chiu, N. Bianco, Radiative properties of irregular open cell solid foams, International Journal of Thermal Sciences 117 (2017) 77–89. [50] R. Coquard, D. Baillis, Radiative characteristics of opaque spherical particle beds: a new method of prediction, Journal of Thermophysics and Heat Transfer 18 (2004) 178–186. [51] R. Coquard, D. Baillis, Radiative characteristics of beds of spheres containing an absorbing and scattering medium, Journal of Thermophysics and Heat Transfer 19 (2005) 226–234. [52] R. Coquard, D. Baillis, E. Maire, Numerical investigation of the radiative properties of polymeric foams from tomographic images, Journal of Thermophysics and Heat Transfer 24 (2010) 647–658. [53] J. Petrasch, P. Wyss, A. Steinfeld, Tomography-based Monte Carlo determination of radiative properties of reticulate porous ceramics, Journal of Quantitative Spectroscopy and Radiative Transfer 105 (2) (2007) 180–197. [54] S. Haussener, P. Coray, W. Lipinski, ´ P. Wyss, A. Steinfeld, Tomography-based heat and mass transfer characterization of reticulate porous ceramics for high-temperature processing, ASME Journal of Heat Transfer 132 (2010) 023305. [55] S. Haussener, W. Lipinski, ´ P. Wyss, A. Steinfeld, Tomography-based analysis of radiative transfer in reacting packed beds undergoing a solid-gas thermochemical transformation, ASME Journal of Heat Transfer 132 (2010) 061201. [56] R. Coquard, B. Rousseau, P. Echegut, D. Baillis, H. Gomart, E. Iacona, Investigations of the radiative properties of Al–NiP foams using tomographic images and stereoscopic micrographs, International Journal of Heat and Mass Transfer 55 (2012) 1606–1619. [57] R. Arambakam, H.V. Tafreshi, B. Pourdeyhimi, Analytical Monte Carlo ray tracing simulation of radiative heat transfer through bimodal fibrous insulations with translucent fibers, International Journal of Heat and Mass Transfer 55 (2012) 7234–7246. [58] A.C. Gladen, S.C. Mantell, J.H. Davidson, A parametric numerical study of optical behavior of thermotropic materials for solar thermal collectors, ASME Journal of Heat Transfer 136 (2014) 072703. [59] K. Ganesan, J.H. Randrianalisoa, W. Lipinski, ´ Effect of morphology on spectral radiative properties of three-dimensionally ordered macroporous ceria packed bed, ASME Journal of Heat Transfer 135 (2013) 122701. [60] J. Marti, M. Roesle, A. Steinfeld, Combined experimental-numerical approach to determine radiation properties of particle suspensions, ASME Journal of Heat Transfer 136 (2014) 092701. [61] C.-A. Wang, Q.-Z. Lai, T.-R. Shen, J.Y. Tan, Numerical investigation of radiation transfer in packed bed of semitransparent coated spherical particles, International Journal of Thermal Sciences 127 (2018) 1–6. [62] C.-A. Wang, L.-X. Ma, J.Y. Tan, L.H. Liu, Study of radiative transfer in 1d densely packed bed layer containing absorbing-scattering spherical particles, International Journal of Heat and Mass Transfer 102 (2016) 669–678. [63] G. Contento, M. Oliviero, N. Bianco, V. Naso, The prediction of radiation heat transfer in open cell metal foams by a model based on the Lord Kelvin representation, International Journal of Heat and Mass Transfer 76 (2014) 499–508. [64] S. Cunsolo, M. Oliviero, W.M. Harris, A. Andreozzi, N. Bianco, W.K.S. Chiu, V. Naso, Monte Carlo determination of radiative properties of metal foams: comparison between idealized and real cell structures, International Journal of Thermal Sciences 87 (2015) 94–102. [65] Y. Li, X.L. Xia, C. Sun, Q. Ai, B. Liu, H.P. Tan, Tomography-based analysis of apparent directional spectral emissivity of high-porosity nickel foams, International Journal of Heat and Mass Transfer 118 (2018) 402–415. [66] Y. Li, X.L. Xia, C. Sun, S.-D. Zhang, H.P. Tan, Volumetric radiative properties of irregular open-cell foams made from semitransparent absorbing-scattering media, Journal of Quantitative Spectroscopy and Radiative Transfer 224 (2019) 325–342. [67] J. Qiu, L.H. Liu, P.-F. Hsu, Thermal radiation in subwavelength aluminum foam structures by finite-difference time-domain method, Journal of Quantitative Spectroscopy and Radiative Transfer 158 (2015) 101–110. [68] V.P. Tishkovets, E.V. Petrova, Coherent backscattering by discrete random media composed of clusters of spherical particles, Journal of Quantitative Spectroscopy and Radiative Transfer 127 (2013) 192–206. [69] V.P. Tishkovets, E.V. Petrova, On applicability of the far-field approximation to the analysis of light scattering by particulate media, Journal of Quantitative Spectroscopy and Radiative Transfer 182 (2016) 24–34. [70] J.-J. Zhao, Y.-Y. Duan, X.-D. Wang, B.-X. Wang, Radiative properties and heat transfer characteristics of fiber-loaded silica aerogel composites for thermal insulation, International Journal of Heat and Mass Transfer 55 (2012) 5196–5204. [71] R. Arambakam, H.V. Tafreshi, B. Pourdeyhimi, Dual-scale 3-D approach for modeling radiative heat transfer in fibrous insulations, International Journal of Heat and Mass Transfer 64 (2013) 1109–1117. [72] D.L. Hardy, M.A. Badri, B. Rousseau, S. Chaupin, D. Rochais, Y. Favennec, 3D numerical modelling of the propagation of radiative intensity through a X-ray tomographied ligament, Journal of Quantitative Spectroscopy and Radiative Transfer 194 (2017) 86–97. [73] P. Ferkl, R. Pokomy, J. Kosek, Multiphase approach to coupled conduction–radiation heat transfer in reconstructed polymeric foams, International Journal of Thermal Sciences 83 (2014) 68–79. [74] R. Viskanta, E.E. Anderson, Heat transfer in semi-transparent solids, in: Advances in Heat Transfer, vol. 11, Academic Press, New York, 1975, pp. 317–441.

466 Radiative Heat Transfer

[75] P. Coray, J. Petrasch, W. Lipinski, ´ A. Steinfeld, Determination of radiative characteristics of reticulate porous ceramics, in: M.P. Mengüç, N. Selçuk (Eds.), Proceedings of the ICHMT Fifth International Symposium on Radiative Transfer, Bodrum, Turkey, 2007. [76] W. Leupacher, A. Penzkofer, Refractive-index measurement of absorbing condensed media, Applied Optics 23 (10) (1984) 1554–1558. [77] Y. Lu, A. Penzkofer, Optical constants measurements of strongly absorbing media, Applied Optics 25 (1) (1986) 221–225. [78] F. Wooten, Optical Properties of Solids, Academic Press, New York, 1972. [79] V.H. Myers, A. Ono, D.P. DeWitt, A method for measuring optical properties of semitransparent materials at high temperatures, AIAA Journal 24 (2) (1986) 321–326. [80] D.L. Stierwalt, Infrared spectral emittance of optical materials, Applied Optics 5 (12) (1966) 1911–1915. [81] K.M. Shvarev, B.A. Baum, P.V. Gel’d, Optical properties of liquid silicon, Soviet Physics. Solid State 16 (11) (May 1975) 2111–2112. [82] A.J. Barker, A compact, windowless reflectance furnace for infrared studies of corrosive melts, Journal of Physics E: Scientific Instruments 6 (1973) 241–244. [83] J.L. Ebert, S.A. Self, The optical properties of molten coal slag, in: Heat Transfer Phenomena in Radiation, Combustion and Fires, vol. HTD-106, ASME, 1989, pp. 123–126. [84] S.B. Gupta, M.F. Modest, Measurement of infrared absorption coefficient of molten LiF and Li2 S, in: 28th AIAA Thermophysics Conference, Orlando, Florida, 1993, AIAA paper no. 93-2760. [85] T. Makino, M. Edamura, A. Kato, A. Yoshida, Thermal radiation properties of molten salt (properties of alkali metal carbonates), Heat Transfer – Japanese Research 21 (4) (1992) 331–339. [86] Z. Zhang, M.F. Modest, S.P. Bharadwaj, Measurement of infrared absorption coefficients of molten glasses, Experimental Heat Transfer 14 (3) (2001) 145–156. [87] J. Hashempour, A. Sharifian, J. Billingsley, Experimental measurement of direct thermal radiation through single-layer square-cell plain woven screens, ASME Journal of Heat Transfer 138 (2016) 012701. [88] J.-J. Zhao, Y.-Y. Duan, X.-D. Wang, X.-R. Zhang, Y.-H. Han, Y.-B. Gao, Z.-H. Lv, H.-T. Yu, B.-X. Wang, Optical and radiative properties of infrared opacifier particles loaded in silica aerogels for high temperature thermal insulation, International Journal of Thermal Sciences 70 (2013) 54–64. [89] B. Dietrich, T. Fischedick, S. Heissler, P.G. Weidler, C. Wöll, M. Kind, Optical parameters for characterization of thermal radiation in ceramic sponges — experimental results and correlation, International Journal of Heat and Mass Transfer 79 (2014) 655–665. ˇ [90] J. Ondruˇska, I. Medved, V. Koˇcí, R. Cerný, Measurement of the contribution of radiation to the apparent thermal conductivity of fiber reinforced cement composites exposed to elevated temperatures, International Journal of Thermal Sciences 100 (2016) 298–304. [91] J. Kuhn, H.P. Ebert, M.C. Arduini-Schuster, D. Buettner, J. Fricke, Thermal transport in polystyrene and polyurethane foam insulations, International Journal of Heat and Mass Transfer 35 (7) (1992) 1795–1801. [92] V.P. Nicolau, M. Raynaud, J.-F. Sacadura, Spectral radiative properties identification of fiber insulating materials, International Journal of Heat and Mass Transfer 37 (1994) 311–324. [93] D. Doermann, J.-F. Sacadura, Heat transfer in open cell foam insulation, ASME Journal of Heat Transfer 118 (1) (1996) 88–93. [94] D. Baillis, M. Raynaud, J.-F. Sacadura, Spectral radiative properties of open-cell foam insulation, Journal of Thermophysics and Heat Transfer 13 (3) (1999) 292–298. [95] R. Mital, J.P. Gore, R. Viskanta, Measurements of radiative properties of cellular ceramics at high temperatures, Journal of Thermophysics and Heat Transfer 10 (1) (January-March 1996) 33–38. [96] T.J. Hendricks, J.R. Howell, Absorption/scattering coefficients and scattering phase functions in reticulated porous ceramics, ASME Journal of Heat Transfer 118 (1) (1996) 79–87. [97] B. Zeghondy, E. Iacona, J. Taine, Experimental and RDFI calculated radiative properties of a mullite foam, International Journal of Heat and Mass Transfer 49 (2006) 3702–3707. [98] G.R. Cunnington, T.W. Tong, P.S. Swathi, Angular scattering of radiation from coated cylindrical fibers, Journal of Quantitative Spectroscopy and Radiative Transfer 48 (4) (1992) 353–362. [99] G.R. Cunnington, S.C. Lee, Radiative properties of fibrous insulations: theory versus experiments, Journal of Thermophysics and Heat Transfer 10 (3) (1996) 460–466. [100] G.R. Cunnington, S.C. Lee, S.M. White, Radiative properties of fiber-reinforced aerogel: theory versus experiment, Journal of Thermophysics and Heat Transfer 12 (1) (1998) 17–22. [101] S.C. Lee, Radiative transfer through a fibrous medium: allowance for fiber orientation, Journal of Quantitative Spectroscopy and Radiative Transfer 36 (3) (1986) 253–263. [102] S.C. Lee, Radiation heat-transfer model for fibers oriented parallel to diffuse boundaries, Journal of Thermophysics and Heat Transfer 2 (4) (Oct 1988) 303–308. [103] S.C. Lee, Effect of fiber orientation on thermal radiation in fibrous media, International Journal of Heat and Mass Transfer 32 (2) (1989) 311–320. [104] S.C. Lee, Scattering phase function for fibrous media, International Journal of Heat and Mass Transfer 33 (10) (1990) 2183–2190.

Chapter 13

Exact Solutions for One-Dimensional Gray Media 13.1 Introduction The governing equation for radiative transfer of absorbing, emitting, and scattering media was developed in Chapter 9, resulting in an integro-differential equation for radiative intensity in five independent variables (three space coordinates and two direction coordinates). The problem becomes even more complicated if the medium is nongray (which introduces an additional variable, such as wavelength or frequency) and/or if other modes of heat transfer are present (which make it necessary to solve simultaneously for overall conservation of energy, to which intensity is related in a nonlinear way). Consequently, exact analytical solutions exist for only a few extremely simple situations. The simplest case arises when one considers thermal radiation in a one-dimensional plane-parallel gray medium that is either at radiative equilibrium (i.e., radiation is the only mode of heat transfer) or whose temperature field is known. Analytical solutions for such simple problems have been studied extensively, partly because of the great importance of one-dimensional plane-parallel media, partly because the simplicity of such solutions allows testing of more general solution methods, and partly because such a solution can give qualitative indications for more difficult situations. In the present chapter we develop some analytical solutions for one-dimensional plane-parallel media and also include a few solutions for one-dimensional cylindrical and spherical media (without development). In general, we shall assume the medium and bounding walls to be gray,  ∞ and all radiative intensity-related quantities are total, i.e., frequency-integrated quantities, for example, Ib = 0 Ibν dν = n2 σT 4 /π. Most relations also hold, on a spectral basis, for nongray media, except for those that utilize the statement of radiative equilibrium, ∇ · q = 0 (since this relation does not hold on a spectral basis).

13.2 General Formulation for a Plane-Parallel Medium The governing equation for the intensity field in an absorbing, emitting, and scattering medium is, from equation (9.21),  σs sˆ · ∇I = κIb − βI + I(ˆs i ) Φ(ˆs i , sˆ ) dΩ i , (13.1) 4π 4π which describes the change of radiative intensity along a path in the direction of sˆ . The formal solution to equation (13.1) is given by equation (9.28) as  τs  −τs S(τs , sˆ ) e−(τs −τs ) dτs , (13.2) I(r, sˆ ) = Iw (ˆs) e + 0

where S is the radiative source term, equation (9.25), S(τs , sˆ )

= (1 −

ω)Ib (τs )

ω + 4π

 4π

I(τs , sˆ i ) Φ(ˆs, sˆ i ) dΩ i ,

(13.3)

s and τs = 0 β(s) ds is optical thickness or optical depth based on extinction coefficient1 measured from a point on the wall (τs = 0) toward the point under consideration (τs = τs ), in the direction of sˆ . For a plane-parallel medium 1. We use here the notation τs to describe optical depth along s so that we will be able to use the simpler τ for optical depth perpendicular z to the plates, i.e., τ = 0 β dz. Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00021-3 Copyright © 2022 Elsevier Inc. All rights reserved.

467

468 Radiative Heat Transfer

FIGURE 13.1 Coordinates for radiative intensities in a one-dimensional plane-parallel medium: (a) upward directions, (b) downward directions.

the change of intensity is illustrated in Fig. 13.1a, measuring polar angle θ from the direction perpendicular to the plates (z-direction), and azimuthal angle ψ in a plane parallel to the plates (x-y-plane): Radiative intensity of strength Iw (ˆs) = Iw (θ, ψ) leaves the point on the bottom surface into the direction of θ, ψ, toward the point under consideration, P. This intensity is augmented by the radiative source (by emission and by in-scattering, i.e., scattering of intensity from other directions into the direction of P). The amount of energy S(τs , θ, ψ) dτs is released over the infinitesimal optical depth dτs and travels toward P. Since this energy also undergoes  absorption and out-scattering along its path from τs to τs , only the fraction e−(τs −τs ) actually arrives at P. In general, the intensity leaving the bottom wall may vary across the bottom surface, and radiative source and medium properties may vary throughout the medium, i.e., in the directions parallel to the plates as well as normal to them. We shall now assume that both plates are isothermal and isotropic, i.e., neither temperature nor radiative properties vary across each plate and properties may show a directional dependence on polar angle θ, but not on azimuthal angle ψ. Thus, the intensity leaving the bottom plate at a certain location is the same for all azimuthal angles and, indeed, for all positions on that plate; it is a function of polar angle θ alone. We also assume that the temperature field and radiative properties of the medium vary only in the direction perpendicular to the plates. This assumption implies that the radiative source at position Q, S(τ , θ), is identical to the one at position  z Qs , S(τs , θ), or any horizontal position with identical z-coordinate τ = 0 β dz (based on extinction coefficient). Therefore, radiative source, S(τ, θ), and radiative intensity, I(τ, θ), both depend only on a single space coordinate plus a single direction coordinate. The radiative source term may be simplified for the one-dimensional case to ω S(τ , θ) = (1 − ω)Ib (τ ) + 4π 





2π

ψi =0



π

θi =0

I(τ , θi ) Φ (θ, ψ, θi , ψi ) sin θi dθi dψi .

(13.4)

For isotropic scattering, Φ ≡ 1, and we find immediately from the definition for incident radiation, G [equation (9.32)], that S(τ ) = (1 − ω)Ib (τ ) +

ω G(τ ). 4π

(13.5)

In other words, the source term does not depend on direction, that is, the radiative source due to isotropic emission and isotropic in-scattering is also isotropic. If the scattering is anisotropic, we may write, from equation (11.99),2 Φ(ˆs · sˆ i ) = 1 +

M 

Am Pm (ˆs · sˆ i ),

m=1

2. In Chapter 11 we used Θ to denote the angle between the incoming and scattered ray and, therefore, cos Θ = sˆ · sˆ i .

(13.6)

Exact Solutions for One-Dimensional Gray Media Chapter | 13 469

where it is assumed that the series may be truncated after M terms. Measuring the polar angle from the z-axis and the azimuthal angle from the x-axis (in the x-y-plane) for both sˆ and sˆ i , we get the direction vectors ˆ sˆ = sin θ(cos ψî + sin ψˆj) + cos θk,

(13.7)

ˆ sˆ i = sin θi (cos ψi î + sin ψi ˆj) + cos θi k, and Φ(θ, ψ, θi , ψi ) = 1 +

M 

(13.8)

Am Pm [cos θ cos θi + sin θ sin θi cos(ψ − ψi )].

(13.9)

m=1

Using a relationship between Legendre polynomials [1], one may separate the directional dependence in the last relationship by Pm [cos θ cos θi + sin θ sin θi cos(ψ − ψi )] = Pm (cos θ)Pm (cos θi ) + 2

m  (m − n)! n=1

(m + n)!

Pnm (cos θ)Pnm (cos θi ) cos m(ψ − ψi ),

(13.10)

where the Pnm are associated Legendre polynomials. Thus, the scattering phase function may be rewritten as Φ(θ, ψ, θi , ψi ) = 1 +

M 

Am Pm (cos θ)Pm (cos θi ) + 2

m=1

M  m 

Am

m=1 n=1

(m − n)! m P (cos θ)Pnm (cos θi ) cos m(ψ − ψi ). (13.11) (m + n)! n

For a one-dimensional plane-parallel geometry, the intensity does not depend on azimuthal angle, and we may carry out the ψi -integration in equation (13.4). This integration leads to a one-dimensional scattering phase function of  2π M  1 Φ(θ, θi ) = Φ(ˆs · sˆ i ) dψi = 1 + Am Pm (cos θ)Pm (cos θi ), (13.12) 2π 0 since

 2π 0

m=1

cos m(ψ − ψi ) dψi = 0. The radiative source then becomes S(τ , θ) = (1 − ω)Ib (τ ) +

ω 2



π

I(τ , θi ) Φ(θ, θi ) sin θi dθi .

(13.13)

0

For linear-anisotropic scattering, with Φ(ˆs · sˆ i ) = 1 + A1 P1 (ˆs · sˆ i ) = 1 + A1 sˆ · sˆ i ,

M = 1,

(13.14)

and, using the definitions for incident radiation and radiative heat flux, equations (9.32) and (9.55), respectively, equation (13.13) reduces to S(τ , θ) = (1 − ω)Ib (τ ) +

 ω  G(τ ) + A1 q(τ ) cos θ . 4π

(13.15)

We may now simplify the equation of radiative transfer, equation (13.1), using the geometric relations τs = τ/ cos θ and τs = τ / cos θ (see Fig. 13.1a), dI dI ω 1 dI = = (1 − ω)Ib − I + = cos θ β ds dτs dτ 2



π

I(τ, θi ) Φ(θ, θi ) sin θi dθi .

(13.16)

0

Similarly, the expression for intensity, equation (13.2), may be simplified to  I+ (τ, θ) = I1 (θ) e−τ/ cos θ + 0

τ



S(τ , θ) e−(τ−τ )/ cos θ

dτ , cos θ

0 π/2) we obtain (see Fig. 13.1b)  τ dτ  I− (τ, θ) = I2 (θ) e(τL −τ)/ cos θ + S(τ , θ) e(τ −τ)/ cos θ cos θ τ  LτL dτ π  , < θ < π, (13.18) S(τ , θ) e(τ −τ)/ cos θ = I2 (θ) e(τL −τ)/ cos θ − cos θ 2 τ where I2 (θ) is the intensity leaving the wall at τ = τL (Wall 2). It is customary (and somewhat more compact) to rewrite equations (13.16) through (13.18) in terms of the direction cosine μ = cos θ, or μ

ω dI + I = (1 − ω)Ib + dτ 2 



1

−1

I(τ, μi ) Φ(μ, μi ) dμi = S(τ, μ),

(13.19)

τ

dτ  , S(τ , μ) e−(τ−τ )/μ μ 0  τL dτ  I− (τ, μ) = I2 (μ) e(τL −τ)/μ − , S(τ , μ) e(τ −τ)/μ μ τ +

−τ/μ

I (τ, μ) = I1 (μ) e

+

0 0,

(15.69b)

m > 0,

(15.69c)

l=1

where the pm are defined as n, j  m pm n, j = p j,n =

1

0

¯ m ¯ μ, ¯ Pnm (μ)P j (μ)d

(15.70)

and the coefficients um , vm , wm are related to them by li li li um li = vm li = wm li =

− pm pm 2l−1,2i−1 2l+1,2i−1

, 2(4l+1) − π2 (2l−m)pm π2 (2l+m)pm 2l−1,2i−1 2l+1,2i−1 2(4l+1) m (2l+m)p2l−1,2i−1 + (2l−m+1)pm 2l+1,2i−1 (4l+1)

(15.71a) ,

(15.71b)

.

(15.71c) m

In equations (15.69) and (15.71) it is implied that coefficients in front of nonsensical In (i.e., |m| > n) and pm with nj m nonsensical subscripts (n < m) are zero. The pn, j may be determined through recursion relationships [18] and are listed in Table 15.2 (scaled by a factor of 10−m ) for up to the P5 -approximation.

532 Radiative Heat Transfer

TABLE 15.2 Half-moments of associated Legendre polynomials, 10−m × pm . n,j m

j

0

n

0

1

2

3

4 5

1 .

2

3

4

5

.

.

.

.

.

.

.

.

0

1.00000

1

0.50000

0.33333

2

0.00000

0.12500

0.20000

3

−0.12500

0.00000

0.12500

0.14286

4

0.00000

−0.02083

0.00000

0.07031

0.11111

5

0.06250

0.00000

−0.03906

0.00000

0.07031

.

.

.

.

.

.

1

.

0.06667

2

.

0.07500

0.12000

3

.

0.00000

0.07500

0.17143

4

.

−0.04167

0.00000

0.14062

0.22222

5

.

0.00000

−0.02344

0.00000

0.14062

2

.

.

0.04800

3

.

.

0.07500

0.17143

4

.

.

0.00000

0.14062

0.40000

5

.

.

−0.06563

0.00000

0.39375

3

.

.

.

0.10286

4

.

.

.

0.19687

0.56000

5

.

.

.

0.00000

0.55125

4

.

.

.

.

0.44800

5

.

.

.

.

0.99225

5

.

.

.

.

. 0.09091

.

.

.

.

.

.

.

.

. . 0.27273

.

.

.

.

.

.

. 0.76364 . . 1.83273 . 3.29891 3.29891

m

It remains to rotate the In in equations (15.69) to global values Inm , which results in ⎧ N−1 ⎫ ⎪ ⎪ 2l 2 ⎪ ⎪   ⎪ ∂ ⎪ ⎨ 0 2l m 0 ¯ 2l m ⎬ ¯ : p2l,2i−1 Δ0,m I2l + vli Δ1,m I2l ⎪ ⎪ ⎪ ⎪ ⎪ ∂τx ⎪ ⎩ l=1 m =−2l ⎭ l=0 m =−2l ⎧ N−1 ⎫ ⎧ N−1 ⎫ ⎪ ⎪ ⎪ ⎪ 2l 2l 2 2 ⎪ ⎪ ⎪    ⎪ ⎪ ∂ ⎪ ∂ ⎪ ⎪ ⎨ ⎬ ⎨ 0 ¯ 2l m 0 ¯ 2l m ⎬ + − = Iw p00,2i−1 , m = 0, v I w I Δ Δ ⎪ ⎪ ⎪   ⎪ ⎪ li −1,m 2l ⎪ li 0,m 2l ⎪ ⎪ ⎪ ⎪ ∂τ y ⎪ ⎩ l=1 m =−2l ⎭ ∂τz ⎪ ⎩ l=0 m =−2l ⎭ ⎧ N−1 ⎫ N−1 ⎪ ⎪ 2l 2l  2 2 ⎪ ⎪      ⎪ ⎪ m m ¯ 2l m ¯ 2l m ⎬ ¯ 2l  I m − ∂ ⎨ Y2i−1 : pm )u − v Δ Δ Δ (1+δ I ⎪ ⎪   m,1 li m−1,m m+1,m 2l 2l,2i−1 m,m 2l li ⎪ ⎪ ⎪ ∂τx ⎪ ⎩ l=0 m =−2l ⎭ l=1 m =−2l ⎧ N−1 ⎫ ⎧ N−1 ⎫ ⎪ ⎪ ⎪ ⎪ 2l  2l 2 2 ⎪ ⎪ ⎪ ⎪      ⎪ ⎪ ⎪ ⎪ ∂ ⎨ ∂ ⎨ m ¯ 2l m ¯ 2l m ⎬ m ¯ 2l m ⎬ − = 0, + )u + v w I Δ Δ Δ (1−δ I ⎪ ⎪ ⎪  m,1 li −(m−1),m m,m 2l ⎪ −(m+1),m li li 2l ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂τ y ⎪ ⎩ l=1 m =−2l ⎭ ∂τz ⎪ ⎩ l=1 m =−2l ⎭ N−1

0 Y2i−1

−m

Y2i−1

2l 2  



⎧ N−1 ⎫ ⎪ ⎪ 2l 2l  2 ⎪ ⎪      ⎪ ⎪ ∂   ⎨ m ¯ 2l m ¯ 2l m ⎬ ¯ 2l  I m − : pm )u − v Δ Δ Δ (1−δ I ⎪ ⎪   m,1 −(m−1),m −(m+1),m 2l,2i−1 −m,m 2l li li 2l ⎪ ⎪ ⎪ ∂τx ⎪ ⎩ l=1 m =−2l ⎭ l=1 m =−2l ⎧ N−1 ⎫ ⎧ N−1 ⎫ ⎪ ⎪ ⎪ ⎪ 2l  2l 2 2 ⎪ ⎪ ⎪ ⎪      ⎪ ⎪ ⎪ ⎪ ∂ ⎨ ∂ ⎨ m ¯ 2l m ¯ 2l m ⎬ m ¯ 2l m ⎬ − = 0, − (1+δ I )u + v w I Δ Δ Δ  ⎪ ⎪ ⎪  m,1 li m−1,m −m,m 2l ⎪ m+1,m li li 2l ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂τ y ⎪ ⎩ l=0 m =−2l ⎭ ∂τz ⎪ ⎩ l=1 m =−2l ⎭

(15.72a)

m > 0, (15.72b)

N−1 2

m > 0. (15.72c)

The Method of Spherical Harmonics (PN -Approximation) Chapter | 15 533

Equations (15.72) are a set of (N+2)(N+1)/2 boundary conditions for N(N+1)/2 variables I2lm (l = 0, 1, ..., (N−1)/2; m = −2l, ..., +2l), containing normal as well as tangential derivatives, or N + 1 too many. Commercial PDE solvers generally allow for boundary conditions containing normal derivatives. In principle, i.e., if the coefficients in front of the I2lm inside the normal derivatives form a nonsingular matrix, linear combination of the boundary conditions leads to a set of “natural” boundary conditions for each variable, or ⎞ ⎛   ∂I2lm ⎟⎟ ⎜⎜ m ∂I2lm ∂I2lm     1 l = 0, ..., 12 (N−1), m = −2l, ..., +2l, = f ⎜⎜⎝I2l , , ; l = 0, ... 2 (N−1); m = −2l , ..., +2l ⎟⎟⎠ , (15.73) ∂τz ∂τx ∂τ y which can be used with FlexPDE [55] and other commercial programs. Modest [18] has shown that such a nonsingular matrix can be found only if, for the largest value of i = 12 (N + 1), only the even values of m are employed (omitting the N+1 odd values). Therefore, the qualifier “all relevant m” in equations (15.68), (15.69), and (15.72) may be restated precisely as ⎧ ⎪ 1 ⎪ ⎨i = 1, 2, ..., 2 (N − 1), all m, All relevant m = ⎪ (15.74) ⎪ ⎩i = 1 (N + 1), all even m, 2

which supplies a consistent set of N(N + 1)/2 boundary conditions for an equal number of variables. Other codes, such as PDE2D [56] or FDEM [57], use derivatives in global coordinates in the boundary conditions. In that case, the transformation to global Inm using equation (15.67) is carried out first, followed by elimination of odd orders. The resulting boundary conditions are given in [13]. Example 15.6. Determine the necessary boundary conditions for the problem of Example 15.4 for the surface location indicated in Fig. 15.6. The surface is black and at temperature Tw . Solution The boundary conditions are usually expressed in terms of local coordinates (i.e., in terms of gradients into the surface m normal and tangential directions), either using local spherical harmonics In , equation (15.69), followed by rotation to m global spherical harmonics In , or by directly applying equation (15.72). We will follow the first track here. With local azimuthal angle ψ defined from the x-axis in the x–y–plane, for this two-dimensional problem independent of y we must m have I(θ, ψ) = I(θ, −ψ) and, therefore, all In with negative m vanish. Thus, from equation (15.69), eliminating all terms with negative m and y-gradients, we obtain ' ) ' ) 0 0 0 1 0 0 ∂ ∂ v011 I2 − w001 I0 + w011 I2 = Ibw p001 , Y1 : p001 I0 + p021 I2 + ∂τx ∂τz ' ) ' ) 1 1 ∂ ∂ 1 1 1 0 1 0 1 2 2u01 I0 + 2u11 I2 − v11 I2 − w111 I2 = 0, Y1 : p21 I2 − ∂τx ∂τz ' ) ' ) 0 0 0 1 0 0 ∂ ∂ v0 I − w0 I + w012 I2 = Ibw p003 , Y3 : p003 I0 + p023 I2 + ∂τx 12 2 ∂τz 02 0 ' ) ' ) 2 2 1 2 ∂ ∂ u212 I2 − w212 I2 = 0. Y3 : p223 I2 − ∂τx ∂τz −1

−2

m

The equations for Y1 and Y3 contain only In with negative m and, thus, vanish identically, leaving us with the proper m , um , vm , and wm are found from Table 15.2 [or, four boundary conditions for the four unknown In . The coefficients pm nj li li li more easily from program pnbcs.f90 in Appendix F] as 1 1 3 1 1 15 , p021 = , p121 = , p003 = − , p023 = , p223 = ; 2 8 4 8 8 2     p1 −p131 p2 −p233 −p111 1 1 2 1 1 120 12 u101 = = − , u111 = 11 = −0 = , u212 = 13 = 0− =− ; 2·1 3 2·5 10 3 15 2·5 10 7 7     0 0 1 1 − 2 · 3p 2 · 3p − 1 · 2p 3 · 4p 3 1 1 1 2 4 11 31 11 31 = − 0 = , v111 = = 12 × − 0 = , v011 = 2·5 5 3 5 2·5 10 3 5   0 0 − 2 · 3p 2 · 3p 3 1 3 33 13 = 0− =− ; v012 = 2·5 5 7 35     0 0 0 + 3p 2p 3p1 + 2p131 1 · p 1 1 2 2 1 2 2 31 11 = , w011 = 11 = −0 = , w111 = 11 = 3× +0 = , w001 = 1 3 5 5 3 15 5 5 3 5 p001 =

534 Radiative Heat Transfer

w002 =

1 · p013 1

= 0, w012 =

2p013 + 3p033 5

=

    4p2 + 1 · p233 1 3 3 1 120 24 0+ = , w212 = 13 = 0+ = . 5 7 35 5 5 7 7

Therefore, after normalization with the leading term, 1

0

Y1 : 1

Y1 : 0 Y3 2

:

Y3 :

0 1 0 2 ∂I2 I0 + I2 + − 4 5 ∂τx ' ) 1 ∂ 8 0 8 0 16 2 I2 + I0 − I2 + I2 − ∂τx 9 45 15 0 I0

−

0

0

2 ∂I0 4 ∂I2 − = Ibw , 3 ∂τz 15 ∂τz 1

8 ∂I2 15 ∂τz

1

0 I2

24 ∂I2 + 35 ∂τx

24 ∂I2 + 35 ∂τz

2

1 ∂I2

2 ∂I2

I2 +

8 35 ∂τx

(15.75a)

= 0,

(15.75b)

= Ibw ,

(15.75c)

= 0.

(15.75d)

0

−

16 35 ∂τz

0

m

Next, the local In must be converted to global Inm with equation (15.67). For n = 0 this simply gives I0 = I00 , i.e., I00 is nondirectional and does not vary with rotation, and we will drop the unnecessary superscript from I0 . Remembering that, in global coordinates, Inm with odd m vanish (as opposed to negative m in local coordinates), for n = 2 this leads to 0 ¯ 2 I −2 + Δ ¯ 2 I0 + Δ ¯ 2 I 2, I2 = Δ 0,−2 2 0,0 2 0,2 2 1

¯ 2 I −2 + Δ ¯ 2 I0 + Δ ¯ 2 I 2, I2 = Δ 1,−2 2 1,0 2 1,2 2 2 ¯ 2 I −2 + Δ ¯ 2 I0 + Δ ¯ 2 I 2. I2 = Δ 2,−2 2 2,0 2 2,2 2

¯ 2  (−γ = − π , −β = − 3π , −α = π − δ) are determined via backward rotation from equation (15.64) with The necessary Δ m,m 2 2 2 ⎧ ⎪ ⎪ −1, ⎪ ⎪ ⎪ ⎪ ⎪ 0,  ⎪  ⎪ π ⎨ =⎪ Ψm − 1, ⎪ ⎪ 2 ⎪ ⎪ ⎪ −1, ⎪ ⎪ ⎪ ⎩ 0, β

m=2 1 0 , −1 −2

⎧ ⎪ ⎪ − cos 2δ, m = 2 ⎪ ⎪ ⎪ ⎪ ⎪ sin δ, 1  ⎪  ⎪ π ⎨ Ψm −δ =⎪ 1, 0 , ⎪ ⎪ 2 ⎪ ⎪ ⎪ cos δ, −1 ⎪ ⎪ ⎪ ⎩ sin 2δ, −2

β

2 ) = − √12 . The dmm and cos( 2 ) = sin( 2 ) = cos(− 3π  follow from equation (15.66) after some painful algebra (or, more easily, 4 by manipulating program Delta.f90 in Appendix F). Finally, 0 1 I2 = −3 sin 2δ I2−2 − I20 − 3 cos 2δ I22 , 2 1

I2 = −2 cos 2δ I2−2 + 2 sin 2δ I22 , 2

I2 =

1 1 1 sin 2δ I2−2 − I20 + cos 2δ I22 . 2 4 2

Sticking this into equation (15.75) delivers the desired local boundary conditions as 0

Y1 :

1

Y1 :

0

Y3 :

 3 1 3 4 ∂  cos 2δ I2−2 − sin 2δ I22 sin 2δ I2−2 − I20 − cos 2δ I22 − 4 8 4 5 ∂τx  2 ∂  5I0 − 6 sin 2δ I2−2 − I20 − 6 cos 2δ I22 = Iw , − 15 ∂τz  8 ∂  −2 2 5I0 − I2 + 6 sin 2δ I2−2 + 6 cos 2δ I22 −2 cos 2δ I2 + 2 sin 2δ I2 + 45 ∂τx  48 ∂  + cos 2δ I2−2 − sin 2δ I22 = 0, 45 ∂τz  ∂  1 48 cos 2δ I2−2 − sin 2δ I22 I0 + 3 sin 2δ I2−2 + I20 + 3 cos 2δ I22 − 2 35 ∂τx ' ) 24 ∂ 1 − 3 sin 2δ I2−2 + I20 + 3 cos 2δ I22 = Iw , 35 ∂τz 2 I0 −

The Method of Spherical Harmonics (PN -Approximation) Chapter | 15 535

2

Y3 :

 1 1 1 16 ∂  sin 2δ I2−2 − I20 + cos 2δ I22 − cos 2δ I2−2 − sin 2δ I22 2 4 2 35 ∂τx  4 ∂  − 2 sin 2δ I2−2 − I20 + 2 cos 2δ I22 = 0. 35 ∂τz

Once all Inm for even n have been determined, the remaining Inm (odd n) may be determined from relations given in Modest and Yang [13]. Normally, only incident radiation G = 4πI0 and radiative flux are of interest, the latter being related to the I1m : comparing equations (15.24), (15.25), and (15.31) and noting that higher-order terms drop out because of the orthogonality of spherical harmonics [14], leads to ⎛ ⎞ ⎜⎜ −I 1 ⎟⎟  ⎜ 1⎟ 4π ⎜⎜⎜ −1 ⎟⎟⎟ ⎜⎜−I ⎟⎟ , (15.76) q(r) = I(r, sˆ ) sˆ dΩ = 3 ⎜⎜⎜ 1 ⎟⎟⎟ 4π ⎝ 0 ⎠ I1 where the I1m are given by [13] I10 = −

0 1 −1 ∂I0 2 ∂I2 3 ∂I2 3 ∂I2 − + + , ∂τz 5 ∂τz 5 ∂τx 5 ∂τ y

(15.77a)

I11 = +

0 1 2 −2 ∂I0 1 ∂I2 3 ∂I2 6 ∂I2 6 ∂I2 − − + + , ∂τx 5 ∂τx 5 ∂τz 5 ∂τx 5 ∂τ y

(15.77b)

0 −1 2 −2 ∂I0 1 ∂I2 3 ∂I2 6 ∂I2 6 ∂I2 − − − + . ∂τ y 5 ∂τ y 5 ∂τz 5 ∂τ y 5 ∂τx

(15.77c)

I1−1 = +

Since equation (15.1) is valid for any coordinate system orientation, equations (15.76) and (15.77) are valid for m both the global coordinate system (x-y-z, Inm ) as well as a local coordinate system at a boundary (x-y-z, In ). Finally, for nonblack surfaces the boundary radiosity Jw = πIw must be related to the wall’s emissive power and/or net radiative flux. From equations (15.1) and (15.76) we have qn =

π 4π 0 [Ibw − Iw ] = I , 1−

3 1

(15.78)

0

where is the surface’s emittance, and with I1 transformed to global I1m through equation (15.67). If the temperature of the surface, Tw , is specified, Iw is determined from   0 4 1 − 1 I1 . (15.79) Iw = Ibw − 3

For three-dimensional geometries, it is obvious that anything but low-order approximations quickly become extremely cumbersome to deal with. Already the P3 -approximation may result in as many as six simultaneous partial differential equations (depending on the symmetry), and it includes cross-derivatives, which do not ordinarily occur in engineering problems (and which complicate numerical solutions). In addition, complicated boundary conditions need to be developed from equation (15.72). As a result of this complexity, very few multidimensional problems have been solved by the P3 -approximation, and apparently none by higher orders. First results using the new elliptic formulation of equations (15.53) and (15.72) have been reported by Modest and coworkers [13,18,52]. We shall limit ourselves here to a simple example for a one-dimensional plane-parallel slab. Example 15.7. Consider an isothermal medium at temperature T, confined between two large, parallel black plates that are isothermal at the (same) temperature Tw . The medium is gray and absorbs and emits, but does not scatter. Determine an expression for the heat transfer rates within the medium using the P3 -approximation. Employ the results from the previous three examples. Solution For such a one-dimensional problem it is, generally, advantageous to choose τz as the (nondimensional) space coordinate between the plates, as was done in Example 15.2, since this will make all Inm vanish with m  0. However, for demonstrative

536 Radiative Heat Transfer

purposes, and to utilize results from the previous three examples, we will choose the global coordinate system of Fig. 15.6, i.e., the problem becomes one-dimensional in the y-direction, with the bottom surface corresponding to δ = 0, and the top to δ = π. Since now we have no x-dependence we must have I(θ, ψ) = I(θ, π − ψ), which implies that we will not have any odd positive or even negative m terms in equation (15.55a). Together with n + m = even (no z-dependence) that reduces the set of equations developed in Example 15.4 to Y00 : Y20 : Y22 :

 d2 2 2 I + dτ2y 5 2  d2 4 2 I + dτ2y 7 2  d2 3 2 I + 2 7 2 dτ y

 1 0 1 I2 − I0 +I0 = Ib , 15 3  5 0 1 I2 − I0 −I20 = 0, 21 3  1 0 1 I2 − I0 −I22 = 0, 21 6

and all terms vanish for the Y2−2 -equation, i.e., we now have three equations in three unknowns (since I2−2 = 0). To exploit the symmetry of the problem, we choose the origin for τ y to be at the midpoint between the two plates. Then the first derivatives of all three unknowns will be zero at the midpoint: dI 0 dI 2 dI0 = 2 = 2 = 0. dτ y dτ y dτ y

τy = 0 :

The necessary second set of boundary conditions follows from Example 15.6 with δ = 0 at τ y = −τL /2 (and τL is the total optical thickness of the medium) as 0

Y1 : 0

Y3 : 2

Y3 :

1 3 2 d I0 − I20 − I22 − 8 4 15 dτ y 1 12 d I0 + I20 + 3I22 − 2 35 dτ y 1 1 4 d − I20 + I22 + 4 2 35 dτ y

  

 5I0 − I20 − 6I22 = Ibw , I20 + 6I22 I20 − 2I22

 

= Ibw , = 0,

1

with all terms in the Y1 boundary condition vanishing. While the given set of three simultaneous ordinary differential equations in I0 , I20 , and I22 , together with their boundary conditions, can be solved as they are, we do know from Section 15.3 that, for a one-dimensional problem, there should be only a single Inm for every n (i.e., In0 ). Inspecting the governing equations and boundary conditions, we find that I20 and I22 always occur in one of two combinations, viz.  1 0 I2 + 6I22 , 2 K2 = I20 − 2I22 , I2 = −

where the factor − 12 was included for convenience (i.e., I2 just so happens to be I20 for the case that the z-axis points from plate to plate). Then 2  I − 15 2 11 − I2 − 21

−

Y00 : Y20 + 6Y22 : Y20 − 2Y22 :

1  I + I 0 = Ib , 3 0 2  I + I2 = 0, 3 0 1  K − K2 = 0, 7 2

where the primes have been introduced as shorthand for d/dτ y . The boundary conditions at τ y = −τL /2 follow as 0

Y1 : 0

Y3 : 2

Y3 :

1 2 4 I0 + I2 − I0 − I2 = Ibw , 4 3 15 24 I0 − I 2 + I2 = Ibw , 35 1 4 K2 + K2 = 0. 2 35

The Method of Spherical Harmonics (PN -Approximation) Chapter | 15 537

It follows that K2 ≡ 0, since both its governing equation and its boundary conditions are homogeneous. I2 can be eliminated from the remaining equations: first we eliminate I2 from the first two equations, leading to − or

9  14 I + I 0 − I 2 = Ib , 55 0 55

I2 = −

9  55 (I0 − Ib ) . I + 14 0 14

Differentiating twice and eliminating I2 from the Y00 equation, we obtain 3 (iv) 6  I − I 0 + I 0 = Ib . 35 0 7 The general solution to the above equation (keeping in mind that Ib = const) is I0 (τ y ) = Ib + (Ibw − Ib )[C1 cosh λ1 τ y + C2 cosh λ2 τ y + C3 sinh λ1 τ y + C4 sinh λ2 τ y ], where the constant factor (Ibw − Ib ) was included to make the Ci dimensionless. The λ1 and λ2 are the positive roots of the equation 3 4 6 2 λ − λ + 1 = 0, 35 7 or λ1 = 1.1613 and λ2 = 2.9413. With τ y = 0 placed at the midpoint between the two plates I0 (0) = I0 (0) = 0 and C3 = C4 = 0. The two needed boundary conditions at one of the plates, say at τ = −τL /2, are found by again eliminating I2 , or ' ' ) ) 0 9 9 1 55 4 55 2 (I0 − Ib ) − I0 − − I0 + − I0 + I0 = Ibw , Y1 : I0 + 4 14 14 3 15 14 14 ' ) ' ) 0 9  55  9  55 24 (I0 − Ib ) − I0 + I0 = Ibw , Y3 : I0 − − I0 + + 14 14 35 14 14 leading to Ibw − Ib =

111 12 9 6 (I0 − Ib ) − I0 − I0 + I0 , 56 7 56 35

Ibw − Ib = −

41 132  9 108  (I0 − Ib ) + I + I − I . 14 49 0 14 0 245 0

Now, substituting the solution for I0 into these boundary conditions leads to 1 = a1 C1 + a2 C2 = b1 C1 + b2 C2 , where

   9 12 τL τL 111 6 − λ2i cosh λi + λi − λ3i sinh λi , i = 1, 2, 56 56 2 7 35 2     9 2 132 τL τL 41 108 3 bi = − − λ cosh λi − λi − λ sinh λi , i = 1, 2. 14 14 i 2 49 245 i 2 ai =



Finally, we get C1 =

b2 − a2 , a1 b2 − a2 b1

C2 =

a1 − b1 . a1 b2 − a2 b1

The heat flux through the medium is determined from equations (15.76) and (15.77) as q(τ y ) = −

    0 4π −1 4π ∂I0 1 ∂I2 6 ∂I22 2 ∂I2 4π ∂I0 I1 = − − − + =− . 3 3 ∂τ y 5 ∂τ y 5 ∂τ y 3 ∂τ y 5 ∂τ y

Substituting for I2 we obtain q(τ y ) = −

  4π  9 11 I0 − I0 + I0 , 3 35 7

538 Radiative Heat Transfer

FIGURE 15.8 Nondimensional wall heat fluxes for an isothermal slab; comparison of P1 - and P3 -approximations with the exact solution.

and the heat flux may be expressed in nondimensional form as Ψ=

q(τ y ) n2 σ(Tw4

− T 4)

=−

2   12  12 10I0 − I0 =− (10λi − λ3i )Ci sinh λi τ y , 35 Ibw − Ib 35 i=1

where, for simplicity, it was assumed that the medium is gray, or Ib = n2 σT 4 /π. The nondimensional heat flux at the top surface (τ y = τL /2) is shown in Fig. 15.8, as a function of optical depth of the slab. The results are compared with those of the P1 - or differential approximation (Example 15.2), and with the exact result, Ψ = 1 − 2E3 (τL ), which is readily found from equation (13.35). For this particular example the P1 -approximation is very accurate (maximum error ∼15%) and, as to be expected, the P3 -approximation performs even better (maximum error ∼7%).

It should be clear from the above example that P3 - and higher-order PN -approximations quickly become very tedious, even for simple geometries. However, P3 results can be substantially more accurate than P1 results, particularly in optically thin media and/or geometries with large aspect ratios. Another example, shown in Fig. 15.5, depicts nondimensional heat flux through a gray, nonscattering medium at radiative equilibrium, confined between infinitely long, concentric, black and isothermal cylinders, in which the P3 -solution of Bayazitoglu ˘ and Higenyi [24] is compared with the P1 -solution (Example 15.3). Observe that the P3 -approximation introduces roughly half the error of the P1 -method, which appears to be approximately true for all problems. One outstanding advantage of the P3 -method is that, once the problem has been formulated (setting up the governing equations suitable for a numerical solution), the increase in computer time required (compared with the P1 -method) is relatively minor. In addition, P3 -calculations are also usually very grid-compatible with conduction/convection calculations, if one must account for combined modes of heat transfer. Ravishankar et al. [58] presented a formulation for solving the P3 equations developed in [13] using the finite volume method on a two-dimensional unstructured (triangular) mesh. The four governing PDEs were solved in a coupled mannerand, and the system was embedded into a reacting flow solver for a laminar flame calculation. Ge and coworkers [59] have incorporated the set of equations (15.53), and their boundary conditions (15.72), into the OpenFOAM open source CFD solver [60], which has gained a lot of popularity during the past few years. Their implementation includes levels up to P7 , and the coupled N(N +1)/2 simultaneous PDEs and their boundary conditions are solved iteratively by the preconditioned conjugate gradient (PCG) algorithm [61]. In this method each of the PDEs is solved sequentially until I00 has converged to prescribed criteria. They also generated a 2D axisymmetric version of the PN system, reducing the number of unknowns and equations to (N+1)2 /4 [62]. This was also implemented in OpenFOAM up to P7 . Finally, Ge and coworkers [63] also formulated a general 2D Cartesian slab version of the PN system, while extending boundary conditions to include mixed diffuse–specular

The Method of Spherical Harmonics (PN -Approximation) Chapter | 15 539

reflecting walls. This was also incorporated into OpenFOAM up to P7 . In all cases it was found that the PCG algorithm increases CPU times substantially more than linearly, i.e., the effort for P7 may be several hundred times that of P1 . However, no more efficient implementation for higher order PN appears to be available yet. Several additional two-dimensional examples will be presented in the final section of this chapter, comparing results from different orders and different schemes of the spherical harmonics method.

15.7 Simplified PN -Approximation As noted in the previous section, higher-order PN -formulations for anything but one-dimensional slabs become extremely cumbersome mathematically, and they also introduce cross-derivatives, which make a numerical solution considerably more involved. Facing these mathematical difficulties Gelbard [5] introduced the Simplified PN -Approximation some 50 years ago, as an intuitive three-dimensional extension to the one-dimensional slab PN -formulation, equation (15.14), and its Marshak boundary conditions, equations (15.21). Gelbard formulated his set of simplified-PN or SPN equations, such that they reduced to the standard PN -approximation for a onedimensional slab and some other narrow circumstances, but the method lacked any theoretical foundation, which impeded its acceptance. Theoretical justifications were found many years later by Larsen et al. [64] (showing SPN to be an asymptotic correction to the diffusion approximation of Section 14.2) and by Pomraning [65] (showing the SPN to be asymptotically related to the PN -equations for the slab geometry). A fine review of the SPN -method has recently been given by McClarren [66]. While the developments of Larsen and Pomraning provide theoretical credentials to the method, they are rather tedious, and we will here only provide the intuitive development of Gelbard, further developed for radiative heat transfer applications by Modest [67]. Depending on whether k is odd or even, Gelbard made the following substitutions in equations (15.14) and (15.21): k odd :

Ik (τ) → Ik (τx , τ y , τz ),

k even :

Ik (τ) → Ik (τx , τ y , τz ),

dIk → ∇τ · Ik , dτ dIk → ∇τ Ik , Ik = dτ Ik =

(15.80a) (15.80b)

i.e., for every odd k the Ik becomes a vector and differentiation is replaced by the divergence operator, while even Ik remain scalars and their differentiation is replaced by the gradient operator. Substituting equations (15.80) into equation (15.14) leads to k = 0, 2, . . . , N − 1 (even) :

k k+1 ∇τ · Ik+1 + ∇τ · Ik−1 + αk Ik = αk Ib δ0k , 2k + 3 2k − 1

(15.81a)

k = 1, 3, . . . , N

k+1 k ∇τ Ik+1 + ∇τ Ik−1 + αk Ik = 0, 2k + 3 2k − 1

(15.81b)

(odd) :

where αk = 1 −

ωAk . 2k + 1

(15.81c)

Solving equation (15.81b) for Ik and substituting the result into (15.81a) produces a set of simultaneous elliptic partial differential equations in the unknown scalars Ik (k even): k = 0, 2, . . . , N − 1 (even) :

    (k + 1)(k + 2) (k + 1)2 1 1 ∇τ · ∇τ · ∇τ Ik+2 + ∇τ Ik (2k + 3)(2k + 5) αk+1 (2k + 3)(2k + 1) αk+1     k(k − 1) 1 1 k2 + ∇τ · ∇τ · ∇τ Ik + ∇τ Ik−2 = αk (Ik − Ib δ0k ). (2k − 1)(2k + 1) αk−1 (2k − 1)(2k − 3) αk−1

(15.82)

540 Radiative Heat Transfer

Similarly, sticking equations (15.80) into the PN boundary conditions, equations (15.21), gives us a consistent set of conditions for the SPN -equations: N−1 

1 Pk (μ)P2i−1 (μ)dμ +

Ik

k even

N 

1 nˆ · Ik

k odd

0

0

Jw Pk (μ)P2i−1 (μ)dμ = π

1 P2i−1 (μ)dμ,

i = 1, 2, . . . , 12 (N + 1),

(15.83)

0

given by equation (15.70), or, with the definition of the Legendre polynomial half-moments pm n, j N−1 

p0k,2i−1 Ik +

k even

N 

p0k,2i−1 nˆ · Ik =

p00,2i−1

k odd

π

Jw ,

i = 1, 2, . . . , 12 (N + 1).

(15.84)

Again, eliminating the odd Ik with equation (15.81b), this set of boundary conditions reduces to N−1  k even

p0k,2i−1 Ik

−

 N p0  k,2i−1 k odd

αk

 p00,2i−1 k k+1 nˆ · ∇τ Ik−1 + nˆ · ∇τ Ik+1 = Jw , 2k − 1 2k + 3 π

i = 1, 2, . . . , 12 (N + 1).

(15.85)

No direct formula for intensity is derived, but one may assume a series of the form I(r, sˆ ) = I0 (r) + I1 (r) · sˆ + I2 (r)P02 (ˆs) + . . . ,

(15.86)

which is no longer a complete series of orthogonal functions and, therefore, is not guaranteed to approach the exact answer in the limit. However, assuming this to be an orthogonal set, we can obtain incident radiation G and radiative flux q from their definitions as  G(r) = I(r, sˆ ) dΩ = 4πI0 (r), (15.87) 4π ' ) 4π 4π 2 q(r) = I1 (r) = − ∇τ I0 + ∇τ I2 . (15.88) I(r, sˆ ) sˆ dΩ = 3 3α1 5 4π While equations (15.82) and (15.85) form a self-consistent set of (N + 1)/2 simultaneous elliptic partial differential equations and their boundary conditions, the problem can be further simplified by recognizing that the combination of variables Jk =

k+1 k+2 Ik + Ik+2 2k + 1 2k + 5

(15.89)

appears repeatedly in both the governing equations and boundary conditions. In addition, inspection of Table 15.2 shows that p0n, j = 0 if n + j = even, with the exception of n = j. Thus we may rewrite equations (15.82) as k = 0, 2, . . . , N − 1 (even) :     k+1 1 1 k ∇τ · ∇τ · ∇τ Jk + ∇τ Jk−2 = αk (Ik − Ib δ0k ), 2k + 3 αk+1 2k − 1 αk−1

(15.90)

and boundary conditions (15.85) as p02i−1,2i−1 α2i−1

N−1

nˆ · ∇τ J2i−2 =

2 

k=0

p02k,2i−1 I2k −

p00,2i−1 π

Jw ,

i = 1, 2, . . . , 12 (N + 1).

(15.91)

The Ik on the right-hand sides may be eliminated by inverting equation (15.89), starting with k = N − 1 (and noting that IN+1 ≡ 0). This results in individual partial differential equations for each Jk , in which Jl (l  k) occur only as source terms without derivatives. Once the Jk have been determined, incident radiation and radiative

The Method of Spherical Harmonics (PN -Approximation) Chapter | 15 541

flux are obtained from equations (15.87) and (15.88) as ' ) 2 8 J4 (r) − + . . . , G(r) = 4π J0 (r) − J2 (r) + 3 15 4π ∇τ J0 (r). q(r) = − 3α1

(15.92) (15.93)

We will demonstrate this by looking in more detail at the SP1 - and SP3 -approximations (even orders, such as SP2 , have also been formulated [68], but—based on the development shown here—appear to be as inappropriate as for the standard PN -method).

SP1 -Approximation With N = 1 we obtain a single equation and a single boundary condition from equations (15.90) and (15.91), i.e.: Governing equation:   1 1 k=0: ∇τ · ∇τ J0 = α0 (I0 − Ib ); (15.94) 3 α1 Boundary condition: p01,1

i=1: With p00,1 =

1 2

and p01,1 =

1 3

α1

nˆ · ∇τ J0 = p00,1 (I0 − Jw /π).

(15.95)

from Table 15.2, and I0 = J0 from equation (15.89), we obtain   1 1 ∇τ · ∇τ J0 = α0 (J0 − Ib ), 3 α1

(15.96)

with boundary condition 1 1 nˆ · ∇τ J0 = (J0 − Jw /π). 3α1 2

(15.97)

Not surprisingly, comparison with equations (15.38) and (15.48) and using G = 4πI0 = 4πJ0 shows that the SP1 -approximation is identical to the P1 -method.

SP3 -Approximation Setting N = 3 we get two simultaneous equations and two boundary conditions: Governing equations: k=0: k=2:

    1 1 2 ∇τ · ∇τ J0 = α0 (I0 − Ib ) = α0 J0 − J2 − Ib , 3 α1 3     3 2 1 1 5 ∇τ · ∇τ J2 + ∇τ · ∇τ J0 = α2 I2 = α2 J2 , 7 α3 3 α1 3

(15.98a) (15.98b)

or, subtracting 2 × equation (15.98a), k=2:

    5 1 4 3 ∇τ · ∇τ J2 = α2 + α0 J2 − 2α0 (J0 − Ib ). 7 α3 3 3

(15.98c)

Boundary conditions: i=1: i=2:

p01,1 α1 p03,3 α3

nˆ · ∇τ J0 = p00,1 (I0 − Jw /π) + p02,1 I2 ,

(15.99a)

nˆ · ∇τ J2 = p00,3 (I0 − Jw /π) + p02,3 I2 .

(15.99b)

542 Radiative Heat Transfer

With p02,1 = p02,3 = 18 , p03,3 = 17 , p00,3 = − 18 , and eliminating the Ik , the boundary conditions become i=1: i=2:

1 nˆ · ∇τ J0 = 12 (J0 − 23 J2 − Jw /π) + 18 53 J2 = 12 (J0 − Jw /π) − 18 J2 , 3α1 1 7 nˆ · ∇τ J2 = − 18 (J0 − 23 J2 − Jw /π) + 18 53 J2 = − 18 (J0 − Jw /π) + 24 J2 . 7α3

(15.99c) (15.99d)

Unlike the regular P3 -approximation, SP3 has only two, and nearly separated, elliptic partial differential equations: equations (15.98a) and (15.99c) for J0 and equations (15.98c) and (15.99d) for J2 , the only connection being the other Jk appearing in source terms. Example 15.8. Repeat Examples 15.4, 15.6, and 15.7 using the SP3 -approximation. Solution For a nonscattering medium without z-dependence equations (15.98) reduce to 1 2 (Lxx + L yy )J0 − J0 = − J2 − Ib , 3 3 1 2 (Lxx + L yy )J2 − J2 = − (J0 − Ib ), 7 3 where we have used the operators defined in equation (15.52) for better comparison with the equivalent P3 set of Example 15.4. The boundary conditions for a general location simplify to 1 ∂J0 1 1 = (J0 − Ib ) − J2 , 3 ∂τz 2 8 1 ∂J2 1 7 J2 . = − (J0 − Ib ) + 7 ∂τz 8 24 Finally, for the one-dimensional case with only y-dependence, and again taking advantage of the symmetry by placing τ y = 0 at the midplane, the equations and boundary conditions further reduce to 1  2 J − J 0 = − J2 − I b , 3 0 3 1  2 J − J2 = − (J0 − Ib ), 7 2 3 J0 = J2 = 0,

τy = 0 :

1  1 1 J = (J0 − Ibw ) − J2 , 3 0 2 8 1  1 7 J = − (J0 − Ibw ) + J2 . 7 2 8 24

τ y = −τL /2 :

The set of two simultaneous equations is readily reduced to one, by solving the first for J2 : J2 =

3 1 (J0 − Ib ) − J0 , 2 2

then substituting for J2 and J2 in the second, or

or

' ) ' ) 1 3  1 (iv) 3 2 1 J0 − J0 − (J0 − Ib ) − J0 = − (J0 − Ib ), 7 2 2 2 2 3 3 (iv) 6  J − J 0 + J 0 = Ib . 35 0 7

The Method of Spherical Harmonics (PN -Approximation) Chapter | 15 543

Similarly, we eliminate J2 from the boundary conditions: J0 = J0 = 0,

τy = 0 :

' ) 1  1 1 3 1 J0 = (J0 − Ibw ) − (J0 − Ib ) − J0 , 3 2 8 2 2 ' ' ) ) 1 3  1  1 7 3 1 J0 − J0 = − (J0 − Ibw ) + (J0 − Ib ) − J0 , 7 2 2 8 24 2 2

τ y = −τL /2 :

leading to 5 2 1 (J0 − Ib ) − J0 + J0 8 3 8 5 12  7  4  J + J − J . Ibw − Ib = − (J0 − Ib ) + 2 7 0 6 0 7 0

τ y = −τL /2 :

Ibw − Ib =

Since the governing fourth-order equation is exactly the same as the one for I0 in Example 15.7, the solution is also the same, J0 (τ y ) = Ib + (Ibw − Ib )[C1 cosh λ1 τ y + C2 cosh λ2 τ y ], (here given right away without the C3 and C4 , which are eliminated through the τ y = 0 boundary condition). Again, C1 =

b2 − a2 , a1 b2 − a2 b1

C2 =

a1 − b1 , a1 b2 − a2 b1

but with the ai and bi replaced by   τL 2 τL 5 1 2 + λi cosh λi + λi sinh λi , ai = 8 8 2 3 2     τL τL 12 5 7 2 4 3 bi = − + λi cosh λi − λi − λi sinh λi , 2 6 2 7 7 2

i = 1, 2, i = 1, 2.

The heat flux through the medium is determined from equation (15.88) as   4π 4π  2  q(τ y ) = − I0 + I2 = − J0 . 3 5 3 Substituting for J0 we may express the heat flux for a gray medium again in nondimensional form as Ψ=

q(τ y ) n2 σ(Tw4 − T 4 )

4 Ci λi sinh λi τ y . 3 i=1 2

=−

As mentioned in the beginning of this section, for a one-dimensional slab the SPN -method reduces to the regular PN solution. Therefore, the solution here must be identical to that of Example 15.7, which can be shown to be true after considerable algebra.

15.8 Other Methods Based on the P1 -Approximation As indicated earlier, the P1 - or differential approximation enjoys great popularity because of its relative simplicity and because of its compatibility with standard methods for the solution of the (overall) energy equation. The fact that the P1 -approximation may become very inaccurate in optically thin media—and thus of limited use—has prompted a number of investigators to seek enhancements or modifications to the differential approximation to make it reasonably accurate for all conditions [69–83]. We shall describe some of them here, giving special attention to the so-called modified differential approximation as well as the advanced differential approximation. The reader is also referred to the M1 -approximation, which was briefly mentioned in Section 14.4. The directional intensity at any given point inside the medium is due to two sources: radiation originating from a surface (due to emission and reflection), and radiation originating from within the medium (due to emission and in-scattering). The contribution due to radiation emanating from walls may display very irregular directional behavior, especially in optically thin situations (due to surface radiosities varying across the enclosure surface, causing irradiation to change rapidly over incoming directions). Intensity emanating from inside the medium generally varies very slowly with direction because emission and isotropic scattering result in an

544 Radiative Heat Transfer

FIGURE 15.9 Radiative intensity within an arbitrary enclosure.

isotropic radiation source. Only for highly anisotropic scattering may the radiation source—and, therefore, at least locally also the intensity—display irregular directional behavior.

The Modified Differential Approximation In what they termed the modified differential approximation (MDA) Olfe [69–72] and Glatt and Olfe [84] separated wall emission from medium emission in simple black and gray-walled enclosures with gray, nonscattering media, evaluating radiation due to wall emission with exact methods, and radiation from medium emission with the differential (or P1 ) approximation. While very accurate, their model was limited to nonscattering media in simple, mostly one-dimensional enclosures. Wu and coworkers [73] demonstrated, for one-dimensional plane-parallel media, that the MDA may be extended to scattering media with reflecting boundaries. Finally, Modest [74] showed that the method can be applied to three-dimensional linear-anisotropically scattering media with reflecting boundaries. While until recently only used in conjunction with the P1 -approximation, higher order PN - and SPN -methods can also benefit from this approach, as recently shown by Modest and Yang [13], who demonstrated the accuracy of a modified P3 -approach. Consider an arbitrary enclosure as shown in Fig. 15.9. The equation of transfer is, from equation (15.4), dI (r, sˆ ) = sˆ · ∇τ I = S(r, sˆ ) − I(r, sˆ ), dτs

(15.100)

where, for linear-anisotropic scattering with a phase function given by equation (15.32), the radiative source term is, from equation (15.33), S(r, sˆ ) = (1 − ω)Ib (r) +

ω [G(r) + A1 q(r) · sˆ ]. 4π

(15.101)

For diffusely reflecting walls, equations (15.100) and (15.101) are subject to the boundary condition I(rw , sˆ ) =

Jw 1−

ˆ w ), (rw ) = Ibw (rw ) − q · n(r π π

(15.102)

where Jw is the surface radiosity related to Ibw and qw = q · nˆ through equation (15.45). We now break up the intensity at any point into two components: one, Iw , which may be traced back to emission from the enclosure wall (but may have been attenuated by absorption and scattering in the medium, and by reflections from the enclosure walls), and the remainder, Im , which may be traced back to the radiative source term (i.e., radiative intensity released within the medium into a given direction by emission and scattering). Thus, we write I(r, sˆ ) = Iw (r, sˆ ) + Im (r, sˆ )

(15.103)

The Method of Spherical Harmonics (PN -Approximation) Chapter | 15 545

and let Iw satisfy the equation dIw (r, sˆ ) = −Iw (r, sˆ ), dτs

(15.104)

Jw (rw ) e −τs , π

(15.105)

leading to Iw (r, sˆ ) =

as indicated in Fig. 15.9. Since for Iw no radiative source within the medium is considered, the radiosity in equation (15.105) is the one caused by wall emission only (with attenuation within the medium). The radiosity variation along the enclosure wall may be determined by invoking the definition of the radiosity as the sum of emission plus reflected irradiation, or  ˆ dΩ Iw (r, sˆ ) |ˆs · n| Jw (r) = πIbw (r) + (1 − ) ˆ sˆ ·n ri ). (d) What is the surface temperature of the sun? 15.15 Repeat Problem 15.14 but replace assumption (iv) by the following: The fusion process may be approximated by assuming that the sun releases heat uniformly throughout its volume corresponding to the total heat loss of the sun. 15.16 Consider a sphere of very hot dissociated gas of radius 5 cm. The gas may be approximated as a gray, linearanisotropically scattering medium with κ = 0.1 cm−1 , σs = 0.2 cm−1 , A1 = 1. The gas is suspended magnetically in vacuum within a large cold container and is initially at a uniform temperature T g = 10,000 K. Using the P1 approximation and neglecting conduction and convection, specify the total heat loss per unit time from the entire sphere at time t = 0. Outline the solution procedure for times t > 0. Hint: Solve the governing equation by introducing a new dependent variable g(τ) = τ(4πIb − G). 15.17 A spherical test bomb of 1 m radius is coated with a nonreflective material and cooled. Inside the sphere is nitrogen mixed with spherical particles at a rate of 108 particles/m3 . The particles have a radius of 300 μm, are diffuse-gray with = 0.5, and generate heat at a rate of 150 W/cm3 of particle volume. Using absorption and scattering coefficients found in Problem 11.12, determine the temperature distribution inside the bomb, using the P1 -approximation and two simplified phase functions:

558 Radiative Heat Transfer

(i) isotropic scattering, and (ii) linear-anisotropic back scattering with A1 = −1. In particular, what is the gas temperature at the center and at the wall? How much do the two scattering treatments differ from one another? 15.18 A revolutionary new fuel is ground up into small particles, magnetically confined to remain within a spherical cloud of radius R. This cloud of particles has a constant, gray absorption coefficient, does not scatter, and releases heat uniformly at Q˙  (W/m3 ). The cloud is suspended in a vacuum chamber, enclosed by a large, isothermal chamber (at     Tw ). Heat transfer is solely by radiation, i.e., ∇ · q = 1/r2 d r2 q /dr = Q˙  . (a) Assuming the P1 -approximation to be valid, set up the necessary equations and boundary conditions to determine the heat transfer rates, and temperature distribution within the spherical cloud. (b) Determine the maximum temperature in the cloud. 15.19 Repeat Problem 15.5 using subroutine P1sor and/or program P1-2D. How do the answers change for a quadratic enclosure (side walls also cold and black)? 15.20 Repeat Problem 15.6 using subroutine P1sor and/or program P1-2D. How do the answers change for a quadratic enclosure (side walls also black, with a linear surface temperature variation from T(x = 0) = T1 to T(x = L) = T2 )? 15.21 Consider a gray, isotropically scattering medium at radiative equilibrium contained between large, isothermal, gray plates at temperatures T1 and T2 , and emittances 1 and 2 , respectively. Determine the radiative heat flux between the plates using the P3 -approximation. Compare the results with the answer from Problem 15.2. 15.22 Do Problem 15.3 using the P3 -approximation with Marshak’s boundary condition. 15.23 A hot gray medium is contained between two concentric black spheres of radius R1 = 10 cm and R2 = 20 cm. The surfaces of the spheres are isothermal at T1 = 2000 K and T2 = 500 K, respectively. The medium absorbs and emits with n = 1, κ = 0.05 cm−1 , but does not scatter radiation. Determine the heat flux between the spheres using the modified differential approximation (MDA). Note: This problem requires the numerical solution of a simple ordinary differential equation. 15.24 Repeat Problem 15.23 for concentric cylinders of the same radii. Compare your result with those of Fig. 15.5. Note: This problem requires the numerical solution of a simple ordinary differential equation.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

J.H. Jeans, The equations of radiative transfer of energy, Monthly Notices of the Royal Astronomical Society 78 (1917) 28–36. V. Kourganoff, Basic Methods in Transfer Problems, Dover Publications, New York, 1963. B. Davison, Neutron Transport Theory, Oxford University Press, London, 1958. R.L. Murray, Nuclear Reactor Physics, Prentice Hall, Englewood Cliffs, NJ, 1957. E.M. Gelbard, Simplified spherical harmonics equations and their use in shielding problems, Technical Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961. A. Sommerfeld, Partial Differential Equations of Physics, Academic Press, New York, 1964. M. Krook, On the solution of the equation of transfer, I, The Astrophysical Journal 122 (1955) 488–497. P. Cheng, Study of the flow of a radiating gas by a differential approximation, Ph.D. thesis, Stanford University, Stanford, CA, 1965. P. Cheng, Dynamics of a radiating gas with application to flow over a wavy wall, AIAA Journal 4 (2) (1966) 238–245. S.C.S. Ou, K.N. Liou, Generalization of the spherical harmonic method to radiative transfer in multi-dimensional space, Journal of Quantitative Spectroscopy and Radiative Transfer 28 (4) (1982) 271–288. D. Condiff, Anisotropic scattering in three dimensional differential approximation of radiation heat transfer, in: Fundamentals and Applications of Radiation Heat Transfer, vol. HTD-72, ASME, 1987, pp. 19–29. H. Brenner, The Stokes resistance of a slightly deformed sphere — II intrinsic resistance operators for an arbitrary initial flow, Chemical Engineering Science 22 (1967) 375. M.F. Modest, J. Yang, Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries, Journal of Quantitative Spectroscopy and Radiative Transfer 109 (2008) 1641–1666. M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover Publications, New York, 1965. J.C. Mark, The spherical harmonics method, Part I, Technical Report Atomic Energy Report No. MT 92, National Research Council of Canada, 1944. J.C. Mark, The spherical harmonics method, Part II, Technical Report Atomic Energy Report No. MT 97, National Research Council of Canada, 1945. R.E. Marshak, Note on the spherical harmonics method as applied to the Milne problem for a sphere, Physical Review 71 (1947) 443–446. M.F. Modest, Further developments of the elliptic PN -approximation formulation and its Marshak boundary conditions, Numerical Heat Transfer – Part B: Fundamentals 62 (2–3) (2012) 181–202.

The Method of Spherical Harmonics (PN -Approximation) Chapter | 15 559

[19] B. Pellaud, Numerical comparison of different types of vacuum boundary conditions for the PN approximation, Transactions of the American Nuclear Society 9 (1966) 434–435. [20] E. Schmidt, E.M. Gelbard, A double PN method for spheres and cylinders, Transactions of the American Nuclear Society 9 (1966) 432–433. [21] T.M. MacRobert, Spherical Harmonics, 3rd ed., Pergamon Press, New York, 1967. [22] M.F. Modest, Photon-gas formulation of the differential approximation in radiative transfer, Letters in Heat and Mass Transfer 3 (1976) 111–116. [23] L.A. Pipes, L.R. Harvill, Applied Mathematics for Engineers and Physicists, McGraw-Hill, New York, 1970. [24] Y. Bayazitoglu, ˘ J. Higenyi, The higher-order differential equations of radiative transfer: P3 approximation, AIAA Journal 17 (1979) 424–431. [25] S.H. Park, S.S. Kim, Thermophoretic deposition of absorbing, emitting and isotropically scattering particles in laminar tube flow with high particle mass loading, International Journal of Heat and Mass Transfer 36 (14) (1993) 3477–3485. [26] H.M. Park, R.K. Ahluwalia, K.H. Im, Three-dimensional radiation in absorbing–emitting–scattering media using the modified differential approximation, International Journal of Heat and Mass Transfer 36 (5) (1993) 1181–1189. [27] K. Kamiuto, S. Saitoh, Combined forced-convection and correlated–radiation heat transfer in cylindrical packed beds, Journal of Thermophysics and Heat Transfer 8 (1) (1994) 119–124. [28] R.K. Ahluwalia, K.H. Im, Spectral radiative heat-transfer in coal furnaces using a hybrid technique, Journal of the Institute of Energy 67 (1994) 23–29. [29] D.A. Kaminski, X.D. Fu, M.K. Jensen, Numerical and experimental analysis of combined convective and radiative heat transfer in laminar flow over a circular cylinder, International Journal of Heat and Mass Transfer 38 (17) (1995) 3161–3169. [30] T. Tsukada, K. Kakinoki, M. Hozawa, N. Imaishi, Effect of internal radiation within crystal and melt on Czochralski crystal growth of oxide, International Journal of Heat and Mass Transfer 38 (1995) 2707–2714. [31] O.A. Ezekoye, Z. Zhang, Convective and radiative coupling in a burner-supported diffusion flame, Journal of Thermophysics and Heat Transfer 11 (2) (1997) 239–245. [32] J.J. Derby, S. Brandon, A.G. Salinger, The diffusion and P1 approximations for modeling buoyant flow of an optically thick fluid, International Journal of Heat and Mass Transfer 41 (11) (1998) 1405–1415. [33] Y. Bayazitoglu, ˘ B.Y. Wang, Wavelets in the solution of nongray radiative heat transfer equation, ASME Journal of Heat Transfer 120 (1) (1998) 133–139. [34] T.H. Lin, C.H. Chen, Influence of two-dimensional gas phase radiation on downward flame spread, Combustion Science and Technology 141 (1) (1999) 83–106. [35] D.C. Kuo, J.C. Morales, K.S. Ball, Combined natural convection and volumetric radiation in a horizontal annulus: spectral and finite volume predictions, ASME Journal of Heat Transfer 121 (1999) 610–615. [36] B. Zheng, C.X. Lin, M.A. Ebadian, Combined laminar forced convection and thermal radiation in a helical pipe, International Journal of Heat and Mass Transfer 43 (7) (2000) 1067–1078. [37] G. Li, M.F. Modest, A method to accelerate convergence and to preserve radiative energy balance in solving the P1 equation by iterative methods, ASME Journal of Heat Transfer 124 (3) (2002) 580–582. [38] F. Liu, J. Swithenbank, E.S. Garbett, The boundary condition of the PN -approximation used to solve the radiative transfer equation, International Journal of Heat and Mass Transfer 35 (1992) 2043–2052. [39] B. Su, More on boundary conditions for differential approximations, Journal of Quantitative Spectroscopy and Radiative Transfer 64 (2000) 409–419. [40] M.P. Mengüç, S. Subramaniam, Radiative transfer through an inhomogeneous fly-ash cloud: effects of temperature and wavelength dependent optical properties, Numerical Heat Transfer – Part A: Applications 21 (3) (1992) 261–273. [41] T. Khan, A. Thomas, Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices, Optics Communications 255 (2005) 130–166. [42] J.-M. Tualle, E. Tinet, Derivation of the radiative transfer equation for scattering media with a spatially varying refractive index, Optics Communications 228 (2003) 33–38. [43] M. Bhuvaneswari, C.Y. Wu, Differential approximations for transient radiative transfer in refractive planar media with pulse irradiation, Journal of Quantitative Spectroscopy and Radiative Transfer 110 (6–7) (2009) 389–401. [44] K.-Y. Zhu, Y. Huang, J. Wang, Combination of spherical harmonics and spectral method for radiative heat transfer in one-dimensional anisotropic scattering medium with graded index, International Journal of Heat and Mass Transfer 62 (2013) 200–204. [45] W. Kofink, Complete spherical harmonics solution of the Boltzmann equation for neutron transport in homogeneous media with cylindrical geometry, Nuclear Science and Engineering 6 (1959) 473–486. [46] T.W. Tong, P.S. Swathi, Radiative heat transfer in emitting–absorbing–scattering spherical media, Journal of Thermophysics and Heat Transfer 1 (2) (1987) 162–170. [47] W. Li, T.W. Tong, Radiative heat transfer in isothermal spherical media, Journal of Quantitative Spectroscopy and Radiative Transfer 43 (3) (1990) 239–251. [48] T.W. Tong, W. Li, Enhancement of thermal emission from porous radiant burners, Journal of Quantitative Spectroscopy and Radiative Transfer 53 (2) (1995) 235–248.

560 Radiative Heat Transfer

[49] J.W. Wu, H.S. Chu, Combined conduction and radiation heat transfer in plane-parallel packed beds with variable porosity, Journal of Quantitative Spectroscopy and Radiative Transfer 61 (4) (1999) 443–452. [50] M.P. Mengüç, R. Viskanta, Radiative transfer in three-dimensional rectangular enclosures containing inhomogeneous, anisotropically scattering media, Journal of Quantitative Spectroscopy and Radiative Transfer 33 (6) (1985) 533–549. [51] M.P. Mengüç, R. Viskanta, Radiative transfer in axisymmetric, finite cylindrical enclosures, ASME Journal of Heat Transfer 108 (1986) 271–276. [52] J. Yang, M.F. Modest, High-order P-N approximation for radiative transfer in arbitrary geometries, Journal of Quantitative Spectroscopy and Radiative Transfer 104 (2) (2007) 217–227. [53] D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, Singapore, 1981. [54] M.A. Blanco, M. Flórez, M. Bermejo, Evaluation of the rotation matrices in the basis of real spherical harmonics, Journal of Molecular Structure: THEOCHEM 49 (1997) 19–27. [55] FlexPDE software, PDE Solutions Inc., Antioch. [56] G. Sewell, PDE2D 9.0, http://www.pde2d.com/. [57] W. Schönauer, FDEM, https://swmath.org/software/10383, 2003. [58] M. Ravishankar, S. Mazumder, A. Kumar, Finite-volume formulation and solution of the P3 equations of radiative transfer on unstructured meshes, ASME Journal of Heat Transfer 132 (2) (2010) 023402. [59] W. Ge, R. Marquez, M.F. Modest, S.P. Roy, Implementation of high order spherical harmonics methods for radiative heat transfer on OpenFOAM, ASME Journal of Heat Transfer 137 (5) (2015) 052701. [60] OpenFOAM website, http://www.opencfd.co.uk/openfoam/. [61] J. Nocedal, S.J. Wright, Numerical Optimization, 2nd ed., Springer Verlag, Berlin, 2006. [62] W. Ge, M.F. Modest, R. Marquez, Two-dimensional axisymmetric formulation of high order spherical harmonics methods for radiative heat transfer, Journal of Quantitative Spectroscopy and Radiative Transfer 156 (2015) 58–66. [63] W. Ge, M.F. Modest, S.P. Roy, Development of high-order pn models for radiative heat transfer in special geometries and boundary conditions, Journal of Quantitative Spectroscopy and Radiative Transfer 172 (2016) 98–109. [64] E. Larsen, J.E. Morel, J. McGhee, Simplified PN approximations to the equations of radiative heat transfer and applications, in: Proceedings of the Joint International Conference on Mathematical Methods and Supercomputing in Nuclear Applications, Portland, Oregon, 1993. [65] G.C. Pomraning, Asymptotic and variational derivations of the simplified Pn equations, Annals of Nuclear Energy 20 (9) (1993) 623–637. [66] R.G. McClarren, Theoretical aspects of the simplified Pn equations, Transport Theory and Statistical Physics 39 (2011) 73–109. [67] M.F. Modest, S. Lei, Simplified spherical harmonics method for radiative heat transfer, in: Proceedings of Eurotherm Seminar 95, Elsevier, Nancy, France, April 2012. [68] E. Larsen, G. Thömmes, A. Klar, M. Seaïd, T. Götz, Asymptotic derivation of the simplified PN equations, Journal of Computational Physics 183 (2002) 652–675. [69] D.B. Olfe, A modification of the differential approximation for radiative transfer, AIAA Journal 5 (4) (1967) 638–643. [70] D.B. Olfe, Application of a modified differential approximation to radiative transfer in a gray medium between concentric sphere and cylinders, Journal of Quantitative Spectroscopy and Radiative Transfer 8 (1968) 899–907. [71] D.B. Olfe, Radiative equilibrium of a gray medium bounded by nonisothermal walls, Progress in Astronautics and Aeronautics 23 (1970) 295–317. [72] D.B. Olfe, Radiative equilibrium of a gray medium in a rectangular enclosure, Journal of Quantitative Spectroscopy and Radiative Transfer 13 (1973) 881–895. [73] C.Y. Wu, W.H. Sutton, T.J. Love, Successive improvement of the modified differential approximation in radiative heat transfer, Journal of Thermophysics and Heat Transfer 1 (4) (1987) 296–300. [74] M.F. Modest, The modified differential approximation for radiative transfer in general three-dimensional media, Journal of Thermophysics and Heat Transfer 3 (3) (1989) 283–288. [75] M.F. Modest, Two-dimensional radiative equilibrium of a gray medium in a plane layer bounded by gray non-isothermal walls, ASME Journal of Heat Transfer 96C (1974) 483–488. [76] M.F. Modest, Radiative equilibrium in a rectangular enclosure bounded by gray non-isothermal walls, Journal of Quantitative Spectroscopy and Radiative Transfer 15 (1975) 445–461. [77] M.F. Modest, D. Stevens, Two dimensional radiative equilibrium of a gray medium between concentric cylinders, Journal of Quantitative Spectroscopy and Radiative Transfer 19 (1978) 353–365. [78] M.F. Modest, The improved differential approximation for radiative transfer in general three-dimensional media, in: Heat Transfer Phenomena in Radiation, Combustion and Fires, vol. HTD-106, ASME, 1989, pp. 213–220. [79] M.F. Modest, The improved differential approximation for radiative transfer in multi-dimensional media, ASME Journal of Heat Transfer 112 (1990) 819–821. [80] M.F. Modest, G. Pal, Advanced differential approximation formulation of the PN method for radiative transfer, in: Proceedings of ASME Summer Heat Transfer Conference, 2009, Paper HT2009-88242. [81] N. Doner, M1 model for radiative heat transfer in absorbing, emitting, and scattering medium, International Journal of Thermal Sciences 79 (2014) 34–39.

The Method of Spherical Harmonics (PN -Approximation) Chapter | 15 561

[82] T. Hanawa, E. Audit, Reformulation of the M1 model of radiative transfer, Journal of Quantitative Spectroscopy and Radiative Transfer 145 (2014) 9–16. [83] G. Pal, M.F. Modest, Advanced differential approximation formulation of the PN method for radiative transfer, ASME Journal of Heat Transfer 137 (July 2015) 072701. [84] L. Glatt, D.B. Olfe, Radiative equilibrium of a gray medium in a rectangular enclosure, Journal of Quantitative Spectroscopy and Radiative Transfer 13 (1973) 881–895. [85] L. Hartung, H.A. Hassan, Radiation transport around axisymmetric blunt body vehicles using a modified differential approximation, Journal of Thermophysics and Heat Transfer 7 (2) (1993) 220–227. [86] C.Y. Wu, N.R. Ou, Transient two-dimensional radiative and conductive heat transfer in a scattering medium, International Journal of Heat and Mass Transfer 37 (17) (1994) 2675–2686. [87] M. Ravishankar, S. Mazumder, M. Sankar, Application of the modified differential approximation for radiative transfer to arbitrary geometry, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (14) (2010) 2052–2069. [88] J. Gérardin, N. Seiler, P. Ruyer, L. Trovalet, P. Boulet, P1 approximation, MDA and IDA for the simulation of radiative transfer in a 3D geometry for an absorbing scattering medium, Journal of Quantitative Spectroscopy and Radiative Transfer 113 (2012) 140–149. [89] J. Gérardin, N. Seiler, P. Ruyer, P. Boulet, Radiative transfer during the reflooding step of a loca, Journal of Quantitative Spectroscopy and Radiative Transfer 128 (2013) 43–51. [90] M. Sankar, S. Mazumder, Solution of the radiative transfer equation in three-dimensional participating media using a hybrid discrete ordinates-spherical harmonics method, ASME Journal of Heat Transfer 134 (2012) 112702. [91] K.V. Deshmukh, M.F. Modest, D.C. Haworth, Higher-order spherical harmonics to model radiation in direct numerical simulation of turbulent reacting flows, Computational Thermal Sciences 1 (2009) 207–230. [92] A. Wang, M.F. Modest, D.C. Haworth, L. Wang, Monte Carlo simulation of radiative heat transfer and turbulence interactions in methane/air jet flames, Journal of Quantitative Spectroscopy and Radiative Transfer 109 (2) (2008) 269–279. [93] G. Pal, A. Gupta, M.F. Modest, D.C. Haworth, Comparison of accuracy and computational expense of radiation models in simulation of nonpremixed turbulent jet flames, in: Proceedings of 2011 ASME/JSME Thermal Engineering Joint Conference, 2011. [94] G. Pal, A. Gupta, M.F. Modest, D.C. Haworth, Comparison of accuracy and computational expense of radiation models in simulation of nonpremixed turbulent jet flames, Combustion and Flame 162 (2015) 2487–2495. [95] C. David, W. Ge, S.P. Roy, M.F. Modest, R. Sankaran, Comparison of radiation models for a turbulent piloted methane/air jet flame: a frozen-field study, in: Proceedings of the ASME 2021 Summer Heat Transfer Conferences SHTC2021, 2021, pp. HT2021–62417. [96] W. Ge, C. David, S.P. Roy, M.F. Modest, R. Sankaran, Comparison of spherical harmonics method and discrete ordinates method for radiative transfer in a turbulent jet flame, Journal of Quantitative Spectroscopy and Radiative Transfer (2021), in preparation at time of press. [97] R.S. Barlow, International Workshop on Measurement and Computation of Turbulent Nonpremixed Flames (TNF), website:: http:// www.sandia.gov/TNF/abstract.html.

Chapter 16

The Method of Discrete Ordinates (SN-Approximation) 16.1 Introduction Like the spherical harmonics method, the discrete ordinates method is a tool to transform the equation of transfer (for a gray medium, or on a spectral basis) into a set of simultaneous partial differential equations. Similar to the PN -method, the discrete ordinates or SN -method may be carried out to any arbitrary order and accuracy, although the mathematical formulation of high-order SN -schemes is considerably less involved. First proposed by Chandrasekhar [1] in his work on stellar and atmospheric radiation, the SN -method originally received little attention in the heat transfer community. Again, like the PN -method, the discrete ordinates method was first systematically applied to problems in neutron transport theory, notably by Lee [2] and Lathrop [3,4]. There were some early, unoptimized attempts to apply the method to one-dimensional, planar thermal radiation problems (Love et al. [5,6], Hottel et al. [7], Roux and Smith [8,9]). Only during the past 40 years, however, has the discrete ordinates method been applied to, and optimized for, general radiative heat transfer problems, primarily through the pioneering works of Fiveland [10–13] and Truelove [14–16]. The discrete ordinates method (DOM) is based on a discrete representation of the directional variation of the radiative intensity. A solution to the transport problem is found by solving the equation of transfer for a preselected set of discrete directions spanning the total solid angle range of 4π. As such, the discrete ordinates method is simply a finite differencing of the directional dependence of the equation of transfer. Integrals over solid angle are approximated by numerical quadrature (e.g., for the evaluation of the radiative source term, the radiative heat flux, etc.). Today, many numerical heat transfer models use finite volumes rather than finite differences. Similarly, one may also use finite solid angles for directional (or angular) discretization. This variation of the discrete ordinates method has been historically known as the finite volume method (for radiative transfer), and enjoys increasing popularity. In this text, it will be referred to as the finite angle method (FAM), to be consistent with its aforementioned philosophy of using finite solid angles (and to avoid confusion with the finite volume method for spatial discretization). As a result of the relatively straightforward formulation of high-order implementations, the DOM and its finite angle cousin, the FAM, have received great attention and today, together with the P1 -approximation, are probably the most popular RTE solvers. Some version of them is incorporated in most commercial computational fluid dynamics (CFD) codes. Detailed reviews of the capabilities and shortcomings of the DOM and FAM have been given by Charest et al. [17] and by Coelho [18]. The latter provides the most complete description of the method for general geometries, far exceeding the details we can provide in this book. In this chapter, we shall first develop the set of partial differential equations for the standard DOM and their boundary conditions. This is followed by a section describing how the method may be applied to onedimensional plane-parallel media, and another dealing with spherical and cylindrical geometries. Then, its application to general multidimensional problems will be outlined, with specific reductions and examples for two-dimensional Cartesian geometries. This is followed by the development and demonstration of the FAM. Finally, the chapter will close with a brief look at other, related methods.

16.2 General Relations The general equation of transfer for an absorbing, emitting, and anisotropically scattering medium is, according to equation (9.21),  σs (r) dI = sˆ · ∇I(r, sˆ ) = κ(r)Ib (r) − β(r)I(r, sˆ ) + I(r, sˆ  ) Φ(r, sˆ  , sˆ ) dΩ . (16.1) ds 4π 4π Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00024-9 Copyright © 2022 Elsevier Inc. All rights reserved.

563

564 Radiative Heat Transfer

Equation (16.1) is valid for a gray medium or, on a spectral basis, for a nongray medium, and is subject to the boundary condition  ρ(rw ) I(rw , sˆ ) = (rw )Ib (rw ) + I(rw , sˆ  ) |nˆ · sˆ  | dΩ , (16.2) π ˆ s 0), and once where it strikes the wall, to be absorbed or reflected (nˆ · sˆ i < 0). The governing equation is first order, requiring only one boundary condition (for the emanating intensity, nˆ · sˆ i > 0). Equations (16.4) together with their boundary conditions (16.5) constitute a set of n simultaneous, first-order, linear partial differential equations for the unknown Ii (r) = I(r, sˆ i ). The solution for the Ii may be found using any standard technique (analytical or numerical). If scattering is present (σs  0), and/or if the bounding walls are reflecting, the equations are coupled in such a way that generally an iterative procedure is necessary. Even in the absence of scattering and surface reflections, the temperature field may not be known, but must be calculated from the intensity field if radiative equilibrium persists, again making iterations necessary. Only in the absence of scattering and wall reflections, and if the temperature field is given, then the solution to the intensities Ii is straightforward (as is the exact solution). Once the intensities have been determined the desired direction-integrated quantities are readily calculated. The radiative heat flux, inside the medium or at a surface, may be found from its definition, equation (9.55),  q(r) =

I(r, sˆ ) sˆ dΩ 4π

n 

wi Ii (r) sˆ i .

(16.6)

i=1

The incident radiation G [and, through equation (9.62), the divergence of the radiative heat flux] is similarly determined as  n  G(r) = I(r, sˆ ) dΩ wi Ii (r). (16.7) 4π

i=1

At a surface the heat flux may also be determined from surface energy balances [equations (4.1) and (3.16)] as    ˆ w ) = (rw ) [πIb (rw ) − H(rw )] (rw ) πIb (rw ) − wi Ii (rw ) |nˆ · sˆ i | . (16.8) q · n(r ˆ si 0

(16.11)

at the enclosure surface. Of course, radiative heat flux and incident radiation are unknowns to be determined from directional intensities from the series in equations (16.6) and (16.7).

Selection of Discrete Ordinate Directions The choice of quadrature scheme is arbitrary, although restrictions on the directions sˆ i and quadrature weights wi may arise from the desire to preserve symmetry and to satisfy certain conditions. It is customary to choose sets of directions and weights that are completely symmetric (i.e., sets that are invariant after any rotation of 90◦ ), and that satisfy the zeroth, first, and second moments, or  dΩ = 4π = 

4π

sˆ dΩ = 0 = 

4π

n 

wi ,

(16.12a)

wi sˆ i ,

(16.12b)

i=1 n  i=1

n  4π sˆ sˆ dΩ = δ = wi sˆ i sˆ i , 3 4π

(16.12c)

i=1

where δ is the unit tensor [cf. equation (15.30)]. Different sets of directions and weights satisfying all these criteria have been tabulated, for example, by Lee [2] and Lathrop and Carlson [19]. Fiveland [12] and Truelove [15] have observed that different sets of ordinates may result in considerably different accuracy. They noted that (i) the intensity may have directional discontinuity at a wall and (ii) the important radiative heat fluxes at the walls are evaluated through a first moment of intensity over a half range of 2π [equation (16.8)]. They concluded that the set of ordinates and weights should also satisfy the first moment over a half range, that is,    nˆ · sˆ dΩ = π = |nˆ · sˆ | dΩ = wi nˆ · sˆ i . (16.13) ˆ s0 n·ˆ

ˆ si >0 n·ˆ

While it is impossible to satisfy equation (16.13) for arbitrary orientations of the surface normal, it can be ˆ Sets of ordinates and weights that satisfy (i) the symmetry satisfied for the principal orientations, if nˆ = î, ˆj, or k. requirement, (ii) the moment equations (16.12), and (iii) the half-moment equation (16.13) (for the three principal ˆ 1 have been given by Lathrop and Carlson [19]. The first four sets labeled S2 -, S4 -, S6 -, and directions of n) S8 -approximation are reproduced in Table 16.1. In the table the ξi , ηi , and μi are the direction cosines of sˆ i , or ˆ ˆ kˆ = ξi î + ηi ˆj + μi k. sˆ i = (ˆs i · î) î + (ˆs i · ˆj) ˆj + (ˆs i · k)

(16.14)

Only positive direction cosines are given in Table 16.1, covering one-eighth of the total range of solid angles 4π. To cover the entire 4π any or all of the values of ξi , ηi , and μi may be positive or negative. Therefore, each row of ordinates contains eight different directions. For example, for the S2 -approximation the different directions are 1. With the exception of the symmetric S2 -approximation.

566 Radiative Heat Transfer

TABLE 16.1 Discrete ordinates for the SN -approximation (N = 2, 4, 6, 8), from [19]. Order of Approximation

Ordinates

Weights

ξ

η

μ

w

S2 (symmetric)

0.5773503

0.5773503

0.5773503

1.5707963

S2 (nonsymmetric)

0.5000000

0.7071068

0.5000000

1.5707963

S4

0.2958759

0.2958759

0.9082483

0.5235987

0.2958759

0.9082483

0.2958759

0.5235987

0.9082483

0.2958759

0.2958759

0.5235987

0.1838670

0.1838670

0.9656013

0.1609517

0.1838670

0.6950514

0.6950514

0.3626469

0.1838670

0.9656013

0.1838670

0.1609517

0.6950514

0.1838670

0.6950514

0.3626469

0.6950514

0.6950514

0.1838670

0.3626469

0.9656013

0.1838670

0.1838670

0.1609517

0.1422555

0.1422555

0.9795543

0.1712359

0.1422555

0.5773503

0.8040087

0.0992284

0.1422555

0.8040087

0.5773503

0.0992284

0.1422555

0.9795543

0.1422555

0.1712359

0.5773503

0.1422555

0.8040087

0.0992284

0.5773503

0.5773503

0.5773503

0.4617179

0.5773503

0.8040087

0.1422555

0.0992284

0.8040087

0.1422555

0.5773503

0.0992284

0.8040087

0.5773503

0.1422555

0.0992284

0.9795543

0.1422555

0.1422555

0.1712359

S6

S8

ˆ sˆ 2 = 0.577350(î+ˆj − k), ˆ . . . , sˆ 8 = −0.577350(î+ˆj + k). ˆ Since the symmetric S2 -approximation sˆ 1 = 0.577350(î+ˆj + k), does not satisfy the half-moment condition, a nonsymmetric S2 -approximation is also included in Table 16.1, as proposed by Truelove [15]. This approximation satisfies equation (16.13) for two principal directions and should be applied to one- and two-dimensional problems, from which the nonsymmetric term drops out (as seen in Example 16.1 in the following section). The name “SN -approximation” indicates that N different direction cosines are used for each principal direction. For example, for the S4 -approximation ξi = ±0.295876 and ±0.908248 (or ηi or μi ). Altogether there are always n = N(N +2) different directions to be considered (because of symmetry, many of these may be unnecessary for one- and two-dimensional problems). Several other quadrature schemes can be found in the literature. Carlson [20] proposed a set with equal weights wi (such as the S2 and S4 sets in Table 16.1). Two more quadratures and a good review of the applicability of all discrete ordinate sets have been given by Fiveland [21]. Other publications documenting procedures for the generation of quadrature sets are those of Sánchez and Smith [22] and El-Wakil and Sacadura [23]. A new family of quadrature sets, like the Sn sets symmetric in 90◦ rotations, but with different arrangement of directions, have been given by Thurgood and coworkers [24], and have been dubbed Tn sets by the authors. These always generate positive weights and are claimed to reduce the so-called “ray effect” (which will be discussed a little later on p. 588). These sets have been further refined by Li and coworkers [25]. A comprehensive review of directional quadrature schemes, including an evaluation of their accuracies, has been given by Koch and Becker [26]. None of the above ordinate sets can treat collimated (i.e., unidirectional) irradiation accurately. To address this problem Li and coworkers [27] developed the ISW scheme adding a single ordinate of “infinitely small weight” to the regular quadrature set.

16.3 The One-Dimensional Slab We will first demonstrate how the SN discrete ordinates method is applied to the simple case of a one-dimensional plane-parallel slab bounded by two diffusely emitting and reflecting isothermal plates. As in previous chapters, we shall limit ourselves to linear-anisotropic scattering, although extension to arbitrarily anisotropic scattering

The Method of Discrete Ordinates (SN -Approximation) Chapter | 16 567

TABLE 16.2 Discrete ordinates for the onedimensional SN -approximation (N = 2, 4, 6, 8). Order of Approximation

Ordinates

Weights

μ

w

S2 (symmetric)

0.5773503

6.2831853

S2 (nonsymmetric)

0.5000000

6.2831853

S4

0.2958759

4.1887902

0.9082483

2.0943951

0.1838670

2.7382012

0.6950514

2.9011752

0.9656013

0.6438068

0.1422555

2.1637144

0.5773503

2.6406988

0.8040087

0.7938272

0.9795543

0.6849436

S6

S8

is straightforward. We avoid it here to make the steps in the development a little easier to follow. If we choose z as the spatial coordinate between the two plates (0 ≤ z ≤ L), and introduce the optical coordinate τ with dτ = β dz (0 ≤ τ ≤ τL ), equation (16.4) is transformed to μi

n   dIi ω  = (1 − ω) Ib − Ii + w j I j 1+A1 (μi μ j +ξi ξ j +ηi η j ) , dτ 4π

i = 1, 2, . . . , n.

(16.15)

j=1

For a one-dimensional slab intensity is independent of azimuthal angle. Since for every ordinate j (with a given μj ) with a positive value for ξ j there is another with the same, but negative, value, and since the intensity is the same for both ordinates, the terms involving ξ j in equation (16.15) add to zero. The same is true for the terms involving η j , but not for those with μ j (since the intensity does depend on polar angle θ, and μ = cos θ). However, the terms involving μ j are repeated several times: Each value of μ (counting positive and negative μ-values separately) shown in one row of Table 16.1 corresponds to four different ordinates (combinations of positive and negative values for ξ and η). In addition, a particular value of μ may occur on more than one line of Table 16.1. If all the quadrature weights corresponding to a single μ-value are added together, equation (16.15) reduces to dIi ω   = (1 − ω) Ib − Ii + w j I j (1 + A1 μi μ j ), dτ 4π N

μi

i = 1, 2, . . . , N,

(16.16)

j=1

where the wj are the summed quadrature weights, and summation to N, i.e., the order of the method or number of different direction cosines. For example, for μ = 0.2958759 in the S4 -approximation the summed quadrature weight is w = 4 × (0.5235987 + 0.5235987) = 4π/3, and so forth. The ordinates and quadrature weights for the one-dimensional slab are listed in Table 16.2. Equation (16.16) could have been found less painfully by using equation (16.10) instead of (16.4), leading directly to μi

dIi ω + Ii = (1 − ω) Ib + (G + A1 qμi ), dτ 4π

i = 1, 2, . . . , N.

(16.17)

Before proceeding to the boundary conditions of equation (16.17) we should recognize that of the N different intensities, half emanate from the wall at τ = 0 (with μi > 0) and the other half from the wall at τ = τL (with μi < 0). Following the notation of Chapter 13, we replace the N different Ii by + I1+ , I2+ , . . . , IN/2

and

− I1− , I2− , . . . , IN/2 .

568 Radiative Heat Transfer

Then equation (16.17) may be rewritten as dIi+

ω (G + A1 qμi ), dτ 4π dI− ω (G − A1 qμi ), −μi i + Ii− = (1 − ω) Ib + dτ 4π i = 1, 2, . . . , N/2; μi > 0. μi

+ Ii+ = (1 − ω) Ib +

(16.18a) (16.18b)

With this notation the boundary conditions for equation (16.18) follow from equations (16.5) or (16.11) as τ=0: τ = τL :

1 − 1 q1 ,

1 π 1 − 2 q2 , Ii− = J2 /π = Ib2 +

2 π i = 1, 2, . . . , N/2,

Ii+ = J1 /π = Ib1 −

(16.19a) (16.19b) μi > 0.

(For the boundary condition at τL the sign switches since nˆ points in the direction opposite to z.) Radiative heat flux q and incident radiation G are related to the directional intensities through equations (16.6) and (16.7), or q=

N/2 

wi μi (Ii+ − Ii− ),

(16.20a)

wi (Ii+ + Ii− ).

(16.20b)

i=1

G=

N/2  i=1

At the two surfaces the radiative heat flux is more conveniently evaluated from equation (16.8) as τ=0:

N/2    q1 = q(0) = 1 Eb1 − wi μi Ii− ,

(16.21a)

i=1

τ = τL :

N/2    q2 = − q(τL ) = − 2 Eb2 − wi μi Ii+ .

(16.21b)

i=1

Example 16.1. Consider two large, parallel, gray-diffuse and isothermal plates, separated by a distance L. One plate is at temperature T1 with emittance 1 and the other is at T2 with 2 . The medium between the two plates is a gray, absorbing/emitting and linear-anisotropically scattering gas (n = 1) with constant extinction coefficient β and single scattering albedo ω. Assuming that radiative equilibrium prevails, determine the radiative heat flux between the two plates using the S2 -approximation. Solution For radiative equilibrium we have, from equation (9.62), Ib = G/4π and q = const; equations (16.18) and (16.19) become dI1+

1 (G + A1 ωμ1 q), 4π 1 + I1− = (G − A1 ωμ1 q), −μ1 dτ 4π μ1

τ=0:

dτ dI1−

+ I1+ =

I1+ = J1 /π, τ = βL = τL :

I1− = J2 /π.

For the S2 -approximation we have only a single ordinate direction μ1 (pointing toward τL for I1+ and toward 0 for I1− ), where μ1 = 0.57735 for the symmetric S2 -approximation, and μ1 = 0.5 for the nonsymmetric S2 -approximation [which satisfies the half-range moment, equation (16.13)]. For the simple S2 -approximation the simultaneous equations (only

The Method of Discrete Ordinates (SN -Approximation) Chapter | 16 569

two in this case) may be separated. We do this here by eliminating I1+ and I1− in favor of G and q. From equation (16.20), with wi = 2π, G = 2π(I1+ + I1− ), q = 2π μ1 (I1+ − I1− ). Therefore, adding and subtracting the two differential equations and multiplying by 2π leads to dq dq + G = G, or = 0, dτ dτ   dG 1 1 dG μ1 + q = A1 ωμ1 q, or = − 2 − A1 ω q. dτ μ1 dτ μ1 The first equation is simply a restatement of radiative equilibrium, while the second may be integrated (since q = const), or   1 G = C − 2 − A1 ω qτ. μ1 This relation contains two unknown constants (C and q), which must be determined from the boundary conditions, that is,   q 1 = J1 /π, G+ τ = 0 : I1+ = 4π μ1   q 1 = J2 /π, G− τ = τL : I1− = 4π μ1 or q q =C+ , τ = 0 : 4J1 = G + μ1 μ1   q q 1 = C − 2 − A1 ω q τL − . τ = τL : 4J2 = G − μ1 μ1 μ1 Subtracting, we obtain, Ψ=

2μ1 q   = , J1 − J2 1 + 1/μ21 − A1 ω μ1 τL /2

from which the radiosities may be eliminated through equation (13.48). For the symmetric S2 -approximation, μ1 = √ 0.57735 = 1/ 3, and with isotropic scattering, A1 = 0, this expression becomes Ψsymmetric = √

1 3/2 + 3τL /4

.

On the other hand, for the nonsymmetric S2 -approximation (μ1 = 0.5), also with isotropic scattering, Ψnonsymmetric =

1 . 1 + τL

Results from the two S2 -approximations are compared in Table 16.3 with those from the P1 -approximation and the exact solution. It is seen that the accuracy of the S2 -method is roughly equivalent to that of the P1 -approximation. The nonsymmetric S2 -approximation is superior to the symmetric one, since the symmetric S2 does not satisfy the half-moment condition, equation (16.13), and causes substantial errors in the optically thin limit.

The S2 -approximation is the same as the two-flux method discussed in Section 14.3, and the nonsymmetric S2 -method is nothing but the Schuster–Schwarzschild approximation. As a second example for the one-dimensional discrete ordinates method we shall repeat Example 15.4, which was originally designed to demonstrate the use of the P3 -approximation. Example 16.2. Consider an isothermal medium at temperature T, confined between two large, parallel black plates that are isothermal at the (same) temperature Tw . The medium is gray and absorbs and emits, but does not scatter. Determine an expression for the heat transfer rates within the medium using the S2 and S4 discrete ordinates approximations.

570 Radiative Heat Transfer

TABLE 16.3 Radiative heat flux through a onedimensional plane-parallel medium at radiative equilibrium; comparison of S2 - and P1 -approximations. Ψ = q/(J1 − J2 ) τL

Exact

S2 (sym)

S2 (nonsym)

P1

0.0

1.0000

1.1547

1.0000

1.0000

0.1

0.9157

1.0627

0.9091

0.9302

0.5

0.7040

0.8058

0.6667

0.7273

1.0

0.5532

0.6188

0.5000

0.5714

5.0

0.2077

0.2166

0.1667

0.2105

Solution For this particularly simple case equations (16.18) reduce to dIi+

+ I i = Ib , dτ − dI − μi i + Ii = Ib . dτ μi

Since Ib = const, these equations may be integrated right away, leading to Ii+ = Ib + C+ e−τ/μi , Ii− = Ib + C− eτ/μi . The integration constants C+ and C− may be found from boundary conditions (16.19) as τ=0:

Ii+ = Ibw = Ib + C+ ,

τ = τL :

Ii− = Ibw = Ib + C− eτL /μi ,

or

C+ = Ibw − Ib ; or

C− = (Ibw − Ib ) e−τL /μi .

Thus, Ii+ = Ib + (Ibw − Ib ) e−τ/μi , Ii− = Ib + (Ibw − Ib ) e−(τL −τ)/μi . The radiative heat flux follows then from equation (16.20) as q=

N/2 

  wi μi (Ibw −Ib ) e−τ/μi − e−(τL −τ)/μi ,

i=1

or, in nondimensional form, Ψ=

q n2 σ(Tw4 −T 4 )

=

N/2  1    −τ/μi wi μi e − e−(τL −τ)/μi . π i=1

For the nonsymmetric S2 -approximation we have w1 = 2π and μ1 = 0.5, or ΨS2 = e−2τ − e−2(τL −τ) . For the S4 -approximation, w1 = 4π/3, w2 = 2π/3, μ1 = 0.2958759, μ2 = 0.9082483, and

5

wi μi = π, so that

    ΨS4 = 0.3945012 e−τ/0.2958759 − e−(τL −τ)/0.2958759 + 0.6054088 e−τ/0.9082483 − e−(τL −τ)/0.9082483 . The results should be compared with those of Examples 15.2 and 15.4 for the P1 - and P3 -approximations. Note that the SN -method goes to the correct optically thick limit (τL → ∞) at the wall, i.e., Ψ → 1 [if the half moment of equation (16.13) is satisfied]. The PN -approximations, on the other hand, overpredict the optically thick limit for this particular example.

The Method of Discrete Ordinates (SN -Approximation) Chapter | 16 571

It should be emphasized that this last example—dealing with a nonscattering, isothermal medium—is particularly well suited for the discrete ordinates method. One should not expect that, for a general problem, the S4 -method to be more accurate than the P3 -approximation. A number of researchers have solved more complicated one-dimensional problems by the discrete ordinates method. Fiveland [12] considered the identical case as presented in this section, but allowed for arbitrarily anisotropic scattering. Solving the system of equations by a finite difference method, he noted that higher-order SN -methods demand a smaller numerical step Δτ, in order to obtain a stable solution. Kumar and coworkers [28] not only allowed arbitrarily anisotropic scattering, but also considered boundaries with specular reflectances as well as boundaries with collimated irradiation (as discussed in Chapter 18). Stamnes and colleagues [29,30] investigated the same problem as Kumar and coworkers but also allowed for variable radiative properties and a general bidirectional reflection function at the surfaces. They decoupled the set of simultaneous equations using methods of linear algebra and found exact analytical solutions in terms of eigenvalues and eigenvectors. Other examples of the use of the one-dimensional discrete ordinates model as a tool to solve more complex problems may be found in [31–40].

16.4 One-Dimensional Concentric Spheres and Cylinders Applying the discrete ordinates method and taking advantage of the symmetries in a one-dimensional problem is considerably more difficult for concentric spheres and cylinders than for a plane-parallel slab. The reason is that the local direction cosines change while traveling along a straight line of sight through such enclosures.

Concentric Spheres Consider two concentric spheres of radius R1 and R2 , respectively. The inner sphere surface has an emittance

1 and is kept isothermal at temperature T1 , while the outer sphere is at temperature T2 with emittance 2 . If the temperature within the medium is a function of radius only, then the equation of transfer is given by equation (13.85), μ or, alternatively,

∂I 1 − μ2 ∂I + + βI = βS, r ∂μ ∂r

(16.22a)

 μ ∂ 2 1 ∂  2 (r I) + ) I + βI = βS, (1 − μ r ∂μ r2 ∂r

(16.22b)

where μ = cos θ is the cosine of the polar angle, measured from the radial direction (see Fig. 13.8). S is the radiative source function, S(r, μ) = (1 − ω) Ib +

ω 2



1

−1

I (r, μ ) Φ(μ, μ ) dμ .

(16.23)

The additional difficulty lies in the fact that equation (16.22) contains a derivative over direction cosine, μ, that is to be discretized in the discrete ordinates method. Applying the SN -method to equation (16.22), we obtain 0 /  μi d 2 1 ∂  2 (r Ii ) + + β Ii = β Si , (1−μ )I r ∂μ r2 dr μ=μi

i = 1, 2, . . . , N,

(16.24)

where Si is readily determined from equation (16.23) (and is independent of ordinate direction unless the medium scatters anisotropically). Equation (16.24) is only applied to the N principal ordinates since, similar to the slab, there is no azimuthal dependence. Since the direction vector μ is discretized, its derivative must be approximated by finite differences. We may write /

 ∂  (1−μ2 )I ∂μ

0 μ=μi

αi+1/2 Ii+1/2 − αi−1/2 Ii−1/2 , wi

(16.25)

572 Radiative Heat Transfer

FIGURE 16.1 Directional discretization and discrete ordinate values for one-dimensional problems.

which is a central difference with the Ii±1/2 evaluated at the boundaries between two ordinates, as shown in Fig. 16.1. Since the differences between any two sequential μi are nonuniform, the geometrical coefficients α are nonconstant and need to be determined. The values of α depend only on the differencing scheme and, therefore, are independent of intensity and may be determined by examining a particularly simple intensity field. For example, if both spheres are at the same temperature, then Ib1 = Ib2 = Ib = const, and also I = Ib = const. This then leads to    ∂ 2 (1 − μ ) = −2 wi μi , i = 1, 2, . . . , N. (16.26) αi+1/2 − αi−1/2 = wi ∂μ μ=μi This expression may be used as a recursion formula for αi+1/2 , if a value for α1/2 can be determined. That value is found by noting that I1/2 is evaluated at μ = −1 (Fig. 16.1), where (1 − μ2 )I = 0 and, therefore, α1/2 = 0. Similarly, IN+1/2 is evaluated at μ = +1 and also αN+1/2 = 0. The finite-difference scheme of equations (16.25) and (16.26) satisfies the relation [4]  +1 +1  ∂  (1 − μ2 )I dμ = (1 − μ2 ) I = 0 −1 −1 ∂μ / 0 N N      ∂  αi+1/2 Ii+1/2 − αi−1/2 Ii−1/2 wi = = (1 − μ2 )I ∂μ μ=μi i=1

i=1

= α3/2 I3/2 −α1/2 I1/2 +α5/2 I5/2 −α3/2 I3/2 +− · · · αN+1/2 IN+1/2 −αN−1/2 IN−1/2 Finally, the intensities at the node boundaries, Ii±1/2 , need to be expressed in terms of node center values, Ii . We shall use here simple, linear averaging, i.e., Ii+1/2 12 (Ii + Ii+1 ). Equation (16.24) may now be rewritten as αi+1/2 Ii+1 + (αi+1/2 − αi−1/2 )Ii − αi−1/2 Ii−1 μi d 2 (r Ii ) + + βIi = βSi , 2 dr 2rwi r

The Method of Discrete Ordinates (SN -Approximation) Chapter | 16 573

or, carrying out the differentiation and using equation (16.26), μi

αi+1/2 Ii+1 − αi−1/2 Ii−1 dIi μi + Ii + + βIi = βSi , dr r 2rwi αi+1/2 = αi−1/2 − 2wi μi ,

α1/2 = αN+1/2 = 0,

(16.27a) i = 1, 2, . . . , N.

(16.27b)

Equations (16.27) constitute a set of N simultaneous differential equations in the N unknown intensities Ii , subject to the boundary conditions [cf. equation (16.19)] 1− 1 q1 ,

1 π 1− 2 q2 , r = R2 : Ii = J2 /π = Ib2 +

2 π

r = R1 : Ii = J1 /π = Ib1 −

N N +1, +2, . . . , N (μi > 0), 2 2 N i = 1, 2, . . . , (μi < 0). 2 i=

(16.28a) (16.28b)

As for the one-dimensional slab the radiative heat flux and incident radiation are evaluated [cf. equations (16.20) and (16.21)] from G(r) =

N 

wi Ii (r),

(16.29a)

wi μi Ii (r),

(16.29b)

i=1

q(r) =

N  i=1

and

N/2    q(R1 ) = q1 = 1 Eb1 + wi μi Ii ,

(16.29c)

i=1 (μi 0)

Example 16.3. Consider a nonscattering medium at radiative equilibrium that is contained between two isothermal, gray spheres. The absorption coefficient of the medium may be assumed to be gray and constant. Using the S2 -approximation determine the radiative heat flux between the two concentric spheres. Solution From equation (16.27) we find, with N = 2, that α1/2 = α5/2 = 0, α3/2 = −2w1 μ1 = 2w2 μ2 = 4πμ (since μ2 = −μ1 > 0; we keep μ = μ2 as a nonnumerical value to allow comparison between the symmetric and nonsymmetric S2 -approximations). For a gray, nonscattering medium at radiative equilibrium we have β = κ and ∇ · q = 0, and the source function is, from equations (9.64) and (16.39), S = Ib = G/4π. i=1:

i=2:

μ 1 G dI1 μ − I 1 + I 2 + I1 = = (I1 + I2 ), dτ τ τ 4π 2 μ  1 dI1 − − (I1 − I2 ) = 0, −μ dτ τ 2 −μ

μ 1 dI2 μ + I2 − I1 + I2 = (I1 + I2 ), dτ τ τ 2 μ  dI2 1 μ − + (I1 − I2 ) = 0. dτ τ 2 μ

While addition of the two equations simply leads to a restatement of radiative equilibrium (as in Example 16.1), subtracting them (and multiplying by wi = 2π) leads to −μ or

d [2π(I1 + I2 )] + 2π(I1 − I2 ) = 0, dτ

574 Radiative Heat Transfer

q τ2 q 1 dG = − 2 = − 2 2. dτ μ μ τ Since for a medium at radiative equilibrium between concentric spheres Q = 4πr2 q = const and, therefore, τ2 q = const, the incident radiation may be found by integration, G(τ) =

τ2 q 1 + C, μ2 τ

where the two constants (τ2 q) and C are still unknown and must be determined from the boundary conditions, equations (16.28): I2 (τ1 ) = J1 /π,

I1 (τ2 ) = J2 /π.

Using the definitions for q and G, equations (16.29), q = 2πμ (I2 − I1 ) or I1 =

and G = 2π(I2 + I1 ),

    q q 1 1 G− , I2 = G+ , 4π μ 4π μ

the boundary conditions may be restated in terms of q and G as τ = τ1 : τ = τ2 :

μ τ1 q1 τ2 q  1 q1 q1 = 2 +C+ = 2 + 2 + C, μ μ μ μ τ1 τ1 μ τ2 q2 τ2 q  1 q2 q2 = 2 +C− = 2 4J2 = G − − 2 + C. μ μ μ μ τ2 τ2 4J1 = G +

Subtracting the second boundary condition from the first we obtain Ψ=

1 τ2 q =   . 2 J − J 2 τ1 1 τ 2 1 τ1 τ1 1 + 12 + 2 1 − 4μ 4μ τ2 τ2

√ For the symmetric S2 -approximation, with μ = 1/ 3, this equation becomes 1 Ψsymmetric = √   , 2 τ τ1 3 3τ1 1 1+ 2 + 1− 4 4 τ2 τ2 and for the nonsymmetric approximation with μ = 0.5, Ψnonsymmetric =

1

 . 1 τ1 1 + 2 + τ1 1 − 2 τ2 τ2 

τ21 

The accuracy of the S2 -approximation is very similar to that of the P1 -approximation, for which ΨP1 =

1   . 2 τ τ1 1 3τ1 1 + 12 + 1− 2 4 τ2 τ2

Note that the method is very accurate for large τ1 (large optical thickness) but breaks down for optically thin conditions (κ → 0), in particular for small ratios of radii, R1 /R2 . In the limit (κ → 0, R1 /R2 → 0) we find ΨP1 = ΨS2 ,nonsym → 2, while the correct limit should go to Ψexact → 1.

Numerical solutions to equations (16.27), allowing for anisotropic scattering, variable properties, and external irradiation, have been reported by Tsai and colleagues [41] using the S8 discrete ordinates method with the equalweight ordinates of Fiveland [12]. The same method was used by Jones and Bayazitoglu ˘ [42,43] to determine the combined effects of conduction and radiation through a spherical shell.

The Method of Discrete Ordinates (SN -Approximation) Chapter | 16 575

Concentric Cylinders The analysis for two concentric cylinders follows along similar lines. Again we consider an absorbing, emitting, and scattering medium contained between two isothermal cylinders with radii R1 (temperature T1 , diffuse emittance 1 ) and R2 (temperature T2 , emittance 2 ), respectively. For this case the equation of transfer is given by equation (13.104), sin θ cos ψ

∂I sin θ sin ψ ∂I − + βI = βS, r ∂r ∂ψ

(16.30)

where polar angle θ is measured from the z-axis, and azimuthal angle ψ is measured from the local radial direction (cf. Fig. 13.9). S is the radiative source function and has been given by equation (16.23). Introducing the direction cosines ξ = sˆ · êz = cos θ, μ = sˆ · ê r = sin θ cos ψ, and η = sˆ · ê ψc = sin θ sin ψ, we may rewrite equation (16.30) as μ ∂ 1 ∂ (rI) − (η I) + βI = βS. r ∂r r ∂ψ

(16.31)

For a one-dimensional cylindrical medium the symmetry conditions are not as straightforward as for slabs and spheres. Here we have I(r, θ, ψ) = I(r, π − θ, ψ) = I(r, θ, −ψ).

(16.32)

Therefore, the intensity is the same for positive and negative values of ξ, as well as for positive and negative values of η. Thus, we only need to consider positive values for ξi and ηi from Table 16.1, leading to Nc = N(N+2)/4 different ordinates for the SN -approximation, with quadrature weights w = 4wi . Equation (16.31) may then be i written in discrete ordinates form as 0 / μi d 1 ∂ (rIi ) − (ηI) + βIi = βSi , i = 1, 2, . . . , Nc . (16.33) r dr r ∂ψ ψ=ψi As for the concentric spheres case the term in braces is approximated as /

0 αi+1/2 Ii+1/2 − αi−1/2 Ii−1/2 ∂ (ηI) , ∂ψ w ψ=ψi i

i = 1, 2, . . . , Ni , ξi fixed.

(16.34)

In this relation the subscript i + 1/2 implies “toward the next higher value of ψi , keeping ξi constant.” The value of Ni depends on the value of ξi . For example, for the S4 -approximation we have from Table 16.1 Ni = 4 for ξi = 0.2958759 (four different values for μi , two positive and two negative) and Ni = 2 for ξi = 0.9082483. In the case of concentric cylinders the recursion formula for α, by letting I = S = const in equation (16.31), is obtained as  ∂η   αi+1/2 − αi−1/2 = w = w i = 1, 2, . . . , Ni , ξi fixed. (16.35) i i μi , ∂ψ ψ=ψi Again, α1/2 = 0 since at that location ψ1/2 = 0 and, therefore, η = 0. Similarly, αNi +1/2 = 0 since ψNi +1/2 = π and η = 0. Finally, using linear averaging for the half-node intensities leads to μi

αi+1/2 Ii+1 − αi−1/2 Ii−1 dIi μi + Ii − + βIi = βSi , dr 2r 2rw i

αi+1/2 = αi−1/2 + w i μi ,

α1/2 = αN+1/2 = 0,

i = 1, 2, . . . , Nc ,

(16.36a)

i = 1, 2, . . . Ni , ξi fixed.

(16.36b)

Equation (16.36) is the set of equations for concentric cylinders, for the Nc = N(N + 2)/4 unknown directional intensities Ii , and is equivalent to the set for concentric spheres, equation (16.27). The boundary conditions

576 Radiative Heat Transfer

for cylinders and spheres are basically identical [equations (16.28)], except for some renumbering, as are the expressions for incident intensity and radiative heat flux [equations (16.29)], that is, r = R1 : r = R2 :

J1 = Ib1 − π J2 = Ib2 + Ii = π Ii =

Nc 1− 1 Nc q1 , i = +1, +2, . . . , Nc (μi > 0),

1 π 2 2 1− 2 Nc q2 , i = 1, 2, . . . , (μi < 0),

2 π 2 Nc  w G(r) = i Ii (r),

(16.37a) (16.37b) (16.37c)

i=1

q(r) =

Nc 

w i μi Ii (r),

(16.37d)

i=1

and

N c /2    q(R1 ) = q1 = 1 Eb1 + w μ I , i i i

(16.37e)

i=1 (μi 0)

An example of the use of the discrete ordinates method in a one-dimensional medium is the work of Krishnaprakas [44], who considered combined conduction and radiation in a gray, constant property medium with various scattering behaviors.

16.5 Multidimensional Problems While the discrete ordinates method is readily extended to multidimensional configurations, the method results in a set of simultaneous first-order partial differential equations that generally must be solved numerically. As for one-dimensional geometries, the equation of transfer is slightly different whether a Cartesian, cylindrical, or spherical coordinate system is employed. We shall first describe the method for Cartesian coordinate systems, followed by a brief survey of the application of the method for multidimensional Cartesian, cylindrical, and spherical geometries.

General Formulation in Cartesian Coordinates For Cartesian coordinates equation (16.4) becomes, using equation (16.14), ξi

∂Ii ∂Ii ∂Ii + ηi + μi + β Ii = β Si , ∂x ∂y ∂z

i = 1, 2, . . . , n,

(16.38)

i = 1, 2, . . . , n.

(16.39)

where Si is again shorthand for the radiative source function ω  w j Φi j I j , 4π n

Si = (1 − ω) Ib +

j=1

Equation (16.38) is subject to the boundary conditions in equation (16.5) along each surface. For example, for a surface parallel to the y-z-plane, with nˆ = î and nˆ · sˆ j = sˆ j · î = ξ j , we have for all i with ξi > 0 (n/2 boundary conditions) Ii = Jw /π = w Ibw +

1 − w π

 ξ j 0 (pointing out of volume element) may actually overlap into the cell. Similarly, it is unlikely to have solid angle boundaries line up perfectly with the solid boundaries everywhere. They improved the accuracy of the method through pixelation , i.e., by breaking up Ω i into smaller pieces, to determine overlap fractions. Also, noting that the standard line-by-line iterative methods lead to unacceptably slow convergence in optically thick situations, they introduced a new scheme, which updates all directional intensities within a cell simultaneously, leading to convergence rates essentially independent of optical thickness [69,70]. Hassanzadeh and coworkers [146] also developed a method to accelerate convergence for optically thick media by carrying out iterations in terms of mean intensity, G/4π, as opposed to all directional intensities. Several other improvements to the method have been suggested. Kim and Huh [147] noted that most researchers broke up the total solid angle of 4π into N × N segments of equal polar angles θ and azimuthal angles ψ. This makes the Ωi very small near the poles (θ = 0, π), and large near the equator (θ = π/2). In their method, known as the FTn finite volume method, they suggest that, for n different polar angles θi , one should pick fewer azimuthal angles near the poles, namely a distribution of 4, 8, ..., 2n − 4, 2n, 2n, 2n − 4, ..., 8, 4 with growing θi . This results in n(n + 2) different solid angles (equal to the number of ordinates in the standard Sn scheme), with all Ωi being roughly equally large. The FTn angular discretization scheme has been explored by other researchers, as well [148,149]. Kamdem [76] suggested that the accuracy of the FAM can be improved by using higher refinement of the azimuthal angle around the end points. For example, if ψ is discretized between 0 and π in a certain two-dimensional problem, higher angular refinement in the vicinity of ψ = 0 and ψ = π is recommended. The finite angle method has been used with unstructured grids to model complex two- and three-dimensional geometries [59,144,150]. In recent years, the method has also been used for nongray media. Sun et al. [151] have demonstrated its use for a three-dimensional medium comprised of molecular gases. They used a hybrid Monte Carlo/FAM method, wherein the Monte Carlo method was used only for treatment of the nongray nature of the medium (wavenumber selection) while the FAM was used for solution of the RTE on an unstructured grid. Liu and coworkers [152] have also shown how the finite angle method with unstructured grids can be parallelized using domain decomposition. The method has also been employed in a number of combined heat transfer problems [153,154] and is included in several important commercial CFD codes, such as FLUENT [155], wherein an unstructured finite volume method is also used. In recent years, the finite element method has been

The Method of Discrete Ordinates (SN -Approximation) Chapter | 16 601

FIGURE 16.11 Temperature distribution for radiative equilibrium in a geometry similar to the one considered in Example 16.9: (a) DOM with S8 approximation, (b) FAM with exact same direction vectors as the S8 approximation. Dark shades indicate hot regions, while light shades indicate cold regions. Results are reproduced from [162].

demonstrated for both spatial and angular discretization of the RTE [156–158]. One major advantage of the FEM for angular discretization, akin to its advantage for spatial discretization, is that it can be elegantly expanded to support angular adaptivity [157]. Like the standard DOM, the FAM has also been formulated and demonstrated for media with spatially varying refractive index (graded media) [159,160]. In particular, Liu [159] formulated the discrete form of the RTE for a graded medium using the FAM for angular discretization and the finite volume method for spatial discretization in 3D. A structured orthogonal mesh and Cartesian coordinates were used. The methodology was validated against published results for a 1D slab and demonstrated for a 2D rectangle. More recently, Asllanaj and Fumeron [160] applied the FAM to complex 2D graded media discretized with an unstructured mesh.

Comparison of DOM with FAM Historically, the finite angle method was developed with the primary goal of conserving radiative energy, much like the finite volume method in space was developed (as a competitor to the finite difference method) to guarantee both local and global conservation of fluxes. Since its inception, a handful of studies [139,150,161,162] have compared the FAM with the standard DOM directly, i.e., using the same number of directions and cells in both computations. These studies have revealed that one of the side benefits of the FAM is that ray effects, which are strongly evident in most DOM results, are somewhat alleviated. Conceptually, integration over solid angles has an averaging effect and, therefore, is expected to mitigate “streaking” of energy along ordinate directions. To investigate ray effects in both methods, Mazumder and coworkers [150,162] performed computations in a twodimensional geometry similar to the one considered in Example 16.9, i.e., a square cavity with cold black walls on all sides except for a heated patch on the bottom wall between −0.05 ≤ x ≤ 0.05. A similar test problem—the heated strip placed between −0.1 ≤ x ≤ 0.1 being the only difference—is now regarded as a benchmark problem in the radiation community, and has been considered by several other researchers [74,143,163,164] and also in Example 16.9. Although conceptually simple, it has been found to be very challenging for most RTE solution methods. Calculations were performed for a range of optical thicknesses in purely isotropically scattering media as well as media with only absorption and emission. If radiative equilibrium prevails an isotropically scattering gray medium behaves identically to a purely absorbing-emitting (nonscattering) gray medium, since local absorption followed by re-emission is tantamount to isotropic scattering. Here, to highlight ray effects, results for radiative equilibrium calculation in a purely absorbing-emitting medium with an optical thickness κL = 0.01 are shown in Fig. 16.11. These results were obtained using a fine (160 × 160) grid. The streaking of energy (ray effect) along ordinate directions is clearly evident in the DOM results. Some evidence of streaking is also evident in the FAM results, especially as one moves away from the hot patch, i.e., closer to the top wall. However, on average, ray effects appear to have been somewhat mitigated by the FAM. Sankar and Mazumder [150]

602 Radiative Heat Transfer

FIGURE 16.12 Wall heat fluxes for the problem considered in Example 16.9 for an optical thickness, τL = 1: (a) top wall, (b) side wall. Results are reproduced from [150]. Th is the temperature of the hot patch at the bottom.

solved the same problem considered in Example 16.9 but with a fine unstructured mesh discretized using 9,492 triangular cells. The step scheme was used for both DOM and FAM for spatial discretization, while the S8 scheme (or an equivalent number of angles in the FAM, i.e., 1 × 40) was used for angular discretization. Figure 16.12 shows that the FAM produces results that are accurate if a fine spatial grid is coupled with a fine angular grid. Although minor oscillations in the predicted heat fluxes are still evident, they are not as pronounced as when a coarse angular grid is used with a fine spatial grid (see Example 16.9). DOM, on the other hand, still exhibits strong ray effects manifested by wild oscillations in the heat fluxes on both walls. In other studies, Liu and coworkers [165] have expressed the RTE in general body-fitted (curvilinear) coordinates [166], and applied both the standard discrete ordinates method and the finite angle method to a number of two- and three-dimensional problems. They found both methods to require similar amounts of CPU time, while the finite angle method was always slightly more accurate. Similar conclusions were drawn by Fiveland and Jessee [167] and by Kim and Huh [168], noting that the finite angle method outperforms standard discrete ordinates particularly in optically thin media, since it is less sensitive to ray effects. Coelho and coworkers [169] compared the performance of the finite angle method with that of the discrete transfer method [88] and, like Selçuk and Kayakol [87], found the finite angle method to be much more economical. Major advantages of the finite angle method are greater freedom to select ordinates, and the fact that the finite angle method conserves radiative energy. In addition, treatment of complex enclosures comes more natural to the finite angle method. For example, Baek and colleagues [170–172] used body-fitted coordinates to investigate radiation in several three-dimensional enclosures with gray, constant-property media.

16.7 The Modified Discrete Ordinates Method It was noted in Section 16.5 that the discrete ordinates method (in its standard or finite angle form) can suffer from ray effects, if directional discretization is coarse compared to spatial discretization, and if the medium contains small sources of strong emission (from walls or from within the medium). This prompted Ramankutty and Crosbie [173,174] to separate boundary emission from medium emission, as is done in the modified differential approximation of Section 15.8, i.e., letting I(r, sˆ ) = Iw (r, sˆ ) + Im (r, sˆ ).

(16.90)

The wall-related intensity field can be solved by any standard method as outlined in Section 16.5, while the RTE and boundary conditions for Im become   dIm σs σs = κIb − βIm (ˆs) + Im (ˆs ) Φ(ˆs , sˆ ) dΩ + Iw (ˆs ) Φ(ˆs , sˆ ) dΩ , (16.91) ds 4π 4π 4π 4π

The Method of Discrete Ordinates (SN -Approximation) Chapter | 16 603

Im (rw , sˆ ) =

1−

π

 ˆ s 0. (16.98)  π ˆ s 0.

References [1] H.C. Hottel, E.S. Cohen, Radiant heat exchange in a gas-filled enclosure: allowance for nonuniformity of gas temperature, AIChE Journal 4 (1958) 3–14. [2] H.C. Hottel, A.F. Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967. [3] V.A. Walther, J. Dörr, E. Eller, Mathematische Berechnung der Temperaturverteilung in der Glasschmelze mit Berücksichtigung von Wärmeleitung und Wärmestrahlung, Glastechnishe Berichte 26 (1953) 133–140. [4] T.H. Einstein, Radiant heat transfer to absorbing gases enclosed between parallel flat plates with flow and conduction, NASA TR R-154, 1963. [5] T.H. Einstein, Radiant heat transfer to absorbing gases enclosed in a circular pipe with conduction, gas flow, and internal heat generation, NASA TR R-156, 1963. [6] M.E. Larsen, J.R. Howell, The exchange factor method: an alternative zonal formulation of radiating enclosure analysis, ASME Journal of Heat Transfer 108 (4) (1985) 936–942. [7] H.P. Liu, J.R. Howell, Measurement of radiation exchange factors, ASME Journal of Heat Transfer 109 (2) (1987) 470–477. [8] S. Maruyama, T. Aihara, Radiation heat transfer of arbitrary three-dimensional absorbing, emitting and scattering media and specular and diffuse surfaces, ASME Journal of Heat Transfer 119 (1) (1997) 129–136. [9] S. Maruyama, Z. Guo, M. Higano, Radiative heat transfer of arbitrary three-dimensional, nongray and anisotropically scattering media and surfaces, in: Proceedings of the 11th International Heat Transfer Conference, Kyongju, Korea, vol. 7, 1998, pp. 457–462. [10] S. Maruyama, Z. Guo, Radiative heat transfer in arbitrary configurations with nongray absorbing, emitting, and anisotropic scattering media, ASME Journal of Heat Transfer 121 (3) (1999) 722–726. [11] W.W. Yuen, E.E. Takara, Development of a general zonal method for analysis of radiative transfer in absorbing and anisotropically scattering media, Numerical Heat Transfer – Part B: Fundamentals 25 (1994) 75–96. [12] A. Ma, Generalized zoning method in one-dimensional participating media, ASME Journal of Heat Transfer 117 (1995) 520–523. [13] J.J. Noble, The zone method: explicit matrix relations for total exchange areas, International Journal of Heat and Mass Transfer 18 (2) (1975) 261–269. [14] M.F. Modest, Radiative equilibrium in a rectangular enclosure bounded by gray non-isothermal walls, Journal of Quantitative Spectroscopy and Radiative Transfer 15 (1975) 445–461. [15] M.F. Modest, D. Stevens, Two dimensional radiative equilibrium of a gray medium between concentric cylinders, Journal of Quantitative Spectroscopy and Radiative Transfer 19 (1978) 353–365. [16] M.H.N. Naraghi, M. Kassemi, Radiative transfer in rectangular enclosures: A discretized exchange factor solution, in: Proceedings of the 1988 National Heat Transfer Conference, vol. HTD-96, ASME, 1988, pp. 259–268. [17] M. Kassemi, M.H.N. Naraghi, Analysis of radiation–natural convection interactions in 1-g and low-g environments using the discrete exchange factor method, International Journal of Heat and Mass Transfer 36 (17) (1993) 4141–4149. [18] M. Kassemi, M.H.N. Naraghi, Application of discrete exchange factor method to combined heat transfer problems in cylindrical media, in: Transport Phenomena in Materials Processing and Manufacturing, vol. HTD-336, ASME, 1996, pp. 151–160.

The Zonal Method Chapter | 17 639

[19] H.A.J. Vercammen, G.F. Froment, An improved zone method using Monte Carlo techniques for the simulation of radiation in industrial furnaces, International Journal of Heat and Mass Transfer 23 (1980) 329–337. [20] M.H.N. Naraghi, B.T.F. Chung, A unified matrix formulation for the zone method: a stochastic approach, International Journal of Heat and Mass Transfer 28 (2) (1985) 245–251. [21] H.C. Hottel, Radiant heat transmission, in: W.H. McAdams (Ed.), Heat Transmission, 3rd ed., McGraw-Hill, New York, 1954, ch. 4. [22] Z. Yin, Y. Jaluria, Zonal method to model radiative transport in an optical fiber drawing furnace, ASME Journal of Heat Transfer 119 (1997) 597–603. [23] Z. Yin, Y. Jaluria, Thermal transport and flow in high-speed optical fiber drawing, ASME Journal of Heat Transfer 120 (4) (1998) 916–930. [24] D.A. Nelson, Radiation heat transfer in molecular-gas filled enclosures, ASME paper no. 77-HT-16, 1977. [25] T.F. Smith, Z.F. Shen, J.N. Friedman, Evaluation of coefficients for the weighted sum of gray gases model, ASME Journal of Heat Transfer 104 (1982) 602–608. [26] T.F. Smith, Z.F. Shen, A.M. Al-Turki, Radiative and conductive transfer in a cylindrical enclosure for a real gas, ASME Journal of Heat Transfer 107 (1985) 482–485. [27] A.J. Sistino, Mean beam length and the zone method (without and with scattering) for a cylindrical enclosure, ASME paper no. 82-HT-3 1982. [28] M.H.N. Naraghi, B.T.F. Chung, B. Litkouhi, A continuous exchange factor method for radiative exchange in enclosures with participating media, ASME Journal of Heat Transfer 110 (2) (1988) 456–462. [29] M.H.N. Naraghi, M. Kassemi, Radiative transfer in rectangular enclosures: a discretized exchange factor solution, ASME Journal of Heat Transfer 111 (4) (1989) 1117–1119.

Chapter 18

Collimated Irradiation and Transient Phenomena 18.1 Introduction In recent years, there has been increasing interest in the analysis of radiative transfer in multidimensional absorbing, emitting, and scattering media with collimated irradiation. By collimated irradiation we mean external radiation that penetrates from the outside into a participating medium (as opposed to emission from a bounding surface), with all light waves being parallel to one another (or approximately so). Typical examples include solar radiation through the atmosphere and into the ocean, laser irradiation of particles or liquids, and so on. With the advent of short-pulsed lasers with pulse durations measured in pico- or even femtoseconds, transient radiation effects have also become of interest. Since, in engineering applications, virtually all transient radiation effects are due to short-pulsed lasers, these two topics are treated jointly in the present chapter. By collimated irradiation we mean that the intensity incident on a surface dA at location rw on the bounding surface of the medium, as shown in Fig. 18.1, may be written as Iow (rw , sˆ ) = qo (rw ) δ [ˆs − sˆ o (rw )]   = qo (rw ) δ μ − μo (rw ) δ[ψ − ψo (rw )], where δ is the Dirac-delta function, which is here defined as1 ⎧ ⎪ 0, |x| > , ⎪ ⎪ ⎨ δ(x) = ⎪ 1 ⎪ ⎪ ⎩lim , |x| < ,

→0 2

 2π  +1  f (ˆs) δ(ˆs − sˆ o ) dΩ = f (μ, ψ) δ(μ − μo ) δ(ψ − ψo ) dμ dψ = f (μo , ψo ), 0

4π

−1

(18.1)

(18.2a)

(18.2b)

and sˆ o = cos θo nˆ + sin θo (cos ψo ˆt1 + sin ψo ˆt2 ),

μo = cos θo ,

(18.3)

is the direction from which the collimated radiation impinges onto the medium (with nˆ the surface normal pointing into the medium and ˆt1 and ˆt2 two orthogonal unit vectors lying on the boundary surface). Equation (18.1) implies that the incident intensity is zero for all directions except for sˆ o , where it is infinitely large. The total heat flux within the collimated irradiation is determined from   ˆ sˆ δ(ˆs − sˆ o ) dΩ = qo sˆ o , qo = Iow (ˆs) s dΩ = qo (18.4) 4π

4π

that is, qo is the total radiative heat flux of the collimated irradiation through a surface normal to the rays. The component penetrating into the medium is then   (18.5) q c = 1 − ρ(rw , sˆ o ) qo sˆ c , where ρ is the reflectance of the interface in the direction of sˆ o . Since the irradiation penetrating into the medium may be refracted, the unit direction vector inside the medium is denoted as sˆ c , which may be different from sˆ o . 1. For a definition of the standard, one-dimensional Dirac-delta function see equation (10.111) in Section 10.9. Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00026-2 Copyright © 2022 Elsevier Inc. All rights reserved.

641

642 Radiative Heat Transfer

FIGURE 18.1 Collimated irradiation impinging on an arbitrary surface: (a) solar irradiation, (b) laser irradiation.

As indicated in the above expressions the magnitude of the irradiation qo , as well as the direction of irradiation, sˆ o , may vary over the surface of the enclosure, while the reflectance of the surface may vary with position and direction. In a strictly mathematical sense equation (18.1) introduces nothing new: Collimated irradiation could simply be treated as “strongly directional emission.” However, the discontinuity of intensity with direction causes problems with analytical as well as numerical solution techniques, thus warranting a separate approach for this type of problem. Most earlier works on collimated radiation dealt with solar radiation and other atmospheric or astrophysical applications. They are, therefore, generally limited to one-dimensional cases with uniform irradiation of a planar medium. For this simple case, some exact and approximate solutions have been given by Irvine [1], who used the Henyey–Greenstein phase function, a scattering phase function that adequately approximates the anisotropic scattering behavior of a large number of media [2], as given by equation (11.95). The identical problem for Rayleigh scattering was treated by Kubo [3] without, however, reporting any results. Armaly and El-Baz [4] found some approximate solutions for isotropic scattering in a finite-thickness slab using the kernel approximation. Their application was in the area of solar collectors. A similar problem was treated by Houf and Incropera [5], who investigated different approximate techniques for solar irradiation of aqueous media. Only with the advent of the laser as a research and manufacturing tool has nonsolar collimated radiation received some research attention. Smith [6] investigated the case of a uniform strip of collimated radiation incident on a semi-infinite medium. The resulting two-dimensional integral equation was reduced to onedimensional form using Fourier transforms. Hunt [7] investigated the effect of a cylindrical collimated beam impinging upon a finite layer. A solution was found for the basic case of Bessel-function varying intensity using Green’s functions. The first ones to apply this theory to laser radiation appear to be Beckett and coworkers [8], who investigated numerically the effect of a cylindrical beam with Gaussian variation penetrating through a finite layer. They showed how a diagnostic laser beam can be used to deduce radiative properties of an optically thick slab, such as single-scattering albedo, extinction, and absorption coefficients. Finally, a number of papers by Crosbie and coworkers [9–12] dealt with exact solutions to the general two-dimensional problem of collimated radiation impinging onto an absorbing–scattering layer. First, they treated collimated strip sources irradiating a semi-infinite body [9]; later, they discussed cylindrical beams falling on a semi-infinite body [10,11]. Collimated irradiation onto a rectangular medium was investigated by Crosbie and Schrenker [12] for isotropic scattering, while Kim and Lee [13] demonstrated the accuracy of the high-order discrete ordinates method by applying it to the same problem with anisotropic scattering. The exact solutions may be used as benchmarks for evaluation of approximate methods and may be necessary in cases where the requirement for highly accurate results justifies going through the trouble usually associated with these methods. In the area of heat transfer, however, approximate solutions often result in acceptable predictions for most practical situations.

Collimated Irradiation and Transient Phenomena Chapter | 18 643

Recently, some more advanced problems have also become of interest. Tan and coworkers considered combined conduction and radiative laser heating of glass [14], while Lacroix and colleagues [15] and Xu and Song [16,17] applied the discrete ordinates method to analyze the interaction of a laser beam with the plume or plasma generated by the laser. El Ammouri et al. [18] showed how laser beam fluctuations, caused by temperature fluctuations, can be employed as a tool to measure turbulence levels. Along the same line Ben-Abdallah [19] and coworkers analyzed the curved beam path that a laser traverses in a gas with varying refractive index. Lasers with ultra-short pulse lengths are utilized heavily in the emerging field of nanotechnology and also in biomedical engineering [20]. The radiative fields generated by such short-pulsed lasers may differ from those discussed in this book in two important aspects: (1) since light travels only 300 μm during a time span of 1 ps, transient effects must be accounted for, and (2) packing a fixed amount of energy into a pulse of extremely short duration leads to temporally extreme intensities. The former requires consideration of the transient term in the RTE, equation (9.20), and has been investigated by a number of researchers [21–32]. At very high intensities, many molecules are promoted to excited levels, which have different absorption behavior, making the absorption coefficient a function of intensity; this is known as saturable absorption [33]. Depending on the relative magnitude of absorption cross-sections, this may lead to bleaching (absorption coefficient decreases with intensity) or darkening (absorption coefficient increases with intensity) [34]. In addition, at very high intensities molecules may absorb more than a single photon at the same time, raising the molecule to an excited electronic state. This is known as multiphoton absorption [35–37]. If the total absorbed photon energy is high enough this, in turn, may lead to ionization or dissociation, which is known as photolysis. Several photochemical and photothermal models have been developed to describe short-pulse laser ablation of materials [38–41]. In biomedical engineering short-pulsed lasers are seen as promising tools for optical imaging (of tumors, etc.) [24,42,43] and for minimally invasive surgery (such as ablation of tumors) [44]. In this chapter we shall describe how problems involving radiative transfer in an absorbing, emitting, and anisotropically scattering medium of arbitrary geometry exposed to arbitrary collimated irradiation2 are dealt with by separating the collimated radiation (as it travels through the medium) from the rest of the radiation field. The problem is thus reduced to one without collimated irradiation, but with a modified radiation source term (now including a source due to the scattered part of the collimated irradiation). We shall see that it is possible to incorporate collimated irradiation readily into well-known approximate methods such as the P1 -approximation. Because of its emerging importance, the chapter also includes a very brief section on transient effects during short-pulsed laser irradiation.

18.2 Reduction of the Problem The equation of transfer for an absorbing, emitting, and anisotropically scattering medium is given by equation (9.18) as  σs sˆ · ∇I(r, sˆ ) = κ Ib (r) − βI(r, sˆ ) + I(r, sˆ  ) Φ(ˆs, sˆ  ) dΩ . (18.6) 4π 4π As usual, the lack of a spectral subscript implies that we deal either with spectral intensity or with a gray medium. We shall limit ourselves here to media with diffusely emitting and reflecting boundaries. Then the boundary condition for equation (18.6) is, for any location rw on the surface,  ρ(rw )   I(rw , sˆ ) = 1 − ρ(rw ) Iow (rw , sˆ ) + (rw )Ibw (rw ) + I(rw , sˆ  ) |nˆ · sˆ  | dΩ . (18.7)  π ˆ s 10−1 cm−1 (and the actual maximum k as the upper limit), and the same g-values are employed for the k-distribution correlation. The * k-values are then calculated from equation (19.142), resulting in almost identical values for the HITEMP and correlation distributions for large g (where the correlation is very accurate), and k HITEMP > kcorrelation for g < 0.8 (where the correlation underpredicts actual k-values); see Fig. 19.11a. Consequently, both SLW simulations yield similar results for small L (where large k dominate), with increasing underprediction by the correlation as L increases. The SLW results oscillate slightly around the LBL data, indicating that the accuracy of the SLW method depends mostly on a wise choice of g-ranges (with different optima for different slab widths L). Here the performance of the basic SLW method with only four gray gases has been demonstrated, while use of the latest versions (discussed in the following section) returns results identical to those of the FSK for homogeneous media. Again, with accurate and fast look-up tables available today, the use of correlations is to be discouraged for the SLW method, as well.

696 Radiative Heat Transfer

19.11 The FSK and SLW Methods for Nonhomogeneous Media Since the FSK and SLW methods are so similar, we will combine the discussion of how these two methods are applied to nonhomogeneous media. This will allow us to clearly highlight differences and commonalities; in fact, we will find that, if the SLW method is used with (the newly proposed [124]) Gaussian quadrature nodes for the complementary ALBDF, * Fn , the methods become almost identical, with only (very slight) differences in how the nongray stretching factor (FSK) or gray gas weight (SLW) an are evaluated. Extension of both methods to nonhomogeneous media with correlated absorption coefficients will require k-distributions or ALBDFs evaluated for different absorption coefficient states φ as well as different Planck function temperatures TP (which may be different from the local gas temperature). It is, therefore, important to understand the relationships between these distributions, as reported by Modest [141] and Solovjov et al. [124]. This summary comparison of methods has very recently been formalized by Wang and coworkers [146], and their development is presented here. For a nonhomogeneous medium the thermodynamic state of the gas enters the definition of the generalized full spectrum k-distribution, now defined as  ∞   1 Ibη (TP ) δ k − κη (η, φ) dη, (19.145) f (TP , φ, k) = Ib 0 which is a function of temperature through the Planck function, and also of φ through the state at which the absorption coefficient κη is evaluated (which includes the gas temperature, which may be different from the Planck function temperature). Following the development for narrow band correlated k-distributions in Section 19.9 [but now with an added Ibη (TP ) inside the integral], we find    ∞   dη  dη  1 1  f (TP , φ0 , k) = Ibη (TP ) δ k − κη (η, φ0 ) dκη = Ibηi (TP )  , (19.146)  dκη  Ib 0 dκη Ib (TP ) κη (ηi ,φ0 )=k

i

and, for the local state φ,

   dη  1  Ibηi (TP )  f (TP , φ, k ) =  dκη  Ib (TP ) κ ∗

i

, η (ηi ,φ)=k

(19.147)

∗

leading to, f (TP , φ, k∗ ) dk∗ = f (TP , φ0 , k) dk. and

 g(TP , φ0 , k) =

0



k

f (TP , φ0 , k) dk =

k∗

f (TP , φ, k∗ ) dk∗ = g(TP , φ, k∗ ),

(19.148)

(19.149)

0

i.e., as for narrow bands the cumulative k-distribution g, while a function of Planck function temperature, remains identical for all gas states (but with different corresponding k-values). This is the definition of full spectrum correlated k-distributions. Similarly, for a correlated absorption cross-section in the SLW method, we obtain from equation (19.136) F(C, φ0 , TP ) = F(C∗ , φ, TP ).

(19.150)

At any given position inside the medium with state φ, and the correlated absorption coefficient (or crosssection) referenced against an arbitrary, but fixed, state φ0 , there are (a minimum of) four different k-distributions or ALBDFs, as depicted in Fig. 19.12. Shown are distributions for mixtures of H2 O and CO2 in nitrogen, with φ0 = (T0 = 1000 K, x0,CO2 = 0.2, x0,H2O = 0.1) and φ = (T = 2000 K, xCO2 = 0.1, xH2O = 0.2)12 ; for the purpose of the present discussion one of the distributions was altered a little bit to make the absorption coefficient perfectly correlated (between the two given states). The two thin lines, k(T0 , φ0 , g0 ) and k(T, φ0 , g0 ) are based on the same absorption coefficient and, thus, have identical ranges for k, but at different values for g, depending on the Planck function temperature, which acts as a stretching factor. Similarly, the two thick lines, based on κη (φ), also have identical ranges of k, similarly stretched in g by the Planck function temperature. For a truly correlated 12. Note that the x and x0 are reversed in order to demonstrate their impact on correlatedness.

Solution Methods for Nongray Extinction Coefficients Chapter | 19 697

FIGURE 19.12 k-distribution equivalence of correlated absorption coefficients for varying states and Planck function temperatures (the indicates altered k-distribution to achieve perfect correlatedness); frame (b) shows a zoomed-in detail.

∗

absorption coefficient the four straight lines form a perfect rectangle 0–1–2–3. Changes of k at constant g imply different states φ for the evaluation of the absorption coefficient, while changes of g at fixed k indicate changes in Planck function temperature T. This implies that the k∗ (φ, k(g0 )) in equation (19.110) may be evaluated anywhere along the line 1–2, or k∗ (φ, k) = k∗ (T, φ, g) = k∗ (T0 , φ, g0 ),

(19.151)

or any Planck function temperature in between. Of course, in reality absorption coefficients (and cross-sections) are never truly correlated and, thus, equation (19.151) is only approximately correct, as seen from Fig. 19.13. Clearly, the depicted mixture of water vapor and carbon is not perfectly correlated, leading to different values of k∗ (or C∗ ), depending on whether point 2 is reached via point 1, or via point 3. Note that, since any evaluation along a horizontal line (1–2 or 0–3) involves only the absorption coefficient (or cross-section) of a single state, it is exact whether a gas is correlated or not. Evaluations along vertical lines, on the other hand (0–1 and 3–2) involve multiple states and are exact only for perfectly correlated gas coefficients. The different possibilities of evaluating k∗ (or C∗ ) as well as gray gas weights a, have given rise to a number of competing schemes, which will be discussed and compared later in this section.

The Full Spectrum Correlated-k (FSCK) Method As was done for the NBCK method of Section 19.9, a reordered RTE is obtained by multiplying equation (19.112) and boundary condition (19.113) by the Dirac-delta function δ(k − kη ), i.e., using the absorption coefficient at a representative reference state φ0 introduced in equation (19.110), but now followed by integration over the entire spectrum. This leads to    σs dIk = k∗ (φ, k) f (T, φ0 , k)Ib − k∗ (φ, k) + σs Ik + Ik (ˆs ) Φ(ˆs, sˆ  ) dΩ (19.152) ds 4π 4π with boundary condition Ik = Iwk = w f (Tw , φ0 , k)Ibw + (1 − w ) and

 Ik = 0

∞

Iη δ(k − kη ) dη.

1 π

 ˆ s 0 (or relatively large Cmin ) can easily account for large parts of the spectrum with negligible absorption coefficient, while with the FSCK scheme use of gmin > 0 or the transformation given by equation (19.143) is less common and was not used. A comparison between the SLW, WSGG and experiment for an oxy-fuel furnace was recently given by Webb et al. [170]. Several extensions to the FSCK schemes have also been given: Pal and Modest [151] demonstrated that the FSCK schemes remain valid at high pressures (where strong line broadening and overlap may destroy the uncorrelateness between species required for mixing models). Cai and Modest [171] and Wang et al. [172] developed different FSCK extensions to include suspended particles (with temperature different from the surrounding gas). Some summary conclusions may be drawn as follows: 1. The historical first comers, i.e., SLW-1 and FSCK-1, do not preserve local emission and are more susceptible to error; therefore, they are not recommended. 2. The scaled versions of the method, FSSK and its SLW counterpart, have been shown to perform well for some 1D problems, but require more research to determine viable scaling functions for general geometries. 3. Given the latest SLW developments [124] (employing predetermined sets of ALBDFs following high-order quadrature rules), SLW and FSCK methods have merged to a point that only quasi-spectral summation/integration remains different, with FSCK (Gaussian quadrature) giving somewhat superior results compared to SLW (trapezoidal integration). 4. All methods FSCK/SLW-n (n = 2, 3, 4) perform almost equally well. Of these FSCK/SLW-2 are perhaps the most accurate, but also require some extra evaluations and, thus, a few percent additional CPU time. 5. All recommended methods FSCK/SLW-n (n = 2, 3, 4) are relatively insensitive to the choice of reference temperature. Optimal values seem to be close to the Planck-mean emission temperature, with little additional errors if a somewhat lower temperature is chosen.

19.12 Evaluation of k-Distributions and ALBDFs Since absorption coefficient κη and absorption cross-section Cη differ only by a constant factor, the evaluation of ALBDFs and cumulative k-distributions is essentially identical, and we will outline only the latter here. Full spectrum k-distributions f (T, k) and cumulative k-distributions g(T, k) are evaluated exactly as outlined in Section 10.9, except that δη/Δη is replaced by Ibη δη/Ib , or    Ibηi (T)  δη    [H(kj + δkj − κη ) − H(kj − κη )]. f (T, kj ) δkj  Ib (T)  δκη i

(19.181)

i

If the simple method described in Section 10.9 is used, the k-distributions can be found simultaneously for any number of temperatures Ti (i = 1, 2, ..., I): The relevant (i.e., contributing) part of the total spectrum is broken up into N equal subintervals δη, the absorption coefficient κη is evaluated at the center of each interval and, if kj ≤ κη ≤ kj+1 , the value of each f (Ti , kj ) δkj is incremented by Ibη (Ti , ηn ) δη/Ib (Ti ). At the end, the cumulative

712 Radiative Heat Transfer

function g(T, k) is again calculated from equation (10.117), or g(Ti , kj+1 ) =

j 

f (Ti , kj ) δkj = g(kj ) + f (Ti , kj ) δkj .

(19.182)

j =1

fskdist is the corresponding Fortran program in Appendix F that evaluates the f (Ti , kj ) and g(Ti , kj ) for a set of temperatures Ti and absorption coefficients kj , as well as the a(Ti , kj ). Figure 19.9 shows the full spectrum k-distributions for 10% CO2 in nitrogen for two temperatures. For efficient integration of equation (19.134) it is desirable to have the function a(T, g) as smooth as possible, which—in turn—depends on the accuracy with which the f (T, k) are evaluated. Different smoothing schemes have been discussed in the original paper by Modest and Zhang [116]. As indicated earlier, assembling narrow band or full spectrum k-distributions from high-resolution databases is a very time-consuming task. First, the absorption coefficient must be calculated at fine spectral resolution for all relevant temperatures, pressures, and concentrations. This was apparently first done by Rivière and coworkers [173] for various gases, using the HITRAN 1992 database together with some proprietary French high-temperature extensions. For repeated calculations absorption coefficients may be precalculated and placed into an absorption coefficient database, such as the one by Wang and Modest [125], which was based on HITEMP 1995 [17] (H2 O and CO), and CDSD-1000 [174] for CO2 . Wang and Modest’s database includes 23 temperatures (300–2500 K), 24 pressures (0.1–30 bar), and 5 concentrations, requiring about 225 GB of storage. (In its present version the database also includes several hydrocarbon species and has been updated to HITRAN 2008 and HITEMP 2010.) Next, in the case of gas mixtures, the absorption coefficients of individual species are added; then the k-distribution is found from equations (19.181) and (19.182). This should be done for closely spaced δkj . Finally, the resulting function is inverted to determine the relevant k for desired quadrature points g. Clearly, this process of assembling k-distributions is too involved to make it part of an overall heat transfer analysis, or even a pure radiation calculation. Rather, they must be available from simple correlations or from databases.

Correlations for the ALBDF and Full Spectrum k-Distributions Denison and Webb [3,97,175] calculated large numbers of ALBDFs for water vapor and carbon dioxide, using the HITRAN92 database [115] together with the high-temperature extrapolation scheme of Hartmann and coworkers [176]. The resulting ALBDFs were then presented in the form of relatively straightforward correlations for engineering use. The correlations were subsequently updated (in terms of cumulative k-distributions) by Modest and coworkers using the then-new HITEMP 1995 database [147,177], and one more time for CO2 using CDSD-1000 [143], after it was recognized that HITEMP 1995 was seriously in error for CO2 above 1000 K. After the appearance of HITEMP 2010 two more sets of ALBDF correlations were developed, first by Liu et al. [178] and then by Pearson et al. [179], the latter also adding one more species, CO, and containing algorithms to use the correlations for nonatmospheric pressures. Recognizing that, in a semi-log sense, the ALBDF (or full spectrum k-distributions) resemble a hyperbolic tangent function (cf. Figs. 19.11 to 19.17), all correlations have the basic form (here written for the FSK) g(TP , Tg , x; k) =

  1 1 + tanh P(TP , Tg , x; k) , 2

(19.183)

with the function P given as P(TP , Tg , x; k) =

3  3  3  l=0 m=0 n=0

 almn

Tg Tref

n '

TP Tref

)m 

 log10

k0 (Tg , k, x) kref

l ,

(19.184)

where the Tref and kref are reference values and almn are correlation constants. With k0 = k equations (19.183) and (19.184) give the cumulative k-distribution for air broadening, i.e., for small amounts of absorbing gas in air (x 0, accounting for collision broadening due to collisions with air molecules; see Section 10.4). If the mole fraction of the absorbing gas is substantial, self-broadening must be accounted for (collisions between two molecules of the absorbing species), resulting in a shift in g. For CO2 this shift is negligible (since CO2 and air molecules have roughly the same size), and k0 = k. However, for water vapor the effect is quite substantial (since

Solution Methods for Nongray Extinction Coefficients Chapter | 19 713

H2 O molecules are much smaller than air molecules), and must be accounted for. Modest and Singh [177] give a correlation for k0 (Tg , k, x) as  log10

      n   2  2  1 Tg l k0 k k = log10 + blmn log10 [x]m+1 . kref kref kref Tref m=0 n=0

(19.185)

l=0

(If a correlation for the ALBDF is desired, one simply replaces the g in equation (19.183) by F, as well as k by C in all three equations.) As an example, results of the correlational fit (19.183) by Modest and Mehta [143] (based on the CDSD-1000 database), for a 10% CO2 –N2 mixture at 1000 K are compared in Fig. 19.11a with the one calculated from the newer HITEMP 2010 (which is based on a CDSD-1000 version), both directly and assembled from the NBKDIR database described in the following section. It is observed that the fit is generally very good for large values of k (> 10−3 cm−1 , the range over which most of the heat transfer takes place in common applications). Extensive tests have shown that Planck-mean absorption coefficients and slab emissivities determined with this correlation are never in error by more than 10% for CO2 and 8% for H2 O, respectively. For the convenience of the reader several Fortran routines are included in Appendix F for the evaluation of equations (19.183) to (19.185). fskdh2o and fskdco2 are for the correlations given by Modest et al. [143,177], while fskdh2odw and fskdco2dw for the older Dennison and Webb correlations [3,97] are also included in Appendix F. The newest correlation by Pearson et al. [179] contains more terms [each of the series in equations (19.184) and (19.185) has one additional term], and may be downloaded from http://albdf.byu.edu. The correlation by Liu et al. [178] has a different form to account for water vapor self-broadening. It should be noted that none of the correlations allow for gas mixtures, i.e., finding the ALBDF or k-distribution for mixtures involves another layer of effort as well as approximation. This, together with the fact that today databases for k-distribution of mixtures are available, and that these can be retrieved faster than from correlations, now limits the usefulness of such correlations.

Narrow Band k-Distribution Databases k-distributions may be databased in narrow band form, which can then be collected into full spectrum versions, or they may be stored directly in full spectrum form. Narrow band k-distributions depend only on the local gas state, i.e., φ = (T, p, x), and—assuming a fixed constant pressure—can be obtained by double interpolation in temperature and mole fraction (albeit for many narrow bands). They have the additional advantages that they lend themselves better to mixing of species (as shown later in this section), and they can be used to obtain wide band k-distributions. Full spectrum k-distributions, on the other hand, also depend on the Planck function temperature, i.e., a triple interpolation is required, and the size of the database becomes correspondingly larger. If gas mixture distributions are needed, size of the database and number of interpolations grow further. As already indicated in Chapter 10, Soufiani and Taine [64] were the first to assemble a narrow band database for H2 O and CO2 , using the HITRAN 1992 database together with some proprietary French high-temperature extensions. This has since been updated by Rivière and Soufiani [93] employing their statistical narrow band database described in Section 10.8. The new EM2C database now contains 51 bands for H2 O, obtained from HITEMP 2010 [2], and 40 bands for CO2 , generated from CDSD-4000 [180], for 16 temperatures at atmospheric pressure, employing the same 7 Gauss-Lobatto quadrature points as the original version. Similarly, the more voluminous high-accuracy narrow band database by Wang and Modest [125], originally based on CDSD1000 [174] (for CO2 ) and HITEMP 1995 [17] (for H2 O), was updated and augmented by Cai and Modest [31] to include additional species (CO, CH4 , and C2 H4 ), as well as larger temperatures (38 temperatures up to 4000 K) and a larger pressure range (34 values between 0.1 and 80 bar). Each k-distribution is given for nth order nested quadrature schemes: the (variable) nth order guarantees 0.5% accuracy, but the nesting allows for the use of lower orders (with fewer quadrature points). This Narrow Band K-Distribution for InfraRed (NBKDIR) database is continuously updated to incorporate the newest spectroscopic data; at the time of print all k-distributions have been obtained from HITEMP 2010 [2] (H2 O, CO2 , and CO) and HITRAN 2008 [105], (CH4 and C2 H4 ). Both the EM2C and NBKDIR databases are included in Appendix F.

714 Radiative Heat Transfer

Full spectrum k-distributions from narrow band data are assembled using the definition of the cumulative k-distribution,  k  ∞   1 g(TP , φ0 , k) = f (TP , φ0 , k) dk = Ibη (TP )H k − κη (φ0 ) dη Ib 0 0  Ibj g j (φ0 , k), (19.186) = Ib j∈[all NB’s]

where H is the Heaviside step function, and Ibj is the Planck function integrated over the narrow band:  Ibj = Ibη dη. (19.187) Δη j

Routines to assemble full spectrum k-distributions from narrow band data are included in NBKDIR.

Full Spectrum k-Distributions for Mixtures Similar to narrow band k-distributions, variable mixtures of different absorbing gases, and perhaps the addition of nonscattering particles, such as soot, pose no additional difficulty, in principle, because the absorption coefficient of all species can simply be added up. In practice, however, because of the considerable effort involved, one would like to precalculate and database all necessary k-distributions, before embarking on detailed heat transfer calculations. Because of the infinite number of possible mixture concentrations this would result in huge databases many GB in size. It is, therefore, desirable to build full spectrum k-distributions for arbitrary gas mixtures from relatively few distributions databased for individual species. Construction of mixture k-distributions can be avoided as long as the absorption coefficient of each species is unaffected by the other species (i.e., collisions with varying amounts of other species have no impact on line broadening), by employing a double integration approach [6,8,12,98], but the integral in equation (19.134) or (19.160) becomes a multiple integral (one for each species). Numerical effort would increase from N RTE evaluations (number of quadrature points), to NM RTE evaluations in a mixture of M absorbing species, quickly eliminating the advantages of the FSK methods. Collecting mixture full spectrum k-distributions is essentially identical to the narrow band mixing case described in Section 10.9, equations (10.123) through (10.138) and, therefore, the discussion here will be very brief. Variable Mole Fraction of a Single Absorbing Gas As in equation (10.123) we consider an absorption coefficient that is linearly dependent on its partial pressure, i.e., a gas whose line broadening is unaffected by its own partial pressure, or κxη (Tg , p, x; η) = xκη (Tg , p; η).

(19.188)

Going through the identical steps as for narrow bands, we obtain fx (TP , Tg , p, x; kx ) =

1 f (TP , Tg , p; kx /x), x

(19.189)

and g(TP , Tg , p; k) = gx (TP , Tg , p, x; kx ),

(19.190)

where the arguments, from the definition of full spectrum k-distributions, now include a Planck function temperature TP . As before, the k vs. g behavior is independent of mole fraction, with kx smaller than k(g) by the multiplicative factor x for any value of g. Equation (19.189) also implies that the nongray stretching factor a remains unaffected if the mole fraction is changed [see equations (19.133) and (19.159)]. Single Absorbing Gas Mixed with Gray Medium Identical to its narrow band equivalent, if κpη (Tg , p, κp ; η) = κη (Tg , p, η) + κp ,

(19.191)

it follows that fp (TP , Tg , p, κp ; kp ) = f (TP , Tg , p; k = kp − κp ) g(TP , Tg , p; k) = gp (TP , Tg , p, κp ; kp = k + κp ).

(19.192) (19.193)

Solution Methods for Nongray Extinction Coefficients Chapter | 19 715

As for narrow bands, for the same g the k-values are displaced by a constant additive factor κp and, as for the variable mole fraction case, equation (19.192) implies that the weight factor a remains unaffected. Superposition of k-Distributions Under certain conditions it may be acceptable to neglect overlap of spectral lines from different species. For example, Bansal and coworkers [119] have shown that radiation in air plasma is dominated by few widely spaced electronic excitation lines of monatomic N and O. Thus, if we consider a mixture of M different absorbing gases, whose absorption coefficients do not overlap each other anywhere across the entire spectrum, then the k-distributions of the individual species are unaffected by the others, i.e., the spectral locations where k = κmη for the mth species remain unaffected by the other gases, and

fmix (TP , Tg , p; k) =

M 

fm (TP , Tg , p; k),

(19.194)

m=1

where the fm are the k-distributions of the individual species. Keeping in mind that, for nonoverlapping absorption coefficients, each species must have large parts of the spectrum with κmη ≡ 0, we integrate equation (19.194) as  1 − gmix (TP , Tg , p; k) =

∞

f (TP , Tg , p; k) dk =

k

M  

gmix (TP , Tg , p; k) =

M 

fm (TP , Tg , p; k) dk =

k

m=1

or

∞

M  

 1 − gm (TP , Tg , p; k)

m=1

gm (TP , Tg , p; k) − M + 1.

(19.195)

m=1

Therefore, the cumulative k-distribution for a nonoverlapping mixture is constructed by simply adding up the individual components. Note that 1 − gm is the (Planck function weighted) part of the spectrum where κmη > 0 and, thus, their sum can never exceed unity. k-Distributions for Random Overlap As pointed out by Taine and Soufiani [181], there is no physical reason why there should be any significant correlation between the spectral variation of absorption coefficients of different gas species. If one treats the absorption coefficients of the M species as statistically independent random variables of wavenumber, the k-distributions are said to be statistically uncorrelated. Using such an argument, Solovjov and Webb [12] postulated that the cumulative k-distributions are multiplicative, or

gmix (TP , Tg , p; k) = g1 (TP , Tg , p; k) × g2 (TP , Tg , p; k) × . . . =

M :

gm (TP , Tg , p; k).

(19.196)

m=1

Statistically Uncorrelated Gas Mixtures Taine and Soufiani [181] argued that if the M species in a gas mixture are statistically uncorrelated, then their transmissivities should be multiplicative, i.e.,

τmix =

M :

τη,m ,

(19.197)

m=1

and this was shown to be true on a narrow band basis by comparison with LBL calculations (see [181] as well as Fig. 10.20). Based on equation (19.197) Modest and Riazzi [148] developed the narrow band mixing scheme in Section 10.9, equations (10.133) through (10.138). The same argument can also be made at the full spectrum level. Defining a full spectrum transmissivity as τ(T, L) =

1 Ib

 0

∞

Ibη (T)e−κη L dη,

(19.198)

716 Radiative Heat Transfer

we can manipulate this expression, using the definition of the Dirac-delta function given by equation (10.111), to obtain  ∞  ∞  ∞  ∞ 1 1 τ(T, L) = Ibη (T)e−κη L δ(k − κη ) dk dη = e−kL Ibη (T)δ(k − κη ) dη dk Ib η=0 Ib η=0 k=0 k=0  1  ∞ e−kL f (T, k) dk = e−kL dg, (19.199) = k=0

g=0

which is identical to equation (10.134). Assuming equation (19.197) to hold, the analysis is identical to the narrow band case, leading to  gmix (T, kmix ) =



1 g1 =0



1 g2 =0

H[kmix − (k1 + k2 )]dg2 dg1 =

1 g1 =0

g2 (kmix − k1 ) dg1

(19.200)

for a two-component mixture, and  gmix (T, kmix ) =



1 g1 =0

....

1 gM =0

H[kmix − (k1 + .... + kM )]dgM ....dg1

(19.201)

for a mixture of M species, but now using full spectrum k-distributions. Mixtures of Gases and Particles If the particles are assumed to be gray, equation (19.193) applies directly, i.e., the full spectrum k-distribution is found for the gas mixture as g(TP , Tg , p; k). The mixture’s k-distribution is then determined by simply adding the particles’ constant absorption coefficient κp for every value of g. If the particles’ absorption coefficient is nongray, mixing must be performed at the narrow band level, assuming that κpη = κp, j is constant across narrow band range j. Any of the narrow band schemes described in Section 10.9 may be employed. For example, with the Modest and Riazzi [148] model, equation (10.138), for each narrow band:  gmix, j (φ0 , k) =



1 g1 =0

....

1 gM =0

H[kmix − (k1 + .... + kM + κp, j )]dgM ....dg1 ,

(19.202)

where φ0 is the (reference) state at which the absorption coefficients of the gas are evaluated, and κp, j has been added to the argument inside the Heaviside function according to equation (10.131). The full spectrum k-distribution is then determined from g(TP , φ0 , k) =

 j∈[all NB’s]

Ibj Ib

(TP ) gmix, j (φ0 , k).

(19.203)

Test Calculations A real gas mixture will, of course, always have some spectral overlap, and the absorption coefficient will never be quite statistically uncorrelated. Figure 19.21 shows the case of a 10% CO2 –20% H2 O–70% N2 mixture, with the absorption coefficient evaluated at a reference temperature of T0 = 1000 K. Full-spectrum k-distributions were evaluated for a number of Planck function temperatures by five methods: (i) the exact k-distribution for the mixture was found from the HITEMP database [2], (ii) individual k-distributions were found for CO2 and H2 O, and a mixture distribution was found from equation (19.195) (superposition, neglecting overlap), (iii) similarly a mixture distribution was determined from equation (19.196) (random overlap), (iv) the Modest and Riazzi full spectrum mixing of equation (19.200) was employed, and finally (v) mixing was done on a narrow band basis, using the Modest and Riazzi scheme, equation (10.137), after which the full spectrum distribution was obtained from equation (19.186). Figure 19.21 shows that all approximate methods predict the correct distribution very well for large values of k. For very small values of k substantial overlap between species is to be expected, and the superposition method fails. The product method, on the other hand, appears to give good accuracy for nearly all conditions. The uncorrelated transmissivity scheme of Modest and Riazzi, when applied on a full-spectrum basis, displays good accuracy similar to the multiplication scheme, but outperforms it for very small values of k. When applied at the narrow band level, the uncorrelated mixing rule is virtually exact, i.e., lines become indistinguishable. This is not true, however, for multiplicative mixing (as seen, for example, in the flame calculation shown later in this section in Fig. 19.23). Note that the k(g) levels decrease with

Solution Methods for Nongray Extinction Coefficients Chapter | 19 717

FIGURE 19.21 Full-spectrum k-distributions for a 10% CO2 –20% H2 O–70% N2 mixture without soot, for various Planck function temperatures (absorption coefficient evaluated at 1000 K).

FIGURE 19.22 Full-spectrum k-distributions for a 10% CO2 –20% H2 O–70% N2 mixture with soot, for various Planck function temperatures (absorption coefficient evaluated at 1000 K).

temperature, because of the strong effect of the rotational band of water vapor at long wavelengths, favoring low temperatures. When applied to evaluate the radiative source within a homogeneous slab [148] (for which directly calculated k-distributions return exact answers), the multiplication scheme incurred errors of 4% and 5% when mixing on narrow band and full spectrum levels, respectively, while the Modest and Riazzi mixing scheme resulted in 0% and 1% error, respectively. These findings were corroborated by Demarco et al. [165], who tested several spectral models (WSGG with parameters from Smith and coworkers [80]; statistical narrow band using the EM2C database [64]; and SLW and FSCK using the EM2C database [64] to assemble full-spectrum k-distributions) as well as mixing models (superposition, multiplication, and uncorrelated mixing), and found that the combination of FSCK with the Modest and Riazzi mixing scheme gave the most accurate results. The methods were also tested for gas mixtures with nonscattering soot, using equation (11.126) with a volume fraction of fv = 5 × 10−6 and a refractive index m = 1.89 − 0.92i. Clearly, none of the first four methods should work terribly well, since strong overlap is assured, and the soot absorption coefficient is anything but random. That Fig. 19.22, nonetheless, shows reasonable agreement is a consequence of the fact that the soot kdistribution dominates the mixture, especially at high Planck function temperatures (favoring short wavelengths

718 Radiative Heat Transfer

with strong soot and weak gas radiation). For the same reason k(g) values now increase with temperature. None of the assumptions underlying scheme (v) are violated, and it again displays superb accuracy (with its line indistinguishable from the exact one). Other work comparing mixing models can be found in [32] (comparing multiplication and Modest and Riazzi mixing for the scaled Sandia D flame considered in [33], both on a narrow band level and full spectrum using correlations) and [155] (same, plus also a radiative heat source within a 3-gas homogeneous slab). Both confirmed the previous conclusions, i.e., while computationally more expensive, the Modest and Riazzi scheme is more accurate, especially when mixing on a narrow band level. When mixing at the full spectrum level, the multiplicative scheme may be preferable, i.e., the slight increase in accuracy does not warrant the extra expense. Most research papers employing mixing, primarily of narrow bands, have employed the Modest and Riazzi scheme, e.g., [35,100,135–139,165,166,182], while most work employing the SLW method (by definition full spectrum) preferred multiplicative mixing.

Full Spectrum k-Distribution and ALBDF Databases Because of the extreme data storage requirements for complete sets of ALBDFs or cumulative k-distributions, these have not been available until only a few years ago. However, today’s computers routinely have many GB of rapidly accessible memory, removing this obstacle. First, a large compilation of ALBDFs for H2 O, CO2 and CO was made available by Pearson et al. [179], in which ALBDFs for individual species are tabulated for 10 pressures (0.1–50 atm), each file containing 28 gas and reference temperatures (300–3,000 K) in ASCII format; for H2 O only (for which self-broadening is important) data are also listed for 10 mole fractions. Since the table predates the latest SLW developments as given by Solovjov et al. [124], the ALBDF F is tabulated for fixed values of absorption cross-section C. Therefore, the table must be inverted for use with the newer SLW-1b, SLW-2 and SLW-3 methods. Files for individual pressures can be downloaded from http://albdf.byu.edu. Since only individual species are databased, the ALBDF for a mixture must be determined with a mixing rule, such as the multiplicative mixing of Solovjov and Webb [12]. Wang and coworkers [142,155–157,183] have generated several full-spectrum cumulative k-distributions, which are listed in Appendix F with instructions how they may be obtained from a dedicated repository. In a first effort [155] they tabulated an optimized set of FSCKs for the three most important species (H2 O, CO2 and CO), assembling 32-point Gauss-Chebyshev quadrature sets of k(TP , φ, g) and corresponding a(TP , φ, g) (for the FSCK-1 and FSCK-2 schemes), requiring 5 GB of storage. They employed an optimized data access method championed by Wang and Modest [125], so that only the data needed by the calculations are loaded into memory, saving both time for reading the whole table and memory when carrying out a specified set of calculations. For this purpose data are stored separated by both pressure and mole fraction for the three species. For a single pressure, required memory for all k-distributions is reduced to 58.4 MB for 32 quadrature points, and 34.2 MB for 16, respectively. The range of included thermodynamic states is listed in Table 19.4. Determination of a kdistribution from the table requires a six-dimensional linear interpolation (Planck temperature, gas temperature, pressure and 3 mole fractions) for both k and a. The accuracy of the six-dimensional linear interpolation was thoroughly tested by comparing the radiative source in a homogeneous slab against direct FSK calculations, with errors never exceeding 2%, rivaling the accuracy of k-distributions assembled from Cai and Modest’s [31] narrow band database together with Modest and Riazzi [148] mixing. A comparison of the CPU times required to assemble k-distributions for random thermodynamic states by various methods is given in Table 19.5. Here the exact FSK is calculated from a LBL database of absorption coefficients, and, while the required CPU time is large, it is considerably lower than if calculated directly from HITEMP 2010. Assembly from a narrow band database is considerably faster; using the multiplicative approach [12] for the required mixing of species is an order faster than the Modest and Riazzi scheme [148], with the latter, however, being much more accurate (see Fig. 19.23). Using the FSK database not only provides the most accurate results, it also allows for fastest retrieval of cumulative k-distribution, even beating out the inaccurate correlations. An example of an application is given in Fig. 19.23 comparing the radiative source for the scaled Sandia flame shown in Fig. 15.14, as calculated by several methods and compared against LBL results. It is seen that the error of the tabulated FSK against LBL is smaller than the maximum 2% interpolation error. Additionally, it is observed that, when assembling the full-spectrum k-distributions from a narrow band database, the Modest and Riazzi mixing rule provides superb accuracy, while multiplicative mixing results in substantial error; similar conclusions were drawn by [32]. And, as expected, using correlations are rather inaccurate regardless of the mixing scheme.

Solution Methods for Nongray Extinction Coefficients Chapter | 19 719

TABLE 19.4 Precalculated thermodynamic states contained in FSK look-up tables of Wang et al.: without soot [142,155], with soot [156,157]. Parameters

Range

Values

Number of points

Species

H2 O, CO2 and CO

Pressure (total)

0.1–0.5 bar

Every 0.1 bar

0.7 bar

0.7 bar

1.0–14.0 bar

Every 1.0 bar

3 34

15.0–80.0 bar

Every 5.0 bar

Gas temperature

300–3,000 K

Every 100 K

28

Reference temperature

300–3,000 K

Every 100 K

28

Mole fractions of H2 O

0.0–0.05

Every 0.01

13

0.10–0.20

Every 0.05

0.25–1.0

Every 0.25

Mole fractions of CO2

0.0–0.05

Every 0.01

0.25–1.0

Every 0.25

Mole fractions of CO

0.0–0.5

[0.0, 0.01, 0.05, 0.10, 0.25, 0.50]

6

Volume fractions of soot

0–10 ppm

[0 ppm, 10 ppm]

6

10

TABLE 19.5 CPU time comparison of generating 10,000 arbitrary k-distributions from different databases. Database

Mixing Model

CPU time (s)

Exact FSK

–

31216

Narrow band

Multiplication

1390

Modest and Riazzi

5904

Correlations Look-up table

Multiplication

0.41

Modest and Riazzi

8.97

–

0.26

FIGURE 19.23 Radiative source for the scaled Sandia D flame with k-distributions assembled in several ways [155].

720 Radiative Heat Transfer

Wang et al. subsequently published several additional extended and improved databases. In [156] soot was added as an additional species; to reduce size and increase efficiency, the nongray stretching factor a is calculated on-the-fly rather than being tabulated, and some unnecessary mole fractions were eliminated, resulting in a database of 3.2 GB in size. In [142] that database was republished for different quadrature points, after it was recognized that a transformation factor of α = 2, as discussed in equation (19.143), leads to better accuracy in most relevant applications. After the FSCK-4 scheme was developed, which requires only two sets of cumulative k-distributions, Wang et al. [157] assembled an additional database for the retrieval of the “correlated k-distributions” for the FSCK-4 scheme of Section 19.11. While that database is also about 5 GB in size, with their memory management scheme only about 35 MB are needed for a three-gas mixture at a single total pressure. Finally, Wang et al. [183] released an FSCK database for use in conjunction with the Monte Carlo method described in Section 20.4. During the past few years machine learning and neural networks have become a popular tool to deal with large datasets. This was first exploited for global spectral models by Sun et al. [184], who used neural networks to regenerate the ALBDFs of Pearson et al. [179] for H2 O and CO2 at atmospheric pressure. Their look-up table is considerably faster, since C(F) can be determined directly without inversion. A somewhat larger effort was made by Zhou and coworkers [185,186], who used machine learning to regenerate the original cumulative kdistribution database of Wang and coworkers [155] for all the conditions given in Table 19.4. They noted that look-up times are essentially unchanged, but their entire database only requires about 35 MB (instead of the original 3.2 GB). In their latest work comparing the various FSCK and SLW methods, Wang and coworkers [146] observed that, when using a database to obtain the necessary k-distributions/ALBDFs, for the FSCK schemes the nongray stretching factor a should also be part of the database: if the a are calculated in situ from a 32-point database, such as the ones of Wang et al. [142,156] (which do not include the a), accuracy may deteriorate to that of the corresponding SLW scheme. They also noted that, since open Gaussian quadrature (one without the values of 0 and 1 as quadrature points) obeys a 3N+1 nesting rule [125], i.e., the 4 quadrature points for N=4 are contained in a 13-point database which, in turn, are included in a 40-pt database, and so on. Consequently, they tested all the FSCK and SLW schemes, as applied to the problem shown in Fig. 19.19, using a 40-pt k-distribution/ALBDF tabulation. They learnt that most of the CPU time is taken up by reading in the tabulation for all possible states (∼0.07 s for SLW, ∼0.14 s for FSCK, the latter including the stretch factors a). If the tabulation is not nested, all interpolations and evaluations for N=4 required 0.06–0.11 s for SLW (including the evaluation of gray gas ¯ and 0.05–0.12 s for FSCK. These CPU times are an order of magnitude lower than the ∼1 s required to weights a) make these calculations using the 5000-pt k-distributions/ALBDFs employed in Fig. 19.19b. If the tabulation is nested, all necessary values for FSCK-1,3,4 can be looked up directly, cutting 5D interpolations by a factor of ten and eliminating 1D interpolations, reducing interpolation and evaluation times to 40 atm) of 80% N2 and 20% CO at 2000 K is contained between two large, parallel, cold black plates, spaced 1 m apart. If the radiative flux to each wall may not exceed 100 kW/m2 , what is the maximum pressure the gas mixture may be raised to? Use the box model together with (a) the P1 -approximation as well as (b) the exact formulation. 19.10 The coal particles of Problem 11.3 are burnt in a long cylindrical combustion chamber of R = 1 m radius. The combustor walls are gray and diffuse, with w = 0.8, and are at 800 K. Since it is well stirred, combustion results in uniform heat generation throughout of Q˙  = 720 kW/m3 . (a) Determine the maximum temperature in the combustor, using the P1 /differential approximation, assuming radiation is the only mode of heat transfer (use κ = 4.5 m−1 and σs = 0.5 m−1 if the results of Problem 11.3 are not available). (b) How will the answer change if, instead, the combustion gas is responsible for the radiation with ⎧ ⎪ ⎪ ⎨10 cm−1 , 4 μm < λ < 5 μm κλ = ⎪ ; σs = 0? ⎪ ⎩0, elsewhere (c) What if both are present? 19.11 Consider a sphere of very hot molecular gas of radius 50 cm. The gas has a single vibration–rotation band at η0 = 3000 cm−1 , is suspended magnetically in a vacuum within a√large cold container and is initially at a uniform temperature Tg = 3000 K. For this gas (ρa α)(T) = 500 cm−2 , ω(T) = 100 T/100 K cm−1 , and β  1. These properties imply that the absorption coefficient may be determined from κη = κ0 e−2|η−η0 |/ω ,

κ0 =

ρa α ω

and the band absorptance from A(s) = ωA∗ = ω[E1 (κ0 s) + ln(κ0 s) + γE ],

γE = 0.577216.

Using the stepwise-gray model together with the P1 -approximation and neglecting conduction and convection, specify the total heat loss per unit time from the entire sphere at time t = 0. Outline the solution procedure for times t > 0. Hint: Solve the governing equation by introducing a new dependent variable g(τ) = τ(4πIb − G). 19.12 Consider a 1 m thick isothermal layer of pure CO2 at a pressure of 100 kPa and a temperature of 1700 K. The gas is confined between two parallel, cold, black plates. Calculate the radiative transfer from the gas to the walls using the exponential wide band model. 19.13 Repeat Problem 19.11 using the exact integral relations together with the exponential wide band model. 19.14 Repeat Problem 19.11 using the weighted-sum-of-gray gases approach together with the P1 -approximation.

Solution Methods for Nongray Extinction Coefficients Chapter | 19 729

19.15 Repeat Problem 19.11 for varying line overlap β, say β = 0.01, 0.1, 1, and 10. Plot heat loss at t = 0 vs. β. Hint: Use Table 10.3 or some other correlation for the band absorptance. 19.16 An infinitely long cylinder of radius R = 10 cm is bounded by a wall that is isothermal at Tw = 1500 K and has a gray emittance of = 0.3. Inside the cylinder there is uniform heat generation of Q˙  = 38,136 W/m3 . The cylinder is filled with a mixture of combustion gases at p = 1 atm, containing 10% by volume CO2 and 20% water vapor. Assuming the gas to be well-stirred (i.e., isothermal) determine the gas temperature using the weighted-sum-of-gray-gases approach, using the data of Table 19.2. This problem will require an iteration and, thus, is most conveniently solved on a computer. (a) Set up all necessary equations and explain the procedure. You may use the exact relations of Section 13.7 or the P1 -approximation. (b) Write a small computer code to find the gas temperature. Note for the P1 -approximation: The solution to the ODE   1 d df r − ν2 f = 0 r dr dr is f (r) = C1 I0 (νr) + C2 K0 (νr), 

where I0 and K0 are modified Bessel functions. Note also that K0 (0) → ∞ and I0 (x) = I1 (x). 19.17 Repeat Problem 19.7 for the case that the medium is a mixture of 30% water vapor in nitrogen, using the SLW method with four gray gases, together with the correlation of Denison and Webb. To determine an appropriate reference temperature, first make a more approximate gray calculation, using a Planck mean absorption coefficient from Fig. 10.33. 19.18 A spherical container of 1 m diameter is filled with pure CO2 and is initially at 2000 K, 1 bar. While the CO2 is continuously stirred (i.e., stays isothermal), the walls of the container are cooled such that the gray, diffuse wall ( w = 0.6) remains at a constant Tw = 400 K. Determine the time it takes for the gas to cool down to 500 K, using the FSK method together with the Denison and Webb correlation. Assume a constant reference condition of Tref = 1000 K, and use the P1 method to solve the RTE. 19.19 Repeat Problem 19.18 adding small gray particles with an absorption coefficient of κp = 0.1 m−1 and an (isotropic) scattering coefficient of σp = 1 m−1 . 19.20 Repeat Problem 19.18 adding H2 O and N2 to the mixture, so that the final mixture has 20% CO2 and 40% N2 (by volume). 19.21 Repeat Problem 19.20 using the WSGG approach together with the correlation of Truelove. 19.22 Repeat Problem 19.20 using the SLW method with four gray gases. Compare with results from the previous problem. 19.23 Repeat Problem 19.20 for the case of radiative equilibrium without stirring. 19.24 Repeat Problem 19.22 for the case of radiative equilibrium without stirring.

References [1] I.E. Gordon, L.S. Rothman, C. Hill, R.V. Kochanov, Y. Tan, P.F. Bernath, M. Birk, V. Boudon, A. Campargue, K.V. Chance, B.J. Drouin, J.-M. Flaud, R.R. Gamache, J.T. Hodges, D. Jacquemart, V.I. Perevalov, A. Perrin, K.P. Shine, M.-A.H. Smith, J. Tennyson, G.C. Toon, H. Tran, V.G. Tyuterev, A. Barbe, A.G. Császár, V.M. Devi, T. Furtenbacher, J.J. Harrison, J.-M. Hartmann, A. Jolly, T.J. Johnson, T. Karman, I. Kleiner, A.A. Kyuberis, J. Loos, O.M. Lyulin, S.T. Massie, S.N. Mikhailenko, N. Moazzen-Ahmadi, H.S.P. Müller, O.V. Naumenko, A.V. Nikitin, O.L. Polyansky, M. Rey, M. Rotger, S.W. Sharpe, K. Sung, E. Starikova, S.A. Tashkun, J. Vander Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, E.J. Zak, The HITRAN 2016 molecular spectroscopic database, Journal of Quantitative Spectroscopy and Radiative Transfer 203 (2017) 3–69. [2] L.S. Rothman, I.E. Gordon, R.J. Barber, H. Dothe, R.R. Gamache, A. Goldman, V.I. Perevalov, S.A. Tashkun, J. Tennyson, HITEMP, the high-temperature molecular spectroscopic database, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (15) (2010) 2139–2150. [3] M.K. Denison, B.W. Webb, An absorption-line blackbody distribution function for efficient calculation of total gas radiative transfer, Journal of Quantitative Spectroscopy and Radiative Transfer 50 (1993) 499–510. [4] M.K. Denison, B.W. Webb, A spectral line based weighted-sum-of-gray-gases model for arbitrary RTE solvers, ASME Journal of Heat Transfer 115 (1993) 1004–1012. [5] M.K. Denison, B.W. Webb, The spectral-line-based weighted-sum-of-gray-gases model in nonisothermal nonhomogeneous media, ASME Journal of Heat Transfer 117 (1995) 359–365.

730 Radiative Heat Transfer

[6] P. Rivière, A. Soufiani, M.-Y. Perrin, H. Riad, A. Gleizes, Air mixture radiative property modelling in the temperature range 10000–40000 K, Journal of Quantitative Spectroscopy and Radiative Transfer 56 (1996) 29–45. [7] L. Pierrot, A. Soufiani, J. Taine, Accuracy of narrow-band and global models for radiative transfer in H2 O, CO2 , and H2 O–CO2 mixtures at high temperature, Journal of Quantitative Spectroscopy and Radiative Transfer 62 (1999) 523–548. [8] L. Pierrot, P. Rivière, A. Soufiani, J. Taine, A fictitious-gas-based absorption distribution function global model for radiative transfer in hot gases, Journal of Quantitative Spectroscopy and Radiative Transfer 62 (1999) 609–624. [9] D. Scutaru, L. Rosenmann, J. Taine, Approximate band intensities of CO2 hot bands at 2.7, 4.3 and 12 μm for high temperature and medium resolution applications, Journal of Quantitative Spectroscopy and Radiative Transfer 52 (1994) 765–781. [10] P. Rivière, A. Soufiani, J. Taine, Correlated-k and fictitious gas model for H2 O infrared radiation in the Voigt regime, Journal of Quantitative Spectroscopy and Radiative Transfer 53 (1995) 335–346. [11] O. Marin, R.O. Buckius, Wide band correlated-k approach to thermal radiative transport in nonhomogeneous media, ASME Journal of Heat Transfer 119 (4) (1997) 719–729. [12] V.P. Solovjov, B.W. Webb, SLW modeling of radiative transfer in multicomponent gas mixtures, Journal of Quantitative Spectroscopy and Radiative Transfer 65 (2000) 655–672. [13] M.F. Modest, H. Zhang, The full-spectrum correlated-k distribution and its relationship to the weighted-sum-of-gray-gases method, in: Proceedings of the IMECE 2000, vol. HTD-366-1, ASME, Orlando, FL, 2000, pp. 75–84. [14] H. Zhang, M.F. Modest, A multi-scale full-spectrum correlated-k distribution for radiative heat transfer in inhomogeneous gas mixtures, Journal of Quantitative Spectroscopy and Radiative Transfer 73 (2–5) (2002) 349–360. [15] H. Zhang, M.F. Modest, Scalable multi-group full-spectrum correlated-k distributions for radiative heat transfer, ASME Journal of Heat Transfer 125 (3) (2003) 454–461. [16] H. Zhang, M.F. Modest, Multi-group full-spectrum k-distribution database for water vapor mixtures in radiative transfer calculations, International Journal of Heat and Mass Transfer 46 (19) (2003) 3593–3603. [17] L.S. Rothman, R.B. Wattson, R.R. Gamache, J. Schroeder, A. McCann, HITRAN, HAWKS and HITEMP high temperature databases, Proceedings of SPIE 2471 (1995) 105–111. [18] C. Park, Nonequilibrium air radiation (NEQAIR) program: User’s manual, NASA/Ames Research Center, Moffett Field, CA 940351000, 1985. [19] D.R. Olynick, W.D. Henline, L. Hartung-Chambers, G.V. Candler, Comparison of coupled radiative flow solutions with project Fire II flight data, Journal of Thermophysics and Heat Transfer 9 (4) (1995) 586–594. [20] D.R. Olynick, Y.-K. Chen, M.E. Tauber, Aerothermodynamics of the Stardust sample return capsule, Journal of Spacecraft and Rockets 36 (3) (1999) 442–462. [21] J. Olejniczak, M. Wright, D. Prabhu, N. Takashima, B.R. Hollis, E.V. Zoby, An analysis of the radiative heating environment for aerocapture at Titan, in: 39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Huntsville, Alabama, 2003, AIAA Paper No. 2003-4953. [22] M.L. da Silva, An adaptive line-by-line–statistical model for fast and accurate spectral simulations in low-pressure plasmas, Journal of Quantitative Spectroscopy and Radiative Transfer 108 (2007) 106–125. [23] A.M. Feldick, M.F. Modest, D.A. Levin, Closely coupled flowfield–radiation interactions during hypersonic reentry, Journal of Thermophysics and Heat Transfer 25 (4) (2011) 481–492. [24] L. Hartung, H.A. Hassan, Radiation transport around axisymmetric blunt body vehicles using a modified differential approximation, Journal of Thermophysics and Heat Transfer 7 (2) (1993) 220–227. [25] H. Chu, F. Liu, H. Zhou, Calculations of gas radiation heat transfer in a two-dimensional rectangular enclosure using the line-by-line approach and the statistical narrow-band correlated-k model, International Journal of Thermal Sciences 59 (2012) 66–74. [26] H. Amiri, K. Lari, P.J. Coelho, Comparison of CK model and line by line method using old and updated parameters/databases, International Journal of Thermal Sciences 118 (2017) 448–460. [27] H. Amiri, K. Lari, Line by line benchmark solutions for radiative heat transfer in 2d irregular enclosures with non-gray media, International Journal of Thermal Sciences 133 (2018) 307–319. [28] F.R. Centeno, R. Brittes, F.H.R. França, O.A. Ezekoye, Evaluation of gas radiation heat transfer in a 2D axisymmetric geometry using the line-by-line integration and WSGG models, Journal of Quantitative Spectroscopy and Radiative Transfer 156 (2015) 1–11. [29] F.R. Centeno, R. Brittes, L.G.P. Rodrigues, F.R. Coelho, F.H.R. França, Evaluation of the WSGG model against line-by-line calculation of thermal radiation in a non-gray sooting medium representing an axisymmetric laminar jet flame, International Journal of Heat and Mass Transfer 124 (2018) 475–483. [30] G. Pal, M.F. Modest, A narrow-band based multi-scale multi-group full-spectrum k- distribution method for radiative transfer in nonhomogeneous gas–soot mixture, ASME Journal of Heat Transfer 132 (2010) 023307. [31] J. Cai, M.F. Modest, Improved full-spectrum k-distribution implementation for inhomogeneous media using a narrow-band database, Journal of Quantitative Spectroscopy and Radiative Transfer 141 (2013) 65–72. [32] J. Cai, R. Marquez, M.F. Modest, Comparisons of radiative heat transfer calculations in a jet diffusion flame using spherical harmonics and k-distributions, ASME Journal of Heat Transfer 136 (2014) 112702. [33] M.F. Modest, J. Cai, W. Ge, E. Lee, Elliptic formulation of the simplified spherical harmonics method in radiative heat transfer, International Journal of Heat and Mass Transfer 76 (2014) 459–466.

Solution Methods for Nongray Extinction Coefficients Chapter | 19 731

[34] W. Ge, R. Marquez, M.F. Modest, S.P. Roy, Implementation of high order spherical harmonics methods for radiative heat transfer on OpenFOAM, ASME Journal of Heat Transfer 137 (5) (2015) 052701. [35] H. Chu, M. Gu, J.-L. Consalvi, F. Liu, H. Zhou, Effects of total pressure on non-grey gas radiation transfer in oxy-fuel combustion using the LBL, SNB, SNBCK, WSGG, and FSCK methods, Journal of Quantitative Spectroscopy and Radiative Transfer 172 (2016) 24–35. [36] H. Chu, J.-L. Consalvi, M. Gu, F. Liu, Calculations of radiative heat transfer in an axisymmetric jet diffusion flame at elevated pressures using different gas radiation models, Journal of Quantitative Spectroscopy and Radiative Transfer 197 (2017) 12–25. [37] H.C. Hottel, Radiant heat transmission, in: W.H. McAdams (Ed.), Heat Transmission, 3rd ed., McGraw-Hill, New York, 1954, ch. 4. [38] H.C. Hottel, A.F. Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967. [39] R.V. Dunkle, Geometric mean beam lengths for radiant heat transfer calculations, ASME Journal of Heat Transfer 86 (1) (1964) 75–80. [40] K.M. Andersen, S. Hadvig, Geometric mean beam lengths for the space between two coaxial cylinders, ASME Journal of Heat Transfer 111 (3) (1989) 811–812. [41] F.M.B. Andersen, Geometric mean beam lengths between two concentric spheres, ASME Journal of Heat Transfer 119 (2) (1997) 379–380. [42] R. Viskanta, Concerning the definitions of the mean absorption coefficient, International Journal of Heat and Mass Transfer 7 (9) (1964) 1047–1049. [43] D. Finkleman, K.Y. Chien, Semigrey radiative transfer, AIAA Journal 6 (4) (1968) 755–758. [44] D. Finkleman, Numerical studies in semigray radiative transfer, AIAA Journal 7 (1969) 1602–1605. [45] D. Finkleman, A note on boundary conditions for use with the differential approximation to radiative transfer, International Journal of Heat and Mass Transfer 12 (1969) 653–656. [46] S.C. Traugott, Radiative heat-flux potential for a nongrey gas, AIAA Journal 4 (3) (1966) 541–542. [47] S.C. Traugott, On grey absorption coefficients in radiative transfer, Journal of Quantitative Spectroscopy and Radiative Transfer 8 (1968) 971–999. [48] M.F. Modest, K.K. Sikka, The application of the stepwise-gray P-1 approximation to molecular gas–particulate mixtures, Journal of Quantitative Spectroscopy and Radiative Transfer 48 (2) (1992) 159–168. [49] M.F. Modest, The weighted-sum-of-gray-gases model for arbitrary solution methods in radiative transfer, ASME Journal of Heat Transfer 113 (3) (1991) 650–656. [50] S. Chandrasekhar, The radiative equilibrium of the outer layers of a star with special reference to the blanketing effects of the reversing layer, Monthly Notices of the Royal Astronomical Society 96 (1935) 21–42. [51] C.E. Siewert, P.F. Zweifel, An exact solution of equations of radiative transfer for local thermodynamic equilibrium in the non-gray case: picket fence approximation, Annals of Physics 36 (1966) 61–85. [52] S.C. Kung, M. Sibulkin, Radiative transfer in a nongray gas between parallel walls, Journal of Quantitative Spectroscopy and Radiative Transfer 9 (1969) 1447–1461. [53] R.J. Reith, C.E. Siewert, M.N. Özi¸sik, Non-grey radiative heat transfer in conservative plane-parallel media with reflecting boundaries, Journal of Quantitative Spectroscopy and Radiative Transfer 11 (1971) 1441–1462. [54] R. Greif, Energy transfer by radiation and conduction with variable gas properties, International Journal of Heat and Mass Transfer 7 (1964) 891–900. [55] M.F. Modest, A simple differential approximation for radiative transfer in non-gray gases, ASME Journal of Heat Transfer 101 (1979) 735–736. [56] S.T. Thynell, Radiation due to CO2 or H2 O and particles in cylindrical media, Journal of Thermophysics and Heat Transfer 4 (4) (1990) 436–445. [57] D.A. Kaminski, X.D. Fu, M.K. Jensen, Numerical and experimental analysis of combined convective and radiative heat transfer in laminar flow over a circular cylinder, International Journal of Heat and Mass Transfer 38 (17) (1995) 3161–3169. [58] T. Seo, M.K. Jensen, D.A. Kaminski, Combined convection and non-gray radiation in simultaneously developing turbulent flow and heat transfer, in: Proceedings of the 11th International Heat Transfer Conference, vol. 7, Kyongju, Korea, 1998, pp. 337–342. [59] S. Mazumder, M.F. Modest, A PDF approach to modeling turbulence–radiation interactions in nonluminous flames, International Journal of Heat and Mass Transfer 42 (1999) 971–991. [60] S. Mazumder, M.F. Modest, Turbulence–radiation interactions in nonreactive flow of combustion gases, ASME Journal of Heat Transfer 121 (1999) 726–729. [61] G.H. Van Wylen, R.E. Sonntag, Fundamentals of Classical Thermodynamics, John Wiley & Sons, New York, 1985. [62] W.L. Grosshandler, Radiative transfer in nonhomogeneous gases: a simplified approach, International Journal of Heat and Mass Transfer 23 (1980) 1447–1457. [63] W.L. Grosshandler, RADCAL: a narrow-band model for radiation calculations in a combustion environment, Technical Report NIST Technical Note 1402, National Institute of Standards and Technology, 1993. [64] A. Soufiani, J. Taine, High temperature gas radiative property parameters of statistical narrow-band model for H2 O, CO2 and CO, and correlated-k model for H2 O and CO2 , International Journal of Heat and Mass Transfer 40 (4) (1997) 987–991. [65] T.K. Kim, J.A. Menart, H.S. Lee, Nongray radiative gas analyses using the S-N discrete ordinates method, ASME Journal of Heat Transfer 113 (4) (1991) 946–952. [66] J.A. Menart, H.S. Lee, T.K. Kim, Discrete ordinates solutions of nongray radiative transfer with diffusely reflecting walls, ASME Journal of Heat Transfer 115 (1) (1993) 184–193.

732 Radiative Heat Transfer

[67] J.A. Menart, H.S. Lee, Nongray gas analyses for reflecting walls utilizing a flux technique, ASME Journal of Heat Transfer 115 (3) (1993) 645–652. [68] M. Cherkaoui, J.-L. Dufresne, R. Fournier, J.-Y. Grandpeix, A. Lahellec, Radiative net exchange formulation within one-dimensional gas enclosures with reflective surfaces, ASME Journal of Heat Transfer 120 (1) (1998) 275–278. [69] F. Liu, O.L. Gülder, G.J. Smallwood, Y. Ju, Non-grey gas radiative transfer analyses using the statistical narrow-band model, International Journal of Heat and Mass Transfer 41 (14) (1998) 2227–2236. [70] Y. Ju, K. Takita, M. Goro, F. Liu, H. Guo, Analyses of extinction and flammability limit of stretched premixed flames using the statistical narrow-band model, in: Proceedings of the 11th International Heat Transfer Conference, vol. 7, Kyongju, Korea, 1998, pp. 301–306. [71] F. Liu, Numerical solutions of three-dimensional non-grey gas radiative transfer using the statistical narrow-band model, ASME Journal of Heat Transfer 121 (1) (1999) 200–203. [72] D.K. Edwards, A. Balakrishnan, Slab band absorptance for molecular gas radiation, Journal of Quantitative Spectroscopy and Radiative Transfer 12 (1972) 1379–1387. [73] D.K. Edwards, Molecular gas band radiation, in: Advances in Heat Transfer, vol. 12, Academic Press, New York, 1976, pp. 115–193. [74] A.T. Wassel, D.K. Edwards, Molecular gas band radiation in cylinders, ASME Journal of Heat Transfer 96 (1974) 21–26. [75] J.R. Hutchison, R.F. Richards, Effect of nongray gas radiation on thermal stability in carbon dioxide, Journal of Thermophysics and Heat Transfer 13 (1) (1999) 25–32. [76] M.F. Modest, Radiative heat transfer in a plane-layer mixture of non-gray particulates and molecular gases, Journal of Quantitative Spectroscopy and Radiative Transfer 26 (1981) 523–533. [77] P.S. Cumber, M. Fairweather, H.S. Ledin, Application of wide band radiation models to non-homogeneous combustion systems, International Journal of Heat and Mass Transfer 41 (11) (1998) 1573–1584. [78] F. Liu, G.J. Smallwood, O.L. Gülder, Application of statistical narrowband model to three-dimensional absorbing–emitting–scattering media, Journal of Thermophysics and Heat Transfer 13 (3) (1999) 285–291. [79] S. Maruyama, Z. Guo, Radiative heat transfer in arbitrary configurations with nongray absorbing, emitting, and anisotropic scattering media, ASME Journal of Heat Transfer 121 (3) (1999) 722–726. [80] T.F. Smith, Z.F. Shen, J.N. Friedman, Evaluation of coefficients for the weighted sum of gray gases model, ASME Journal of Heat Transfer 104 (1982) 602–608. [81] I.H. Farag, T.A. Allam, Gray-gas approximation of carbon dioxide standard emissivity, ASME Journal of Heat Transfer 103 (1981) 403–405. [82] J.S. Truelove, The zone method for radiative heat transfer calculations in cylindrical geometries, HTFS Design Report DR33 (Part I: AERE-R8167), Atomic Energy Authority, Harwell, 1975. [83] N.W. Bressloff, The influence of soot loading on weighted sum of grey gases solutions to the radiative transfer equation across mixtures of gases and soot, International Journal of Heat and Mass Transfer 42 (18) (1999) 3469–3480. [84] P.B. Taylor, P.J. Foster, Some gray gas weighting coefficients for CO2 –H2 O–soot mixtures, International Journal of Heat and Mass Transfer 18 (1975) 1331–1332. [85] C. Yin, L.C.R. Johansen, L.A. Rosendahl, S.K. Kær, New weighted sum of gray gases model applicable to computational fluid dynamics (CFD) modeling of oxy-fuel combustion: derivation, validation, and implementation, Energy & Fuels 24 (2010) 6275–6282. [86] R. Johansson, K. Andersson, B. Leckner, H. Thunman, Models for gaseous radiative heat transfer applied to oxy-fuel conditions in boilers, International Journal of Heat and Mass Transfer 53 (2010) 220–230. [87] T. Kangwanpongpan, F.H.R. França, R.C. da Silva, P.S. Schneider, H.J. Krautz, New correlations for the weighted-sum-of-gray-gases model in oxy-fuel conditions based on HITEMP 2010 database, International Journal of Heat and Mass Transfer 55 (2012) 7419–7433. [88] L.J. Dorigon, G. Duciak, R. Brittes, F. Cassol, M. Galarça, F.H.R. França, WSGG correlations based on HITEMP2010 for computation of thermal radiation in non-isothermal, non-homogeneous h2 o/co2 mixtures, International Journal of Heat and Mass Transfer 64 (2013) 863–873. [89] M.H. Bordbar, G. Wecel, T. Hyppänen, A line by line based weighted sum of gray gases model for inhomogeneous CO2 –H2 O mixture in oxy-fired combustion, Combustion and Flame 161 (2014) 2435–2445. [90] F. Cassol, R. Brittes, F.H.R. França, O.A. Ezekoye, Application of the weighted-sum-of-gray-gases model for media composed of arbitrary concentrations of H2 O, CO2 and soot, International Journal of Heat and Mass Transfer 79 (2014) 796–806. [91] M.H. Bordbar, G.C. Fraga, S. Hostikka, An extended weighted-sum-of-gray-gases model to account for all CO2 –H2 O molar fraction ratios in thermal radiation, International Communications in Heat and Mass Transfer 110 (2020) 104400. [92] S. Shan, B. Qian, Z. Zhou, Z. Wang, K.-F. Cen, New pressurized wsgg model and the effect of pressure on the radiation heat transfer of H2O/CO2 gas mixtures, International Journal of Heat and Mass Transfer 121 (2018) 999–1010. [93] P. Rivière, A. Soufiani, Updated band model parameters for H2 O, CO2 , CH4 and CO radiation at high temperature, International Journal of Heat and Mass Transfer 55 (2012) 3349–3358. [94] F.R. Coelho, F.H.R. França, WSGG correlations based on HITEMP2010 for computation of thermal radiation in non-isothermal, non-homogeneous H2 O/CO2 mixtures, International Journal of Heat and Mass Transfer 127 (2018) 105–114. [95] Y. Zhou, R. Duan, X. Zhu, J. Wu, J. Ma, X. Li, Q. Wang, An improved model to calculate radiative heat transfer in hot combustion gases, Combustion Theory and Modelling 24 (5) (2020) 829–851. [96] M.K. Denison, B.W. Webb, k-distributions and weighted-sum-of-gray gases: a hybrid model, in: Tenth International Heat Transfer Conference, Taylor & Francis, 1994, pp. 19–24.

Solution Methods for Nongray Extinction Coefficients Chapter | 19 733

[97] M.K. Denison, B.W. Webb, Development and application of an absorption line blackbody distribution function for CO2 , International Journal of Heat and Mass Transfer 38 (1995) 1813–1821. [98] M.K. Denison, B.W. Webb, The spectral-line weighted-sum-of-gray-gases model for H2 O/CO2 mixtures, ASME Journal of Heat Transfer 117 (1995) 788–792. [99] M.H. Bordbar, K. Myöhänen, T. Hyppänen, Coupling of a radiative heat transfer model and a three-dimensional combustion model for a circulating fluidized bed furnace, Applied Thermal Engineering 76 (2015) 344–356. [100] V. Kez, F. Liu, J.-L. Consalvi, J. Ströhle, B. Epple, A comprehensive evaluation of different radiation models in a gas turbine combustor under conditions of oxy-fuel combustion with dry recycle, Journal of Quantitative Spectroscopy and Radiative Transfer 172 (2016) 121–133. [101] V. Kez, J.-L. Consalvi, F. Liu, J. Ströhle, B. Epple, Assessment of several gas radiation models for radiative heat transfer calculations in a three-dimensional oxy-fuel furnace under coal-fired conditions, International Journal of Thermal Sciences 120 (2017) 289–302. [102] T. Gronarz, J. Schulze, M. Laemmerhold, P. Graeser, J. Gorewoda, V. Kez, M. Habermehl, M. Schiemann, J. Ströhle, B. Epple, V. Scherer, R. Kneer, Quantification of the influence of parameters determining radiative heat transfer in an oxy-fuel operated boiler, Fuel Processing Technology 157 (2017) 76–89. [103] X. Yang, Z. He, S. Dong, H.P. Tan, Evaluation of the non-gray weighted sum of gray gases models for radiative heat transfer in realistic non-isothermal and non-homogeneous flames using decoupled and coupled calculations, International Journal of Heat and Mass Transfer 134 (2019) 226–236. [104] V.P. Solovjov, D. Lemonnier, B.W. Webb, The SLW-1 model for efficient prediction of radiative transfer in high temperature gases, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (7) (2011) 1205–1212. [105] L.S. Rothman, I.E. Gordon, A. Barbe, D.C. Benner, P.F. Bernath, M. Birk, V. Boudon, L.R. Brown, A. Campargue, J.-P. Champion, K. Chance, L.H. Coudert, V. Dana, V.M. Devi, S. Fally, J.-M. Flaud, R.R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W.J. Lafferty, J.-Y. Mandin, S.T. Massie, S.N. Mikhailenko, C.E. Miller, N. Moazzen-Ahmadi, O.V. Naumenko, A.V. Nikitin, J. Orphal, V.I. Perevalov, A. Perrin, A. Predoi-Cross, C.P. Rinsland, M. Rotger, M. Simeckova, M.A.H. Smith, K. Sung, S.A. Tashkun, J. Tennyson, R.A. Toth, A.C. Vandaele, J.V. Auwera, The HITRAN 2008 molecular spectroscopic database, Journal of Quantitative Spectroscopy and Radiative Transfer 110 (2009) 533–572. [106] P.Y.C. Lee, G.D. Raithby, K.G.T. Hollands, The “reordering” concept of the absorption coefficient for modelling nongray gases, in: Radiative Heat Transfer: Current Research, vol. HTD-276, ASME, 1994, pp. 21–30. [107] P.Y.C. Lee, K.G.T. Hollands, G.D. Raithby, Reordering the absorption coefficient within the wide band for predicting gaseous radiant exchange, ASME Journal of Heat Transfer 118 (2) (1996) 394–400. [108] G. Parthasarathy, J.C. Chai, S.V. Patankar, A simple approach to nongray gas modeling, Numerical Heat Transfer 29 (1996) 394–400. [109] M.K. Denison, W.A. Fiveland, A correlation for the reordered wave number of the wideband absorptance of radiating gases, ASME Journal of Heat Transfer 119 (1997) 853–856. [110] J. Ströhle, Assessment of the re-ordered wide band model for non-grey radiative transfer calculations in 3D enclosures, Journal of Quantitative Spectroscopy and Radiative Transfer 109 (10) (2008) 1622–1640. [111] O. Marin, R.O. Buckius, Wide band correlated-k method applied to absorbing, emitting and scattering media, Journal of Thermophysics and Heat Transfer 10 (1996) 364–371. [112] O. Marin, R.O. Buckius, A model of the cumulative distribution function for wide band radiative properties, Journal of Quantitative Spectroscopy and Radiative Transfer 59 (1998) 671–685. [113] O. Marin, R.O. Buckius, A simplified wide band model of the cumulative distribution function for water vapor, International Journal of Heat and Mass Transfer 41 (1998) 2877–2892. [114] O. Marin, R.O. Buckius, A simplified wide band model of the cumulative distribution function for carbon dioxide, International Journal of Heat and Mass Transfer 41 (1998) 3881–3897. [115] L.S. Rothman, R.R. Gamache, R.H. Tipping, C.P. Rinsland, M.A.H. Smith, D.C. Benner, V.M. Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S.T. Massie, L.R. Brown, R.A. Toth, The HITRAN molecular database: editions of 1991 and 1992, Journal of Quantitative Spectroscopy and Radiative Transfer 48 (5/6) (1992) 469–507. [116] M.F. Modest, H. Zhang, The full-spectrum correlated-k distribution for thermal radiation from molecular gas–particulate mixtures, ASME Journal of Heat Transfer 124 (1) (2002) 30–38. [117] D.T. Pawlak, E.E. Clothiaux, M.F. Modest, J.N.S. Cole, Full spectrum correlated-k for shortwave atmospheric radiative transfer, Journal of the Atmospheric Sciences 61 (2004) 2588–2601. [118] R.J. Hogan, The full-spectrum correlated-k method for longwave atmospheric radiative transfer using an effective Planck function, Journal of the Atmospheric Sciences 67 (2010) 2086–2100. [119] A. Bansal, M.F. Modest, D.A. Levin, Multigroup correlated-k distribution method for nonequilibrium atomic radiation, Journal of Thermophysics and Heat Transfer 24 (3) (2010) 638–646, https://doi.org/10.2514/1.46641. [120] A. Bansal, M.F. Modest, D.A. Levin, Multi-scale k-distribution model for gas mixtures in hypersonic nonequilibrium flows, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (7) (2011) 1213–1221. [121] A. Bansal, M.F. Modest, Narrow-band k-distribution database for atomic radiation in hypersonic nonequilibrium flows, ASME Journal of Heat Transfer 133 (2011) 122701. [122] A. Maurente, F.H.R. França, K. Miki, J.R. Howell, Application of approximations for joint cumulative k-distributions for mixtures to FSK radiation heat transfer in multi-component high temperature non-LTE plasmas, Journal of Quantitative Spectroscopy and Radiative Transfer 113 (2012) 1521–1535.

734 Radiative Heat Transfer

[123] R.M. Goody, Y.L. Yung, Atmospheric Radiation – Theoretical Basis, 2nd ed., Oxford University Press, New York, 1989. [124] V.P. Solovjov, F. André, D. Lemonnier, B.W. Webb, The rank correlated SLW model of gas radiation in non-uniform media, Journal of Quantitative Spectroscopy and Radiative Transfer 197 (2017) 26–44. [125] A. Wang, M.F. Modest, High-accuracy, compact database of narrow-band k-distributions for water vapor and carbon dioxide, Journal of Quantitative Spectroscopy and Radiative Transfer 93 (2005) 245–261. [126] J. Cai, M.F. Modest, Improved full-spectrum k-distribution implementation for inhomogeneous media using a narrow-band database, in: 22nd National and 11th International ISHMT-ASME Heat and Mass Transfer Conference 2013, Kharagpur, India, 2013. [127] S. Dembele, A. Delmas, J.-F. Sacadura, A method for modeling the mitigation of hazardous fire thermal radiation by water spray curtains, ASME Journal of Heat Transfer 119 (1997) 746–753. [128] S. Dembele, J.X. Wen, Investigation of a spectral formulation for radiative heat transfer in a one-dimensional fires and combustion system, International Journal of Heat and Mass Transfer 43 (2000) 4019–4030. [129] K.C. Tang, M.Q. Brewster, Analysis of molecular gas radiation: real gas property effects, in: 7th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, vol. HTD-357-1, ASME, 1998, pp. 23–32. [130] F. Liu, G.J. Smallwood, O.L. Gülder, Application of the statistical narrow-band correlated-k method to low-resolution spectral intensity and radiative heat transfer calculations – effects of the quadrature, International Journal of Heat and Mass Transfer 43 (2000) 3119–3135. [131] F. Liu, G.J. Smallwood, An efficient approach for the implementation of the SNB based correlated-k method and its evaluation, Journal of Quantitative Spectroscopy and Radiative Transfer 84 (2004) 465–475. [132] L. Wang, N.E. Endrud, S.R. Turns, M.D. D’Agostini, A.G. Slavejkov, A study of the influence of oxygen index on soot, radiation, and emissions characteristics of turbulent jet flames, Combustion Science and Technology 174 (8) (2002) 45–72. [133] F. Nmira, J.-L. Consalvi, F. André, Pressure effects on radiative heat transfer in hydrogen/air turbulent diffusion flames, Journal of Quantitative Spectroscopy and Radiative Transfer 220 (2018) 172–179. [134] F. Nmira, Y. Liu, J.-L. Consalvi, F. André, F. Liu, Pressure effects on radiative heat transfer in sooting turbulent diffusion flames, Journal of Quantitative Spectroscopy and Radiative Transfer 245 (2020) 106906. [135] J.-L. Consalvi, F. Nmira, Transported scalar PDF modeling of oxygen-enriched turbulent jet diffusion flames: soot production and radiative heat transfer, Fuel 178 (2016) 37–48. [136] J.-L. Consalvi, F. Nmira, Effects of soot absorption coefficient-Planck function correlation on radiative heat transfer in oxygen-enriched propane turbulent diffusion flame, Journal of Quantitative Spectroscopy and Radiative Transfer 172 (2016) 50–57. [137] J.-L. Consalvi, F. Nmira, W. Kong, On the modeling of the filtered radiative transfer equation in large eddy simulations of lab-scale sooting turbulent diffusion flames, Journal of Quantitative Spectroscopy and Radiative Transfer 221 (2018) 51–60. [138] F. Nmira, D. Burot, J.-L. Consalvi, Stochastic Eulerian field method for radiative heat transfer in a propane oxygen-enhanced turbulent diffusion flame, Combustion Theory and Modelling 21 (2017) 62–78. [139] F. Nmira, D. Burot, J.-L. Consalvi, Soot emission radiation-turbulence interactions in diffusion jet flames, Combustion Science and Technology 191 (2019) 126–136. [140] L. Tessé, F. Dupoirieux, B. Zamuner, J. Taine, Radiative transfer in real gases using reciprocal and forward Monte Carlo methods and a correlated-k approach, International Journal of Heat and Mass Transfer 45 (2002) 2797–2814. [141] M.F. Modest, Narrow-band and full-spectrum k-distributions for radiative heat transfer—correlated-k vs. scaling approximation, Journal of Quantitative Spectroscopy and Radiative Transfer 76 (1) (2003) 69–83. [142] C. Wang, M.F. Modest, B. He, Improvement of full-spectrum k-distribution method using quadrature transformation, International Journal of Thermal Sciences 108 (2016) 100–107. [143] M.F. Modest, R.S. Mehta, Full spectrum k-distribution correlations for CO2 from the CDSD-1000 spectroscopic databank, International Journal of Heat and Mass Transfer 47 (2004) 2487–2491. [144] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover Publications, New York, 1965. [145] W.H. Press, B.P. Flannery, S.A. Tenkolsky, W.T. Vetterling, Numerical Recipes – The Art of Scientific Computing, 1st ed., 1989, Cambridge, New York. [146] C. Wang, M.F. Modest, T. Ren, J. Cai, B. He, Comparison and refinement of the various full-spectrum k-distribution and spectral-linebased-weighted-sum-of-gray-gases models for nonhomogeneous media, Journal of Quantitative Spectroscopy and Radiative Transfer 271 (2021) 107695. [147] H. Zhang, M.F. Modest, Full-spectrum k-distribution correlations for carbon dioxide mixtures, Journal of Thermophysics and Heat Transfer 17 (2) (2003) 259–263. [148] M.F. Modest, R.J. Riazzi, Assembly of full-spectrum k-distributions from a narrow-band database; effects of mixing gases, gases and nongray absorbing particles, and mixtures with nongray scatterers in nongray enclosures, Journal of Quantitative Spectroscopy and Radiative Transfer 90 (2) (2005) 169–189. [149] L. Wang, M.F. Modest, Narrow-band based multi-scale full-spectrum k-distribution method for radiative transfer in inhomogeneous gas mixtures, ASME Journal of Heat Transfer 127 (2005) 740–748. [150] L. Wang, J. Yang, M.F. Modest, D.C. Haworth, Application of the full-spectrum k-distribution method to photon Monte Carlo solvers, Journal of Quantitative Spectroscopy and Radiative Transfer 104 (2) (2007) 297–304. [151] G. Pal, M.F. Modest, k-distribution methods for radiation calculations in high pressure combustion, Journal of Thermophysics and Heat Transfer 27 (3) (2013) 584–587.

Solution Methods for Nongray Extinction Coefficients Chapter | 19 735

[152] M.F. Modest, The treatment of nongray properties in radiative heat transfer — from past to present, ASME Journal of Heat Transfer 135 (6) (2013) 4023596. [153] G. Pal, A. Gupta, M.F. Modest, D.C. Haworth, Comparison of accuracy and computational expense of radiation models in simulation of nonpremixed turbulent jet flames, Combustion and Flame 162 (2015) 2487–2495. [154] J. Cai, M. Handa, M.F. Modest, Eulerian–Eulerian multi-fluid methods for pulverized coal flames with nongray radiation, Combustion and Flame 162 (2015) 1550–1565. [155] C. Wang, W. Ge, M.F. Modest, B. He, A full-spectrum k-distribution look-up table for radiative transfer in nonhomogeneous gaseous media, Journal of Quantitative Spectroscopy and Radiative Transfer 168 (2016) 46–56. [156] C. Wang, M.F. Modest, B. He, Full-spectrum k-distribution look-up table for nonhomogeneous gas–soot mixtures, Journal of Quantitative Spectroscopy and Radiative Transfer 176 (2016) 129–136. [157] C. Wang, B. He, M.F. Modest, T. Ren, Efficient full-spectrum correlated-k-distribution look-up table, Journal of Quantitative Spectroscopy and Radiative Transfer 219 (2018) 108–116. [158] S. Mazumder, M.F. Modest, Application of the full spectrum correlated-k distribution approach to modeling non-gray radiation in combustion gases, Combustion and Flame 129 (4) (2002) 416–438. [159] B.W. Webb, V.P. Solovjov, F. André, The spectral line weighted-sum-of-gray-gases (SLW) model for prediction of radiative transfer in molecular gases, in: Advances in Heat Transfer, vol. 51, 2019, pp. 207–298, Ch. 4. [160] V.P. Solovjov, B.W. Webb, F. André, The rank correlated FSK model for prediction of gas radiation in non-uniform media, and its relationship to the rank correlated SLW model, Journal of Quantitative Spectroscopy and Radiative Transfer 214 (2018) 120–132. [161] R.S. Barlow, International Workshop on Measurement and Computation of Turbulent Nonpremixed Flames (TNF), website: http:// www.sandia.gov/TNF/abstract.html. [162] W. Ge, M.F. Modest, R. Marquez, Two-dimensional axisymmetric formulation of high order spherical harmonics methods for radiative heat transfer, Journal of Quantitative Spectroscopy and Radiative Transfer 156 (2015) 58–66. [163] R. Porter, F. Liu, M. Pourkashanian, A. Williams, D. Smith, Evaluation of solution methods for radiative heat transfer in gaseous oxy-fuel combustion environments, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (14) (2010) 2084–2094. [164] A.G. Clements, R. Porter, A. Pranzitelli, M. Pourkashanian, Evaluation of FSK models for radiative heat transfer under oxyfuel conditions, Journal of Quantitative Spectroscopy and Radiative Transfer 151 (2015) 67–75. [165] R. Demarco, J.-L. Consalvi, A. Fuentes, S. Melis, Assessment of radiative property models in non-gray sooting media, International Journal of Thermal Sciences 50 (2011) 1672–1684. [166] R. Demarco, F. Nmira, J.-L. Consalvi, Influence of thermal radiation on soot production in laminar axisymmetric diffusion flames, Journal of Quantitative Spectroscopy and Radiative Transfer 120 (2013) 52–69. [167] H. Chu, F. Liu, J.-L. Consalvi, Relationship between the spectral line based weighted-sum-of-gray-gases model and the full spectrum k-distribution model, Journal of Quantitative Spectroscopy and Radiative Transfer 143 (2014) 111–120. [168] J.-L. Consalvi, F. Liu, Radiative heat transfer in the core of axisymmetric pool fires - I: evaluation of approximate radiative property models, International Journal of Thermal Sciences 84 (2014) 104–117. [169] J.-L. Consalvi, F. Liu, Radiative heat transfer through the fuel-rich core of laboratory-scale pool fires, Combustion Science and Technology 186 (2014) 475–489. [170] B.W. Webb, J. Ma, J.T. Pearson, V.P. Solovjov, Slw modeling of radiation transfer in comprehensive combustion predictions, Combustion Science and Technology 190 (2018) 1392–1408. [171] J. Cai, M.F. Modest, Absorption coefficient regression scheme for splitting radiative heat sources across phases in gas-particulate mixtures, Powder Technology 265 (2014) 76–82. [172] C. Wang, B. He, M.F. Modest, Full-spectrum correlated-k-distribution look-up table for radiative transfer in nonhomogeneous participating media with gas-particle mixtures, International Journal of Heat and Mass Transfer 137 (2019) 1053–1063. [173] P. Rivière, A. Soufiani, J. Taine, Correlated-k and fictitious gas methods for H2 O near 2.7 μm, Journal of Quantitative Spectroscopy and Radiative Transfer 48 (1992) 187–203. [174] S.A. Tashkun, V.I. Perevalov, A.D. Bykov, N.N. Lavrentieva, J.-L. Teffo, Carbon Dioxide Spectroscopic databank (CDSD), available from ftp://ftp.iao.ru/pub/CDSD-1000, 2002. [175] M.K. Denison, B.W. Webb, The absorption-line blackbody distribution function at elevated pressure, in: M.P. Mengüç (Ed.), Proceedings of the First International Symposium on Radiation Transfer, Begell House, 1996, pp. 228–238. [176] J.-M. Hartmann, R. Levi Di Leon, J. Taine, Line-by-line and narrow-band statistical model calculations for H2 O, Journal of Quantitative Spectroscopy and Radiative Transfer 32 (2) (1984) 119–127. [177] M.F. Modest, V. Singh, Engineering correlations for full spectrum k-distribution of H2 O from the HITEMP spectroscopic databank, Journal of Quantitative Spectroscopy and Radiative Transfer 93 (2005) 263–271. [178] F. Liu, H. Chu, H. Zhou, G.J. Smallwood, Evaluation of the absorption line blackbody distribution function of CO2 and H2O using the proper orthogonal decomposition and hyperbolic correlations, Journal of Quantitative Spectroscopy and Radiative Transfer 128 (2013) 27–33. [179] J.T. Pearson, B.W. Webb, V.P. Solovjov, J. Ma, Efficient representation of the absorption line blackbody distribution function for H2 O, CO2 , and CO at variable temperature, mole fraction, and total pressure, Journal of Quantitative Spectroscopy and Radiative Transfer 138 (2014) 82–96.

736 Radiative Heat Transfer

[180] S.A. Tashkun, V.I. Perevalov, CDSD-4000: high-resolution, high-temperature carbon dioxide spectroscopic databank, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (9) (2011) 1403–1410, available from ftp://ftp.iao.ru/pub/CDSD-4000. [181] J. Taine, A. Soufiani, Gas IR radiative properties: from spectroscopic data to approximate models, in: Advances in Heat Transfer, vol. 33, Academic Press, New York, 1999, pp. 295–414. [182] J.-L. Consalvi, F. Nmira, Absorption turbulence–radiation interactions in sooting turbulent jet flames, Journal of Quantitative Spectroscopy and Radiative Transfer 201 (2017) 1–9. [183] C. Wang, M.F. Modest, B. He, Full-spectrum correlated-k-distribution look-up table for use with radiative Monte Carlo solvers, International Journal of Heat and Mass Transfer 131 (2019) 167–175. [184] Y. Sun, J. Ma, Y. Yu, B. Ye, C. Gao, Efficient SLW models for water vapor and carbon dioxide based on neural network method, Journal of Quantitative Spectroscopy and Radiative Transfer 236 (2019) 106600. [185] Y. Zhou, C. Wang, T. Ren, A machine learning based efficient and compact full-spectrum correlated k-distribution model, Journal of Quantitative Spectroscopy and Radiative Transfer 254 (2020) 107199. [186] Y. Zhou, C. Wang, T. Ren, C. Zhao, A machine learning based efficient and compact full-spectrum correlated k-distribution model, Journal of Quantitative Spectroscopy and Radiative Transfer 268 (2021) 107628. [187] L. Wang, M.F. Modest, Treatment of wall emission in the narrow-band based multi-scale full-spectrum k-distribution method, ASME Journal of Heat Transfer 129 (6) (2007) 743–748. [188] G. Pal, M.F. Modest, A multi-scale full-spectrum k-distribution method for radiative transfer in nonhomogeneous gas–soot mixture with wall emission, Computational Thermal Sciences 1 (2009) 137–158. [189] G. Pal, M.F. Modest, L. Wang, Hybrid full-spectrum correlated k-distribution method for radiative transfer in strongly nonhomogeneous gas mixtures, ASME Journal of Heat Transfer 130 (2008) 082701. [190] H. Chang, T.T. Charalampopoulos, Determination of the wavelength dependence of refractive indices of flame soot, Proceedings of the Royal Society (London) A 430 (1880) (1990) 577–591. [191] G. Pal, M.F. Modest, k-distribution methods for radiation calculations in high pressure combustion, in: 50th Aerospace Sciences Meeting, 2012, Paper No. AIAA-2012-0529. [192] J.H. Kent, D. Honnery, Modeling sooting turbulent jet flames using an extended flamelet technique, Combustion Science and Technology 54 (1987) 383–397.

Chapter 20

The Monte Carlo Method for Participating Media 20.1 Introduction In Chapter 7 we first introduced the Monte Carlo method for the evaluation of radiative exchange between surfaces. While statistical in nature, in this method we are tracing physically meaningful photons from their point of emission to their point of absorption, or their exit from the enclosure (albeit only a tiny, but statistically relevant sample). Therefore, the method is immediately applicable to participating media: we simply need to add statistical algorithms for the emission of photons from a gas, particles, or a semitransparent medium, as well as rules for the interaction of streaming photons with the medium, i.e., volumetric absorption and scattering. We have observed in the previous chapters that the radiative transfer equation (RTE) is a five-dimensional integro-differential equation, which is extremely difficult and expensive to solve. In fact, while the spherical harmonics and discrete ordinates methods and, to a lesser extent, the zonal method each enjoy a certain popularity and can be applied to fairly general problems, to this day no truly satisfactory RTE solution method has emerged. The problem is exacerbated by strong spectral variations of radiative properties (gases as well as particulates), so up to one million RTE evaluations are needed to achieve acceptable accuracy. These challenges, combined with the fact that Monte Carlo methods, unlike conventional RTE solvers, are ideal candidates for parallel computing, have led to their rapidly increasing popularity during the past few years. In fact, stochastic modeling, i.e., Monte Carlo methods, are rapidly being adopted in many engineering disciplines, such as turbulence and combustion modeling. Therefore, to distinguish the Monte Carlo methods for radiative transfer from others, we will sometimes use the terminology photon Monte Carlo or PMC. On the downside, Monte Carlo simulations demand large computer resources (quickly becoming less of an issue today, with modern computers, even desktops, being equipped with many processors as well as vast memory). Furthermore, they can be challenging to integrate with standard finite volume codes (i) because they are statistical in nature and (ii) perhaps more importantly, because they tend to be zeroth order, i.e., properties are assumed constant across cells, and there is no connectivity between cells.

20.2 Heat Transfer Relations for Participating Media If the enclosure is filled with an absorbing, emitting, and/or scattering medium, equations (7.14) through (7.21) for the evaluation of surface heat fluxes must be augmented by a term to account for emission from within the medium, and the definition for the generalized radiation exchange factor must be altered to allow for absorption and/or scattering. Again assuming a refractive index of unity, from equation (9.57) the total emission per unit volume is 4κP σT 4 and, therefore,   dFdA →dA dFdV →dA q(r) = (r)σT 4 (r) − dA − dV  ,

(r ) σT 4 (r ) 4κP (r )σT 4 (r ) (20.1) dA dA A V where κP (r ) dFdV →dA

= =

local Planck-mean absorption coefficient of the medium at r , generalized radiation exchange factor between volume elements dV  and surface element dA.

Equation (7.15) still applies to all exchange factors, including dFdV →dA , with the added stipulation that energy bundles may be attenuated by absorption and/or redirected by scattering. Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00028-6 Copyright © 2022 Elsevier Inc. All rights reserved.

737

738 Radiative Heat Transfer

A similar equation is needed to describe the net amount of radiative energy deposited (or withdrawn) per unit volume of the medium, i.e., the divergence of the radiative heat flux. From equations (9.62) and (10.197), it follows that  ∞ κη Gη dη, (20.2) ∇ · q = 4κP σT 4 − 0 1

where Gη is the spectral incident radiation. The first term in equation (20.2) describes emission from within the volume, and the second term gives the absorbed fraction per unit volume of all radiation incident on the element. For Monte Carlo calculations this term may be replaced by an expression similar to the one in equation (20.1), i.e.,   dFdV →dV 4  4  dFdA →dV  dA − dV  .

(r )σT (r ) 4κP (r )σT 4 (r ) (20.3) ∇ · q = 4κP σT − dV dV A V For numerical calculations it is again necessary to break up the enclosure into a number J finite subsurfaces and K finite subvolumes, transforming equations (20.1) and (20.3) to Qi =

i σTi4 Ai

−

J 

j σTj4 A j Fj→i − qext As Fs→i −

j=1

 Vl

∇ · q dV = 4κPl σTl4 Vl −

K 

4κPk σTk4 Vk Fk→i ,

i = 1, 2, . . . , J,

(20.4)

k=1 J 

j σTj4 A j Fj→l − qext As Fs→l −

j=1

K 

4κPk σTk4 Vk Fk→l ,

l = 1, 2, . . . , K,

(20.5)

k=1

where the Tk are suitably defined average temperatures within the medium  1 κPk σTk4 = κP σT 4 dV. V k Vk

(20.6)

Using this formulation, all generalized exchange factors may then be evaluated through equation (7.21). As for surface transport, the exchange factor approach is often bypassed in favor of direct calculation of surface fluxes and internal radiative source terms, ∇ · q. In all cases the total emission from surfaces and volume must be distributed over the total number of energy bundles to be traced. While separate ΔE may be chosen for individual subsurfaces or subvolumes, as shown in Section 7.3, it is generally a good idea for better statistical convergence to employ values of ΔE within a small range.

20.3 Random Number Relations for Participating Media Besides the random number relations established in Chapter 7 we need to find additional expressions for emission from within the volume, for absorption by the medium, and for scattering.

Points of Emission within Medium The total emission from a subvolume Vk is given by  Ek = 4κP σT 4 dV.

(20.7)

Vk

Using Cartesian coordinates with dV = dx dy dz, equation (20.7) may be rewritten as   X  Y  Z  X 4 4κP σT dz dy dx = Ek (x) dx. Ek = 0

0

0

(20.8)

0

1. In previous chapters we noted that for participating gases wavenumber η is generally the preferred spectral variable, while wavelength λ is more popular for particulate media and some other fields. For consistency we will only use η here: conversion to wavelength simply requires replacing subscript η by λ everywhere for the remainder of this chapter.

The Monte Carlo Method for Participating Media Chapter | 20 739

FIGURE 20.1 Vectorial representation of emission point inside a tetrahedron DABC.

Then, following the development of equations (7.30) through (7.35), points of emission may be related to random numbers through . X  Y  Z  x Y Z  x 1 Ek dx = κP σT 4 dz dy dx κP σT 4 dz dy dx, (20.9a) Rx = Ek 0 0 0 0 0 0 0 . Y  Z  y Z 4 Ry = κP σT dz dy κP σT 4 dz dy, (20.9b) 0

 Rz =

0

0

.

z

κP σT 4 dz 0

0

Z

κP σT 4 dz,

(20.9c)

0

or x = x(Rx ),

y = y(R y , x),

z = z(Rz , x, y).

(20.10)

Again, choices for x, y, and z become independent of one another if the emission term is separable, e.g., for an isothermal medium with uniform absorption coefficient. In modern-day engineering computations, the computational domain is usually discretized by a mesh that is generated by a mesh generator. The resulting control volumes or cells may be of arbitrary polyhedral shape, and properties are generally assumed constant throughout the cell. In 3D, an arbitrary polyhedron can be dissected into a set of tetrahedra, e.g., a hexahedron (or brick) can be dissected into five nonoverlapping tetrahedra. Following equation (7.41), the emission point inside a tetrahedral cell DABC (see Fig. 20.1) may be written as re = rD + R1 a + R2 b + R3 c,

if R1 + R2 + R3 ≤ 1,

(20.11)

where R1 , R2 , and R3 are three random numbers between 0 and 1, and a = rA − rD , b = rB − rD , and c = rC − rD . Since the coordinates of the vertices of the tetrahedron are known from the mesh generator, these three vectors can be readily calculated. Following the procedure for surface emission, three random numbers are drawn and it is checked to see if their sum is less than unity. If the criterion is satisfied, the point of emission is calculated using equation (20.11). Else, the random numbers are discarded, and a fresh set of three random numbers are drawn. For computations in 2D planar geometries, equation (7.41) can be directly used to find the emission point inside a triangular cell.

Wavenumbers for Emission from within Medium As for surface emission, the choice of emission wavenumber (or wavelength), in general, depends on emission location (x, y, z), unless the volume is isothermal with constant absorption coefficient. From equation (9.57) and the definition of the Planck-mean absorption coefficient it follows immediately that  η π κη Ibη dη, (20.12) Rη = κP σT 4 0 and, after inversion,

740 Radiative Heat Transfer

η = η(Rη , x, y, z).

(20.13)

Directions for Emission from within Medium Under local thermodynamic equilibrium conditions emission within a participating medium is isotropic, i.e., all possible directions are equally likely for the emission of a photon. All possible directions from a point within  2π  π the medium are contained within the solid angle of 4π = 0 0 sin θ dθ dψ, where polar angle θ and azimuthal angle ψ are measured from arbitrary reference axes. Thus, since the integrand is separable, ψ , or ψ = 2πRψ , 2π  θ 1 1 Rθ = sin θ dθ = (1 − cos θ), 2 0 2

Rψ =

or

θ = cos−1 (1 − 2Rθ ) .

(20.14a) (20.14b) (20.14c)

If the polar angle is measured from the z-axis and ψ from the x-axis, then the unit direction vector for emission is given by equation (1.27) as ˆ sˆ = sin θ(cos ψî + sin ψj) + cos θk.

(20.15)

Absorption within Medium When radiative energy travels through a participating medium, the energy is attenuated by absorption and scattered. Equation (9.5) gives the absorptivity for a photon path of length lκ as    lκ αη = 1 − exp − 0 κη ds . (20.16) Therefore, the fraction of energy penetrating through a layer of thickness lκ is    lκ Rκ = exp − 0 κη ds .

(20.17)

Note that lκ does not have to be a straight path, i.e., the number of photons absorbed from a energy bundle depends only on the number of absorbing molecules encountered along its path. This implies that, if the total radiative energy is divided into bundles of equal energy content, the fraction Rκ will be transmitted over a distance lκ or farther, either along a straight path or a zigzagging one (i.e., after being scattered and/or reflected one or more times). Thus, we may relate the distance that any one bundle travels before absorption to a random number by inverting equation (20.17). For the case of media with nonuniform refractive index (or graded media), the photon bundles follow curved trajectories (see Fig. 13.6), and while equation (20.17) remains valid, the evaluation of ds must be done together with equation (13.79) (see also the brief discussion in Section 20.9). This inversion is readily obtained if the absorption coefficient does not vary throughout the medium (κη = const). Under these conditions lκ =

1 1 ln , κη Rκ

(20.18)

and the bundle is allowed to travel a total distance lκ through the medium before being absorbed (unless it is absorbed by a surface before traveling this far). If the absorption coefficient is not uniform (because of temperature dependence or because of a nonisotropic medium), inversion of equation (20.17) is considerably more difficult. Usually, the optical path is evaluated by breaking the volume up into K subvolumes with constant absorption coefficient. Then  s  κη ds κηk sk , (20.19) 0

k

The Monte Carlo Method for Participating Media Chapter | 20 741

where the summation is over those subvolumes (k) through which the bundle has traveled, and sk is the geometric distance the bundle travels through these elements. As long as 



s

lκ

κη ds < 0

κη ds = ln

0

1 , Rκ

(20.20)

the bundle is not absorbed and is allowed to travel on. This standard absorption scheme is occasionally also referred to as “collision-based absorption” [1], i.e., a photon (bundle) travels until it collides with, and is absorbed by, a gas molecule or a particle. Modest [2] noted that, as for surface transport, one may also employ the energy partitioning scheme presented in Chapter 7, i.e., the gradual depletion of a photon bundle’s energy. Since each photon bundle contains many billion individual photons, this implies that a fraction of its photons is absorbed according to the optical thickness of its path. Let Qkin, j be the energy of the k-th photon bundle as it enters cell j. Then, from equation (20.17),    lk j Qkout, j = Qkin, j exp − 0 κη ds = Qkin, j+1 (20.21) is the bundle’s energy leaving cell j, entering the adjacent cell j + 1, and lk j is the length of its path through the cell. In cell-based Monte Carlo simulations, the absorption coefficient is usually considered constant across the cell (but does not have to be), and energy deposited into cell j by energy bundle k is '   )   lk j Qkabs, j = Qkin, j − Qkout, j = Qkin, j 1 − exp − 0 κη ds = Qkin, j 1 − e−κη, j lk j .

(20.22)

This energy partitioning scheme has also been called “absorption suppression” [3] and “pathlength method” [1]. Unlike in collision-based simulations, the bundle is never destroyed (absorbed in its totality) and must be traced until it leaves the enclosure or until its energy is depleted to a tiny prescribed fraction of its original energy which, for conservation of energy, is usually deposited in the final cell visited.

Scattering within Medium Attenuation by scattering obeys the same relationships as for absorption, with the absorption coefficient replaced by the scattering coefficient. Thus, lσ =

1 1 ln σsη Rσ

(20.23)

is the distance a bundle travels in a medium with uniform scattering coefficient before being scattered, or 



s

σsη ds < 0

lσ

σsη ds = ln

0

1 Rσ

(20.24)

for a medium with variable scattering coefficient. Alternatively, the combined event of absorption and scattering can be modeled by determining an extinction distance, lβ =

1 1 ln . βη Rβ

(20.25)

When the location lβ is reached, a second random number Rω is required to decide whether the bundle is absorbed (Rω > ω) or scattered (Rω < ω), with ω being the single scattering albedo. Once a photon bundle is scattered, it will travel on into a new direction. The probability that the scattered bundle will travel within a cone of solid angle dΩ around the direction sˆ  , after originally traveling in the direction sˆ , is P(ˆs ) dΩ = Φ(ˆs · sˆ  ) dΩ ,

742 Radiative Heat Transfer

FIGURE 20.2 Local coordinate system for scattering direction.

where Φ is the scattering phase function. Therefore, we may establish polar and azimuthal angles for scattering as . 2π  π  ψ  π Rψ = Φ(ˆs · sˆ  ) sin θ dθ dψ Φ(ˆs · sˆ  ) sin θ dθ dψ , (20.26a) 0

0

and

 Rθ =

θ

0

. 



Φ(ˆs · sˆ ) sin θ dθ

0



π

0

Φ(ˆs · sˆ  ) sin θ dθ .

(20.26b)

0

For linear anisotropic scattering, from equation (11.99), Φ(ˆs · sˆ  ) = 1 + A1 sˆ · sˆ  = 1 + A1 cos θ ,

(20.27)

where it is assumed that the polar angle θ is measured from an axis pointing into the sˆ -direction, and the azimuthal angle ψ is measured in a plane normal to sˆ . Equations (20.26a) and (20.26b), then, reduce for linear anisotropic scattering to ψ , or ψ = 2πRψ , 2π   1 A1 sin2 θ . 1 − cos θ + Rθ = 2 2

Rψ =

(20.28a) (20.28b)

For isotropic scattering (A1 ≡ 0) these relations are identical to those for (by nature isotropic) emission, equations (20.14). The new direction vector, sˆ  , must then be found by introducing a local coordinate system at the point of scattering, with sˆ pointing into its z-direction (i.e., from where the polar angle θ is measured), as shown in Fig. 20.2. The local x-direction (from where ψ is measured) and y-direction are given by ê1 = a × sˆ / |a × sˆ |,

ê2 = sˆ × ê1 ,

(20.29)

where a is any arbitrary vector. The first of equations (20.29) ensures that the local x-axis is perpendicular to sˆ , and the second makes the coordinate system right-handed. Similar to equation (20.15), the new direction vector may now be expressed as sˆ  = sin θ (cos ψ ê1 + sin ψ ê2 ) + cos θ sˆ .

(20.30)

If scattering is isotropic the scattering direction does not depend on the original path sˆ (all directions are equally likely). In that case, the choice of a local coordinate is totally arbitrary, and equation (20.15) may be used directly. Example 20.1. Consider again the geometry of Example 7.2. The medium within the diffuser is gray with absorption and scattering coefficients of κr0 = 1 and σs r0 = 2, respectively, and an anisotropy factor of A1 = 1 (strong forward scattering). How far will the energy bundle of Example 7.3 travel before being absorbed and/or scattered, if random numbers Rκ =

The Monte Carlo Method for Participating Media Chapter | 20 743

0.200 and Rσ = 0.082 are drawn? If scattering occurs, determine the energy bundle’s new direction after the scattering event, for Rψ = 0.25 and Rθ = 0.13. Solution From equations (20.18) and (20.23) lκ = r0 ln(1/0.20) = 1.61r0 , and lσ = (r0 /2) × ln(1/0.082) = 1.25r0 . From Example 7.3 we know that the bundle must travel a distance of D = (2r0 ) cos 0/0.8 = 2.5r0 before hitting the diffuser. Since lσ < lκ < D, this implies that the bundle will scatter before hitting the diffuser, after which it will travel another distance of lκ − lσ = 0.36r0 before being absorbed (over which distance it may be scattered again or hit a diffuser wall). The location at which the scattering occurs is, from equation (7.63), x = xe +lσ sˆ ·î = 1.25r0 ×0.8 = 1.0r0 , y = 0, z = 1.25r0 ×0.6 = 0.75r0 . √ √ From equations (20.28) we find ψ = 2π × 0.25 = π/2 and cos θ = 2 1 − Rθ − 1 = 2 1 − 0.13 − 1 = 0.8655, or θ = 30◦ . Here the polar angle θ is measured from the direction of sˆ = 0.8î + 0.6kˆ and ψ in the plane normal to it. At the scattering point we may introduce a local coordinate system with, say, a = j, or B ˆ ê2 = sˆ × ê1 = j, ê1 = j × sˆ |j × sˆ | = 0.6î − 0.8k, and, from equation (20.30), sˆ  = 12 (0 + 1 × j) +

1 2

√

√ √ ˆ = 0.4 3î + 0.5j − 0.3 3k. ˆ 3(0.8î − 0.6k)

20.4 Treatment of Spectral Line Structure Effects If the participating medium contains an absorbing/emitting molecular gas, the gas will have a number of vibration–rotation bands, which in turn consist of thousands of overlapping spectral lines (cf. the discussion on gas properties in Chapter 10). The absorption coefficient becomes a strongly gyrating function of wavenumber (cf. Fig. 10.13), making the use of equation (20.12) (emission wavenumber) and equation (20.17) (absorption location, requiring the determination of spectral absorption coefficients along the path) difficult, if not impractical: (i) many digits of accuracy are required in the evaluation of η to ascertain whether emission occurs near a line center (with large κη ) or between lines (small κη ), and (ii) accurate knowledge of the spectral variation of κη was not known until recently. A first attempt to include line structure effects was made by Modest [4], employing the narrow band models described in Chapter 10. With the advent of high-power computers as well as high-resolution spectroscopic databases line-by-line accurate Monte Carlo solutions have recently become reality. And, finally, with modern k-distributions rapidly replacing band models, the Monte Carlo method may also be used in conjunction with them.

Narrow Band Model Monte Carlo In order to find statistically meaningful emission wavenumbers using the statistical narrow band models of Chapter 10, the absorption coefficient is first split into two components, κη = κpη + κ gη ,

(20.31)

where κpη is the (spectrally smooth) absorption coefficient of other participating material (such as particles or ions), and κ gη is the rapidly varying gas absorption coefficient. Taking a narrow band average over the Planck function-weighted absorption coefficient leads to   η   η  η 1  κ gη Ibη dη = κ gη Ibη dη dη κ gη Ibη dη, δη δη 0 0 0 where κ gη = (S/δ)η is the narrow band average of the gas absorption coefficient. The wavenumber of emission is determined with equation (20.12) from  η  π  Rη =  κpη + κ gη Ibη dη, (20.32) κpP +κ gP σT 4 0 and again, after inversion,

744 Radiative Heat Transfer

η = η(Rη , x, y, z).

(20.33)

Application of the narrow band model to find the location of absorption within the participating medium is somewhat more complicated. The random number relations are different for photon bundles emitted from a surface (with spectrally smooth emittance η ), as opposed to bundles emitted from within the medium (with strongly varying absorption coefficient κη ). Bundles emitted from a wall are equally likely to have wavenumbers close to the center of a line or the gap between two lines, causing them to travel a certain distance before absorption. Bundles emitted from within the medium are likely to have wavenumbers for which κη is large [as easily seen by looking at equations (9.57) or (20.7) on a spectral basis], making them much more likely to be absorbed near the point of emission. We will limit our discussion here to the case of a spatially constant absorption coefficient, i.e., κη = κη (η). The more general case of a spatially varying (i.e., temperature- and/or concentration-dependent) absorption coefficient may be found in the original paper of Modest [4]. The amount of energy emitted by a surface element dA over a wavenumber range dη into a pencil of rays dΩ is

η Ibη dAp dη dΩ, where dAp = dA|nˆ · sˆ | is the projected area normal to the pencil of rays. Of this, the amount

η Ibη dAp dη dΩ e−κη l penetrates a distance l into the medium. Taking a narrow band average of both expressions leaves the first one untouched while the second becomes    1

η Ibη dAp dη dΩ e−κη l dη = η Ibη dAp dη dΩ 1 − αη , δη δη where αη is the narrow band average of the spectral absorptivity. The ratio of the two expressions gives the fraction of energy traveling a distance l. Thus, using one of the narrow band models summarized in Table 10.2, we find   W , (20.34) Rκ = 1 − αη exp −κpη l − δ with W/δ from equation (10.77) (Elsasser model) or equations (10.86) through (10.88) (statistical models), for which τ = κ gη l and β is the line overlap parameter. In the high-pressure limit (strong line overlap with β → ∞) equation (20.34) reduces to equation (20.17) for all narrow band models. Explicit inversion of equation (20.34) is possible only for the Malkmus model (unless κpη = 0). If emission is from a volume element, we have for a volume dV, a wavenumber range dη, and a pencil of rays dΩ, the total emitted energy κη Ibη dV dη dΩ, of which the amount κη Ibη dV dη dΩ e−κη l is transmitted over a distance of l. Taking the narrow band average of both expressions and dividing the second by the first gives the transmitted fraction as .   1 1 −κη l Rκ = κη e dη κη dη δη δη δη δη   d W    κpη + dαη dl δ d 1 1 1 −κη l = e dη = αη . (20.35) =− κpη + κ gη dl δη δη κpη + κ gη dl κpη + κ gη

The Monte Carlo Method for Participating Media Chapter | 20 745

Again, equation (20.35) reduces to equation (20.17) for β → ∞. All other random number relations, since they do not involve the spectral absorption coefficient, are unaffected by spectral line effects. Example 20.2. Consider a photon bundle traveling through a molecular gas. The wavenumber of the bundle is such that κ gη = 1 cm−1 and β = 0.1. Drawing a random number of Rκ = 0.200, how far will the bundle travel before absorption, if it was emitted (a) by a gray wall, (b) from within the gas? Use the Goody statistical model. Solution (a) If the bundle originates from a wall, we have from equation (20.34)  Rκ = 0.200 = exp −

W δ



⎞ ⎛   ⎟⎟ ⎜⎜ τ τ ⎟ ⎜ . = exp ⎝⎜− % ⎟⎠ = exp − √ 1 + τ/β 1 + 10τ

By trial and error (or solution of a quadratic equation), it follows that τ = 25.9 and l = τ/κ gη = 25.9 cm. (b) For medium emission, equation (20.35) is applicable, and ⎞ ⎛     ⎟⎟ ⎜⎜ β+τ/2 τ 1 d W d W 1 ⎟⎟ ⎜ Rκ = 0.200 = exp ⎜⎝− % = = % ⎠ dτ δ κ gη dl δ 1+τ/β 1+τ/β β+τ = √

1+5τ −τ/ √1+10τ e , 1+10τ 1+10τ 1

or τ 0.48 and l = 0.48 cm. Therefore, as expected, the bundle travels much farther if emitted from a wall. For comparison, in a gray medium the bundle would have traveled l=

1 1 1 1 ln = 1.61 cm = ln −1 κ Rκ 0.200 1 cm

for both cases.

Some Monte Carlo results for gas–particulate mixtures with line structure effects are shown in Chapter 19, in Fig. 19.3.

Line-by-Line Monte Carlo As indicated in the beginning of this chapter, Monte Carlo methods tend to be CPU time intensive, but they can be applied to very advanced problems without drastically increasing computational effort. Therefore, unlike line-by-line (LBL) accurate conventional RTE solutions (requiring upwards of 1 million spectral RTE solutions), LBL-accurate Monte Carlo calculations can essentially be had for the price of a gray simulation, provided (i) emission wavenumbers and spectral absorption coefficients can be determined efficiently, and (ii) the data storage requirements do not become excessive. The first LBL Monte Carlo scheme was implemented by Wang and Modest [5], who considered mixtures of CO2 and H2 O at combustion conditions. Total local emission per unit volume is determined from  ∞ κpη Ibη dη. (20.36) Etot = 0

Considering that the absorption coefficients of individual species in a mixture are additive, κη =

 i

κη,i =

 i

κpη,i pi

and

κpη = κη /p =

 i

xi κpη,i ,

(20.37)

746 Radiative Heat Transfer

FIGURE 20.3 Random number and absorption coefficient distributions in a small spectral interval [5].

where κpη is the pressure-based absorption coefficient, xi = pi /p is the mole fraction of species i, and p is the total pressure of the mixture, one can obtain the random-number relation for the gas mixture as Rη = =



π κp σT4 σT

where

η Rη,i =

 0∞ 0

κpη Ibη dη =

0

π 5 4 i

η

 xi κp,i

 xi

κη,i Ibη dη

i

 xi κp,i

κpη,i Ibη dη =

0

i

κη,i Ibη dη

η

σT

π 5 4

η

=

κpη,i Ibη dη  0∞ κpη,i Ibη dη 0

η

0





xi κpη,i Ibη dη

i

xi κp,i Rη,i

i

;

xi κp,i ,

(20.38)

i

π = κp,i σT4



η

κpη,i Ibη dη,

(20.39)

0

is the emission wavenumber random number for species i, and κp,i is the pressure-based Planck-mean absorption coefficient. Equation (20.38) establishes a direct relation between the mixture random number Rη and species random numbers Rη,i . Since the relation between the mixture random number Rη and the corresponding wavenumber η is a complicated implicit expression, the emission wavenumber is found by trial-and-error. First an emission wavenumber is guessed, and then the species random numbers Rη,i are determined, followed by the calculation of Rη through equation (20.38). Figure 20.3 shows the random number and corresponding absorption coefficient distributions of a gas mixture in a small spectral interval. Although the random number is a monotonically increasing function, it has strongly varying gradients even in such a small interval. A small error in random number may result in a significant deviation in absorption coefficient. Therefore, common root-finding techniques relying on smooth gradients, such as the Newton-Raphson method, cannot be used here to invert random numbers; instead, a bisectional search algorithm was employed. For absorption calculations, the desired mixture absorption coefficient κη at a given wavenumber can be directly calculated from species pressure-based absorption coefficients κpη,i through equation (20.37). Therefore, a database tabulating both Rη,i –η and κpη,i –η relations of each species can be utilized to determine emission wavenumbers and absorption coefficients for the mixture, and such a database can be constructed once and for all. If the total pressure is fixed, both the species random number, as in equation (20.39) and the pressure-based absorption coefficient are functions of wavenumber, temperature, and species concentration only, i.e., Rη,i = fR,i (η, T, xi ),

κpη,i = fκ,i (η, T, xi ),

i = 1, 2, ..., I,

(20.40)

The Monte Carlo Method for Participating Media Chapter | 20 747

where I is the number of species. Rη,i and κpη,i are functions of temperature and wavenumber only, and a 3D interpolation scheme is sufficient for the database and the computational effort increases only linearly with increasing number of species. Several 2D axisymmetric example problems were calculated using the mocacyl.f code of Appendix F, leading to two important conclusions: 1. Very respectable LBL accuracy can be obtained with very few photon bundles; e.g., as few as 30,000 bundles produced relative standard deviations of < 2% (vs. the need of 1 million or so conventional RTE solutions). Apparently (in optically thin to intermediate media), spectral regions with large absorption coefficients contribute most to the heat transfer, and such wavenumbers are chosen preferentially by the Monte Carlo method. 2. Computer time was dominated by spectral property calculations (primarily determination of emission wavenumbers), indicating the need to make these evaluations as efficient as possible. Ozawa et al. [6] generated a LBL-accurate Monte Carlo scheme for nonequilibrium plasmas found in hypersonic atmospheric entry of space vehicles. They realized that overlap between species is of no consequence for emission, i.e., one may statistically choose the emitting species before applying equation (20.38). Potentially, this eliminates the need for costly trial-and-error solutions, at least for equilibrium applications (such as combustion). In nonequilibrium applications the emission term [cf. equation (9.20)] can depend on many parameters (four temperatures, electronic level populations, number densities of ions, electrons, and neutrals), and a trial-anderror procedure was used by Ozawa et al. [6]. In follow-up work by Feldick and Modest [7] it was recognized that, within a given species, emission from individual lines is also independent of overlap, and the choice can be refined by statistically choosing an individual electronic transition, then a vibrational one, and finally an individual line, for which equation (20.38) can be inverted analytically (for Lorentz and Doppler line shapes). Ren and Modest [8] ported Ozawa et al.’s wavenumber selection scheme to equilibrium applications; because of the hundreds of millions of rovibrational transitions for combustion gas species, it is not practical to separate transition types and particular transitions, and they limited their scheme to only consider emission from different species separately, i.e., equation (20.36) is rewritten as Etot =

∞  ns 0

ns  

∞

κη,i Ibη dη =

i=1

κη,i Ibη dη.

(20.41)

i=1 0

Equation (20.41) indicates that total emission is due to contributions from the individual species and emissions from individual species are independent of another, i.e., there are no overlap effects for emission. Rather than looking for the appropriate wavenumber for the mixture, one may first determine the emitting species s, and only then the appropriate wavenumber. Following this idea, Equation (20.38) can be reformulated as s−1 5

Rη =

i=1

η Ei

Etot

+

κη,s Ibη dη

0

Etot

,

(20.42)

where s is the number index of the emitting species. Ei is the total emission from species i in the given cell, i.e., ∞ Ei =

κη,i Ibη dη.

(20.43)

0

In Ren and Modest’s improved wavenumber selection scheme [8], first a random number for emission wavenumber, Rη , is drawn, and the emitting species s is determined from j−1 5

s= j

if

i=1 ns 5 i=1

5j

Ei < Rη ≤ Ei

i=1 ns 5 i=1

Ei . Ei

(20.44)

748 Radiative Heat Transfer

This ensures that the fraction Ei /Etot of random numbers is employed to pick emission wavenumbers for species i, in accordance with the fractional emission from the species. Once the emitting species is found, Rη is rescaled according to Rη Etot − 0 ≤ Rη,s =

Ej

j−1 5 i=1

Ei < 1.

(20.45)

In the original scheme of Wang and Modest [5], databases tabulating both Rη,i –η and κη,i –η relations for each species were established. When a random number is chosen, the appropriate wavenumber can be found by searching through the mixture random-number relations. In the new wavenumber selection scheme, first the emitting species is determined, and then the rescaled random number Rη,s selects photons of equal strength from species s, with wavenumber found from the probability density function, equation (20.39). Instead of tabulating fractions Rη,i vs. η through equation (20.39), an inverted relation η–Rη,i may be tabulated. Once the emitting species s is determined from equation (20.44), the emission wavenumber can then be found directly (or by linear interpolation) from the η–Rη,s database. By eliminating expensive computational searching, the new scheme with the new database should be more efficient; tests showed a factor of 20 speedup over the original scheme of Wang and Modest. However, Ren and Modest noticed that temperature and concentration interpolation of η = fη,i (Rη,i , T, xi ) can lead to large inaccuracies, while interpolation of Rη,i = fR,i (η, T, xi ) does not. Therefore, they recommend the use of the original tabulation; i.e., first choosing the emitting species, followed by determining the emission wavenumber by searching through a single species Rη,i = fR,i (η, T, xi ) tabulation. This compromise method reduces computational effort by a factor of 10. In follow-up work Ren and Modest [9] also assembled a line-by-line database for Monte Carlo calculation with roughly the same entries as the ones listed in Table 19.4, and also including values for CH4 , C2 H4 , and soot (but coarser mole fraction resolution). This database is available from a repository, as outlined in Appendix F.

FSCK Monte Carlo The Monte Carlo method may also be combined with the full-spectrum k-distribution (FSCK) model of Chapter 19, as was first done by Wang et al. [10], while the related ADF method was employed by Maurente and coworkers [11]. If one compares the general RTE, e.g., equation (9.21), with the transformed FSCK RTE, equation (19.155), it is apparent that the emission term κη Ibη is replaced by k∗ (g0 )a(T, T0 , g0 )Ib (T), where g0 is the new spectral variable, and the absorption coefficient becomes k∗ (g0 ). Thus, a Monte Carlo simulation can be done by simply replacing actual spectral data by reordered emission and absorption as a function of the new spectral variable. Proper emission rescaled “wavenumbers” g0 are then found from equation (20.12) as  g0  g0 ∗ k (g )a(T, T , g )I (T) dg k∗ (g0 )a(T, T0 , g0 ) dg0 0 0 0 0 b R g = 0 1 = 0 1 , (20.46) ∗ (g )a(T, T , g )I (T) dg ∗ (g )a(T, T , g ) dg k k 0 0 0 0 0 0 0 0 b 0 0 (since Ib does not depend on g0 ), with k∗ (g0 ) evaluated according to the chosen FSCK scheme of Section 19.11. The denominator is the local Planck-mean absorption coefficient. In contrast to the oscillatory variation of the spectral absorption coefficient, e.g., Fig. 10.13, the variations of k∗ and a with g0 are much smoother as shown, e.g., in Fig. 19.11. A number of data points on the order of 100 is sufficient to represent these smooth k- and a-distributions. Memory requirements are thus no longer an issue if the FSCK method is used in the Monte Carlo simulation. In addition, the number of photons needed to resolve the spectral variation during one Monte Carlo trial is somewhat reduced. On the negative side, mixture k-distributions must be preassembled from databases for all possible states, as described in Chapter 19, as well as the partially integrated R g = f (g0 , φ, T0 ) given by equation (20.46), which is no trivial task. Unlike for LBL Monte Carlo, absorption coefficients of different species can no longer be separated, and g0 (R g , φ) and k∗ (g0 , φ) must be found from I+2-order interpolation in a mixture with I radiating species. Wang and coworkers [12] assembled such a database for g0 (R g , φ) and k∗ (g0 , φ) to use with the FSCK-4 method of Section 19.11, and for the identical range of thermodynamic states. Their results indicate that compared to benchmark LBL calculations, almost the same accuracy can be achieved using the FSCK/PMC method with the FSCK-4 look-up table and the g0 (R g , φ) random-number database, while the ray selection/tracing process becomes roughly 20% more efficient (compared against the improved LBL scheme

The Monte Carlo Method for Participating Media Chapter | 20 749

of Ren and Modest [8,9]), while memory requirements are also greatly reduced. They noted two important applications for using the FSCK/PMC method: (1) problems where the savings in memory and (relatively minor) CPU time are important, and (2) as a benchmark to test the accuracy of conventional RTE solvers paired with the FSCK spectral model. This g0 (R g , φ) random-number database is also available from a repository, as outlined in Appendix F. Much greater speed-ups of up to a factor of 20 can be obtained in nonequilibrium applications, where radiative property evaluations are exceedingly expensive.

20.5 Overall Energy Conservation The temperature field within the medium is determined from overall conservation of energy, as given by equation (9.75). In the absence of conduction and convection, i.e., if radiative equilibrium prevails, this equation reduces to the simple form of ∇ · qR = 0, where qR is the radiative heat flux. Whether a deterministic technique or a Monte Carlo method is used, the solution is simplest for a gray medium at radiative equilibrium, followed by the case of radiative equilibrium in a nongray medium and, finally, the gray and nongray medium in the presence of conduction and/or convection.

Gray Medium at Radiative Equilibrium Radiative equilibrium implies that anywhere within the medium the material absorbs precisely as much radiative energy as it emits. Therefore, for every photon bundle absorbed at location r, another photon bundle of the same strength must be emitted at the same location. The direction of the new photon is determined from equations (20.14). We note that these relations are identical to those for isotropic scattering, equations (20.28), since emission is always isotropic. The wavenumber of the newly emitted energy bundle may be determined from equation (20.12) and depends on the local temperature. However, if the medium and the walls are gray, then the wavenumber of the bundle is irrelevant (indeed, does not have to be determined). Thus, if absorption and scattering coefficients are independent of temperature, knowledge of the temperature field is not required to find the solution: Energy bundles are emitted from the bounding walls (according to their temperatures) and are followed until they are absorbed by a wall (after perhaps numerous scattering and absorption–re-emission events inside the medium). Numerically, the process is identical to a purely scattering medium, with the extinction coefficient β = κ + σs replaced by an effective scattering coefficient σs = β. The temperature field inside the medium is determined by keeping track of the total re-emitted energy from a control volume Vi : Qabs,i =

Ni 

Qi j = Qem,i = 4σκi Ti4 Vi ,

(20.47)

j=1

or

⎛ ⎞1/4 Ni ⎜⎜ ⎟⎟ + ⎟ ⎜ Qi j 4σκi Vi ⎟⎟⎟ , Ti = ⎜⎜⎜ ⎠ ⎝

(20.48)

j=1

where the Qi j are the amounts of energy carried by the Ni photon bundles that have been absorbed within Vi (after emission from a wall and, possibly, re-emission from within the medium). This solution is limited to the case of constant properties, since absorption and scattering locations depend on local values of absorption and scattering coefficients. If these properties depend on temperature, a temperature field must be guessed to determine them, and an iteration becomes necessary.

Nongray Medium at Radiative Equilibrium If the medium is nongray, the wavenumber of each re-emitted bundle must be determined from equation (20.12), requiring knowledge of the temperature field. Therefore, the solution becomes an iterative process: First a temperature field is guessed, and employing this guess, the solution proceeds similar to the one described above for a gray medium, after which local temperatures are recalculated from equation (20.48), etc., until the solution converges. There is another way to obtain a solution. Based on the guess of the temperature field we “know” how much energy is emitted from each subvolume. We may therefore separate the emission and absorption processes:

750 Radiative Heat Transfer

Photon bundles are emitted not only by the walls, but also by the medium, and they are then traced until they are absorbed by either wall or medium (i.e., there is no re-emission in this method). This leads to different values for Qabs,i and Qem,i in equation (20.47), which may be used to update the temperature field. This method of solution is usually inferior since emission depends very strongly on the (unknown) temperature field, while nongray behavior is only implicitly influenced by the temperature.

Coupling with Conduction and/or Convection If conduction and/or convection are of importance the radiation problem must be solved simultaneously with overall conservation of energy, equation (9.75). Since the energy equation is usually solved by conventional numerical methods (although a Monte Carlo solution is, in principle, possible; see, e.g., Haji-Sheikh [13]), an iteration in the temperature field is necessary: Similar to radiative equilibrium in a nongray medium a temperature field is guessed and used to solve the radiation problem, leading to volume emission rates, Qem,i , and absorption rates, Qabs,i , for each subvolume. The net radiative source is then 

  1  ∇ · qR i = Qem,i − Qabs,i , Vi

(20.49)

which is substituted into the solution for equation (9.75) to predict an updated temperature field. Because of the statistical uncertainties in the Monte Carlo calculations, this may lead to instabilities. The tight convergence standards normally applied to finite difference/volume iterations must be loosened considerably. If quasi-steady turbulence is treated through stochastic particle fields (i.e., by a Monte Carlo method applied to the flow), radiation Monte Carlo schemes blend naturally with the turbulence model and can be very efficient [14,15] (see following section). Tight coupling with a quasi-steady fully finite-volume flow code can also be achieved through time blending (limited sampling during a given iteration blended with the solution from previous iterations) [16], as detailed in Section 22.2.

20.6 Discrete Particle Fields In modern combustion simulations it is becoming common to represent pulverized coal and fuel sprays through Lagrangian discrete particle fields, e.g., [17,18]. Turbulent combustion models use stochastic probability density function (PDF) models to resolve the nonlinear turbulence–chemistry interaction term, in which the fluid is represented by a large number of notional point-masses (see discussion in Chapter 22). To simulate the radiative transfer process by ray tracing in a discrete particle field, the interaction between infinitesimal point-masses and infinitesimally thin photon rays needs to be modeled. This can be done by assigning effective volumes to the point-masses, by assigning an influence volume to the ray’s trajectory, or a combination of both. In this section, several particle models and ray models are developed, as well as photon emission and absorption algorithms based on these models. More detail can be found in the original papers of Wang and Modest [19,20]. Their work dealt with stochastic particles used in turbulence modeling, and we will describe the method in this context. However, with very minor modifications the model is also directly applicable to physical particle fields. This was explored by Marquez and coworkers [21] for mixtures of gases and multiple solid particle phases, and by Roy et al. [22] for spray combustion.

Particle and Photon Ray Models Point Particle Model (PPM) In this model, particles are treated as point-masses, i.e., they carry an amount of mass without a specific shape at a certain spatial location as shown in Fig. 20.4a, which is a 2D particle field. The only geometric information known about the particles is their position vector ri . However, particles do have a nominal volume, which may be calculated from their thermophysical properties such as pressure and temperature. For example, for stochastic fluid particles, if the ideal gas assumption is adopted, the nominal volume may be computed as Vi =

mi RTi , pi

(20.50)

The Monte Carlo Method for Participating Media Chapter | 20 751

FIGURE 20.4 Discrete particle field representations of a 2D medium: (a) PPM representation; (b) SPM/CDS representation of a sub-region in (a); and (c) cone–PPM scheme.

where mi is the mass carried by particle i, Ti is its temperature, pi is its total pressure, and R is the gas constant. To enforce consistency in the discrete particle representation of the medium, the overall nominal volume of all particles should be the same as the actual geometric expanse of the medium. As a consequence, one may regard the nominal volume of a particle as its real volume. The Point Particle Model only contains the particle information that the original discrete particle field contains. It does not employ any other assumption and, therefore, it will not induce any inconsistency. The disadvantage of this model is that it is difficult to determine the interaction of a photon ray with a volume without shape. Spherical Particle Model (SPM) In this method, each point-mass mi has a spherical influence region Ωi , surrounding it as shown in Fig. 20.4b. The mass is distributed to its influence region according to a density ˆ Ri is its influence radius, and ρi is the nominal density calculated by profile ρ, ρi =

pi mi = , Vi RTi

(20.51)

so that the total mass in the influence region is equal to the point-mass. In this method, particles are assigned a spherical volume (influence region) with varying density, and overlapping other particles in the domain. Here we consider only the case of constant density spheres with a radius determined by their nominal volumes, Ri =



3Vi 4π

1/3

,

(20.52)

termed the Constant Density Sphere (CDS) model. The overall density at an arbitrary position is the sum of density contributions from all nearby particles. Some locations may be influenced by more than one particle, while some other locations may not be in the influence region of any particle, i.e., there is a void in these places. Therefore, this model cannot recover a continuous density medium as shown in Fig. 20.4b, which is a small portion of the CDS representation of the 2D field given in Fig. 20.4a (if variable densities were employed, the Ri would be larger, resulting in substantial overlap, even in this region of few particles). A location with lots of void space was chosen for better readability. In order to show particle locations in a plane, a 2D rather than 3D particle field is depicted. Line Ray Model In this model, a ray is simply treated as a volume-less line and energy propagates onedimensionally along the line. This is the standard model for ray tracing in continuous media. Since such rays are not designed to have a specific volume, they are not able to interact with point-masses. Therefore, this model requires volumetric particle models for radiative transfer simulations. Cone Ray Model Physically, a photon bundle consists of many millions of individual photons, occupying a small solid angle. Thus, to model the volume of a ray, one may assign a small solid angle to the ray and treat it as a cone. Energy is assumed to propagate axisymmetrically along the cone, with its strength decaying in the radial direction normal to the cone axis, similar to the varying particle density in the spherical particle model. For a ray emitted at ro into a direction given by a unit direction vector sˆ , the intensity at location r within the ray’s cone can then be modeled as I(s, r) = Io (s)wc (r/Rc (s)),

(20.53)

752 Radiative Heat Transfer

where s = (r − ro ) · sˆ is the distance from the emission location to a point on the ray axis, r is the distance from a point to the ray axis on a plane normal to the axis, Io (s) is the intensity at the ray center, Rc (S) is the local influence radius of the cross-section as depicted in Fig. 20.4c, and wc is a normalized two-dimensional center-symmetric profile, which satisfies  Rc 2 wc (r)r dr = 1. (20.54) R2c 0 Again, many weight functions are possible, ranging from wc = 1 to Gaussian decay. A popular Gaussian-like weight function is given by [23]. Physically, the distribution of energy emitted from a point is isotropic in all directions. Different rays from the same point may overlap if rays have a volume. The Gaussian decay of energy along the radial direction provides a smoother overlap than a uniform energy distribution across the cone crosssection. Since in this model the ray has a specific volume, volume-less particles can be intercepted by the ray, and this model can work together with the Point Particle Model.

Emission from a Particle A small gas volume emits energy uniformly into all directions. In Monte Carlo simulations, the total energy is divided into a number of photon bundles (rays) which are released into random directions. In a physical gas volume, the emitted energy comes from every point in the volume. If the medium is represented by discrete particles, emission takes place inside these particles. Thus, depending on the optical thickness of the particle, and the point and direction of emission, some of the emitted energy may not escape from the particle due to self-absorption. If the particle is optically thin, the self-absorption of emission is negligible and the total emission from particle i is calculated from equation (9.57): Qem,i = 4κρ,i mi σTi4 ,

(20.55)

where κρ,i is the density-based Planck-mean absorption coefficient at particle temperature Ti . If self-absorption is considered and the particle is assumed to be a constant density sphere, the total emission from a sphere is obtained from Example 9.3 as ⎫ ⎧ ⎪ ⎪ 1  ⎨ −2τi ⎬ (20.56) 1 − Qem,i = 4πR2i σTi4 ⎪ )e 1 − (1 + 2τ ⎪ i ⎭, ⎩ 2τi2 where τi = ρi κρ,i Ri is the optical thickness of the spherical volume based on the nominal radius. In the Point Particle Model, the shape of a particle is arbitrary, but equation (20.56) is still a good approximation of total emission from such a particle. If more than one ray is emitted from a particle, the sum of initial energy carried by all rays must be equal to the total emission calculated from equation (20.55) or equation (20.56), depending on whether self-absorption is neglected. The number of rays emitted by a specific particle should be determined by the total emission of the particle, guided by the average value of energy that the rays carry, i.e., Qavg =

Np 

; Qem,i Nr ,

(20.57)

i=1

where Np is the total number of particles in the computational domain and Nr is the prescribed total number of rays to trace. The range of ray energy [Qmin , Qmax ] can be chosen around the average ray energy, Qmin < Qavg < Qmax ,

(20.58)

since the total emission from a particle cannot be expected to be an integer multiple of the average ray energy. If the total emission of a particle is in the range defined in equation (20.58), its total energy will be lumped into one ray. However, particles in hot zones of the medium tend to emit more energy, and if the total emission of particle i exceeds the maximum ray energy, it needs to emit more than one ray in order for each ray to obey equation (20.58). The number of rays emitted by particle i can be determined from Nr,i = Qem,i /Qavg + 0.5,

(20.59)

The Monte Carlo Method for Participating Media Chapter | 20 753

with x being the largest integer ≤ x. The individual bundles’ strengths leaving particle i as ray j are then Qi, j = Qem,i /Nr,i .

(20.60)

Because the energy of each ray should also satisfy equation (20.58), a requirement of choosing the ray energy range is obtained as Qmax ≥ 2Qmin .

(20.61)

One convenient choice is Qmin =

2 Qavg 3

and

Qmax =

4 Qavg . 3

(20.62)

In cold zones particles emit little energy and, for increased efficiency, it is advantageous to combine the emission of several particles into one ray. To be meaningful, a low-emission particle should be combined with particles in its close proximity. In modeling of combustion flows a finite-volume mesh is often used to control the particle number density and resolve different levels of gradients. Particle size and other properties tend to be relatively uniform in a single finite-volume cell, which means that a low-emission particle tends to be surrounded by other low-emission particles. Therefore, the finite-volume mesh can be utilized to search lowemission particles and combine their emission. The emission point of the resultant ray is then determined as ;  r= Qem,c rc Qem,c , (20.63) c

c

where the subscript c denotes those particles combined together. Equation (20.61) also guarantees that the resultant ray energy falls into the prescribed ray energy range during the particle emission combination process.

Absorption Models The basic task of simulating the absorption of a photon bundle in a medium described by a point particle field is the evaluation of the optical thickness that a ray traverses along its path. This is achieved by modeling the interaction between the ray and the particles that it encounters. Based on different models employed for rays and particles, several schemes for absorption simulation may be obtained. Line–SPM Scheme In this scheme, the ray is treated as a line and the Spherical Particle Model (SPM) is employed for the particles as shown in Fig. 20.4b. If the Constant Density Sphere (CDS) model is employed, the mass of the particle is distributed uniformly across its influence region and the optical thickness that ray j passes through is computed as ( Δτi j = 2ρi κρ,i R2i − r2i j , (20.64) where ri j is the distance from the center of particle i to ray j, as indicated in Fig. 20.4b. The total optical thickness that ray j passes through is simply the summation of the contributions from the individual particles it interacts with,  τj = Δτi j , (20.65) i∈Ij

where Ij denotes all the particles intersected by ray j. Cone–PPM Scheme If the ray is modeled as a cone, it is possible to let it interact with point particles. The energy change of a conical ray when it traverses over a small distance ds in a continuous medium is 

Rc

dE(s) = − 0



Rc

κ dsI(r)2πr dr = −κ ds

I(r)2πr dr = −E(s)κ ds,

(20.66)

0

where E(s) is the plane-integrated energy over the cone cross-section at axial location s, κ(s) is the local absorption coefficient, κ(s) is the plane-averaged absorption coefficient, and Rc (s) is the local radius of the cone’s cross-

754 Radiative Heat Transfer

section. The plane-averaged absorption coefficient can be derived as  Rc 0

κ= R c 0

 Rc

κIr dr

κwc r dr 2 = 0 R = 2 c R c Ir dr wc r dr 0



Rc

κwc r dr.

(20.67)

0

Limiting ourselves again to constant weights (wc = 1), 2 R2c

κ=



1

κr dr.

(20.68)

0

Therefore, the total optical thickness that ray j passes through along S is τ = − ln

E(S) =− E(0)

 S

dE = E



  κ ds = S

S

0

Rc

κ 2πr dr ds = πR2c

 Vj

κ dV, πR2c

(20.69)

where Vj is the volume that the ray covers in its path. In discrete particle fields as shown in Fig. 20.4c, the absorption coefficient is represented by a set of Dirac delta functions,2   κ= κi Vi δ(r − ri ) = κi Vi δ(x − xi ) δ(y − yi ) δ(z − zi ). (20.70) i

i

Integration over Vj yields τ=

 κV  κρ,i mi i i = , πR2c,i πR2c,i i∈I i∈I

(20.71)

where I denotes all the particles enclosed by the cone. Cone–SPM Scheme In the most advanced scheme, the ray is treated as a cone, and the particle is given a specific shape and a density distribution may exist across its volume, as described in [19]. All three absorption models were found to be roughly equally accurate, with the Cone–SPM scheme slightly better, but somewhat more involved and expensive.

Implementation Considerations In order to evaluate and compare the performance of the different schemes for Monte Carlo ray tracing in media represented by statistical (or physical) particles, one-dimensional radiative heat transfer problems in a nonscattering gray gaseous medium were studied. Two media were considered: a 1D gas slab bounded by two infinitely large, parallel, cold, black walls and a gas sphere surrounded by a cold black wall. The thickness of the slab and the radius of the sphere were fixed, while temperature and density (or absorption coefficient) were varied across the slab thickness or along the sphere radial direction. The resulting radiative heat fluxes at the boundary were compared with exact values found through numerical integration. In the slab problem, the 1D medium was simulated by repeating a gas cube, each with equal side lengths in the two infinite dimensions. A single gas cube is then taken as the computational domain in the Monte Carlo simulation. In the sphere problem, the computational domain is the gas sphere itself. The continuous gas medium in both problems is represented by a number of discrete gas particles randomly placed inside the computational domain. The mass of particles can be equally sized or have a distribution function. For computational efficiency, a mesh of cubic cells is laid on top of the computational domain because the ray-tracing algorithm on smaller cubic cells is simpler and more efficient. The same cubic-cell mesh is used for the sphere problem as well. In the slab problem each of the cells contains a number of gas particles, while in the sphere problem some cells at the corners of the mesh may contain no particles, because they may be outside the spherical computational domain. If the Point Particle Model (PPM) is employed, it can be assumed that each particle is completely enclosed by a single cell, since the shape of particles is not specified. However, if the Spherical Particle Model (SPM) is employed, the cells contain not only the particles with their center in it, but also parts of particles from 2. For a definition of one- and multidimensional Dirac-delta functions see equations (10.111) and (18.1).

The Monte Carlo Method for Participating Media Chapter | 20 755

FIGURE 20.5 Figure of merit (FoM) of Cone Ray Models at different cone opening angles; 50 × 10,000 equally-sized particles; 1 ray/particle; homogeneous medium.

neighboring cells. Thus, a scheme must be developed to avoid having the ray interact with a single particle more than once, since a single particle may belong to multiple cells. When the Cone Ray Model is adopted for ray tracing, the opening angle (the angle between the cone axis and its lateral surface) needs to be chosen. Larger opening angles result in more particles caught by the ray, requiring more CPU time per ray. At the same time, larger opening angles reduce the statistical scatter (i.e., reduce the number of required photon bundles for a given desired standard deviation), while also smoothing out gradients that may exist in the solution. For example, in turbulent flow fields large opening angles may smooth out the turbulence. The “figure of merit” (FoM) of a Monte Carlo simulation is defined as [24] 1 FoM = 2 ,

t

⎞1/2 ⎛ S ⎟⎟ ⎜⎜ 1  (qs /q0 − 1)2 ⎟⎟⎟⎠

= ⎜⎜⎜⎝ S

(20.72)

s=1

where is the root-mean-square (RMS) relative error of the simulation and t is the simulation time. Here, the error of 50 simulations was employed for (S = 50). qs is the simulation result of radiative flux at the boundary and q0 is the exact solution. A good Monte Carlo simulation should have a high FoM score. Figure 20.5 shows FoM scores for different opening angles. The gas slab or sphere was represented by 10,000 randomly distributed, equally sized particles, each of which emits all its energy into a single random direction. Temperature and absorption coefficient are uniform and, thus, the smoothing effect of larger cone angles is not an issue. The mesh in use contains 5 × 5 × 5 = 125 cubic cells. As seen from Fig. 20.5, for this one-dimensional problem 1◦ is the optimal opening angle, which can achieve high accuracy as well as high computational efficiency. Although smaller opening angles required less computational time, their errors were larger, because they could not interact with enough particles. Similar results were also obtained for other temperature and absorption coefficient profiles. Another factor that can affect the simulation speed is the number of particles per cell. When a ray is traced, the cells that it travels through are identified first. Then all particles in those cells are checked for interaction with the ray. For a finer mesh, the number of particles per cell is smaller and, thus, a smaller number of particles are checked during ray tracing. However, more cells must be searched. Thus, finer meshes tend to reduce the time spent on checking particles for their interaction with a ray, but increase the overhead related to cell searching and recording. It was found that no optimal cell size exists for the Line–CDS scheme; the computational time decreases consistently with decrease of cell size. For Cone schemes, however, an optimal value was found to be around 50 particles/cell in both the slab and the sphere problems.

756 Radiative Heat Transfer

Media Containing Solid Particles or Droplets Marquez and coworkers [21] extended the spectral Monte Carlo approach to media containing large numbers of small coal and/or ash particles, such as fluidized bed or pulverized coal combustion. This allowed them to represent the particles by continuum relationships, with their radiative properties obtained from the Buckius and Hwang [25] correlations of Chapter 11.11. Since both gas and particles are treated as continuum, standard line ray tracing is invoked with energy splitting across phases for both emission and absorption. Emission of photon bundles is considered on a cell-by-cell basis; thus for cell i Qemi,i = Qemi,g,i +

Ms 

Qemi,s,m,i ,

(20.73)

m=1

where Qemi,g,i is emission by the gas phase, while Qemi,s,m,i is emission from solid phase m, and each solid phase (coal, ash, and/or particles of different sizes separated into groups) has its own temperature and radiative properties. Emission location (within cell i), direction, and wavenumber are determined by standard means. To choose an emission wavenumber the LBL scheme of Ren and Modest [8] is used for the gas [equations (20.42) through (20.45)], and for particles a spectral model conforming with the Buckius and Hwang [25] correlations was developed. Employing the energy partitioning scheme of equation (20.22), the energy of the k-th ray absorbed by cell j is   Qkabs, j = Qkin, j 1 − exp(−κη, j lk j ) , (20.74) where κη, j is the absorption coefficient of the cell’s multi-phase mixture, κη, j = κη,g, j +

Ns 

κη,s,m, j .

(20.75)

m=1

This energy must be distributed over the various phases. Invoking physical constraints Marquez et al. [21] developed a weighted absorption splitting scheme as Qkabs,(g)or(s,m), j = Qkabs, j wη,(g)or(s,m), j

(20.76)

with weights wη,(g)or(s,m), j =

κη,(g)or(s,m), j . 5 s κη,g, j + N κ m=1 η,s,m, j

(20.77)

The multi-phase Monte Carlo approach was applied to a pulverized coal simulation by embedding the code into the open source multi-phase flow solver MFIX [18]. An alternative simplified Monte Carlo code for fluidized beds with concentrated solar radiation was developed by Zedtwitz et al. [26]. Roy et al. [22] developed a Monte Carlo code for spray combustion. Their model is embedded into the OpenFOAM open source CFD solver [17], which has built-in spray modeling capabilities. In OpenFOAM, sprays are numerically represented as a set of ‘parcels,’ and each parcel contains several droplets of identical properties. In this description of spray, all droplets are spatially co-located at one single coordinate for the entire parcel. Two different models were developed, one Eulerian, the other Lagrangian. In the Eulerian model radiative properties of the Lagrangian spray parcels are converted to an equivalent Eulerian phase for PMC calculations. Since the Lagrangian identity is lost for ray tracing purposes, line rays are used in this model, and the droplet properties are “distributed” over the entire computational cell. Therefore, each ray interacts with all droplets present in the cell. In the Lagrangian Monte Carlo version the carrier gas phase is treated in a Lagrangian manner using stochastic PDF particles (a “transported PDF” method [27]), as described earlier in this section). In this approach, each spray parcel retains its Lagrangian identity, and hence a (small) finite volume and a specific location. Since the volume of a parcel is very small compared to the volume of a computational finite-volume cell, they may be treated as point-masses, and the cone ray–point particle mass (cone-PPM) scheme is invoked. In both, Eulerian and Lagrangian models, emission from the fuel spray is neglected due to its generally low temperature. Energy accounting is similar to the one of Marquez and coworkers [21], employing

The Monte Carlo Method for Participating Media Chapter | 20 757

the same weighted absorption scheme. Both models were tested against exact solutions for a one-dimensional slab with cold, black walls and various mixtures of gas and spray, with results from both methods closely matching the exact solutions. Since cone-PPM tracing involves a more expensive search than line ray tracing, the Lagrangian Monte Carlo is expected to be more costly: for the test configurations, for the same number of rays, the Lagrangian PMC required approximately 12 times the CPU time of the Eulerian version. The models were also applied to important applications, such as a high-pressure combustion bomb and a Diesel engine at industry-relevant operating conditions.

20.7 Backward Monte Carlo The Monte Carlo scheme, as presented so far, is a “forward” method, i.e., a photon bundle is emitted and we then follow its progress until it is absorbed or until it leaves the system. The method can easily simulate problems of great complexity and, for the majority of problems where overall knowledge of the radiation field is desired, the method is reasonably efficient. However, if only the radiative intensity hitting a small spot and/or over a small range of solid angles is required, the method can become terribly inefficient. Consider, for example, a small detector (maybe 1 mm × 1 mm in size) with a small field of view (capturing only photons hitting it from within a small cone of solid angles) monitoring the radiation from a large furnace filled with an absorbing, emitting, and scattering medium. In a standard Monte Carlo simulation, we would emit many photon bundles within the furnace and would trace the path of each of these photons, even though only the tiniest of fractions will hit the detector. It may take many billion bundles before a statistically meaningful result is achieved—at the same time the intensity field is being calculated everywhere (without need); clearly a very wasteful procedure. Obviously, it would be much more desirable if one could just trace those photon bundles that eventually hit the detector. This idea of a backward tracing solution, sometimes also called reverse Monte Carlo, has been applied by several investigators [28–39], all based on the principle of reciprocity described by Case [40]. This principle states that if Iη1 and Iη2 are two different solutions to the radiative transfer equation for a specific medium, sˆ · ∇Iη j (r, sˆ ) = Sη j (r, sˆ ) − βη (r)Iη j (r, sˆ ) +

σsη (r)



4π

Iη j (r, sˆ  )Φη (r, sˆ  , sˆ ) dΩ,

j = 1, 2,

(20.78)

4π

subject to the boundary condition Iη j (rw , sˆ ) = Iwη j (rw , sˆ ),

j = 1, 2,

(20.79)

then these two solutions are related by the following identity:   A

n·ˆ ˆ s>0



 Iwη2 (rw , sˆ )Iη1 (rw , −ˆs) − Iwη1 (rw , sˆ )Iη2 (rw , −ˆs) (nˆ · sˆ ) δΩ dA     = Iη2 (r, −ˆs)Sη1 (r, sˆ ) − Iη1 (r, sˆ )Sη2 (r, −ˆs) dΩ dV, (20.80) V

4π

where A and V denote integration over enclosure surface area and enclosure volume, respectively, and nˆ · sˆ > 0 indicates that the integration is over the hemisphere on a point on the surface pointing into the medium. In the backward Monte Carlo scheme, the solution to Iη1 (r, sˆ ) [with specified Sη1 (r, sˆ ) and Iwη1 (rη1 , sˆ )] is found from the solution to a much simpler problem Iη2 (r, sˆ ). In particular, if we desire the solution to Iη1 at location ri (say, a detector at the wall) into direction −ˆsi (pointing out of the medium into the surface), we choose Iη2 to be the solution to a collimated point source of unit strength located also at ri , but pointing into the opposite direction, +ˆsi . Mathematically, this can be expressed as Iwη2 (rw , sˆ ) = 0, Sη2 (r, sˆ ) = δ(r − ri ) δ(ˆs − sˆ i ),

(20.81a) (20.81b)

758 Radiative Heat Transfer

FIGURE 20.6 Typical ray path in a backward Monte Carlo simulation.

where the δ are Dirac-delta functions for volume and solid angles.3 If the infinitesimal cross-section of the source, normal to sˆ i , is dAi , then this results in an Iη2 intensity at ri of Iη2 (ri , sˆ ) =

δ(ˆs − sˆ i ) . dAi

(20.82)

As the Iη2 light beam travels through the absorbing and/or scattering medium, it will be attenuated accordingly. Sticking equations (20.81) into equation (20.80) yields the desired intensity as     Iwη1 (rw , sˆ )Iη2 (rw , −ˆs)(nˆ · sˆ ) dΩ dA + Sη1 (r, sˆ )Iη2 (r, −ˆs) dΩ dV. (20.83) Iη1 (ri , −ˆsi ) = A

n·ˆ ˆ s>0

V

4π

While the Iη2 problem is much simpler to solve than the Iη1 problem, it remains quite difficult if the medium scatters radiation, making a Monte Carlo solution desirable. Therefore, we will approximate Iη1 as the statistical average over N distinct paths that a photon bundle emitted at ri into direction sˆ i traverses, as schematically shown in Fig. 20.6, or Iη1 (ri , −ˆsi ) =

N 1  Iη1n (ri , −ˆsi ), N

(20.84)

n=1

where the solution for each Iη1n is found for its distinct statistical path (with absorption and scattering occurrences chosen exactly as in the forward Monte Carlo method). Along such a zigzag path of total length l from ri to rw , consisting of several straight segments pointing along a local direction sˆ  (r ), Iη2 is nonzero only over an infinitesimal volume along the path, dV = dAi l, and an infinitesimal solid angle centered around the local direction vector −ˆs = sˆ  (r ). At its final destination on the enclosure surface, the beam of cross-section dAi    illuminates an area of only dA = dAi / −ˆs (rw )·nˆ , so that equation (20.83) simplifies to   l     l           Iη1n (ri , −ˆsi ) = Iwη1 rw , −ˆs (rw ) exp − κη (r ) dl + Sη1 r , −ˆs (r ) exp − 0

0

l

 

κη (r ) dl



dl ,

(20.85)

0

 l where 0 dl indicates integration along the piecewise straight path, starting at ri . It is seen that Iη1n (ri , −ˆsi ) consists of intensity emitted at the wall into the direction of sˆ  (rw ) (i.e., along the path toward ri ), attenuated by absorption along the path, and by emission along the path due to the source Sη1 , in the direction of −ˆs (r ) (also 3. For a definition of one- and multidimensional Dirac-delta functions see equations (10.111) and (18.1).

The Monte Carlo Method for Participating Media Chapter | 20 759

along the path toward ri ), and attenuated by absorption along the path, between the point of emission, r , and ri . This result is intuitively obvious since it is the same as equation (9.28), except that we here have a zigzag path due to scattering and/or wall reflection events. If we trace a photon bundle back toward its point of emission, allowing for intermediate reflections from the enclosure wall (as indicated in Fig. 20.6), then, at the emission point rw , Iwη1 = η Ibη (rw ). And, if the internal source of radiation is due to isotropic emission, then, comparing equations (9.21) and (20.78) we find Sη1 (r , −ˆs ) = κη (r )Ibη (r ). Thus,   l     l Iηn (ri , −ˆsi ) = η (rw )Ibη (rw ) exp − κη (r ) dl + κη (r )Ibη (r ) exp − 0

l

 κη (r ) dl dl ,

(20.86)

0

0

where the subscript “1” has been dropped since it is no longer needed. Equation (20.86) may be solved via a standard Monte Carlo simulation or using the energy partitioning scheme of the previous section. For the standard method scattering lengths lσ are chosen from equation (20.24) as well as an absorption length lκ from equation (20.20). The bundle is then traced backward from ri unattenuated [i.e., the exponential decay terms in equation (20.86) are dropped], until the total path length equals lκ or until emission location rw is reached (whichever comes first). Thus, ⎧  lκ ⎪ ⎪ ⎪ ⎪ ⎪ κ (r )Ibη (r ) dl , ⎪ ⎪ ⎨ 0 η Iηn (ri , −ˆsi ) = ⎪  l ⎪ ⎪ ⎪ ⎪ ⎪ κη (r )Ibη (r ) dl , ⎪ ⎩ η (rw )Ibη (rw ) +

lκ < l, (20.87) lκ ≥ l.

0

If energy partitioning is used only scattering lengths are chosen from equation (20.24) and Iηn is found directly from equation (20.86). Radiative Fluxes If radiative flux onto a surface at location ri over a finite range of solid angles is desired, the absorbed incoming flux needs to be computed as in equation (3.17), using the statistical data obtained for Iηn (ri , −ˆsi ). This is best done by the method described in Section 7.2, equation (7.10). For example, for a detector located at ri with opening angle θmax one obtains 

2π

qdet = 0

=

1 2

 0



θmax

0 2π  1

η (θ, ψ)Iη (π − θ, ψ) cos θ sin θ dθ dψ

cos2 θ

η (θ, ψ)Iη (π − θ, ψ) d(cos2 θ) dψ max

π(1 − cos2 θmax )

N 

η (ˆsin )Iηn (−ˆsin ),

(20.88)

n=1

where the directions sˆ in need to be picked uniformly from the interval 0 ≤ ψ ≤ 2π, cos2 θmax ≤ cos2 θ ≤ 1. The azimuthal angle ψn is found from equation (7.51), while θn is found from 1 cos2 θn

Rθ =  1

dζ

cos2 θmax

dζ

=

1 − cos2 θn sin2 θn = , 2 1 − cos θmax sin2 θmax

or

θn = sin−1

%

 Rθ sin θmax .

(20.89)

If the detector is of finite dimension, points distributed across the surface are chosen like in a forward Monte Carlo simulation. Collimated Irradiation Backward Monte Carlo is extremely efficient if radiative fluxes onto a small surface and/or over a small solid angle range are needed. Conversely, forward Monte Carlo is most efficient if the radiation source is confined to a small volume and/or solid angle range. Both methods become extremely inefficient, or fail, if radiation from a small source intercepted by a small detector is needed. For collimated irradiation (and similar problems) backward Monte Carlo can be made efficient by separating intensity into a

760 Radiative Heat Transfer

direct (collimated) and a scattered part, as outlined in Chapter 18. Thus, comparing equations (20.78) and (18.12) we find, assuming volumetric emission to be negligible,   lc  qcoll (rw )  exp − (κη + σsη ) dlc Φ(r, sˆ 0 , sˆ ), Sη1 (r, sˆ ) = σs (r) 4π 0

(20.90)

where qcoll is the collimated flux entering the medium at rw , traveling a distance of lc toward r in the direction of sˆ 0 , and Φ(r, sˆ 0 , sˆ ) indicates the amount of collimated flux arriving at r from sˆ 0 , being scattered into the direction of sˆ . Therefore, the diffuse component of the intensity at ri is found immediately from equation (20.85) as  Iηn (ri , −ˆsi ) =

l

  Sη1 (r , −ˆs ) exp − 

0



l



κη dl



dl ,

(20.91)

0

with Sη1 from equation (20.90). As before, equation (20.91) may be solved using standard tracing [picking absorption length lκ , and dropping the exponential attenuation term in equation (20.91)] or energy partitioning [using equation (20.91) as given].

20.8 Efficiency/Accuracy Considerations Monte Carlo calculations in the presence of a participating medium are generally even more computationally intensive than those for surface exchange, making efficiency considerations all the more important. Computational efficiency and accuracy of Monte Carlo calculations go hand in hand, and are generally conflicting. For example, the energy partitioning scheme of equations (20.21) and (20.22) results in much more accurate solutions (with reduced statistical errors) than the collision-based Monte Carlo approach with the same number of photon bundles. However, in this method, since each bundle is traced until its energy is completely depleted, tracing is more expensive. Ultimately, the merit of the Monte Carlo method has to be measured by considering accuracy and efficiency simultaneously. This was already discussed in Section 20.6, and the figure of merit, equation (20.72), was introduced [24] as FoM =

1 , σ2m tCPU

(20.92)

where we have replaced the RMS error by the variance, since the exact answer is rarely known. Sometimes, the figure of merit is also defined as the reciprocal of equation (20.92), e.g., Farmer and Howell [1]. All efficiency improvements introduced in Chapter 7 continue to hold in participating media, such as inversion of random number relations in terms of look-up tables and interpolation from precalculated databases, energy partitioning, etc. Smoothing algorithms similar to those presented in Chapter 7 can also be applied to exchange areas for participating media, as used in the zonal method of Chapter 17 [41,42]. A simple, yet very effective smoothing scheme for energy deposition into volumetric cells has been proposed by Fippel and Nüsslin [43], by minimizing local second derivatives. Wu and coworkers [44] developed perhaps the only higher-order Monte Carlo scheme, evaluating local emission via Lagrangian interpolation of varying order, and similarly distributing absorbed energies across adjacent nodal points. For Monte Carlo calculations in nongray media it appears intuitive to assign multiple wavenumbers (for LBL) or multiple values of reordered spectral variable g0 (for FSCK) to a single photon bundle, in order to reduce ray tracing effort. This was first proposed by Wang and Modest [5] for both LBL-PMC and FSCK-PMC. They noted, however, that assigning several wavenumbers (or g0 ) to photon bundles did not significantly improve CPU times. Similar tests (and conclusions) were made by Maurente and França [45], also for both LBL and FSCK. It is worth noting that the reason for this appeared to be the fact that determination of emission wavenumbers (or g0 ) was consuming most of the CPU time for the calculations. However, Wang and Modest [5] employed the early selection scheme of equations (20.38) through (20.40), while Maurente and França [45] used even slower in situ selection. With the 10 times faster new wavenumber selection scheme of Ren and Modest [8,9] wavenumber selection has become a small part of the simulation, and the idea of multi-spectral photon bundles for LBL-PMC may be worthy of reconsideration. Besides the aforementioned schemes there are many other ways to make a particular Monte Carlo simulation computationally more efficient. Some of the more important ones will be presented in some detail below. Others

The Monte Carlo Method for Participating Media Chapter | 20 761

include, for example, Farmer and Howell [46,47], who mitigated the standard method’s inefficiency in optically thick media by using hybrid approaches, employing the diffusion approximation of Chapter 14 for optically thick volume elements, and a regular Monte Carlo simulation for the rest. A similar hybrid, separating near-opaque wavenumbers, for which they used the P1 -method of Chapter 15, was proposed by Feldick and coworkers [48]. A number of formal variance (statistical error) reduction techniques have been used for Monte Carlo calculations in other disciplines, the two most commonly used techniques being importance sampling [49,50] and the use of control variates [51–54]. In thermal radiation importance sampling has been employed to overcome the dilemma of optically thick regions: photon bundles emitted in near-opaque regions are given larger weights (and, thus, are chosen less often) [55,56], etc. Control variates are applied, for example, by the Deviational Monte Carlo (DMC) technique, described in a bit more detail below.

Energy Partitioning In optically thick media, bundles emitted in the interior rarely travel far enough (before absorption) to hit a bounding surface, although it is often the surface heat fluxes that are of primary interest. For such problems it is advantageous to employ the energy partitioning scheme. An illustration of this method is included with Example 20.4 (for, both, forward and backward Monte Carlo) in the following section. The method has also been described by Walters and Buckius [3], who called it “absorption suppression,” and by Farmer and Howell [1], who named it “pathlength method.” Both references also discuss several variations to the method. If an optically thin medium is externally irradiated, it is the lack of substantial absorption that causes the method to become inefficient. Energy partitioning can also be used to increase the efficiency for such problems, either in the way described in equations (20.21) and (20.22), or through a variation called “forced collisions” by Walters and Buckius [3]. Wong and Mengüç [57] systematically compared the efficiency of different tracing schemes for irradiated slabs and found energy partitioning to be more efficient for large optical thicknesses and for strong scatterers (ω > 0.5); however, they did not investigate near-transparent media, for which the method should also improve convergence. Energy partitioning can result in considerable improvement in the statistical accuracy (and, thus, computer time savings) for optically thick and thin media, but its performance must be assessed with the FoM (note that Table 20.1 only lists number of bundles, not CPU times). The method is limited to media with known (or iterated) temperature field (i.e., it cannot be applied to the standard method for radiative equilibrium, where photon bundles are absorbed and re-emitted at selected locations).

Direct Exchange Monte Carlo As noted in Section 20.8, standard Monte Carlo implementations become inefficient in optically thick media, but that can be mitigated by using the energy partitioning approach. Another difficulty arises in near-isothermal enclosures: while emission from and absorption by a hot cell can be substantial, the net heat transfer between two cells may be very small, i.e., emission is nearly balanced by absorption of incoming radiation. Thus, a small percentage of uncertainty in emitted and absorbed energies may lead to huge uncertainty in the radiative source ∇ · q, the difference between emission and absorption. In the Direct Exchange Monte Carlo scheme the energy exchange between any two cells is formulated in terms of volume integrals (and also a spectral integral if the medium is nongray), as done in the Zonal Method (Chapter 17). However, as opposed to the Zonal Method, the exchange integrals are evaluated stochastically rather than deterministically, using the methods of Section 7.2. The Direct Exchange Monte Carlo method differs from the traditional photon Monte Carlo method in that, rather than tracing statistically meaningful photon bundles that directly contribute to energy exchange budgets, the Monte Carlo method is simply used to compute multidimensional integrals. For example, the net energy exchange between two homogeneous cells Vi and Vj , in the absence of scattering and wall reflections, is, from Section 17.3 Qi→ j = −Q j→i =

∞  0

Ibη,i − Ibη, j

 

e−



κη dS

κη,i κη, j S2

dV j dVi dη,

(20.93)

Vi V j

 where S is the distance between any two points within Vi and Vj , and κη dS is the absorption coefficient integrated over that path. Computation of the last two integrals in equation (20.93) requires sampling points by

762 Radiative Heat Transfer

assigning probability density functions for points within Vi and Vj (such as pi = 1/Vi for uniform probability),  − κη dS and then connecting pairs of points by line segments since evaluation of both S and e require two points. The procedure is conceptually similar to either tracing a set of rays from Vi to V j or from V j to Vi or a combination thereof. For wavenumber selection, equation (20.93) is first evaluated at each wavenumber using estimates of the various quantities, i.e.,   κη,i κη, j V j Vi , (20.94) Qi→ j,η = −Q j→i,η = Ibη,i − Ibη, j e−κη S S2 where κη is the path-averaged value of κη between Vi and V j , and κη,i and κη, j are average values of κη within Vi and V j , respectively. Next, an integration over the whole spectrum is performed on the right-hand side of equation (20.94) to estimate the total energy exchanged between Vi and V j . Once the spectral and total exchanged energies have been estimated, the wavenumber can be selected using the same procedure as described in equation (20.12). Based on this procedure, the wavenumbers selected for the evaluation of each direct exchange factor will be different. However, the procedure ensures that, for the evaluation of each exchange factor, samples are drawn from wavenumbers that cause the most energy exchange between Vi and V j . This method was first introduced by Cherkaoui et al. for a one-dimensional slab of a nonscattering medium contained between black [58] and reflecting [59] plates. They noted that CPU requirements for an isothermal slab were orders of magnitude lower than for standard Monte Carlo. Tessé [60,61] and coworkers have conceptually extended the method to nonscattering media in three-dimensional enclosures, but only 1D results were reported. A 2D axisymmetric solution for a sooty flame (i.e., without surface reflections) was also reported [62].

Reciprocity Monte Carlo The principle of reciprocity, as described by equation (20.80), must be obeyed in any radiation exchange calculation. However, neither the forward Monte Carlo nor the backward Monte Carlo method, by themselves, obey reciprocity. The Direct Exchange Monte Carlo method, on the other hand, does obey reciprocity, as evident from equation (20.93). While this makes the method attractive from an accuracy standpoint, it still suffers from computational inefficiency. The number of exchange factors that need to be independently evaluated in this method is approximately (NV + NS )2 /2 [58], where NV and NS are the number of volume and surface elements in the computational domain. Each evaluation may require several hundred samples to accurately compute the triple-integral shown in equation (20.93), rendering the procedure computationally expensive even by Monte Carlo standards. In practice, rather than compute each exchange factor independently, a large number of photon bundles (or rays) may be emitted from a given volume element and their contributions to the calculation of the integrals [appearing in equation (20.93)] of all receiving volume or surface elements may be tallied simultaneously. Conversely, rays may be traced back to a given volume element from all other volume or surface elements. The former procedure is similar in principle to the forward Monte Carlo method while the latter to the backward Monte Carlo method, with the notable difference that the ray tracing procedure is used here only to compute the multidimensional integrals. Instead of connecting a pair of preselected points, a ray is emitted from one point, and points along the ray’s path receive contributions toward the integral calculation. Based on this broad idea, Tessé et al. [60] formulated two different ways of computing the direct exchange factors. In the Emission Reciprocity Method (ERM), rays are traced forward from an emitting volume, while in the Absorption Reciprocity Method (ARM) rays are traced back to a given volume element from all other volume and surface elements. In both methods, reciprocity is enforced. Since the sampled rays are different in the two methods, the computed exchange factors by the two methods are also different. In principle, the two approaches may also be combined in a single radiative heat transfer calculation. Dupoirieux et al. [61] used a combination of ARM and ERM in a 1D slab. In their method—the so-called Optimized Reciprocity Method (ORM)—whether an exchange factor is computed using ERM or ARM is determined by examining the statistical error. Of course, to implement the method, one must be able to estimate the statistical % error, σm , of either method a priori. Dupoirieux et al. estimated the ratio of statistical errors as: σm,ERM /σm,ARM = Ib,i /Ib, j . In other words, ERM is preferable over ARM if V j is hotter than Vi and vice versa. In a recent work [63], rather than making a choice between ERM and ARM to compute a given exchange factor, the two methods were combined, resulting in the so-called Bidirectional Reciprocity Method (BRM). The weight used to combine the two methods was derived from minimizing the estimate of the statistical error in the computed result. The method was exercised for a 1D slab, and showed improved efficiency over the ORM without any penalty on accuracy.

The Monte Carlo Method for Participating Media Chapter | 20 763

Deviational Monte Carlo Use of this method for surface-to-surface Monte Carlo calculations was described in Section 7.7. It was shown that the method is particularly beneficial in situations where the temperature range is relatively “small” or the enclosure is near-isothermal. As discussed earlier, standard Monte Carlo methods for participating media, either forward or backward, may produce large statistical errors or variance in near-isothermal situations. The deviational Monte Carlo method is expected to be beneficial in such scenarios. In order to apply the DMC method to radiation calculations in participating media, the intensity is first decomposed into two parts: Iη = IηC + Iη ,

(20.95)

where IηC and Iη are the control intensity and deviational intensity, respectively. If this decomposition is substituted into the RTE and the two parts are separated, the result will be two RTEs—one for IηC and the other for Iη . At first glance, no benefit would be derived in doing so. However, if the solution to the RTE for IηC is trivial, e.g., if IηC is already known, and the other RTE can be solved more efficiently and/or accurately using the same Monte Carlo procedure, the final outcome would be beneficial. In keeping with these stipulations, it is common practice [51,64] to choose IηC = Ibη (TC ), where Ibη (TC ) is the Planck function evaluated at a fixed, but arbitrary, control temperature TC . Substituting equation (20.95) with IηC = Ibη (TC ) into equation (9.21), and noting that Ibη (TC ) is independent of both direction and spatial location, we obtain dIη ds

= sˆ ·

∇Iη



= κη Ibη − Ibη (TC ) −

Iη



−

σsη Iη

+

σsη 4π

 4π

Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i .

(20.96)

Equation (20.96) is identical to equation (9.21), except that Ibη in the emission term has been replaced by Ibη −Ibη (TC ). Likewise, making the same substitution in the boundary condition for the RTE—the spectral counterpart of equation (9.36), for example—we obtain   ρη (rw )  Iη (rw , sˆ  ) |nˆ · sˆ  | dΩ . Iη (rw , sˆ ) = η (rw ) Ibη (rw ) − Ibη (TC ) +  π ˆ s L). If the bundle strikes the bottom surface (z = L), incidence angle (ˆs · kˆ > cos θmax ?) and location (x, y on detector?) are checked and a detector hit is recorded, if appropriate. Results are shown in Fig. 20.11. As the detector’s acceptance angle increases, more photon bundles are captured. Obviously, this results in a larger detector-absorbed flux. However, it also increases the fraction of statistically meaningful samples, decreasing the variance of the results or the number of required photon bundles to achieve a given variance. All calculations were carried out until the variance fell below 2% of the calculated flux, and the necessary number of bundles is also included in the figure. For the chosen variance about 4 × 106 bundles are required for large acceptance angles, rising to 512 × 106 for θmax = 10◦ . Results are difficult to obtain for θmax < 10◦ . Similar remarks can be made for detector area: as the detector area decreases, the necessary number of bundles increases. Modeling a more typical detector 1 mm × 1 mm in size would almost be impossible. Backward Monte Carlo In this case no direct radiation hits the detector (x0 > R), and the scattered irradiation is calculated from equations (20.91) and (20.90) with qcoll = Q/πR2 as  l  σs Q −σs z  In (ri , −ˆsi ) = e H R − r(l ) dl , 2 R2 4π 0 where l consists of a number of straight-line segments, for which dl = dz/cos θ, and H is Heaviside’s unit step function.4 Therefore,  σs Q  z2j −σs z dz Q  e−σs z1j − eσs z2j In (ri , −ˆsi ) = e = , (20.98) 4π2 R2 j z1j szj 4π2 R2 j szj where szj = cos θj is the z-component of the direction vector for the jth segment, and z1j and z2j are the z-locations between which the segment lies within the cylindrical column r ≤ R (note that some segments may lie totally inside this column, some partially, and some not at all). 4. For its definition see equation (10.115) in Section 10.9.

768 Radiative Heat Transfer

FIGURE 20.11 Detector fluxes and required number of photon bundles for Example 20.3.

As in forward Monte Carlo a starting point on the detector is chosen from equation (7.40), and a direction for the backward trace is picked from equations (20.89) and (7.51). Again, a scattering distance is found from equation (20.23), after which the bundle is scattered into a new direction found from equations (20.28). However, rather than having fixed energy, the backward-traveling bundles accumulate energy according to equation (20.98) as they travel through regions with a radiative source. The total flux hitting the detector is calculated by adding up bundle energies according to equation (20.88). Results are included in Fig. 20.11, and are seen to coincide with forward Monte Carlo results to about one variance or better (discrepancy being larger at large θmax , since the absolute variance increases). However, the number of required bundles remains essentially independent of opening angle at about 20,000 (and, similarly independent of detector area). Since the tracing of a photon bundle requires essentially the same CPU time for forward and backward tracing, for the problem given here the backward Monte Carlo scheme is up to 25,000 times more efficient than forward Monte Carlo. Fortran90 codes used for this example are included in Appendix F as RevMCcs and FwdMCcs. Example 20.4. Repeat the previous example, for an acceptance angle of θmax =10◦ , assuming that the medium absorbs as well as scatters radiation, using absorption coefficients of κη = 1 m−1 and κη = 5 m−1 . Use forward as well as backward Monte Carlo, and also both standard ray tracing as well as energy partitioning. Solution Forward Monte Carlo—standard ray tracing The solution proceeds as in the previous example, except that also an absorption length lκ is chosen, from equation (20.18). If the sum of all scattering paths exceeds lκ , the bundle is terminated. Forward Monte Carlo—energy partitioning The solution proceeds as in the previous example, except the energy of each bundle hitting the detector is attenuated by a factor of exp(−κl), where l is the total (scattered) path that the bundle travels through the layer before hitting the detector. Backward Monte Carlo—standard ray tracing The solution proceeds as in the previous example, except for two changes. First, the local scattering source must be attenuated by absorption of the direct beam, and equation (20.98) becomes  σs Q  z2j −(κ+σs )z dz ωQ  e−βz1j − e−βz2j In (ri , −ˆsi ) = e = , (20.99) 2 2 4π R j z1j szj 4π2 R2 j szj where ω and β are scattering albedo and extinction coefficient, as usual. And again, an absorption length lκ is chosen, and the addition in equation (20.99) is stopped as soon as the total path reaches lκ or the bundle leaves the layer (whichever comes first). Backward Monte Carlo—energy partitioning Again, the scattering source must be attenuated as in equation (20.99), but the exponential attenuation term in equation (20.91) must also be retained. Thus, σs Q In (ri , −ˆsi ) = 4π2 R2



l

    e−βz(l )−κl H R − r(l ) dl ,

0

where the integrand contributes only where the source is active (r ≤ R), but attenuation of the bundle takes place everywhere (l = total distance along path from ri to r ). With l = l1j + (z − z1j )/szj , dl = dz/szj , and l2j = l1j + (z2j − z1j )/szj ,

The Monte Carlo Method for Participating Media Chapter | 20 769

TABLE 20.1 Comparison between four different Monte Carlo implementations to calculate irradiation onto a detector from a collimated source. κ (m−1 )

Forward MC—Standard

Forward MC—Energy partitioning

Backward MC—Standard

Backward MC—Energy partitioning

N×10−6

Qdet

N×10−6

Qdet

N×10−6

Qdet

N×10−6

0

9.22×10−4

512

9.22×10−4

512

9.17×10−4

0.02

9.17×10−4

0.02

1

2.66×10−4

512

2.70×10−4

512

2.56×10−4

0.08

2.59×10−4

0.02

5

2.54×10−6

16,384*

2.93×10−6

512

2.77×10−6

5.12

2.79×10−6

0.02

Qdet

* Variance

of 5% (all other data have variance of 2%).

where l1j and l2j are total path lengths of the bundle until the beginning and end of segment j, respectively, this becomes In (ri , −ˆsi ) = = =

 σs Q  −κl1j z2j −βz−κ(z−z1j )/sz j dz e e 4π2 R2 j szj z1j σs Q  −κl1j e−βz1j − e−βz2j −κ(z2j −z1j )/sz j e 4π2 R2 j β + κ/szj

  Q  σs e−βz1j −κl1j − e−βz2j −κl2j . 2 2 4π R j β + κ/szj

The rest of the simulation remains as in the previous example. Results are summarized in Table 20.1. As expected, if standard ray tracing is employed, the number of required bundles grows astronomically if the absorption coefficient becomes large, both for forward and backward Monte Carlo. While backward Monte Carlo retains its advantage (indeed, the forward Monte Carlo simulation for κη = 5 m−1 could only be carried out to a variance of 5%), the relative growth of required bundles appears to be worse for backward Monte Carlo. If energy partitioning is employed, the number of bundles remains unaffected by the absorption coefficient for both, forward and backward Monte Carlo. All four Fortran90 codes used for this example have also been included in Appendix F.

It was demonstrated in the last two examples that in media with large optical thickness based on absorption coefficient, energy partitioning is vastly more efficient than the standard method. And in problems to find irradiation onto small surfaces and/or small solid angles, backward Monte Carlo strongly outperforms forward Monte Carlo. As seen in the last example, employing backward Monte Carlo with energy partitioning may reduce CPU time by a factor of 1,000,000 or more!

Problems Because of the nature of the Monte Carlo technique, most of the following problems require the development of a small computer code. However, all problem solutions can be outlined by giving relevant relations, equations, and a detailed flow chart. 20.1 Consider the (highly artificial) absorption coefficient of Problem 10.23. Find narrow band averages for the absorption coefficient and the transmissivity using Monte Carlo integration (use mcint.f90 or write your own code). Compare with answers from Problem 10.23. 20.2 Consider radiative equilibrium in a plane-parallel medium between two isothermal, diffusely emitting and reflecting gray plates (T1 = 300 K, 1 = 0.5, T2 = 2000 K, 2 = 0.8) spaced L = 1 m apart. The medium has constant absorption and scattering coefficients (κ = 0.01 cm−1 , σs = 0.04 cm−1 ), and scattering is linear-anisotropic with A1 = 0.5. Calculate the radiative heat flux and the temperature distribution within the medium by the Monte Carlo method. Compare with results from the P1 -approximation. 20.3 Consider an isothermal plane-parallel slab (T = 1000 K) between two cold, gray, diffuse surfaces ( = 0.5). The medium absorbs and emits but does not scatter. Prepare a standard Monte Carlo solution to obtain the radiative heat loss from the medium for optical thickness κL = 0.2, 1, 5, 10. Compare with the exact solution. 20.4 Repeat Problem 20.3 using energy partitioning. Compare the efficiency of the two methods.

770 Radiative Heat Transfer

20.5 A molecular gas is confined between two parallel, black plates, spaced 1 m apart, that are kept isothermal at T1 = 1200 K and T2 = 800 K, respectively. The (hypothetical) gas has a single vibration–rotation band in the infrared, with an average absorption coefficient of κ¯ gη =

  S δ

=

η

α −2|η−η0 |/ω e , ω

η0 = 3000 cm−1 ,

ω = 200 cm−1

and a line overlap parameter of β (see the discussion of narrow band and wide band models in Chapter 10). Assuming convection and conduction to be negligible, determine the radiative heat flux between the two plates, using the Monte Carlo method. Carry out the analysis for variable values of (α/ω) and β, and plot nondimensional radiative heat flux vs. (S/δ)0 L with β as a parameter. 20.6 Consider a sphere of very hot molecular gas of radius 50 cm. The gas has a single vibration–rotation band at η0 = 3000 cm−1 , is suspended magnetically in a vacuum within a large, √ cold container, and is initially at a uniform temperature T g = 3000 K. For this gas, ρa α(T) = 500 cm−2 , ω = 100 T/100 K cm−1 , β  1. This implies that the absorption coefficient may be determined from κη = κ0 e−2|η−η0 |/ω ,

κ0 =

ρa α ω

and the band absorptance from A(s) = ωA∗ = ω[E1 (κ0 s) + ln(κ0 s) + γE ]. Find the total heat loss from the sphere and its temperature distribution by the Monte Carlo method (including t > 0). 20.7 Consider a sphere of very hot dissociated gas of radius 5 cm. The gas may be approximated as a gray, isotropically scattering medium with κ = 0.1 cm−1 , σs = 0.2 cm−1 . The gas is suspended magnetically in a vacuum within a large, cold container and is initially at a uniform temperature T g = 10,000 K. Using the Monte Carlo method and neglecting conduction and convection, specify the total heat loss per unit time from the entire sphere at t = 0. Outline the solution for times t > 0. 20.8 Consider an absorbing–scattering slab irradiated by a short-pulsed laser, as described in Example 18.3. Prepare a transient Monte Carlo code to predict the flux exiting the slab as a function of time into either direction (transmissivity and reflectivity).

References [1] J.T. Farmer, J.R. Howell, Comparison of Monte Carlo strategies for radiative transfer in participating media, in: J.P. Hartnett, T.F. Irvine (Eds.), Advances in Heat Transfer, vol. 31, Academic Press, New York, 1998. [2] M.F. Modest, Radiative heat transfer fluxes through the exit of GE combustor transition piece, Technical Report (private communication to General Electric Co.), 1980. [3] D.V. Walters, R.O. Buckius, Monte Carlo methods for radiative heat transfer in scattering media, in: Annual Review of Heat Transfer, vol. 5, Hemisphere, New York, 1992, pp. 131–176. [4] M.F. Modest, The Monte Carlo method applied to gases with spectral line structure, Numerical Heat Transfer – Part B: Fundamentals 22 (3) (1992) 273–284. [5] A. Wang, M.F. Modest, Spectral Monte Carlo models for nongray radiation analyses in inhomogeneous participating media, International Journal of Heat and Mass Transfer 50 (2007) 3877–3889. [6] T. Ozawa, M.F. Modest, D.A. Levin, Spectral module for photon Monte Carlo calculations in hypersonic nonequilibrium radiation, ASME Journal of Heat Transfer 132 (2010) 023406. [7] A.M. Feldick, M.F. Modest, An improved wavelength selection scheme for Monte Carlo solvers applied to hypersonic plasmas, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (2011) 1394–1401. [8] T. Ren, M.F. Modest, Hybrid wavenumber selection scheme for line-by-line photon Monte Carlo simulations in high-temperature gases, ASME Journal of Heat Transfer 135 (8) (2013) 084501. [9] T. Ren, M.F. Modest, Line-by-line random-number database for photon Monte Carlo simulations of radiation in participating media, ASME Journal of Heat Transfer 141 (2) (2019) 0227019. [10] L. Wang, J. Yang, M.F. Modest, D.C. Haworth, Application of the full-spectrum k-distribution method to photon Monte Carlo solvers, Journal of Quantitative Spectroscopy and Radiative Transfer 104 (2) (2007) 297–304. [11] A. Maurente, H.A. Vielmo, F.H.R. França, Regularized variable metric method versus the conjugate gradient method in solution of radiative boundary design problem, Journal of Quantitative Spectroscopy and Radiative Transfer 108 (2) (2007) 295–307. [12] C. Wang, M.F. Modest, B. He, Full-spectrum correlated-k-distribution look-up table for use with radiative Monte Carlo solvers, International Journal of Heat and Mass Transfer 131 (2019) 167–175.

The Monte Carlo Method for Participating Media Chapter | 20 771

[13] A. Haji-Sheikh, Monte Carlo methods, in: Handbook of Numerical Heat Transfer, John Wiley & Sons, New York, 1988, pp. 673–722. [14] R.S. Mehta, A. Wang, M.F. Modest, D.C. Haworth, Modeling of a turbulent ethylene/air flame using hybrid finite volume/Monte Carlo methods, Computational Thermal Sciences 1 (2009) 37–53. [15] G. Pal, A. Gupta, M.F. Modest, D.C. Haworth, Comparison of accuracy and computational expense of radiation models in simulation of nonpremixed turbulent jet flames, in: Proceedings of 2011 ASME/JSME Thermal Engineering Joint Conference, 2011. [16] A.M. Feldick, M.F. Modest, A spectrally accurate tightly-coupled 2-D axisymmetric photon Monte-Carlo RTE solver for hypersonic entry flows, ASME Journal of Heat Transfer 134 (12) (2012) 122701. [17] OpenFOAM website, http://www.opencfd.co.uk/openfoam/. [18] M. Syamlal, W. Rodgers, T. O’Brien, MFIX documentation: Theory guide, Technical Note, DOE/METC-94/1004 1993. [19] A. Wang, M.F. Modest, Photon Monte Carlo simulation for radiative transfer in gaseous media represented by discrete particle fields, ASME Journal of Heat Transfer 128 (2006) 1041–1049. [20] A. Wang, M.F. Modest, An adaptive emission model for Monte Carlo ray-tracing in participating media represented by statistical particle fields, Journal of Quantitative Spectroscopy and Radiative Transfer 104 (2) (2007) 288–296. [21] R. Marquez, M.F. Modest, J. Cai, Spectral photon Monte Carlo with energy splitting across phases for gas-particle mixtures, ASME Journal of Heat Transfer 137 (2015) 121012. [22] S.P. Roy, J. Cai, M.F. Modest, Development of a multiphase photon Monte Carlo method for spray combustion and its application in high-pressure conditions, International Journal of Heat and Mass Transfer 115, Part A (2017) 453–466. [23] G.R. Liu, M.B. Liu, Smoothed Particle Hydrodynamics – a Meshfree Particle Method, World Scientific Publishing Co. Pte. Ltd., Singapore, 2003. [24] E.E. Lewis, W.F.J. Miller, Computational Methods of Neutron Transport, John Wiley & Sons, Inc., 1984. [25] R.O. Buckius, D.C. Hwang, Radiation properties for polydispersions: application to coal, ASME Journal of Heat Transfer 102 (1980) 99–103. [26] P. von Zedtwitz, W. Lipinski, ´ A. Steinfeld, Numerical and experimental study of gas–particle radiative heat exchange in a fluidized-bed reactor for steam-gasification of coal, Chemical Engineering Science 62 (2007) 599–607. [27] D.C. Haworth, Progress in probability density function methods for turbulent reacting flows, Progress in Energy and Combustion Science 36 (2010) 168–259. [28] H.R. Gordon, Ship perturbation of irradiance measurements at sea. 1: Monte Carlo simulations, Applied Optics 24 (1985) 4172–4182. [29] D.G. Collins, W.G. Blättner, M.B. Wells, H.G. Horak, Backward Monte Carlo calculations of the polarization characteristics of the radiation emerging from spherical-shell atmospheres, Applied Optics 11 (1972) 2684–2696. [30] C.N. Adams, G.W. Kattawar, Radiative transfer in spherical shell atmospheres—I. Rayleigh scattering, Icarus 35 (1978) 139–151. [31] T. Nishita, Y. Miyawaki, E. Nakamae, A shading model for atmospheric scattering considering luminous intensity distribution of light sources, Computer Graphics 21 (1987) 303–310. [32] P. Sabella, A rendering algorithm for visualizing 3D scalar fields, Computer Graphics 22 (1988) 51–58. [33] D.K. Edwards, Numerical methods in radiation heat transfer, in: T.M. Shih (Ed.), Proc. Second National Symposium on Numerical Properties and Methodologies in Heat Transfer, Hemisphere, 1983, pp. 479–496. [34] D.V. Walters, R.O. Buckius, Rigorous development for radiation heat transfer in nonhomogeneous absorbing, emitting and scattering media, International Journal of Heat and Mass Transfer 35 (1992) 3323–3333. [35] Y. Shuai, S.K. Dong, H.P. Tan, Simulation of the infrared radiation characteristics of high-temperature exhaust plume including particles using the backward Monte Carlo method, Journal of Quantitative Spectroscopy and Radiative Transfer 95 (2) (2005) 231–240. [36] L.M. Ruan, H.P. Tan, Y.Y. Yan, A Monte Carlo (MC) method applied to the medium with nongray absorbing-emitting-anisotropic scattering particles and gray approximation, Numerical Heat Transfer – Part A: Applications 42 (3) (2002) 253–268. [37] X. Lu, P.-F. Hsu, Reverse Monte Carlo simulations of light pulse propagation in nonhomogeneous media, Journal of Quantitative Spectroscopy and Radiative Transfer 93 (1–3) (2005) 349–367. [38] F. Wang, D. Liu, K. Cen, J.-H. Yan, Q.-X. Huang, Y. Chi, Efficient inverse radiation analysis of temperature distribution in participating medium based on backward Monte Carlo method, Journal of Quantitative Spectroscopy and Radiative Transfer 109 (12–13) (2008) 2171–2181. [39] X. Sun, P.J. Smith, A parametric case study in radiative heat transfer using the reverse Monte-Carlo ray-tracing with full-spectrum k-distribution method, ASME Journal of Heat Transfer 132 (2010) 024501. [40] K.M. Case, Transfer problems and the reciprocity principle, Reviews of Modern Physics 29 (1957) 651–663. [41] H.A.J. Vercammen, G.F. Froment, An improved zone method using Monte Carlo techniques for the simulation of radiation in industrial furnaces, International Journal of Heat and Mass Transfer 23 (1980) 329–337. [42] M.E. Larsen, J.R. Howell, Least-squares smoothing of direct-exchange areas in zonal analysis, ASME Journal of Heat Transfer 108 (1) (1986) 239–242. [43] M. Fippel, F. Nüsslin, Smoothing Monte Carlo calculated dose distributions by iterative reduction of noise, Physics in Medicine and Biology 48 (2003) 1289–1304. [44] Y. Wu, M.F. Modest, D.C. Haworth, A high-order photon Monte Carlo method for radiative transfer in direct numerical simulation of chemically reacting turbulent flows, Journal of Computational Physics 223 (2) (2007) 898–922. [45] A. Maurente, F.H.R. França, A multi-spectral energy bundle method for efficient Monte Carlo radiation heat transfer computations in participating media, International Journal of Heat and Mass Transfer 90 (2015) 351–357.

772 Radiative Heat Transfer

[46] J.T. Farmer, J.R. Howell, Hybrid Monte Carlo/diffusion method for enhanced solution of radiative transfer in optically thick non-gray media, in: Y. Bayazitoglu, ˘ et al. (Eds.), Radiative Transfer: Current Research, ASME, 1994. [47] J.T. Farmer, J.R. Howell, Monte Carlo algorithms for predicting radiative heat transport in optically thick participating media, in: Proc. 10th International Heat Transfer Conference, Brighton, 1994, pp. 37–42. [48] A.M. Feldick, A. Bansal, M.F. Modest, Variance reduction techniques for Monte Carlo solution of radiative transfer in hypersonic flows: hybrid P-1-Monte Carlo, in: 49th AIAA Aerospace Sciences Conference, 2011, AIAA Paper 2011-0249. [49] A. Haghighat, J.C. Wagner, Monte Carlo variance reduction with deterministic importance functions, Progress in Nuclear Energy 42 (2003) 25–53. [50] A.M. Feldick, M.F. Modest, Importance sampling in Monte-Carlo ray tracing solutions applied to radiation in hypersonic entry flows, in: ASME/JSME 8th Thermal Engineering Joint Conference, Honolulu, Hawaii, March 2011, Paper No. AJTEC2011-44487. [51] J.P. Peraud, N.G. Hadjiconstantinou, Efficient simulation of multidimensional phonon transport using energy-based variance-reduced Monte Carlo formulations, Physical Review B 84 (2011) 205331. [52] L.L. Baker, N.G. Hadjiconstantinou, Variance reduction for Monte Carlo solutions of the Boltzmann equation, Physics of Fluids 17 (2005) 051703. [53] G.A. Radtke, N.G. Hadjiconstantinou, Variance reduced particle simulation of the Boltzmann transport equation in the relaxation-time approximation, Physical Review E 79 (2009) 056711. [54] L. Szalmas, Variance-reduced DSMC for binary gas flows as defined by the McCormack kinetic model, Journal of Computational Physics 231 (2012) 3723–3738. [55] A. de Lataillade, J.-L. Dufresne, M. El Hafi, V. Eymet, R. Fournier, A net-exchange Monte Carlo approach to radiation in optically thick systems, Journal of Quantitative Spectroscopy and Radiative Transfer 74 (5) (2002) 563–584. [56] A.M. Feldick, M.F. Modest, Variance reduction techniques for Monte Carlo solution of radiative transfer in hypersonic flows: importance sampling, in: 49th AIAA Aerospace Sciences Conference, 2011, AIAA Paper 2011-0245. [57] B.T. Wong, M.P. Mengüç, Comparison of Monte Carlo techniques to predict the propagation of a collimated beam in participating media, Numerical Heat Transfer – Part B: Fundamentals 42 (2002) 119–140. [58] M. Cherkaoui, J.-L. Dufresne, R. Fournier, J.-Y. Grandpeix, A. Lahellec, Monte Carlo simulation of radiation in gases with a narrow-band model and a net-exchange formulation, ASME Journal of Heat Transfer 118 (2) (1996) 401–407. [59] M. Cherkaoui, J.-L. Dufresne, R. Fournier, J.-Y. Grandpeix, A. Lahellec, Radiative net exchange formulation within one-dimensional gas enclosures with reflective surfaces, ASME Journal of Heat Transfer 120 (1) (1998) 275–278. [60] L. Tessé, F. Dupoirieux, B. Zamuner, J. Taine, Radiative transfer in real gases using reciprocal and forward Monte Carlo methods and a correlated-k approach, International Journal of Heat and Mass Transfer 45 (2002) 2797–2814. [61] F. Dupoirieux, L. Tessé, S. Avila, J. Taine, An optimized reciprocity Monte Carlo method for the calculation of radiative transfer in media of various optical thicknesses, International Journal of Heat and Mass Transfer 49 (2006) 1310–1319. [62] L. Tessé, F. Dupoirieux, J. Taine, Monte Carlo modeling of radiative transfer in a turbulent sooty flame, International Journal of Heat and Mass Transfer 47 (2004) 555–572. [63] H.-F. Sun, F.-X. Sun, X.L. Xia, Bidirectionally weighted Monte Carlo method for radiation transfer in the participating media, Numerical Heat Transfer – Part B: Fundamentals 71 (2017) 202–215. [64] L. Soucasse, P. Rivière, A. Soufiani, Monte Carlo methods for radiative transfer in quasi-isothermal participating media, Journal of Quantitative Spectroscopy and Radiative Transfer 128 (2013) 34–42. [65] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, Society for Industrial and Applied Mathematics, 1992. [66] I.M. Sobol, Uniformly distributed sequences with an additional uniform property, USSR Computation Mathematics and Mathematical Physics 16 (1976) 236–242. [67] J. Halton, On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numerische Mathematik 2 (1960) 84–90. [68] S. Tezuka, Polynomial arithmetic analogue of Halton sequences, ACM Transactions on Modeling and Computer Simulation 3 (1993) 99–107. [69] H. Niederreiter, Point sets and sequences with small discrepancy, Monatshefte für Mathematik 104 (1987) 273–337. [70] H. Niederreiter, Low-discrepancy and low-dispersion sequences, Journal of Number Theory 30 (1988) 51–70. [71] H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, American Mathematical Society 84 (1978) 957–1041. [72] D.M. O’Brien, Accelerated quasi Monte Carlo integration of the radiative transfer equation, Journal of Quantitative Spectroscopy and Radiative Transfer 48 (1992) 41–59. [73] A. Kersch, W. Morokoff, A. Schuster, Radiative heat transfer with quasi-Monte Carlo methods, Transport Theory and Statistical Physics 23 (1994) 1001–1021. [74] A.J. Marston, K.J. Daun, M.R. Collins, Geometric optimization of radiant enclosures containing specularly-reflecting surfaces through quasi-Monte Carlo simulation, Numerical Heat Transfer – Part A: Applications 59 (2011) 81–97. [75] Z. Wang, S. Cui, J. Yang, H. Gao, C. Liu, Z. Zhang, A novel hybrid scattering order-dependent variance reduction method for Monte Carlo simulations of radiative transfer in cloudy atmosphere, Journal of Quantitative Spectroscopy and Radiative Transfer 189 (2017) 283–302. [76] X. Lu, P.-F. Hsu, Reverse Monte Carlo method for transient radiative transfer in participating media, ASME Journal of Heat Transfer 126 (4) (2004) 621–627.

The Monte Carlo Method for Participating Media Chapter | 20 773

[77] J.A. Farmer, S.P. Roy, An efficient Monte Carlo-based solver for thermal radiation in participating media, in: 4th Thermal and Fluids Engineering Conference, ASTFE, Las Vegas, NV, 2019. [78] J.A. Farmer, S. Roy, A quasi-Monte Carlo solver for thermal radiation in participating media, Journal of Quantitative Spectroscopy and Radiative Transfer 242 (2020) 106753. [79] L. Palluotto, N. Dumont, P. Rodrigues, O. Gicquel, R. Vicquelin, Assessment of randomized quasi-Monte Carlo method efficiency in radiative heat transfer simulations, Journal of Quantitative Spectroscopy and Radiative Transfer 236 (Oct 2019) 106570. [80] L. Qian, G. Shi, Y. Huang, Runge-Kutta ray-tracing technique for radiative transfer in a three-dimensional graded-index medium, Journal of Thermophysics and Heat Transfer 32 (2018) 747–755. [81] S.M.H. Sarvari, Multi-grid Monte Carlo method for radiative transfer in multi-dimensional graded index media with diffuse-speculargray boundaries, Journal of Quantitative Spectroscopy and Radiative Transfer 219 (2018) 61–73. [82] C.-H. Wang, Q. Ai, H.-L. Yi, H.P. Tan, Transient radiative transfer in a graded index medium with specularly reflecting surfaces, Numerical Heat Transfer – Part A: Applications 67 (2015) 1232–1252. [83] C.-H. Wang, Y. Zhang, H.-L. Yi, H.P. Tan, Transient radiative transfer in two-dimensional graded index medium by Monte Carlo method combined with the time shift and superposition principle, Numerical Heat Transfer – Part A: Applications 69 (2016) 574–588.

Chapter 21

Radiation Combined with Conduction and Convection 21.1 Introduction In our analyses of radiative transfer in participating media we have, up to this point, always assumed that there was no interaction with other modes of heat transfer, i.e., we have limited ourselves to cases of radiative equilibrium and cases of specified temperature fields. In practical systems, it is nearly always the case that radiation occurs in conjunction with conduction and/or convection, and two or three heat transfer modes must be accounted for simultaneously. In such cases overall conservation of energy, equation (9.75), needs to be solved. Many important applications that involve interactions between radiation and other modes of heat transfer have been reported in the literature. Discussion of all of these could easily, by itself, fill a book as voluminous as this one. In a broad sense, however, these applications may be classified into two categories. The first category, in which surface-to-surface radiation is coupled with other modes of heat transfer, has already been covered in Chapter 8. In the second category, the medium may be radiatively participating, and consequently, radiation is coupled to the other modes not only at the boundaries but also within the medium itself. Examples include industrial infrared heaters, gas turbine and rocket exhaust plumes, flow in the exhaust manifold of automobiles, steam turbines, Czochralski crystal growth, and solar thermal reactors, among others. In this chapter, we will present generic scenarios of coupling with the following objectives: (i) to show the basic trends of how the different modes of heat transfer interact with one another and (ii) to outline some of the numerical schemes that may be used to address such coupling. We will begin this chapter with two sections that present the formulations underlying combined radiation and conduction in participating media, the latter one including change-of-phase effects. Combined radiation and convection is treated in the subsequent section, and includes discussion of radiative coupling with external forced convection, internal forced convection, and both external and internal natural convection. The chapter concludes with a section in which numerical procedures for coupling between the various modes of heat transfer are presented in a general framework. A separate chapter (Chapter 22) will be devoted to coupling of radiation with other modes of heat transfer in chemically reacting flows, with a special emphasis on turbulence–radiation interactions.

21.2 Combined Radiation and Conduction We begin this section by discussing the interaction between radiation and conduction in a stationary, radiatively participating medium. Since we are primarily interested in general trends and in evaluation methods, we will limit ourselves here to the relatively simple example of steady-state heat transfer through a one-dimensional, absorbing–emitting (but not scattering) gray medium, confined between two parallel, isothermal, gray, diffusely emitting and reflecting plates. The energy equation for simultaneous conduction and radiation in a participating medium is, from equation (9.75), ρcv

∂T = ∇ · (k∇T) + Q˙  − ∇ · q R . ∂t

(21.1)

For a one-dimensional, planar medium at steady state and without internal heat generation, this reduces to equation (9.77), or   d dT (21.2) − qR = 0, k dz dz Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00029-8 Copyright © 2022 Elsevier Inc. All rights reserved.

775

776 Radiative Heat Transfer

subject to the boundary conditions z=0:

T(0) = T1 ,

(21.3a)

z=L:

T(L) = T2 .

(21.3b)

The radiative heat flux, or its divergence dqR = κ(4Eb − G), dz

(21.4)

may be obtained by any of the methods discussed in the preceding chapters. For simplicity, we will assume that the thermal conductivity k is a constant. Introducing the nondimensional variables and parameters ξ=

z , L

θ=

T , T1

τL = κL,

ΨR =

θL =

T2 , T1

qR n2 σT14

,

N=

g=

G ; 4n2 σT14

kκ , 4n2 σT13

(21.5)

reduces equations (21.2) through (21.4) to d2 θ 1 dΨR , = 2 4N dτ dτ dΨR = 4(θ 4 − g), dτ θ(0) = 1, θ(τL ) = θL .

(21.6) (21.7) (21.8)

Here τL is the optical thickness of the medium, and N is known as the conduction-to-radiation parameter. For optically thick slabs (τL  1), N provides a good estimate of the relative importance of conductive and radiative heat fluxes: From equations (14.19) and (14.20), introducing the radiative conductivity, τL  1 :

qC kκ k −k ∂T/∂z 3 = = = , qR 4 4n2 σT 4 −kR ∂T/∂z kR

which provides the ratio of heat fluxes in terms of a local temperature. The situation is a little more complicated for optically thin situations (τL 1), for which the temperature field of the entire enclosure must be considered. For example, for an optically thin slab bounded by two black walls at T1 and T2 , respectively, from equation (14.7), and introducing the radiative heat transfer coefficient [employing a suitable average temperature, such as Tav = 12 (T1 +T2 )], τL 1 :

3 qR n2 σ(T14 −T24 ) = 4n2 σTav (T1 −T2 ) = hR (T1 −T2 ),

∂T k(T1 −T2 )/L, ∂z qC 1 kκ k = . 3 qR hR L τL 4n2 σTav qC = −k

If, in an optically thin slab, emission from within the slab (rather than from its boundaries) dominates the radiative heat flux, then qR becomes proportional to κ [cf. equation (9.57)], and   qC N =O 2 . τL 1 (emission dominated) : qR τL Equations (21.6) and (21.7) represent two coupled nonlinear equations (because of the T 4 -dependence for the radiative heat flux). Therefore, iterative solution is warranted. Iterative coupling of the two equations may be

Radiation Combined with Conduction and Convection Chapter | 21 777

performed using one of two procedures that have already been presented in Chapter 8 in the context of boundary conditions: explicit and semi-implicit. Here, the same two procedures are revisited in the context of coupling the radiative source term with the energy equation: • Explicit Coupling: In this procedure, the temperature θ is first guessed. Let us denote this guess by θ∗ . The RTE is next solved using a method of choice to determine the incident radiation, which we shall denote by g∗ . The superscript “*” indicates that the solution to the RTE is based on an old (guessed) temperature. The right-hand side of equation (21.7) can be computed as 4(θ∗4 − g∗ ), and the equation is substituted explicitly into the right-hand side of equation (21.6), such that d2 θ 1 ∗4 (θ − g∗ ), = N dτ2 which can then be solved to yield a new temperature, denoted by θnew . In the next iteration, θnew replaces the guessed θ∗ , and the procedure is repeated until convergence. Due to the strong nonlinearity (fourth power of θ) of the energy equation, such an iterative procedure may often be unstable, and one might have to “slow down” the change in θ from iteration to iteration using under-relaxation: instead of replacing θ∗ by θnew , we replace it by ωu θnew + (1 − ωu )θ∗ , where ωu is the so-called under-relaxation factor, such that 0 ≤ ωu ≤ 1. Note that ωu = 1 implies replacing θ∗ directly by θnew . As we shall see in later examples, sometimes very low values of ωu may be needed to attain convergence. • Semi-Implicit Coupling: As in the explicit procedure, here too, as the first step, the temperature is guessed and the RTE is solved to yield g∗ . However, in the next step, the right-hand side of equation (21.7) is not computed explicitly. Instead, only the g∗ is substituted into the right-hand side of equation (21.6) to yield d2 θ 1 4 (θ − g∗ ). = 2 N dτ The rationale behind such a treatment is that the energy equation, shown above, ought to have the same temperature on both sides of the equation for consistency. The above equation is semi-implicit (as opposed to fully implicit) since the incident radiation is still based on an old value of temperature. Next, the right-hand side of the above equation is re-written using a linear Taylor series approximation: θ4 ≈ θ∗4 + 4θ∗3 (θ − θ∗ ). This type of linearization was proposed as early as 1983 by Smith and coworkers [1]. Substitution of the linear approximation into the governing energy equation yields d2 θ 4 1 (−3θ∗4 − g∗ ). − θ∗3 θ = N dτ2 N It is worth pointing out that, if the θ − θ∗ term of the Taylor series is dropped, the formulation defaults to the explicit formulation. The above modified energy equation is next solved to yield the new temperature θnew , which replaces θ∗ in the next iteration. The process is then repeated until convergence. Under-relaxation may still be used, as needed. The semi-implicit coupling procedure generally yields more stable and rapid convergence than the explicit method, as will be demonstrated in some of the examples to follow. As representative examples of combined radiation and conduction in a slab, next, we will discuss solutions for the radiative heat flux using the exact integral formulation (as presented in Chapter 13) and the differential or P1 -approximation (described in Sections 14.4 and 15.5). Similarly, equation (21.2) may be solved by a variety of numerical techniques. For illustrative purposes, we will limit ourselves here to a finite-difference solution of equations (21.2) and (21.3).

Exact Formulation The exact formulation for incident radiation G and radiative heat flux qR for a one-dimensional slab with specified temperature distribution has been given by equations (13.55) and (13.56). For a nonscattering medium the radiative source term reduces to S(τ) = Ib (τ) = n2 σT 4 (τ)/π [as given by equation (13.54)], and the radiative heat flux, as given by equation (13.56), becomes, in nondimensional form, / 0  τ  τL 4    4    (21.9) ΨR (τ) = 2 J1 E3 (τ) − J2 E3 (τL −τ) + θ (τ )E2 (τ−τ ) dτ − θ (τ )E2 (τ −τ) dτ , 0

τ

778 Radiative Heat Transfer

where we have introduced the nondimensional radiosities Ji = Ji /n2 σT14 . Equation (21.9) may be integrated by parts, using the recursion relations of Appendix E, leading to /  4 ΨR (τ) = 2 (J1 −1)E3 (τ) − (J2 −θL )E3 (τL −τ) −

τ 0

dθ 4  (τ )E3 (τ−τ ) dτ − dτ

 τ

τL

0 dθ 4    (τ )E3 (τ −τ) dτ , dτ

(21.10)

and, using Leibniz’s rule [2], as given by equation (3.114), / 0  τ  τL dΨR dθ 4  dθ 4      = 2 (1−J1 )E2 (τ) + (θL4 −J2 )E2 (τL −τ) + (τ )E (τ−τ ) dτ − (τ )E (τ −τ) dτ . 2 2  dτ dτ τ 0 dτ

(21.11)

Equation (21.11) must be solved simultaneously with equation (21.6) and its boundary conditions (21.8). For nonblack surfaces, two additional relations are required for the determination of the radiosities J1 and J2 . These may be obtained by applying equation (21.10) (evaluation of the radiative heat flux in terms of radiosities and medium temperature) at the two boundaries, eliminating the radiative heat flux through equation (13.48) (relating heat flux to radiosity and surface temperature). For the illustrative purposes of our present discussion, we will limit ourselves to black surfaces, i.e., J1 = 1 and J2 = θL4 , and dΨR =2 dτ

/ 0

τ

dθ 4  (τ )E2 (τ−τ ) dτ − dτ

 τ

τL

0 dθ 4    (τ )E2 (τ −τ) dτ . dτ

(21.12)

For this simple case, substitution of equation (21.12) into (21.6) gives a single nonlinear integro-differential equation for the unknown temperature, θ. Once the temperature field has been determined, the total heat flux follows as q = −k

dT + qR = const, dz

or, in nondimensional form, Ψ=

q n2 σT14

= −4N

dθ + ΨR = const. dτ

(21.13)

Example 21.1. An absorbing–emitting medium is contained between two large, parallel, isothermal, black plates at temperatures T1 and T2 = 0.5 T1 , respectively. Determine the steady-state temperature distribution within the medium and the total heat flux between the two plates, if heat is transferred by conduction and radiation. Discuss the influence of the conduction-to-radiation parameter, N, and of the optical thickness of the layer, τL . Solution The numerical solution to the governing equation may be found in a number of ways. We will employ here J + 1 equally spaced nodes τ = 0, Δτ, 2Δτ, . . . , JΔτ = τL with nodal temperatures θi (i = 0, 1, 2, . . . , J) and simple finite-differencing for the conduction term, d2 θ θi+1 − 2θi + θi−1 + O(Δτ2 ), dτ2 Δτ2 with a truncation error of order Δτ2 . The divergence of the radiative heat flux, equation (21.12), will be calculated by approximating the emissive power, θ 4 , by a spline function, followed by analytical evaluation of the piecewise integrals. In order to obtain the same truncation error as for the conduction term, O(Δτ2 ), the prediction of dθ 4 /dτ must be accurate to O(Δτ) [since the piecewise integration decreases the truncation error by O(Δτ)]. Thus, for the emissive power a linear spline is sufficient, or θ 4 (τ) = θ4i + Bi (τ−τi ) + O(Δτ2 ) = θ 4 − θi4 dθ 4 (τ) = i+1 + O(Δτ), dτ Δτ

4 θ4i (τi+1 −τ) + θi+1 (τ−τi )

Δτ

τi < τ < τi+1 ,

+ O(Δτ2 ),

i = 0, 1, 2, . . . , J − 1.

Radiation Combined with Conduction and Convection Chapter | 21 779

Substituting this into equation (21.12) leads to 

dΨR dτ

 2

i θ4 − θ4   j j−1

i

j=1

Δτ

τj τj−1

E2 (τi −τ ) dτ − 2

4  J  θj4 − θj−1 j=i+1

Δτ

τj

τj−1

E2 (τ −τi ) dτ

J i     2  4 2  4 4 4 = θj − θj−1 E3 (τi −τj ) − E3 (τi −τj−1 ) + θj − θj−1 E3 (τj −τi ) − E3 (τj−1 −τi ) Δτ j=1 Δτ j=i+1

=

J      2  4 4 θj − θj−1 E3 |i− j|Δτ − E3 |i+1− j|Δτ . Δτ j=1

Equating both sides of equation (21.6), we find θi−1 − 2θi + θi+1 = θ0 = 1,

J      Δτ  4 4 (θ − θj−1 ) E3 |i− j|Δτ − E3 |i+1− j|Δτ , 2N j=1 j

i = 1, 2, . . . , J − 1,

θ J = θL .

If N is relatively large (N > 0.1), heat transfer is dominated by conduction, and the solution proceeds as follows: 1. A temperature profile is guessed (e.g., the linear profile for pure conduction), and the (dΨR /dτ)i are calculated based on these temperatures. 2. A new temperature profile is determined by inverting the simple tridiagonal matrix for θ. 3. The temperature profile is iterated on, using the explicit (with under-relaxation, as needed) coupling procedure outlined earlier. If N is small, radiation dominates, and the process should be reversed: 1. A temperature profile is guessed, the conduction contribution is calculated, and an emissive power field is determined by inverting the full matrix for the θi4 on the right-hand side. 2. A new temperature profile is deduced from the emissive powers, etc. Once the temperature profile is known, the total heat flux follows from equations (21.10) and (21.13) as ⎧ i ⎫ J ⎪    τj    τj ⎪ ⎪ 2N 2 ⎪ ⎨  4 4   4 4  ⎬ (θi+1 − θi−1 ) − Ψi − θj − θj−1 θj − θj−1 E3 (τi −τ ) dτ + E3 (τ −τi ) dτ ⎪ ⎪ ⎪ ⎪ ⎭ Δτ Δτ ⎩ j=1 τj−1 τj−1 j=i+1 ⎧ ⎫ J i      ⎪ ⎪ ⎪ 2N 2 ⎪ ⎨  4 ⎬ 4 4 4 (θi+1 − θi−1 ) − =− , θj − θj−1 E4 (τi −τj ) − E4 (τi −τj−1 ) − θj − θj−1 E4 (τj −τi ) − E4 (τj−1 −τi ) ⎪ ⎪ ⎪ ⎩ ⎭ Δτ Δτ ⎪ j=1

j=i+1

i = 1, 2, . . . , J − 1. This value for the nondimensional heat flux should be the same for all nodes. Representative results are shown in Figs. 21.1 and 21.2. Figure 21.1 shows the nondimensional temperature variation within the slab for an intermediate optical thickness of τL = 1, calculated by two different methods: by the integral formulation of the present example, and by the P1 -approximation. For N = 0 there is no conduction, and the temperature profile is discontinuous at the walls, as first indicated in Fig. 13.3. For very small values of N the temperature profile remains similar except near the walls, where the medium temperature must rapidly approach the surface temperatures. As N increases, the influence of conduction increases, and the temperature profile rapidly becomes linear. For optically thin situations (not shown) the effect is even more pronounced: larger temperature jumps at the wall for N = 0 and an already near-linear temperature profile for N = 0.01. This behavior may be explained by noting that—for small τL —little emission and absorption takes place inside the medium; radiative heat flux travels directly from surface to surface. Representative nondimensional heat fluxes are shown in Fig. 21.2 and are compared with approximate methods, which will be discussed a little later. Since the optical thickness of a slab acts as a radiative barrier between two surfaces at different temperatures, the net heat flux increases with decreasing τL . That q/n2 σT14 increases with increasing N may be interpreted in two opposite ways: If the increase of N is due to an increase in thermal conductivity k, then the conductive and total heat fluxes increase. However, if the increase in N is due to a decrease in T1 , the radiative and total heat fluxes decrease due to the decreasing temperature levels (since q/n2 σT14 increases less rapidly than N).

780 Radiative Heat Transfer

FIGURE 21.1 Nondimensional temperature distribution for combined radiation and conduction across a gray slab of optical thickness τL = 1, bounded by black plates with a temperature ratio of θL = T2 /T1 = 0.5.

FIGURE 21.2 Nondimensional total heat flux for combined radiation and conduction across a gray slab, bounded by black plates with a temperature ratio of θL = T2 /T1 = 0.5.

Simple combined conduction–radiation problems such as this were first treated by Viskanta and Grosh [3,4] and Lick [5]. More recent investigations for nonscattering media have looked at laser flash diffusivity measurements of semitransparent materials [6], and several nongray problems such as heat transfer through aerogels [7], plastics [8], and combustion gases [9]. Several other one-dimensional investigations have also used exact radiation formulations in the presence of isotropic [10–12] and even anisotropic scattering [13], all using gray and constant radiation properties. Two-dimensional problems have been considered by Wu and Ou [14], who looked at a gray rectangular medium with isotropic scattering, and by Tuntomo and Tien [15], who applied Maxwell’s equations to small metallic particles irradiated by a laser. A comprehensive review of combined conduction–radiation heat transfer investigations has been given by Siegel [16].

P1 -Approximation The governing equations for the P1 -approximation and their boundary conditions have been given by equations (14.44) through (14.46) for the one-dimensional slab, and by equations (15.49) through (15.51) for general

Radiation Combined with Conduction and Convection Chapter | 21 781

geometries. Wang and Tien [17] apparently were the first ones to employ the P1 - or differential approximation for combined radiation and conduction. For a one-dimensional, gray, nonscattering slab between two gray-diffuse surfaces, the relations may be summarized as dq = 4πIb − G, dτ dG = −3q, dτ τ=0: τ = τL :

1 (4πIb1 − G), 2 − 1

2 − 2q = 4J2 − G = (4πIb2 − G), 2 − 2 2q = 4J1 − G =

(21.14) (21.15) (21.16a) (21.16b)

or, in nondimensional form (as given at the beginning of this section), dΨR = 4(θ 4 − g), dτ dg 3 = − ΨR , dτ 4 τ=0: τ = τL :

2 1 (1 − g), 2 − 1 2 2 − ΨR = 2( J2 − g) = (θ 4 − g). 2 − 2 L ΨR = 2( J1 − g) =

(21.17) (21.18)

(21.19a) (21.19b)

The radiative heat flux, ΨR , may be eliminated from equations (21.17) through (21.19), leading to d2 g + 3(θ 4 − g) = 0, dτ2 τ=0: τ = τL :

dg 3 1 + (1 − g) = 0, dτ 2 2 − 1 dg 3 2 − (θ 4 − g) = 0. dτ 2 2 − 2 L

(21.20)

(21.21a) (21.21b)

This second-order differential equation for the incident radiation is connected to the overall energy equation by combining equations (21.6) and (21.7), or d2 θ 1 = (θ 4 − g), 2 N dτ

(21.22)

with its boundary condition (21.8). Since the two coupled equations, namely, equations (21.20) and (21.22), are nonlinear, iterations are unavoidable when it comes to their solution. The solution may be obtained using one of two iterative coupling procedures already described earlier. Once the solution for θ and g have been obtained, the net heat flux may be calculated from equation (21.13) after evaluation of the radiative heat flux from equation (21.18), or ΨR = −

4 dg . 3 dτ

(21.23)

The two coupling procedures are demonstrated next. Example 21.2. Repeat the previous example, employing the P1 -approximation. Solution We will use a simple finite-difference method for the solution of overall energy as well as the P1 -approximation. As before, we will break up the optical thickness τL into J + 1 equally spaced nodes: i = 0, 1, . . . , J with τi = iΔτ and Δτ = τL /J. Thus,

782 Radiative Heat Transfer

equation (21.22) becomes θ0 = 1, Δτ2 4 (θi − gi ), N θ J = θL .

θi−1 − 2θi + θi+1 =

i = 1, 2, . . . , J − 1,

Similarly, equation (21.20) transforms to gi−1 − (2 + 3Δτ2 ) gi + gi+1 = −3Δτ2 θi4 ,

i = 1, 2, . . . , J − 1.

Two more discrete equations are needed to determine g. These come from the two boundary conditions for g, which are of the third kind, i.e., they contain both the dependent variable and its normal derivative. In order to derive a finite difference equation for the node on the left boundary, we perform a forward Taylor series expansion as follows:     dg Δτ2 d2 g + + .... g1 = g0 + Δτ dτ 0 2 dτ2 0 Substituting the left boundary condition for g, given by equation (21.21), into the above equation, we obtain   ' ) 3 Δτ2 d2 g g1 = g0 + Δτ (g0 − 1) + + .... 2 2 dτ2 0 0 Rearrangement of the above equation to obtain an expression for the second derivative, discarding higher order terms, and substitution into the governing equation, given by equation (21.20), yields − [2 + 3Δτ(1 + Δτ)] g0 + 2g1 = −3Δτ(1 + Δτ). Similarly, at the other boundary, 2gN−1 − [2 + 3Δτ(1 + Δτ)] gN = −3Δτ(1 + Δτ) θL4 . In the explicit method, the two sets of tridiagonal equations for gi and θi , just derived, are solved sequentially within an outer iteration loop, as described earlier. For this particular example, 101 nodes are used. Two separate residuals are computed in order to monitor convergence: ⎡ J−1  2 ⎤1/2 ⎥⎥ ⎢⎢ gi−1 − 2gi + gi+1 4 ⎥⎥ , + 3(θ − g ) R g = ⎢⎢⎣⎢ i ⎥⎦ i 2 (Δτ) i=1

⎡ J−1  2 ⎤1/2 ⎥⎥ ⎢⎢ θi−1 − 2θi + θi+1 1 4 ⎢ Rθ = ⎢⎢⎣ − (θi − gi ) ⎥⎥⎥⎦ . 2 (Δτ) N i=1 Convergence is deemed to have been reached when both R g and Rθ drop by at least 6 orders of magnitude. Once convergence is reached, the net heat flux is obtained from Ψi =

2N 2 (θi−1 − θi+1 ) + (gi−1 − gi+1 ). Δτ 3Δτ

Some sample results are included in Figs. 21.1 and 21.2 for comparison with the exact results. It is observed that the accuracy of the temperature profile is as expected from the differential approximation (cf. Chapters 14 and 15). Also as expected, the accuracy improves with increasing N, i.e., when conduction dominates more and more over radiation. Similar observations hold true for the evaluation of net heat fluxes. Table 21.1 summarizes the performance of the explicit coupling procedure for various values of the conduction-to-radiation parameter. The quantity shown in brackets in the third column is the value of the under-relaxation factor for which optimum convergence was obtained. As shown in Table 21.1, for N = 0.001, convergence cannot be attained even for extremely small values of ωu . + 4θ∗3 (θi − θ∗i ), is first substituted into the governing equation In the semi-implicit method, the approximation, θ4i ≈ θ∗4 i i for θ. The resulting equation is then rearranged to yield   2 4 ∗3 1 1 1 θi − θ + θi−1 − θi+1 = (3θ∗i 4 + gi ), i = 1, 2, . . . , J − 1. (Δτ)2 N i (Δτ)2 (Δτ)2 N

Radiation Combined with Conduction and Convection Chapter | 21 783

TABLE 21.1 Iterations needed by the explicit and semi-implicit coupling procedures to reduce residuals by 6 orders of magnitude. N

Explicit with ωu = 0.5

Explicit with optimum ωu

10

22

7 (1.0)

5

1

21

13 (0.7)

12

Semi-implicit

0.1

diverged

79 (0.16)

19

0.01

diverged

641 (0.019)

22

0.001

diverged

diverged

23

The boundary conditions remain unchanged. This new tridiagonal equation for θ is solved in a sequential manner along with the tridiagonal set for g presented for the explicit coupling procedure. However, in contrast to the explicit coupling procedure, no under-relaxation is needed here. The definition of the residuals remains unchanged, and the same convergence criterion as for the explicit coupling procedure is used. Table 21.1 also shows the performance of the semi-implicit coupling procedure. It is seen that convergence is attained with this coupling procedure irrespective of the value of N. Furthermore, it is seen that the number of iterations needed for the semi-implicit coupling procedure is smaller than those for the optimized explicit coupling procedure. The Fortran90 program for this example problem, CpldP1En1D.f90, is provided in Appendix F.

Additive Solutions Since the evaluation of simultaneous heat transfer by conduction and radiation is rather cumbersome, it is tempting to treat each mode of energy transfer separately (as if the other one weren’t there), followed by adding the two resulting heat fluxes. This simple method gives the correct heat flux for the two limiting situations (when only a single mode of heat transfer is present). The question is, how accurate is the method for intermediate situations? The energy flux by pure steady-state conduction through a one-dimensional slab of thickness L is given by qC = k

T1 − T2 , L

(21.24)

while the radiative heat flux for a gray, nonscattering medium at radiative equilibrium, confined between two isothermal black plates is, from Example 14.5, qR =

n2 σ(T14 − T24 ) 1 + 34 τL

,

(21.25)

where we have used the result obtained from the differential approximation, in order to make a closed-form expression possible. Adding these two heat fluxes yields the approximate net heat flux, which, in nondimensional form, may be written as Ψ=

q n2 σT14



1 − θL4 4N (1 − θL ) + , τL 1 + 34 τL

(21.26)

which is also included in Fig. 21.2. It is observed that the additive solution is surprisingly accurate. Einstein [18] and Cess [19] have shown that the method is within 10% of exact results for black plates, although somewhat larger errors are observed for strongly reflecting surfaces. Zeng and coworkers [20] have applied the method to somewhat nongray materials, and Howell [21] has demonstrated the relative accuracy of the method for concentric cylinders. Since the method has no physical foundation, it is impossible to predict its accuracy for general geometries. In addition, the method cannot be used to predict the temperature field, since pure conduction and pure radiation each predict their own—conflicting—profiles. Therefore, this method should be used with great caution, if at all; perhaps mostly for postprocessing estimates.

784 Radiative Heat Transfer

Other Work Since the early 1960s numerous articles on combined conduction–radiation problems have appeared in the literature. In this section, only a small subset of these articles—ones that may be considered representative—are briefly discussed. For additional material on these topics, the reader is referred to the references cited within these articles. Most of the early papers dealt with very simple one-dimensional problems, e.g. [3–5,22–25]. A number of investigations dealt with the effects of scattering in a one-dimensional slab, e.g. [26–40]; others considered spectral/nongray effects in varying degrees of sophistication, e.g. [35,37,38,41–49]. The effects of external irradiation on the combined-mode heat transfer in a one-dimensional slab have been discussed in various investigations, e.g. [6,44,50–58] and the influence of transient conduction in others, e.g. [6,33,40,55– 66]. Others considered variable property effects (thermal conductivity and/or radiative properties) [34,39,67], some studied ultrafast effects (hyperbolic conduction) [58,66], and others again applied inverse analysis to infer properties from experimental measurements [13,68,69]. Various numerical schemes for the solution of the governing nonlinear integro-differential equation have been employed, such as collocation with B-spline trial functions [70], collocation with Chebyshev polynomials [71], Galerkin methods [59,72], integral transform method [73], and finite-element methods [59]. In addition to the “exact” integral expressions, a number of different approximate methods were used to evaluate the radiative heat flux, such as the diffusion method [36, 47,54,74,75], the two-flux method [27,35,38,57,63,76,77], the exponential kernel approximation [5,40,57,78], the PN -approximation or variations of it [14,17,31,39,60,68,79], the discrete ordinates method, e.g. [33,34,37,47,66, 67,69,79–83], the zonal method [45], the Monte Carlo method [83–87], and others. In recent years, combined conduction-radiation-mass-diffusion one-dimensional models have been used for the design of clothing for fire fighters [88]. The few available experimental measurements of conduction–radiation interaction demonstrate the validity of theoretical models for glass [82,89,90], aerogel [7,91], glass particles [71], fiberglass [92], porous media [93,94], packed spheres [37], and gases [95]. While the majority of investigations have dealt with the interaction in a one-dimensional slab, other geometries have been increasingly considered, such as one-dimensional spheres, e.g. [21,96–99], one-dimensional cylinders [77,87,100–102], and rectangular and other two- and three-dimensional configurations [14,69,75,80–84, 103–109], including solution of the transient conduction-radiation problem [110], with particular focus on laser processing for engineering and biomedical applications [111,112].

21.3 Melting and Solidification with Internal Radiation Melting and solidification of materials is of importance in many applications and has been studied for over a century. Until the 1950s attention had been focused exclusively on melting and solidification of opaque materials, i.e., situations where the influence of internal radiative heat transfer may be neglected. Early investigations into the effects of radiation have assumed that, as in the case of opaque bodies, there is a distinct interface between liquid and solid zones [113–123], even though meteorologists had already realized that internal melting may occur within ice (e.g., [124,125]). Chan and coworkers [126] postulated that there exists a two-phase zone between the pure liquid and pure solid zones, as shown schematically in Fig. 21.3. The existence of such a two-phase layer in the presence of an internal radiation field may be explained as follows. Consider the melting of a semi-infinite solid, which is initially isothermal at its melting temperature Tm . A constant radiative heat flux is supplied to the face of the solid, as indicated in Fig. 21.3. If the material is opaque, the incident heat flux is absorbed by a thin surface layer at x = 0, and heat transfer inside the medium is by conduction alone. Melting then proceeds with a distinct interface as indicated in Fig. 21.3a, and as described in many papers and textbooks, e.g., [127]. If the material is semitransparent the external radiation penetrates deep into the solid, and some of the energy is absorbed internally, say, in the strip dx. This absorbed energy cannot be conducted away (the solid is isothermal at Tm ), nor can it raise the sensible heat without first melting the solid within the layer. Since the amount of energy absorbed over a short period of time cannot be sufficient to melt all of the material within the layer dx instantaneously, only gradual—and, therefore, partial—melting can be expected. As the amount of absorbed energy decreases for increasing distance away from the surface, the melt fraction will decrease along with it. For the more general case, if there is solid at temperatures below the melting point, absorbed radiative energy will be used first to raise the sensible heat of the material, resulting in a purely solid zone. Similar conclusions about the existence of a two-phase zone or “mushy zone” can be reached by replacing the external heat flux by a hot surface (with its surface emission), or by considering solidification rather than melting.

Radiation Combined with Conduction and Convection Chapter | 21 785

FIGURE 21.3 Melting zones within a semi-infinite body: (a) opaque medium, (b) semitransparent medium.

For the illustrative purposes of the present section, we will limit our consideration to a semi-infinite body, which is originally liquid and isothermal at temperature T∞ (T∞ > Tm , the melting temperature of the medium). For times t > 0 the temperature of the face at x = 0 is changed to, and kept at, a temperature Tw , which is lower than the melting/solidification temperature Tm . This results in a three-layer system with a qualitative temperature distribution as shown in Fig. 21.4. To keep the analysis simple, we will further assume that liquid and solid have identical and constant properties (kl = ks = k, κl = κs = κ, etc.), that the medium does not scatter, and that the face is black ( w = 1). Consideration of variable properties, different boundary conditions, different geometry, and/or melting instead of freezing is straightforward (but very tedious) and will not be discussed here. In the following pages we will set up the relevant energy equations governing the three zones, and the boundary conditions that they require, following the development of Chan and coworkers [126]. Pure Solid Region If, at t = 0, the temperature of the face is lowered instantaneously to Tw < Tm , this requires the instantaneous formation of an (infinitesimally thin) layer of pure solid, which will grow with time. The governing equation for the temperature within the solid zone follows from equation (9.75) as ρc

∂T ∂2 T dqR , =k 2 − dx ∂t ∂x

FIGURE 21.4 Solidification of a semitransparent liquid at T∞ , subjected to a cold boundary Tw (Tw < Tm < T∞ ).

(21.27)

786 Radiative Heat Transfer

which—assuming for now the location of the solid–mushy zone interface X1 (t) to be known—requires an initial condition and two boundary conditions, that is, t=0: x=0: x = X1 (t) :

T(x, 0) = T∞ , T(0, t) = Tw , T(X1 , t) = Tm .

(21.28a) (21.28b) (21.28c)

We defer, for the moment, the evaluation of the radiative heat flux since this is done in the same way for all three zones. Two-Phase Region (Mushy Zone) In the presence of a two-phase region, at least a part of the solidification takes place over a finite volume (rather than only at a distinct interface). Since during solidification the medium releases heat in the amount of L J/kg (where L is the heat of fusion), this gives rise to a volumetric heat source in the amount of ˙  Q˙  = L m˙  s = L ρs V s = L ρs

∂ fs , ∂t

(21.29)

˙  where m˙  s and Vs are the mass and volume of solid formed per unit time and volume, respectively, ρs is the density of the pure solid, and fs is the local solid fraction. Thus, with this heat source the energy equation (9.75) becomes ρc

∂ fs ∂T ∂2 T dqR =k 2 − + ρL , ∂t dx ∂t ∂x

(21.30)

where we have omitted the subscript s from ρs in the heat source term, since we assume that ρs = ρl = ρ = const. Since everywhere within the two-phase zone liquid and solid coexist and are assumed to be in local thermodynamic equilibrium, this implies that the temperature in the mushy zone is uniformly at the melting point, and there can be no sensible heat change (∂T/∂t = 0) and no conduction (∂2 T/∂x2 = 0). Thus, the energy equation simply becomes a relationship for the determination of the solid fraction, or ∂ fs 1 dqR , = ρL dx ∂t

(21.31)

subject to the initial condition t=0:

fs (x, 0) = 0.

(21.32)

Pure Liquid Region The energy equation for the pure liquid region is identical to the one for the solid, but with different boundary conditions since the zone extends from x = X2 (t) to x → ∞: ρc

∂2 T dqR ∂T =k 2 − , ∂t dx ∂x

t=0: x = X2 (t) : x→∞:

T(x, 0) = T∞ , T(X2 , t) = Tm , T(∞, t) = T∞ .

(21.33) (21.34a) (21.34b) (21.34c)

Radiative Heat Flux The radiative heat flux within a semitransparent, semi-infinite medium bounded by a black wall, as well as its divergence, are readily found from equations (13.56) through (13.36):    τ  ∞ (21.35) qR (τ) = 2 Ebw E3 (τ) + Eb (τ )E2 (τ−τ ) dτ − Eb (τ )E2 (τ −τ) dτ , 0 τ    ∞ dqR    (τ) = 4Eb (τ) − 2 Ebw E2 (τ) + Eb (τ )E1 (|τ−τ |) dτ , (21.36) dτ 0 where τ = κx is the usual optical coordinate, and we assume here that the absorption coefficient is constant and the same for both liquid and solid. Note that qR (τ) and dqR /dτ are continuous everywhere, including interfaces1 1. This is also true for variable/different absorption coefficients, for which equations (21.35) and (21.36) continue to hold with τ =

x 0

κ(x) dx.

Radiation Combined with Conduction and Convection Chapter | 21 787

FIGURE 21.5 Energy balance at the moving interface between solid and mushy zones.

(which is not true for the divergence of the conductive heat flux, as we will see from the interface conditions below). Interface Conditions Finally, we need two conditions for the determination of the location of the two interfaces between solid and mushy zones, X1 (t), and between mushy zone and pure liquid, X2 (t). These are obtained by performing energy balances over infinitesimal volumes adjacent to the interface, as depicted in Fig. 21.5. Consider a volume of thickness dX1 at the solid–mushy zone interface, dX1 being the thickness that becomes purely solid over a time period dt. An energy balance gives energy conducted in at X1 (t) + energy radiated in at X1 (t) + energy released during dt = energy conducted out at X1 (t+dt) + energy radiated out at X1 (t+dt), or   ∂T  ∂T    dt + qR (X1 ) dt + ρL(1− fs ) dX1 = −k dt + qR (X1 +dX1 ) dt, −k ∂x X1 −0 ∂x X1 +dX1 +0

(21.37)

where the subscripts ±0 imply locations on the left of the interface (−0), i.e., in the solid, and on the right of the interface (+0), i.e., in the mushy zone. The heat release term contains the factor (1 − fs ) because the fraction fs is already solid. Noting that T = Tm = const inside the mushy zone, it follows that ∂T/∂x|X1 +dX1 +0 = 0. The radiative heat flux, on the other hand, is continuous and cancels out from the interface condition once dt and dX1 are shrunk to zero, and equation (21.37) becomes simply x = X1 (t) :

−k

 ∂T  dX1  = 0, + ρL(1− fs ) dt ∂x X1 −0

(21.38)

subject to t=0:

X1 (0) = 0.

(21.39)

Note that there does not appear to be any requirement of fs → 1 at the interface (smooth transition from mushy zone to pure solid). Similar to equation (21.37) we find for the mushy zone–liquid interface   ∂T  ∂T    dt + qR (X2 ) dt + ρL fs dX2 = −k dt + qR (X2 +dX2 ) dt, −k ∂x X2 −0 ∂x X2 +dX2 +0

(21.40)

788 Radiative Heat Transfer

FIGURE 21.6 Solidification of a semi-infinite, semitransparent medium initially isothermal at melting temperature: development of solid 3 = 0.75, Ste = L/cT = 500, t∗ = 2κn2 σ(T 4 −T 4 )t/ρL. and mushy zones; θw = Tw /Tm = 0.9, N = kκ/4n2 σTm m m w

where (1 − fs ) is replaced by fs since the fraction fs solidifies from pure liquid. Upon shrinking dt and dX2 , the qR cancel again, and the conduction term within the mushy zone vanishes, or  dX2 ∂T   = −k ρL fs . (21.41) dt ∂x X2 +0 Now, in order for freezing to occur, we must have dX2 /dt > 0 and ∂T/∂x ≥ 0. Since the solid fraction must be nonnegative, this implies that the left-hand side of equation (21.41) should be positive and the right-hand side should be negative. This apparent contradiction can be overcome only if both sides of equation (21.41) are identically equal to zero, or x = X2 (t) :

fs (X2 , t) = 0,

∂T (X2 , t) = 0. ∂x

(21.42)

This implies that there is no distinct interface between mushy zone and liquid: Temperature, heat flux, and solid fraction are continuous across this “interface.” Mathematically, one distinguishes between mushy zone and pure liquid, since in the mushy zone fs is the unknown variable (T = Tm is known), and in the liquid zone the temperature is unknown ( fs = 0 is known). The location of interface X2 is found implicitly by evaluating fs (x, t) and determining the location where fs = 0. In summary, in order to predict the solidification of a semitransparent solid, it is necessary to simultaneously solve equations (21.27) and (21.28) (solid), equations (21.31) and (21.32) (mushy zone), and equations (21.33) and (21.34) (liquid), together with the interface conditions, equations (21.38) and (21.42). Note that—for an opaque medium—the radiative source within the medium vanishes (qR = 0) and, from equations (21.31) and (21.32), fs (x, t) = 0; that is, the mushy zone shrinks to a point, collapsing the two interfaces as expected for pure conduction. This system of equations is nonlinear, even in the absence of radiation, making exact analytical solutions impossible to find. Chan and coworkers [126] have presented approximate results for a few simple situations. For example, Fig. 21.6 shows the development of the solid and mushy zones for the case of a liquid that is initially uniform at melting temperature. Example 21.3. Consider a large (i.e., semi-infinite) block of clear ice exposed to solar radiation on one of its faces. The ice is initially at a uniform 0◦ C, i.e., at its melting temperature. Heat transfer from the surfaces of the ice (except the solar irradiation) may be neglected, as may the radiative emission from within the ice. Determine the development of the mushy zone for small times. Indicate how the movement of the liquid–mushy zone interface may be calculated. Solution Since the side walls are insulated, the problem is one-dimensional; and since the block is “very large,” we may assume that it is essentially a semi-infinite body with solar irradiation on its (otherwise insulated) left face at x = 0. Since, in this

Radiation Combined with Conduction and Convection Chapter | 21 789

example, we consider the melting of a solid, the order of zones is reversed, i.e., we have pure liquid for 0 ≤ x ≤ X1 , the mushy zone for X1 < x < X2 , and pure solid for x > X2 . In the present example X2 → ∞, since the ice is everywhere at the melting point. Also, since the face temperature is not increased abruptly, there is no instantaneous formation of a pure liquid layer and X1 = 0 for some time t > 0. The solar irradiation is not absorbed by the surface but penetrates into the ice, causing a local radiative heat flux—if emission from and scattering by the ice is neglected—of qR (x) = qsol e−κx , where qsol is the strength of solar irradiation penetrating into the ice (after losing some of its strength due to reflection at the interface at x = 0) (see Chapter 18). The purely liquid zone is essentially described by equations (21.27) and (21.28): ρc

∂2 T ∂T = k 2 + qsol κ e−κx , ∂t ∂x

t = t0 : x=0: x = X1 (t) :

T(x, t0 ) = Tm , ∂T (x, t) = 0, ∂x T(X1 , t) = Tm ,

where t0 is the time at which a purely liquid zone starts to exist, and the boundary condition at x = 0 has been replaced to reflect the lack of heat transfer at the surface. The heat generation term of equation (21.29) becomes a sink, and, while the expression is correct as is, it appears more logical to work with a liquid fraction, fl = 1 − fs , in the case of melting. Thus, equation (21.31) becomes qsol κ −κx ∂ fl 1 dqR =− = e , ∂t ρL dx ρL t=0:

fl (0) = 0.

Finally, the interface equation at x = X1 (t) must be rewritten as  dX1 ∂T   , = ρL(1− fl ) −k ∂x X1 −0 dt t = t0 :

X1 = 0,

where fs has been replaced by fl , and L by −L (since melting requires heat rather than releasing it). Since ∂T/∂x = 0 at x = 0, no liquid layer can grow until fl = 1 at x = 0. After this has taken place (at time t = t0 ) the temperature may rise at x = 0, and ∂T/∂x becomes negative at x = X1 − 0; therefore, fl (X1 ) must diminish again, and dX1 /dt > 0. For times t < t0 , the equation for the mushy zone is readily solved, leading to fl (x, t) =

qsol κt −κx e , ρL

0 = X1 (t) < x < ∞.

From this relationship it follows that a purely liquid zone starts at t0 =

ρL , qsol κ

that is, when fl = 1 at x = 0. For times larger than t0 , the relation for the liquid fraction, fl (x, t), within the mushy zone continues to hold, but only for x ≥ X1 > 0. The temperature profile within the liquid zone and the location of its interface must be determined by simultaneously solving the conduction and interface equations (with known values of fl ).

Since the original postulation by Chan and coworkers [126], the notion of a mushy zone has found widespread acceptance among other researchers [65,128–131]. Recently, Łapka and Furmanski ´ [132,133] have proposed a model that uses a fixed Cartesian grid for modeling of solidification processes of semi-transparent materials. Their model accounts for refraction and partially specular reflection at solid-liquid interfaces and has been validated against experimental results. The effect of variable refractive index (space and time dependent) has also been considered in a recent study [134] by Tan and coworkers.

790 Radiative Heat Transfer

FIGURE 21.7 Laminar flow of an absorbing/emitting fluid over an isothermal gray-diffuse plate.

21.4 Combined Radiation and Convection Many engineering applications involve flow of radiatively participating fluids. In this section, the fundamental physics of how radiation interacts with convection and ultimately alters the temperature distribution and resulting heat fluxes is discussed. We begin with a discussion of radiation-convection interactions in a flat-plate boundary layer. This is followed by a discussion of such interactions in internal flows in channels and tubes and flows driven by natural convection.

Thermal Boundary Layer In this section, we will briefly discuss how at high temperatures the presence of thermal radiation affects the temperature distribution in a thermal boundary layer and, therefore, the heat transfer rate to or from a wall. Again, since we are mainly interested in the basic nature of interaction between convective and radiative heat transfer, we will limit ourselves to a single simple case, laminar flow over a flat plate. Consider steady, laminar flow of a viscous, compressible, absorbing/emitting (but not scattering) gray fluid over an isothermal gray-diffuse plate, as illustrated in Fig. 21.7. Making the standard boundary layer assumptions [135], conservation of mass, momentum, and energy follow as ∂ ∂ (ρu) + (ρv) = 0, ∂x ∂y     dp ∂u ∂ ∂u ∂u +v = μ − , ρ u ∂x ∂y ∂y ∂y dx  2     ∂qR ∂u ∂T ∂T ∂ ∂T ρcp u +v +μ , = k − ∂x ∂y ∂y ∂y ∂y ∂y

(21.43) (21.44) (21.45)

subject to the boundary conditions x=0: y=0: y→∞:

u(0, y) = u∞ , T(0, y) = T∞ ; u(x, 0) = v(x, 0) = 0, T(x, 0) = Tw ; T(x, ∞) = T∞ . u(x, ∞) = u∞ ,

(21.46a) (21.46b) (21.46c)

Equations (21.44) and (21.45) incorporate the standard boundary layer assumptions of ∂u/∂y  ∂u/∂x and ∂T/∂y  ∂T/∂x (momentum and heat transfer rates across the boundary layer are much larger than along the plate, which is dominated by convection), as well as the simplified dissipation function (∂u/∂y)2 . Similarly, one may drop the x-wise radiation term in favor of the radiative heat flux across the boundary layer. This is readily justified by using the diffusion approximation to get an order-of-magnitude estimate for the radiative heat flux: From equation (14.22), qR = −kR ∇T, and—since ∂T/∂y  ∂T/∂x—radiation along the plate may be neglected as compared to radiation across the boundary layer. Therefore, assuming that the radiative heat flux y is one-dimensional, qR may be approximated from equation (13.56) (with τ = 0 κ dy and τL → ∞) as2 

τ

qR (x, y) = 2Jw (x)E3 (τ) + 2 0



Eb (x, τ )E2 (τ−τ ) dτ − 2

τ

∞

Eb (x, τ )E2 (τ −τ) dτ ,

and 2. Equations (21.47) and (21.48) are approximate since they assume that the local value of Eb is independent of x.

(21.47)

Radiation Combined with Conduction and Convection Chapter | 21 791

∂qR 1 ∂qR (x, y) = = 4Eb (x, τ) − 2Jw E2 (τ) − 2 κ ∂y ∂τ



∞

Eb (x, τ )E1 (|τ−τ |) dτ .

(21.48)

0

Alternatively, the radiative heat flux may be evaluated from any of the approximate methods discussed in Chapter 14. It should be remembered that photons carry momentum, thus causing radiation pressure and radiation stress (cf. Section 1.8), and that a control volume stores radiative energy [cf. equation (9.20) and Section 9.8]. However, these effects are generally negligible except at extremely high temperatures (> 50,000 K at 1 atm pressure) [136,137] and will not be included here. To improve the clarity of development, we will make the additional assumptions of constant fluid properties (ρ, cp , μ, k, κ = const), negligible dissipation term, a black plate [ w = 1, or Jw = Eb (Tw ) = Ebw ], and constant free stream values (u∞ , T∞ = const). Then equations (21.43) through (21.45) and (21.48) reduce to ∂u ∂v + = 0, ∂x ∂y u

(21.49)

∂u ∂2 u ∂u +v = ν 2, ∂x ∂y ∂y

(21.50)

∂T ∂T ∂2 T 1 ∂qR +v =α 2 − , ∂x ∂y ρcp ∂y ∂y  ∞   ∂qR = 2κ 2Eb (x, τ) − Ebw E2 (τ) − Eb (x, τ )E1 (|τ − τ |) dτ , ∂y 0 u

(21.51) (21.52)

subject to boundary conditions (21.46). Here ν = μ/ρ is the kinematic viscosity, and α = k/ρcp is the thermal diffusivity. Introducing the stream function ψ as u=

∂ψ , ∂y

v=−

∂ψ , ∂x

(21.53)

eliminates the continuity equation and transforms the momentum and energy equations to ∂ψ ∂2 ψ ∂ψ ∂2 ψ ∂3 ψ − = ν , ∂y ∂x∂y ∂x ∂y2 ∂y3

(21.54)

∂ψ ∂T ∂ψ ∂T ∂2 T 1 ∂qR − =α 2 − . ∂y ∂x ∂x ∂y ρcp ∂y ∂y

(21.55)

Making the standard3 coordinate transformation from x and y to the nondimensional ξ and η, where ξ=

3 κx 4n2 σT∞ , ρcp u∞

η=



u∞ νx

1/2 (21.56)

y,

and introducing new nondimensional dependent variables f =

ψ , (νu∞ x)1/2

θ=

T , T∞

ΨR =

qR 4 n2 σT∞

(21.57)

reduces the momentum and energy equations to d3 f 1 d 2 f + f = 0, dη3 2 dη2

(21.58)

  1 ∂2 θ 1 ∂θ d f ∂θ 1 ξ 1/2 ∂ΨR f = ξ + . + Pr ∂η2 2 ∂η dη ∂ξ 4 N Pr ∂η

(21.59)

3. Except for the nondimensionalization factor for ξ.

792 Radiative Heat Transfer

In this equation Pr = ν/α = μcp /k is the Prandtl number of the fluid, and N is the conduction-to-radiation parameter previously introduced as N≡

kκ . 3 4n2 σT∞

(21.60)

Sometimes, a convection-to-radiation parameter, or Boltzmann number, is also introduced, which is defined as ρcp u∞

Bo ≡

3 n2 σT∞

=4



NRex Pr ξ

1/2

,

(21.61)

where Rex = u∞ x/ν is the local Reynolds number. Very similar to the conduction-to-radiation parameter N, the Boltzmann number provides a qualitative measure of the relative magnitudes of convective and radiative heat fluxes. Equation (21.58) contains no ξ-derivative since η turns out to be a similarity variable, i.e., no term in the equation (except the ξ-derivative) contains ξ, and the boundary conditions for f do not depend on ξ, collapsing to η=0:

f =

df = 0, dη

η→∞:

df = 1. dη

(21.62)

Thus, equation (21.58) is an ordinary differential equation for the unknown f , which is a function of the similarity variable η alone. Equation (21.58) and its solution was first given by Blasius and is well documented in fluid mechanics texts, such as [138]. The energy equation (21.59) is a partial differential equation for the unknown θ, subject to the boundary conditions η=0: ξ=0:

Tw = θw , T∞ θ = 1.

θ=

η→∞:

θ = 1,

(21.63a) (21.63b)

Since the boundary conditions at x = 0 correspond to both ξ = 0 and η → ∞, equation (21.59) can also reduce to a similarity solution, but only if ΨR ∝ ξ−1/2 . This is not the case if ΨR is evaluated from equation (21.52) or most approximate methods discussed in Chapter 14. However, if the thermal boundary layer is optically very thick, so that the diffusion approximation becomes applicable, one finds from equation (14.22) ΨR = −

4 ∂θ 4 4 ∂θ 4 =− . 3κ ∂y 3(N Pr ξ)1/2 ∂η

(21.64)

This expression is substituted into equation (21.59), resulting in the ordinary differential equation 1 d2 θ 1 dθ 1 d2 θ 4 f = − + , Pr dη2 2 dη 3N Pr dη2

(21.65)

since then θ is a function of the similarity variable η only. The interaction of radiation and convection in an optically thick laminar boundary layer of a gray gas was first investigated by Viskanta and Grosh [139] and others [140–143]. Figure 21.8 shows the similarity profile for the nondimensional temperature, as obtained using the diffusion approximation [139], for a number of different values for the conduction-to-radiation parameter N. For N = 10 the temperature profile was found to be within 2% of the pure convection case (which numerically corresponds to N → ∞). When radiation is present, the thermal boundary layer was always found to thicken, which may be explained by the fact that radiation provides an additional means to diffuse energy. Even for strong radiation (large T∞ ) the thickening of the thermal boundary layer may be limited if the fluid is optically thick (large κ). However, if the absorption coefficient is small (optically thin fluid), the thickening of the thermal boundary may become so large as to invalidate the basic boundary layer assumptions (i.e., the neglect of conduction and radiation in the x-direction).

Radiation Combined with Conduction and Convection Chapter | 21 793

FIGURE 21.8 Similarity profiles for nondimensional temperature profiles across an optically thick laminar boundary over a flat plate; Pr = 1: (a) θw = Tw /T∞ = 0.5, (b) θw = 2.

FIGURE 21.9 Comparison of conductive, radiative, and total heat fluxes for a laminar boundary layer over a flat plate: optically thin solution from [19], optically thick solution from [139], and exact solution from [144]; N = 0.1, Pr = 1.0, θw = 0.1.

Figure 21.9 shows nondimensional radiative, conductive, and total surface heat fluxes along the plate for a representative case as evaluated by three different methods. The radiative heat flux is evaluated according to the definition in equation (21.57), and the conductive heat flux is defined as     ∂T  + 2 4 N 1/2 ∂θ    , n σT∞ = −4 ΨC = −k ∂y  y=0 Pr ξ ∂η η=0

(21.66)

and Ψ = ΨC + ΨR .

(21.67)

The “exact” results are a numerical solution of equation (21.59) with the radiation term evaluated from equation (21.52), as obtained by Zamuraev [144] (and reported by Viskanta [145]). In the optically thick solution ΨR is evaluated from equation (21.64) as  ∂θ 4  4 ΨR = −  , 3(N Pr ξ)1/2 ∂η η=0

(21.68)

794 Radiative Heat Transfer

FIGURE 21.10 Thermally developing Poiseuille flow of a gray, absorbing, and emitting fluid between gray-diffuse plates.

and displays a simple ξ−1/2 dependence. The optically thin solution has been taken from Cess [19,146], who postulated a two-region temperature field consisting of a very thin conventional thermal boundary layer (in which radiation is neglected in favor of conduction) and an outer region with slowly changing temperature (in which conduction is neglected). As seen from Fig. 21.9, the diffusion approximation predicts the wall heat flux accurately over the entire length of the plate, while the optically thin approximation fails a short distance away from the leading edge (apparently since downstream the boundary layer grows too thick to neglect radiation and/or the outer layer becomes too nonisothermal to neglect conduction). Other early optically thin models have been reported by Smith and Hassan [147] and Tabaczynski and Kennedy [148]; Pai and Tsao [149] used the exponential kernel approach, and Oliver and McFadden [150] solved the “exact” relations, equation (21.52), by the method of successive approximations, stopping after three iterations. Dissipation effects [146,151–153] as well as hypersonic conditions [152,154–156] have been considered by a number of investigators, including the treatment of the interactions between non-gray radiation and a hypersonic boundary layer [157]. The influences of scattering [158,159], nongray radiation properties [160–162], external irradiation [163,164], turbulent boundary layers [165–167], as well as laminar flow across cylinders [162] and spheres [168] have also been addressed.

Poiseuille Flow In this section we will examine the interaction of radiation and convection for a radiatively participating medium flowing through a duct. Specifically, we consider hydrodynamically fully-developed laminar flow of an incompressible, constant-property fluid through a parallel-plate channel. This is commonly referred to as Poiseuille flow. We will assume that the fluid is gray, absorbing, and emitting (but not scattering), and that the plates are gray and diffuse, a distance H apart, and isothermal, as indicated in Fig. 21.10. The fully-developed velocity distribution for Poiseuille flow follows readily from equations (21.49) and (21.50), setting u = u(y), as u = 6um

y y  1− , H H

v = 0,

(21.69)

where um is the mean velocity across the duct. Thus, the energy equation (21.51) reduces to u(y)

∂T ∂2 T 1 ∂qR =α 2 − , ∂x ρcp ∂y ∂y

(21.70)

if again we limit ourselves to the case in which conduction and radiation in the flow direction (along x) are negligible as compared to their transverse values (along y). This is generally a good assumption for channel locations that are a few channel heights H removed from the inlet [169]. Equation (21.70) is subject to the boundary conditions x=0: y = 0, H :

T = Ti ,

(21.71a)

T = Tw ,

(21.71b)

Radiation Combined with Conduction and Convection Chapter | 21 795

and the radiative heat flux may be obtained from equation (13.56) as4  τ  qR (x, y) = 2Jw (x) [E3 (τ) − E3 (τH −τ)] + 2 Eb (x, τ )E2 (τ−τ ) dτ − 2

τH

τ

0

Eb (x, τ )E2 (τ −τ) dτ ,

(21.72)

(where the optical coordinate τ = κy was used for the y-direction). The radiative heat flux can, of course, also be evaluated by any of the methods discussed in the preceding chapters. Here, we consider the P1 approximation in addition to the exact solution with the objective of testing its accuracy for such problems. The governing P1 equation and its boundary conditions for one-dimensional media have already been presented earlier [equations (21.20) and (21.21)] and may be used here directly without additional alterations. Introducing similar nondimensional variables and parameters as in the previous section, qR T , ΨR = 2 4 , Tw n σTw ; y x um H ν x = , η= , ξ= H Re Pr H ν α H kκ N = 2 3 , τH = κH, 4n σTw θ=

(21.73a) τ = κy,

(21.73b) (21.73c)

transforms equations (21.69) through (21.72) to 6η(1−η)

∂θ ∂2 θ τH dΨR , = − 2 4N dη ∂ξ ∂η ξ=0:

  ΨR = 2 E3 (τ) − E3 (τH −τ) +

(21.74)

θ = Ti /Tw = θi , τ

η = 0, 1 : 







θ (ξ, τ )E2 (τ−τ ) dτ − 4

0

τ

τH

θ = 1,

(21.75) 







θ (ξ, τ )E2 (τ −τ) dτ , 4

(21.76)

where, for simplicity, we have limited ourselves to black channel walls. The divergence of the heat flux, dΨR /dη, in equation (21.74), can be obtained directly from equation (21.12) for each axial location ξ. If the P1 approximation is used, the divergence of the nondimensional radiative heat flux, appearing in equation (21.74), is written, following equation (21.17), as dΨR = 4τH (θ4 − g). dη

(21.77)

Equation (21.74) and its boundary conditions must be solved either after substituting equation (21.12) directly (for exact solution) or simultaneously with equations (21.20) and (21.21) (for P1 approximation). Prior to their numerical solution, they must be discretized. As in preceding sections, here too we use simple finite-difference approximations on an equally spaced mesh, comprising of K + 1 nodes in the x-direction and M + 1 nodes in the y-direction, such that the grid spacings are Δξ = [(L/H)/Re Pr]/K and Δη = 1/M. For a generic interior node (i, j), the following finite difference approximations are employed for the terms in equation (21.74):   θi+1, j − θi, j ∂θ ≈ , (21.78a) ∂ξ Δξ ∂2 θ θi, j+1 − 2θi, j + θi, j−1 ≈ . (21.78b) ∂η2 (Δη)2 All boundary nodes on the inlet or on either of the two walls have prescribed fixed temperatures and require no special treatment. Equation (21.74) is a parabolic differential equation allowing a straightforward numerical solution technique, marching forward from ξ = 0 (i = 0). For the case when the exact solution of the RTE is used to evaluate the radiative source term, the solution to the coupled set of nonlinear equations was first obtained 4. Again using an approximate, i.e., one-dimensional, solution by neglecting the x-wise variation of emissive power in the evaluation of qR .

796 Radiative Heat Transfer

FIGURE 21.11 Comparison of exact solution and P1 approximation for Poiseuille flow—heated wall: (a) temperature distributions at various axial locations for τH = 5 and N = 0.05, (b) local Nusselt number development.

by Kurosaki [170]. Starting from a guessed temperature profile at the first node downstream of the inlet (i = 1), the divergence of the radiative flux is first calculated using the discretized form of equation (21.12) shown in Example 21.1. This is substituted into the right-hand side of the discretized form of equation (21.74). The resulting tridiagonal set of equations is solved to obtain a new temperature distribution at that axial location. The procedure is repeated until convergence at that axial location. This converged temperature distribution at i = 1 is used as initial guess for the next downstream node. The procedure is repeated for all downstream axial nodes until the last node (i = K + 1) is reached. For the case when the P1 approximation is used to solve the RTE, the discrete forms of the energy equation and the P1 equation are coupled and solved using one of the two coupling procedures described at the beginning of Section 21.2 and also demonstrated in Example 21.2. The present calculations are performed using a very fine mesh with K = 2000, and M = 200 for both methods. Figure 21.11 shows a comparison of the results obtained by the two methods for the case of θi = 0.5 (cold fluid, hot wall). The two methods yield almost identical temperature profiles very close to the inlet (ξ = 0.0005), as seen in Fig. 21.11a. However, as the flow moves downstream, the small errors incurred by the P1 approximation accumulate since energy (temperature) is convected downstream by the flow. Consequently, further away from the inlet, for example at ξ = 0.01, the differences in temperature determined using the two methods are larger. The axial development of the local Nusselt number is also shown in Fig. 21.11b, with the Nusselt number defined as Nux (ξ) =

qw H , k [Tw − Tm (ξ)]

(21.79)

where qw = qC + qR is total heat flux per unit area at the wall, by radiation and conduction. In terms of nondimensional quantities, the local Nusselt number becomes   τH 1 ∂θ + ΨR Nux (ξ) = . (21.80) − 1 − θm (ξ) ∂η 4N η=0 It is apparent from Fig. 21.11b that—due to the nonlinear radiative contribution—no fully developed temperature profile, and consequently no asymptotic Nusselt number, develops. Rather, the Nusselt number goes through a minimum at a certain downstream location, beyond which it tends to increase again. This phenomenon may be explained as follows: downstream from the inlet the convective heat flux always decreases more rapidly than the temperature difference, Tw − Tm (ξ), causing a steady decrease in the convective contribution to the Nusselt number; the radiative heat flux, on the other hand, decreases less with x than the temperature difference, leading to the observed behavior. The same trend is observed irrespective of the method used to solve the RTE.

Radiation Combined with Conduction and Convection Chapter | 21 797

The Nusselt numbers obtained using the P1 approximation overpredict those obtained using the exact formulation of the RTE by a maximum of 11%, with the largest differences being at downstream locations where, as discussed, the differences in temperature are also the largest. The Nusselt numbers shown in Fig. 21.11b for the exact solution are smaller by about 20–25% than those calculated by Kurosaki [170]. While the exact reason is not clear, grid size could be a possible source of discrepancy.

Overview of Laminar and Turbulent Duct Flow Heat transfer in ducts in the presence of radiation has been studied extensively over the past several decades. In this section, we present a very brief overview of selected works of historical importance, and some recent works that shed light on certain “new” physical phenomena that have not been discussed in the early literature, especially in the context of turbulent flow. Qualitatively, the Nusselt number development for other duct flows with heated walls is similar to that of Poiseuille flow (regardless of geometry, turbulence, presence of scattering, nongrayness, etc.). The Nusselt number decreases monotonically near the inlet, somewhat similar to the pure convection case, and eventually appears to either increase (as in Fig. 21.11b) or reach an asymptotic value. In the presence of thermal radiation this “fully developed” case where an asymptotic value is ultimately reached does not occur until the bulk temperature is essentially equal to the wall temperature (note that, for pure convection, the mean (or bulk) temperature changes only by ≈ 20% of maximum when fully developed conditions have been reached), which is practically possible only in an extremely long duct. Therefore, it may be concluded that no thermally fully developed conditions can exist for forced convection in duct flow combined with appreciable thermal radiation (i.e., radiative heat fluxes too large to be approximated by a linear expression in temperature). This fact was not realized by a number of early investigations on the subject, which employed “thermally developed” conditions to obtain relatively simple results, e.g., [171–174]. Azad and Modest [175] have shown how temperature level and optical thickness influence the variation of Nusselt number and bulk temperature in tube flow. Reducing N/τR (which does not depend on absorption coefficient and, for a given medium and tube radius, implies raising temperatures) for a constant optical thickness results in increased heat transfer rates due entirely to an increase in radiative heat flux. The radiative heat flux goes through a maximum at an intermediate optical thickness, τR 1 (for constant N/τR ). This is readily explained by examining the optical limits. In the optically thin limit the medium does not emit or absorb any radiation, resulting in purely convective heat transfer. On the other hand, in the optically thick limit any emitted radiation is promptly absorbed again in the immediate vicinity of the emission point, again reducing radiative heat flux to zero. A simple one-dimensional temperature profile does exist in the case of Couette flow (two infinite parallel plates moving at different velocities), since the entire problem becomes one-dimensional. The analysis for this case reduces to the same equations arrived at in the previous section for combined conduction and radiation, which have been solved numerically by Goulard and Goulard [176] and Viskanta and Grosh [177]. As indicated earlier, the Poiseuille flow problem was originally investigated by Kurosaki [170], using the exact integral relations for the radiative heat flux. Echigo and coworkers extended this work by performing calculations for values of N as small as 10−6 [178], and flow through cylindrical tubes [179]. The problem had been addressed a little earlier by Timofeyev and coworkers [180], using the two-flux method. The case of slug flow between parallel plates, with rigorous modeling of the radiative heat flux, has been treated by Lii and Özi¸sik [181]. The influence of scattering on Poiseuille flow has been discussed by a number of investigators [182–185]. Yener and coworkers [186,187] examined the same problem for turbulent flow conditions, while Echigo and Hasegawa [188] addressed a laminar, scattering gas–particulate mixture. All of these publications neglected axial radiation. Two-dimensional radiation for Poiseuille flow has been studied by Einstein [18] (nonscattering fluid) and Kassemi and Chung [189] (isotropically scattering fluid), using the zonal method, and by Kim and Lee [190] (anisotropically scattering fluid), using the discrete ordinates method. Other investigations on turbulent tube flows with gray media, also using the discrete ordinates method, include those of Kim and Baek [191,192] (two-dimensional radiation without scattering) and Krishnaprakas and coworkers [193] (onedimensional radiation with linear-anisotropic scattering). Combined convection and radiation in thermally developing tube flow appears to have been investigated first by Einstein [194], deSoto [169], and Echigo and coworkers [179], considering two-dimensional (axial and radial) radiation, while Bergero and colleagues [195] considered developing flow and three-dimensional, gray radiation in a laminar rectangular duct, using the finite volume method (for radiation). In an important recent study, Taine and coworkers [196] have demonstrated

798 Radiative Heat Transfer

FIGURE 21.12 Effect of radiation on the temperature distribution in a turbulent boundary layer: (a) optically thick medium (40 atm pressure), (b) optically thin medium (1 atm pressure); from [196]. T+ is the nondimensional temperature, while y+ is the nondimensional distance from the wall.

how radiation alters the turbulent boundary layer structure. In this study, turbulent convection in a channel was modeled using Direct Numerical Simulation (DNS), while radiation was modeled using the Monte Carlo method. The spectral nature of gas radiation was addressed using a narrow-band based correlated-k model. The inclusion of radiation not only enhances heat transfer due to the contribution of radiation itself, but also alters the convective heat fluxes by altering the near-wall temperature distribution, as shown in Fig. 21.12. They attributed this alteration to a combined effect of radiative interactions within the medium (referred to as gas–gas interactions) and radiative interactions between the medium and the adjacent wall (referred to as gas–surface interactions). It is shown that these two interactions result in conflicting effects on the near-wall temperature distribution. As expected, the effect of gas-gas interactions and of radiation overall on the near-wall temperature distribution is relatively weaker in optically thin situations, as is evident by comparing the two figures in Fig. 21.12. Based on the results of this study, a new wall-function (a modified law of the wall) for combined turbulent convection– radiation was proposed by the same researchers [197]. A set of criteria that ought to be met to warrant use of such a modification has also been proposed [198]. The effects of nongray molecular gas radiation on laminar tube and channel flows, employing the exponential wide band model, have been studied by a number of investigators [199–202]. Similar calculations for turbulent flows have also been carried out using fully developed flow and simple algebraic expressions for the eddy diffusivity for heat [174,200,203–205], while Smith and coworkers [206] used the two-dimensional zonal method and weighted-sum-of-gray-gases approach. More accurate analyses, using the statistical narrow band model, narrow band k-distributions, and the global ADF model for radiation calculations, have been carried out by the group around Soufiani and Taine for laminar [161,207,208] and turbulent [209,210] tube and channel flows, the latter using the k–ε turbulence model. The general trends are similar to flows of gray media, i.e., strong radiation effects are evidenced by the much faster development of the temperature profiles (resulting in larger Nusselt numbers), regardless of whether the gas is heated or cooled. However, comparison with wide band model results showed that the latter can produce significant errors in predicting temperature fields and radiative fluxes. Comparison with experiment [207], on the other hand, showed excellent agreement with temperature fields predicted with the narrow band model. The coupling of mixed free and forced convection with nongray radiation has also been modeled recently in inclined ducts using the FSK model [211]. Gas–particulate suspension flows were first addressed by Echigo and colleagues [212,213] for laminar and turbulent flow of nonscattering media, respectively. Anisotropic scattering in tube suspension flows has been treated by Modest and coworkers for gray [175,214] and nongray [205] carrier gases. Nongray effects in suspension flows have also been studied by Al-Turki and Smith [215], using the zonal method, while two-dimensional, gray particle radiation was considered by Park and Kim [216], using the P1 -approximation. Recently, direct numerical simulation of particle laden turbulent flow and its interactions with radiation has been modeled [217] by

Radiation Combined with Conduction and Convection Chapter | 21 799

Boyd and coworkers. In their study, the RTE was solved using the Monte Carlo method. Coupled convection– radiation analysis has also been conducted for determining the temperature and heat flux inside a turbofan engine [218]. Radiation effects in liquid glass jets were investigated by Yin and Jaluria [219,220] and by Song et al. [221], both using a two-dimensional stepwise gray approach together with the zonal [219,220] or discrete ordinates method [221]. Finally, there have been several attempts at modeling radiation interactions with flow through porous media [222–224] and packed beds [225–227]. A general (but somewhat dated) overview of the literature has been given by Viskanta [228].

Free Convection The effects of radiation are often even more important when combined with free convection rather than forced convection. The radiation effects on a vertical free-convection boundary layer have been modeled by Cess [229] for the optically thin case and by Arpaci [230] for the optically thin and thick cases, while Cheng and Özi¸sik [231] and Desrayaud and Lauriat [232] looked at isotropic scattering effects, and Krishnaprakas et al. [233] considered linear anisotropic scattering. Hossain et al. [234] used the diffusion approximation to deal with an optically thick gas next to a porous vertical plate with suction. Webb and Viskanta [235] investigated the effects of external irradiation, verifying their model with experiment [236], and a vertical square duct was studied by Yan and Li [237, 238]. Careful experimental work by Lacona and Taine [239] verified standard (no-radiation) prediction models, and showed that radiation can strongly modify free convection temperature profiles. They used holographic interferometry and laser deflection techniques to measure temperatures in nitrogen (suppression of radiation) and pure carbon dioxide (strong radiation effects). Thermal stability of horizontal layers with radiation has also attracted early attention [240–242] as has combined radiation and free convection within enclosed, particularly square cavities, e.g. [243–247], and parallel vertical plates [248]. In addition, horizontal [249,250] and vertical annuli [251] and cubical cavities [252] have been studied. The interaction between free convection and radiation in liquids was studied by Derby and coworkers [253], investigating a cylindrical container with molten glass, and by Tsukada and colleagues [254–256] and Fang et al. [257], who considered internal radiation during Czochralski crystal growth. Most of the above studies have been limited to the simple case of constant, gray radiation properties. Exceptions are the studies of Mesyngier and Farouk [258], who considered a H2 O–CO2 mixture in a square enclosure, using the discrete ordinates method and the weighted-sum-of-gray-gases approach, and of Bdéoui and coworkers [259], who studied water vapor radiation effects on Rayleigh–Bénard convection, using an exact formulation together with the ADF method. In addition, Colomer et al. [245] studied square cavities filled with H2 O–CO2 mixtures using the SLW method and showed that nongray gas properties have very strong impact on temperature distributions in such flows.

21.5 General Formulations for Coupling In the preceding several sections, the coupling between radiation and other modes of heat transfer was discussed with the objective of divulging the specific role of radiation in commonly studied problems in heat transfer. In many of the cases discussed, a one-dimensional geometry was considered, making both the solution of the RTE, as well as the coupling somewhat straightforward. For multidimensional problems, neither the energy equation nor the RTE can be solved analytically, and one must resort to numerically solving both equations. In this section, the “best practices” of robust coupling between the energy equation and RTE are systematically discussed under the premise that both equations will be solved numerically. In particular, we will discuss the semi-implicit coupling procedure that was first introduced in the context of boundary conditions in Chapter 8, and later revisited in Section 21.2 for coupling of the radiation source term, albeit for a one-dimensional problem. A brief summary of fully implicit coupling is also given toward the end of the section. The reader is forewarned that proper understanding of the contents of this section requires a basic knowledge of numerical discretization and solution of partial differential equations using standard deterministic techniques such as the finite difference (FD), the finite volume (FV), or the finite element (FE) method. We begin our discussion by considering the overall energy conservation equation. Irrespective of the discretization method used (FD, FV, or FE method), the discretized form of the energy equation may be written for

800 Radiative Heat Transfer

FIGURE 21.13 Illustration of stencils showing a set of nodes (left) or cells (right), as typically used in either FD or FV formulations.

a node or cell P as A TP + T P

Nnb 

  ATP,nb TP,nb = SR,P + ST,P = − ∇ · qR P + ST,P ,

(21.81)

nb=1

where SR,P is the radiative source term, and ST,P represents any other source term in the energy equation (such as viscous dissipation, heat generation, external work, etc.). ATP is the diagonal element and ATP,nb are the off-diagonal elements in the coefficient matrix for the discrete equation corresponding to the node or cell center, P. These coefficients are often referred to as link coefficients. The superscript ‘T’ indicates that these link coefficients are for the energy (or T) equation. TP is the temperature of the node or cell for which equation (21.81) has been written, and TP,nb are temperatures of the nodes or cells that are neighbors to the node or cell, P. For clarity, a representative set of control volumes (cells) and nodes are shown in Fig. 21.13, and the subsequent discussion is applicable to both the FD method and the FV method. Nnb is the number of neighboring cells or nodes of cell or node P. The other symbols carry their usual meanings. The coupling of equation (21.81) with the RTE using the semi-implicit coupling procedure is independent of the RTE solver used, and entails the following steps: 1. The temperature field is first guessed within the entire computational domain. This is denoted by T∗ . If nonisothermal boundaries are present, the temperatures at the boundaries are also guessed, and are denoted by TB∗ . 2. The temperature dependent radiative properties are computed. 3. The RTE is solved using a method of choice: PN approximation, SN approximation, the Zonal method, or the Monte Carlo method. The outputs of this calculation are the incident radiation (gray or spectral) and the divergence of the radiative heat flux. Since the RTE is solved using a guessed (or old) temperature field, the outputs are denoted by G∗λ (or G∗ in the gray case). The radiative source in equation (21.81) is then evaluated as   SR,P = −[∇ · qR ]P = − 4κ∗P Eb (T) − 0

∞

 κ∗λ G∗λ dλ

,

(21.82)

P

where the Planck-mean absorption coefficient computed using the old temperature field is denoted by κ∗P , while the blackbody emissive power is denoted by Eb (T). Equation (21.82) is semi-implicit since it uses the current temperature only in the emission term, while all other quantities are evaluated using the old (previous iteration) values of temperature. Using a Taylor series expansion for the blackbody emissive power and retaining only up to the linear term, i.e., Eb (T) ≈ Eb (T∗ ) + [dEb /dT]∗ (T − T∗ ), we obtain 

 ∗

SR,P = − 4κP



dEb Eb (T ) + dT ∗

∗

  (T − T ) −

∞

∗

0

 κ∗λ G∗λ

,

dλ P

(21.83)

Radiation Combined with Conduction and Convection Chapter | 21 801

which may be rearranged further as  SR,P = −[∇ · q∗R ]P − 4κ∗P,P

dEb dT

∗

(TP − TP∗ ) = −[∇ · q∗R ]P − 16κ∗P,P n2 σTP∗3 (TP − TP∗ ).

(21.84)

P

The first term on the right-hand side of equation (21.84) is the radiative source computed using the old (previous iteration) temperature field and, as discussed earlier, is a standard output from any RTE solver, deterministic or stochastic. Substituting equation (21.84) into equation (21.81) for cell P, followed by rearrangement, yields Nnb      T ∗ 2 ∗3 ATP,nb TP,nb = − ∇ · q∗R P + 16κ∗P,P n2 σTP∗4 + ST,P . AP + 16κP,P n σTP TP +

(21.85)

nb=1

Prior to solving equation (21.85), its boundary conditions must be formulated. For prescribed-temperature boundaries, no special treatment is necessary since the temperature is already known. For other boundaries (either second or third kind), an energy balance at the boundary must be performed to compute the local temperature and/or the heat flux at the boundary. For illustrative purposes, let us consider a boundary subjected to Newton cooling externally with heat transfer coefficient h and ambient temperature T∞ . An energy balance at the boundary yields [−k∇T + qR,B ] · nˆ = h(T∞ − TB ).

(21.86)

For a first-order approximation on an orthogonal mesh5 , the normal derivative of the temperature at the boundary may be written as [∇T] · nˆ ≈ (TP − TB )/δ, where δ is either the distance between two nodes (in the FD method) or the distance between the cell center adjacent to the boundary and the boundary face center (in the FV method), as shown in Fig. 21.14. Substituting this approximation and equation (8.11) into equation (21.86) yields k

(TB − TP∗ ) + q∗R,B + 4 i σTB∗3 (TB − TB∗ ) = h(T∞ − TB ), δ

(21.87)

ˆ and the superscript “*” indicates previous iteration values. Equation (21.87) may be where qR,B = qR,B · n, rearranged to solve for the boundary temperature:  .   kT∗ k TB = −q∗R,B + P + hT∞ + 4 i σTB∗4 + h + 4 i σTB∗3 . (21.88) δ δ In the FD method, nodes are located on the boundary. Therefore, once the boundary temperature has been computed, the boundary condition is readily applied. In the FV method, the flux at boundaries is needed, and this can be obtained by substituting the computed value of the boundary temperature into either the left-hand side or the right-hand side of equation (21.87). 4. The discrete semi-implicit form of the energy equation is next solved. The solution to the energy equation yields the new temperature field, T. 5. The old temperature, T∗ , is replaced by the new temperature T, and Steps 2–4 are repeated—referred to as outer iterations—until both the RTE and the energy equation are simultaneously satisfied, i.e., the residuals for both equations reach a prescribed small tolerance. The semi-implicit coupling procedure has the benefit that it is relatively straightforward to implement since the equations are still solved in a segregated manner. For example, if new bands (nongray RTEs) are added or if new directions are added (as in the discrete ordinates method), it is simply a matter of adding these new equations within the outer iteration loop. Furthermore, since each equation is solved sequentially, the memory that is allocated for storing the coefficient matrix may be reused. This is a major advantage of this approach 5. for high-order approximations and/or nonorthogonal mesh, the reader is referred to texts on numerical solution of partial differential equations, such as [260].

802 Radiative Heat Transfer

FIGURE 21.14 Nodes (left) or a cell (right) on or adjacent to the boundary.

for large-scale computations. Finally, advances in efficient iterative solution of a discretized partial differential equation, i.e., solution of the resulting linear system, can be capitalized upon. Example 21.4. A gas with constant thermophysical properties flows through a short tube of circular cross-section with diameter, D = 0.2 m and length, L = 1 m. The flow is hydrodynamically full-developed—also known as Poiseuille flow (see Section 21.4). In this scenario, the axial component of the velocity is given by u(r) = 2um [1 − (2r/D)2 ], while the radial component is zero. The gas enters at a temperature of Ti = 2000 K, and the wall of the tube has a fixed temperature of Tw = 1000 K. The gas is gray and absorbing-emitting, but not scattering. All surfaces of the tube, including the inlet and the outlet, are black. The thermophysical properties of the gas and its mean velocity, um , are adjusted such that the following conditions are met: (a) the Péclet number, defined as the product of the Reynolds and Prandtl number, Pe = Re Pr = 100, (b) the optical thickness, κD = 1, and (c) the conduction-to-radiation parameter, N = kκ/(4σTw3 ) is in the range 0.01–1, i.e., N is treated as a parameter in the problem. Determine the temperature distribution in the tube at steady state for N equal to 0.01, 0.1, and 1. For these values of N, determine also the local Nusselt number distribution along the wall of the tube. For simplicity, neglect axial conduction. Perform radiation calculations using the P1 approximation. To investigate the possible importance of axial radiation, perform two sets of calculations; one in which axial radiation is neglected (1D), and another in which both axial and radial transport of radiation are accounted for (2D). Compare the results for these two sets of calculations. Solve the problem using the semi-implicit coupling procedure. Solution Assuming that viscous dissipation, compressibility effects, and pressure work are negligible, the overall energy equation at steady state, following equation (9.75), may be written as ρcp v · ∇T = ∇ · (k ∇T) − ∇ · qR . where v is the flow velocity. Since the medium is gray, ∇ · qR = κ(4Eb − G), and the resulting energy equation in cylindrical coordinates (assuming axisymmetric conditions) is   α ∂ ∂T 1 ∂T = r − u(r) κ(4Eb − G). ∂x r ∂r ∂r ρcp In the above equation, axial conduction has been neglected. Thus, heat is carried axially only by the flow, i.e., advection. Conduction and radiation then assist in transporting that heat radially from the hot fluid to the cold walls. The boundary conditions for the above energy equation may be written as Inlet: Axis of symmetry: Tube Wall:

T(0, r) = Ti ,   ∂T = 0, ∂r (x,0) T(x, D/2) = Tw .

Only one boundary condition is needed in the x-direction since the governing energy equation only has a first derivative in x. In order to discretize the governing equations, we will employ the finite difference method. Consequently, the duct is discretized using a uniform 801 (K = 800) × 81 (M = 80) mesh, such that Δx = L/K and Δr = (D/2)/M. For reference, a schematic illustration of the mesh is shown in Fig. 21.15, along with the nodal indexing being used. For a generic interior node (i, j), the following finite difference approximations are written:   Ti,j − Ti−1,j ∂T u ≈ uj , ∂x Δx       Ti,j+1 − 2Ti,j + Ti,j−1 1 ∂ ∂T 1 Ti,j+1 − Ti,j−1 ∂2 T 1 ∂T + r = 2 + ≈ . r ∂r ∂r ∂r r ∂r (Δr)2 rj 2Δr

Radiation Combined with Conduction and Convection Chapter | 21 803

FIGURE 21.15 Schematic illustration of geometry, boundary conditions, and finite-difference mesh for problem considered in Example 21.4.

For the advective term shown in the first equation, an upwind difference scheme [260] is used as opposed to the central difference scheme used for the diffusion terms (last two equations), as is conventionally done in computational fluid dynamics to enhance the stability of the resulting equations during iterative solution. Substituting the above finite-difference approximations, along with the semi-implicit treatment of the radiative source term suggested by equation (21.84) into the above energy equation, we obtain an equation of the same form as equation (21.85). This equation is written as ⎤ ⎡ ∗3 ∗4 16κσTi,j 12κσTi,j ⎥⎥ ⎢⎢ κ T T ⎥⎥ T + AT T ⎢⎢AT + Gi,j + ; ⎥⎦ i,j ⎢⎣ P W i−1,j + AN Ti,j+1 + AS Ti,j−1 = ρcp ρcp ρcp ATP =

uj T uj 2α α α α α ; A = − ; ATN = − ; AT = − . + − + (Δr)2 Δx W Δx (Δr)2 2r j Δr S (Δr)2 2r j Δr

The nodes at the inlet and the tube wall need no special treatment, since the temperatures at these nodes are already known. At the axis of symmetry, the term (∂T/∂r)/r is undefined since both ∂T/∂r and r are equal to zero at r = 0. To circumvent this problem, we take the limit of this term as r tends to zero. Using L’Hospital’s rule, this yields (∂2 T/∂r2 ). To express this in finite-difference form at the centerline, we perform a Taylor series expansion at r = 0 (or j = 0):  Ti,1 = Ti,0 + Δr

∂T ∂r

 + i,0

  (Δr)2 ∂2 T + .... 2 ∂r2 i,0

Substituting the centerline condition [∂T/∂r]i,0 = 0 into the above equation, neglecting higher order terms and rearranging, we obtain  2   ∂T 2  ≈ Ti,1 − Ti,0 . ∂r2 i,0 (Δr)2 All other terms in the governing energy equation for the nodes on the axis can be approximated using the finite-difference approximations provided above for interior nodes. This concludes our discussion on discretization of the energy equation. The RTE, simplified using the P1 approximation, given by equation (15.39) with ω = 0 for a nonscattering medium with constant absorption coefficient, may be written in cylindrical coordinates as ∇2 G =

∂2 G ∂2 G 1 ∂G = −3κ2 (4σT4 − G), + 2 + ∂x2 ∂r r ∂r

subject to the boundary conditions  Inlet:  Outlet:  Axis of symmetry:  Tube Wall:

∂G ∂x ∂G ∂x ∂G ∂r ∂G ∂r

 

+

3 i κ[4σTi4 − G(0, r)] = 0, 2 2 − i

−

3 o κ[4σT(L, r)4 − G(L, r)] = 0, 2 2 − o

(0,r)



(L,r)



(x,0)

= 0, − (x,D/2)

3 w κ[4σTw4 − G(x, D/2)] = 0. 2 2 − w

804 Radiative Heat Transfer

For the 1D approximation, the term ∂2 G/∂x2 in the governing equation is ignored, and boundary conditions need not be applied at the inlet and outlet. Applying the same finite difference approximations provided earlier for the second derivatives of temperature to second derivatives of G, and applying the same linearization procedure that was used to derive equation (21.84), the governing equation for incident radiation in 2D may be written as   ∗3 ∗4 Ti,j − 36κσTi,j ; AGP + 3κ2 Gi,j + AGE Gi+1,j + AGW Gi−1,j + AGN Gi,j+1 + AGS Gi,j−1 = 48κσTi,j AGP =

2 2 1 1 1 1 1 1 ; AG = − . + , AG = − ; AG = − ; AG = − − + (Δx)2 (Δr)2 E (Δx)2 W (Δx)2 N (Δr)2 2r j Δr S (Δr)2 2r j Δr

For the 1D approximation, the resulting finite difference equations are   ∗3 ∗4 AGP + 3κ2 Gi,j + AGN Gi,j+1 + AGS Gi,j−1 = 48κσTi,j Ti,j − 36κσTi,j , AGP =

2 1 1 1 1 ; AG = − . , AG = − − + (Δr)2 N (Δr)2 2r j Δr S (Δr)2 2r j Δr

For the nodes on the axis, the boundary condition is applied in exactly the same manner as described earlier for the energy equation. For the nodes on the other three boundaries, the same procedure needs to be followed. For example, for the top boundary (tube wall), an expression for the second derivative in the radial direction needs to be rederived at the boundary. To do so, we first perform a backward Taylor series expansion as follows:     (Δr)2 ∂2 G ∂G + + .... Gi,M−1 = Gi,M − Δr 2 ∂r i,M ∂r2 i,M Substituting the boundary condition at the top wall into the above equation, followed by rearrangement and neglecting higher order terms yields  2  ∂G 2 3 w = [Gi,M−1 − Gi,M ] + κ[4σTw4 − Gi,M ]. Δr 2 − w ∂r2 i,M (Δr)2 For the first derivative with respect to r, the boundary condition can be substituted directly. The second derivative in the x-direction needs no special treatment at the top boundary. Using the same procedure, the nodal equations for nodes residing on the other boundaries may also be readily derived. All these equations may finally be written in the general form shown above for the interior nodes, with the link coefficients determined after comparison of the general form with the nodal equation in question. The finite difference equations presented earlier for the 1D approximation are parabolic in nature. This means that no downstream information is needed to compute upstream solutions. For example, the discrete form of the energy equation shown earlier may be written for the second column of nodes (i = 1) as ⎤ ⎡ ∗3 ∗4 16κσT1,j 12κσT1,j ⎥⎥ ⎢⎢ κ T T ⎥⎥ T + AT T ⎢⎢AT + + A T = −A T + G + . 1,j+1 1,j−1 0,j 1,j ⎥⎦ 1,j ⎢⎣ P N S W ρcp ρcp ρcp The above equation is tridiagonal and can be solved readily. T0,j represents temperature values at the inlet nodes. Therefore, all terms on the right-hand side of the above equation are known. The 1D version of the discretized P1 equation also has the same tridiagonal form. Thus, solution of both the energy and P1 equations requires just one sweep through the computational domain starting from i = 1 and marching forward to i = K. Iterations are only necessary to couple the two equations with each other and due to the fact that the energy equation is nonlinear. Spatial coupling of the nodes does not require any iterations. In contrast, if axial radiation is taken into account, the governing P1 equation is elliptic. This is indicated by the fact that Gi+1,j appears as an unknown in the finite-difference equations. To solve the 2D finite-difference equations, an iterative solver that couples the nodes spatially is needed. One simple way to attain such coupling is to perform the inlet-to-outlet sweep described earlier for the 1D equations multiple times—often referred to as inner iterations. Since only the P1 equation is elliptic, such inner iterations—typically a few—are only required for this equation. For the energy equation, which is still parabolic in the absence of axial conduction, one sweep is sufficient. The two equations are solved sequentially within an outer iteration loop, as described at the beginning of this section. Since the diameter of the tube is prescribed, the value of κ is adjusted to meet the prescribed value of the optical thickness (=1 in this particular example). Likewise, the value of the thermal conductivity is adjusted to meet the prescribed value of N by inverting its definition, i.e., k = 4NσTw3 /κ. Figure 21.16a shows the nondimensional mean temperature as a function of the axial location for various values of N. For small values of N, radiation is dominant and the mean temperature rapidly approaches the wall temperature. On the other hand, for N = 1, the mean temperature decreases much more gradually, and the distribution resembles one

Radiation Combined with Conduction and Convection Chapter | 21 805

FIGURE 21.16 Comparison of 1D and 2D solutions for the problem considered in Example 21.4: (a) mean temperature distributions, (b) local Nusselt number development.

TABLE 21.2 Iterations needed by the semi-implicit coupling procedure to reduce residuals by 6 orders of magnitude for the problem considered in Example 21.4. N

Iterations for 1D

1

6

24454

0.1

8

28744

0.01

14

39499

Iterations for 2D

that is observed for a case with pure convection (not shown). When 2D effects are accounted for, the medium close to the inlet receives significant radiation from the inlet and heats up. Consequently, the mean temperature near the inlet is larger. The effect is more pronounced for smaller values of N, when radiation is more dominant. One of the important contributions of axial transport of radiation is that it alters the so-called thermal entry length drastically. This is the length from the inlet that the fluid travels before its temperature is within a certain percentage of the wall temperature. It is evident from the two curves for N = 0.01 that the nondimensional mean temperature reaches a value below 0.05 (1050 K) at nondimensional distances of approximately 0.01 and 0.02 for 1D and 2D cases, respectively. Since the Péclet number in this example is 100, this implies that the fluid travels approximately one diameter and two diameters for 1D and 2D cases, respectively, before its temperature is within 5% of the wall temperature. In other words, inclusion of axial radiation transport (2D effect) doubles the thermal entry length. The local Nusselt number distributions for both with (2D) and without (1D) axial radiation are shown in Fig. 21.16b. For all values of N, inclusion of axial radiation increases the local Nusselt number. This is partly because radiation from the hot black inlet directly impinges on the tube walls, and partly because the temperature of the fluid adjacent to the walls is increased due to absorption of radiation from the inlet, resulting in a larger temperature difference between the fluid and the wall. This, in turn, increases the radiation flux. Also, it is observed that the Nusselt number increases near the outlet, particularly for lower values of N. This is because extra radiation from the outlet (which is hotter than the wall and is black) impinges on the sections of the wall closer to the outlet. This effect is also present at larger values of N. However, since the radiation flux is much smaller than the conductive flux at the wall, the observable increase in the Nusselt number is only minor. The local Nusselt number values shown in Fig. 21.16 for the 1D case are approximately twice as large as those shown in Fig. 21.11. For example, for optical thickness of unity and N = 0.1, at a nondimensional axial location of 0.0005, the computed Nusselt number for a tube is 31.42, while that for a planar channel is 15.68. For a planar channel, the Nusselt number is based on the height of the channel. If instead, the hydraulic diameter of the channel, which is twice the channel height, were to be used, the Nusselt number would be 31.36, i.e., Nusselt numbers for parallel ducts and cylindrical tubes are very similar. The computational performance of the semi-implicit coupling procedure is summarized in Table 21.2. As discussed earlier, in the 1D case, the problem is parabolic and the purpose of iterations is only to couple the energy equation with the P1 equation and to address nonlinearities. In contrast, in the 2D case, the problem is elliptic, and additional iterations

806 Radiative Heat Transfer

are needed to couple spatial nodes. Hence, 1D calculations require few iterations, while 2D calculations require a larger number of iterations since a 801 × 81 mesh is used in this particular case. In both cases, as radiation becomes dominant over conduction and the problem becomes increasingly nonlinear, more iterations are needed to attain convergence. The Fortran90 program for the 2D solution to this example problem, CpldP1En2D.f90, is also provided in Appendix F.

Although fully implicit coupling to solve partial differential equations has found prolific use in other disciplines such as computational fluid dynamics (especially for reacting flows) and computational electromagnetics, only a handful of works are found in the literature when it comes to coupling of the overall energy equation with the RTE. This can only be implemented once both equations have been discretized and recast into their algebraic forms. Therefore, the exact formulation for coupling is very much dependent upon the RTE solver being used. In the late 1990s, Murthy and Mathur [261,262] proposed coupled solutions of the directional RTEs resulting from the discrete ordinates method and the energy equation—a method referred to by the authors as the coupled ordinates method (or COMET). In their approach, a block Gauss-Seidel procedure was used as a smoother within a multigrid algorithm. The method is attractive because of the inter-equation coupling manifested by the “block” formulation, and quick information propagation through the grid because of the multigrid algorithm. Mazumder [263] demonstrated fully implicit coupling of the energy equation with nongray radiation in a two-dimensional pipe flow using the block-Alternating Direction Implicit (block-ADI) method. Mazumder and coworkers [264] also demonstrated fully-implicit coupling of the energy equation with the P3 equations on an unstructured two-dimensional mesh. In their work, Krylov subspace solvers were used to solve the linear system of coupled equations.

Problems 21.1 A vat of molten glass is heated from below by a gray, diffuse surface with T = 1800 K and = 0.8. The glass layer is 1 m thick, and its top is exposed to free convection and radiation with an ambient space at 1000 K (heat transfer coefficient for free convection = 5 W/m2 K). Neglecting convection within the melt, estimate the temperature distribution within the glass, using the radiative properties of glass as given in Figs. 1.17 and 3.16. What is the total heat loss from the bottom surface? 21.2 Estimate the total heat flux for Problem 21.1, as well as the glass–air interface temperature, by using the additive solution method. 21.3 A glass sphere (D = 4 cm) initially at uniform temperature Ti = 300 K is placed into a furnace, whose walls and inert gas are at a uniform T g = Tw = 1500 K. Assuming the glass to be gray and nonscattering (κ = 1 cm−1 , n = 1.5, k = 1.5 W/m K) and a sphere/furnace gas heat transfer coefficient of 10 W/m2 K, determine the sphere’s temperature distribution as a function of time. 21.4 A 1 cm thick quartz window (assumed gray with κ = 1 cm−1 and n = 1.5) forms the barrier between a furnace and the ambient, resulting in face temperatures of T1 = 800 K and T2 = 400 K. Estimate the conductive, radiative, and total heat fluxes passing through the window (k = 1.5 W/m K). 21.5 Repeat Problem 5.36 for the case in which a gray, isotropically scattering, stationary gas (κ = 2 cm−1 , k = 0.04 W/m K) is filling the 1 cm thick gap between surface and shield. 21.6 A sheet of ice 20 cm thick is lying on top of black soil. Initially, ice and soil are at −10◦ C when the sun begins to shine, hitting the top of the ice with a strength of 800 W/m2 (normal to the rays), at an off-normal angle of 30◦ . Assume the ground to be insulated, ice and water to have constant and equal properties (k, ρ, cp ), a gray absorption coefficient (for solar light) of κ 1 cm−1 , and a gray reflectance of 0.02. Neglecting emission from and scattering by the ice, as well as convection losses/gains at the surface, determine the transient temperature distribution within the ice/water until the time when all ice has melted. 21.7 Consider a gray medium separating an axle from its bearing. The gap is so narrow that the movement between axle and bearing may be approximated by Couette flow (two infinite parallel plates, one stationary, and the other moving at constant velocity U). The movement is so rapid that viscous dissipation must be considered [Φ = (∂u/∂y)2 , where u = u(y) is the velocity at a distance y from the lower, stationary plate]. The medium is gray and nonscattering with a constant absorption coefficient, and both surfaces are isothermal (at different temperatures) and gray-diffuse. Set up the necessary equations and boundary conditions to calculate the net heat transfer rates on the two surfaces.

Radiation Combined with Conduction and Convection Chapter | 21 807

21.8 Consider a solar water heater as shown in the adjacent sketch. A 5 mm thick layer of water is flowing down a black, insulated plate as shown while exposed to sunshine. The water is seeded with a fine powder that gives it a gray absorption coefficient of κ = 5 cm−1 . The top of the water layer loses heat by free convection (h = 10 W/m2 K) to the ambient at Tamb = 300 K. At the top of the collector (x = 0) the water enters at a uniform temperature of T0 = 300 K. The velocity profile may be considered fully developed everywhere. Determine the cumulative collected solar energy as a function of x. 21.9 Consider a gas–particulate mixture flowing through an isothermal tube ( w = 1, Tw = 400 K). The gas is radiatively nonparticipating and has constant velocity u across the tube cross-section such that Pe = Re Pr = uD/α = 30,000. The ˙ p )particles /(mc ˙ p )gas = 2. particles are very small, gray, and uniformly distributed such that κp R = 5 (no scattering) and (mc The particles are so small that they are essentially at the same temperature as the gas surrounding them. Using the diffusion approximation for the radiative heat transfer, set up the relevant equations and boundary conditions for the calculation of local bulk temperature and local total heat flux. Obtain a numerical solution (after neglecting axial conduction and radiation), and prepare a plot for local mean temperature and Nusselt number. 21.10 Consider a long straight semitransparent cable of circular cross-section connected between two walls. For simplicity, it may be assumed that the length of the cable is very large compared to its diameter, i.e., L/D  1. The two ends of the cable have fixed temperatures equal to T1 and T2 . The outer surface of the cable is exposed to ambient conditions with a fixed convective heat transfer coefficient ho and temperature To . The thermal conductivity of the cable is denoted by k, and may be assumed to be constant. The cable may be treated as nonscattering and gray, with absorption coefficient κ and refractive index of m n = 1.5. The cable is coated with a material, which is highly reflective on its inside, so that radial radiation maybe neglected, and has an emittance of o on its outside facing the ambient. (a) Using an energy balance, derive an appropriate differential equation that will allow you to determine the temperature distribution along the length of the cable at steady state. State the boundary conditions that you will use to obtain a solution to this equation. (b) Develop a computer program to determine the temperature distribution along the cable using the P1 approximation to treat radiation. You may use the program CpldP1En1D.f90, provided in Appendix F, as a starting point. Use the following numerical values for your computer program: D = 10 mm, L = 1 m, k = 2 W/mK, T1 = 400 K, T2 = 350 K, o = 0.2, ho = 1 W/m2 K, and To = 290 K. The two ends may be assumed black. Adjust the value of κ to vary the optical thickness (= κD). Perform the same calculation for optical thicknesses of 0.001, 0.01, and 0.1, and plot the temperature distributions on the same graph for the three cases. How will the optical thickness of the cable alter the heat loss from the cable? (c) Perform the calculations for Part (b) using both the explicit and the implicit coupling procedures, and compare and contrast the convergence of the two methods. 21.11 Consider an absorbing–emitting gas confined between two black infinite parallel plates, separated by a distance L. Assume that conduction and radiation are the only two modes of heat transfer in the gas. The thermal conductivity of the gas is denoted by k, and its absorption coefficient by κ. One of the plates is held at a constant temperature T1 , while the other plate is exposed to the ambient at temperature To . The external convective heat transfer coefficient is ho . Neglect external radiation. Using the finite angle method (FAM; Section 16.6) to solve the RTE, develop a set of discrete equations starting from the governing RTE and overall energy equation after breaking up the space between the plates into N control volumes. You may use Example 16.7 as a guide to develop the FAM equations with just two solid angles. Write a computer program to find the steady-state temperature distribution, T(z), in the gas layer for values of the optical thickness, κL, equal to 0 (nonparticipating gas), 0.1, 1, and 10. Use T1 = 1000 K, To = 300 K, ho = 100 W/m2 K, k = 0.03 W/mK, and L = 1 m for numerical calculations. For κL = 0, make qualitative sketches of how the temperature distribution may progress with time if the initial temperature of the gas is To . Compare the evolution of the temperature distribution with and without radiation and comment on your results. 21.12 Consider a long commercial cooking oven of square cross-section (L × L in the xy-plane). The gas within has thermal conductivity, k, absorption coefficient, κ, and you may neglect any scattering in the oven. The outside of the oven is exposed on three sides (top, and all sides) to the ambient at temperature, To . The external convective heat transfer coefficient is ho , and external radiation may be neglected. The emittances of the internal walls of the oven, which are diffuse, are denoted by i . The oven is heated by coils in its base, which may be approximated by using a constant uniform heat flux, qw . Assuming that the oven may be modeled as a 2D planar system (cross-sectional plane), write down the governing overall energy equation and the RTE, along with associated boundary conditions that you will need to determine the steady state temperature distribution, T(x, y), within the oven. Assume that the flow distribution, caused by free convection within the oven, is known and does not change, and the thickness of the oven walls is negligible. Using the finite angle method (FAM; Section 16.6) for angular discretization of the RTE and the finite-volume method for spatial discretization of both the RTE and the energy equation, derive a set of discrete equations that are needed to develop a computer program to solve this problem. Note that your

808 Radiative Heat Transfer

discrete equations must incorporate appropriate boundary conditions. You may use Example 16.8 to guide the development of the FAM equations. Briefly discuss the solution procedure for the equations that you have developed. Specifically, discuss how you will couple the discrete form of the RTE with the discrete form of the overall energy equation. 21.13 Repeat Example 21.4 but with constant heat flux of qw = 5000 W/m2 being removed at the tube wall (rather than fixed temperature). You may either develop your own computer program from ground up, or use CpldP1En2D.f90, provided in Appendix F, as a starting point. 21.14 Consider a nonscattering molten salt with absorption coefficient, κ, and thermal conductivity, k, contained in a tall vessel of negligible wall thickness, which may be modeled as a 1D slab of thickness 2L. The internal surface of the vessel has an emittance equal to i and is cooled externally by convection with heat transfer coefficient ho to an ambient temperature To . External radiation may be neglected. During electrolysis, current transport through the salt causes heat generation at a uniform rate of Q˙  . Any motion of the molten salt due to natural convection may be neglected. (a) Write down the governing energy equation and boundary conditions needed to determine the steady state temperature distribution in the slab. Also, write down the governing equation and boundary conditions for radiative transfer using the P1 approximation. You may invoke symmetry at the center to simplify the problem. (b) Convert your governing equations to nondimensional form using the parameters defined in Section 21.2. Hint: In addition to the parameters already defined, the Biot number, Bi = ho L/k, should also appear as a parameter. (c) Write a computer program to obtain the temperature distribution in the slab for various values of i and Biot number. You may use the program CpldP1En1D.f90, provided in Appendix F, as a guide or as a starting point. Use a nondimensional heat generation rate Q˙  L/σTo4 = 1 for your calculations. Also use a conduction-to-radiation parameter of N = 1, as defined in equation (21.5), and an optical thickness of κL = 1.

References [1] Z.F. Shen, T.F. Smith, P. Hix, Linearization of the radiation terms for improved convergence by use of the zone method, Numerical Heat Transfer 6 (1983) 377–382. [2] C.R. Wylie, Advanced Engineering Mathematics, 5th ed., McGraw-Hill, New York, 1982. [3] R. Viskanta, R.J. Grosh, Effect of surface emissivity on heat transfer by simultaneous conduction and radiation, International Journal of Heat and Mass Transfer 5 (1962) 729–734. [4] R. Viskanta, R.J. Grosh, Heat transfer by simultaneous conduction and radiation in an absorbing medium, ASME Journal of Heat Transfer 84 (1963) 63–72. [5] W. Lick, Energy transfer by radiation and conduction, in: Proceedings of the Heat Transfer and Fluid Mechanics Institute, Stanford University Press, Palo Alto, California, 1963, pp. 14–26. [6] S. André, A. Degiovanni, A theoretical study of the transient coupled conduction and radiation heat transfer in glass: phonic diffusivity measurements by the flash technique, International Journal of Heat and Mass Transfer 38 (1995) 3401–3412. [7] U. Heinemann, R. Caps, J. Fricke, Radiation–conduction interaction: an investigation on silica aerogels, International Journal of Heat and Mass Transfer 39 (1996) 2115–2130. [8] S.S. Manohar, A.K. Kulkarni, S.T. Thynell, In-depth absorption of externally incident radiation in nongray media, ASME Journal of Heat Transfer 117 (1) (1995) 146–151. [9] A. Soufiani, J.-M. Hartmann, J. Taine, Validity of band-model calculations for CO2 and H2 O applied to radiative properties and conductive–radiative transfer, Journal of Quantitative Spectroscopy and Radiative Transfer 33 (1985) 243–257. [10] C. Yao, B.T.F. Chung, Transient heat transfer in a scattering–radiating–conducting layer, Journal of Thermophysics and Heat Transfer 13 (1) (1999) 18–24. [11] E.M. Abulwafa, Conductive–radiative heat transfer in an inhomogeneous slab with directional reflecting boundaries, Journal of Physics D: Applied Physics 32 (1999) 1626–1632. [12] H.P. Tan, L.M. Ruan, X.L. Xia, T.W. Tong, Transient coupled radiative and conductive heat transfer in an absorbing, emitting and scattering medium, International Journal of Heat and Mass Transfer 42 (1999) 2967–2980. [13] N. Ruperti Jr, M. Raynaud, J.-F. Sacadura, A method for the solution of the coupled inverse heat conduction–radiation problem, ASME Journal of Heat Transfer 118 (1996) 10–17. [14] C.Y. Wu, N.R. Ou, Transient two-dimensional radiative and conductive heat transfer in a scattering medium, International Journal of Heat and Mass Transfer 37 (17) (1994) 2675–2686. [15] A. Tuntomo, C.L. Tien, Transient heat transfer in a conducting particle with internal radiant absorption, ASME Journal of Heat Transfer 114 (1992) 304–309. [16] R. Siegel, Transient thermal effects of radiant energy in translucent materials, ASME Journal of Heat Transfer 120 (1) (1998) 5–23. [17] L.S. Wang, C.L. Tien, Study of the interaction between radiation and conduction by a differential method, in: Proceedings of the Third International Heat Transfer Conference, vol. 5, Hemisphere, Washington, D.C., 1966, pp. 190–199.

Radiation Combined with Conduction and Convection Chapter | 21 809

[18] T.H. Einstein, Radiant heat transfer to absorbing gases enclosed between parallel flat plates with flow and conduction, NASA TR R-154, 1963. [19] R.D. Cess, The interaction of thermal radiation with conduction and convection heat transfer, in: Advances in Heat Transfer, vol. 1, Academic Press, New York, 1964, pp. 1–50. [20] S.Q. Zeng, A.J. Hunt, R. Greif, W. Cao, Approximate formulation for coupled conduction and radiation through a medium with arbitrary optical thickness, ASME Journal of Heat Transfer 117 (1995) 797–799. [21] J.R. Howell, Determination of combined conduction and radiation of heat through absorbing media by the exchange factor approximation, Chemical Engineering Progress Symposium Series 61 (59) (1965) 162–171. [22] L.S. Wang, C.L. Tien, A study of various limits in radiation heat-transfer problems, International Journal of Heat and Mass Transfer 10 (1967) 1327–1338. [23] Y.P. Chang, A potential treatment of energy transfer by conduction, radiation, and convection, AIAA Journal 5 (1967) 1024–1026. [24] C.L. Tien, P.S. Jagannathan, B.F. Armaly, Analysis of lateral conduction and radiation along two parallel long plates, AIAA Journal 7 (1969) 1806–1808. [25] E.E. Anderson, R. Viskanta, W.H. Stevenson, Heat transfer through semitransparent solids, ASME Journal of Heat Transfer 95 (2) (1973) 179–186. [26] R. Viskanta, Heat transfer by conduction and radiation in absorbing and scattering materials, ASME Journal of Heat Transfer 87 (1965) 143–150. [27] J.B. Bergquam, R.A. Seban, Heat transfer by conduction and radiation in absorbing and scattering materials, ASME Journal of Heat Transfer 93 (1971) 236–238. [28] A. Dayan, C.L. Tien, Heat transfer in a gray planar medium with linear anisotropic scattering, ASME Journal of Heat Transfer 97 (1975) 391–396. [29] J.A. Roux, A.M. Smith, Combined conductive and radiative heat transfer in an absorbing scattering infinite slab, ASME Journal of Heat Transfer 100 (1) (1978) 98–104. [30] W.W. Yuen, L.W. Wang, Heat transfer by conduction and radiation in a one-dimensional absorbing, emitting and anisotropically scattering medium, ASME Journal of Heat Transfer 102 (1980) 303–307. [31] A.C. Ratzel, J.R. Howell, Heat transfer by conduction and radiation in one-dimensional planar medium using differential approximation, ASME Journal of Heat Transfer 104 (1982) 388–391. [32] C.H. Ho, M.N. Özi¸sik, Combined conduction and radiation in a two-layer planar medium with flux boundary condition, Numerical Heat Transfer 11 (3) (1987) 321. [33] J.H. Tsai, J.D. Lin, Transient combined conduction and radiation with anisotropic scattering, Journal of Thermophysics and Heat Transfer 4 (1) (1990) 92–97. [34] K. Kamiuto, M. Iwamoto, Y. Nagumo, Combined conduction and correlated–radiation heat transfer in packed beds, Journal of Thermophysics and Heat Transfer 7 (3) (1993) 496–501. [35] R. Siegel, C.M. Spuckler, Approximate solution methods for spectral radiative transfer in high refractive index layers, International Journal of Heat and Mass Transfer 37 (1994) 403–413. [36] D. Doermann, J.-F. Sacadura, Heat transfer in open cell foam insulation, ASME Journal of Heat Transfer 118 (1996) 88–93. [37] P.D. Jones, D.G. McLeod, D.E. Dorai-Raj, Correlation of measured and computed radiation intensity exiting a packed bed, ASME Journal of Heat Transfer 118 (1996) 94–102. [38] R. Siegel, Radiative exchange in a parallel-plate enclosure with translucent protective coatings on its walls, International Journal of Heat and Mass Transfer 42 (1) (1999) 73–84. [39] J.W. Wu, H.S. Chu, Combined conduction and radiation heat transfer in plane-parallel packed beds with variable porosity, Journal of Quantitative Spectroscopy and Radiative Transfer 61 (4) (1999) 443–452. [40] M. Lazard, S. André, A.D. Maillet, Transient coupled radiative–conductive heat transfer in a gray planar medium with anisotropic scattering, Journal of Quantitative Spectroscopy and Radiative Transfer 69 (2001) 23–33. [41] R. Greif, Energy transfer by radiation and conduction with variable gas properties, International Journal of Heat and Mass Transfer 7 (1964) 891–900. [42] W. Lick, Transient energy transfer by radiation and conduction, International Journal of Heat and Mass Transfer 8 (1965) 119–127. [43] R. Echigo, S. Hasegawa, Y. Miyazaki, Composite heat transfer with thermal radiation in nongray medium: Part I: interaction of radiation with conduction, International Journal of Heat and Mass Transfer 14 (1971) 2001–2015. [44] R. Viskanta, D.M. Kim, Heat transfer through irradiated semi-transparent layer at high temperature, ASME Journal of Heat Transfer 102 (1980) 388–390. [45] T.F. Smith, A.M. Al-Turki, K.H. Byun, T.K. Kim, Radiative and conductive transfer for a gas/soot mixture between diffuse parallel plates, Journal of Thermophysics and Heat Transfer 1 (1) (1987) 50–55. [46] K. Kamiuto, Combined conduction and nongray radiation heat transfer in carbon dioxide, Journal of Thermophysics and Heat Transfer 10 (4) (1996) 701–704. [47] K.H. Lee, R. Viskanta, Comparison of the diffusion approximation and the discrete ordinates method for the investigation of heat transfer in glass, Glass Science and Technology 72 (1999) 254–265. [48] T.J. Moore, M.R. Jones, Analysis of the conduction-radiation problem in absorbing, emitting, non-gray planar media using an exact method, International Journal of Heat and Mass Transfer 73 (2014) 804–809.

810 Radiative Heat Transfer

[49] Y. Sun, X. Zhang, J.R. Howell, Non-gray combined conduction and radiation heat transfer by using FVM and SLW, Journal of Quantitative Spectroscopy and Radiative Transfer 197 (2017) 51–59. [50] B.C.H. Wendlandt, Temperature in an irradiated thermally conducting medium, Journal of Physics D: Applied Physics 6 (1973) 657–660. [51] R.R. Gilpin, R.B. Roberton, B. Singh, Radiative heating in ice, ASME Journal of Heat Transfer 99 (1977) 227–232. [52] R. Viskanta, E.D. Hirleman, Combined conduction–radiation heat transfer through an irradiated semitransparent plate, ASME Journal of Heat Transfer 100 (1978) 169–172. [53] R.A. Zakhidov, S.Y. Bogomolov, D.A. Kirgizbaev, S.I. Klychev, Determination of the temperature field of semitransparent materials when they are heated with optical radiant, Applied Solar Energy 23 (6) (1987) 36. [54] G.J. Kowalski, Transient response of an optically thick medium exposed to short pulses of laser radiation, in: Fundamentals and Applications in Radiation Heat Transfer, vol. HTD-72, ASME, 1987, pp. 67–74. [55] H.P. Tan, B. Maestre, M. Lallemand, Transient and steady-state combined heat transfer in semi-transparent materials subjected to a pulse or a step irradiation, ASME Journal of Heat Transfer 113 (1) (1991) 166–173. [56] O. Hahn, F. Raether, M.C. Arduini-Schuster, Transient coupled conductive/radiative heat transfer in absorbing, emitting and scattering media: application to laser-flash measurements on ceramic materials, International Journal of Heat and Mass Transfer 40 (1997) 689–698. [57] S. André, A. Degiovanni, A new way of solving transient radiative–conductive heat transfer problems, ASME Journal of Heat Transfer 120 (4) (1998) 943–955. [58] L.H. Liu, H.P. Tan, T.W. Tong, Non-Fourier effects on transient temperature response in semitransparent medium caused by laser pulse, International Journal of Heat and Mass Transfer 44 (2001) 3335–3344. [59] R. Fernandes, J. Francis, J.N. Reddy, A finite-element approach to combined conductive and radiative heat transfer in a planar medium, in: Heat Transfer and Thermal Control, Progress in Aeronautics and Astronautics, vol. 78, AIAA, New York, 1981, pp. 92–109. [60] A.S. Hazzak, J.V. Beck, Unsteady combined conduction–radiation energy transfer using a rigorous differential method, International Journal of Heat and Mass Transfer 13 (1970) 517–522. [61] C.C. Lii, M.N. Özi¸sik, Transient radiation and conduction in an absorbing, emitting, scattering slab with reflective boundaries, International Journal of Heat and Mass Transfer 15 (1972) 1175–1179. [62] K.C. Weston, J.L. Hauth, Unsteady, combined radiation and conduction in an absorbing, scattering, and emitting medium, ASME Journal of Heat Transfer 95 (1973) 357–364. [63] L.K. Matthews, R. Viskanta, F.P. Incropera, Combined conduction and radiation heat transfer in porous materials heated by intense solar radiation, Solar Energy 107 (1) (1985) 29. [64] D.E. Glass, M.N. Özi¸sik, D.S. McRae, Hyperbolic heat conduction with radiation in an absorbing and emitting medium, Numerical Heat Transfer 12 (3) (1987) 321. [65] B.W. Webb, R. Viskanta, Crystallographic effects during radiative melting of semitransparent materials, Journal of Thermophysics and Heat Transfer 1 (4) (1987) 313–320. [66] L.H. Liu, H.P. Tan, Non-Fourier effects on transient coupled radiative–conductive heat transfer in one-dimensional semitransparent medium subjected to periodic irradiation, Journal of Quantitative Spectroscopy and Radiative Transfer 71 (2001) 11–24. [67] C.J. Tseng, H.S. Chu, Transient combined conduction and radiation in an absorbing, emitting and anisotropically-scattering medium with variable thermal conductivity, International Journal of Heat and Mass Transfer 35 (1992) 1844–1847. [68] H.Y. Li, Estimation of thermal properties in combined conduction and radiation, International Journal of Heat and Mass Transfer 42 (3) (1999) 565–572. [69] H.M. Park, T.Y. Yoon, Solution of the inverse radiation problem using a conjugate gradient method, International Journal of Heat and Mass Transfer 43 (10) (2000) 1767–1776. [70] T.C. Chawla, S.H. Chan, Solution of radiation–conduction problems with collocation method using B-splines as approximating functions, International Journal of Heat and Mass Transfer 22 (12) (1979) 1657–1667. [71] K. Kamiuto, M. Iwamoto, Combined conductive and radiative heat transfer through a glass particle layer, in: Proceedings of the Second ASME/JSME Conference, vol. 4, 1987, pp. 77–84. [72] S.T. Wu, R.E. Ferguson, L.L. Altigilbers, Application of finite-element techniques to the interaction of conduction and radiation in a participating medium, in: Heat Transfer and Thermal Control, Progress in Aeronautics and Astronautics, vol. 78, AIAA, New York, 1981, pp. 61–69. [73] I.F. Pinheiro, L.A. Sphaier, L.S. de, B. Alves, Integral transform solution of integro-differential equations in conduction-radiation problems, Numerical Heat Transfer – Part A: Applications 73 (2018) 94–114. [74] V.A. Petrov, Combined radiation and conduction heat transfer in high temperature fiber thermal insulation, International Journal of Heat and Mass Transfer 40 (1997) 2241–2247. [75] F.T. Lentes, N. Siedow, Three-dimensional radiative heat transfer in glass cooling processes, Glass Science and Technology: Glastechnische Berichte 72 (6) (1999) 188–196. [76] A. Campo, A. Tremante, Two-flux model applied to combined conduction–radiation in a gray planar medium, Wärme- und Stoffübertragung 21 (4) (1987) 221. [77] A. Tremante, F. Malpica, Analysis of the temperature profile of ceramic composite materials exposed to combined conduction–radiation between concentric cylinders, Journal of Engineering for Gas Turbines and Power 120 (2) (1998) 271–275.

Radiation Combined with Conduction and Convection Chapter | 21 811

[78] J.R. Hutchison, R.F. Richards, Effect of nongray gas radiation on thermal stability in carbon dioxide, Journal of Thermophysics and Heat Transfer 13 (1) (1999) 25–32. [79] Y. Sun, X. Zhang, J.R. Howell, Evaluation of three different radiative transfer equation solvers for combined conduction and radiation heat transfer, Journal of Quantitative Spectroscopy and Radiative Transfer 184 (2016) 262–273. [80] S.W. Baek, T.Y. Kim, J.S. Lee, Transient cooling of a finite cylindrical medium in the rarefied cold environment, International Journal of Heat and Mass Transfer 36 (1993) 3949–3956. [81] M. Sakami, A. Charette, V. Le Dez, Application of the discrete ordinates method to combined conductive and radiative heat transfer in a two-dimensional complex geometry, Journal of Quantitative Spectroscopy and Radiative Transfer 56 (4) (1996) 517–533. [82] K.H. Lee, R. Viskanta, Transient conductive–radiative cooling of an optical quality glass disk, International Journal of Heat and Mass Transfer 41 (1998) 2083–2096. [83] S.C. Mishra, M.Y. Kim, S. Maruyama, Performance evaluation of four radiative transfer methods in solving multidimensional radiation and/or conduction heat transfer problems, International Journal of Heat and Mass Transfer 55 (2012) 5819–5835. [84] M. Mishkin, G.J. Kowalski, Application of Monte Carlo techniques to the steady state radiative and conductive heat transfer through a participating medium, ASME Paper No. 83-WA/HT-27, 1983. [85] N.M. Kholodov, Z.H. Flom, P.S. Koltun, Calculation of radiative conductive transfer in a semitransparent plate by the Monte Carlo method, Journal of Engineering Physics 42 (1982) 333–338. [86] A.A. Abed, J.-F. Sacadura, A Monte Carlo–finite difference method for coupled radiation–conduction heat transfer in semi-transparent media, ASME Journal of Heat Transfer 105 (4) (1983) 931. [87] T. Götz, Coupling heat conduction and radiative transfer, Journal of Quantitative Spectroscopy and Radiative Transfer 72 (2002) 57–73. [88] P. Furmanski, ´ P. Łapka, Evaluation of a human skin surface temperature for the protective clothing – skin system based on the protective clothing-skin imitating material results, International Journal of Heat and Mass Transfer 114 (2017) 1331–1340. [89] M. Nishimura, M. Hasatani, S. Sugiyama, Simultaneous heat transfer by radiation and conduction: high-temperature one-dimensional heat transfer in molten glass, International Chemical Engineering 8 (1968) 739–745. [90] N.D. Eryou, L.R. Glicksman, An experimental and analytical study of radiative and conductive heat transfer in molten glass, ASME Journal of Heat Transfer 94 (2) (1972) 224–230. [91] P. Scheuerpflug, R. Caps, D. Buettner, J. Fricke, Apparent thermal conductivity of evacuated, SiO2 aerogel tiles under variation of radiative boundary conditions, International Journal of Heat and Mass Transfer 28 (12) (1985) 2299–2306. [92] R.L. Houston, S.A. Korpela, Heat transfer through fiberglass insulation, in: Proceedings of the Seventh International Heat Transfer Conference, Munich, vol. 2, Hemisphere, Washington, D.C., 1982, pp. 499–504. [93] K. Kamiuto, Y. Miyoshi, I. Kinoshita, S. Hasegawa, Combined conductive and radiative heat transfer in an optically thick porous body (case of cordierite porous bodies), Bulletin of the JSME 27 (1984) 1136. [94] S. Ackermann, M. Takacs, J. Scheffe, A. Steinfeld, Reticulated porous ceria undergoing thermochemical reduction with high-flux irradiation, International Journal of Heat and Mass Transfer 107 (2017) 439–449. [95] W.P. Schimmel, J.L. Novotny, F.A. Olsofka, Interferometric study of radiation–conduction interaction, in: Proceedings of the Fourth International Heat Transfer Conference, Elsevier, New York, September 1970. [96] R. Viskanta, R.L. Merriam, Heat transfer by combined conduction and radiation between concentric spheres separated by radiating medium, ASME Journal of Heat Transfer 90 (1968) 248–256. [97] S.T. Thynell, Interaction of conduction and radiation in anisotropically scattering, spherical media, Journal of Thermophysics and Heat Transfer 4 (3) (1990) 299–304. [98] G. Miliauskas, Regularities of unsteady radiative–conductive heat transfer in evaporating semitransparent liquid droplets, International Journal of Heat and Mass Transfer 44 (2001) 785–798. [99] L.H. Liu, H.P. Tan, T.W. Tong, Transient coupled radiation–conduction in semitransparent spherical particle, Journal of Thermophysics and Heat Transfer 16 (1) (2002) 43–49. [100] F. Gordoninejad, J. Francis, A finite difference solution to transient combined conductive and radiative heat transfer in an annular medium, ASME Journal of Heat Transfer 106 (4) (1984) 888–891. [101] T.K. Kim, T.F. Smith, Radiative and conductive transfer for a real gas cylinder enclosure with gray walls, International Journal of Heat and Mass Transfer 28 (12) (1985) 2269–2277. [102] J.R. Tsai, M.N. Özi¸sik, Transient combined conduction and radiation in an absorbing, emitting, and isotropically scattering solid cylinder, Journal of Applied Physics 64 (8) (1988) 3820. [103] D.W. Amlin, S.A. Korpela, Influence of thermal radiation on the temperature distribution in semi-transparent solid, ASME Journal of Heat Transfer 102 (1980) 76–80. [104] A. Yücel, M.L. Williams, Heat transfer by combined conduction and radiation in axisymmetric enclosures, Journal of Thermophysics and Heat Transfer 1 (4) (1987) 301–306. [105] C.H. Ho, M.N. Özi¸sik, Combined conduction and radiation in a two-dimensional rectangular enclosure, Numerical Heat Transfer 13 (2) (1988) 229. [106] W.W. Yuen, E.E. Takara, Analysis of combined conductive–radiative heat transfer in a two-dimensional rectangular enclosure with a gray medium, ASME Journal of Heat Transfer 110 (2) (1988) 468–474. [107] S.W. Baek, T.Y. Kim, The conductive and radiative heat transfer in rectangular enclosure using the discrete ordinates method, in: Proceedings of the Ninth International Heat Transfer Conference, Hemisphere, Washington, D.C., 1990, pp. 433–438.

812 Radiative Heat Transfer

[108] P.-F. Hsu, Z.M. Tan, Radiative and combined-mode heat transfer within L-shaped nonhomogeneous and nongray participating media, Numerical Heat Transfer – Part A: Applications 31 (8) (1997) 819–835. [109] H.M. Park, T.H. Kim, J.H. Lee, Dynamic simulation of thermal radiation in participating media by means of mode reduction, Journal of Quantitative Spectroscopy and Radiative Transfer 62 (1999) 141–161. [110] Y.-S. Sun, J. Ma, B.-W. Li, Chebyshev collocation spectral method for three-dimensional transient coupled radiative-conductive heat transfer, ASME Journal of Heat Transfer 134 (2012) 092701. [111] S.C. Mishra, R. Muthukumaran, S. Maruyama, Comparison of the thermal effects of the transport of a short-pulse laser and a multi-pulse laser through a participating medium, International Journal of Heat and Mass Transfer 55 (2012) 5583–5596. [112] J.H. Randrianalisoa, L.A. Dombrovsky, W. Lipinski, ´ V. Timchenko, Effects of short-pulsed laser radiation on transient heating of superficial human tissues, International Journal of Heat and Mass Transfer 78 (2014) 488–497. [113] I.S. Habib, Solidification of semi-transparent materials by conduction and radiation, International Journal of Heat and Mass Transfer 14 (1971) 2161–2164. [114] I.S. Habib, Solidification of a semi-transparent cylindrical medium by conduction and radiation, ASME Journal of Heat Transfer 95 (1973) 37–41. [115] M. Abrams, R. Viskanta, The effects of radiative heat transfer upon the melting and solidification of semi-transparent crystals, ASME Journal of Heat Transfer 96 (1974) 184–190. [116] R. Viskanta, E.E. Anderson, Heat transfer in semi-transparent solids, in: Advances in Heat Transfer, vol. 11, Academic Press, New York, 1975, pp. 317–441. [117] C. Cho, M.N. Özi¸sik, Effects of radiation on melting of a semi-transparent, semi-infinite medium, in: Proceedings of the Sixth International Heat Transfer Conference, vol. 3, Hemisphere, Washington, D.C., 1978, pp. 373–378. [118] N. Seki, M. Sugawara, S. Fukusako, Radiative melting of horizontal clear ice layer, Wärme- und Stoffübertragung 11 (1978) 207–216. [119] N. Seki, M. Sugawara, S. Fukusako, Radiative melting of ice layer adhering to a vertical surface, Wärme- und Stoffübertragung 12 (1979) 137–144. [120] N. Seki, M. Sugawara, S. Fukusako, Back melting of a horizontal cloudy ice layer with radiative heating, ASME Journal of Heat Transfer 101 (1979) 90–95. [121] L.A. Diaz, R. Viskanta, Melting of a semitransparent material by irradiation from an external radiation source, in: Spacecraft Radiative Heat Transfer and Temperature Control, vol. 83, AIAA, New York, 1982, pp. 38–60. [122] L.A. Diaz, R. Viskanta, Radiation induced melting of a semitransparent phase change material, AIAA Paper No. 82-0848, 1982. [123] R. Viskanta, X. Wu, Effect of radiation on the melting of glass batch, Glastechnische Berichte 56 (1983) 138–147. [124] N.E. Dorsey, Properties of Ordinary Water Substance, Hafner, New York, 1963, p. 404. [125] C.A. Knight, The Freezing of Supercooled Liquids, Van Nostrand, Princeton, 1967, p. 125. [126] S.H. Chan, D.H. Cho, G. Kocamustafaogullari, Melting and solidification with internal radiative transfer – a generalized phase change model, International Journal of Heat and Mass Transfer 26 (4) (1983) 621–633. [127] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford University Press, 1959. [128] F.O. Oruma, M.N. Özi¸sik, M.A. Boles, Effects of anisotropic scattering on melting and solidification of a semi-infinite, semi-transparent medium, International Journal of Heat and Mass Transfer 28 (2) (1985) 441–449. [129] A.L. Burka, N.A. Rubtsov, N.A. Savvinova, Nonsteady-state radiant–conductive heat exchange in a semitransparent medium with phase transition, Journal of Applied Mechanics and Technical Physics 28 (1) (1987) 91. [130] S.H. Chan, K.Y. Hsu, The mushy zone in a phase change model of a semitransparent material with internal radiative transfer, ASME Journal of Heat Transfer 110 (February 1988) 260–264. [131] C. Yao, B.T.F. Chung, G.X. Wang, Mushy zone equilibrium solidification of a semitransparent layer subject to radiative and convective cooling, International Journal of Heat and Mass Transfer 45 (2002) 2397–2405. [132] P. Łapka, P. Furmanski, ´ Fixed cartesian grid based numerical model for solidification process of semi-transparent materials I: modelling and verification, International Journal of Heat and Mass Transfer 55 (2012) 4941–4952. [133] P. Łapka, P. Furmanski, ´ Fixed cartesian grid based numerical model for solidification process of semi-transparent materials II: reflection and refraction or transmission of the thermal radiation at the solid-liquid interface, International Journal of Heat and Mass Transfer 55 (2012) 4953–4964. [134] H.-L. Yi, C.-H. Wang, H.P. Tan, W. Zhou, Radiative heat transfer in semitransparent solidifying slab considering space-time dependent refractive index, International Journal of Heat and Mass Transfer 55 (2012) 1724–1731. [135] W.M. Kays, M.E. Crawford, Convective Heat and Mass Transfer, 3rd ed., McGraw-Hill, 1993. [136] S.I. Pai, Inviscid flow of radiation gas dynamics, Journal of Mathematical and Physical Sciences 39 (1969) 361–370. [137] M.N. Özi¸sik, Radiative Transfer and Interactions with Conduction and Convection, John Wiley & Sons, New York, 1973. [138] H. Schlichting, Boundary Layer Theory, 7th ed., McGraw-Hill, New York, 1979. [139] R. Viskanta, R.J. Grosh, Boundary layer in thermal radiation absorbing and emitting media, International Journal of Heat and Mass Transfer 5 (1962) 795–806. [140] A.N. Rumynskii, Boundary layers in radiating and absorbing media, American Rocket Society Journal 32 (1962) 1135–1138. [141] R. Goulard, The transition from black body to Rosseland formulations in optically thick flows, International Journal of Heat and Mass Transfer 7 (1964) 1145–1146. [142] J.L. Novotny, K.T. Yang, The interaction of thermal radiation in optically thick boundary layers, ASME Paper No. 67-HT-9, 1967.

Radiation Combined with Conduction and Convection Chapter | 21 813

[143] S.I. Pai, A.P. Scaglione, Unsteady laminar boundary layers of an infinite plate in an optically thick radiating gas, Applied Scientific Research 22 (1970) 97–112. [144] V.P. Zamuraev, Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki 3 (1964) 73. [145] R. Viskanta, Radiation transfer and interaction of convection with radiation heat transfer, in: Advances in Heat Transfer, vol. 3, Academic Press, New York, 1966, pp. 175–251. [146] R.D. Cess, Radiation effects upon boundary layer flow of an absorbing gas, ASME Journal of Heat Transfer 86C (1964) 469–475. [147] A.M. Smith, H.A. Hassan, Nongray radiation effects on the boundary layer at low Eckert numbers, ASME Paper No. 66-WA/HT-35 1966. [148] R.J. Tabaczynski, L.A. Kennedy, Thermal radiation effects in laminar boundary-layer flow, AIAA Journal 5 (1967) 1893–1894. [149] S.I. Pai, C.K. Tsao, A uniform flow of a radiating gas over a flat plate, in: Proceedings of the Third International Heat Transfer Conference, vol. 5, 1966, pp. 129–137. [150] C.C. Oliver, P.W. McFadden, The interaction of radiation and convection in the laminar boundary layer, ASME Journal of Heat Transfer C 88 (1966) 205–213. [151] A.N. Rumynskii, Boundary layer with an opaque underlayer, American Rocket Society Journal 32 (1962) 1139–1140. [152] J.C.Y. Koh, C.N. DeSilva, Interaction between radiation and convection in the hypersonic boundary layer on a flat plate, American Rocket Society Journal 32 (1962) 739–743. [153] Y. Taitel, J.P. Hartnett, Equilibrium temperature in a boundary layer flow over a flat plate of absorbing–emitting gas, ASME Paper No. 66-WA/HT-48 1966. [154] M.N. Robin, R.I. Souloukhin, I.B. Yutevich, Influence of reflection of radiation on radiative–convective heat exchange during hypersonic flow over a blunt body, Journal of Applied Mechanics and Technical Physics 21 (1980) 239–245. [155] V.N. Golubkin, On the asymptotic theory of the three-dimensional flow of a hypersonic stream of radiating gas around a body, Applied Mathematics and Mechanics 47 (1983) 493. [156] S.N. Tiwari, K.Y. Szema, J.N. Moss, S.V. Subramanian, Convective and radiative heating of a Saturn entry probe, International Journal of Heat and Mass Transfer 27 (1984) 191–206. [157] L. Soucasse, J.B. Scoggins, P. Rivière, T.E. Magin, A. Soufiani, Flow-radiation coupling for atmospheric entries using a hybrid statistical narrow band model, Journal of Quantitative Spectroscopy and Radiative Transfer 180 (2016) 55–69. [158] L.A. Dombrovsky, Radiation–convection heat transfer by an optically thick boundary layer on a plate, High Temperature 19 (1981) 100–109. [159] A. Yücel, Y. Bayazitoglu, ˘ Radiative heat transfer in absorbing, emitting and anisotropically scattering boundary layer, AIAA Paper No. 83-1504, 1983. [160] A. Yücel, R.H. Kehtarenavaz, Y. Bayazitoglu, ˘ Interaction of radiation with the boundary layer: Nongray media, ASME Paper No. 83-HT-33, 1983. [161] A. Soufiani, J. Taine, Application of statistical narrow-band model to coupled radiation and convection at high temperature, International Journal of Heat and Mass Transfer 30 (3) (1987) 437–448. [162] D.A. Kaminski, X.D. Fu, M.K. Jensen, Numerical and experimental analysis of combined convective and radiative heat transfer in laminar flow over a circular cylinder, International Journal of Heat and Mass Transfer 38 (17) (1995) 3161–3169. [163] C.A. Fritsch, R.J. Grosh, M.W. Wild, Radiative heat transfer through an absorbing boundary layer, ASME Journal of Heat Transfer 86 (4) (1966) 296–304. [164] W.G. Houf, F.P. Incropera, R. Viskanta, Thermal conditions in irradiated, slowly moving liquid layers, ASME Journal of Heat Transfer 107 (1) (1985) 92–98. [165] J.M. Elliott, R.I. Vachon, D.F. Dyer, J.R. Dunn, Application of the Patankar–Spalding finite difference procedure to turbulent radiating boundary layer flow, International Journal of Heat and Mass Transfer 14 (1971) 667–672. [166] D.Y. Goswami, R.I. Vachon, Turbulent boundary layer flow of absorbing, emitting and axisymmetrically scattering gaseous medium, AIAA Paper No. 80-1518, 1980. [167] V.I. Naidenov, S.A. Shindini, Interaction of radiation with turbulent fluctuations in a boundary layer, High Temperature 19 (1981) 106–109. [168] P.D. Jones, Y. Bayazitoglu, ˘ Radiation, conduction and convection from a sphere in an absorbing, emitting, gray medium, ASME Journal of Heat Transfer 114 (1) (1992) 250–254. [169] S. DeSoto, Coupled radiation, conduction and convection in entrance region flow, International Journal of Heat and Mass Transfer 11 (1968) 39–53. [170] Y. Kurosaki, Heat transfer by simultaneous radiation and convection in an absorbing and emitting medium in a flow between parallel plates, in: Proceedings of the Fourth International Heat Transfer Conference, vol. 3, No. R2.5, Elsevier, New York, 1970. [171] R. Viskanta, Interaction of heat transfer by conduction, convection, and radiation in a radiating fluid, ASME Journal of Heat Transfer 85 (1963) 318–328. [172] J.C. Chen, Simultaneous radiative and convective heat transfer in an absorbing, emitting, and scattering medium in slug flow between parallel plates, AIChE Journal 10 (2) (1964) 253–259. [173] D.K. Edwards, A. Balakrishnan, Nongray radiative transfer in a turbulent gas layer, International Journal of Heat and Mass Transfer 16 (1973) 1003–1015. [174] A.T. Wassel, D.K. Edwards, Molecular radiation in a laminar or turbulent pipe flow, ASME Journal of Heat Transfer 98 (1976) 101–107.

814 Radiative Heat Transfer

[175] F.H. Azad, M.F. Modest, Combined radiation and convection in absorbing, emitting and anisotropically scattering gas–particulate tube flow, International Journal of Heat and Mass Transfer 24 (1981) 1681–1698. [176] R. Goulard, M. Goulard, Energy Transfer in the Couette Flow of a Radiant and Chemically Reacting Gas, Stanford University Press, Stanford, CA, 1959, pp. 126–139. [177] R. Viskanta, R.J. Grosh, Temperature distribution in Couette flow with radiation, American Rocket Society Journal 31 (1961) 839–840. [178] R. Echigo, K. Kamiuto, S. Hasegawa, Analytical method on composite heat transfer with predominant radiation—analysis by integral equation and examination on radiation slip, in: Proceedings of the Fifth International Heat Transfer Conference, vol. 1, JSME, Japan, 1974, pp. 103–107. [179] R. Echigo, S. Hasegawa, K. Kamiuto, Composite heat transfer in a pipe with thermal radiation of two-dimensional propagation, International Journal of Heat and Mass Transfer 18 (1975) 1149–1159. [180] V.N. Timofeyev, F.R. Shklyar, V.M. Malkin, K.H. Berland, Combined heat transfer in an absorbing stream moving in a flat channel. Parts I, II, and III, Heat Transfer, Soviet Research 1 (6) (November 1969) 57–93. [181] C.C. Lii, M.N. Özi¸sik, Heat transfer in an absorbing, emitting and scattering slug flow between parallel plates, ASME Journal of Heat Transfer 95C (1973) 538–540. [182] J.B. Bergquam, N.S. Wang, Heat transfer by convection and radiation in an absorbing, scattering medium flowing between parallel plates, ASME Paper No. 76-HT-50, 1976. [183] T.C. Chawla, S.H. Chan, Spline collocation solution of combined radiation–convection in thermally developing flows with scattering, Numerical Heat Transfer 3 (1980) 47–76. [184] T.C. Chawla, S.H. Chan, Combined radiation and convection in thermally developing Poiseuille flow with scattering, ASME Journal of Heat Transfer 102 (1980) 297–302. [185] M.P. Mengüç, Y. Yener, M.N. Özi¸sik, Interaction of radiation in thermally developing laminar flow in a parallel plate channel, ASME Paper No. 83-HT-35, 1983. [186] Y. Yener, B. Shahidi-Zandi, M.N. Özi¸sik, Simultaneous radiation and forced convection in thermally developing turbulent flow through a parallel plate channel, ASME Paper No. 84-WA/HT-15, 1984. [187] Y. Yener, M.N. Özi¸sik, Simultaneous radiation and forced convection in thermally developing turbulent flow through a parallel plate channel, ASME Journal of Heat Transfer 108 (4) (1986) 985–987. [188] R. Echigo, S. Hasegawa, Radiative heat transfer by flowing multiphase medium—part I: an analysis on heat transfer of laminar flow between parallel flat plates, International Journal of Heat and Mass Transfer 15 (1972) 2519–2534. [189] M. Kassemi, B.T.F. Chung, Two-dimensional convection and radiation with scattering from a Poiseuille flow, Journal of Thermophysics and Heat Transfer 4 (1) (1990) 98–105. [190] T.K. Kim, H.S. Lee, Two-dimensional anisotropic scattering radiation in a thermally developing Poiseuille flow, Journal of Thermophysics and Heat Transfer 4 (3) (1990) 292–298. [191] T.Y. Kim, S.W. Baek, Thermal development of radiatively active pipe flow with nonaxisymmetric circumferential convective heat loss, International Journal of Heat and Mass Transfer 39 (14) (1996) 2969–2976. [192] S.S. Kim, S.W. Baek, Radiation affected compressible turbulent flow over a backward facing step, International Journal of Heat and Mass Transfer 39 (16) (1996) 3325–3332. [193] C.K. Krishnaprakas, K.B. Narayana, P. Dutta, Combined convective and radiative heat transfer in turbulent tube flow, Journal of Thermophysics and Heat Transfer 13 (3) (1999) 390–394. [194] T.H. Einstein, Radiant heat transfer to absorbing gases enclosed in a circular pipe with conduction, gas flow, and internal heat generation, NASA TR R-156, 1963. [195] S. Bergero, E. Nannei, R. Sala, Combined radiative and convective heat transfer in a three-dimensional rectangular channel at different wall temperatures, Wärme- und Stoffübertragung 36 (6) (1999) 443–450. [196] Y.F. Zhang, R. Vicquelin, O. Gicquel, J. Taine, Physical study of radiation effects on the boundary layer structure in a turbulent channel flow, International Journal of Heat and Mass Transfer 61 (2013) 654–666. [197] Y.F. Zhang, R. Vicquelin, O. Gicquel, J. Taine, A wall model for LES accounting for radiation effects, International Journal of Heat and Mass Transfer 67 (2013) 712–723. [198] Y.F. Zhang, R. Vicquelin, O. Gicquel, J. Taine, Practical indicators for assessing the magnitudes of wall radiative flux and of coupling effects between radiation and other heat transfer modes on the temperature law-of-the wall in turbulent gaseous boundary layers, International Journal of Heat and Mass Transfer 120 (2018) 76–85. [199] D.M. Kim, R. Viskanta, Interaction of convection and radiation heat transfer in high pressure and temperature steam, International Journal of Heat and Mass Transfer 27 (1984) 939–941. [200] C. Gau, D.C. Chi, A simple numerical study of combined radiation and convection heat transfer in the entry region of a circular pipe flow, in: Proceedings of the Second ASME/JSME Conference, vol. 3, 1987, pp. 635–643. [201] K. Kamiuto, Combined laminar forced convection and nongray-radiation heat transfer to carbon dioxide flowing in a nonblack plane-parallel duct, Numerical Heat Transfer – Part A: Applications 28 Part A (1995) 575–587. [202] C. Mesyngier, B. Farouk, Convection–nongray gas radiation interactions in a channel flow, in: Proceedings of the 1996 Heat Transfer Conference, vol. HTD-325 No. 3, ASME, 1996, pp. 103–113. [203] I.S. Habib, R. Greif, Heat transfer to a flowing non-gray radiating gas: an experimental and theoretical study, International Journal of Heat and Mass Transfer 13 (1970) 1571–1582.

Radiation Combined with Conduction and Convection Chapter | 21 815

[204] Z. Chiba, R. Greif, Heat transfer to steam flowing turbulently in a pipe, International Journal of Heat and Mass Transfer 16 (1973) 1645–1648. [205] S. Tabanfar, M.F. Modest, Combined radiation and convection in tube flow with non-gray gases and particulates, ASME Journal of Heat Transfer 109 (1987) 478–484. [206] T.F. Smith, Z.F. Shen, A.M. Al-Turki, Radiative and convective transfer in a cylindrical enclosure for a real gas, ASME Journal of Heat Transfer 107 (2) (1985) 482–485. [207] A. Soufiani, J. Taine, Experimental and theoretical studies of combined radiative and convective transfer in CO2 and H2 O laminar flows, International Journal of Heat and Mass Transfer 32 (3) (1989) 477–486. [208] E. Sediki, A. Soufiani, M.S. Sifaoui, Spectrally correlated radiation and laminar forced convection in the entrance region of a circular duct, International Journal of Heat and Mass Transfer 45 (2002) 5069–5081. [209] A. Soufiani, P. Mignon, J. Taine, Radiation–turbulence interaction in channel flows of infrared active gases, in: Proceedings of the Ninth International Heat Transfer Conference, vol. 6, Hemisphere, Washington, D.C., 1990, pp. 403–408. [210] A. Soufiani, P. Mignon, J. Taine, Radiation effects on turbulent heat transfer in channel flows of infrared active gases, in: Proceedings of the 1990 AIAA/ASME Thermophysics and Heat Transfer Conference, vol. HTD-137, ASME, 1990, pp. 141–148. [211] M. Atashafrooz, S.A.G. Nassab, K. Lari, Coupled thermal radiation and mixed convection step flow of nongray gas, ASME Journal of Heat Transfer 138 (2016) 072701. [212] R. Echigo, S. Hasegawa, H. Tamehiro, Radiative heat transfer by flowing multiphase medium—part II: an analysis on heat transfer of laminar flow in an entrance region of circular tube, International Journal of Heat and Mass Transfer 15 (1972) 2595–2610. [213] H. Tamehiro, R. Echigo, S. Hasegawa, Radiative heat transfer by flowing multiphase medium—part III: an analysis on heat transfer of turbulent flow in a circular tube, International Journal of Heat and Mass Transfer 16 (1973) 1199–1213. [214] M.F. Modest, B.R. Meyer, F.H. Azad, Combined convection and radiation in tube flow of an absorbing, emitting and anisotropically scattering gas–particulate suspension, ASME Paper No. 80-HT-27, 1980. [215] A.M. Al-Turki, T.F. Smith, Radiative and convective transfer in a cylindrical enclosure for a gas/soot mixture, ASME Journal of Heat Transfer 109 (1) (1987) 259. [216] S.H. Park, S.S. Kim, Thermophoretic deposition of absorbing, emitting and isotropically scattering particles in laminar tube flow with high particle mass loading, International Journal of Heat and Mass Transfer 36 (14) (1993) 3477–3485. [217] E. Farbar, I.D. Boyd, M. Esmaily-Moghadam, Monte Carlo modeling of radiative heat transfer in particle-laden flow, Journal of Quantitative Spectroscopy and Radiative Transfer 184 (2016) 146–160. [218] W. Huang, H. Ji, Effect of emissivity and reflectance on infrared radiation signature of turbofan engine, Journal of Thermophysics and Heat Transfer 31 (2017) 39–47. [219] Z. Yin, Y. Jaluria, Zonal method to model radiative transport in an optical fiber drawing furnace, ASME Journal of Heat Transfer 119 (1997) 597–603. [220] Z. Yin, Y. Jaluria, Thermal transport and flow in high-speed optical fiber drawing, ASME Journal of Heat Transfer 120 (4) (1998) 916–930. [221] M. Song, K.S. Ball, T.L. Bergman, A model for radiative cooling of a semitransparent molten glass jet, ASME Journal of Heat Transfer 120 (4) (1998) 931–938. [222] T.W. Tong, S.B. Sathe, R.E. Peck, Improving the performance of porous radiant burners through use of sub-micron size fibers, International Journal of Heat and Mass Transfer 33 (6) (1990) 1339–1346. [223] A.R. Martin, C. Saltiel, J.C. Chai, W. Shyy, Convective and radiative internal heat transfer augmentation with fiber arrays, International Journal of Heat and Mass Transfer 41 (1998) 3431–3440. [224] Y. Mahmoudi, Effect of thermal radiation on temperature differential in a porous medium under local thermal non-equilibrium condition, International Journal of Heat and Mass Transfer 76 (2014) 105–121. [225] B.P. Singh, M. Kaviany, Modelling radiative heat transfer in packed beds, International Journal of Heat and Mass Transfer 35 (1992) 1397–1405. [226] J.D. Lu, G. Flamant, B. Variot, Theoretical study of combined conductive, convective and radiative heat transfer between plates and packed beds, International Journal of Heat and Mass Transfer 37 (5) (1994) 727–736. [227] K. Kamiuto, S. Saitoh, Combined forced-convection and correlated–radiation heat transfer in cylindrical packed beds, Journal of Thermophysics and Heat Transfer 8 (1) (1994) 119–124. [228] R. Viskanta, Overview of convection and radiation in high temperature gas flows, International Journal of Engineering Science 36 (1998) 1677–1699. [229] R.D. Cess, The interaction of thermal radiation with free convection heat transfer, International Journal of Heat and Mass Transfer 9 (1966) 1269–1277. [230] V.S. Arpaci, Effect of thermal radiation on the laminar free convection from a heated vertical plate, International Journal of Heat and Mass Transfer 11 (1968) 871–881. [231] E.H. Cheng, M.N. Özi¸sik, Radiation with free convection in an absorbing, emitting and scattering medium, International Journal of Heat and Mass Transfer 15 (1972) 1243–1252. [232] G. Desrayaud, G. Lauriat, Natural convection of a radiating fluid in a vertical layer, ASME Journal of Heat Transfer 107 (3) (1985) 710–712.

816 Radiative Heat Transfer

[233] C.K. Krishnaprakas, K.B. Narayana, P. Dutta, Interaction of radiation with natural convection, Journal of Thermophysics and Heat Transfer 13 (3) (1999) 387–390. [234] M.A. Hossain, M.A. Alim, D.A.S. Rees, The effect of radiation on free convection from a porous vertical plate, International Journal of Heat and Mass Transfer 42 (1) (1999) 181–191. [235] B.W. Webb, R. Viskanta, Analysis of radiation-induced natural convection in rectangular enclosures, Journal of Thermophysics and Heat Transfer 1 (2) (1987) 146–153. [236] B.W. Webb, Interaction of radiation and free convection on a heated vertical plate: experiment and analysis, Journal of Thermophysics and Heat Transfer 4 (1) (1990) 117–120. [237] W.M. Yan, H.Y. Li, Radiation effects on laminar mixed convection in an inclined square duct, ASME Journal of Heat Transfer 121 (1) (1999) 194–200. [238] W.M. Yan, H.Y. Li, Radiation effects on mixed convection heat transfer in a vertical square duct, International Journal of Heat and Mass Transfer 44 (2001) 1401–1410. [239] E. Lacona, J. Taine, Holographic interferometry applied to coupled free convection and radiative transfer in a cavity containing a vertical plate between 290 and 650K, International Journal of Heat and Mass Transfer 44 (2001) 3755–3764. [240] M. Epstein, F.B. Cheung, T.C. Chawla, G.M. Hauser, Effective thermal conductivity for combined radiation and free convection in an optically thick heated fluid layer, ASME Journal of Heat Transfer 103 (1981) 114–120. [241] S. Bakan, Thermal stability of radiating fluids: the scattering problem, Physics of Fluids 27 (12) (1984) 2969. [242] W.M. Yang, Thermal instability of a fluid layer induced by radiation, Numerical Heat Transfer – Part A: Applications 17 (1990) 365–376. [243] L.C. Chang, K.T. Yang, J.R. Lloyd, Radiation–natural convection interactions in two-dimensional enclosures, ASME Journal of Heat Transfer 105 (1) (1983) 89–95. [244] T. Fusegi, B. Farouk, A computational and experimental study of natural convection and surface/gas radiation interactions in a square cavity, ASME Journal of Heat Transfer 112 (1990) 802–804. [245] G. Colomer, R. Cònsul, A. Oliva, Coupled radiation and natural convection: different approaches of the SLW model for a non-gray gas mixture, Journal of Quantitative Spectroscopy and Radiative Transfer 107 (1) (2007) 30–46. [246] Y. Sun, X. Zhang, J.R. Howell, Assessment of different radiative transfer equation solvers for combined natural convection and radiation heat transfer problems, Journal of Quantitative Spectroscopy and Radiative Transfer 194 (2017) 31–46. [247] M. Parmananda, R. Thirumalaisamy, A. Dalal, G. Natarajan, Investigations of turbulence–radiation interaction in non-OberbeckBoussinesq buoyancy-driven flows, International Journal of Thermal Sciences 134 (2018) 298–316. [248] Y. Yamada, Combined radiation and free convection heat transfer in a vertical channel with arbitrary wall emissivities, International Journal of Heat and Mass Transfer 31 (2) (1988) 429–440. [249] D.C. Kuo, J.C. Morales, K.S. Ball, Combined natural convection and volumetric radiation in a horizontal annulus: spectral and finite volume predictions, ASME Journal of Heat Transfer 121 (1999) 610–615. [250] Y. Sun, X. Zhang, J.R. Howell, Combined natural convection and non-gray radiation heat transfer in a horizontal annulus, Journal of Quantitative Spectroscopy and Radiative Transfer 206 (2018) 242–250. [251] A. Campo, U. Lacoa, Influence of thermal radiation on natural convection inside vertical annular enclosures, in: Proceedings of the 1988 National Heat Transfer Conference, vol. HTD-96, ASME, 1988, pp. 219–226. [252] T. Fusegi, K. Ishii, B. Farouk, K. Kuwahara, Three-dimensional study of convection–radiation interactions in a cubical enclosure field with a non-gray gas, in: Proceedings of the Ninth International Heat Transfer Conference, Hemisphere, Washington, D.C., 1990, pp. 421–426. [253] J.J. Derby, S. Brandon, A.G. Salinger, The diffusion and P1 approximations for modeling buoyant flow of an optically thick fluid, International Journal of Heat and Mass Transfer 41 (11) (1998) 1405–1415. [254] T. Tsukada, K. Kakinoki, M. Hozawa, N. Imaishi, Effect of internal radiation within crystal and melt on Czochralski crystal growth of oxide, International Journal of Heat and Mass Transfer 38 (1995) 2707–2714. [255] K. Abe, Y. Nagato, K. Sugioka, M. Kubo, T. Tsukada, S. Maruyama, Development of a global model of heat transfer in the Czochralski furnace taking into account specular reflection at the crystal surface, Numerical Heat Transfer – Part A: Applications 67 (2015) 1311–1323. [256] K. Abe, K. Sugioka, M. Kubo, T. Tsukada, S. Maruyama, Radiation heat transfer analysis in a semitransparent single crystal with specular surfaces: application of REM2 , Numerical Heat Transfer – Part A: Applications 63 (2013) 1–13. [257] H.S. Fang, Q.J. Zhang, J. Tian, S. Wang, R.J. Ma, Study of internal radiation with solute inclusions during Czochralski sapphire crystal growth, International Journal of Heat and Mass Transfer 79 (2014) 783–789. [258] C. Mesyngier, B. Farouk, Turbulent natural convection–nongray gas radiation analysis in a square enclosure, Numerical Heat Transfer – Part A: Applications 29 (7) (1996) 671–687. [259] F. Bdéoui, A. Soufiani, P.L. Quéré, A numerical study of Rayleigh–Benard convection in radiating gases, in: Proceedings of the 11th International Heat Transfer Conference, vol. 7, Kyongju, Korea, 1998, pp. 261–266. [260] S. Mazumder, Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods, Academic Press, 2016. [261] S.R. Mathur, J.Y. Murthy, Coupled ordinates method for multigrid acceleration of radiation calculations, Journal of Thermophysics and Heat Transfer 13 (4) (1999) 467–473.

Radiation Combined with Conduction and Convection Chapter | 21 817

[262] S.R. Mathur, J.Y. Murthy, Acceleration of anisotropic scattering computations using coupled ordinates method (Comet), ASME Journal of Heat Transfer 123 (3) (1999) 607–612. [263] S. Mazumder, A new numerical procedure for coupling radiation in participating media with other modes of heat transfer, ASME Journal of Heat Transfer 127 (9) (2005) 1037–1045. [264] M. Ravishankar, S. Mazumder, A. Kumar, Finite-volume formulation and solution of the P3 equations of radiative transfer on unstructured meshes, ASME Journal of Heat Transfer 132 (2) (2010) 023402.

Chapter 22

Radiation in Chemically Reacting Systems 22.1 Introduction Chemically reacting systems constitute important applications for radiative heat transfer. In combustion systems, chemical reactions produce high temperature, often making radiation the dominant mode of heat transfer. Furthermore, common products of combustion, such as carbon dioxide, water vapor, and soot are radiatively participating, and affect radiative transport and the ensuing flame structure in complex ways. In most flames and fires, upstream propagation of radiation causes preheating of the fuel (via absorption) and is an important mechanism for making the flame or fire stable. In the atmosphere, absorption of solar radiation plays an important role in assisting chemical reactions that affect the amount of ozone. Solar radiation is also being used now to convert the products of hydrocarbon combustion into useful fuels—a process wherein thermal radiation assists chemical reactions. In addition to the challenges outlined in the preceding chapter for coupling radiation with the other modes of heat transfer, chemically reacting systems pose additional challenges. In chemically reacting systems, the entities that alter the radiation field, i.e., the radiatively participating chemical species, are rarely spatially or temporally uniform. They are created and destroyed locally by chemical reactions. Local creation or destruction is often accompanied by production or removal of heat due to the reactions being exothermic or endothermic. Hence, in order to couple radiation with the other modes of heat transfer in such systems, one must account for (1) spatio-temporal variations of species and their effects on local instantaneous radiative properties and (2) strong inhomogeneities in the temperature field, which in turn strongly affect both direct emission and radiative properties. In other words, interactions that are absent or relatively weak in nonreacting systems become critical. In this chapter, we focus on these new interactions. In a turbulent flow field, the aforementioned spatio-temporal variations in species concentrations and temperature occur at length and time scales that may span several orders of magnitude. Fluctuations in temperature result in fluctuating extinction coefficients as well as fluctuating intensities. Since these turbulent fluctuations occur over a wide spectrum of length and time scales, they cannot be always resolved with a spatio-temporal mesh, and a model that averages these fluctuations over certain length/time scales may need to be resorted to. For capturing the interactions between the fluctuating extinction coefficient and the radiation intensity—referred to as turbulence–radiation interactions (TRI)—special models are necessary. A large subsection of this chapter is devoted to the discussion of TRI—specifically, under what scenarios TRI may become important and what role it plays in altering the temperature field and radiative heat fluxes. Models and measurement techniques for quantifying TRI are also discussed, along with an overview of past and ongoing research in this field. The final section of the chapter is dedicated to the discussion of solar thermal and solar thermochemical reactors. These types of reactors constitute an important emerging application of radiative heat transfer wherein solar radiation is strongly coupled with the other modes of heat transfer within a chemically reacting environment.

22.2 Coupling Considerations Continuing from the last section of the preceding chapter on the theme of coupling radiation with other modes of heat transfer, we begin with a brief overview of special considerations for reacting systems. Combining high-level models for reacting flows and radiation requires great care to avoid instabilities, lack of convergence, and/or exorbitant computer memory and time requirements since nonlinearities may now arise due to radiation as well as chemical reactions, and these nonlinearities may interact. The overall coupling procedure is similar to Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00030-4 Copyright © 2022 Elsevier Inc. All rights reserved.

819

820 Radiative Heat Transfer

the procedure outlined in the preceding chapter for nonreacting systems, with a few additional considerations, and generally consists of the following steps: 1. The temperature and the species concentration (partial pressure) fields are first guessed. This includes nonisothermal boundaries. 2. The absorption and scattering coefficients are calculated as a function of pressure, temperature, and species concentrations using their guessed values. For nongray calculations, appropriate spectrally grouped or averaged radiative properties need to be calculated and, often, stored in a database for the sake of computational efficiency. Such databases are generally constructed off-line prior to actual radiation calculations (cf. Chapter 19), and in this step, the appropriate radiative properties are simply retrieved from the database. 3. For the given temperature and extinction coefficient field, the RTE is solved to obtain the radiative source = −∇ · qR , and the radiative heat flux at the boundaries. Simple or sophisticated RTE solvers may term, Q˙  R be employed (P1 -approximation, different levels of the discrete ordinates or finite angle methods, etc.), and primitive or advanced spectral models may be used (gray gas, wide band models, narrow band models, k-distributions, etc.). It is also possible to start out with relatively crude RTE solutions, moving toward more accurate models as the overall solution progresses. 4. The radiative source and the radiative heat flux at nonisothermal boundaries are substituted into the overall energy equation, and the flow field is calculated, including velocity, pressure, temperature (or enthalpy), and species concentrations. This may require several inner iterations for complicated flow fields. Likewise, computation of the mass and heat sources due to chemical reactions in the species mass conservation and energy equations, respectively, may require special time integration methods that account for the extreme nonlinearity and stiffness of the governing reaction rate (chemical kinetics) equations. When substituting the radiative source term and the radiative heat fluxes at boundaries into the overall energy equation, linearization may be employed (this is the semi-implicit coupling procedure described in Chapter 21) to stabilize the iterative algorithm. 5. Steps 2 through 4 are repeated until some overall convergence criteria are met. When the flow field is turbulent, extremely fine grids often have to be used near the walls in order to resolve the turbulent boundary layer and/or flow separation. On the other hand, such fine grids are not always necessary for radiation calculations, especially if the optical thickness is small. In an effort to improve the computational efficiency of coupled turbulent convection-radiation calculations, some researchers [1–4] have proposed and demonstrated the use of separate grids for flow and radiation calculations. However, such a procedure introduces additional interpolation errors into the solution, in addition to convergence difficulties. Three-dimensional calculations in complex geometries using multiple meshes (which may be of different topologies) are extremely tedious, and the computational overheads of interpolation and book-keeping may often be prohibitively large. Depending on the fidelity of the spectral model and the order of the RTE solver (N in PN or SN ) being used, radiation calculations in reacting systems can sometimes be much more time consuming than solving the same problem without radiation. Evidence to this effect is provided in Section 22.5. To mitigate this challenge, researchers have proposed solving the RTE every few iterations or time steps [5–9]. This strategy is particularly effective in steady-state calculations or in transient calculations if the time-step size is “small.” If the Monte Carlo method is used for solution of the RTE, tracing sufficient number of bundles within each time step or iteration to arrive at an acceptably low statistical error is computationally expensive to the point that even skipping a few time steps or iterations is prohibitive. Furthermore, completely replacing a Monte Carlo solution with a new one can often lead to difficulties in convergence because of the inherent statistical error in the individual solutions. To circumvent these difficulties, blending or tempered averaging of the statistics may be used [10,11]. In this method, at each time step (in a time-marching code) or iteration (in a steady-state code) a relatively small number of photon bundles, N, (perhaps 10 times the number of cells) is traced to compute a rough estimate of the negative radiative source ∇·q(n) . These sources are then blended according to (n)

∇·q

(n−1)

= α∇ · q(n) + (1 − α)∇ · q

,

(22.1)

where α is the (usually small) blending factor. Expanding equation (22.1) (n)

∇·q

 6 7 = α∇ · q(n) + (1 − α) α∇ · q(n−1) + (1 − α) α∇ · q(n−2) + . . . ,

(22.2)

Radiation in Chemically Reacting Systems Chapter | 22 821

(n)

shows that ∇·q is a weighted average of many subsamples, with decreasing weights for those further removed from the current step n. The sum of weight factors over m steps is readily evaluated as Σm = α + (1 − α)[α + (1 − α){α + (1 − α)α + . . . }] = 1 − (1 − α)m .

(22.3)

(n)

Thus, for example, if α = 0.02 then 99% of all contributions to ∇·q come from the most recent m = 228 computational steps. Section 22.5 presents concrete evidence of the benefits of using such multi time step blending schemes in the context of using the Monte Carlo method for radiation calculations in reacting systems.

22.3 Combined Radiation and Laminar Combustion Thermal radiation from gases and particulates is an important, and often the dominant heat transfer mechanism during the burning of fuel. Therefore, inclusion of an adequate radiation model is essential to the success of a mathematical model of the combustion process, particularly in large systems (with larger optical thickness). The description of the burning process is an extremely difficult task even in the absence of radiation: “complete” chemical reaction mechanisms can involve hundreds of chemical species and thousands of elementary reactions [12], modeled by a nonlinear, stiff set of simultaneous differential equations. Furthermore, the combustion process is generally accompanied by multidimensional (perhaps two-phase) convection involving all species, as well as by turbulent mixing. Comprehensive reviews of the pertinent literature up to 1986 [13] and 2004 [14] have been given by Viskanta and Mengüç. Here we will briefly discuss the particularly simple case of a vertical porous flat plate burner producing a laminar buoyant diffusion flame, using a simple fuel (methane, CH4 ), a simple global reaction mechanism, CH4 + 2O2 → CO2 + 2H2 O

(22.4)

(neglecting multistep chemistry and intermediate species generation), and a simple reaction rate model (assuming an infinitely fast reaction wherever methane and oxygen come into contact). Such analyses were carried out in early work by Negrelli and coworkers [15] for the lower stagnation region of a horizontal cylinder, and by Liu and colleagues [16] for a vertical flat plate burner. Results for combustion–radiation interaction in a simple, laminar diffusion flame are very characteristic for all reacting flows and can, qualitatively, be applied to fairly general combustion systems. For such a buoyant laminar diffusion flame, with gravity pointing in the negative x-direction, the governing conservation equations, namely equations (21.43) through (21.46) are changed to Overall mass: x − Momentum: Energy: Species mass:

∂ ∂ (ρu) + (ρv) = 0, ∂x ∂y     ∂u ∂u ∂ ∂u +v ρ u = μ − g(ρ∞ − ρ), ∂x ∂y ∂y ∂y     ∂qR ∂T ∂T ∂ ∂T ρcp u +v + Q˙  = k − ch , ∂x ∂y ∂y ∂y ∂y     ∂Yi ∂Yi ∂ ∂Yi +v ρ u = ρD + m˙  i , i = species, ∂x ∂y ∂y ∂y

(22.5) (22.6) (22.7) (22.8)

where, for free convection, the pressure gradient term in the x–momentum equation has been replaced by a buoyancy term, and viscous dissipation has been neglected. The energy equation now has a heat source term (due to the release of chemical energy), and equations of mass conservation, written in terms of the mass fractions Yi of all species, must be added. In early work it was common practice to further simplify the problem by assuming a single mass diffusivity, D, for all species and to only consider fuel (methane, F), oxidizer (oxygen, O), and products (H2 O and CO2 , P) as independent “species.” Furthermore, the fuel is assumed to be injected uniformly through the porous vertical plate. The above equations invoke the so-called boundary layer approximation, which shows [17] that the momentum in the y-direction is significantly smaller that the momentum in the x-direction. Hence, the y–momentum equation is not necessary. Similarly, diffusive transport of mass, momentum, or energy in the x-direction (along the burner surface) is negligible compared to the transport of the same quantities in the y-direction (normal to the burner surface). The system of equations is closed with the ideal gas law, or ρT = const

822 Radiative Heat Transfer

FIGURE 22.1 Experimental and theoretical temperature profiles for a laminar methane buoyant diffusion flame; from [15].

(assuming constant pressure), while the sources Q˙  and m˙  are calculated from the reaction kinetics. Finally, i ch the boundary conditions are replaced by x=0: y=0: y→∞:

u(0, y) = 0, T(0, y) = T∞ , YF = YP = 0, YO = YO∞ ; u(x, 0) = 0, v(x, 0) = vw , T(x, 0) = Tw , YF = 1, YP = YO = 0; u(x, ∞) = 0, T(x, ∞) = T∞ , YF = YP = 0, YO = YO∞ ,

(22.9a) (22.9b) (22.9c)

where x = 0 denotes the leading edge of the vertical porous plate, and y = 0 its surface. For the radiation term, both Negrelli [15] and Liu [16] used an exact 1D-solution of the RTE together with the wide band model to simulate the nongray radiation from the absorbing/emitting combustion gases (CH4 , CO2 , and H2 O). The above set of equations was solved by both Negrelli and coworkers [15] and by Liu and colleagues [16] in a semianalytical way and compared with experimental data. Both teams found reasonable agreement between theory and experiment, especially in light of the somewhat primitive models. Figure 22.1 shows an example of the results of Negrelli et al. [15], who performed their calculations also for the cases of a transparent gas (no radiation) and a gray gas (using a Planck-mean absorption coefficient based on local partial pressures). Comparison with the solution without radiation makes it evident that radiation lowers the temperatures in the high-temperature region of the boundary layer (by more than 100◦ C), and raises them in the cooler region near the outer edge of the boundary layer. Obviously, radiation’s “action at a distance” allows energy to travel directly from the hot zone to the colder parts. It is also observed that radiation increases the thermal boundary layer thickness, for the same reasons. On the other hand, using a gray gas approximation severely over-predicts the effect of radiation on flame temperature and heat loss from the flame. The nongray gas emits and absorbs radiation across spectral lines that may be optically very thick, i.e., the emitted energy is reabsorbed in the immediate vicinity of the emission point; little emission occurs over vast parts of the spectrum (“spectral windows” with near-zero absorption coefficient). The gray approximation replaces the nongray absorption coefficient by a single, intermediate value, which predicts the correct overall emission, but (for large enough flames) strongly underpredicts reabsorption of this emission. More recent investigations of laminar, methane diffusion flames have used more sophisticated reaction kinetics together with the CHEMKIN software [18,19], and employed the statistical narrow band model for radiation [20–23]. The influence of soot radiation on laminar diffusion flames has been studied for ethylene [24– 29] and acetylene flames [30]. Kaplan and coworkers [24] also assessed the importance of radiation by comparing with calculations, in which radiation was ignored. Figure 22.2 is an example of their work, which clearly indicates that ignoring radiation, with its over-prediction of temperature levels, leads to grossly over-predicted soot levels. Similar conclusions about the importance of radiative heat transfer can be drawn with respect to high-temperature production of trace pollutants, such as NOx [31–33]. The older investigations used simple

Radiation in Chemically Reacting Systems Chapter | 22 823

FIGURE 22.2 Theoretical soot level profiles for a laminar acetylene diffusion flame; from [24].

one-step kinetics and assumed the absorption coefficient to be gray (assuming radiation to be dominated by the near-gray soot), but used different soot nucleation, growth, coagulation and oxidation models as well as different RTE solvers. The more recent ones used full chemistry and the statistical narrow band model together with nongray soot for radiation. In a very recent work, Wu and Zhao [34] investigated the effect of radiation on both steady and unsteady (flickering) laminar nonpremixed flames. A detailed line-by-line Monte Carlo solver was used to account for the nongray nature of gas radiation and soot. It was found that nongray radiation and how accurately it is treated, plays a significant role in the prediction of the structure and dynamics of unsteady laminar flames, especially when soot is considered. A number of studies have investigated the effect of radiation on laminar flames under microgravity conditions. Liu and coworkers [27] found that radiation effects become much stronger under microgravity conditions. This finding was also corroborated by Consalvi and coworkers [22], who recently modeled 24 different laminar diffusion flame configurations under both terrestrial and microgravity conditions with the specific goal to identify the role of thermal radiation on soot formation and vice versa. Tang et al. [35] have investigated the accuracy of various radiation models—both gray and nongray—on the evolution of a spherical flame under microgravity conditions by comparing with their own experiments. Figure 22.3 shows the evolution of the flame radius with time using various radiation models. It is evident that the statistical narrow band model, which is the most sophisticated spectral model used in this study, best replicates the experimental behavior. This result leads to the conclusion that detailed spectral modeling of gas radiation is warranted for flame calculations in microgravity environments. In fact, both the “gray gas + optically thin model” and the “wide band model,” although they replicate the experimental data well for a short period of time, ultimately lead to premature extinction of the flame, as indicated by the incomplete curves for these two models. Although most practical applications of combustion involve turbulent flow fields, laminar flames continue to be investigated, with the primary focus on spectral models for combustion gases and soot, and how they alter the flame speed and structure. Such studies have included use of correlated-k models [22,36,37], the FSCK model [11,36,38,39], and the MSFSK model [36,38], among others. In an extensive study that included 24 different laminar diffusion flame configurations with different hydrocarbon fuels, Consalvi and coworkers [22] have concluded that soot may be treated as gray but gases cannot be treated as gray in laminar diffusion flame calculations. Very recently, line-by-line calculations have also been conducted [36,40] to quantify how radiation from the products of combustion preheats the reactants (fuel). Over the past decade, experimental investigations [41–43] have also been undertaken to directly measure radiation intensities and temperatures in laminar diffusion flames in an effort to quantify the impact of radiation on such flames.

824 Radiative Heat Transfer

FIGURE 22.3 Comparison between experimental data and simulation results for flame radius evolution of a spherical ethylene diffusion flame in diluted air. Results of four different radiation models are shown: a) adiabatic; b) gray gas + optically thin model; c) wide band model; and d) statistical narrow band model; from [35].

22.4 Combined Radiation and Turbulent Combustion The past two decades have seen tremendous advances in the modeling of turbulent flows and chemical reactions, as well as in the field of multidimensional, nongray radiation modeling, each requiring their own sophisticated and time-consuming algorithms to produce accurate results. Today, the literature on the interaction of radiative heat transfer in turbulent combustion applications is growing at a rapid pace, including investigations on turbulent jet diffusion flames [44–53], flame and fire spread along vertical surfaces [54–57], droplet [58–60] and fluidized bed combustion [61–68], simulations of fires [69], furnaces [2,70–77], gas turbine combustors [78, 79], internal combustion engines [39,80] and, recently, of future oxy-fuel combustors [33,81–83] (designed for carbon capture). Several experimental studies have been conducted to characterize radiation in laboratory-scale turbulent non-premixed flames and pool fires with and without soot [84–87]. Combined experimental and modeling studies have also been conducted recently to quantify the effect of radiation in aerospace applications, such as exhaust plumes [88,89] and scramjets [90]. The results from all of these studies are consistent with the qualitative behavior for laminar flames described in the preceding section. Also notable is the finding that nongray soot modeling can be of equal or greater importance than nongray gas modeling in sooty turbulent flames, with gray soot models producing large errors [52].

Turbulence–Radiation Interactions (TRI) While the development of modern large eddy simulations (LES) and direct numerical simulations (DNS) is progressing at a rapid pace, most computational fluid dynamics (CFD) calculations for practical applications still rely on the Reynolds-averaged Navier-Stokes (RANS) equations for the computation of turbulent flows. Given that both DNS and LES are still prohibitively expensive even for nonreacting flows of practical interest, the popularity of RANS-based calculations for reacting flows is likely to continue in the foreseeable future. In RANS calculations, the Navier-Stokes equations are solved in terms of time-averaged means, with all turbulence effects being modeled. While turbulence–convection interaction is always accounted for in these schemes (with eddy diffusivities or more advanced models), the interactions between the turbulent flow fields and fluctuating radiation intensities have generally been neglected. In this subsection, we first briefly describe how extra terms, representing the interactions between turbulence and radiation (i.e., TRI), appear when applying the Reynolds averaging procedure to the RTE. This description is followed by a detailed discussion of TRI and various models to treat them.

Radiation in Chemically Reacting Systems Chapter | 22 825

During the development of the radiative transfer equation (RTE) in Chapter 9, we noted that heat transfer due to thermal radiation is essentially instantaneous, depending on the temporal temperature distribution as well as the temporal concentration field of the absorbing, emitting, and/or scattering medium. In a turbulent flow, the temperature field and, for mixtures, the concentration fields undergo rapid and irregular local oscillations (but still slow compared with the response time of thermal radiation). The governing equations, such as equations (21.43) through (21.45) or equations (22.5) through (22.7), are then rewritten in terms of time-averaged quantities (denoted by an overbar), e.g., 1 ρ(x, y) = δt

 δt

ρ(x, y, t) dt,

(22.10)

where δt is the (small) time interval used for averaging. Commonly, the so-called Favre averaging (or massweighted averaging), denoted by a tilde, is also employed for compressible flows, that is, + * = ρφ ρ, φ

(22.11)

where φ is the quantity to be averaged. Next, we present the governing conservation equations that need to be solved to model a turbulent reacting flow with radiation in the RANS framework. For simplicity, we will assume that the mean flow is 2D and steady. Under these assumptions, the governing equations are [91] Overall mass: x − Momentum: y − Momentum: Energy (enthalpy):

∂   ∂   ρ* u + ρ* v = 0, (22.12) ∂x ∂y       dp ∂* u ∂* u ∂* u ∂* u ∂ ∂ (22.13) +* v ρ * u ρ(ν + νt ) ρ(ν + νt ) = + − , dx ∂x ∂y ∂x ∂x ∂y ∂y       dp ∂* v ∂* v ∂* v ∂ ∂ ∂* v +* v , (22.14) ρ * ρ(ν + νt ) ρ(ν + νt ) = + − u ∂x ∂y ∂x ∂x ∂y ∂y dy ⎤ ⎤ ⎡ ⎡ ⎞ ⎛ s−1 s−1 ⎥⎥ ⎥⎥ ⎜⎜ ∂* ∂ ⎢⎢⎢ ∂ ⎢⎢⎢ h ∂* h ⎟⎟⎟ ∂* h  ∂* h  ⎥ ⎜ u +* v ⎟⎠ = + + ρ ⎜⎝* jx,i hi ⎥⎥⎦ + j y,i hi ⎥⎥⎥⎦ + Q˙  , ⎢⎢⎣ρ(α + αt ) ⎢⎢⎣ρ(α + αt ) R ∂x ∂y ∂x ∂x ∂y ∂y i=1

Species mass:

⎞ ⎡ ⎤ ⎡ ⎤ ⎛ *i *i ⎟⎟ *i ⎥⎥ *i ⎥⎥ ⎜⎜ ∂Y ∂ ⎢⎢⎢ ∂ ⎢⎢⎢ ∂Y ∂Y ∂Y ⎟ ⎥ ⎥⎥ + m˙  , ⎜ u +* v ρ ⎝⎜* ⎟= ⎢ρ(Di + Dt ) ⎥+ ⎢ρ(Di + Dt ) i ∂x ∂y ⎠ ∂x ⎣ ∂x ⎦ ∂y ⎣ ∂y ⎦

i=1

(22.15) i = 1, . . . , s − 1, (22.16)

where the energy equation has been written in terms of enthalpy rather than temperature (see equation (22.7) for comparison), as is customary in multi-species reacting flow computations for enhanced numerical stability [91]. T In equation (22.15), hi denotes the enthalpy of species i, and is defined as hi = h0f,i + T cp,i (T ) dT , where T0 is the 0 so-called standard state or reference temperature, and h0f,i is the enthalpy of formation at the standard state. The specific heat capacity of species i at constant pressure is denoted by cp,i . The mixture enthalpy, appearing in the other terms of the energy equation, is related to the enthalpies of the individual species through the relationship 5 h = s−1 i=1 Yi hi . In equations (22.13) through (22.16), νt , αt , and Dt are turbulent viscosity and heat and mass diffusivity, respectively. The determination of these quantities requires turbulence models. In equation (22.15), the quantities, jx,i and j y,i are the x- and y-components of the mass diffusion flux of species i, respectively. If Fick’s law of diffusion is used, these flux components may be written as jx,i = ρDi

∂Yi ∂Yi ; j y,i = ρDi , ∂x ∂y

(22.17)

where Di is the molecular (or laminar) mass diffusivity of species i. Equation (22.17) has, in fact, been used in equation (22.16) in a Favre-averaged sense. In the enthalpy formulation of the energy equation presented here, the heat source due to chemical reaction, Q˙  , as appearing in the energy equation written in terms of temperature, ch

826 Radiative Heat Transfer

equation (22.7), is not explicitly manifested because the enthalpy includes the enthalpy of formation. It can be computed using the relationship Q˙  ch

⎞ ⎞ ⎛ s−1 ⎛ s−1 ⎟⎟ ⎟⎟ ∂ ⎜⎜⎜ ∂ ⎜⎜⎜ ⎟ 0 0 ⎜⎜ ⎜⎜ = jx,i h f,i ⎟⎟⎠ + j y,i h f,i ⎟⎟⎟⎠ . ⎝ ⎝ ∂x ∂y i=1

(22.18)

i=1

The source terms, Q˙  , and m˙  are strongly nonlinear functions of the s composition variables, collected into R i a vector φ (φi = Yi , i = 1, . . . , s − 1; φs = h). To determine their time-averaged values, as appearing in the governing equations, one must resort to additional turbulence models. While the determination of Q˙  is the R

involves resolving the interactions between species mass central topic of discussion here, determination of m˙  i fraction fluctuations as well as between temperature fluctuations and species mass fraction fluctuations since (φ)  m˙  (φ). These nonlinear interactions are referred to as turbulence-chemistry interactions. Likewise, the m˙  i i terms jx,i hi and j y,i hi also require resolution of the aforementioned interaction between temperature fluctuations and species mass fraction fluctuations since the diffusion fluxes are functions of species mass fractions, while enthalpy is a function of temperature. Turbulence modeling is a field of great complexity and research interest that has seen dramatic progress during recent years. Reynolds-averaged Navier–Stokes (RANS)-based turbulence models are the most popular today, in particular the ubiquitous k–ε model [92], and a number of more accurate models are also available. The interaction between turbulence and chemistry has received considerable attention, resulting in flamelet models [93–95] and PDF (probability density function) methods [96]. While very relevant for the modeling of turbulence–radiation interactions, these models go much beyond the scope of this book, and the reader is referred to the relevant literature [97–102]. Experimental evidence supporting the idea that turbulence and radiation interact in unique ways can be traced back to the early works of Foster and coworkers [103–105], who measured the mean transmissivity from several propane jet flames. Formal systematic study of radiation and TRI in turbulent flames was first undertaken by Faeth [106–110] and continued subsequently by Gore and coworkers [111–120]. Their combined experimental and theoretical investigations have indicated that, depending on the fuel used and other conditions, radiative emission from a flame may be as much as 50%–300% higher than would be expected based on mean values of temperature and absorption coefficient. Other studies that have experimentally quantified TRI in turbulent flames include those by Krebs et al. [121,122] and Fischer et al. [123]. Recently, Gore and coworkers have developed a new infrared imaging-based technique [124] for the measurement of radiation from flames. This new technique allowed, for the first time, measurement of space–time cross correlations of the intensity fluctuations in turbulent jet flames [84–86], and buoyant flames [125]. To theoretically account for the interaction between turbulence and radiation (TRI), the time-averaged radiative source must be evaluated, or ∞    ˙ QR = −∇ · qR = − κη 4πIbη − 0





∞

Iη dΩ dη = −

4π

  4πκη Ibη −

0

 κη Iη dΩ dη.

(22.19)

4π

Because of their nonlinear dependence on composition variables these terms cannot be determined based on mean values. Thus, two turbulence moments or correlations are required: the correlations between absorption coefficient and Planck function, κη Ibη , and between absorption coefficient and radiative intensity, κη Iη . The former correlation is termed Emission TRI :

* bη (* κη Ibη  κη (φ)I T),

(22.20)

* η (φ). * κη Iη  κη (φ)I

(22.21)

while the latter is known as Absorption TRI :

Absorption TRI is particularly difficult to evaluate because the fluctuations of the local intensity may be influenced by property fluctuations from everywhere in the medium. On the other hand, in some early work Kabashnikov and coworkers [126–128] have suggested that, if the mean free path of radiation is much larger than the turbulence eddy length scale lt , then the local radiative intensity is only weakly correlated with the local

Radiation in Chemically Reacting Systems Chapter | 22 827

absorption coefficient, i.e., Absorption TRI :

κη Iη κη Iη .

(22.22)

This expression, valid if κη lt 1, and commonly known as the (optically) thin eddy approximation, or optically thin fluctuation approximation (OTFA), simplifies the evaluation of Q˙  considerably, since the remaining correlations R

κη and κη Ibη can be constructed from single-point statistics of the composition variables. Note that, in order to invoke this approximation, one must have κη lt 1 for all wavenumbers. While this condition is generally violated by combustion gases for very small parts of the spectrum (see, e.g., Fig. 10.6), and also for extremely sooty flames, it is justifiable in the vast majority of applications, as discussed further later. To date, most predictions of TRI (turbulence–radiation interactions) have employed the OTFA. Very similar to the time-averaged chemical source term, evaluation of the remaining correlations, κη and κη Ibη , requires equations or models for the correlations between any two composition variables, for a total of s2 moments [129]; this task is clearly not feasible with traditional RANS-based models. Because of these difficulties radiation and turbulence have traditionally been treated as independent phenomena, i.e., the influence of turbulent fluctuations on the composition variables (that determine the local values of radiative properties, blackbody intensity and, therefore, the local radiative intensity) have been neglected. If effects of radiation are considered at all, the calculations are generally based on mean (time-averaged) composition variables. In an old study, Cox [130] has shown that emission from a hot medium increases dramatically due to turbulence, simply by expanding the emissive power into a Taylor series. For example, for a simple, gray medium with constant absorption coefficient, emission TRI reduce to κIb = κIb = κEb /π, where  1 Eb (x, y, t) = n2 σT 4 (x, y, t) dt. (22.23) δt δt If one writes temperature and its fluctuations in terms of a time average, T(x, y, t) = T(x, y, t) + T (x, y, t),

T = 0,

(22.24)

then Eb (x, y, t) can be approximated by a truncated Taylor series as Eb (T) Eb (T) + T and

d2 Eb 1 + (T )2 2 + . . . , 2 dT dT

dEb

⎡ ⎤ 4⎢ (T )2 ⎥⎥⎥ 1 d2 Eb  2 ⎢⎢ Eb (T) Eb (T) + (T ) = σT ⎢⎣1 + 6 2 ⎥⎦ . 2 dT2 T

(22.25a)

(22.25b)

Equation (22.25b) shows that the so-called temperature self-correlation (time-averaged emissive power) is always positive, resulting in enhanced emission due to turbulence–radiation interactions. In the present case (gray, constant-property medium) temperature fluctuations of ± 30% would likely increase emission by more than 50%! Coelho [131] has expanded the analysis to include both temperature and absorption coefficient fluctuations, and has shown that if terms of order two or higher are neglected, then ⎤ ⎡  )2  T ⎥ ⎢⎢ (T κ ⎥⎥ κEb (T) κEb (T) ⎢⎢⎣1 + 6 2 + 4 ⎥⎦ . κT T

(22.26)

Equation (22.26) shows that if the correlation between absorption coefficient and temperature (third term) is neglected, TRI always result in increased emission since the second term is always positive. However, the third term cannot be neglected in a turbulent flow in general—reacting or nonreacting. This two-point correlation is expected to be problem dependent, and may enhance emission by the medium if it is positive or may suppress emission if it is negative. Based on previous results [132,133], it appears that in nonreacting flow of combustion gases, this term suppresses emission. All these conclusions also hold for nongray calculations, in which case, κ is simply replaced by the Planck-mean absorption coefficient κP , as given by equation (10.197). In a recent theoretical study [134] of turbulent methane–air diffusion flames, it was shown that neglecting absorption

828 Radiative Heat Transfer

coefficient fluctuations overestimates emission by the flame, which suggests that even in reacting flows, this term suppresses emission, i.e., it is negative. The accurate treatment of the emission TRI term has been a topic of intense research over the past several decades. As in the modeling of turbulence–chemistry interactions, two very different approaches have been pursued. The first approach, specifically applicable to the RANS-based modeling framework and henceforth referred to as the mean field approach, is to compute the temperature self-correlation and the temperature– absorption coefficient two-point correlation [see equation (22.26)] using certain assumptions pertaining to the nature of the fluctuations in composition (temperature and concentrations). These assumptions are necessary to attain closure, and are incorporated in the form of so-called assumed probability density functions (PDFs). If the PDFs of temperature and concentration fluctuations are assumed, it is possible, at least in principle, to compute the aforementioned two correlations directly since the absorption coefficient is an explicit function of temperature and concentrations. The procedure to do so is outlined in the monograph by Coelho [131], along with associated challenges. Early works by Faeth and Gore [106,108–110,112–116,135] have used this approach extensively, and it continues to be used in more recent studies [134,136–139] on emission TRI. Clearly, in this approach, the accuracy of prediction of the emission TRI term is largely dependent on the PDF being assumed. Commonly used PDFs for the composition fluctuations are Gaussian, clipped or truncated Gaussian, and the β-PDF. In an effort to quantify the effect of the assumed PDF on the accuracy of the emission TRI term, Cumber and Onokpe [140] have compared results generated by the β-PDF with a clipped Gaussian PDF. In addition to finding that the β-PDF yields more accurate results, they found that wall heat fluxes are quite sensitive to the assumed PDF, especially if TRI are taken into account. In the model for emission TRI given by equation (22.26), higher-order terms have been neglected. If the fluctuations are large, this assumption may introduce additional errors. In an effort to include the neglected higher-order terms, Snegirev [141] proposed a modification to equation (22.26): ⎤ ⎡ ⎢⎢ (T )2 κ T ⎥⎥⎥ ⎢ κEb (T) κEb (T) ⎢⎣1 + 6CTRI,1 2 + 4CTRI,2 ⎥⎦ , κT T

(22.27)

where CTRI,1 and CTRI,2 are new unknown constants, to be determined by fitting the model predictions to experimental data. Initially, Snegirev proposed a value of 1.25 for CTRI,1 based on consideration of data for jet flames, but later revised the value to 2.5 based on consideration of additional data for buoyant flames. The value of the second constant, CTRI,2 , remains in question since it is strongly problem dependent. Direct evaluation of κ T using assumed PDFs is quite challenging since κ may be a complicated nonlinear function of temperature. In an effort to avoid direct evaluation of κ T , researchers have often expressed it as κ T ≈ (T )2 (dκ/dT)T . Using this approximation is advantageous since (T )2 can be determined readily. However, determination of the derivative of the absorption coefficient comes with its own set of challenges, depending on the spectral model being used. Furthermore, in gaseous mixtures, the absorption coefficient is also a function of species concentrations and, therefore, expanding its temperature dependence alone as a Taylor series is not particularly beneficial [129]. In order to circumvent these difficulties, in a recent study, Fraga et al. [142] expressed (fitted) the higher-order correlations in terms of the lower-order correlations by employing time-resolved data obtained from LES of pool fires. Although the inclusion of higher-order terms resulted in improved accuracy of emission TRI by about 15%, it is not clear if the proposed curve fits are applicable to all flame types. For further reading on the aforementioned issues, and others pertaining to such RANS-based modeling of the emission TRI term, the reader is referred to [134,142,143]. The second approach to modeling emission TRI is to compute κEb directly without decomposing either the absorption coefficient or the emissive power into their mean and fluctuating components. However, this is possible only if the instantaneous κ and Eb are available, which essentially implies having the composition field available at all instances of time, since κ and Eb are functions of temperature and species concentrations. Henceforth, we will refer to this approach as the instantaneous field approach to modeling TRI. Here again, two different approaches have been pursued: (a) stochastic approach in which the instantaneous flow and/or composition fields are represented by stochastic variables, and (b) deterministic approach, in which the instantaneous flow and/or composition fields are computed directly by solving the governing conservation equations. In the stochastic approach, the first step is to convert the governing conservation equations into a hyper-dimensional partial differential equation that describes the evolution of the joint probability density function of the turbulent

Radiation in Chemically Reacting Systems Chapter | 22 829

flow and scalar fields. Two methods have been reported in the literature for solving this hyper-dimensional partial differential equation. The traditional and more established method is to transform this equation into a set of Lagrangian stochastic differential equations, tracking notional particles as a function of time to determine the evolution of the aforementioned joint PDF. Since the resulting equations are Lagrangian, in principle, no grid is necessary to resolve the length scales of the turbulent field. However, this approach requires stochastic models for the description of molecular mixing. A recent alternative approach to determine the same PDF is the so-called stochastic Eulerian field approach, wherein a system of Eulerian fields is constructed such that its one-point, one-time joint PDF evolves in exactly the same manner as the joint PDF of the turbulent reacting flow field. This method is amenable to implementation in standard frameworks for solving partial differential equations since the solution is obtained on a grid. For a better understanding of these and other approaches, the reader is referred to the review monograph by Haworth [102]. In the deterministic approach, on the other hand, no models are necessary to predict the turbulent field. However, in this case, the challenge is to use a grid that is fine enough to resolve all the relevant length scales. Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) of turbulence belong in the second category, while transported PDF methods belong in the first category. We will begin our discussion with the first category, and eventually move on to the second. The transported PDF method for modeling turbulent combustion can be traced back to the early works of Pope [96], who specifically proposed it for addressing turbulence–chemistry interactions. The philosophy of the transported probability density function (PDF) approach is to consider the dependent variables describing the reacting turbulent flow field (v, h, Y, etc.) as stochastic variables and consider the transport of their PDFs. The great advantage of PDF methods is that once the PDF has been determined by solving the PDF transport equation, the mean for any quantity, say a source Q˙  , can be evaluated directly from the PDF, provided Q˙  is a function of local composition variables φ only. This leads to ∞ f (ψ) Q˙ (ψ) dψ,

Q˙  =

(22.28)

0

where ψ represents the sample space for the composition variables φ (for example, 0 ≤ ψs < ∞ is the range of values that the last composition variable, φs = h, can attain); f (ψ) is the probability density of the compound event of φ = ψ (i.e., φ1 = ψ1 , φ2 = ψ2 , . . . , φs = ψs ), so that f (ψ) dψ = probability(ψ ≤ φ ≤ ψ + dψ).

(22.29)

In recognition of the fact that modeling TRI encompasses a similar set of challenges as modeling turbulence– chemistry interactions, the first study on modeling TRI using the transported PDF method was conducted by Mazumder and Modest [132,144]. They considered a methane–air diffusion flame and a nonreacting combustion gas mixture, respectively. The governing transport equation for the velocity-composition joint PDF was solved using the Monte Carlo method. The optically thin fluctuation approximation was invoked, i.e., absorption TRI were neglected. Using the box model of Section 19.4 for radiation, they were able to evaluate turbulence–radiation interactions without any additional approximation. Studying nonreacting flows [132], it was confirmed that, indeed, TRI are seldom of great importance in nonreacting flows, never changing radiative sources and fluxes by more than 10%. On the other hand, in the methane–air flame, heat loss rate was found to increase by up to 75% [144] due to emission TRI effects. A recent DNS study of a nonreacting jet of water vapor [145] has confirmed the decades-old finding that TRI only have minor influence on net radiative emission in the absence of chemical reactions. The study also showed that radiation actually assists in damping out the fluctuations in temperature, and the damping is caused by the larger energy-containing eddies rather than the smallest ones.

Nonpremixed Jet Flames Since the early 1990s, the importance of radiation in turbulent nonpremixed jet and bluff-body flames has been investigated by a large number of researchers [44–52,84,86,146–148]. Although the early studies neglected TRI, they consistently revealed, much like in laminar nonpremixed flames, two important effects when radiation is taken into account: (1) the flame becomes colder, and (2) the flame becomes shorter. Early studies attributed much of these effects to the presence of soot in the flame since combustion gases were usually modeled as

830 Radiative Heat Transfer

optically thin and gray. More recent studies that account for the nongray nature of molecular gases, however, have revealed that radiation plays a similar role even in nonsooting flames. A systematic analysis of turbulence–radiation interactions in two-dimensional, axisymmetric, nonluminous jet diffusion flames was first carried out by Li and Modest [149–151]. They employed a hybrid approach, using a commercial finite volume code (Fluent [152]) together with the composition PDF method [96,102], and also invoked the thin eddy approximation (OTFA). The composition PDF is the simplest form of the PDF methods since it carries information for the composition variables only, collected in the vector φ, which contains the s−1 mass fractions Y and the enthalpy h. The transport equation for the composition PDF for radiating reactive flow has been developed by Li and Modest [149], also based on the extensive work of Pope [96]. This resulted in a partial differential equation in s + 4 independent variables (time, space, and composition variable space), which—because of its high dimensionality—is generally solved using stochastic particle tracing (or particle Monte Carlo) methods [96,102,153,154]. The composition PDF carries no information on the velocity field and, therefore, must be combined with a deterministic RANS solver to provide the solutions to the mean momentum equations as well as a turbulence model (such as k–ε). A similar approach, referred to as the Stochastic Eulerian Field approach, combines steady flamelet models with a PDF for the mixture fraction and enthalpy, and a RANS solver for the velocity field. It has been used extensively by Consalvi and coworkers [148,155–158] for the study of TRI in turbulent flames. Li and Modest employed a simple single-step mechanism for chemistry, and the FSK method of Section 19.10 together with the P1 -approximation for the evaluation of thermal radiation from the combustion gases (CO2 , H2 O, and CH4 ). Flames were characterized through nondimensional parameters, namely, Reynolds number Re (describing jet velocity, flame size, turbulence level), optical thickness τL (flame size), Damköhler number Da (flow time scale vs. chemical reaction time scale), and Froude number Fr (buoyancy effects), and their impact on turbulence–radiation interactions was assessed. Their base configuration was Sandia Flame D [159], for which an abundance of experimental measurements is available (including radiation data). However, Sandia D is a small laboratory flame (as are most experimentally documented flames) with, therefore, relatively little radiation. Thus, Li and Modest also studied flames scaled up by factors of 2 and 4 to determine radiation and TRI effects in larger flames. It was found that TRI affect the flame in two ways: (1) emission from and self-absorption by the flame are both strongly, and about equally, increased and (2) the additional net heat loss causes the flame to cool (and this, in turn, can substantially lower emission as well as chemical reaction rates to the point of flame extinction). Not surprisingly, the strength of TRI is most strongly sensitive to the flame’s optical thickness. Optically thin flames have little emission and, thus, lose relatively little heat by radiation; TRI cause this loss to increase by a substantial 50%, but decrease flame temperature only by a small amount (maybe 20◦ C). This additional heat loss causes optically thick flames to cool down substantially (by 100◦ C and more), resulting in a sharp drop in emissive power, and overall heat loss rates are only increased by a few percent. To isolate the importance of the various terms that contribute to the total TRI effect, Li and Modest [150] focused on “frozen” composition variable fields for several flames (using the converged temperature and species mass fraction fields for the flame with fully considered TRI). They determined the various radiative contributions to flame emission and self-absorption under a number of different scenarios. It was found that, on a percentage basis, the increase in radiative heat losses due to TRI is essentially independent of optical thickness: for all three flames, both emission and self-absorption are consistently increased by about 60%. However, in optically thin flames this translates into a net additional loss of 50%, since temperature levels (in an “unfrozen” field) decrease by only 20◦ C or so. In optically thick flames, TRI reduce temperature levels by more than 100◦ C, and the net heat loss is hardly increased at all. The different underlying TRI mechanisms display similarly consistent trends: if only the Planck function self-correlation is considered, emission and absorption increase by roughly 35% for a gray medium. However, if the nongrayness of the combustion gases is accounted for, this increase is less than 10%, again regardless of optical thickness (absorption lagging behind emission, since it is a response to the raised emission level). The reason is that the gas radiates only over the fairly narrow absorption–emission bands, across which the nonlinearity of the Planck function is much less severe. Even for a gray medium, for which the Planck function self-correlation is the most important driving force of the TRI, it by no means dominates the interaction. The strongest contributions to TRI always come from the correlation between absorption coefficient and Planck function fluctuations. These findings are qualitatively consistent with the findings of Coelho [131,134] and other researchers [137,138,143], who arrived at similar conclusions using RANS-based models combined with assumed PDFs.

Radiation in Chemically Reacting Systems Chapter | 22 831

FIGURE 22.4 Comparison of centerline temperatures in an axisymmetric methane–air jet diffusion flame, with and without turbulence– radiation interaction; from Wang and coworkers [10].

The effect of pressure on emission TRI has been studied in sooting turbulent diffusion flames by Nmira et al. [160]. They found that emission TRI due to the gas are always positive and increases with pressure. On the other hand, emission TRI due to soot are negative, it being dominated by the temperature–soot–volume fraction correlation (resulting from the time-average of the product of temperature and absorption coefficient of soot). Mehta et al. [161,162], who studied turbulent jet flames under atmospheric pressure conditions, showed that the temperature–soot–volume fraction correlation is generally negative in the high-temperature regions of the flame, although their studies showed that the temperature self-correlation, which is always positive, generally dominates the emission TRI term for soot, making it positive, although not as strongly positive as its gas radiation counterpart. As to how exactly the temperature–soot–volume fraction correlation behaves— magnitude wise—depends strongly on the chemical reaction mechanism being used to predict soot formation and burn-out. Nonetheless, both studies appear to suggest that this correlation is generally negative. Nmira et al. further contend that at high pressure, when soot radiation can be substantial, the gain in gas emission TRI can be partially offset by the loss in soot emission TRI, thereby limiting the increased heat loss by the flame due to TRI effects. In contrast, in nonsooting methane–hydrogen–air flames, Yang et al. [147] found that the radiative heat loss increased strongly with increase in pressure due to increased emission TRI. Several investigators have investigated Sandia Flame D in the context of TRI [5,10,163–165], most of them at a lesser level of sophistication than the work of Li and Modest [149,150], but all providing consistent answers for the quantitative importance of radiation and turbulence–radiation interactions. The most advanced and accurate model to date of Flame D is the one by Wang and coworkers [10], using models similar to those of Li and Modest, but employing a more advanced composition PDF code, a more realistic chemical reaction mechanism, and the line-by-line accurate photon Monte Carlo scheme described in Section 20.6, which was specifically developed for the stochastic media employed in transported PDF methods. As can be seen from Fig. 22.4, their model predicts centerline temperatures very well, but also that radiation (with or without TRI) has little influence in a small, optically thin flame. Wang and coworkers also scaled up the flame by factors of 2 (kL2) and 4 (kL3) (but in a different way from Li and Modest to preserve realistic chemistry). Consistent with Li and Modest’s observation, they noted that increasing flame size, and thus its optical thickness, increases radiative heat loss while also reducing temperature levels. The relative importance of TRI was found to be independent of optical thickness (roughly 30% for all flames). While temperature levels in optically thin flames are only weakly dependent on radiation, pollutant levels tend to be a strong function of temperature. Pal et al. [5] used Wang and coworkers’ code to investigate NO

832 Radiative Heat Transfer

FIGURE 22.5 Radial profiles of NO mass fraction at various axial locations of Flame D: (a) mean values, (b) RMS fluctuations; from Pal et al. [5].

levels in Sandia D, as shown in Fig. 22.5. Radiation is seen to decrease NO levels appreciably (due to the slightly lower temperatures). The agreement between experiment and theory is rather encouraging, and Fig. 22.5 clearly demonstrates the importance of radiation and TRI on mean pollutant levels and their turbulent fluctuations: Radiation lowers temperatures in the center of the flame (lowering NO levels), but heats colder regions further away (increasing NO). Radiation’s “action at a distance” decreases RMS fluctuations everywhere, except for colder regions that have no NO at all without radiation. These effects are amplified in larger flames, with greater influence of radiation and TRI on temperatures: predicted NO levels decrease by orders of magnitude when radiation and TRI are taken into account. The first attempt to quantify absorption TRI was made by Tessé and coworkers [166], who investigated a small sooting (luminous) ethylene flame, using detailed chemistry and a sophisticated soot model [167], together with a Lagrangian solver to obtain the composition PDF. They then constructed many homogeneous turbulence structures from this PDF and determined the thermal radiation with a photon Monte Carlo scheme together with the narrow band k-distribution model of Soufiani and Taine [168]. They found emission to increase by 30%, and also found absorption TRI to be appreciable (5% of total emission) for this luminous flame, indicating eddies of appreciable optical thickness. The first ones to assess absorption TRI from basic principles (i.e., without the assumptions for turbulence structures made by Tessé) were Wang and coworkers [10], who used a transported composition PDF to determine composition variables and their turbulence moments, together with Wang’s [169,170] LBL-accurate photon Monte Carlo scheme for stochastic particles. This radiation solver was specifically developed to determine a PDF for photons, providing full compatibility with the stochastic turbulence model. With their model Wang and coworkers [10] provided proof that absorption TRI are negligible for Sandia D and, indeed, also for large nonluminous flames. The method was further employed to investigate the influence of TRI in sooting flames: Mehta et al. [161,162,171] modeled six sooting flames [172–174] using Wang and coworkers’ [10] schemes together with a sophisticated soot model [175], to assess the importance of both emission and absorption TRI in such systems. They found emission TRI (30% to 60%) and heat losses from the flame (increases of 45% to 90%) to be stronger than in nonluminous flames. However, in contrast to Tessé’s [166] observations, absorption TRI were found to be negligible for all six laboratory-scale flames, despite the soot. Only when scaling up the sootiest flame [173] by a factor of 32 did absorption TRI become appreciable (6% of total emission). Similar observations were made by Consalvi and coworkers [156], who found that absorption TRI are dominated by soot, but even in extreme cases (artificially scaling the size of a laboratory-scale flame by a factor of 50), its effect on the radiative source is no more than 10%. With increasing availability of supercomputing resources, the second category of the instantaneous field approaches, namely, DNS and LES of turbulent combustion, is slowly becoming realizable. In particular, LES of turbulent combustion is being conducted by many research groups worldwide. As discussed earlier, turbulence– radiation interactions may also be assessed using LES and DNS. Chandy and coworkers [176] were the first to

Radiation in Chemically Reacting Systems Chapter | 22 833

FIGURE 22.6 Radiative heat source from LES/FDF computation of a methane-air jet flame at an axial location of x/d jet = 15; reproduced from [179].

study TRI using LES together with a filtered density function (FDF) for the composition variables (the LES equivalent of the transported PDF method but used only at the subfilter or subgrid scale, or SGS), looking at an idealized luminous flame with a primitive soot model. They concluded that, while emission TRI are always important at the SGS level, absorption TRI at the SGS level can always be neglected. Roger et al. [177,178] also showed SGS absorption TRI to be negligible by using DNS of stationary isotropic turbulence. Similar conclusions were drawn by Gupta et al. [179], who used a similar LES/FDF approach, but coupled with Wang’s [169,170] LBL-accurate photon Monte Carlo scheme. Figure 22.6 shows the radiative heat source reported in their study for the Sandia D methane-air jet flame configuration scaled up by a factor of 4 and with soot. The filter size for these LES calculations was chosen such that 84% of the turbulent kinetic energy was resolved by the grid. Several conclusions may be drawn from Fig. 22.6. First, the inclusion of TRI significantly alters the radiative source. Second, emission TRI at the SGS level have significant impact on the radiative source. Third, absorption TRI at the SGS level appear to have no impact on the results. Although not shown here, the impact of emission TRI at the SGS level on the flame temperature was estimated to be as high as 60 K. Studies by Coelho [180] and by Consalvi et al. [181] also confirm that the OTFA can be more reliably used at the SGS level in LES calculations than in RANS calculations. Poitou et al. [182] conducted LES of turbulent propane-air flames with two-step chemistry. The RTE was solved using the discrete ordinates method, and the SNB-FSCK model was used for spectral modeling of the combustion gas mixture. Both emission and absorption TRI were included, although subgrid scale absorption TRI effects were considered negligible. Their results showed that TRI increase radiative loss from the flame by about 7.4%. The data obtained from the LES were also analyzed posteriori, and the role of various turbulent moments that arise in RANS calculations were quantified. Although there is unanimity in the finding that SGS emission TRI are non-negligible, one of the critical issues in computing TRI in LES of reacting flows is the effect of filter size on the computed SGS TRI. In a study, which extended to turbulent planar jet flames, Roger et al. [183] attempted to address this important issue by quantifying SGS emission TRI contributions using data obtained from their own DNS calculations and various filter sizes typically used in LES. Figure 22.7 shows a sample result. In the planar jet flame considered in this study, the jet is centered at y/H = 3, and its two edges are approximately at y/H = 1 and y/H = 5. TRI-related quantities were computed as a postprocessing step using temperature and concentration fields obtained using DNS. A line-of-sight ray tracing scheme was combined with the correlated-k distribution model for spectral modeling. Only the effect of carbon dioxide on radiative properties was considered. Figure 22.7a shows the mean temperature distribution, which highlights that the center of the jet flame (at y/H = 3) reaches 1400 K, while its fringes (y/H ≤ 1) are cold. Figure 22.7b shows the ratio of SGS emission TRI to total emission TRI for

834 Radiative Heat Transfer

FIGURE 22.7 Results reproduced from a turbulent planar jet flame [183]: (a) mean temperature in K and (b) ratio of SGS emission TRI to total emission (includes mean as well as TRI contributions at all scales), denoted by R.

various filter sizes. The quantity δ is the DNS grid size, while the length scale for dissipation of the turbulent fluctuations is approximately 4δ. Most reacting flow LES calculations, on the other hand, are conducted using filter size in the range 8δ–16δ. According to Fig. 22.7b, for this filter size range, SGS emission TRI peak at the fringes of the flame rather than at the centerline, and is substantial. If a filter size close to 4δ is used instead, SGS emission TRI may be neglected altogether. A second conclusion that may be drawn from these results is that since the peaks occur in cold regions of the flame (Fig. 22.7a) where emission is weak to begin with, SGS emission TRI may be considered negligible irrespective of the filter size used. In contrast, other studies [176,179] suggest that SGS level emission TRI cannot be neglected. It is difficult to directly compare the results reported in various studies. For example, it is not clear how the filter size chosen by Gupta et al. [179] correlates with those considered by Roger et al. [183]. Furthermore, Roger et al. considered only carbon dioxide in their calculations, while Gupta et al. considered all combustion gases, as well as soot. A limited number of DNS studies of turbulence–radiation interactions have also been conducted. Early on, the group around Haworth and Modest [184–188] performed DNS calculations for a number of artificial scenarios. In a recent study, Kang [189] simulated a premixed flame using DNS. However, the absorption coefficient of the combustion gas mixture was assumed to be gray (but still a function of temperature and concentrations). TRI effects, in particular the effect of the temperature self-correlation term, were assessed and compared against the results presented by Wu et al. [184], and showed good agreement. Rejeb and Echekki [190] also assessed TRI in a gray medium similar to the one considered by Kang [189]. They developed a model that combined LES for turbulence with the Monte Carlo method for photon tracing. The model results were assessed against their own DNS calculations and showed reasonable agreement. A survey of the literature reveals that although LES of turbulent flames with the inclusion of TRI is becoming increasingly popular, to date, DNS simulations of radiation and TRI in reacting systems are limited to system sizes in the mm range, i.e., ranges over which combustion media are optically extremely thin, and consequently, radiation is not altered by the medium at all. Today, the study of turbulence–radiation interactions remains an extremely active field of research. For further reading, the reader is directed toward several review articles by Modest [191–193], a recent text book by Modest and Haworth [194], and a recent review article and an exhaustive monograph by Coelho [131,195].

Confined Flames Study of the role of radiation in turbulent flames in confined spaces, such as internal combustion engines and gas turbine combustors have received little attention to date. In such applications, because of the confinement, there is strong recirculation and large-scale mixing, making the distributions of species concentrations and temperature quite complex, which, in turn also affects the distribution of radiative properties.

Radiation in Chemically Reacting Systems Chapter | 22 835

FIGURE 22.8 Effect of TRI on radiative source in a diesel engine; from [80]: (a) part load, and (b) full load engine operating condition. Computations were performed using a line-by-line photon Monte Carlo radiation solver coupled to the transported PDF method for reacting flow.

The earliest study of radiation in internal combustion engines can be traced back to Mengüç et al. [196]. In their study, the RTE was solved using both the P1 and P3 approximations. The gases were considered optically thin, but absorption and scattering by the fuel droplets was considered. The contribution by radiation was estimated as a post-processing step with “frozen” flow, temperature, and species distributions. The effect of scattering by the droplets was found to be negligible. Abraham and Magi [197] used the discrete ordinates method (DOM) to compute radiative heat loss in a Diesel engine. Gray radiative properties appear to have been used in this study, which concluded that radiation lowered NOx concentrations (due to reduction of the flame temperature), but soot concentrations remained almost unchanged. Wiedenhoefer and Reitz [198] also computed radiative heat transfer contributions in a Diesel engine using the DOM. They used a wide band model for molecular gases and soot, and concluded—based on comparison with experimental data—that the low-order S2 DOM is adequate for predicting the radiative source for the operating conditions considered in their study. Yoshikawa and Reitz [199] investigated the effect of radiation on soot and NOx formation using a gray gas assumption. They concluded, contrary to studies before them, that radiation has negligible influence on both NOx and soot. The aforementioned investigations used simplistic models for spectral radiative properties and the treatment of turbulence and TRI. In recent years, more sophisticated approaches have been used in some studies. Bolla et al. [200] investigated the influence of radiation with and without inclusion of turbulent fluctuations on NO formation in Diesel engine spray combustion environments. They used the transported PDF model as well as a simplified mixing model for the treatment of turbulence–chemistry interactions. Their studies showed that inclusion of turbulent fluctuations, which affect both turbulence–radiation and turbulence–chemistry interactions, reduces NO concentrations by 5–10%. However, no experimental data were available to validate their claim. Paul et al. [39] conducted RANS-based CFD calculations in an engine-like configuration along with a variety of spectral models and RTE solvers to gain a quantitative understanding of computational efficiency vs. accuracy of each radiation model. Their calculations showed that a simple stepwise-gray spectral model coupled with a P1 -based RTE solver yielded results within 10% of the most sophisticated model (Monte Carlo for the RTE coupled with a line-by-line spectral model) but at only one-thirtieth of the computational cost. Finally, in a very comprehensive investigation of the effect of radiation and TRI on in-cylinder combustion in Diesel engines, Paul et al. [80] explored several different spectral radiation models and RTE solvers. It was found that TRI effects amount to less than 10% of the total radiation source even when temperature fluctuations are as high as 100 K, as shown in Fig. 22.8. Furthermore, the effect of radiation itself on emissions coming out of the engine and heat loss was found to be less than 10% of their values computed without radiation. In gas turbine combustion computations to date, typically radiation has either been neglected or addressed using simplified models that generally treat the medium as gray [78,201,202]. In a recent study, Ren et al. [79]

836 Radiative Heat Transfer

conducted line-by-line calculations in a gas turbine combustor using the photon Monte Carlo method. A detailed chemical reaction mechanism that included nitrogen chemistry was used so that the effect of radiation on NOx formation could also be investigated. Temperature, CO2 , and H2 O distributions with and without radiation were found to be within a few percent of each other. However, that slight difference in temperature was large enough to manifest itself in significant differences (few tens of percent) in the predicted NOx concentrations, indicating that taking radiation into account in gas turbine combustor calculations is important if the goal is to predict pollutant formation accurately. This particular study neglected the effects of TRI, which could further alter the results. Kez et al. [203] assessed the accuracy and efficiency of several state-of-the-art spectral models applied to CFD calculations of a gas turbine combustor under oxygen-rich conditions and high pressure. These include the statistical narrow-band based correlated-k (SNBCK) model, the full-spectrum correlated-k (FSCK) model, the wide-band based correlated-k (WBCK) model, and the weighted-sum-of-gray-gases (WSGG) model. It was found that the FSCK model offered the best compromise between accuracy and computational efficiency. TRI were also neglected in this study.

Multiphase Combustion In many combustion systems, the fuel enters the combustor either in the liquid phase as droplets or in the solid phase as tiny particles. Radiation modeling in such systems is rendered complex by the fact that, in addition to absorption and emission by gases (products of combustion), one has to account for absorption and strong scattering by the second phase. Multiple scattering and absorption events prevent the emitted radiation from escaping the combustion zone (to the walls) and often helps sustain combustion. Radiation in spray and coal combustion systems has been studied since the late 1980s. Early investigations primarily focused on how radiation interacts with a single spherical droplet or particle, much of which has been discussed in Chapters 11 and 12. Later investigations [58–60,204–211] included radiation in CFD calculations of the combustor. However, in the overwhelming majority of these studies, the effect of the liquid phase (droplets) on radiation (or multiphase radiation) was not investigated. In a first attempt, Kurose and coworkers [210] considered scattering by droplets in their simulation of a spray combustor with n-decane as the fuel, but neglected absorption and emission by the droplets. A very simple model that uses average volume fraction and scattering efficiency of a cloud of droplets was used to compute a gray scattering coefficient. Scattering was assumed to be isotropic. The same model was used in another study [207] by the same group that conducted DNS of a turbulent spray flame, also with n-decane as the fuel. Multiphase radiation was considered in a comprehensive study by Roy et al. [208]. The droplets—Diesel, in this particular case—were assumed to be nonemitting, but nongray absorption and scattering were accounted for. The spectral properties of the droplets were computed using a wavelengthdependent complex index of refraction. In addition, two other limiting scenarios—fully transparent and fully opaque droplets—were also considered. The RTE was solved using the Monte Carlo method, which was also developed as part of their study, and integrated with a detailed CFD calculation of a combustor. Under the high pressure operating conditions considered in this investigation, the effect of multiphase radiation was found to be significant only on trace species, such as C2 H2 . Major species, such as carbon dioxide and water vapor, as well as temperature distributions were mostly unaffected when multiphase radiation was considered. Mukut and Roy [211] attempted to isolate multiphase radiation effects by conducting calculations with and without multiphase radiation, with detailed spectral modeling of the gas and soot included in both cases. Once again, it was found that inclusion of multiphase radiation only affects the temperature distribution marginally; local hot and cold spot distributions change slightly. As a result, the impact of multiphase radiation on NO formation is more pronounced, since NO formation is sensitive to local temperature. A number of investigations have also been conducted to elucidate the role of radiation in combustion environments that involve coal and other solid particulates [61–68,212]. Most notably, Selçuk and coworkers [63– 67] focused on coal combustion in circulating fluidized bed combustors under both air- and oxygen-fired conditions and compared their predictions against their own experimental data. In most of their studies, nongray gas radiation was modeled either using the SLW model or the SNBCK model, while radiative properties of particles (coal and fly ash) were modeled using models of varying degrees of sophistication. In some cases, geometrical optics was used to compute the scattering coefficient, while in other cases Lorenz-Mie theory was used. The following main conclusions were reached based on these studies: (a) the gray particle assumption is quite accurate for heat flux predictions, but produces non-negligible errors in radiative source term predictions; (b) considering nongray particle properties is important if the composition of the particle (such as presence of

Radiation in Chemically Reacting Systems Chapter | 22 837

iron oxides) is to be considered, as the presence of such chemicals alters the complex index of refraction of the base material; (c) results of forward scattering lie between those of “no scattering” and isotropic scattering; (d) the isotropic scattering assumption leads to underprediction of both heat flux and radiative source term; and (e) particle size distributions of coal have a significant impact on the predicted radiative source term. Wu et al. [68] studied the effect of particulate radiation in a coal-fired jet flame. They concluded that the gray particle approximation for coal considerably underpredicts the radiative source term (although less than the effect of a gray gas approximation), while its effect, when applied to fly ash, is negligible.

Buoyant Flames and Fires Buoyant flames and pool fires, in particular, are often caused by oil or chemical spills or by incendiary solid fuels (such as dry vegetation). Therefore, such flames constitute an important application for health and safety. A distinguishing feature of such flames is that air is entrained into the flame and the degree of combustion is limited by the supply of air. As such, such flames usually burn under fuel-rich conditions, thereby producing a lot of unburnt hydrocarbon particles and gases, soot, and smoke. The study of radiation in fires can be traced back to the early 1970s [213–217]. Most notably, in a combined study, Modak [217] presented a closed-form analytical model to predict the burning rate and net radiative power from a pool fire and compared the predictions against experimental measurements. In his model, the entire flame was treated as homogeneous (uniform temperature and species concentrations) and gray radiation only from soot was considered. It was found that the predicted radiative heat fluxes on the surface of the fuel pool agreed reasonably well with measured data, thereby asserting the claim that soot may be treated as gray in radiation models for such flames and fires. In the 1980s, several experimental studies were conducted and semi-empirical models were proposed to predict the “effective emissive power” of large-scale pool fires [218–226]. These studies contend that the effective emissive power decreases with increasing pool size. In one of the first studies that specifically focused on the role of turbulence in large-scale pool fires, Grosshandler [123] measured the fluctuating components of velocity, temperature, and species concentrations in an ethanol pool fire of 0.5 m diameter. Line-of-sight intensity calculations showed that the use of time-averaged temperature and concentrations underpredicted the measured intensity—one of the earliest known evidences of TRI in fires. Over the past two decades, a number of studies have contributed to better understanding of the role of radiation with or without TRI in large-scale fires [36,38,87,136,227–240]. Chatterjee et al. [236] conducted LES of a heptane pool fire, wherein radiative properties, including those of soot, were assumed to be gray. TRI were taken into account and the model proposed by Snegirev [141] was used for closure of the subgrid scale TRI terms. Although spatial distributions of predicted radiative fluxes and temperature were found to be in good agreement with experimental data, the roles of radiation and TRI were not isolated. Consalvi [136] studied TRI in a small-scale nonsooting methane pool fire for which experimental data are also available. The FSCK model was used to treat nongray radiation in gases, and absorption TRI were neglected. A RANS-based model was used to simulate the pool fire. Overall, the findings of this study are consistent with those of nonpremixed flames: TRI significantly enhance the radiative loss from the flame, and emission TRI are equally influenced by both the temperature self-correlation and the temperature–absorption coefficient correlation. Fraga et al. [238] investigated the role of TRI in both ethanol and methanol pool fires using LES. Nongray gas radiation was treated using the WSGG model. One unique contribution of this study is that it isolated the effect of temperature and species concentration fluctuations on TRI; it was found that temperature fluctuations play a more important role in accurately determining TRI as opposed to species concentration fluctuations. Recently, Sikic et al. [239] and Snegirev et al. [237] also performed LES of buoyant flames including TRI, although the former study neglected the temperature–absorption coefficient correlation. In both studies, the WSGG model was used for nongray gas radiation. Snegirev’s study found that soot accounts for about two-third of the emission from large-scale heptane buoyant flames. Wu et al. [240] used a detailed photon Monte Carlo based line-by-line solver for the RTE in conjunction with an LES code for flow and chemistry. Their computations exhibited excellent match with experimental measurements, including spectral line-of-sight intensities, even though TRI were neglected in this study.

Oxy–Fuel Combustion Oxy-fuel combustion is being touted as a promising technology for carbon sequestration and capture in power plants, particularly those that are coal fired. In this technology, the nitrogen in air is replaced by a flue gas

838 Radiative Heat Transfer

that is comprised of the recycled products of combustion, namely carbon dioxide and water vapor. Since the carbon dioxide is recycled, its concentration is high, making it easier to sequester. Furthermore, the incoming oxygen concentration can be controlled independently by altering the amount of recirculated gas (as opposed to oxygen having a fixed mole fraction in air). From a radiation perspective, prevalence of high concentrations of carbon dioxide and water vapor translates to more pronounced gas radiation effects. Also, excess amounts of carbon containing compounds within the combustor can potentially lead to more soot formation. Over the past decade or so, a number of investigations have been conducted to understand the effect of increased combustion gas concentrations and/or soot on radiative transport [63,82,203,241–253]. Leckner and coworkers [241,242] conducted measurements of radiation intensity in propane and lignite-fired flames under both air and oxy-fired conditions. In the case of propane flames, significant differences were observed between the two conditions. Under oxy-fired conditions, radiation intensities were higher even though the temperatures in oxy-fired flames were slightly lower than air-fired flames, presumably due to the dominance of soot radiation under oxygen-rich conditions. In the case of lignite-fired flames, however, they noted that if the conditions could be adjusted to keep the temperature distributions in the two cases similar, the radiation intensities were also similar even though the two flames had markedly different carbon dioxide amounts, probably because the radiation from the burning particles dominated the intensity signature. Additional investigations [245], which included modeling studies using the statistical narrow-band model, confirmed the importance of soot radiation in the propane flames and the role of particle radiation in lignite flames vis á vis gas radiation. In realization of the fact that CO2 /H2 O molar ratios can be significantly different in oxy-fuel combustion compared to traditional air-fuel combustion, previously used nongray models for gas radiation have been revisited. Yin et al. [244,246], Krishnamoorthy et al. [243], and Johansson et al. [248,254] independently proposed revised sets of parameters for the WSGG model that accounts for oxy-fuel conditions, and used their resulting model within RANS-based CFD calculations of full-scale oxy-fired combustors. Soot and lignite particles were assumed to be gray in these calculations. Edge et al. [247] conducted both RANS-based CFD and LES of an oxy-fired pulverized coal combustor using both the WSGG model (unrevised version) as well as the FSK model. Particles were assumed to be gray. Comparison to experimental data showed best agreement when LES was used in conjunction with the FSK model. Johansson et al. [82] used the Malkmus statistical narrow band (SNB) model for computing gas radiative properties and Mie theory for computing particle properties. Bordbar et al. [249] proposed a new set of parameters for the WSGG model that were derived directly using the HITEMP2010 database, rather than the exponential wide band model used by Yin et al. [244,246] or the SNB model used by Johansson et al. [248,254] to compute total emissivity and fit the parameters. The accuracy of their model appears to be similar to the one proposed by Johansson and coworkers. Ozen et al. [63] have used the SLW model to model gas radiation inside a circulating fluidized bed combustor under both air and oxy-fired conditions.

22.5 Comparison of RTE Solvers for Reacting Systems Challenges associated with coupling the RTE to reacting flow solvers have been summarized in Section 22.2. In this section, progress and challenges in solving the RTE for chemically reacting systems are discussed, with consideration of issues such as accuracy, computational efficiency, and ease of implementation of the various RTE solvers and spectral models employed by researchers over the years. Earliest efforts at benchmarking RTE solvers for chemically reacting systems can be traced back to a paper [255] that summarizes the results of comparing solutions to the RTE for a benchmark problem—a mixture of carbon particles, nitrogen and carbon dioxide confined within a box of size L × H × W. In Part 1 of the problem, two of the dimensions of the box were considered infinitely long making it a one-dimensional slab problem, while in Part 2, the full three-dimensional (3D) solution was considered. In Part 3, carbon particle concentration and temperature were varied spatially, i.e., the medium was made nonhomogeneous. The spectral absorption coefficient of carbon dioxide was specified (Elsasser narrow-band model with specified parameters; temperature dependence of the absorption coefficient was ignored), as was the scattering phase function. Seven groups generated results using the Monte Carlo method [256], the YIX method [257], the Discrete Exchange Factor method [258], and the Zonal method [259]. The results highlighted significant differences between the various methods and it was concluded that although significant progress had been made in RTE solutions for 3D media, accuracy remained a major concern especially for nonhomogeneous media. No attempt was made to compare computational efficiencies of the various methods. To this day, benchmarking RTE solvers for reacting flows remains a worthwhile activity for the thermal radiation community. In 2016, a workshop on radiative heat transfer generated a new set of

Radiation in Chemically Reacting Systems Chapter | 22 839

TABLE 22.1 Summary of studies dedicated to comparison of RTE solvers and spectral models in reacting systems. Topic

References

RTE solver

Spectral model

Laminar flame

[7,11,22,28,29,42]

PMC, DOM, FAM, P1 , MDA, SP3 , SP5

LBL, FSCK, WSGG, SLW, RADCAL, SNBCK

Turbulent flame, laboratory-scale

[5,6,8,9,146]

PMC, DOM, FAM, P1 , P3

LBL, WSGG, RADCAL, FSCK, MSFSK, MSMGFSK

Combustors, furnaces, and engines (turbulent)

[39,80,196,203,251–253,261]

PMC, P1 , DOM, FAM

LBL, SNB, FSCK, SNBCK, WBM (Box), WBCK, WSGG

Pool fire and buoyant flames (turbulent)

[36,38,239]

FAM

LBL, FSCK, MSFSK, WBM, NBCK, WSGG, SNB

Multiphase (spray, fluidized bed, etc.)

[62]

DOM

SLW, SNBCK

Other

[250,255,261–264]

PMC, DOM, FAM, P1 , YIX, Zonal, Hybrid

LBL, SLW, FSCK, WBCK, SNBCK, WSGG

benchmark problems that have been summarized in a “challenge paper” by Howell and Mengüç [260]. The problems proposed include a buoyant laminar flame, a buoyant turbulent flame, a high-temperature porous burner, and a steel reheating furnace. Solutions to these challenge problems have yet to be published. Beyond these formal efforts at benchmarking RTE solvers, numerous validation and verification efforts have been directed toward assessing the accuracy and computational cost of various RTE solvers and spectral models used in the context of reacting systems. A high-level summary is provided in Table 22.1. A handful of studies conducted such investigations for laminar flames. Notably, Consalvi and coworkers [22, 29,42] explored various RTE solvers and spectral models for the computation of axisymmetric laminar diffusion flames under a wide range of operating conditions including microgravity and with various hydrocarbon fuels. These include the standard DOM (with T3 quadrature), as well as the FAM, in conjunction with the FSCK and the SNBCK models for gas radiation and both gray and nongray models for soot. Although the two RTE solvers were not compared in any single study, comparison of the two spectral models showed that the FSCK model can yield results almost as accurate as the SNBCK model but with almost a factor of 20 reduction in computational time. For the FAM calculations, fine angular discretizations, such as 36×48 or 24×32 were used, although no comparison was made between various angular resolutions. Cai et al. [11] studied laminar hydrogen-air diffusion flames under atmospheric and high-pressure conditions. Benchmark RTE solutions were obtained using LBL absorption coefficient data for water vapor coupled to a photon Monte Carlo solver (LBL-PMC). The RTE was solved using the P1 , SP3 , and SP5 approximations, and the FSCK model was used as the nongray model. It was found that SP3 and SP5 did not offer any particular advantage over P1 in terms of accuracy, except at high pressure (30 bar) wherein SP5 outperformed the other two solvers. The computational cost incurred in solving the additional partial differential equations in either SP3 or SP5 was found to be marginal. The LBL-PMC calculations required approximately twice as much computational time as the FSCK computations with any of the three deterministic solvers. In the FSCK computations, the majority of the computational time was spent in assembling the k-distributions, and the actual solution time of the RTE was found to be a small fraction of the total time. In contrast, in the PMC method, most of the computational time was spent in sampling and tracing. In the PMC implementation, so-called “tempered averaging” (or time blending), equation (22.1), was used. As a result, approximately hundred times fewer bundles need to be traced per time step or iteration, resulting in significant improvement in the computational efficiency of the PMC method. Time blending has been found to be effective mostly for calculations where the steady-state solution is sought (as opposed to transient calculations). With modern techniques for assembling and using k-distributions [265], the computational overhead of the FSCK method can be reduced dramatically as well, as discussed in Section 19.12. Hence, although the PMC method produces the most accurate results, the accuracy comes at a significantly larger computational cost compared to the deterministic RTE solution techniques. In another study, Garten et al. [7] conducted computations in which the WSGG and SLW models were used in conjunction with the P1 approximation, the MDA, and the DOM to compute the radiation field in a laminar methane partial oxidation flame. The MDA combined with the SLW model appeared to provide the best compromise between accuracy and efficiency, although it was noted that computing the wall-emitted component of the intensity in the MDA is computationally expensive, and a

840 Radiative Heat Transfer

reasonable time-saving strategy is to compute it only once or a few times if the problem at hand permits that, i.e., if the temperature of the walls does not change substantially. While numerous studies have been performed on the role of radiation and TRI in laboratory-scale turbulent diffusion flames, surprisingly few studies have been dedicated to comparative analysis of RTE solvers and spectral models. Traditionally, RTE solvers and spectral models have been assessed using two strategies for turbulent reacting systems. The first is one where the temperature and concentration fields are first calculated without radiation, and subsequently assumed to remain unchanged or “frozen” for radiation calculations. The radiation source (∇ · qR ) is then calculated using various radiation models and compared. This strategy ensures that for radiation calculations, the inputs for each radiation model, namely, temperature and species concentration distributions, are identical and any discrepancies in results are due solely to the radiation model. Furthermore, the tedious task of computing temperature and concentration fields for a turbulent reacting flow needs to be conducted only once. The shortcoming of this approach is that quantities, such as temperature and species concentrations, that are typically measured, cannot be used to validate the predictions. The second approach is one where the RTE is fully (two-way) coupled with the flow solver, and the predicted temperature and concentrations of various species are then compared. While this approach does not precisely isolate and identify pros and cons of the radiation model per se, it provides a more complete picture of the sensitivity of the computed radiation source in predicting measurable quantities. Also, it reveals bottlenecks in the computational aspects of a radiation model since full two-way coupling is far more computationally challenging. As such, both strategies have been used extensively and have been found to be useful. Modest and coworkers [5,6] have investigated the performance of the FAM, P1 , and P3 solvers paired with sophisticated nongray models such as the FSCK, MSFSK, and MSMGFSK. Sandia Flame D, both unscaled and scaled by a factor of 4, were used as test cases for these investigations. Benchmark results for the radiative source (with frozen temperature and composition) were obtained using a PMC solver that uses LBL data directly (LBL–PMC). The study concluded that for optically thin flames (Sandia Flame D without scaling), the P1 solver coupled with a gray (Planck mean) model offers almost the same accuracy as the benchmark solution, but at a computational cost that is several orders of magnitude lower. For larger flames, the accuracy depends on the choice of the RTE solver and the spectral model; the P3 solver coupled with the MSFSK spectral model offered the best accuracy, while being computationally comparable in cost to the LBL–PMC model for stationary (steady state) calculations in which time blending was effectively used for LBL–PMC. For transient calculations, wherein time blending is not as effective for LBL–PMC, the highest fidelity deterministic solver outperformed the PMC solver in terms of computational efficiency by more than an order of magnitude. In another study, Orbegoso et al. [146] compared results (fully coupled) obtained using the DOM and two different variations of the WSGG model and RADCAL with equivalent computations conducted by Wang et al. [51] (using a P1 solver coupled with the FSK spectral model), as well as against experimental data for a turbulent soot-laden diffusion flame [266]. In some cases, the WSGG results were found to better match experimental data than the FSK model. However, it is not clear if the discrepancies were a result of different flow solvers and ensuing temperature/composition fields used in the two calculations or due to differences in the spectral models and/or RTE solvers. In an ongoing study, Ge et al. [8,9] compared PN and FAM solvers of various orders/angular resolution, coupled to the FSCK-2 model and the fast look-up table of Wang et al. [265] (see Section 19.12), for computation of Sandia Flame D scaled by a factor of 4. In their first effort they used a snapshot (frozen field) of the composition variable field obtained by Pal et al. [6], for which they also obtained benchmark results using a LBL-accurate PMC solver. The radiation sources predicted by various radiation models are shown in Fig. 22.9. For this situation the PN solvers consistently overpredict the radiation source, with P1 having the worst accuracy. The improvement in accuracy from P1 to P3 is fairly substantial, while the accuracy improvement exhibits diminishing returns in going from P3 to P7 . Based on these results, it may be concluded that P3 offers a reasonable compromise between accuracy and computational efficiency. Overall, the FAM solvers predict the radiation sources with better accuracy compared to the PN solvers at both axial locations. In [9] Ge et al. obtained fully coupled solutions without considering turbulence–radiation interaction for the same flame (while, however, employing a simpler turbulence–chemistry interaction model than the one with which the frozen field in [8] was generated). Sample results for radial distributions of mean temperature and mean NO at a particular axial location are shown in Fig. 22.10. For temperature predictions, no observable difference is noted between the P3 , P5 , and P7 solvers, and their difference with P1 is also quite small. Likewise, minor differences are observed between the various FAM solvers. The PN solvers appear to slightly underpredict the mean temperature, which manifests itself in significant underprediction (about 20%) in mean NO amounts. With the exception of the lowest resolution FAM solver (2×4), the FAM solvers predict both mean

Radiation in Chemically Reacting Systems Chapter | 22 841

FIGURE 22.9 Comparison of various RTE solvers for the radiative source computation of a turbulent diffusion flame (Sandia Flame D scaled up by a factor of 4) using frozen temperature and concentration fields; from [8]. Radial distributions for two axial locations from the inlet are shown: (a) z/d ∼ 30 and (b) z/d ∼ 45; d = 28.8 mm is the diameter of the fuel jet.

FIGURE 22.10 Comparison of various RTE solvers for the computation of a turbulent diffusion flame (Sandia Flame D scaled up by a factor of 4) with full two-way coupling; from [9]: (a) mean temperature and (b) mean mass fraction of NO. Radial distributions are shown at a distance of 45 fuel jet diameters from the fuel inlet.

temperature and mean NO very accurately, especially considering the complexity of such calculations. Similar trends (not shown) were observed at other spatial locations. For fully coupled solutions, since local conditions change with the radiation model being used, rather than considering local radiation sources, perhaps a more meaningful avenue is to examine the global energy budget of the flame at stationary state. If Q˙ C is the net heat released by combustion per unit time, Q˙ em the net radiation emitted by the flame per unit time, and Q˙ r the net radiation lost by the flame per unit time, then the ratios Q˙ r /Q˙ em and Q˙ r /Q˙ C portray the overall accuracy of the radiation model (combination of RTE solver and spectral model) for both absorption and emission. Table

842 Radiative Heat Transfer

TABLE 22.2 Summary of global (entire flame) energy budget predicted by various radiation models and computational times, as reported in [9]; other than the LBL–PMC model, the spectral model used in each case is the FSCK-2 model. The computational times are normalized by the same quantity without radiation. Radiation model

˙ em (%) ˙ r /Q Q

˙ C (%) ˙ r /Q Q

Relative computational time

LBL–PMC

30.2

28.0

1.061, 1.122

P1

36.7

33.4

1.000, 1.183

P3

32.8

30.9

1.012, 1.280

P5

32.4

30.7

1.024, 1.659

P7

32.2

30.6

1.037, 2.000

FAM 2 × 4

24.0

32.0

1.024, 1.354

FAM 4 × 4

24.7

26.6

1.024, 1.463

FAM 4 × 8

27.4

25.8

1.049, 1.732

FAM 8 × 8

27.7

27.8

1.061, 2.171

22.2 summarizes the predictions of the various models considered in [9] for these two ratios. It shows that the PN solvers consistently overpredict Q˙ r /Q˙ em , while the FAM solvers consistently underpredict the same ratio. As the fidelity of each solver type is increased—order for PN and number of angles in FAM—the solution approaches the LBL–PMC solution. For Q˙ r /Q˙ C , the highest fidelity FAM solver (8×8) appears to produce slightly more accurate results than the highest fidelity PN solver (P7 ). Overall, the PN solvers appear to predict slightly higher radiation loss than the FAM solvers, which are consistent with the lower temperatures predicted by the PN solvers (Fig. 22.10). Table 22.2 also reports the computational times required by the various models relative to the case without radiation. For each model, two different numbers are reported. For the LBL–PMC model, the first number is the average computational time required with tracing of 5000 photon bundles per time step, while the second is with 10,000 photon bundles per time step. In both cases, a time blending factor of α = 0.02 [cf. equation (22.1)] was used, implying averaging over ∼ 200 time steps. As is evident, the LBL–PMC model is computationally quite efficient if time blending is employed. For the deterministic solvers, the first number represents the relative computational time if the RTE is solved once every 250 time steps (cheapest option), while the second number represents the same quantity if the RTE is solved every time step (most expensive option). Clearly, with the highest order deterministic methods (P7 or 8×8 FAM), solving the RTE every time step is quite expensive—the extra computational time in solving the RTE being equal to or greater than the solution time without radiation. At the other extreme end (updating radiation only every 250 steps), even the highest resolution method requires no more than 6% additional (over the solution without radiation) computational time, while the time required for the P1 -approximation is essentially negligible (even though it is coupled with the sophisticated FSCK-2 spectral model with 8 quadrature points). No observable difference was found between updating the radiation field after each vs. after every 250 time steps. Hence, for situations where steady state solutions are sought, updating the RTE solution infrequently (about 100 time steps or more may be optimal) is warranted. Furthermore, significant advances in the solution of linear algebraic equations (such as multigrid solvers) may make deterministic RTE solvers even more attractive than what has been demonstrated thus far [267]. The computational times required by the RTE solver alone were also compiled (not shown in Table 22.2). The highest order PN solver, namely P7 , required about 18% less computational time than the highest resolution FAM, namely 8×8. The P1 solver, however, was found to be significantly more efficient than any of the other solvers, being about 2.5 times as efficient as the lowest resolution FAM (2×4). Based on the results of this study, one may conclude that if only the mean temperature is of interest, then using the P1 solver along with the FSCK model is, arguably, the best choice. However, if predicting pollutants is of interest, a slightly more computationally expensive RTE solver, such as FAM or a higher-order PN solver (P3 being the best compromise between accuracy and efficiency) is desirable, with the FSCK model offering adequate spectral accuracy in all cases. Aside from laboratory-scale turbulent flames, some comparative studies have also been conducted in other application areas of turbulent combustion, as listed in Table 22.1. However, most of these studies dwell on comparison of nongray radiation models (for both gas and particulates), with almost no comparison between various RTE solvers. Consalvi and Liu [36,38] investigated the accuracy of several state-of-the-art spectral gas

Radiation in Chemically Reacting Systems Chapter | 22 843

FIGURE 22.11 Accuracy of various spectral models in the computation of the radiative source along the centerline of two different pool fires; from [38]: (a) small (34 kW) and (b) large (176 kW).

radiation models in the simulation of pool fires using frozen temperature and concentration fields. These include the NBCK, FSCK, MSFSK, and wide-band models. The accuracy of each model was assessed by comparing the results against LBL calculations. In addition, both gray and nongray models for soot were considered. The FAM was the chosen RTE solver in all cases. Figure 22.11 shows the radiative source, ∇·qR , computed using various spectral models along the centerline of two different pool files—one small (34 kW) and one large (176 kW). Similar trends (not shown here) were observed in the radial direction, as well. Although the absolute errors are comparable for the two flames, since the magnitude of the radiative source is smaller for the larger flame, the errors are higher for the larger flame, percentage wise. The errors produced by the wide-band model (WBM) are unacceptably large. At the opposite end, both the NBCK and the MSFSK models offer almost LBL accuracy. The FSCK model produces slightly larger errors than these two models, but at significantly lower computational cost, arguably making it the most attractive choice for large-scale pool fire computations, as was also noted in [6] for turbulent diffusion flames. Consalvi, Liu, and coworkers have also investigated several state-of-the-art nongray gas radiation models for oxy-fired conditions [203,250,251]. These include the SNB model, the SNBCK model, the FSCK model, the WBCK model, and the WSGG model with the parameters prescribed by Bordbar et al. [249]. LBL calculations were used as the reference to assess the accuracy of each model. Once again, the FAM was used for solution of the RTE. In [250], the models were assessed for a one-dimensional isothermal and homogeneous plane-parallel medium, and it was found that the FSCK model is the most accurate over the entire pressure range considered, while also being computationally very efficient. Calculations in a gas turbine combustor [203] revealed that the SNBCK and FSCK models are comparable in accuracy, while the WSGG model (with Bordbar parameters) is not as accurate. The WBCK model is the least accurate. Similar calculations in a coal-fired combustor operating under oxygen-rich conditions also yielded similar results. In the studies conducted by Consalvi, Liu, and coworkers, soot and particles were ignored and, therefore, while the findings shed significant light on the accuracy of gas radiation models, their implications on net radiation fluxes and sources remain unknown. In summary, based on a survey of the literature, it appears that the two most widely used RTE solvers for reacting systems are the DOM (and its popular variant, the FAM), and the P1 approximation. The popularity of the Zonal method appears to have faded considerably in the past two decades, probably because the scalability of the method is poor (the number of exchange factors that need to be evaluated scales as the square of the number of cells or grid points), making it computationally very inefficient for large-scale 3D calculations. The DOM (and FAM) is attractive because the method is easily extendable to higher order. While higher-order PN appear equally promising from an accuracy standpoint, extension of P1 to higher orders requires significant additional code development. Of the two methods, the DOM is computationally more expensive as the number of directional RTEs can quickly grow if used for 3D calculations. In an effort to exploit the efficiency of the P1 approximation, while retaining the directional accuracy of the DOM, hybrid methods have also been proposed and used for reacting systems [262,263], wherein the P1 solver is used for optically thick gray gases, and the DOM

844 Radiative Heat Transfer

(or FAM) is used for optically thin gray gases within the context of the WSGG or SLW models, both of which require one RTE solution per gray gas. Sun and Zhang [263] demonstrated a 31% reduction in computational time (with marginal compromise on accuracy) if 4 out of the 6 gray gases employed are solved using P1 rather than the FAM/DOM. In addition to these two methods, the Monte Carlo method (PMC) continues to grow in popularity, and not just to generate benchmark solutions. It is relatively trivial to combine a PMC solver with a LBL-accurate spectral model. It was shown that, with time blending, the method is competitive with (and more accurate than) higher-order conventional RTE solvers when applied to steady state (laminar) or statistically stationary (turbulent) flows with full two-way coupling. Furthermore, the PMC method is the only method to date to fully resolve turbulence–radiation interactions (TRI), i.e., without invoking the optically thin fluctuation assumption (OTFA). On the downside, the solutions generated by PMC have inherent statistical uncertainties, and coupling the solution to a deterministic flow solver may adversely affect convergence. It is clear from the above discussion that for reacting systems at quasi-steady state, with time steps used to resolve chemical reactions and turbulent eddies generally small compared to the time scale over which the mean temperature changes, performing radiation calculations every 100+ time steps when using conventional RTE solvers should be sufficient. Other recommendations include using lower-order approximations (for PN and SN ) during the initial stages of convergence and transitioning to their higher-order counterparts as convergence is approached. Equivalently, when using Monte Carlo the use of time blending (with parameters as used for the results given in Table 22.2) should suffice. Here also one may use a smaller number of photon bundles per time step during the early stages, and more as the solution progresses. Similar conclusions should also be true for transient solutions, although how often the RTE needs to be updated for deterministic solvers, or the appropriate blending factor and bundles/time step for Monte Carlo solutions, is unknown at the present time and, in all likelihood, problem dependent.

22.6 Radiation in Concentrating Solar Energy Systems Radiative heat transfer plays an important role in the harnessing of concentrated solar radiation. Applications include solar thermal power [268,269], solar thermochemistry [270–272], and concentrating photovoltaics [273– 275], in which solar radiation is converted to thermal, chemical, and electrical energy, respectively. Radiative fluxes that can be obtained with optical concentrators vary between few kW/m2 and several MW/m2 . Concentrating solar systems are characterized by the solar concentration ratio, defined as the ratio of the concentrated solar flux to solar irradiation of 1 kW/m2 . High temperatures can be achieved by increasing the concentration ratio to limit the reradiation losses from a receiver [276]. While high temperatures are targeted in solar thermal power and thermochemical systems to increase their efficiency, they are unwanted in photovoltaic cells due to cell efficiency decreasing with temperature. Thus, concentrating photovoltaics systems typically utilize lower fluxes and research efforts are focused on cell thermal management [273]. High-temperature solar thermal systems often include solid-gas heterogeneous media at temperatures exceeding several hundred degrees Celsius, and in some applications reaching more than 2000◦ C. Such media serve multiple purposes. They absorb high-flux irradiation (absorption is predominantly by the solid phase as direct gas absorption is ineffective for length scales of a solar device) and transfer the heat to a working gas in a solar thermal receiver driving a thermodynamic power cycle, and/or to a chemical reaction in a solar thermochemical reactor. Depending on the specific application, the solid phase may either be in the form of a suspension (tiny solid particles suspended in the flowing gas, i.e., a fluidized bed) [277], a porous medium [278], or a multichannel honeycomb monolith whose inner walls are selectively coated to enhance solar absorption [279]. In directly irradiated receivers/reactors radiation is absorbed by a solid that is in direct contact with the working gas or provides surface to a chemical reaction, respectively (Fig. 22.12a). In indirectly irradiated receivers/reactors radiation is absorbed by a solid, and then transferred to a gas or to a chemical reaction by conduction, convection, and/or radiation through an intermediate heat transfer medium (solid, fluid, or multiphase medium, Fig. 22.12b).

Radiation in Solar Thermal Receivers The design of a receiver depends on the type of concentrator, the working fluid, and the operating ranges of temperature, pressure, and radiative flux. A comprehensive review of solar receivers up to 1998 was given by Karni et al. [281]. More recent reviews of volumetric receivers for solar thermal power plants with a central receiver are given by Ávila-Marín [282] and Blanco and Santigosa [283].

Radiation in Chemically Reacting Systems Chapter | 22 845

FIGURE 22.12 Examples of high-temperature devices utilizing concentrated solar radiation: (a) a directly irradiated solar thermochemical test reactor for thermal decomposition of methane, featuring a flow of methane laden with carbon particles exposed to concentrated solar radiation [277]; and (b) an indirectly irradiated solar receiver prototype featuring an annular layer of reticulated porous ceramics (RPC) bounded by two concentric cylinders: concentrated solar radiation passes through a compound parabolic concentrator (CPC), and is absorbed by the inner cylindrical cavity, and converted into heat, which is further transferred by conduction, radiation, and convection to the pressurized air flowing across the layer of RPC [280].

An early radiative heat transfer analysis in a volumetric solar absorber was presented by Flamant [284] for solar fluidized beds of silicon carbide, chamotte, zirconia, and silica particles. Temperature profiles, total emissivity, heat flux distribution, and effective mean penetration distance were determined and compared to experimental data. Combined radiative, conductive, and forced convective heat transfer in a volumetric selective solar absorber containing a packed bed of two spectrally dissimilar slabs of particles was analyzed by Flamant et al. [285] using the two-flux approximation. The model accounted for the variation of absorption and scattering of the layers in the visible and infrared spectral ranges, and its predictions were validated experimentally using a bed of glass and silicon carbide particles that were heating the gas phase. An array of irradiated fin-pins exposed to a gas flow was studied experimentally in a solar furnace by Karni et al. [281]. A two-dimensional steady-state heat transfer model coupling radiation, conduction, and convection was developed for a novel design of a high-temperature pressurized-air receiver for power generation via combined Brayton–Rankine cycles (see Fig. 22.12) [280]. The model employs separate energy equations for solid and gas phases in the annular layer of reticulated porous ceramics saturated with pressurized air: solid : air :

      1 ∂ ∂Ts ∂Ts ∂ rks + ks + Q˙  R = sh Ts − Tf , r ∂r ∂r ∂z ∂z   ∂Tf = sh Ts − Tf , ρcp v ∂z

(22.30a) (22.30b)

where s is the specific surface area of the solid–fluid interface and h is the heat transfer coefficient between particles and air. Radiative transfer in the receiver cavity was modeled using enclosure theory. The Rosseland diffusion approximation, the P1 -approximation, and the Monte Carlo method were employed as alternative methods to study radiative transfer in the porous layer, which was assumed to be gray and isotropically scattering: diffusion, equation (14.18) : P1 , equation (15.49a) : Monte Carlo, equation (20.49) :

  1 ∂ 16σTs3 ∂Ts  ˙ QR = r , r ∂r 3β ∂r Q˙  =κ(G − 4σTs4 ), R

Q˙  R

δQa − 4κσTs4 , = δV

(22.31a) (22.31b) (22.31c)

where the radiative power δQa absorbed by the volume δV is directly computed by the Monte Carlo method. Figure 22.13 shows the radial distributions of the radiative source term for the three solution methods at a selected location z/L = 0.12. The P1 results were found to agree reasonably well with those obtained by the Monte Carlo method, while the Rosseland approximation led to inaccurate results due to the relatively small optical thickness of the porous layer made of reticulated porous ceramics (τRPC = 3). P1 was found to be the most appropriate

846 Radiative Heat Transfer

FIGURE 22.13 Radial distribution of radiative source term within the RPC of the receiver shown in Fig. 22.12b, at a selected axial location z/L = 0.12 (for receiver length of 65 mm, outer radius of SiC tube of 20 mm, and a total solar power input of 1 kW) [280].

method as it simultaneously led to good accuracy and short computational times. The P1 -approximation has also been used by other researchers [286–288]. A combined heat transfer numerical study using the discrete ordinates method for a solid-particle receiver was presented in [289]. Martinek and Weimar [290] investigated the accuracy of the FAM (Chapter 16) in two different closed-cavity solar receiver configurations by comparing against the Monte Carlo (MC) method. Meshes with number of cells ranging from 2,300 to 133,000 were considered along with 3 different angular discretizations: 5×5, 15×15, and 25×25. Both collimated and diffuse solar flux profiles at the receiver aperture were considered. Overall, the FAM solutions were found to be within acceptable accuracy only when a fine spatial mesh was used along with a fine angular mesh. The solutions were found to be least accurate when the cavity is highly specularly reflective or if the absorber area is small, and tended to improve as the solar flux was changed from collimated to diffuse. Based on these results, a hybrid MC/FAM solution technique was also proposed, in which incident solar energy was tracked using the MC method, while emitted energy inside the cavity receiver was tracked using the FAM. Another model for porous cavity receivers, similar to the one in [280], was proposed by Tan and coworkers [278,291]. In this model, convection in both the axial and radial directions within the porous medium were considered. In [291], the space between the aperture and the porous receiver was modeled using the Monte Carlo method, while in [278], a surface-to-surface radiation exchange model was used. Radiation transport inside the porous cavity receiver was predicted with the Rosseland approximation, and a simple gray model for both the absorption coefficient and the scattering coefficient, that accounted for the porosity of the medium, was used. Good agreement between experimental measurements of temperature and model results was found [278]. Lougou et al. [288] also compared the accuracy of the P1 approximation and the FAM in porous cavity receivers, and found marginal differences in predicted temperature: maximally 7% and an average of 2%. However, as expected, the computational time required by the FAM was almost four times that of P1 . Cavity receivers are often enclosed by semitransparent windows to separate the hot gas inside the receiver from a cold ambient atmosphere. Radiative heat transfer in a cavity-receiver containing a windowed aperture was analyzed by Maag et al. [292] for quartz and sapphire windows using the band approximation of Chapter 6. Radiative heat transfer in a solar cavity receiver with a plano-convex window was studied by Yong et al. [293] with the Monte Carlo method.

Radiation in Solar Thermochemical Reactors The use of concentrated solar radiation in chemistry dates back to the 18th century, when Antoine Lavoisier conducted combustion experiments in a solar furnace consisting of two concentric lenses [294]. Pioneering work on solar processes and reactors was done by Trombe and Foex [295], Nakamura [296], Fletcher and Moen [276], and others. A comprehensive review (up to 2016) of the research challenges in combustion and gasification processes in directly irradiated solar thermochemical reactors may be found in Nathan et al. [297]. Other recent reviews include those by Romero and Steinfeld [298] and Lipinski ´ et al. [299]. In a solar thermochemical reactor the incident solar radiation enters a reactor cavity through an aperture, which may be windowed, and is absorbed by reactants and cavity walls. Reactor design and optimization is

Radiation in Chemically Reacting Systems Chapter | 22 847

typically guided by thermochemical models coupling radiation, conduction, and convection to the chemical kinetics [300,301]. Radiation analyses range from models with surface radiative exchange to more sophisticated models, in which medium composition and phases vary with time as chemical reactions progress. Solar-driven redox thermochemical cycles have been investigated to produce H2 and CO from H2 O and CO2 , respectively. A two-step cycle for a generic metal oxide Mx O y can be written as [302]: solar, endothermic step: nonsolar, exothermic step: or

1 1 Mx O y−δox → Mx O y−δred + 0.5O2 , Δδ Δδ 1 1 Mx O y−δred + H2 O → Mx O y−δox + H2 , Δδ Δδ 1 1 Mx O y−δred + CO2 → Mx O y−δox + CO, Δδ Δδ

(22.32) (22.33) (22.34)

where δox and δred are the nonstoichiometric coefficients of the reduced and oxidized forms of the metal oxide, and Δδ = δred − δox . Lipinski ´ et al. [303] studied the decomposition of micrometer-sized zinc oxide particles in a stationary particle suspension under direct high-flux irradiation. A numerical model coupling transient radiative heat transfer to chemical kinetics accounted for time-dependent radiative properties due to decreasing particle sizes, computed from Mie theory at each time step of the transient solution. Abanades et al. [304] developed a multiphase model coupling fluid flow, heat and mass transfer, and chemical kinetics of the zinc oxide decomposition reaction, treating the particles as opaque spheres. Transient radiative heat transfer in directly irradiated solar reactors containing packed beds of zinc oxide particles was numerically analyzed using the Rosseland diffusion approximation by Müller et al. [305] and Schunk and coworkers [306]. A diffusion-based model of internal radiative transport in the packed bed of zinc oxide was also proposed by Dombrovsky et al. [307]. The numerically determined temperature profiles reported in [306,307] were found to be in good agreement with those measured in a solar-driven thermogravimeter. Radiative heat transfer in a solar thermochemical reactor for the reduction of cerium dioxide was analyzed using the Monte Carlo method by Villafán-Vidales et al. [308]. The participating medium was a nonisothermal, nongray, absorbing, emitting, and anisotropically scattering suspension of particles with properties obtained from Mie theory. Radiative characteristics of novel cerium dioxide-based materials for applications in nonstoichiometric redox cycles were studied by Liang et al. [309], Ganesan et al. [310,311], and Haussener and Steinfeld [312]. Very recently, Lipinski ´ and coworkers [301,313] studied the decomposition of manganese oxide and iron-manganese oxide in an indirectly irradiated tubular fluidized or packed bed reactor contained in a solar cavity receiver. A transient 3D multiphase heat and mass transport model including chemical reactions was employed to model the reactor, and radiation transport was modeled using the Monte Carlo method. Within the fluidized/packed bed, the absorption and scattering coefficients, as well as the scattering phase function were determined as a function of the particle size distribution using Mie theory. Numerical results for both radiative heat fluxes as well as temperature showed excellent agreement with experimental data that were also collected as part of the study. Directly irradiated particles of carbonaceous materials are encountered in several solar thermochemical processes including steam gasification of coal and methane decomposition, CHx O1−y + yH2 O = (x/2 + y)H2 + CO, CH4 → C + 2H2 .

(22.35) (22.36)

Transient radiative heat transfer in directly irradiated stationary suspensions of coal particles undergoing steam gasification was studied numerically using the Monte Carlo method and Mie theory by Lipinski ´ and Steinfeld [314] and geometric optics by Lipinski ´ et al. [315]. The Monte Carlo method and geometric optics were also applied in a simulation of a solid–gas fluidized bed reactor for coal gasification. Maag and coworkers [277] developed a transient combined convective–radiative heat transfer model of directly irradiated CH4 flow laden with carbon particles. Mie theory was applied to obtain radiative properties of particles growing due to deposition of carbon from the decomposition reaction. A combined radiative–conductive–convective heat transfer model of an entrained-flow reactor for methane decomposition was developed by Maag et al. [316]. The net radiation method was applied to a cavity with opaque walls and a semi-transparent aperture. Thermal decomposition of calcium carbonate has been studied for the solar production of lime and cement, as well as solar thermochemical CO2 capture. In these models for reacting packed beds CaCO3 particles were

848 Radiative Heat Transfer

assumed to be in the size range of geometric optics. The Rosseland diffusion approximation was applied in a transient combined radiation–conduction model [317], while spectral characteristics of the refracting and absorbing semitransparent particles were accounted for in another study [318].

References [1] T. Badinand, T. Fransson, Improvement of the finite volume method for coupled flow and radiation calculations by the use of two grids and rotational periodic interface, in: M.P. Mengüç, N. Selçuk (Eds.), Radiative Transfer 2001 — The Third International Symposium on Radiative Transfer, Begell House, 2001. [2] T. Omori, S. Yamaguchi, T. Fusegi, Computational heat transfer analysis of a furnace using the WSGG model, in: Proceedings of 2000 IMECE, vol. HTD-366-1, ASME, 2000, pp. 103–108. [3] E. Sediki, A. Soufiani, M.S. Sifaoui, Spectrally correlated radiation and laminar forced convection in the entrance region of a circular duct, International Journal of Heat and Mass Transfer 45 (2002) 5069–5081. [4] L.H. Howell, R.B. Pember, P. Colella, J.P. Jessee, W.A. Fiveland, A conservative adaptive-mesh algorithm for unsteady, combined-mode heat transfer using the discrete ordinates method, Numerical Heat Transfer – Part B: Fundamentals 35 (4) (1999) 407–430. [5] G. Pal, A. Gupta, M.F. Modest, D.C. Haworth, Comparison of accuracy and computational expense of radiation models in simulation of nonpremixed turbulent jet flames, in: Proceedings of 2011 ASME/JSME Thermal Engineering Joint Conference, 2011. [6] G. Pal, A. Gupta, M.F. Modest, D.C. Haworth, Comparison of accuracy and computational expense of radiation models in simulation of nonpremixed turbulent jet flames, Combustion and Flame 162 (2015) 2487–2495. [7] B. Garten, F. Hunger, D. Messig, B. Stelzner, D. Trimis, C. Hasse, Detailed radiation modeling of a partial-oxidation flame, International Journal of Thermal Sciences 87 (2015) 68–84. [8] C. David, W. Ge, S.P. Roy, M.F. Modest, R. Sankaran, Comparison of radiation models for a turbulent piloted methane/air jet flame: a frozen-field study, in: Proceedings of the ASME 2021 Summer Heat Transfer Conferences SHTC2021, 2021, pp. HT2021–62417. [9] W. Ge, C. David, S.P. Roy, M.F. Modest, R. Sankaran, Comparison of spherical harmonics method and discrete ordinates method for radiative transfer in a turbulent jet flame, Journal of Quantitative Spectroscopy and Radiative Transfer (2021), in preparation. [10] A. Wang, M.F. Modest, D.C. Haworth, L. Wang, Monte Carlo simulation of radiative heat transfer and turbulence interactions in methane/air jet flames, Journal of Quantitative Spectroscopy and Radiative Transfer 109 (2) (2008) 269–279. [11] J. Cai, S. Lei, A. Dasgupta, M.F. Modest, D.C. Haworth, Radiative heat transfer in high-pressure laminar hydrogen–air diffusion flames, Combustion Theory and Modelling 18 (6) (2014) 607–626. [12] H.J. Curran, P. Gaffuri, W.J. Pitz, C.K. Westbrook, Comprehensive modeling study of iso-octane oxidation, Lawrence Livermore National Laboratories, Livermore, CA, 2000. [13] R. Viskanta, M.P. Mengüç, Radiation heat transfer in combustion systems, Progress in Energy and Combustion Science 13 (1987) 97–160. [14] R. Viskanta, Radiative Transfer in Combustion Systems: Fundamentals and Applications, Begell House, New York, 2005. [15] D.E. Negrelli, J.R. Lloyd, J.L. Novotny, A theoretical and experimental study of radiation–convection interaction in a diffusion flame, ASME Journal of Heat Transfer 99 (1977) 212–220. [16] K.V. Liu, J.R. Lloyd, K.T. Yang, An investigation of a laminar diffusion flame adjacent to a vertical flat plate burner, International Journal of Heat and Mass Transfer 24 (12) (1981) 1959–1970. [17] W.M. Kays, M.E. Crawford, Convective Heat and Mass Transfer, 3rd ed., McGraw-Hill, 1993. [18] R.J. Kee, F.M. Rupley, J.A. Miller, CHEMKIN-II: a Fortran chemical kinetics package for the analysis of gas-phase chemical kinetics, Technical Report SAND89-8009B, Sandia National Laboratories, 1989. [19] R.J. Kee, G. Dixon-Lewis, J. Warnatx, M.E. Coltrin, J.A. Miller, A Fortran computer code package for the evaluation of gas-phase, multicomponent transport properties, Technical Report SAND86-8246, Sandia National Laboratory, 1986. [20] T. Daguse, T. Croonenbroek, J.C. Rolon, N. Darabiha, A. Soufiani, Study of radiative effects on laminar counterflow H2 /O2 /N2 diffusion flames, Combustion and Flame 106 (1996) 271–287. [21] J. Ruan, H. Kobayashi, T. Niioka, Y. Ju, Combined effects of nongray radiation and pressure on premixed CH4 /O2 /CO2 flames, Combustion and Flame 124 (2001) 225–230. [22] R. Demarco, F. Nmira, J.-L. Consalvi, Influence of thermal radiation on soot production in laminar axisymmetric diffusion flames, Journal of Quantitative Spectroscopy and Radiative Transfer 120 (2013) 52–69. [23] H. Abdul-Sater, G. Krishnamoorthy, An assessment of radiation modeling strategies in simulations of laminar to transitional, oxymethane, diffusion flames, Applied Thermal Engineering 61 (2013) 507–518. [24] C.R. Kaplan, S.W. Baek, E.S. Oran, J.L. Ellzey, Dynamics of a strongly radiating unsteady ethylene jet diffusion flame, Combustion and Flame 96 (1994) 1–21. [25] F. Liu, H. Guo, G.J. Smallwood, O.L. Gülder, Effects of gas and soot radiation on soot formation in a coflow laminar ethylene diffusion flame, Journal of Quantitative Spectroscopy and Radiative Transfer 73 (2002) 409–421. [26] F. Liu, H. Guo, G.J. Smallwood, M. El Hafi, Effects of gas and soot radiation on soot formation in counterflow ethylene diffusion flames, Journal of Quantitative Spectroscopy and Radiative Transfer 84 (2004) 501–511. [27] F. Liu, G.J. Smallwood, W. Kong, The importance of thermal radiation transfer in laminar diffusion flames at normal and microgravity, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (7) (2011) 1241–1249.

Radiation in Chemically Reacting Systems Chapter | 22 849

[28] R.R. Dobbins, R.J. Hall, S. Cao, B.A.V. Bennett, M.B. Colket, M.D. Smooke, Radiative emission and reabsorption in laminar, ethylenefueled diffusion flames using the discrete ordinates method, Combustion Science and Technology 187 (2015) 230–248. [29] F. Liu, J.-L. Consalvi, A. Fuentes, Effects of water vapor addition to the air stream on soot formation and flame properties in a laminar coflow ethylene/air diffusion flame, Combustion and Flame 161 (2014) 1724–1734. [30] O.A. Ezekoye, Z. Zhang, Convective and radiative coupling in a burner-supported diffusion flame, Journal of Thermophysics and Heat Transfer 11 (2) (1997) 239–245. [31] S.R. Turns, Understanding NOx formation in nonpremixed flames: experiments and modeling, Progress in Energy and Combustion Science 21 (1995) 361–385. [32] R.S. Barlow, N.S.A. Smith, J.Y. Chen, R.W. Bilger, Comparison of CMC and PDF modeling predictions with experimental nitric oxide LIF/Raman measurements in a turbulent H2 jet flame, Combustion Science and Technology 105 (1995) 357–375. [33] E. Alemi, M.R. Zargarabadi, Effects of jet characteristics on NO formation in a jet-stabilized combustor, International Journal of Thermal Sciences 112 (2017) 55–67. [34] B. Wu, X.Y. Zhao, Effects of radiation models on steady and flickering laminar non-premixed flames, Journal of Quantitative Spectroscopy and Radiative Transfer 253 (2020) 107103. [35] S. Tang, M.K. Chernovsky, H.G. Im, A. Atreya, A computational study of spherical diffusion flames in microgravity with gas radiation Part I: model development and validation, Combustion and Flame 157 (2010) 118–126. [36] J.-L. Consalvi, F. Liu, Radiative heat transfer through the fuel-rich core of laboratory-scale pool fires, Combustion Science and Technology 186 (2014) 475–489. [37] C.H. Sohn, Z. Chen, Y. Ju, Effects of radiation on the uncertainty of flame speed determination using spherically propagating flames with CO/CO2/H2O dilutions at elevated pressures, International Journal of Heat and Mass Transfer 86 (2015) 820–825. [38] J.-L. Consalvi, F. Liu, Radiative heat transfer in the core of axisymmetric pool fires - I: evaluation of approximate radiative property models, International Journal of Thermal Sciences 84 (2014) 104–117. [39] C. Paul, D.C. Haworth, M.F. Modest, A simplified cfd model for spectral radiative heat transfer in high-pressure hydrocarbon-air combustion systems, Proceedings of the Combustion Institute 37 (2019) 4617–4624. [40] Q. Binauld, P. Rivière, A. Soufiani, A note on radiation preheating of some hydrocarbons by combustion products, Combustion and Flame 194 (2018) 128–134. [41] B.A. Rankin, D.L. Blunck, V.R. Katta, S.D. Stouffer, J.P. Gore, Experimental and computational infrared imaging of bluff body stabilized laminar diffusion flames, Combustion and Flame 159 (2012) 2841–2843. [42] A. Fuentes, R. Henriquez, F. Nmira, F. Liu, J.-L. Consalvi, Experimental and numerical study of the effects of the oxygen index on the radiation characteristics of laminar coflow diffusion flames, Combustion and Flame 160 (2013) 786–795. [43] B. Ma, G. Wang, G. Magnotti, R.S. Barlow, M.B. Long, Intensity-ratio and color-ratio thin-filament pyrometry: uncertainties and accuracy, Combustion and Flame 161 (2014) 908–916. [44] M.A. Delichatsios, L. Orloff, M.M. Delichatsios, The effects of fuel sooting tendency and the flow on flame radiation in luminous turbulent jet flames, Combustion Science and Technology 84 (1/6) (1992) 199–215. [45] L. Orloff, J. de Ris, M.A. Delichatsios, Radiation from buoyant turbulent diffusion flames, Combustion Science and Technology 84 (1/6) (1992) 177–186. [46] H. Ramamurthy, S. Ramadhyani, R. Viskanta, A two-dimensional axisymmetric model for combusting, reacting and radiating flows in radiant tubes, Journal of the Institute of Energy 67 (1994) 90–100. [47] N.W. Bressloff, J.B. Moss, P.A. Rubini, CFD prediction of coupled radiation heat transfer and soot production in turbulent flames, in: Proceedings of Twenty-Sixth Symposium (International) on Combustion, vol. 2, The Combustion Institute, 1996, pp. 2379–2386. [48] M.J. Zimberg, S.H. Frankel, J.P. Gore, Y.R. Sivathanu, A study of coupled turbulent mixing, soot chemistry, and radiation effects using the linear eddy model, Combustion and Flame 113 (1998) 454–469. [49] S.J. Brookes, J.B. Moss, Predictions of soot and thermal radiation properties in confined turbulent jet diffusion flames, Combustion and Flame 116 (1999) 486–503. [50] S.J. Brookes, J.B. Moss, Measurements of soot production and thermal radiation from confined turbulent jet diffusion flames of methane, Combustion and Flame 116 (1999) 49–61. [51] L. Wang, D.C. Haworth, S.R. Turns, M.F. Modest, Interactions among soot, thermal radiation, and NOx emissions in oxygen-enriched turbulent nonpremixed flames: a CFD modeling study, Combustion and Flame 141 (1–2) (2005) 170–179. [52] L. Wang, M.F. Modest, D.C. Haworth, S.R. Turns, Modeling nongray soot and gas-phase radiation in luminous turbulent nonpremixed jet flames, Combustion Theory and Modelling 9 (3) (2005) 479–498. [53] B.W. Webb, J. Ma, J.T. Pearson, V.P. Solovjov, SLW modeling of radiation transfer in comprehensive combustion predictions, Combustion Science and Technology 190 (2018) 1392–1408. [54] J.S. Kim, S.W. Baek, C.R. Kaplan, Effect of radiation on diffusion flame behavior over a combustible solid, Combustion Science and Technology 88 (1/2) (1993) 133–150. [55] S. Bhattacharjee, R.A. Altenkirch, K. Sacksteder, The effect of ambient pressure on flame spread over thin cellulosic fuel in a quiescent, microgravity environment, ASME Journal of Heat Transfer 118 (1996) 181–190. [56] H.Y. Wang, P. Joulain, J.M. Most, Modeling on burning of large-scale vertical parallel surfaces with fire-induced flow, Fire Safety Journal 32 (3) (1999) 241–247.

850 Radiative Heat Transfer

[57] T.H. Lin, C.H. Chen, Influence of two-dimensional gas phase radiation on downward flame spread, Combustion Science and Technology 141 (1) (1999) 83–106. [58] S.W. Baek, J.H. Park, C.E. Choi, Investigation of droplet combustion with nongray gas radiation effects, Combustion Science and Technology 142 (1) (1999) 55–79. [59] R. Duval, A. Soufiani, J. Taine, Coupled radiation and turbulent multiphase flow in an aluminised solid propellant rocket engine, Journal of Quantitative Spectroscopy and Radiative Transfer 84 (2004) 513–526. [60] D.Y. Byun, S.W. Baek, Numerical investigation of combustion with non-gray thermal radiation and soot formation effect in a liquid rocket engine, International Journal of Heat and Mass Transfer 50 (2007) 412–422. [61] A.A. Mohamad, S. Ramadhyani, R. Viskanta, Modelling of combustion and heat transfer in a packed bed with embedded coolant tubes, International Journal of Heat and Mass Transfer 37 (8) (1994) 1181–1191. [62] G. Ozen, N. Selçuk, Sensitivity of radiation modeling to property estimation techniques in the freeboard of lignite-fired bubbling fluidized bed combustors (BFBCs), Combustion Science and Technology 186 (2014) 684–697. [63] G. Ozen, F. Aydin, N. Selçuk, Radiative heat transfer in the dilute zone of an air-fired circulating fluidized bed combustor and its oxy-fired retrofit, Combustion Science and Technology 188 (2016) 730–744. [64] C. Ates, G. Ozen, N. Selçuk, G. Kulah, Radiative heat transfer in strongly forward scattering media of circulating fluidized bed combustors, Journal of Quantitative Spectroscopy and Radiative Transfer 182 (2016) 264–276. [65] C. Ates, N. Selçuk, G. Kulah, Significance of particle concentration distribution on radiative heat transfer in circulating fluidized bed combustors, International Journal of Heat and Mass Transfer 117 (2018) 58–70. [66] C. Ates, O. Sen, N. Selçuk, G. Kulah, Influence of spectral particle properties on radiative heat transfer in optically thin and thick media of fluidized bed combustors, International Journal of Thermal Sciences 122 (2017) 266–280. [67] C. Ates, N. Selçuk, G. Kulah, Influence of fly ash composition on non-gray particle radiation in combusting systems, Journal of Quantitative Spectroscopy and Radiative Transfer 215 (2018) 25–40. [68] B. Wu, S.P. Roy, X.Y. Zhao, M.F. Modest, Effect of multiphase radiation on coal combustion in a pulverized coal jet flame, Journal of Quantitative Spectroscopy and Radiative Transfer 197 (2017) 154–165. [69] J.-F. Sacadura, Radiative heat transfer in fire safety science, Journal of Quantitative Spectroscopy and Radiative Transfer 93 (2005) 5–24. [70] E.H. Chui, P.M.J. Hughes, G.D. Raithby, Implementation of the finite volume method for calculating radiative transfer in a pulverized fuel flame, Combustion Science and Technology 92 (4/6) (1993) 225–242. [71] S.P. Visona, B.R. Stanmore, 3-D modelling of NOx formation in a 275 MW utility boiler, Journal of the Institute of Energy 69 (1996) 68–79. [72] J. Yuan, V. Semião, M.G. Carvalho, Predictions of particulate formation, oxidation and distribution in a three-dimensional oil-fired furnace, Journal of the Institute of Energy 70 (1997) 57–70. [73] G. Song, T. Bjorge, J. Holen, B.F. Magnussen, Simulation of fluid flow and gaseous radiation heat transfer in a natural gas-fired furnace, International Journal of Numerical Methods for Heat & Fluid Flow 7 (2–3) (1997) 169–180. [74] F. Liu, H.A. Becker, Y. Bindar, A comparative study of radiative heat transfer modelling in gas-fired furnaces using the simple grey gas and the weighted-sum-of-grey-gases models, International Journal of Heat and Mass Transfer 41 (22) (1998) 3357–3371. [75] E.P. Keramida, H.H. Liakos, M.A. Founti, Radiative heat transfer in natural gas-fired furnaces, International Journal of Heat and Mass Transfer 43 (10) (2000) 1801–1809. [76] N. Crnomarkovich, M. Sijercic, S. Belosevic, D. Tucakovic, T. Zivanovic, Numerical investigation of processes in the lignite-fired furnace when simple gray gas and weighted sum of gray gases models are used, International Journal of Heat and Mass Transfer 56 (2013) 197–205. [77] H. Sayah, M. Nemer, W. Nehme, D. Clodic, Modeling of a self-recuperative burner and development of a simplified equivalent radiative model, ASME Journal of Heat Transfer 134 (2012) 121201. [78] P. Ghose, J. Patra, A. Datta, A. Mukhopadhyay, Prediction of soot and thermal radiation in a model gas turbine combustor burning kerosene fuel spray at different swirl levels, Combustion Theory and Modelling 20 (2016) 457–485. [79] T. Ren, M.F. Modest, S. Roy, Monte Carlo simulation for radiative transfer in a high-pressure industrial gas turbine combustion chamber, Journal of Engineering for Gas Turbines and Power 140 (5) (2018) 051503. [80] C. Paul, S.F. Fernandez, D.C. Haworth, S. Roy, M.F. Modest, A detailed modeling study of radiative heat transfer in a heavy-duty diesel engine, Combustion and Flame 200 (2019) 325–341. [81] P. Edge, M. Gharebaghi, R. Irons, R. Porter, R.T.J. Porter, M. Pourkashanian, D. Smith, P. Stephenson, A. Williams, Combustion modelling opportunities and challenges for oxy-coal carbon capture technology, Chemical Engineering Research and Design 89 (2011) 1470–1493. [82] R. Johansson, B. Leckner, K. Andersson, F. Johnsson, Influence of particle and gas radiation in oxy-fuel combustion, International Journal of Heat and Mass Transfer 65 (2013) 143–152. [83] X.Y. Zhao, D.C. Haworth, T. Ren, M.F. Modest, A transported probability density function/photon Monte Carlo method for hightemperature oxy–natural gas combustion with spectral gas and wall radiation, Combustion Theory and Modelling 17 (2) (2013) 354–381. [84] B.A. Rankin, G. Magnotti, R.S. Barlow, J.P. Gore, Radiation intensity imaging measurements of methane and dimethyl ether turbulent nonpremixed and partially premixed jet flames, Combustion and Flame 161 (2014) 2849–2859.

Radiation in Chemically Reacting Systems Chapter | 22 851

[85] R.K. Kapaku, B.A. Rankin, J.P. Gore, Quantitative imaging of radiation from soot and carbon dioxide in a turbulent ethylene jet diffusion flame, Combustion and Flame 162 (2015) 3704–3710. [86] B.A. Rankin, M. Ihme, J.P. Gore, Quantitative model-based imaging of mid-infrared radiation from a turbulent nonpremixed jet flame and plume, Combustion and Flame 162 (2015) 3704–3710. [87] J. Zhao, H. Huang, G. Jomaas, M. Zhong, R. Yang, Experimental study of the burning behaviors of thin-layer pool fires, Combustion and Flame 193 (2018) 327–334. [88] D.L. Blunck, M.E. Harvazinski, B.A. Rankin, C.L. Merkle, J.P. Gore, Turbulent radiation statistics of exhaust plumes exiting from a subsonic axisymmetric nozzle, Journal of Thermophysics and Heat Transfer 26 (2012) 286–293. [89] D.L. Blunck, M.E. Harvazinski, C.L. Merkle, J.P. Gore, Influence of turbulent fluctuations on the radiation intensity emitted from exhaust plumes, Journal of Thermophysics and Heat Transfer 26 (4) (2012) 581–589. [90] A.J. Crow, I.D. Boyd, V.E. Terrapon, Radiation modeling of a hydrogen fueled scramjet, Journal of Thermophysics and Heat Transfer 27 (2013) 11–21. [91] R.O. Fox, Computational Models for Turbulent Reacting Flows, Cambridge University Press, Cambridge, 2003. [92] W.P. Jones, B.E. Launder, The prediction of laminarization with a two-equation model of turbulence, International Journal of Heat and Mass Transfer 15 (1972) 301–304. [93] G.F. Carrier, F.E. Fendell, F.E. Marble, The effect of strain rate on diffusion flames, SIAM Journal of Applied Mathematics 28 (1975) 463. [94] F.A. Williams, Recent advances in theoretical descriptions of turbulent diffusion flames, in: S.N.B. Murphy (Ed.), Turbulent Mixing in Nonreactive and Reactive Flows, Plenum Press, New York, 1975, pp. 189–208. [95] N. Peters, Laminar diffusion flamelet models in nonpremixed turbulent combustion, Progress in Energy and Combustion Science 10 (1984) 319. [96] S.B. Pope, PDF methods for turbulent reactive flows, Progress in Energy and Combustion Science 11 (1985) 119–192. [97] R.W. Bilger, Turbulent flows with nonpremixed reactants, in: Turbulent Reacting Flows, Springer Verlag, 1980, pp. 65–113. [98] K.K. Kuo, Principles of Combustion, Wiley Interscience, New York, 1986. [99] M. Lesieur, Turbulence in Fluids, 3rd ed., Kluwer Academic Publishers, 1996. [100] N. Peters, Turbulent Combustion, Cambridge University Press, Cambridge, 2000. [101] S.B. Pope, Turbulent Flows, Cambridge University Press, Cambridge, 2000. [102] D.C. Haworth, Progress in probability density function methods for turbulent reacting flows, Progress in Energy and Combustion Science 16 (2010) 168–259. [103] P.J. Foster, Relation of time-mean transmission of turbulent flames to optical depth, Journal of the Institute of Fuel 42 (340) (1969) 179. [104] M. Amin, P.J. Foster, Fluctuations in the transmittance of a turbulent propane jet flame, in: F.J. Weinberg (Ed.), Combustion Institute European Symposium, Academic Press, London, U.K., 1973, pp. 530–535. [105] E. Tan, P.J. Foster, Radiation through a turbulent medium, in: J.T. Rogers (Ed.), Proceedings of the 6th International Heat Transfer Conference, vol. 3, Hemisphere, Washington, D.C., 1978, pp. 403–408. [106] S.M. Jeng, M.C. Lai, G.M. Faeth, Nonluminous radiation in turbulent buoyant axisymmetric flames, Combustion Science and Technology 40 (1984) 41–53. [107] S.M. Jeng, G.M. Faeth, Radiative heat fluxes near turbulent buoyant methane diffusion flames, ASME Journal of Heat Transfer 106 (1984) 886–888. [108] G.M. Faeth, Heat and mass transfer in flames, in: Proceedings of the Eighth International Heat Transfer Conference, Hemisphere, Washington, D.C., 1986, pp. 151–160. [109] Y.R. Sivathanu, M.E. Kounalakis, G.M. Faeth, Soot and continuous radiation statistics of luminous turbulent diffusion flames, in: Twenty-Third Symposium (International) on Combustion, The Combustion Institute, 1990, pp. 1543–1550. [110] M.E. Kounalakis, Y.R. Sivathanu, G.M. Faeth, Infrared radiation statistics of nonluminous turbulent diffusion flames, ASME Journal of Heat Transfer 113 (2) (1991) 437–445. [111] J.P. Gore, G.M. Faeth, Structure and spectral radiation properties of turbulent ethylene/air diffusion flames, in: Twenty-First Symposium (International) on Combustion, The Combustion Institute, 1986, pp. 1521–1531. [112] J.P. Gore, S.M. Jeng, G.M. Faeth, Spectral and total radiation properties of turbulent carbon monoxide/air diffusion flames, AIAA Journal 25 (2) (1987) 339–345. [113] J.P. Gore, G.M. Faeth, Structure and spectral radiation properties of luminous acetylene/air diffusion flames, ASME Journal of Heat Transfer 110 (1988) 173–181. [114] M.E. Kounalakis, J.P. Gore, G.M. Faeth, Turbulence/radiation interactions in nonpremixed hydrogen/air flames, in: Twenty-Second Symposium (International) on Combustion, The Combustion Institute, 1988, pp. 1281–1290. [115] M.E. Kounalakis, J.P. Gore, G.M. Faeth, Mean and fluctuating radiation properties of nonpremixed turbulent carbon monoxide/air flames, ASME Journal of Heat Transfer 111 (1989) 1021–1030. [116] G.M. Faeth, J.P. Gore, S.G. Chuech, S.M. Jeng, Radiation from turbulent diffusion flames, in: Annual Review of Numerical Fluid Mechanics and Heat Transfer, vol. 2, Hemisphere, Washington, D.C., 1989, pp. 1–38. [117] J. Ji, Y.R. Sivathanu, J.P. Gore, Measurements and stochastic time and space series simulations of spectral radiation in a turbulent non-premixed flame, in: Twenty-Eighth Symposium (International) on Combustion, The Combustion Institute, 2000, pp. 391–398.

852 Radiative Heat Transfer

[118] Y. Zheng, Y.R. Sivathanu, J.P. Gore, Measurements and stochastic time and space series simulations of spectral radiation in a turbulent non-premixed flame, in: Twenty-Ninth Symposium (International) on Combustion, The Combustion Institute, 2002, pp. 1957–1963. [119] Y. Zheng, R.S. Barlow, J.P. Gore, Measurements and calculations of spectral radiation intensities for turbulent non-premixed and partially premixed flames, ASME Journal of Heat Transfer 125 (2003) 678–686. [120] Y. Zheng, J.P. Gore, Measurements and inverse calculations of spectral radiation intensities of a turbulent ethylene/air jet flame, in: Thirtieth Symposium (International) on Combustion, The Combustion Institute, 2005, pp. 727–734. [121] W. Krebs, R. Koch, H.-J. Bauer, R. Kneer, S. Wittig, Effect of turbulence on radiative heat transfer inside a model combustor, in: Proceedings of Eurotherm Seminar No. 37—Heat Transfer in Radiating and Combusting Systems 2, 1994, pp. 349–362. [122] W. Krebs, R. Koch, B. Ganz, L. Eigenmann, S. Wittig, Effect of temperature and concentration fluctuations on radiative heat transfer in turbulent flames, in: Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, 1996, pp. 2763–2770. [123] S.J. Fischer, B. Hardoiun-Duparc, W.L. Grosshandler, The structure and radiation of an ethanol pool fire, Combustion and Flame 70 (1987) 291–306. [124] B.A. Rankin, D.L. Blunck, J.P. Gore, Infrared imaging and spatiotemporal radiation properties of a turbulent nonpremixed jet flame and plume, ASME Journal of Heat Transfer 135 (2013) 021201. [125] A.S. Newale, B.A. Rankin, H.U. Lalit, J.P. Gore, R.J. McDermott, Quantitative infrared imaging of impinging turbulent buoyant diffusion flames, Proceedings of the Combustion Institute 35 (2015) 2647–2655. [126] V.P. Kabashnikov, G.I. Kmit, Influence of turbulent fluctuations of thermal radiation, Journal of Applied Spectroscopy 31 (1979) 963–967. [127] V.P. Kabashnikov, Thermal radiation of turbulent flows in the case of large fluctuations of the absorption coefficient and the Planck function, Journal of Engineering Physics 49 (1) (1985) 778–784. [128] V.P. Kabashnikov, G.I. Myasnikova, Thermal radiation in turbulent flows—temperature and concentration fluctuations, Heat Transfer, Soviet Research 17 (6) (1985) 116–125. [129] S. Mazumder, Numerical study of chemically reactive turbulent flows with radiative heat transfer, Ph.D. thesis, The Pennsylvania State University, University Park, PA, 1997. [130] G. Cox, On radiant heat transfer from turbulent flames, Combustion Science and Technology 17 (1977) 75–78. [131] P.J. Coelho, Numerical simulation of the interaction between turbulence and radiation in reactive flows, Progress in Energy and Combustion Science 33 (2007) 311–383. [132] S. Mazumder, M.F. Modest, Turbulence–radiation interactions in nonreactive flow of combustion gases, ASME Journal of Heat Transfer 121 (1999) 726–729. [133] A. Gupta, M.F. Modest, D.C. Haworth, Large-eddy simulation of turbulence–radiation interactions in a turbulent planar channel flow, ASME Journal of Heat Transfer 131 (2009) 061704. [134] P.J. Coelho, A theoretical analysis of the influence of turbulence on radiative emission in turbulent diffusion flames of methane, Combustion and Flame 160 (2013) 610–617. [135] J.P. Gore, G.M. Faeth, Structure and spectral radiation properties of turbulent ethylene/air diffusion flames, in: Proceedings of the Twenty-First Symposium (International) on Combustion, 1986, pp. 1521–1531. [136] J.-L. Consalvi, Influence of turbulence–radiation interactions in laboratory-scale methane pool fires, International Journal of Thermal Sciences 60 (2012) 122–130. [137] P.S. Cumber, Validation study of a turbulence radiation interaction model: weak, intermediate and strong TRI in jet flames, International Journal of Heat and Mass Transfer 79 (2014) 1034–1047. [138] P.S. Cumber, Efficient modeling of turbulence-radiation interaction in subsonic hydrogen jet flames, Numerical Heat Transfer – Part B: Fundamentals 63 (2013) 85–114. [139] J.-L. Consalvi, R. Demarco, A. Fuentes, Modelling thermal radiation in buoyant turbulent diffusion flames, Combustion Theory and Modelling 16 (2012) 817–841. [140] P.S. Cumber, O. Onokpe, Turbulent radiation interaction in jet flames: sensitivity to the pdf, International Journal of Heat and Mass Transfer 57 (2013) 250–264. [141] A.Y. Snegirev, Statistical modeling of thermal radiation transfer in buoyant turbulent diffusion flames, Combustion and Flame 136 (2004) 51–71. [142] G.C. Fraga, P.J. Coelho, A.P. Petry, F.H.R. França, Development and testing of a model for turbulence-radiation interaction effects on the radiative emission, Journal of Quantitative Spectroscopy and Radiative Transfer 245 (2020) 106852. [143] F.R. Centeno, R. Brittes, F.H.R. França, C.V. da Silva, Application of the WSGG model for the calculation of gas-soot radiation in a turbulent non-premixed methane-air flame inside a cylindrical combustion chamber, International Journal of Heat and Mass Transfer 93 (2016) 742–753. [144] S. Mazumder, M.F. Modest, A PDF approach to modeling turbulence–radiation interactions in nonluminous flames, International Journal of Heat and Mass Transfer 42 (1999) 971–991. [145] J.M. Armengol, R. Vicquelin, A. Coussement, R.G. Santos, O. Gicquel, Study of turbulence-radiation interactions in a heated jet using direct numerical simulation coupled to a non-gray Monte Carlo solver, International Journal of Heat and Mass Transfer 162 (2020) 120297. [146] E.M. Orbegoso, L.F. Figueira da Silva, R. Serfaty, Comparative study of thermal radiation properties models in turbulent non-premixed sooting combustion, Numerical Heat Transfer – Part A: Applications 69 (2016) 166–179.

Radiation in Chemically Reacting Systems Chapter | 22 853

[147] X. Yang, Z. He, S. Dong, H.P. Tan, Prediction of turbulence radiation interactions of CH4 –H2 /air turbulent flames at atmospheric and elevated pressures, International Journal of Hydrogen Energy 43 (2018) 15537–15550. [148] F. Nmira, J.-L. Consalvi, F. André, Pressure effects on radiative heat transfer in hydrogen/air turbulent diffusion flames, Journal of Quantitative Spectroscopy and Radiative Transfer 220 (2018) 172–179. [149] G. Li, M.F. Modest, Application of composition PDF methods in the investigation of turbulence–radiation interactions, Journal of Quantitative Spectroscopy and Radiative Transfer 73 (2–5) (2002) 461–472. [150] G. Li, M.F. Modest, Importance of turbulence–radiation interactions in turbulent diffusion jet flames, ASME Journal of Heat Transfer 125 (2003) 831–838. [151] G. Li, Investigation of turbulence–radiation interactions by a hybrid FV/PDF Monte Carlo method, Ph.D. thesis, The Pennsylvania State University, University Park, PA, 2002. [152] FLUENT Computational Fluid Dynamics Software, Version 5, Fluent Corp., New Hampshire, 1998. [153] S.B. Pope, On the relationship between stochastic Lagrangian models of turbulence and second-moment closures, Physics of Fluids 6 (1994) 973–985. [154] S.B. Pope, Particle method for turbulent flows: integration of stochastic model equations, Journal of Computational Physics 117 (1995) 332–349. [155] J.-L. Consalvi, F. Nmira, Effects of soot absorption coefficient-Planck function correlation on radiative heat transfer in oxygen-enriched propane turbulent diffusion flame, Journal of Quantitative Spectroscopy and Radiative Transfer 172 (2016) 50–57. [156] J.-L. Consalvi, F. Nmira, Absorption turbulence–radiation interactions in sooting turbulent jet flames, Journal of Quantitative Spectroscopy and Radiative Transfer 201 (2017) 1–9. [157] F. Nmira, D. Burot, J.-L. Consalvi, Stochastic Eulerian field method for radiative heat transfer in a propane oxygen-enhanced turbulent diffusion flame, Combustion Theory and Modelling 21 (2017) 62–78. [158] F. Nmira, D. Burot, J.-L. Consalvi, Soot emission radiation-turbulence interactions in diffusion jet flames, Combustion Science and Technology 191 (2019) 126–136. [159] R.S. Barlow, International Workshop on Measurement and Computation of Turbulent Nonpremixed Flames (TNF), website: http:// www.sandia.gov/TNF/abstract.html. [160] F. Nmira, Y. Liu, J.-L. Consalvi, F. André, F. Liu, Pressure effects on radiative heat transfer in sooting turbulent diffusion flames, Journal of Quantitative Spectroscopy and Radiative Transfer 245 (2020) 106906. [161] R.S. Mehta, A. Wang, M.F. Modest, D.C. Haworth, Modeling of a turbulent ethylene/air flame using hybrid finite volume/Monte Carlo methods, Computational Thermal Sciences 1 (2009) 37–53. [162] R.S. Mehta, M.F. Modest, D.C. Haworth, Radiation characteristics and turbulence–radiation interactions in sooting turbulent jet flames, Combustion Theory and Modelling 14 (1) (2010) 105–124. [163] P.J. Coelho, Detailed numerical simulation of radiative transfer in a nonluminous turbulent jet diffusion flame, Combustion and Flame 136 (2004) 481–492. [164] X. Xu, Y. Chen, H. Wang, Detailed numerical simulation of thermal radiation influence in Sandia flame D, International Journal of Heat and Mass Transfer 49 (2006) 2347–2355. [165] A. Habibi, B. Merci, D. Roekaerts, Turbulence radiation interaction in Reynolds-averaged Navier-Stokes simulations of nonpremixed piloted turbulent laboratory-scale flames, Combustion and Flame 151 (2007) 303–320. [166] L. Tessé, F. Dupoirieux, J. Taine, Monte Carlo modeling of radiative transfer in a turbulent sooty flame, International Journal of Heat and Mass Transfer 47 (2004) 555–572. [167] B. Zamuner, F. Dupoirieux, Numerical simulation of soot formation in a turbulent flame with a Monte-Carlo PDF approach and detailed chemistry, Combustion Science and Technology 158 (2000) 407–438. [168] A. Soufiani, J. Taine, High temperature gas radiative property parameters of statistical narrow-band model for H2 O, CO2 and CO, and correlated-k model for H2 O and CO2 , International Journal of Heat and Mass Transfer 40 (4) (1997) 987–991. [169] A. Wang, M.F. Modest, An adaptive emission model for Monte Carlo ray-tracing in participating media represented by statistical particle fields, Journal of Quantitative Spectroscopy and Radiative Transfer 104 (2) (2007) 288–296. [170] A. Wang, M.F. Modest, Spectral Monte Carlo models for nongray radiation analyses in inhomogeneous participating media, International Journal of Heat and Mass Transfer 50 (2007) 3877–3889. [171] R.S. Mehta, D.C. Haworth, M.F. Modest, Composition PDF/photon Monte Carlo modeling of moderately sooting turbulent jet flames, Combustion and Flame 157 (2010) 982–994. [172] A. Coppalle, D. Joyeux, Temperature and soot volume fraction in turbulent diffusion flames: measurements of mean and fluctuating values, Combustion and Flame 96 (1994) 275–285. [173] J.H. Kent, D. Honnery, Modeling sooting turbulent jet flames using an extended flamelet technique, Combustion Science and Technology 54 (1987) 383–397. [174] N.E. Endrud, Soot, radiation and pollutant emissions in oxygen-enhanced turbulent jet flames, M.S. thesis, The Pennsylvania State University, University Park, PA, 2000. [175] R.S. Mehta, D.C. Haworth, M.F. Modest, An assessment of gas-phase thermochemistry and soot models for laminar atmosphericpressure ethylene–air flames, Proceedings of the Combustion Institute 32 (2009) 1327–1334. [176] A.J. Chandy, D.J. Glaze, S.H. Frankel, A general semicausal stochastic model for turbulence/radiation interactions in flames, ASME Journal of Heat Transfer 113 (3) (1997) 509–516.

854 Radiative Heat Transfer

[177] M. Roger, C.B.D. Silva, P.J. Coelho, Analysis of the turbulence–radiation interactions for large eddy simulations of turbulent flows, International Journal of Heat and Mass Transfer 52 (2009) 2243–2254. [178] M. Roger, P.J. Coelho, C.B. da Silva, The influence of the non-resolved scales of thermal radiation in large eddy simulation of turbulent flows: a fundamental study, International Journal of Heat and Mass Transfer 53 (2010) 2897–2907. [179] A. Gupta, D.C. Haworth, M.F. Modest, Turbulence–radiation interactions in large-eddy simulations of luminous and nonluminous nonpremixed flames, Proceedings of the Combustion Institute 34 (2013) 1281–1288. [180] P.J. Coelho, Approximate solutions of the filtered radiative transfer equation in large eddy simulations of turbulent reactive flows, Combustion and Flame 156 (2009) 1099–1110. [181] J.-L. Consalvi, F. Nmira, W. Kong, On the modeling of the filtered radiative transfer equation in large eddy simulations of lab-scale sooting turbulent diffusion flames, Journal of Quantitative Spectroscopy and Radiative Transfer 221 (2018) 51–60. [182] D. Poitou, J. Amaya, M. El Hafi, B. Cuénot, Analysis of the interaction between turbulent combustion and thermal radiation using unsteady coupled LES/DOM simulations, Combustion and Flame 159 (2012) 1605–1618. [183] M. Roger, P.J. Coelho, C.B. da Silva, Relevance of the subgrid-scales for large eddy simulations of turbulence–radiation interactions in a turbulent plane jet, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (2011) 1250–1256. [184] Y. Wu, D.C. Haworth, M.F. Modest, B. Cuénot, Direct numerical simulation of turbulence/radiation interaction in premixed combustion systems, Proceedings of the Combustion Institute 30 (2005) 639–646. [185] Y. Wu, M.F. Modest, D.C. Haworth, A high-order photon Monte Carlo method for radiative transfer in direct numerical simulation of chemically reacting turbulent flows, Journal of Computational Physics 223 (2) (2007) 898–922. [186] K.V. Deshmukh, D.C. Haworth, M.F. Modest, Direct numerical simulation of turbulence–radiation interactions in a statistically homogeneous nonpremixed combustion system, Proceedings of the Combustion Institute 31 (2007) 1641–1648. [187] K.V. Deshmukh, M.F. Modest, D.C. Haworth, Direct numerical simulation of turbulence–radiation interactions in statistically onedimensional nonpremixed combustion systems, Journal of Quantitative Spectroscopy and Radiative Transfer 109 (14) (2008) 2391–2400. [188] K.V. Deshmukh, M.F. Modest, D.C. Haworth, Higher-order spherical harmonics to model radiation in direct numerical simulation of turbulent reacting flows, Computational Thermal Sciences 1 (2009) 207–230. [189] S.H. Kang, Direct numerical simulation of the turbulent premixed flame propagation with radiation effects, International Journal of Heat and Mass Transfer 102 (2016) 323–330. [190] S.B. Rejeb, T. Echekki, Thermal radiation modeling using the LES-ODT framework for turbulent combustion flows, International Journal of Heat and Mass Transfer 104 (2017) 1300–1316. [191] M.F. Modest, G. Li, Turbulence radiation interactions, in: B. Sunden, ´ M. Faghri (Eds.), Modelling and Simulation of Turbulent Heat Transfer, in: Developments in Heat Transfer, WIT Press, Southampton, England, 2004. [192] M.F. Modest, Multiscale modeling of turbulence, radiation, and combustion interactions in turbulent flames, International Journal for Multiscale Computational Engineering 3 (2) (2005) 85–106. [193] M.F. Modest, Radiative heat transfer in fire modeling, in: B. Sunden, ´ M. Faghri (Eds.), Transport Phenomena in Fires, in: Developments in Heat Transfer, WIT Press, Southampton, England, 2006. [194] M.F. Modest, D.C. Haworth, Radiative Heat Transfer in Turbulent Combustion: Theory and Applications, Springer, 2016. [195] P.J. Coelho, Turbulence–radiation interaction: from theory to application in numerical simulations, ASME Journal of Heat Transfer 134 (2012) 031001. [196] M.P. Mengüç, R. Viskanta, C. Ferguson, Multidimensional modeling of radiative heat transfer in Diesel engines, SAE Technical Paper, no. 850503, SAE International, 1985. [197] J. Abraham, V. Magi, Modeling radiant heat loss characteristics in a Diesel engine, SAE Technical Paper, no. 970888, SAE International, 1997. [198] J.F. Wiedenhoefer, R.D. Reitz, A multidimensional radiation model for diesel engine simulation with comparison to experiment, Numerical Heat Transfer – Part A: Applications 44 (2003) 665–682. [199] T. Yoshikawa, R.D. Reitz, Effect of radiation on Diesel engine combustion and heat transfer, Journal of Thermal Science and Technology 4 (2009) 86–97. [200] M. Bolla, M.A. Chishty, E.R. Hawkes, C.N. Qing, S. Kook, Influence of turbulent fluctuations on radiation heat transfer, NO and soot formation under ecn spray a conditions, Proceedings of the Combustion Institute 36 (2017) 3551–3558. [201] W.P. Jones, M.C. Paul, Combination of DOM with LES in a gas turbine combustor, International Journal of Engineering Science 43 (5–6) (2005) 379–397. [202] G. Bulat, W.P. Jones, A.J. Marquis, NO and CO formation in an industrial gas-turbine combustion chamber using LES with the Eulerian sub-grid PDF method, Combustion and Flame 161 (7) (2014) 1804–1825. [203] V. Kez, F. Liu, J.-L. Consalvi, J. Ströhle, B. Epple, A comprehensive evaluation of different radiation models in a gas turbine combustor under conditions of oxy-fuel combustion with dry recycle, Journal of Quantitative Spectroscopy and Radiative Transfer 172 (2016) 121–133. [204] C.E. Choi, S.W. Baek, Numerical analysis of a spray combustion with nongray radiation using weighted sum of gray gases model, Combustion Science and Technology 115 (1996) 297–315. [205] T. Furuhata, S. Tanno, T. Miura, Y. Ikeda, T. Nakajima, Performance of numerical spray combustion simulation, Energy Conversion and Management 38 (1997) 1111–1122.

Radiation in Chemically Reacting Systems Chapter | 22 855

[206] H.A. El-Asrag, A.C. Iannetti, S.V. Apte, Large eddy simulations for radiation-spray coupling for a lean direct injector combustor, Combustion and Flame 161 (2014) 510–524. [207] A. Fujita, H. Watanabe, R. Kurose, S. Komori, Two-dimensional direct numerical simulation of spray flames–part 1: effects of equivalence ratio, fuel droplet size and radiation, and validity of flamelet model, Fuel 104 (2013) 515–525. [208] S.P. Roy, J. Cai, M.F. Modest, Development of a multiphase photon Monte Carlo method for spray combustion and its application in high-pressure conditions, International Journal of Heat and Mass Transfer 115, Part A (2017) 453–466. [209] F.R. Westlye, Experimental study of liquid fuel spray combustion, Ph.D. thesis, Technical University of Denmark, Kys. Lyngby, Denmark, 2016. [210] H. Watanabe, R. Kurose, S. Komori, H. Pitsch, Effects of radiation on spray flame characteristics and soot formation, Combustion and Flame 152 (2008) 2–13. [211] K.M. Mukut, S.P. Roy, Effect of O2 concentration in ambient mixture and multiphase radiation on pollutant formation in ECN spray-A, Combustion Theory and Modelling 24 (3) (2020) 549–572. [212] L.A. Dombrovsky, D.L. Reviznikov, A.V. Sposovin, Radiative heat transfer from supersonic flow with suspended particles to a blunt body, International Journal of Heat and Mass Transfer 93 (2016) 853–861. [213] W. May, W.G. McQueen, Radiation from large liquified natural gas fires, Combustion Science and Technology 7 (1973) 51–56. [214] A. Dayan, C.L. Tien, Radiant heating from a cylindrical fire column, Combustion Science and Technology 9 (1974) 41–47. [215] P.G. Seeger, On the combustion and heat transfer in fires of liquid fuels in tanks, in: P.L. Blackshear (Ed.), Heat Transfer in Fires, Scripta Book Company, 1974, pp. 95–126. [216] D. Burgess, M. Hertzberg, Radiation in pool flames, in: N.H. Afgan (Ed.), Heat Transfer in Flames, Scripta Book Company, 1974, pp. 413–430. [217] A.T. Modak, Thermal radiation from pool fires, Combustion and Flame 29 (1977) 177–192. [218] G.A. Mizner, J.A. Eyre, Radiation from liquified gas fires on water, Combustion Science and Technology 35 (1983) 33–57. [219] K.S. Mudan, P.A. Croce, A thermal radiation model for LNG trench fires, in: Proceedings of the ASME, ASME, 1984, Paper number 84-WA/HT-75. [220] P.K. Raj, Analysis of JP-4 fire test data and development of a simple fire model, in: Proceedings of the ASME, ASME, 1981, Paper number 81-HT-17. [221] F.D. Wayne, K. Kinsella, Spectral emission characteristics of large hydrocarbon pool fires, in: Proceedings of the ASME, ASME, 1984, Paper number 84-WA/HT-74. [222] G. Heskestad, Luminous height of turbulent diffusion flames, Fire Safety Journal 5 (1983) 103–108. [223] C.L. Beyler, Fire plumes and ceiling jets, Fire Safety Journal 11 (1986) 53–75. [224] V. Babrauskas, Estimating large pool fire burning rates, Fire Technology 19 (1983) 251–261. [225] K.S. Mudan, Thermal radiation hazards from hydrocarbon pool fires, Progress in Energy and Combustion Science 10 (1984) 59–80. [226] M. Shokri, C.L. Beyler, Radiation from large pool fires, Journal of Fire Protection Engineering 1 (1989) 141–150. [227] M. Klassen, J.P. Gore, Structure and radiation properties of pool fires, Technical Report NIST-GCR-94-651, NIST, Gaithersburg, MD, 1994. [228] S.P. Burns, Turbulence radiation interaction modeling in hydrocarbon pool fire simulations, Technical Report SAND 99-3190, Sandia National Laboratory, Albuquerque, NM, 1999. [229] J.L. De Ris, P.K. Wu, G. Heskestad, Radiation fire modeling, in: Proceedings of the Combustion Institute, vol. 28, 2000, pp. 2751–2759. [230] S. Hostikka, K.B. McGrattan, A. Hamins, Numerical modeling of pool fires using LES and finite volume method for radiation, in: D.D. Evans (Ed.), Proceedings of the Seventh International Symposium on Fire Safety Science, International Association for Fire Safety Sciences, 2003, pp. 383–394. [231] P.K. Raj, Large LNG fire thermal radiation-modeling issues and hazard criteria revisited, Process Safety Progress 24 (3) (2005) 192–202. [232] K. Wakatsuki, High temperature radiation absorption of fuel molecules and an evaluation of its influence on pool fire modeling, Ph.D. thesis, University of Maryland, College Park, MD, 2005. [233] J.B.M. Pierce, J.B. Moss, Smoke production, radiation heat transfer and fire growth in a liquid-fuelled compartment fire, Fire Safety Journal 42 (2007) 310–320. [234] A. Collin, P. Boulet, Evaluation of simple models of flame radiation in the frame of fire propagation, International Journal of Heat and Mass Transfer 59 (2013) 83–92. [235] B. Sun, K. Guo, V.K. Pareek, Computational fluid dynamics simulation of LNG pool fire radiation for hazard analysis, Journal of Loss Prevention in the Process Industries 29 (2014) 92–102. [236] P. Chatterjee, Y. Wang, K.V. Meredith, S.B. Dorofeev, Application of a subgrid soot-radiation model in the numerical simulation of a heptane pool fire, Proceedings of the Combustion Institute 35 (2015) 2055–2074. [237] A.Y. Snegirev, E. Markus, E. Kuznetsov, J. Harris, T. Wu, On soot and radiation modeling in buoyant turbulent diffusion flames, Heat and Mass Transfer 54 (2018) 2275–2293. [238] G.C. Fraga, F.R. Centeno, A.P. Petry, P.J. Coelho, F.H.R. França, On the individual importance of temperature and concentration fluctuations in the turbulence-radiation interaction in pool fires, International Journal of Heat and Mass Transfer 136 (2019) 1079–1089. [239] I. Sikic, S. Dembele, J.X. Wen, Non-grey radiative heat transfer modelling in LES-CFD simulated methanol pool fires, Journal of Quantitative Spectroscopy and Radiative Transfer 234 (2019) 78–89. [240] B. Wu, S.P. Roy, X.Y. Zhao, Detailed modeling of a small-scale turbulent pool fire, Combustion and Flame 214 (2020) 224–237.

856 Radiative Heat Transfer

[241] K. Andersson, R. Johansson, F. Johnsson, B. Leckner, Radiation intensity of propane-fired oxy-fuel flames: implications for soot formation, Energy & Fuels 22 (2008) 1535–1541. [242] K. Andersson, R. Johansson, S. Hjärtstam, F. Johnsson, B. Leckner, Radiation intensity of lignite-fired oxy-fuel flames, Experimental Thermal and Fluid Science 33 (2008) 67–76. [243] G. Krishnamoorthy, M. Sami, S. Orsino, A. Perera, M. Shahnam, E.D. Huckaby, Radiation modelling in oxy-fuel combustion scenarios, International Journal of Computational Fluid Dynamics 24 (2010) 69–82. [244] C. Yin, L.C.R. Johansen, L.A. Rosendahl, S.K. Kær, New weighted sum of gray gases model applicable to computational fluid dynamics (CFD) modeling of oxy-fuel combustion: derivation, validation, and implementation, Energy & Fuels 25 (2010) 6275–6282. [245] K. Andersson, R. Johansson, F. Johnsson, Thermal radiation in oxy-fuel flames, International Journal of Greenhouse Gas Control 55 (2011) 558–565. [246] C. Yin, L.A. Rosendahl, S.K. Kær, Chemistry and radiation in oxy-fuel combustion: a computational fluid dynamics modeling study, Fuel 90 (2011) 2519–2529. [247] P. Edge, S.R. Gubba, L. Ma, R. Porter, M. Pourkashanian, A. Williams, LES modelling of air and oxy-fuel pulverised coal combustion— impact on flame properties, Proceedings of the Combustion Institute 33 (2011) 2709–2716. [248] S. Hjärtstam, R. Johansson, K. Andersson, F. Johnsson, Computational fluid dynamics modeling of oxy-fuel flames: the role of soot and gas radiation, Energy & Fuels 26 (2012) 2786–2797. [249] M.H. Bordbar, G. Wecel, T. Hyppänen, A line by line based weighted sum of gray gases model for inhomogeneous CO2 –H2 O mixture in oxy-fired combustion, Combustion and Flame 161 (2014) 2435–2445. [250] H. Chu, M. Gu, J.-L. Consalvi, F. Liu, H. Zhou, Effects of total pressure on non-grey gas radiation transfer in oxy-fuel combustion using the LBL, SNB, SNBCK, WSGG, and FSCK methods, Journal of Quantitative Spectroscopy and Radiative Transfer 172 (2016) 24–35. [251] V. Kez, J.-L. Consalvi, F. Liu, J. Ströhle, B. Epple, Assessment of several gas radiation models for radiative heat transfer calculations in a three-dimensional oxy-fuel furnace under coal-fired conditions, International Journal of Thermal Sciences 120 (2017) 289–302. [252] R. Porter, F. Liu, M. Pourkashanian, A. Williams, D. Smith, Evaluation of solution methods for radiative heat transfer in gaseous oxy-fuel combustion environments, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (14) (2010) 2084–2094. [253] M.A. Rajhi, R. Ben-Mansour, M.A. Habib, M.A. Nemitallah, K. Andersson, Evaluation of gas radiation models in CFD modeling of oxy-combustion, Energy Conversion and Management 81 (2014) 83–97. [254] R. Johansson, B. Leckner, K. Andersson, F. Johnsson, Account for variations in the H2 O to CO2 molar ratio when modelling gaseous radiative heat transfer with the weighted-sum-of-grey-gases model, Combustion and Flame 158 (2011) 893–901. [255] T.W. Tong, R.D. Skocypec, Summary on comparison of radiative heat transfer solutions for a specified problem, in: Developments in Radiative Heat Transfer, vol. 203, ASME HTD, 1992. [256] J.T. Farmer, J.R. Howell, Monte Carlo prediction of radiative heat transfer in inhomogeneous, anisotropic, nongray media, Journal of Thermophysics and Heat Transfer 8 (1) (1994) 133–139. [257] Z.M. Tan, J.R. Howell, A new numerical method for radiation heat transfer in nonhomogeneous participating media, Journal of Thermophysics and Heat Transfer 4 (4) (1990) 419–424. [258] M.H.N. Naraghi, B. Litkouhi, Discrete exchange factor solution of radiative heat transfer in three-dimensional enclosures, in: Heat Transfer Phenomena in Radiation, Conduction, and Fire, vol. 106, ASME HTD, 1989, pp. 221–229. [259] W.W. Yuen, E.E. Takara, Superposition technique for radiative equilibrium in rectangular enclosures with complex boundary conditions, International Journal of Heat and Mass Transfer 33 (5) (1990) 901–916. [260] J.R. Howell, M.P. Mengüç, Challenges for radiative transfer 1: towards the effective solution of conjugate heat transfer problems, Journal of Quantitative Spectroscopy and Radiative Transfer 221 (2018) 253–259. [261] S. Mazumder, M.F. Modest, Application of the full spectrum correlated-k distribution approach to modeling non-gray radiation in combustion gases, Combustion and Flame 129 (4) (2002) 416–438. [262] R. Yadav, A. Kushari, A.K. Verma, V. Eswaran, Weighted sum of gray gas modeling for nongray radiation in combusting environment using the hybrid solution methodology, Numerical Heat Transfer – Part B: Fundamentals 64 (2013) 174–197. [263] Y. Sun, X. Zhang, Contributions of gray gases in SLW for non-gray radiation heat transfer and corresponding accuracies of FVM and P1 method, International Journal of Heat and Mass Transfer 121 (2018) 819–831. [264] R. Demarco, J.-L. Consalvi, A. Fuentes, S. Melis, Assessment of radiative property models in non-gray sooting media, International Journal of Thermal Sciences 50 (2011) 1672–1684. [265] C. Wang, W. Ge, M.F. Modest, B. He, A full-spectrum k-distribution look-up table for radiative transfer in nonhomogeneous gaseous media, Journal of Quantitative Spectroscopy and Radiative Transfer 168 (2016) 46–56. [266] L. Wang, N.E. Endrud, S.R. Turns, M.D. D’Agostini, A.G. Slavejkov, A study of the influence of oxygen index on soot, radiation, and emissions characteristics of turbulent jet flames, Combustion Science and Technology 174 (8) (2002) 45–72. [267] S. Mazumder, Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods, Academic Press, 2016. [268] K. Lovegrove, A. Luzzi, Solar thermal power systems, in: R.A. Meiers (Ed.), Encyclopedia of Physical Science and Technology, vol. 15, Academic Press, San Diego, 2001, pp. 223–235. [269] D. Barlev, R. Vidu, P. Stroeve, Innovation in concentrated solar power, Solar Energy Materials and Solar Cells 95 (2011) 2703–2725. [270] E.A. Fletcher, Solarthermal processing: a review, ASME Journal of Solar Energy Engineering 123 (2001) 63–74.

Radiation in Chemically Reacting Systems Chapter | 22 857

[271] A. Steinfeld, R. Palumbo, Solar thermochemical process technology, in: R. Meiers (Ed.), Encyclopedia of Physical Science and Technology, vol. 15, Academic Press, San Diego, 2001, pp. 237–256. [272] T. Kodama, High-temperature solar chemistry for converting solar heat to chemical fuels, Progress in Energy and Combustion Science 29 (2003) 567–597. [273] A. Royne, C.J. Dey, D.R. Mills, Cooling of photovoltaic cells under concentrated illumination: a critical review, Solar Energy Materials and Solar Cells 86 (2005) 451–483. [274] A. Vossier, D. Chemisana, G. Flamant, A. Dollet, Very high fluxes for concentrating photovoltaics: considerations from simple experiments and modeling, Renewable Energy 16 (2012) 31–39. [275] H. Cotal, C. Fetzer, J. Boisvert, G. Kinsey, R. King, P. Hebert, H. Yoon, N. Karam, III–V multijunction solar cells for concentrating photovoltaics, Energy & Environmental Science 5 (2003) 174–192. [276] E.A. Fletcher, R.L. Moen, Hydrogen and oxygen from water, Science 197 (1977) 1050–1056. [277] G. Maag, W. Lipinski, ´ A. Steinfeld, Particle–gas reacting flow under concentrated solar irradiation, International Journal of Heat and Mass Transfer 52 (2009) 4997–5004. [278] B.G. Lougou, Y. Shuai, R. Pan, G. Chaffa, H.P. Tan, Heat transfer and fluid flow analysis of porous medium solar thermochemical reactor with quartz glass cover, International Journal of Heat and Mass Transfer 127 (2018) 61–74. [279] A.J. Guadarrama-Mendoza, H.I. Villafán-Vidales, P.J. Valadés-Pelayo, C.A. Arancibia-Bulnes, D. Riveros-Rosas, H. Romero-Paredes, Radiative analysis in a multichanneled monolith solar reactor coated with ZnFe2 O4 thin film, International Journal of Thermal Sciences 132 (2018) 275–284. [280] I. Hischier, D. Hess, W. Lipinski, ´ M.F. Modest, A. Steinfeld, Heat transfer analysis of a novel pressurized air receiver for concentrated solar power via combined cycles, Journal of Thermal Science and Engineering Applications 1 (2009) 041002. [281] J. Karni, A. Kribus, R. Rubin, P. Doron, The “Porcupine”: a novel high-flux absorber for volumetric solar receivers, ASME Journal of Solar Energy Engineering 120 (1998) 85–95. [282] A.L. Ávila Marín, Volumetric receivers in solar thermal power plants with central receiver system technology: a review, Solar Energy 85 (5) (2011) 891–910. [283] M. Blanco, L.R. Santigosa, Advances in Concentrating Solar Thermal Research and Technology, Woodhead Publishing, Oxford, 2016. [284] G. Flamant, Theoretical and experimental study of radiant heat transfer in a solar fluidized-bed receiver, AIChE Journal 18 (1982) 529–535. [285] G. Flamant, T. Menigault, D. Schwander, Combined heat transfer in a semitransparent multilayer packed bed, ASME Journal of Heat Transfer 110 (2) (1988) 463–467. [286] X. Chen, X.L. Xia, H. Liu, Y. Li, B. Liu, Heat transfer analysis of a volumetric solar receiver by coupling the solar radiation transport and internal heat transfer, Energy Conversion and Management 114 (2016) 20–27. [287] B.G. Lougou, Y. Shuai, H. Xing, Y. Yuan, H.P. Tan, Thermal performance analysis of solar thermochemical reactor for syngas production, International Journal of Heat and Mass Transfer 111 (2017) 410–418. [288] B.G. Lougou, Y. Shuai, X. Chen, Y. Yuan, H.P. Tan, H. Xing, Analysis of radiation heat transfer and temperature distributions of solar thermochemical reactor for syngas production, Frontiers in Energy 11 (2017) 480–492. [289] T. Tan, Y. Chen, Review of study on solid particle solar receivers, Renewable and Sustainable Energy Reviews 14 (1) (2010) 265–276. [290] J. Martinek, A.W. Weimar, Evaluation of finite volume solutions for radiative heat transfer in a closed cavity solar receiver for high temperature solar thermal processes, International Journal of Heat and Mass Transfer 58 (2013) 585–596. [291] F. Wang, Y. Shuai, H.P. Tan, C. Yu, Thermal performance analysis of porous media receiver with concentrated solar irradiation, International Journal of Heat and Mass Transfer 62 (2013) 247–254. [292] G. Maag, C. Falter, A. Steinfeld, Temperature of a quartz/sapphire window in a solar cavity-receiver, ASME Journal of Solar Energy Engineering 133 (1) (2011) 014501. [293] S. Yong, F.-Q. Wang, X.L. Xia, H.P. Tan, Y.-C. Liang, Radiative properties of a solar cavity receiver/reactor with quartz window, International Journal of Hydrogen Energy 36 (19) (2011) 12148–12158. [294] W.H. Brock, The Norton History of Chemistry, W. W. Norton & Company, Inc., New York, 1993. [295] F. Trombe, M. Foex, Essai de metallurgie du chrome par l’hydrogene au four solaire, Revue de Métallurgie 48 (1951) 359–362. [296] T. Nakamura, Hydrogen production from water utilizing solar heat at high temperatures, Solar Energy 19 (5) (1977) 467–475. [297] G.J. Nathan, B.B. Daly, Z.T. Alwahaby, P.J. van Eyk, M. Jafarian, P.J. Ashman, Research challenges in combustion and gasification arising from emerging technologies employing directly irradiated concentrating solar thermal radiation, Proceedings of the Combustion Institute 36 (2017) 2055–2074. [298] M. Romero, A. Steinfeld, Concentrating solar thermal power and thermochemical fuels, Energy & Environmental Science 5 (2012) 9234–9245. [299] W. Lipinski, ´ J.H. Davidson, S. Haussener, J.F. Klausner, A.M. Mehdizadeh, J. Petrasch, A. Steinfeld, L. Venstrom, Review of heat transfer research for solar thermochemical applications, ASME Journal of Thermal Science and Engineering Applications 5 (2013) 021005. [300] R. Palumbo, M. Keunecke, S. Möller, A. Steinfeld, Reflections on the design of solar thermal chemical reactors: thoughts in transformation, Energy 29 (2004) 727–744. [301] B. Wang, L. Li, J.J. Pottas, R. Bader, P.B. Kreider, V.M. Wheeler, W. Lipinski, ´ Thermal model of a solar thermochemical reactor for metal oxide reduction, ASME Journal of Solar Energy Engineering 142 (2020) 51002.

858 Radiative Heat Transfer

[302] J. Lapp, J.H. Davidson, W. Lipinski, ´ Efficiency of two-step solar thermochemical partial redox cycles with heat recovery, Energy 37 (2012) 591–600. [303] W. Lipinski, ´ D. Thommen, A. Steinfeld, Unsteady radiative heat transfer within a suspension of ZnO particles undergoing thermal dissociation, Chemical Engineering Science 61 (2006) 7029–7035. [304] S. Abanades, P. Charvin, G. Flamant, Design and simulation of a solar chemical reactor for the thermal reduction of metal oxides: case study of zinc oxide dissociation, Chemical Engineering Science 62 (22) (2007) 6323–6333. [305] R. Müller, W. Lipinski, ´ A. Steinfeld, Transient heat transfer in a directly-irradiated solar chemical reactor for the thermal dissociation of ZnO, Applied Thermal Engineering 28 (2008) 524–531. [306] L.O. Schunk, W. Lipinski, ´ A. Steinfeld, Ablative heat transfer in a shrinking packed-bed of ZnO undergoing solar thermal dissociation, AIChE Journal 55 (2009) 1659–1666. [307] L.A. Dombrovsky, L.O. Schunk, W. Lipinski, ´ A. Steinfeld, An ablation model for the thermal decomposition of porous zinc oxide layer heated by concentrated solar radiation, International Journal of Heat and Mass Transfer 52 (2009) 2444–2452. [308] H.I. Villafán-Vidales, C.A. Arancibia-Bulnes, U. Dehesa-Carrasco, H. Romero-Paredes, Monte Carlo radiative transfer simulation of a cavity solar reactor for the reduction of cerium oxide, International Journal of Hydrogen Energy 34 (1) (2009) 115–124. [309] Z. Liang, W.C. Chueh, K. Ganesan, S.M. Haile, W. Lipinski, ´ Experimental determination of transmittance of porous cerium dioxide media in the spectral range 300–1,100 nm, Experimental Heat Transfer 24 (2011) 285–299. [310] K. Ganesan, W. Lipinski, ´ Experimental determination of spectral transmittance of porous cerium dioxide in the range 900–1,700 nm, ASME Journal of Heat Transfer 133 (2011) 104501. [311] K. Ganesan, L.A. Dombrovsky, W. Lipinski, ´ A novel methodology to determine spectral radiative properties of ceria ceramics, in: P. Boulet, D. Lacroix (Eds.), Proceedings of the Eurotherm Seminar 95—Computational Thermal Radiation in Participating Media IV, Nancy, 18–20 April, 2012, pp. 18–20. [312] S. Haussener, A. Steinfeld, Effective heat and mass transport properties of anisotropic porous ceria for solar thermochemical fuel generation, Materials 5 (2012) 192–209. [313] B. Wang, L. Li, F. Schäfer, J.J. Pottas, A. Kumar, V.M. Wheeler, W. Lipinski, ´ Thermal reduction of iron-manganese oxide particles in a high-temperature packed-bed solar thermochemical reactor, Chemical Engineering Journal 412 (2021) 128255. [314] W. Lipinski, ´ A. Steinfeld, Transient radiative heat transfer within a suspension of coal particles undergoing steam gasification, Heat and Mass Transfer 41 (2005) 1021–1032. [315] W. Lipinski, ´ A. Z’Graggen, A. Steinfeld, Transient radiation heat transfer within a nongray nonisothermal absorbing-emittingscattering suspension of reacting particles undergoing shrinkage, Numerical Heat Transfer – Part B: Fundamentals 47 (2005) 443–457. [316] G. Maag, S. Rodat, G. Flamant, A. Steinfeld, Heat transfer model and scale-up of an entrained-flow solar reactor for the thermal decomposition of methane, International Journal of Hydrogen Energy 35 (24) (2010) 13232–13241. [317] W. Lipinski, ´ A. Steinfeld, Heterogeneous thermochemical decomposition under direct irradiation, International Journal of Heat and Mass Transfer 47 (2004) 1907–1916. [318] L.A. Dombrovsky, W. Lipinski, ´ Transient temperature and thermal stress profiles in semi-transparent particles under high-flux irradiation, International Journal of Heat and Mass Transfer 50 (2007) 2117–2123.

Chapter 23

Inverse Radiative Heat Transfer 23.1 Introduction Up to this point we have concerned ourselves with radiative heat transfer problems, where the necessary geometry, temperatures, and radiative properties are known, enabling us to calculate the radiative intensity and radiative heat fluxes in such enclosures. Such cases are sometimes called “direct” heat transfer problems. However, there are many important engineering applications where knowledge of one or more input parameters is desired that cause a certain radiative intensity field. For example, it may be desired to control the temperatures of heating elements in a furnace, in order to achieve a specified temperature distribution or radiative heat load on an object being heated. Or the aim may be to deduce difficult to measure parameters (such as radiative properties, temperature fields inside a furnace, etc.) based on measurements of radiative intensity or radiative flux. Such calculations are known as inverse heat transfer analyses. One of the difficulties associated with inverse heat transfer analyses is the fact that they tend to be ill-posed (unlike direct heat transfer problems, which are nearly always well-posed). The conditions for a problem to be well-posed were first postulated by Hadamard [1] as • the solution to the problem must exist, • the solution must be unique, and • the solution must be stable (i.e., small changes of problem parameters cause only small changes in the solution). Only in rare instances can the solution to an inverse problem be proven to be unique. For example, while a given parameter field will produce, say, a unique radiative flux at a given location (direct problem), the measured radiative flux at a certain location, on the other hand, can be caused by various parameter fields governing the system (inverse problem). Moreover, inverse problems tend to be very sensitive to disturbances in the parameter field, such as random errors attached to experimental data. This generally necessitates special solution techniques to satisfy stability requirements, by adding additional information to the analysis based on prior knowledge of the true (or desired, in the case of design) solution attributes. While rudimentary attempts at inverse heat transfer solutions have been around for many years, formal methods to convert unstable inverse problems into approximate, well-posed problems through different types of regularization or stabilization techniques are only 40 to 50 years old, notably Tikhonov’s regularization procedure [2], and Beck’s function estimation technique [3]. The earliest works on inverse heat transfer problems date back to about 1960 [4–8], all on inverse heat conduction. The first investigations on inverse radiative heat transfer appeared in the early 1990s, mostly by Özi¸sik and coworkers [9–16]. Interestingly, almost all of these papers concerned themselves with radiative transfer within participating media, and there were only a few treatments dealing with inverse surface radiation before the turn of the century [17,18]. The last decade has seen a veritable explosion in research on inverse radiation, which will be summarized after a brief outline is given of the nature of inverse radiation problems, and after some of the more basic and popular solution methods are explained. The reader interested in conducting serious research in this field should consult the various books on inverse heat transfer [19–24] and solution methods for ill-posed problems [2,3,25–32]. Reviews of inverse radiation tools and research have been given by França and coworkers [33] and Daun and Howell [34,35] (inverse surface radiation problems), and by Charette et al. [36] (optical tomography). A very detailed chapter discussing inverse methods and past research in radiative heat transfer was recently given by Daun [37].

23.2 Solution Methods The solution to an inverse problem is usually found by minimizing an objective function, F, using a stabilization technique in the minimization procedure. Here we will briefly discuss a few of the most popular methods, such Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00031-6 Copyright © 2022 Elsevier Inc. All rights reserved.

859

860 Radiative Heat Transfer

the truncated singular value decomposition (TSVD), Tikhonov regularization, Newton’s method, the Levenberg– Marquardt method, and conjugate gradient techniques. Others can be found in the books cited above, as well as in the various research papers in the field. Suppose the radiative intensity or radiative flux, etc., is known for a number of directions, and/or wavelengths. These measured data values (for deducing parameters inside or on the surface of the radiative enclosure) or desired values (for control of parameters) will be denoted by the data vector y (with elements Y1 , Y2 , . . . , YI ). These data need to be compared with corresponding values estimated from a direct analysis, based on an optimized set of the unknown parameters to be determined, denoted by the vector i (with elements I1 , I2 , . . . , II ). If J different parameters are chosen for the inverse problem, these values form a parameter vector p (with elements p1 , p2 , . . . , p J ), and the estimated solutions Ii are a function of this vector. For example, if it is desired to estimate the Planck function (or temperature) distribution within a participating medium, one may postulate the Planck function field to be approximated by Ibη (r, η)

J 

  p j fj r, η ,

(23.1)

j=1

where the fj are known specified basis functions (polynomials, splines, etc.), and the best values for the p j are to be found. If all the data points have statistically equal error values, or if all desired values have equal importance, then the objective function to be minimized is the ordinary least squares norm:1 F=

I 

CC CC2 (Ii − Yi )2 = (i − y) · (i − y) = Ci − yC .

(23.2)

i=1

If the data points are very close together then the summation in equation (23.2) may be replaced by an integral. In many applications the statistical uncertainty of data points, or their variance, σi2 , may be known and may be different for individual data points. In that case it is preferable to define the objective function as a weighted least squares norm F=

I    Ii − Yi 2 = (i − y) · W·(i − y), σi

(23.3)

i=1

where W is a diagonal weighting matrix, ⎛ ⎜⎜ 1/σ12 ⎜⎜ ⎜⎜ ⎜⎜ 0 W = ⎜⎜⎜⎜ . ⎜⎜⎜ .. ⎜⎜ ⎝ 0

0

···

0

1/σ22 .. .

··· .. .

0 .. .

0

1/σI2

⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ . ⎟⎟ ⎟⎟⎟ ⎟⎟ ⎠

(23.4)

In control applications, where y is the desired effect, rather than a vector of measured data, the factors 1/σi function as importance factors attached to individual control values Yi . Equation (23.3) reduces to equation (23.2) if W is equal to the unit tensor δ (with Kronecker’s delta function δi j as elements, i.e., a diagonal matrix with all nondiagonal elements zero, and all diagonal elements Wii = 1). Minimization of the objective function in terms of the parameter vector p requires that the derivatives of F with respect to each of the parameters p j be zero, i.e., ∂F ∂F ∂F = = ··· = = 0, or ∇p F(p) = 0, ∂p1 ∂p2 ∂p J

(23.5)

1. We will follow here again the matrix notation introduced in Chapter 17, i.e., vectors are written as bold lowercase letters, two-dimensional tensors as bold uppercase letters, and dot products imply summation over the closest indices on both sides of the dot. See also footnotes on p. 620.

Inverse Radiative Heat Transfer Chapter | 23 861

where the ∇p F(p) represents the gradient of F(p) with respect to the vector of parameters. Carrying out the differentiation of equation (23.3) with respect to p leads to  I − Y ∂I ∂F i i i =2 = 0, 2 ∂p j ∂p σi j I

j = 1, 2, . . . , J,

(23.6)

i=1

or, in matrix notation, ∇p F(p) = 2(i − y) · W · X = 0,

(23.7)

where X is known as the sensitivity matrix, or Jacobian, with elements ⎛ ⎜⎜ ⎜⎜ ⎜⎜ ⎜⎜ ⎜⎜ ⎜⎜ ⎜⎜ ⎜ X = ∇p i = ⎜⎜⎜⎜ ⎜⎜ ⎜⎜ ⎜⎜ ⎜⎜ ⎜⎜⎜ ⎜⎝

∂I1 ∂p1

∂I1 ∂p2

∂I2 ∂p1 .. . ∂II ∂p1

∂I2 ∂p2 .. . ∂II ∂p2

··· ··· ..

.

···

⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ∂I2 ⎟⎟⎟ ⎟⎟ ∂p J ⎟⎟⎟ , ⎟ .. ⎟⎟⎟⎟ . ⎟⎟⎟ ⎟⎟ ∂II ⎟⎟⎟ ⎠ ∂p J ∂I1 ∂p J

(23.8)

for J unknown parameters and I measured (or defined) data points. If the sensitivity matrix is independent of p the problem is called linear. For example, if the general radiative transfer equation (9.21) is to be solved with the temperature field given by equation (23.1) (and there are no other unknown parameters), differentiation of the intensity field with respect to any parameter p j makes ∂I/∂p j independent of all parameters p. For such a case i=X·p

(23.9)

and equation (23.7) may be restated in standard matrix form as A · p = b,

with A = XT · W · X, b = XT · W · y,

(23.10)

which are known as the normal equations, and its solution may be formally written as p = A−1 · b = (XT · W · X)−1 · XT · W · y.

(23.11)

If the problem is nonlinear, i.e., the sensitivity coefficients Xi j are functions of p, then the problem must be linearized and solved iteratively. This is usually done by expanding i(p) into a truncated Taylor series around the current solution pk at iteration k, or i(p) i(pk ) + Xk · (p − pk ),

(23.12)

where i(pk ) and Xk are the values for estimated intensity and the sensitivity matrix after the kth iteration. Sticking this into equation (23.7) leads to an iterative procedure with updated parameter vector as pk+1 = pk + (XTk · W · Xk )−1 · XTk · W · (y − ik ),

(23.13)

which is known as the Gauss (or Gauss-Newton) method. The linear solution, equation (23.11), and the linearized iterative procedure, equation (23.13), both require   the matrix XT · W · X to be nonsingular, i.e., its determinant may not be zero, or XT · W · X  0. If the value of this determinant is close to zero the problem is called ill-conditioned and, unfortunately, inverse heat transfer problems tend to be very ill-conditioned. This is best understood by visually comparing the solutions to two

862 Radiative Heat Transfer

FIGURE 23.1 Plot of the residual norm for (a) a well-conditioned and (b) an ill-conditioned matrix equation.

very simple well-conditioned and ill-conditioned matrix equations, such as 

well-conditioned: ill-conditioned:

     p1 1 = = b, A·p= · p2 1       p1 1 2 2.1 · A·p= = = b. p2 2 2 1 2 3 3 3

The well-conditioned matrix equation has a well-defined solution (p1 = p2 = minimizes the square residual ⎛ ⎞2 2 ⎜ 2  ⎟⎟ ⎜ ⎟⎟ ⎜⎜⎜ ||A · p − b||2 = A p − b ⎟⎟ , i j j i ⎜⎝ ⎠ i=1

1 5

(23.14a) (23.14b)

in the present case), which

(23.15)

j=1

as shown in a contour plot in Fig. 23.1a. Equation (23.14b) also has a single, unique solution (as is guaranteed for all nonsingular linear problems, with p1 = 12 , p2 = 0 in this particular case), but, as seen from the contour plot in Fig. 23.1b, there is also a substantially extended range of locations along the valley floor (as compared to Fig. 23.1a) that makes the residual very small. In optimization problems, this leads to numerical difficulty in finding the minimum (or solution) exactly, since the residual can be reduced below a small prescribed tolerance by multiple solutions (different sets of p), all of which are very close to the minimum. Similarly, when trying to deduce a property field, many approximate solutions exist that nearly satisfy the experimental data (but may yield property fields with wide fluctuations). In addition, ill-conditionedness makes the solution highly susceptible to small perturbations in experimental data as well as to numerical artifacts, such as roundoff error. Before discussing methods to solve ill-conditioned problems it is instructive to investigate the matrices that need to be inverted in the analysis, such as X in equation (23.9) or XT · W · X in equation (23.10). The properties of an arbitrary matrix A with M×N elements may be diagnosed through Singular Value Decomposition (SVD) [30,38], by decomposing it into a product of an M × N column-orthogonal matrix U, an N × N diagonal matrix S with only positive or zero elements (its singular values, usually placed in descending order), and the transpose of an N × N orthogonal matrix VT , i.e., A = U · S · VT . The beauty of orthogonal matrices is that its inverse is simply its transpose, while the inverse of a diagonal matrix (i.e., a matrix where only the diagonal terms are nonzero) is another diagonal matrix whose elements are the reciprocals of Skk . Therefore, the inverse of A follows as A−1 = V · S−1 · UT ,

(23.16a)

Inverse Radiative Heat Transfer Chapter | 23 863

FIGURE 23.2 Geometry for Example 23.1.

which has elements (A−1 )i j =

N 

Vik

k=1

1 U jk . Skk

(23.16b)

Inspection of equation (23.16b) shows that the only thing that can go wrong with the inversion of (a nonsingular) A is that the inversion gets greatly impacted by roundoff error caused by very small singular values. The condition number of a matrix is the ratio of the largest over the smallest Skk ; the matrix is singular if the condition number is infinite, and is ill-conditioned if its condition number is too large. Example 23.1. Consider two long parallel plates of width w as shown in Fig. 23.2. Both plates have a gray, diffuse emittance of ; they are separated by a distance h and are placed in a large, cold environment. The bottom plate is insulated, and it is desired to keep this plate at an isothermal temperature T∗ across its width through radiative heating from the top plate. Determine the necessary temperature distribution T2 (x2 ) to achieve this result. Solution Direct Solution. From equation (5.27) we find that J1 (x1 ) = σT14 (x1 ) for an insulated surface. Thus, from equation (5.24), with dFdi−di = 0, we obtain  σT14 (x1 ) =

w

J2 (x2 ) dFd1−d2 , 0



J2 (x2 ) = σT24 (x2 ) + (1 − )

w 0

σT14 dFd2−d1 .

The necessary view factors have already been obtained in Example 5.10. Also, using similar nondimensionalization as in that example, i.e., W = w/h, ξ = x/h, Θ(x) = [T(x)/T∗ ]4 , and J(ξ2 ) = J2 (x2 )/σT∗4 , these equations become  Θ1 (ξ1 ) =

W

J(ξ2 ) f (ξ1 − ξ2 ) dξ2 ,

0



J(ξ2 ) = Θ2 (ξ2 ) + (1 − ) f (ξ1 − ξ2 ) =

1 2



W

Θ1 (ξ1 ) f (ξ1 − ξ2 ) dξ1 , 0

1 + (ξ1 − ξ2 )2

−3/2

.

The desired result is Θ1 (ξ1 ) = 1 = const, for which the necessary Θ2 (ξ2 ) needs to be found. We will approximate both plates by N constant temperature (and radiosity) strips of width Δξ = W/N that have nondimensional temperatures Θ1i , Θ2i , i = 1, 2, . . . , N, with optimum values of the parameters Θ2i to be determined (and, for illustrative purposes, without taking advantage of the obvious symmetry across x = w/2). Thus, approximating the integrals by summation over the strips, and assuming view factors to be constant across the width of a strip [evaluated between ξ1i = (i − 12 )Δξ and ξ2j = ( j − 12 )Δξ], we obtain

864 Radiative Heat Transfer

FIGURE 23.3 Singular values for the matrix in Example 23.1.

Θ1i = Δξ

N 

Jj fij ,

i = 1, 2, . . . , N,

j=1

Jj = Θ2j + (1 − )Δξ

N 

Θ1k fjk ,

j = 1, 2, . . . , N,

k=1

or Θ1i =

N  j=1

Θ2j yij +

N 

Θ1k zik ,

yij = Δξ fij ,

zik = (1 − )Δξ2

N 

fij f jk ,

i = 1, 2, . . . , N.

(23.17)

j=1

k=1

Inverse Problem. Since we have chosen equal numbers of strips on the bottom surface (design points) and top heater (parameter vector p), equation (23.17) constitutes a set of linear equations (with given desired Θ1i ) that can be solved directly for the N unknown Θ2j , i.e., by direct solution of equation (23.9). In general, however, the number of strips may be different for each plate and we prefer to minimize the objective function. Since we want each design point to achieve T1 = T∗ (or Θ1 = 1), the objective function becomes, assuming constant weights (σi = 1), F = Θ1 − 12 =

N 

(Θ1i − 1)2 ,

i=1

where 1 is a unity vector (all elements equal 1). The sensitivity matrix is obtained by differentiating equation (23.17) with respect to (p j =) Θ2j , or Xij = yij +

J 

zik Xk j ,

k=1

which is solved by successive approximation. We note that the sensitivity matrix is independent of Θ2j . Thus, the problem is linear (as we noticed already). For a direct solution, or to diagnose the problem’s ill-posedness, we may perform an SVD on the matrix A = XT · X. Once the U, S, and V matrices have been determined, the solution for p (with elements pi = Θ2i ) is then found from equations (23.10) and (23.16b) as Θ2i =

N 

(A−1 )ij b j =

j=1

N N  Vik  U jk b j , Skk j=1

with b = XT · y = XT · 1,

(23.18a)

k=1

or b j =

N  l=1

Xlj yl =

N 

Xlj .

(23.18b)

l=1

The singular values for the present problem, with w/h = 5, = 0.5, and N = 20 (as calculated with the Numerical Recipes routine svdcmp [38]) are shown in Fig. 23.3. It can be seen that the singular values decay rapidly from a maximum

Inverse Radiative Heat Transfer Chapter | 23 865

value of 0.44 down to 3 × 10−10 , with a condition number exceeding 109 , making the problem very ill-conditioned. The solution to equation (23.18a), when calculated in single precision (with about six digits of accuracy) yields oscillatory (and meaningless) values for nondimensional heater emissive power Θ2 varying between approximately −300 and +300 (not shown here).

The matrix A in equation (23.10) contains the square of the sensitivity matrix X, and thus also the square of its condition number, making the solution even more susceptible to round-off error. An alternative, and often preferred, technique involves solving a truncated version of equation (23.7), by setting i − y = 0,

or

X · p = y,

(23.19)

which also finds a (different) least-squares minimum [38]. However, equation (23.19) is overdeterminate if there are more data points than parameters (I > J). From the above example it can be seen that it is generally necessary to apply stabilizing methods even to the solution of linear inverse problems, such as the techniques discussed below. In practice, equation (23.18a) is rarely used to compute p (= Θ2 ), and the inverse of A is not calculated directly; instead, the linear system A · p = b is solved. The various techniques to solve ill-conditioned problems may be loosely collected under the titles regularization, gradient-based optimization, and metaheuristics, and some of the most common methods will be briefly discussed below. More detail can be found in books on the subject, e.g., Hansen [30], as well as several review articles [33,34].

23.3 Regularization We saw in the previous section that an ill-conditioned matrix has a large condition number, i.e., some of the singular values are very small, causing the solution to become unstable. Decreasing the condition number of a matrix A by modifying it (or its inverse) is, in the present context, known as regularization. We will briefly describe here the simple truncated singular value decomposition and the perhaps most popular Tikhonov regularization methods.

Truncated Singular Value Decomposition (TSVD) The simplest form of regularization consists of simply omitting parts of the inverse of A corresponding to the (offending) smallest singular values. This is justified by the fact that the higher terms in the series correspond to “high frequency” components, and often have less physical significance. Our prior knowledge (or desire) of a smooth solution is used as justification for truncation [30]. The matrix A, as given in the normal equation set (23.10), is first singular value decomposed as given by equation (23.16b). The full solution to equation (23.10) would then be obtained from equation (23.18a). Eliminating the largest values of 1/Skk is achieved by keeping only the first K terms in equation (23.18a) (i.e., dropping terms with k > K, thereby decreasing the condition number) p∗i

N K  Vik  = U jk b j , Skk

i = 1, . . . , N,

(23.20)

j=1

k=1

where p∗ is the regularized solution. The proper value for K must be determined through external, often subjective criteria. Large values of K force the result vector i (e.g., the achieved nondimensional temperature of the bottom plate in Example 23.1) to more closely follow the prescribed data vector y (desired temperature), but may result in strongly oscillatory and/or unphysical parameter vectors p (power setting on heater plate). Small values of K, on the other hand, lead to a smooth variation for p, but the result vector i may depart substantially from the desired value y. Example 23.2. Repeat the control problem Example 23.1 using truncated singular value decomposition (TSVD). Solution The solution proceeds exactly as in Example 23.1, but the series in equation (23.18a) is truncated to give nondimensional heater temperatures as Θ2i =

K N  Vik  U jk b j , Skk j=1 k=1

866 Radiative Heat Transfer

FIGURE 23.4 Predicted top surface temperatures and recovery of desired bottom surface temperatures for Example 23.2.

and the resulting design surface temperatures are found from equation (23.9) Θ1i =

N 

Xij Θ2j .

j=1

Figure 23.4 shows the results, again for w/h = 5, = 0.5, and N = 20 strips on each plate, for several odd values of K (even values produce essentially identical results as the next lower K because of symmetry). It is observed that retaining a single singular value (K = 1) results in a very smooth heater setting, and also a smooth design surface temperature (but departing substantially from the desired value of “1”). Larger values of K bring the design plate temperatures closer to the desired value (albeit with slight oscillations), but at a cost of oscillatory heater settings. Values of K > 7 result in some strips having negative emissive power (cooling), which would be undesirable at best.

Tikhonov Regularization Most regularization methods transform an ill-posed inverse problem into a well-behaved one by adding auxiliary information based on desired or assumed solution characteristics [34,39]: F = (i − y) · W·(i − y) + λ2 Ω(p),

(23.21)

where Ω(p) is an arbitrary (positive) function and λ is the (positive) regularization parameter. One of the earliest and most popular examples is Tikhonov regularization [2], employing Ω = p · L · p, where L is an operator. In the simplest 0th order discrete Tikhonov regularization we have L = δ and Ω = p · p. Thus, equation (23.10) is changed to (A + λδ) · p = b,

with A = XT · W · X, b = XT · W · y,

(23.22)

where δ is again an Nth order unity tensor. Many different and higher order versions of Tikhonov’s regularization exist, and the reader is referred to [32,39]. The regularization parameter determines the smoothness of the solution: a small value of λ implies little regularization, while a large λ prioritizes some presumed information, which in the case of standard Tikhonov forces the solution vector toward zero. Several schemes exist to find an optimal value of λ. Numerical Recipes [38] suggests a starting value for λ of λ Tr(A)/N,

(23.23)

where Tr is the trace of the matrix (sum of the N diagonal elements), giving both parts in the minimization equal weights. An optimum value for λ is then found by trial and error. More sophisticated schemes include construction of a so-called L-curve, which leads to a semi-quantitative determination of λ [30,40]. Example 23.3. Repeat Example 23.1 using 0th order discrete Tikhonov regularization.

Inverse Radiative Heat Transfer Chapter | 23 867

FIGURE 23.5 Predicted top surface temperatures and recovery of desired bottom surface temperatures for Example 23.3.

Solution As in the previous example we calculate A = XT · X and b = X · 1. Before inverting A we modify the matrix to A∗ = A + λδ,

or

A∗ij = Aij + λδij ,

i.e., all diagonal elements are incremented by λ, which is evaluated as λ=C

N 1  Aii , N i−1

where C is a constant whose optimal value is to be found by trial and error. Heater emissive powers Θ2 and design surface emissive powers Θ1 are then determined from Θ2i =

N    A∗−1 b j , j=1

Θ1i =

N 

ij

bj =

N 

Xk j ,

k=1

Xij Θ2j .

j=1

Results for Tikhonov regularization are shown in Fig. 23.5, again for w/h = 5, = 0.5, and N = 20 strips on each plate, for five fractional values of C = 2−(5−k) , with larger C implying more regularization. It is seen from the figure that the Numerical Recipes’ suggested value (C = 1) gives a reasonable (perhaps slightly over-regularized) solution with smoothly varying heater values, but with design surface temperatures dropping near the edges of the plate. Smaller values of the regularization parameter lead to somewhat better design surface temperatures, at the cost of stronger heater surface variations. In general, it appears that Tikhonov regularization gives better results than TSVD, at least for the present problem.

23.4 Gradient-Based Optimization In optimization the objective function F, most often using least square norms as given by equations (23.2) or (23.3), is minimized in an iterative process. Iteration is always necessary for nonlinear problems, but may also be employed for linear ones to overcome ill-conditioning, which in optimization manifests itself in the form of a difficult objective function topography having a minimum (or several minima in nonlinear problems) surrounded by a long, shallow valley, as shown in Fig. 23.1b. Many different optimization schemes have been developed to minimize F. When F is continuously differentiable over the feasible region of p, it is generally best to use analytically defined search directions, with gradient-based methods being used most often [41]. In all schemes, during each iteration a step of appropriate size is taken along a direction of descent, which is based on

868 Radiative Heat Transfer

the local curvature of the objective function at the previous iteration. Thus, after the kth iteration a new solution vector is found from pk+1 = pk + βk dk ,

(23.24)

where βk is the search step size, and dk is the direction of descent. The main difference between gradient minimization techniques is how the search direction is chosen, which usually is how they got their name. As indicated by Daun and coworkers [41], whose development we will follow here, nearly all of the methods require first-order curvature information as contained in the gradient vector, 

∂F ∂F ∂F , ,··· , g(p) = ∇p F(p) = ∂p1 ∂p2 ∂p J

T = 2(i − y) · W · X,

(23.25)

where equation (23.7) has been invoked. Some methods also use second-order curvature information contained in the Hessian matrix ⎞ ⎛ ⎜⎜ ∂2 F ∂2 F ∂2 F ⎟⎟⎟ ⎜⎜ ··· ⎟ ⎜⎜ ∂p2 ∂p1 ∂p2 ∂p1 ∂p J ⎟⎟⎟⎟ ⎜⎜ 1 ⎟⎟ ⎜⎜ ⎟ ⎜⎜ ∂2 F ∂2 F ⎟⎟⎟ ⎜⎜ ∂2 F ⎟⎟ ⎜⎜ · · · ⎜ ∂p2 ∂p J ⎟⎟⎟ . ∂p22 (23.26) H(p) = ∇p ∇p F(p) = ⎜⎜⎜ ∂p2 ∂p1 ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ . . . . .. .. .. .. ⎟⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ 2 2 ⎟⎟ ⎜⎜ ∂2 F ∂ F F ∂ ⎟⎟⎠ ⎜⎜⎝ ··· 2 ∂p J ∂p1 ∂p J ∂p2 ∂p J Some of the more common gradient minimization techniques are steepest descent, Newton and quasi-Newton methods, the Levenberg–Marquardt method, and conjugate gradient methods. Steepest descent is the simplest, but has a linear or even slower convergence rate and is, therefore, not recommended. The other four are briefly described below.

Newton’s Method In Newton’s method the direction of descent is calculated using both first- and second-order curvature information, by expanding the objective function into a second-order Taylor series. Assuming the desired parameter vector p∗ is a distance sk away from the latest approximation for pk , i.e., p∗ = pk + sk , the gradient vector of the objective function can be written as a two-term Taylor expansion g(p∗ ) = g(pk + sk ) g(pk ) + skT · H(pk ),

(23.27)

which is exact with constant Hessian, H, if the objective function is quadratic (which tends to be approximately true, if p∗ is reasonably close to pk ). Since F has a global minimum at p∗ all elements of the gradient vector g(p∗ ) are equal to zero, and sk is determined from sk −H(pk )−1 · g(pk ).

(23.28)

In Newton’s method, dk is set equal to sk , which is called Newton’s direction (with an implied step size βk = 1). While the Hessian matrix is generally not constant near the minimum, using Newton’s direction results in much better convergence (typically quadratic), compared with the steepest descent method. However, calculating the Hessian matrix at each iteration tends to require significant extra CPU time, which can make Newton’s method actually less efficient than the steepest descent method. Thus, Newton’s method should only be used when the second derivatives can be calculated easily.

The Quasi-Newton Method The quasi-Newton method avoids calculating the Hessian matrix by approximating it using only first-order curvature data collected at previous iterations. At each iteration, the search direction dk = sk is calculated from

Inverse Radiative Heat Transfer Chapter | 23 869

equation (23.28) with an approximate Hessian B as dk = −(Bk )−1 · g(pk ).

(23.29)

Initially, (Bk )−1 is set equal to the identity matrix δ (which makes it the search direction for the steepest descent method) times an appropriate step size β0 [usually found from a single-value minimization of F(p0 − β0 g0 )]. At each subsequent iteration, the approximation of the Hessian matrix is improved upon by adding an update matrix, Uk , Bk = Bk−1 + Uk ,

(23.30)

and Uk is determined using only values of the objective function and gradient vectors from previous iterations. The most common quasi-Newton scheme is the Broyden-Fletcher-Goldfarb-Shanno (BFGS) scheme [31]; in this method, the update matrix is calculated from Uk =

zk · zk Bk−1 · dk−1 · dk−1 · Bk−1 − , where dk−1 = pk − pk−1 , zk = g(pk ) − g(pk−1 ), zk · dk−1 dk−1 · Bk−1 · dk−1

or, in expanded notation Uikj = 5

5

zki zkj

k k−1 m zm dm

−

5 k−1 k−1 Bk−1 dk−1 p dp Bp j il l 5 5 k−1 k−1 k−1 q p dq Bqp dp

l

(23.31a)

(23.31b)

Since it takes a few iterations for B to accurately approximate the Hessian matrix, the convergence rate of the quasi-Newton scheme is less than the Newton’s method, requiring a few more iterations to find the global minimum for F. However, since no second derivatives are needed, the quasi-Newton scheme is usually computationally more efficient. Here we will illustrate the method by presenting a very simple example, this time a problem to infer radiative properties of a participating medium through intensity measurements. Extension to more complicated geometries and/or radiative property fields affects only the direct-solution part of the problem, which has been discussed extensively in previous chapters. Example 23.4. Consider a one-dimensional, absorbing–emitting (but not scattering) slab of width L, bounded by two cold, black walls. The temperature distribution within the slab is unknown, and is to be estimated with the quasi-Newton method, by measuring exit intensities on both bounding walls for various angles. The absorption coefficient of the medium at the detector wavelength, κ, is known and constant. Solution Direct Problem. The direct solution for this simple problem is immediately found from equation (13.20) as  I(x, μ) = −  =

L



Ib (x ) eκ(x −x)/μ κ

x x



Ib (x ) e−κ(x−x )/μ κ

0

dx , μ

dx , μ

μ < 0, μ > 0,

with I1 = I2 = 0 (cold walls) and S = Ib (no scattering). Letting τL = κL, ξ = x/L, and evaluating only the necessary intensities exiting from the faces at ξ = 0, 1, leads to  τL 1 Ib (ξ) eτL ξ/μ dξ, I(0, μ) = − μ 0  τL 1 I(1, μ) = Ib (ξ) e−τL (1−ξ)/μ dξ, μ 0

μ < 0, μ > 0.

Inverse Problem. We will assume that the unknown Planck function field Ib (ξ) can be approximated by a simple Nth order polynomial, or Ib (ξ) =

N  n=0

pn ξn .

870 Radiative Heat Transfer

(Power series, while simple and adequate for the present example, are generally not a good practice because the coefficients will vary over a wide range of magnitudes [24]). Substituting this into the direct solution for exiting intensity gives 

 τL , μ < 0, pn fn I(0, μ) = − μ n=0   N  τL −τL /μ , μ > 0, I(1, μ) = e pn fn μ n=0  1 n fn (τ) = τ ξn eτξ dξ = eτ − fn−1 (τ). τ 0 N 

(23.32a)

(23.32b)

Since the temperature (or Planck function) is to be found by measuring I(0, μ) and I(1, μ) for a set of I exit angles −1 < μi < +1, and assuming constant weights, the objective function becomes F=

I 

(Ii − Yi )2 ,

i=1

where the Ii are evaluated from equation (23.32a) or (23.32b), depending on whether μi is negative or positive, and the Yi are the corresponding experimental data. The sensitivity matrix is readily found by differentiating equations (23.32a) and (23.32b) with respect to pn , leading to ⎧   ⎪ τL ⎪ ⎪ ⎪ − f , μi < 0, ⎪ n ⎪ μi ⎨ Xin = ⎪   ⎪ ⎪ ⎪ τL ⎪ ⎪ , μi > 0, ⎩ e−τL /μi fn μi and Ii =

N 

pn Xin ,

(23.33)

n=0

since the problem is linear. In order to use the quasi-Newton method, we first need to calculate the gradient vector from equation (23.25), or, assuming unity weights W = δ, gkn = 2

I  

Iik − Yi

 ∂Iik

i=1

∂pn

=2

I  

 Iik − Yi Xin .

(23.34)

i=1

In the first iteration we set B−1 = δ, and p1 = p0 − β0 g0 , using a first guess for p of pn = δn0 (constant temperature slab). The proper step size β0 is found by minimizing F with respect to β0 , i.e., by setting  ∂Ii ∂F (Ii − Yi ) 0 = 0, =2 0 ∂β ∂β i=1 I

or

⎛ N ⎞⎛ N ⎞ I   ⎜⎜  ⎟⎟ ⎜⎜  ⎟⎟ 0 0 0 0 ⎜ ⎜ ⎟ ⎜⎜ pn − β gn Xin − Yi ⎟⎟⎠ ⎜⎜⎝− gn Xin ⎟⎟⎟⎠ = 0, 2 ⎝ i=1

n=0

n=0

⎛ N ⎞⎛ N ⎞ ⎛ N ⎞2 I I   ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ 0 0 0 0 ⎜ ⎜⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ pn Xin − Yi ⎟⎠ ⎜⎝ gn Xin ⎟⎠ − β gn Xin ⎟⎟⎟⎠ = 0 ⎜⎝ ⎜⎝ i=1

n=0

n=0

and, finally I 5

β0 =

i=1

 (Xi0 − Yi ) I 5



N 5

i=1 n=0

N 5 n=0

g0n Xin

i=1

 g0n Xin 2

.

n=0

Inverse Radiative Heat Transfer Chapter | 23 871

FIGURE 23.6 Planck function distribution for Example 23.4 predicted by the quasi-Newton method.

For all following iterations we need to update Bk according to equations (23.30) and (23.31a). Since we are only interested in the inverse of Bk , it is usually more efficient to calculate it directly from the Sherman–Morrison formula [31]:  −1    −1 −1  dk−1 ·zk + zk · Bk−1 ·zk dk−1 dk−1 −1  −1  Bk−1 ·zk dk−1 + dk−1 zk · Bk−1 k k−1 = B + − , (23.35a) B 2 dk−1 ·zk (dk−1 ·zk ) or, in expanded notation,  −1  −1 Bk = Bk−1 + ij

ij

5 l

dk−1 zkl + l

5 5 l

k k−1 −1 k )lm zm m zl (B

5

l

dk−1 zkl l

2



dk−1 dk−1 i j

5 −

l

(Bk−1 )−1 zk dk−1 +dk−1 i il l j 5 k−1 k d l l zl

5

k k−1 −1 )lj l zl (B

.

(23.35b)

After each iteration the objective function is recalculated, and the procedure is stopped when F no longer decreases (substantially). Figure 23.6 shows the simulation results for a Planck function field of Ib (ξ) = 1 + 3ξ2 − 4ξ4 for various optical thicknesses, and using 20 equally spaced measurement directions. For errorless measurements Yi , the exact result is recovered for all optical thicknesses. Figure 23.6 shows the estimated Planck function field for measurements that have been given a random Gaussian error, with a relative variance of 3%. It is seen that the Planck function field recovery is rather poor for optically thin slabs, getting more and more accurate as the optical thickness increases (up to a point: at very large τL the exiting intensities become independent of the internal temperature field and, thus, the temperature field cannot be recovered).

The Levenberg–Marquardt Method The Levenberg–Marquardt method was originally devised for nonlinear parameter estimation problems, but has also proved useful for the solution of ill-conditioned linear problems [3,25,42,43]. In this method the problem of inverting a near-singular matrix is avoided by increasing the value of each diagonal term in the matrix, i.e., by regularizing the Gauss-Newton method of equation (23.13) to pk+1 = pk + (XTk · W · Xk + μk Ω k )−1 · XTk · W · (y − ik ),

(23.36)

where μk is a positive scalar called the damping parameter, and Ω k is a diagonal matrix. In this equation the inverse is an approximation of the Hessian matrix, and the remainder is the negative of the gradient vector, as given by equation (23.7). Levenberg suggested several choices for the diagonal matrix Ω k , among them Ω k = δ (each diagonal term is increased by a fixed amount μk ) and Ωkii = (XTk · W · Xk )ii (each diagonal term is increased by a fixed percentage). As with regularization, large values for μk dampen out oscillations in the ill-conditioned

872 Radiative Heat Transfer

system, but also change the solution. Thus, after starting the iteration with a relatively large value of μk , its value is gradually decreased as the iteration approaches convergence. Comparison with equation (23.21) shows that the method is related to Tikhonov regularization, but using a gradually decreasing regularization parameter. Different versions of the Levenberg–Marquardt method have been incorporated into various numerical libraries, such as the Numerical Recipes [38] and IMSL routines [44].

The Conjugate Gradient Method The conjugate gradient method is another simple and powerful iterative technique to solve linear and nonlinear minimization problems, which is a further development of the method of steepest descent. The method is explained in detail in a number of books, such as [21,24,45–47]. In the method of steepest descent the new search direction vector is simply derived from the local gradient as dk = −g(pk ) = −∇p F(pk ),

(23.37)

with g(pk ) evaluated from equation (23.25). However, the search direction vectors in successive iterations can be shown to always be orthogonal to each other, i.e., dk−1 ·dk = 0 [48]. This implies that the path toward convergence is a zigzag (or staircase) path. This leads to the need for a large number of iterations (direction changes) to reach the minimum of the function, making the method of steepest descent quite inefficient. The conjugate gradient method improves on that by finding the direction of descent as a conjugate of the gradient direction and the previous direction of descent, or dk = −g(pk ) + γk dk−1 ,

(23.38)

where γk is the conjugation coefficient. The search step size βk is taken as the value that minimizes the objective function at the next iteration, F(pk+1 ): using equations (23.3) and (23.24) together with the Taylor expansion, equation (23.12), leads to     F(pk+1 ) = i(pk + βk dk ) − y · W · i(pk + βk dk ) − y     i(pk ) − y + βk Xk · dk · W · i(pk ) − y + βk Xk · dk . (23.39) Differentiating with respect to βk , setting ∂Fk+1 /∂βk = 0, and solving for βk results in βk =

(Xk · dk ) · W · (y − ik ) , (Xk · dk ) · W · (Xk · dk )

(23.40a)

or, in expanded notation, J I Yi − I k 5 5 i

Xikj djk 2 σ j=1 i βk = ⎛ ⎞2 . J I 1 ⎜5 ⎟⎟ 5 ⎜⎜ ⎜ Xikj djk ⎟⎟⎠ 2 ⎝ i=1 σi j=1 i=1

(23.40b)

Several different expressions are in use for the conjugation coefficient γk . We mention here only the simple Fletcher–Reeves expression CC CC2 Cgk C k γ =C , k = 1, 2, . . . , (23.41a) CC k−1 CCC2 g = 0,

k = 0.

(23.41b)

CC CC2 In expanded notation Cgk C becomes, from equation (23.25), ⎛ I ⎞2 J  CC CC2 ⎜⎜ Iik − Yi ⎟⎟ k k ⎜ ⎜⎜ Cg C = 4 Xi j ⎟⎟⎟⎠ . 2 ⎝ σ i j=1 i=1

(23.42)

Inverse Radiative Heat Transfer Chapter | 23 873

TABLE 23.1 Recovery of slab temperature distribution using various inversion techniques. Quasi-Newton with BFGS without line search with line search

Conjugate Gradient

Steepest Descent

Tikhonov

τL

iterations

time (ms)

iterations

time (ms)

0.1

17

0.88

17

1.02

6

0.45

4,446

280

0.34

0.5

18

0.91

20

1.13

5

0.47

51,914

4,330

0.44

1.0

22

1.00

20

1.12

5

0.47

28,286

1,800

0.47

2.0

10

0.75

19

1.14

5

0.46

40,779

2,750

0.48

4.0

11

0.77

19

1.11

5

0.48

30,282

1,990

0.48

iterations time (ms)

iterations

time (ms)

time (ms)

Example 23.5. Repeat Example 23.4 using the conjugate gradient method. Solution The solution proceeds exactly as in the previous example up to and including the evaluation of the gradient vector. But, in order to use the conjugate gradient method the γk and βk coefficients need to be calculated from equations (23.40) through (23.42), i.e., ⎛ I ⎞2 N   N   CC CC2  ⎜⎜  ⎟⎟ 2 k k k k ⎜ ⎜⎜ Cg C = gn = 4 Ii − Yi Xin ⎟⎟⎟⎠ , ⎝ n=0

β = k

I   i=1

n=0

Yi −

Iik

N  n=0

i=1

k Xin

dkn

⎛ N ⎞2 . I ⎜⎜ ⎟ k k⎟ ⎜⎜ Xin dn ⎟⎟⎟⎠ . ⎜⎝ i=1

n=0

The calculation proceeds as follows: 1. Since the problem is linear, the sensitivity matrix is precalculated once and for all. 2. An initial guess is made for the parameter vector (such as pn = 0, all n), and the iteration counter is set to k = 0. 3. The direct solution Iik is found from equation (23.33), and the objective function F k is calculated; if it meets certain stopping criteria, the iteration is terminated. CC CC2 4. The gradient of F k is found from equation (23.34); γk is calculated by division with the previous value of Cgk C (for the first iteration, the “old” value is set to a very large number to force γ0 = 0). A new search direction dk is set from equation (23.38). 5. The search step size is determined from equation (23.40), and the parameter vector is updated with equation (23.24). The calculation returns then to step 3 above (alternatively, the step size βk , or the change in the parameter vector can also be used as stopping criteria). The simulation results for the same field as in Example 23.4, again using 20 equally spaced measurement directions, give essentially identical results when using the conjugate gradient approach, i.e., for errorless measurements the exact result is recovered for all optical thicknesses, and for random Gaussian error are similar to those of Fig. 23.6. The problem was also solved using various other inversion techniques, viz., quasi-Newton BFGS with line search (i.e., BFGS with βk  1 found from the relation for β0 in Example 23.4, with g0n replaced by −dkn ), Tikhonov regularization, and the method of steepest descent. All methods return very similar temperature profiles. The number of iterations and CPU times required for the different methods is compared in Table 23.1. Tikhonov regularization does not require any iteration (for this linear problem) and is, together with the conjugate gradient method, the fastest. Of the iterative techniques the conjugate gradient method requires the fewest iterations and is thus the fastest, while BFGS with line search does not appreciably increase convergence, thus taking a little longer than BFGS without it. Not surprisingly, the method of steepest descent requires many more iterations.

We conclude our discussion of gradient-based optimization methods with one simple, nonlinear example. Example 23.6. Repeat Example 23.5 for the case that the absorption coefficient is also unknown and, thus, must be estimated, as well. Compare performance and effort of the quasi-Newton, Levenberg–Marquardt, and conjugate gradient methods. Solution The solution is identical to the previous example, only now the parameter vector p has one additional member, κ, or equivalently, τL . The sensitivity matrix is identical to the one of Example 23.4, except that it has one additional row,

874 Radiative Heat Transfer

FIGURE 23.7 Absorption coefficient and Planck function distribution for Example 23.6 as predicted by the conjugate gradient method.

TABLE 23.2 Recovery of slab temperature distribution and absorption coefficient using various inversion techniques. Quasi-Newton with BFGS without line search with line search iterations time (ms) iterations time (ms)

τL

Conjugate Gradient iterations

time (ms)

Steepest Descent iterations

time (ms)

Levenberg–Marquardt iterations

time (ms)

0.1

–

––

9

5.69

2334

6.94

57,118

33,200

13

0.97

0.5

24

2.03

22

4.36

716

2.30

130,001

61,080

4

0.52

1.0

20

1.80

19

3.13

235

0.96

34,101

16,350

3

0.45

2.0

21

1.88

20

2.94

350

1.31

13,512

5,760

3

0.59

4.0

23

1.98

24

3.57

752

2.61

19,435

8,770

6

0.69

namely      N  1 τL 1 τL ∂Ii + , =− an fn fn+1 ∂τL τ μ μ μ L i i i n=0       N  1 τL τL 1 1 an − = e−τL /μi fn + fn+1 , τ μ μ μ μ L i i i i n=0

Xi,N+1 =

μi < 0, μi > 0.

The problem is now nonlinear, since all Xin contain the unknown parameter τL , and Xi,N+1 also contains the an . This causes no problem in the conjugate gradient method, except that the sensitivity matrix now has to be evaluated anew after each iteration (i.e., in the calculation procedure of Example 23.5 steps 1 and 2 are interchanged, and the iteration always repeats from step 2). Results for the conjugate gradient method are shown in Fig. 23.7. Again, the exact relations are recovered for undisturbed measurements, and the cases shown are for measurements with a random Gaussian error with 3% relative variance. Results are very similar to Example 23.3, perhaps just a little worse, and recovery of the absorption coefficient is well within the variance of the data, except for Levenberg–Marquardt, which incurs errors up to 5% for small and large τL (not shown). On the other hand, Levenberg–Marquardt also is the fastest of the different methods for this problem, as seen in Table 23.2, which shows the time requirements for the different methods.

23.5 Metaheuristics Conventional gradient-based optimization methods minimize the objective function gradually by iteration. The ill-posed nature of the inverse problem may cause slow convergence and the results may also depend on the initial guess for the parameter vector p to determine the solution vector i in equation (23.2). Most importantly, during each iteration, forward calculations have to be conducted over and over again to provide predicted data sets. This makes conventional gradient-based methods very inefficient when applied to high-dimensional problems with high-resolution (and perhaps multidimensional) spectral measurements.

Inverse Radiative Heat Transfer Chapter | 23 875

Metaheuristics also belong to the family of optimization. They received their name because they are not based on a mathematically rigorous minimization formulation—in contrast to gradient-based methods, which usually approximate the objective function as locally quadratic, and then find the minimum via a Taylor series expansion. The algorithms of many metaheuristics are inspired by physics or biology (genetic algorithms and swarm algorithms are important examples of biomimicry). By their nature metaheuristics are inevitably less efficient than gradient-based methods at finding the few local minima in relatively simple problems. Therefore, they should only be used when gradient-based methods are unreliable or impractical as, for example, highdimensional problems with many local minima.

Simulated Annealing One popular algorithm is simulated annealing, which is based on the changing arrangement of atoms in metals. The simulated annealing algorithm is analogous to nature, where the objective function is the lattice energy, and the design parameters specify the lattice arrangement [35,49,50]. The Second Law of Thermodynamics drives a system toward a lower energy state, so the atoms in a metal will preferentially move into lower energy configurations, but can spontaneously move into a high energy configuration. The same idea applies in metaheuristics, and the nomenclature “annealing schedule,” “temperature,” etc. carries over. At each iteration a candidate step is proposed, analogous to atoms randomly moving. A new candidate objective function is generated and compared to the present one. If the new objective function is lower, the candidate step is always accepted (probability of unity). If the new objective function is larger, the candidate step is accepted with a probability proportional to exp(−ΔF/T), where the “annealing temperature” is defined in terms of the iteration number k. Thus, higher T make uphill steps more likely (smaller k) and, as temperature decreases (cooling the metal), accepting an uphill step becomes increasingly improbable (large k). As in actual metal annealing, the underlying idea behind simulated annealing is that the method allows the design parameters to transition through a temporary higher energy state (a crest in the objective function topography) in their quest for the lowest energy level (global minimum).

Machine Learning Another promising heuristic solution technique is the machine learning method. Machine learning is a field of computer science that gives computer systems the ability to find relations between inputs and outputs even if they are impossible to be represented by explicit algorithms. Their algorithms enable computers to learn from experiences without actually modeling the physical and chemical laws that govern the system [51]. The major focus of machine learning is to extract information from data automatically by computational and statistical methods, which may provide global solution models for nonlinear inverse problems when relations between dependent and independent variables are not clear [52,53]. Due to its ability for predicting and forecasting, machine learning has found many applications in energy systems. Inspired by biological neural network information processes, artificial neural networks are a group of algorithms used for machine learning that model data processing by artificial neurons [54]. By training with a dataset consisting of a given set of inputs and corresponding outputs, a model is generated, which can be used to predict proper results from input that has the same features as the training set. The multi-layer perceptron (MLP) neural network is one of the most popular types of artificial neural networks in machine learning [55]. The MLP consists of an input layer, one or more hidden layers, and an output layer. Each layer comprises several nodes called neurons. Figure 23.8 shows a representative MLP neural network architecture for temperatures and species concentration retrieval from infrared spectral emission measurements [56]. The leftmost layer, known as the input layer, consists of a set of neurons representing the input features (infrared spectral intensities). Each neuron in the hidden layers transforms the values from the previous layer with a weighted linear summation, followed by a nonlinear activation function. The output layer receives the values from the last hidden layer and transforms them into output values (temperatures/concentrations). The numbers of neurons in the input and output layers are determined by the input and output dimensions, respectively. There is no specific approach to determine the number of hidden layers and their neurons for different problems, the choice is usually made by trial and error [51]. Training neural networks is done by adjusting appropriate weights W between neurons to minimize the error of the object function, in which the output values generated by the network are compared to the corresponding actual values.

876 Radiative Heat Transfer

FIGURE 23.8 Schematic of a representative MLP neural network architecture for temperatures and species concentration retrieval from infrared spectral emission measurements of combustion gases [56].

The objective function for the MLP is F(Zp , Z, W) = ||Zp − Z|| + α||W||,

(23.43)

where Zp is the vector of predicted scalar values by the neural network, Z are the “ground truth” values, i.e., the set of desired output parameters (temperature and concentrations in Fig. 23.8) that would cause the input signal (spectral measurements here), and MLP uses the parameter α for regularization to avoid overfitting by penalizing weights with large magnitudes. Learning is an iterative process, starting with a random set of weights, and uses a relatively large number of samples, which should contain information spread evenly over the entire range of the system, allowing to obtain a sufficiently low error of the objective function. After training, the model can be directly used to predict new outputs by feeding new inputs. Another class of machine-learning-based optimization methods is Bayesian optimization, which attempts to find the global optimum in a minimum number of steps. Bayesian optimization proceeds by maintaining a probabilistic belief about F and designing a so-called acquisition function for direct sampling in areas where an improvement over the current best observation is likely [57–59]. Example 23.7. In Ref. [56] Ren et al. showed how temperature and mean species concentrations can be deduced from spectral radiative intensities obtained from the well-characterized NPL standard flame on a Hencken burner [60]. Spectral measurements were carried out at 4 cm−1 and 8 cm−1 resolutions at 10 mm and 20 mm heights above the center of the burner (HAB) for combustion with three different equivalence ratios, resulting in highly stable data with good long-term reproducibility. The temperature fields for the standard flame were previously measured also at NPL with Rayleigh scattering thermometry [60]. The species concentrations in the post-flame region are fairly uniform. To make accurate predictions based on the MLP neural network, the network must be trained with a large number of possible temperature and concentration distributions, using the same spectral resolution as the measured data. Temperature profiles should have the basic features reported for the NPL standard flame, which are flat in the middle with large gradients near the edges of the burner, which may not be strictly symmetric. Therefore, 10,000 nonsymmetric temperature profiles were generated with an example shown in Fig. 23.9, with values for T0,1 , T0,2 , ΔT1 , ΔT2 chosen from random number relationships, which allow generating temperature ranges from 500 K to 3000 K, and are sufficient to cover the flame temperature range. Due to the uniformity of the species compositions for the flame, uniform concentrations for CO2 , H2 O, and CO were randomly generated, with values between 0% and 20%. These were employed together with the random temperature distributions to generate the training spectral intensities. Since experimental data always have noise, Gaussian random noise of 3% was added to all generated intensity spectra. After training of the neural networks, the model was used to predict temperatures and species concentrations from the measured spectral intensities. Two different spectral intervals were chosen for the predictions: Interval I has a range of 1800–2500 cm−1 , where all three species have strong bands, and Interval II with a range of 3000–4200 cm−1 , where weaker bands are located (see Table 10.4). One would expect that Interval I would provide better answers for optical paths with smaller pressure path lengths of the species, and Interval II for optical paths with larger pressure path lengths (i.e., in both cases leading to optically intermediate conditions). As an example, results for the flame with equivalence ratio φ = 0.8, using spectral data with 4 cm−1 resolution are shown in Fig. 23.10. It is seen that both spectral interval models predict the temperature distribution very accurately, with

Inverse Radiative Heat Transfer Chapter | 23 877

FIGURE 23.9 Example of temperature profile used for generating training data sets.

FIGURE 23.10 Predicted temperatures and species concentrations of the NPL standard flame at HAB = 10 mm for φ = 0.8, using spectral data with a resolution of 4 cm−1 .

Interval I doing a slightly better job. No concentration measurements are available, so the retrieved values are compared against chemical equilibrium values. Concentrations of CO2 and H2 O were retrieved accurately when compared against chemical equilibrium values. Since combustion may not reach chemical equilibrium at HAB = 10 mm, the retrieved CO values are higher than the near-zero chemical equilibrium values.

23.6 Summary of Inverse Radiation Research Inverse Surface Radiation While inverse radiation problems involving a participating medium received the earliest attention, more recently a number of researchers have concerned themselves with inverse surface radiation problems. Harutunian et al. [61], Fedorov et al. [62], Jones [17], Ertürk et al. [63], and França et al. [64] were the first to recognize the potential of inverse radiation analysis for control: they investigated the needed energy input into a heating element, in order to achieve a prespecified result at a design surface. This was followed with considerable more work by the group around Howell [35,41,49,65–67] and a few others [68]. That inverse analysis can also be used to deduce surface reflectances was demonstrated by Wu and Wu [18]. Various solution techniques were employed. For example, TSVD was used by França et al. [64,66] to predict heater performance in the presence of convection, and by Daun and coworkers [35] for 3D surface heating;

878 Radiative Heat Transfer

the latter also used Tikhonov regularization, quasi-Newton and conjugate gradient techniques (optimization), and simulated annealing (metaheuristics). The conjugate gradient method was also used by Ertürk et al. [65], who optimized transient heating control of a furnace, while Porter and Howell [49] used metaheuristic methods (simulated annealing and tabu search) to control a surface heater. Daun and coworkers [41,69] and Leduc et al. [68] performed geometric optimization of radiant enclosures using Tikhonov regularization [68], the quasi-Newton method [41], and Kiefer-Wolfowitz stochastic programming (a variation on the steepest descent scheme) [69]. The only work reporting experimental verification seems to be the one by Ertürk et al. [67], who investigated radiative heating control of silicon wafers. They found that accurate knowledge of radiative properties is crucial, and obtained wafer temperatures to within 3% of the target value. A few studies used several inversion techniques to allow for comparison. Daun and coworkers [35], in order to investigate surface heater control in a 3D furnace, used five different inversion techniques, viz., TSVD and Tikhonov regularization, two optimizations (the quasiNewton and conjugate gradient methods), and one metaheuristic scheme (simulated annealing). They found that all techniques predicted solutions within acceptable accuracy, but the methods in some cases provided widely different distributions that achieve the same final result. The regularization, conjugate gradient, and simulated annealing methods provided smooth distributions of heater inputs across the heater surface, whereas the quasi-Newton technique tended to give uneven distributions.

Inverse Radiation in Participating Media Most research to date on inverse radiation within a participating medium has centered around the retrieval of temperature distributions, with some also deducing various radiative properties, such as surface reflectances, scattering albedos, and phase functions. Much of the work dealt with pure radiation in mostly gray [9–16,70–81], and a few nongray [82,83], constant-property, one-dimensional media. Others have dealt with multidimensional geometries [84–98], and interactions between conduction and radiation have also received growing attention [89,90,99–102], along with, to a lesser extent, inverse radiation combined with convection [103]. Most of these investigations have concentrated on developing an inverse method using artificial data. Only a few experiments have been combined with inverse analysis to measure particle distributions and scattering properties of pulverized coal [104,105], and to infer temperature and concentration distributions in axisymmetric flames [106–112]. Most of these determined spatial averages [106] or used Abel’s transformation [107–111] (reconstruction from spatial scans). However, it has been shown that these profiles can also be determined from a single transmission measurement through spectrometry (reconstruction from spectral scan) [83,111–113]. As for surface radiation problems, several different inverse methodologies have been employed, such as TSVD [82], Tikhonov regularization [40,114], Tikhonov regularization plus Kalman filtering (to connect information from transient signals) [115], conjugate gradient methods [81,94,116–123], and metaheuristics [80,83,93,94,97]. A comparison of methods was carried out by Deiveegan et al. [80], who retrieved surface emittances and gas properties in gray participating media, using the Levenberg–Marquardt method, and several metaheuristics schemes, i.e., genetic algorithms, artificial neural networks, and Bayesian statistics. They found that all methods gave acceptable results, with Bayesian statistics being least susceptible to random noise.

Optical Tomography More recently, there has been growing interest in optical tomography, the reconstruction of property fields based on radiative field measurements. This can be achieved by simultaneously measuring emission or absorption at many different angles, directions, and/or wavelengths with low-dimensional sensors, using laser sources together with single-color imaging detectors, or with a single camera with the ability to capture intensity, directional, and/or spectral information of emitted light from the target. Two areas of interest have been identified. One is the detection of internal tumors in biomedical applications, generally using ultrafast lasers with transient radiation effects (see also Chapter 18) [118–123]. A review of that application with many references has been given by Charette and colleagues [36]. Today, optical tomography is being increasingly applied to the diagnosis of combustion systems, with a more detailed discussion below.

Multi-projection Measurements Laser absorption tomography [40,114,115,124–128] obtains spatially resolved planar or volumetric measurements by combining an arrangement of multiple laser beams with reconstruction methods. The method is beginning

Inverse Radiative Heat Transfer Chapter | 23 879

to mature as a technique for the simultaneous imaging of temperature and species concentrations, and is experiencing a surge of interest due to progress in laser technology, spectroscopy, and theoretical developments in nonlinear tomography techniques. The most recent review by Cai and Kaminski [129] provides an excellent overview of laser absorption tomography and its application to combustion studies. Radiation image-based diagnostics do not require external laser sources but rely on detection of chemiluminescence or infrared emission from the flames and, therefore, the sampling system is relatively simple. In such a system, the time and spatial resolution of radiation sampling can be greatly improved by configuring a highly integrated CCD or CMOS camera. However, a single camera is mostly used in axisymmetric flames [130]. For asymmetrical or turbulent flames, multiple camera systems are required for simultaneous measurement of flame emission at different angles and directions [131], which inevitably increases the complexity of the measurement system. Huang et al. [132] have reconstructed three-dimensional flame shapes from two-dimensional images of multiple projection measurements with a convolutional neural network. Jin et al. [133] have used 12 CCD cameras to measure the chemiluminescence of flames from different projections, and they achieved rapid reconstruction of threedimensional mole fraction fields of CH∗ and C∗2 combustion radicals, also with a convolutional neural network.

Light Field Imaging Light field cameras add a series of micro-lenses in front of the sensor, which are able to capture and record multiangle radiative intensity information of a flame through a single shot. After postprocessing and integration of the flame information, three-dimensional reconstruction of the temperature [134–137] and radiative properties [138– 140] can be achieved. Compared to multi-projection tomography, the light field camera approach reduces the complexity of the measurement system and eliminates the use of multiple detectors. Li et al. [139] have developed a theoretical model to reconstruct flame radiative properties from a single light-field camera. Huang et al. [136] combined the light-field imaging technique and the Landweber inverse method to reconstruct 3D temperature distributions in absorbing media theoretically and experimentally.

Hyperspectral Imaging A hyperspectral imaging detector consists of a 2D array of pixels providing spatial and spectrally resolved images, and each pixel collects a large amount of continuous spectral information. Hyperspectral measurements provide multiple spectral measurements from each pixel, significantly reducing the number of required projections [141–147]. Ren et al. employed the machine learning inversion method detailed in Section 23.5 to retrieve temperature and species concentration profiles from hyperspectral measurements of laminar flames, such as the ones of Rhoby and coworkers [141], for 2D [146] and 3D [147] reconstructions. They compared their results against OH-laser absorption measurements, as well as reconstructions made by Rhoby et al. using conventional gradient-based optimization algorithms. Accuracy of the machine learning predictions was found to be excellent and, while up-front training of the model can be computationally expensive (up to several cpu-hours), computational effort of individual reconstructions was reduced by 4 (2D) and 5 (3D) orders of magnitude, respectively, as compared to gradient-based methods.

Problems 23.1 Repeat Example 23.2, but determine the necessary heat flux distribution, q2 (x2 ), along the plate. 23.2 Consider a one-dimensional, absorbing–emitting (but not scattering) slab of width L, bounded by two cold, black walls. The temperature distribution within the slab is unknown, and is to be estimated by measuring spectral exit heat fluxes on both bounding walls for various wavenumbers in a range over which the absorption coefficient of the medium, κ, is known, is linearly proportional to wavenumber, and is spatially constant. Use the P1 -approximation and Tikhonov regularization. Hint: Set up a 1D finite difference solution for the P1 -approximation by breaking up the slab into N isothermal layers; then determine M > N wall fluxes in terms of the Ibη (Ti ). 23.3 Repeat Problem 23.2 for a medium that also scatters radiation isotropically, with a gray scattering coefficient. 23.4 Repeat Problem 23.2 for the case of an unknown absorption coefficient (except for the fact that it is linearly proportional to wavenumber). Use the P1 -approximation together with the quasi-Newton algorithm.

880 Radiative Heat Transfer

23.5 A black plate of width w is irradiated by two line sources as shown. The plate is insulated at the bottom, while the top loses heat by radiation to the (cold) environment. Ideally, the plate should be at a uniform temperature of 500 K. Breaking up the plate into four equally wide segments, determine the optimal heater powers (without exploiting the symmetry): (a) using TSVD on the direct equations (23.19), (b) using TSVD and the normal equations (23.10), (c) using Tikhonov regularization and the normal equations.

23.6 Soot volume fraction and temperature are to be determined by measuring the transmissivity of a gas–soot layer for several wavelengths. Consider a homogeneous layer of thickness L = 0.2 m, whose absorption coefficient obeys equation (11.126), where C0 is a known function of wavelength and temperature, such that C0 (λ, T) = 5[1 + aλ(T − T0 )],

T0 = 300 K,

a = 0.01 (μmK)−1 .

Transmissivity measurements are conducted at four wavelengths as shown in the table: one set of data has been taken with high precision (i.e., zero error), and the other has some noise in the data. Determine soot volume fraction and Wavelength λ

1 μm

2 μm

3 μm

4 μm

High-fidelity data

0.6065

0.7788

0.8465

0.8825

Noisy data

0.617

0.763

0.826

0.891

temperature using Tikhonov regularization. 23.7 Repeat Problem 23.6 using the quasi-Newton method. 23.8 In laser absorption tomography, the concentration of a target species (e.g., gas or soot) is inferred from the transmittance of multiple lasers passing through the flow field. If the domain is split into n regions in each of which the concentration is assumed uniform, the Beer-Lambert law along the ith beam becomes ln(I0i /Ii ) =

n 

Aij κ j ,

j=1

where Aij is the chord length of the ith beam subtended by the jth element. Writing this equation for m beams results in an m × n matrix equation, A · p = b, which relates the beam transmittance data, b, to the unknown species concentration (through the absorption coefficient, p = κ), equivalent to equation (23.19). However, even if n = m the matrix is ill-conditioned, and its inversion must be regularized to suppress measurement noise amplification. Consider the axisymmetric problem shown to the right. Laser transmittance measurements made along the center of each annular element are summarized in the table below. It is known that each data point is contaminated by normally distributed error having a standard deviation of 0.025. y (cm)

0

0.3158

0.6316

0.9474

1.2632

1.5789

1.8947

2.2105

2.5263

2.8421

ln(I0i /Ii )

0.6258

0.5494

0.4652

0.2883

0.1183

0.0831

0.0171

0.0259

–0.0179

–0.0056

(a) Derive the A matrix and perform a singular value decomposition. What do the singular values imply about this problem? (b) Attempt to recover p using no regularization, and plot the values as a function of y. Comment on the solution.

Inverse Radiative Heat Transfer Chapter | 23 881

(c) Use first-order Tikhonov regularization to recover the solution. The truncated equation (23.19) for first-order Tikhonov becomes ⎞ ⎛ ⎟⎟ ⎜⎜ 1 −1 ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ 1 −1 ⎟⎟ ⎜ ⎜⎜   ⎟⎟ . (A + λL ) · p = b, where L = ⎜⎜ ⎟⎟ .. .. ⎜⎜ ⎟⎟ . . ⎟⎟ ⎜⎜⎜ ⎠ ⎝ 1 −1 Attempt to recover the solution using different values of λ. What is the optimal level of regularization?

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

J. Hadamard, Lectures on Cauchy’s Problem in Linear Differential Equations, Yale University Press, New Haven, CT, 1923. A.N. Tikhonov, V.Y. Arsenin, Solution on Ill-Posed Problems, Winston & Sons, Washington, DC, 1977. J.V. Beck, K.J. Arnold, Parameter Estimation in Engineering and Science, Wiley Interscience, New York, 1977. G. Stolz, Numerical solutions to an inverse problem of heat conduction for simple shapes, ASME Journal of Heat Transfer 82 (1960) 20–26. J.V. Beck, Calculation of surface heat flux from an internal temperature history, ASME paper no. 62-HT-46, 1962. I. Frank, An application of least squares method to the solution of the inverse problem of heat conduction, ASME Journal of Heat Transfer 85C (1963) 378–379. O.R. Burggraf, An exact solution of the inverse problem in heat conduction theory and applications, ASME Journal of Heat Transfer 86C (1964) 373–382. E.M. Sparrow, A. Haji-Sheikh, T.S. Lundgren, The inverse problem in transient heat conduction, ASME Journal of Applied Mechanics 86E (1964) 369–375. C.H. Ho, M.N. Özi¸sik, Inverse radiation problem in inhomogeneous media, Journal of Quantitative Spectroscopy and Radiative Transfer 40 (1988) 553–560. C.H. Ho, M.N. Özi¸sik, An inverse radiation problem, International Journal of Heat and Mass Transfer 32 (2) (1989) 335–342. H.Y. Li, M.N. Özi¸sik, Estimation of the radiation source term with a conjugate–gradient method of inverse analysis, Journal of Quantitative Spectroscopy and Radiative Transfer 48 (1992) 237–244. H.Y. Li, M.N. Özi¸sik, Identification of temperature profile in an absorbing, emitting and isotropically scattering medium by inverse analysis, ASME Journal of Heat Transfer 114 (1992) 1060–1063. J.-F. Sacadura, V.P. Nicolau, Spectral radiative properties identification of semi-transparent porous media, in: 3rd UK National & 1st European Conference Thermal Sciences, Birmingham, UK, 1992, pp. 717–723. H.Y. Li, M.N. Özi¸sik, Inverse radiation problem for simultaneous estimation of temperature profile and surface reflectivity, Journal of Thermophysics and Heat Transfer 7 (1) (1993) 88–93. M.P. Mengüç, S. Manickavasagam, Inverse radiation problem in axisymmetric cylindrical scattering media, Journal of Thermophysics and Heat Transfer 7 (3) (1993) 479–486. J. Bokar, M.N. Özi¸sik, An inverse problem for estimation of radiation temperature source term in a sphere, Inverse Problems in Engineering 1 (1995) 191–205. M.R. Jones, Inverse analysis of radiative heat transfer systems, ASME Journal of Heat Transfer 121 (1999) 481–484. C.Y. Wu, S.H. Wu, A new application of successive approximation to radiative exchange among surfaces: direct and inverse problems, International Journal of Heat and Mass Transfer 42 (1999) 2255–2267. J.V. Beck, B. Blackwell, C.R.St. Clair, Inverse Heat Conduction: Ill-Posed Problems, Wiley Interscience, 1985. J.V. Beck, B. Blackwell, Inverse problems, in: W.J. Minkowycz, E.M. Sparrow, G.E. Schneider, R.H. Pletcher (Eds.), Handbook of Numerical Heat Transfer, Wiley Interscience, New York, 1988. O.M. Alifanov, Inverse Heat Transfer Problems, Springer Verlag, 1994. K. Kurpisz, A.J. Nowak, Inverse Thermal Problems, WIT Press, Southampton, UK, 1995. O.M. Alifanov, E. Artyukhin, A. Rumyantsev, Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems, Begell House, 1995. M.N. Özi¸sik, H.R.B. Orlande, Inverse Heat Transfer, Taylor & Francis, New York, 2000. Y.B. Bard, Nonlinear Parameter Estimation, Academic Press, New York, 1974. P.C. Sabatier, Applied Inverse Problems, Springer Verlag, Berlin, 1978. V.A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer Verlag, 1984. E. Hensel, Inverse Theory and Applications for Engineers, Prentice Hall, Englewood Cliffs, NJ, 1991. D.A. Murio, The Mollification Method and the Numerical Solution of Ill-Posed Problems, Wiley Interscience, New York, 1993. P.C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM, Philadelphia, 1998. J. Nocedal, S.J. Wright, Numerical Optimization, 2nd ed., Springer Verlag, Berlin, 2006. J. Idier (Ed.), Bayesian Approach to Inverse Problems, Digital Signal and Image Processing Series, Wiley, New York, 2008.

882 Radiative Heat Transfer

[33] F.H.R. França, J.R. Howell, O.A. Ezekoye, J.C. Morales, Inverse design of thermal systems, in: Advances in Heat Transfer, vol. 36, Elsevier, New York, 2002, pp. 1–110. [34] K.J. Daun, J.R. Howell, Inverse design methods for radiative transfer systems, Journal of Quantitative Spectroscopy and Radiative Transfer 93 (1–3) (2005) 43–60. [35] K.J. Daun, F.H.R. França, M.E. Larsen, G. Leduc, J.R. Howell, Comparison of methods for inverse design of radiant enclosures, ASME Journal of Heat Transfer 45 (2006) 537–552. [36] A. Charette, J. Boulanger, H.K. Kim, An overview on recent radiation transport algorithm development for optical tomography imaging, Journal of Quantitative Spectroscopy and Radiative Transfer 109 (17–18) (2008) 2743–2766. [37] K.J. Daun, Inverse problems in radiative transfer, in: F.A. Kulacki (Ed.), Handbook of Thermal Science and Engineering, Springer International Publishing, 2017, pp. 1–51. [38] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes – The Art of Scientific Computing, 3rd ed., Cambridge University Press, Cambridge, 2007. [39] S.A. Rukolaine, Concept of regularization, in: Thermopedia, 2007, http://thermopedia.com/content/238/?tid=104&sn=74, https://doi. org/10.1615/thermopedia.000238. [40] E.O. Åkesson, K.J. Daun, Parameter selection methods for axisymmetric flame tomography through Tikhonov regularization, Applied Optics 47 (3) (2008) 407–416. [41] K.J. Daun, J.R. Howell, D.P. Morton, Geometric optimization of radiative enclosures through nonlinear programming, Numerical Heat Transfer – Part B: Fundamentals 43 (3) (2003) 203–219. [42] K. Levenberg, A method for the solution of certain non-linear problems in least squares, Quarterly of Applied Mathematics 2 (1944) 164–168. [43] D.W. Marquardt, An algorithm for least squares estimation of nonlinear parameters, Journal of the Society for Industrial and Applied Mathematics 11 (1963) 431–441. [44] IMSL Math/Library, 1st ed., IMSL, Houston, TX, 1989. [45] F.S. Beckman, The solution of linear equation by conjugate gradient method, in: A. Ralston, H.S. Wilf (Eds.), Mathematical Methods for Digital Computers, Wiley Interscience, New York, 1960. [46] J.W. Daniel, The Approximate Minimization of Functionals, Prentice Hall, Englewood Cliffs, NJ, 1971. [47] M.N. Özi¸sik, Heat Conduction, 2nd ed., Wiley Interscience, New York, 1993. [48] S. Mazumder, Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods, Academic Press, 2016. [49] J.M. Porter, M.E. Larsen, J.W. Barnes, J.R. Howell, Metaheuristic optimization of a discrete array of radiant heaters, ASME Journal of Heat Transfer 45 (2006) 537–552. [50] A. Tajouri, K. El Khoury, M. Nemer, Inverse estimation of gray-band emissivity in a three-dimensional enclosure using combined simulated annealing and replating algorithm, Numerical Heat Transfer – Part A: Applications 68 (2015) 268–287. [51] F.T. Scafati, M. Lavorgna, E. Mancaruso, B.M. Vaglieco, Artificial intelligence for modeling and control of nonlinear phenomena in internal combustion engines, in: Nonlinear Systems and Circuits in Internal Combustion Engines, Springer, 2018, pp. 1–19. [52] E. Alpaydin, Machine Learning: The New AI, MIT Press, 2016. [53] C. Voyant, G. Notton, S. Kalogirou, M.-L. Nivet, C. Paoli, F. Motte, A. Fouilloy, Machine learning methods for solar radiation forecasting: a review, Renewable Energy 105 (2017) 569–582. [54] K. Gurney, An Introduction to Neural Networks, CRC Press, 2014. [55] R.C. Deo, M.A. Ghorbani, S. Samadianfard, T. Maraseni, M. Bilgili, M. Biazar, Multi-layer perceptron hybrid model integrated with the firefly optimizer algorithm for windspeed prediction of target site using a limited set of neighboring reference station data, Renewable Energy 116 (2018) 309–323. [56] T. Ren, M.F. Modest, A. Fateev, G. Sutton, W. Zhao, F. Rusu, Machine learning applied to retrieval of temperature and concentration distributions from infrared emission measurements, Applied Energy 252 (2019) 113448. [57] J. Mockus, Bayesian Approach to Global Optimization: Theory and Applications, vol. 37, Springer Science & Business Media, 2012. [58] F. Nogueira, Bayesian Optimization: open source constrained global optimization tool for Python, 2014. [59] J. Snoek, H. Larochelle, R.P. Adams, Practical Bayesian optimization of machine learning algorithms, in: F. Pereira, C.J.C. Burges, L. Bottou, K.Q. Weinberger (Eds.), Advances in Neural Information Processing Systems 25, Curran Associates, Inc., 2012, pp. 2951–2959. [60] G. Sutton, A. Fateev, M.A. Rodríguez-Conejo, J. Meléndez, G. Guarnizo, Validation of emission spectroscopy gas temperature measurements using a standard flame traceable to the International Temperature Scale of 1990 (ITS-90), International Journal of Thermophysics 40 (11) (2019) 1–36. [61] V. Harutunian, J.C. Morales, J.R. Howell, Radiation exchange within an enclosure of diffuse-gray surfaces: the inverse problem, in: W.J. Bryan, J.V. Beck (Eds.), Proceedings of the 30th 1995 National Heat Transfer Conference, vol. HTD-312, ASME, 1995, pp. 133–140. [62] A.G. Fedorov, K.H. Lee, R. Viskanta, Inverse optimal design of the radiant heating in materials processing and manufacturing, Journal of Materials Engineering and Performance 7 (6) (1998) 719–726. [63] H. Ertürk, O.A. Ezekoye, J.R. Howell, Inverse solution of radiative heat transfer in two-dimensional irregularly shaped enclosures, in: Proceedings of 2000 IMECE, vol. HTD-366-1, ASME, 2000, pp. 109–117. [64] F.H.R. França, O.A. Ezekoye, J.R. Howell, Inverse boundary design combining radiation and convection heat transfer, ASME Journal of Heat Transfer 123 (2001) 884–891.

Inverse Radiative Heat Transfer Chapter | 23 883

[65] H. Ertürk, O.A. Ezekoye, J.R. Howell, The application of an inverse formulation in the design of boundary conditions for transient radiating enclosures, ASME Journal of Heat Transfer 124 (2002) 1095–1102. [66] A.C. Mossi, H.A. Vielmo, F.H.R. França, J.R. Howell, Inverse design involving combined radiative and turbulent convective heat transfer, International Journal of Heat and Mass Transfer 51 (2008) 3217–3226. [67] H. Ertürk, M. Gamba, O.A. Ezekoye, J.R. Howell, Validation of inverse boundary condition design in a thermometry test bed, Journal of Quantitative Spectroscopy and Radiative Transfer 109 (2008) 317–326. [68] G. Leduc, F. Monchoux, F. Thellier, Inverse radiative design in human thermal environment, International Journal of Heat and Mass Transfer 47 (2004) 3291–3300. [69] K.J. Daun, D.P. Morton, J.R. Howell, Geometric optimization of radiant enclosures containing specular surfaces, ASME Journal of Heat Transfer 125 (2003) 845–851. [70] S. Subramaniam, M.P. Mengüç, Solution of the inverse radiation problem for inhomogeneous and anisotropically scattering media using a Monte Carlo technique, International Journal of Heat and Mass Transfer 34 (1) (1991) 253–266. [71] T.J. Hendricks, J.R. Howell, Absorption/scattering coefficients and scattering phase functions in reticulated porous ceramics, ASME Journal of Heat Transfer 118 (1) (1996) 79–87. [72] F. Yousefian, M. Lallemand, Temperature and species concentration profiles using high resolution infrared transmission data by inverse radiative analysis, in: M.P. Mengüç (Ed.), Proceedings of the Second International Symposium on Radiation Transfer, Begell House, Kusadasi, Turkey, 1997, pp. 329–340. [73] H.Y. Li, C.Y. Yang, A genetic algorithm for inverse radiation problems, International Journal of Heat and Mass Transfer 40 (7) (1997) 1545–1549. [74] L.H. Liu, H.P. Tan, Q.Z. Yu, Simultaneous identification of temperature profile and wall emissivities in semitransparent medium by inverse radiation analysis, Numerical Heat Transfer 36A (1999) 511–525. [75] H.C. Zhou, P. Yuan, F. Sheng, C.G. Zheng, Simulation of coal combustion by AUSM turbulence–chemistry char combustion model and a full two-fluid model, International Journal of Heat and Mass Transfer 43 (2000) 4361–4364. [76] L.H. Liu, H.P. Tan, Q.Z. Yu, Inverse radiation problem in one-dimensional semitransparent plane-parallel media with opaque and specularly reflecting boundaries, Journal of Quantitative Spectroscopy and Radiative Transfer 64 (2000) 395–407. [77] D. Baillis, M.C. Arduini-Schuster, J.-F. Sacadura, Identification of spectral radiative properties of polyurethane foam from hemispherical and bi-directional transmittance and reflectance measurements, in: M.P. Mengüç, N. Selçuk (Eds.), Radiative Transfer 2001 — The Third International Symposium on Radiative Transfer, Begell House, 2001. [78] L.H. Liu, H.P. Tan, Q.Z. Yu, Inverse radiation problem of sources and emissivities in one-dimensional semitransparent media, International Journal of Heat and Mass Transfer 44 (2001) 63–72. [79] E.S. Chalhoub, H.F.C. Velho, Estimation of the optical properties of seawater from measurements of exit radiance, Journal of Quantitative Spectroscopy and Radiative Transfer 72 (2002) 551–565. [80] M. Deiveegan, C. Balaji, S.P. Venkateshan, Comparison of various methods for simultaneous retrieval of surface emissivities and gas properties in gray participating media, ASME Journal of Heat Transfer 128 (2006) 829–837. [81] W. An, L.M. Ruan, H. Qi, Inverse radiation problem in one-dimensional slab by time-resolved reflected and transmitted signals, Journal of Quantitative Spectroscopy and Radiative Transfer 107 (1) (2007) 47–60. [82] E.I. Vitkin, O. Zhdanovich, V. Tamanovich, V. Senchenko, V. Dozhdikov, M. Ignatiev, I. Smurov, Determination of the temperature and concentrations for the products of combustion of a hydrocarbon fuel on the basis of their infrared self-radiation, International Journal of Heat and Mass Transfer 45 (2002) 1983–1991. [83] H.K. Kim, T.H. Song, Determination of the gas temperature profile in a large-scale furnace using a fast/efficient inversion scheme for the SRS technique, Journal of Quantitative Spectroscopy and Radiative Transfer 93 (2005) 369–381. [84] F.H.R. França, M. Oguma, J.R. Howell, Inverse radiation heat transfer within enclosures with nonisothermal, non-gray participating media, in: R.A.J. Nelson, T. Chopin, S.T. Thynell (Eds.), Proceedings of the ASME Heat Transfer Division - 1998, vol. HTD-361-5, ASME, Anaheim, California, 1998, pp. 145–151. [85] L.H. Liu, H.P. Tan, Q.Z. Yu, Inverse radiation problem of temperature field in three-dimensional rectangular furnaces, International Communications in Heat and Mass Transfer 26 (1999) 239–248. [86] H.C. Zhou, F. Sheng, S.D. Han, C.G. Zheng, A fast algorithm for calculation of radiative energy distributions received by pinhole image-formation process from 2D rectangular enclosures, Numerical Heat Transfer – Part A: Applications 38 (2000) 757–773. [87] L.H. Liu, J. Jiang, Inverse radiation problem for reconstruction of temperature profile in axisymmetric free flames, Journal of Quantitative Spectroscopy and Radiative Transfer 70 (2001) 207–215. [88] L.H. Liu, H.P. Tan, Inverse radiation problem in three-dimensional complicated geometric systems with opaque boundaries, Journal of Quantitative Spectroscopy and Radiative Transfer 68 (2001) 559–573. [89] H.M. Park, D.H. Yoo, A multidimensional inverse radiation problem of estimating the strength of a heat source in participating media, International Journal of Heat and Mass Transfer 44 (2001) 2949–2956. [90] H.M. Park, W.J. Lee, The solution of inverse radiation problems using an efficient computational technique, Journal of Quantitative Spectroscopy and Radiative Transfer 73 (2002) 41–54. [91] H. Ertürk, O.A. Ezekoye, J.R. Howell, Comparison of three regularized solution techniques in a three-dimensional inverse radiation problem, Journal of Quantitative Spectroscopy and Radiative Transfer 73 (2002) 307–316.

884 Radiative Heat Transfer

[92] H.C. Zhou, S.D. Han, F. Sheng, C.G. Zheng, Visualization of three-dimensional temperature distributions in a large-scale furnace via regularized reconstruction from radiative energy images: numerical studies, Journal of Quantitative Spectroscopy and Radiative Transfer 72 (2002) 361–383. [93] J. Wang, N. Zabaras, Using Bayesian statistics in the estimation of heat source in radiation, International Journal of Heat and Mass Transfer 48 (2005) 15–29. [94] K.W. Kim, S.W. Baek, Efficient inverse radiation analysis in a cylindrical geometry using a combined method of hybrid genetic algorithm and finite-difference Newton method, Journal of Quantitative Spectroscopy and Radiative Transfer 108 (3) (2007) 423–439. [95] S.A. Rukolaine, Regularization of inverse boundary design radiative heat transfer problems, Journal of Quantitative Spectroscopy and Radiative Transfer 104 (1) (2007) 171–195. [96] F. Kowsary, K. Pooladvand, A. Pourshaghaghy, Regularized variable metric method versus the conjugate gradient method in solution of radiative boundary design problem, Journal of Quantitative Spectroscopy and Radiative Transfer 108 (2) (2007) 277–294. [97] K.H. Lee, S.W. Baek, K.W. Kim, Inverse radiation analysis using repulsive particle swarm optimization algorithm, International Journal of Heat and Mass Transfer 51 (2008) 2772–2783. [98] F. Wang, D. Liu, K. Cen, J.-H. Yan, Q.-X. Huang, Y. Chi, Efficient inverse radiation analysis of temperature distribution in participating medium based on backward Monte Carlo method, Journal of Quantitative Spectroscopy and Radiative Transfer 109 (12–13) (2008) 2171–2181. [99] N. Ruperti Jr, M. Raynaud, J.-F. Sacadura, A method for the solution of the coupled inverse heat conduction–radiation problem, ASME Journal of Heat Transfer 118 (1996) 10–17. [100] H.Y. Li, Estimation of thermal properties in combined conduction and radiation, International Journal of Heat and Mass Transfer 42 (3) (1999) 565–572. [101] F.H.R. França, O.A. Ezekoye, J.R. Howell, Inverse heat source design combining radiation and conduction heat transfer, in: L.C. Witte (Ed.), Proceedings of the ASME Heat Transfer Division - 1999, vol. HTD-364-1, ASME, Nashville, Tennessee, 1999, pp. 45–52. [102] H.M. Park, T.Y. Yoon, Solution of the inverse radiation problem using a conjugate gradient method, International Journal of Heat and Mass Transfer 43 (10) (2000) 1767–1776. [103] Y.K. Hong, S.W. Baek, K.W. Kim, Inverse radiation analysis using repulsive particle swarm optimization algorithm, Numerical Heat Transfer – Part A: Applications 57 (2010) 315–330. [104] M.P. Mengüç, P. Dutta, Scattering tomography and its application to sooting diffusion flames, ASME Journal of Heat Transfer 116 (1994) 144–151. [105] M.P. Mengüç, S. Manickavasagam, D.A. Dsa, Determination of radiative properties of pulverized coal particles from experiments, Fuel 73 (4) (1994) 613–625. [106] P.R. Solomon, P.E. Best, R.M. Carangelo, J.R. Markham, P.L. Chien, R.J. Santoro, H.G. Semerjian, FT-IR emission/transmission spectroscopy for in situ combustion diagnostics, in: Twenty-First Symposium (International) on Combustion, 1986, pp. 1763–1771. [107] J.Q. Zhang, J.S. Cheng, Determination of the temperature profile of axisymmetric combustion-gas flow from infrared spectral measurements, Combustion and Flame 65 (1986) 163–176. [108] R.J. Hall, P.A. Bonczyk, Sooting flame thermometry using emission/absorption tomography, Applied Optics 29 (31) (1990) 4590–4598. [109] M. Sakami, M. Lallemand, Retrieval of absorption and temperature profiles in axisymmetric and non-axisymmetric emitting–absorbing media by inverse radiative methods, inverse problems in engineering, theory and practice, in: The First Conference in a Series on Inverse Problems in Engineering, Palm Coast, FL, 1993, pp. 259–266. [110] M. Sakami, M. Lallemand, Retrieval of absorption and temperature profiles in a premixed flame by inverse radiative methods, in: M.P. Mengüç (Ed.), First International Symposium Radiative Heat Transfer, Begell House, 1995, pp. 555–576. [111] F. Yousefian, M. Sakami, M. Lallemand, Recovery of temperature and species concentration profiles in flames using low-resolution infrared spectroscopy, ASME Journal of Heat Transfer 121 (2) (1999) 268–279. [112] C.F. Mallery, S.T. Thynell, Line-of-sight variations of temperature and species in solid propellant flames, Journal of Propulsion and Power 16 (3) (2000) 505–512. [113] C.F. Mallery, S.T. Thynell, Line-of-sight temperature and species profiles determined from spectral transmittances, Journal of Thermophysics and Heat Transfer 11 (3) (1997) 367–374. [114] K.J. Daun, Infrared species limited data tomography through Tikhonov reconstruction, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (1) (2010) 105–115. [115] K.J. Daun, S.L. Waslander, B.B. Tulloch, Infrared species tomography of a transient flow field using Kalman filtering, Applied Optics 50 (6) (2011) 891–900. [116] H.Y. Li, Estimation of thermal properties in combined conduction and radiation, International Journal of Heat and Mass Transfer 42 (3) (1999) 565–572. [117] H.M. Park, T.Y. Yoon, Solution of the inverse radiation problem using a conjugate gradient method, International Journal of Heat and Mass Transfer 43 (10) (2000) 1767–1776. [118] A.D. Klose, A.H. Hielscher, Optical tomography using the time-independent equation of radiative transfer — part 2: inverse model, Journal of Quantitative Spectroscopy and Radiative Transfer 72 (2002) 715–732. [119] H.K. Kim, A.H. Hielscher, A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer, Inverse Problems 25 (1) (2009) 1–20.

Inverse Radiative Heat Transfer Chapter | 23 885

[120] J. Boulanger, A. Charette, Numerical developments for short-pulsed near infra-red laser spectroscopy. Part II: inverse treatment, Journal of Quantitative Spectroscopy and Radiative Transfer 91 (2005) 297–318. [121] J. Boulanger, A. Charette, Reconstruction optical spectroscopy using transient radiative transfer equation and pulsed laser: a numerical study, Journal of Quantitative Spectroscopy and Radiative Transfer 93 (2005) 325–336. [122] J. Boulanger, A. El Akel, A. Charette, F. Liu, Direct imaging of turbid media using long-time back-scattered photons, a numerical study, International Journal of Thermal Sciences 45 (2006) 537–552. [123] H.K. Kim, A. Charette, A sensitivity function-based conjugate gradient method for optical tomography with the frequency-domain equation of radiative transfer, Journal of Quantitative Spectroscopy and Radiative Transfer 104 (1) (2007) 24–39. [124] I. Ayrancı, R. Vaillon, N. Selçuk, F. André, D. Escudié, Determination of soot temperature, volume fraction and refractive index from flame emission spectrometry, Journal of Quantitative Spectroscopy and Radiative Transfer 104 (2) (2007) 266–276. [125] V.L. Kasyutich, P.A. Martin, Towards a two-dimensional concentration and temperature laser absorption tomography sensor system, Applied Physics B 102 (1) (2011) 149–162. [126] M.G. Twynstra, K.J. Daun, Laser-absorption tomography beam arrangement optimization using resolution matrices, Applied Optics 51 (29) (2012) 7059–7068. [127] C. Wei, D.I. Pineda, C.S. Goldenstein, R.M. Spearrin, Tomographic laser absorption imaging of combustion species and temperature in the mid-wave infrared, Optics Express 26 (16) (2018) 20944–20951. [128] Z. Wang, Y. Deguchi, T. Kamimoto, K. Tainaka, K. Tanno, Pulverized coal combustion application of laser-based temperature sensing system using computed tomography–tunable diode laser absorption spectroscopy (CT-TDLAS), Fuel 268 (2020) 117370. [129] W. Cai, C.F. Kaminski, Tomographic absorption spectroscopy for the study of gas dynamics and reactive flows, Progress in Energy and Combustion Science 59 (2017) 1–31. [130] G. Liu, D. Liu, Direct simultaneous reconstruction for temperature and concentration profiles of soot and metal-oxide nanoparticles in nanofluid fuel flames by a CCD camera, International Journal of Heat and Mass Transfer 124 (2018) 564–575. [131] G. Liu, D. Liu, Simultaneous reconstruction of temperature and concentration profiles of soot and metal-oxide nanoparticles in asymmetric nanofluid fuel flames by inverse analysis, Journal of Quantitative Spectroscopy and Radiative Transfer 219 (2018) 174–185. [132] J. Huang, H. Liu, W. Cai, Online in situ prediction of 3-D flame evolution from its history 2-D projections via deep learning, Journal of Fluid Mechanics 875 (2019). [133] Y. Jin, W. Zhang, Y. Song, X. Qu, Z. Li, Y. Ji, A. He, Three-dimensional rapid flame chemiluminescence tomography via deep learning, Optics Express 27 (19) (2019) 27308–27334. [134] J. Sun, C. Xu, B. Zhang, M.M. Hossain, S. Wang, H. Qi, H.P. Tan, Three-dimensional temperature field measurement of flame using a single light field camera, Optics Express 24 (2) (2016) 1118–1132. [135] J. Sun, M.M. Hossain, C. Xu, B. Zhang, Investigation of flame radiation sampling and temperature measurement through light field camera, International Journal of Heat and Mass Transfer 121 (2018) 1281–1296. [136] X. Huang, H. Qi, X.-L. Zhang, Y.-T. Ren, L.M. Ruan, H.P. Tan, Application of Landweber method for three-dimensional temperature field reconstruction based on the light-field imaging technique, ASME Journal of Heat Transfer 140 (8) (2018). [137] T. Li, Y. Yuan, B. Zhang, J. Sun, C. Xu, Y. Shuai, H.P. Tan, Experimental verification of three-dimensional temperature field reconstruction method based on Lucy-Richardson and nearest neighbor filtering joint deconvolution algorithm for flame light field imaging, Applied Thermal Engineering 162 (2019) 114235. [138] X. Huang, H. Qi, C. Niu, L.M. Ruan, H.P. Tan, J. Sun, C. Xu, Simultaneous reconstruction of 3D temperature distribution and radiative properties of participating media based on the multi-spectral light-field imaging technique, Applied Thermal Engineering 115 (2017) 1337–1347. [139] T.-J. Li, S.-N. Li, Y. Yuan, F.-Q. Wang, H.P. Tan, Light field imaging analysis of flame radiative properties based on Monte Carlo method, International Journal of Heat and Mass Transfer 119 (2018) 303–311. [140] S. Wen, H. Qi, S.-B. Liu, Y.-T. Ren, J.-W. Shi, X. Huang, A hybrid LSQP algorithm for simultaneous reconstruction of the temperature and absorption coefficient field from the light-field image, Infrared Physics & Technology 105 (2020) 103196. [141] M.R. Rhoby, D.L. Blunck, K.C. Gross, Mid-IR hyperspectral imaging of laminar flames for 2-D scalar values, Optics Express 22 (18) (2014) 21600–21617. [142] H. Liu, S. Zheng, H. Zhou, Measurement of soot temperature and volume fraction of axisymmetric ethylene laminar flames using hyperspectral tomography, IEEE Transactions on Instrumentation and Measurement 66 (2) (2016) 315–324. [143] H. Liu, S. Zheng, H. Zhou, C. Qi, Measurement of distributions of temperature and wavelength-dependent emissivity of a laminar diffusion flame using hyper-spectral imaging technique, Measurement Science and Technology 27 (2) (2015) 025201. [144] K. Wu, Y. Feng, G. Yu, L. Liu, J. Li, Y. Xiong, F. Li, Development of an imaging gas correlation spectrometry based mid-infrared camera for two-dimensional mapping of CO in vehicle exhausts, Optics Express 26 (7) (2018) 8239–8251. [145] M. Si, Q. Cheng, Q. Zhang, D. Wang, Z. Luo, C. Lou, Study of temperature, apparent spectral emissivity, and soot loading of a single burning coal particle using hyper-spectral imaging technique, Combustion and Flame 209 (2019) 267–277. [146] T. Ren, H. Li, M.F. Modest, C. Zhao, Efficient two-dimensional scalar fields reconstruction of laminar flames from infrared hyperspectral measurements with a machine learning approach, Journal of Quantitative Spectroscopy and Radiative Transfer 271 (2021) 107724. [147] T. Ren, H. Li, M.F. Modest, C. Zhao, Machine learning applied to retrieval of three-dimensional scalar fields of laminar flames from hyperspectral measurements, submitted to International Journal of Heat and Mass Transfer (2021).

Chapter 24

Nanoscale Radiative Transfer 24.1 Introduction In the last chapter of this book we will provide a brief introduction to radiative heat transfer in geometries where the pertinent dimensions are “small.” By “small,” we mean dimensions that are comparable to or smaller than the wavelength of radiation, estimated to be approximately 1–20 μm for most applications by the laws of blackbody radiation. Radiation transport in this regime is referred to as near-field or nanoscale radiation, while its counterpart for “large” length scales (subject of majority of this text) is referred to as far-field radiation. Most radiation is incoherent (multispectral, as well as random in polarization and direction) in the far field, and the radiative transfer equation (RTE) and its solution methods described over the previous chapters are only valid for such incoherent radiation. We noticed in Chapters 2 and 3 (optically smooth surfaces) and Chapter 11 (small particles) that, when distances of the order of the wavelength λ are relevant, radiative transfer must be calculated from the full Maxwell’s equations presented in Chapter 2. However, Maxwell’s equations do not include any radiative emission sources, which must be modeled via what is known as fluctuational electrodynamics, pioneered by Rytov [1,2]. Research in the field of nanoscale energy transfer has exploded during the past few years, leading to fascinating new problems and devices in microelectronics and microfabrication technology, such as quantum structures, optoelectronics, molecular- and atomic-level imaging techniques, etc. In the following we will give very brief introductory accounts of some interesting radiative phenomena that are observed at the nanoscale, culminating in the prediction of radiative flux between two plates, spaced a tiny distance apart. The reader interested in detailed knowledge of the subject area should consult the books by Chen [3], Novotny and Hecht [4], and Zhang [5], review articles by Zhang and coworkers [6–8], Song et al. [9], and Edalatpour et al. [10], as well as the large number of recent research papers in the field.

24.2 Coherence of Light No radiation source is perfectly coherent, i.e., perfectly monochromatic and unidirectional, not even lasers or emission from single atoms. On the other hand, no source is truly incoherent: even the most chaotic blackbody radiation has a small coherence length, which is related to the distance the wave travels within a coherence time [11]. If the wave nature of light is completely preserved, we speak of coherent light. If light travels longer than the coherence time, or a distance larger than the coherence length, fluctuations in the waves will diminish wave interference effects (see Fig. 2.13 and the discussion of reflection from a thin layer). The coherence of light in space and time (or, equivalently, frequency) is measured by the mutual coherence function of any two waves, defined as E(r1 , t)E∗ (r2 , t) , where the angular brackets denote time-averaging, and the r1 and r2 are two different locations; the electric field can be expressed in either the frequency domain, or time domain [11]. For our purposes we simply note that the coherence length of random blackbody radiation is about λ/2 [4,12], and longer for more coherent sources.

24.3 Evanescent Waves We observed in Section 2.5, equation (2.100), that at an interface between two dielectrics total reflection takes place if light attempts to enter a less dense material (n2 < n1 ) at an incidence angle θ1 larger than the critical angle n2 (24.1) sin θ1 > sin θc = , n1 with no energy penetrating into Medium 2 (see Fig. 24.1a). This is true as far as far-field radiation is concerned, and also for net (time- and space-averaged) energy. However, if one carefully inspects the electromagnetic Radiative Heat Transfer. https://doi.org/10.1016/B978-0-12-818143-0.00032-8 Copyright © 2022 Elsevier Inc. All rights reserved.

887

888 Radiative Heat Transfer

FIGURE 24.1 Total internal reflection and evanescent waves: (a) propagation of waves at critical angle of incidence, (b) evanescent wave propagating along x-direction and exponentially decaying in −z-direction.

wave theory relationships, one observes that a wave traveling parallel to the interface enters Medium 2, with its strength decreasing exponentially away from the interface, known as an evanescent wave (from Latin for “vanishing”). To simplify the analysis we will, without loss of generality, consider here only the case of a parallel polarized (TM) wave (E⊥ = 0), and only concern ourselves with the electric field. Then, from equations (2.73) and (2.75), we have E c1 = Ei eˆ i e−2πi(wi ·r−νt) + Er eˆ r e−2πi(wr ·r−νt) , −2πi(wt ·r−νt)

E c2 = Et eˆ t e

,

(24.2a) (24.2b)

and the wave vector w, as defined1 by equation (2.31) has x- and z-components ˆ w = η0 nˆs = wx î + wz k.

(24.3)

Since the tangential components of the electrical field must be conserved, equation (2.67), we have wxi = wxr = wxt = wx , and wx = η0 n1 sin θ1 = η0 n2 sin θ2 ,

(24.4)

which is Snell’s law. If θ1 exceeds the critical angle, then n1 sin θ1 > n1 sin θc > n2 , and the z-component of the transmitted wave becomes ( (  2 wzt = η0 n2 − w2x = iη0 (n1 sin θ1 )2 − n22 = i|wzt | = iη0 n2 | cos θ2 |,

(24.5)

(24.6)

i.e., wzt and cos θ2 are purely imaginary (and |wzt | and | cos θ2 | are their magnitudes). Substituting this into equation (24.2b) we have E c2 = Et kˆ e−2π|wzt |+2πiνt ,

(24.7)

with the magnitude of |wzt | = O{η0 = 1/λ0 }, i.e., we have a wave inside Medium 2 traveling along the interface, exponentially decaying in strength over the distance of one wavelength or so (depending on θ1 ). This is depicted in Fig. 24.1b. Performing the same analysis for the magnetic field (with H = 0), it is easy to show that the z-component of the time-averaged Poynting vector, see equation (2.42), is zero, i.e., no net energy crosses the interface [3]. However, if the instantaneous Poynting vector is examined, one finds that there is periodic in- and outflow of energy carried by the evanescent field. 1. Recall that this book’s definition of the wave vector differs by a factor of 2π and in name from the definition k = 2πw in most optics texts in order to conform with our definition of wavenumber.

Nanoscale Radiative Transfer Chapter | 24 889

FIGURE 24.2 Photon tunneling through a layer of lesser refractive index, adjacent to two optically denser materials.

24.4 Radiation Tunneling We have seen in the previous section that, if a radiative wave train is reflected at the interface to an optically less dense medium, an evanescent wave exists within the optically rarer medium with exponentially decaying strength away from the interface. Furthermore, the evanescent wave does not carry any net (time-averaged) energy into the direction normal to the surface. However, if a second denser medium is brought into close proximity to the first, net energy can be transported across the gap or intermediate layer. This phenomenon is known as radiation tunneling (or sometimes as photon tunneling, or frustrated total internal reflection [11]), and is very important for heat transfer between two media a distance of a wavelength or less apart, as schematically shown in Fig. 24.2. While this phenomenon has been known since Newton’s time, in the heat transfer area it was probably first discovered by Cravalho and coworkers [13], who investigated closely spaced cryogenic insulation. Enhancement of photon tunneling is the key to increasing near-field radiative heat transfer between two objects [5]. For example, Zhao and Zhang [14] have demonstrated that heterostructures consisting of a monolayer graphene on a hexagonal boron nitride film can greatly enhance photon tunneling and outperform individual structures made of either graphene or hexagonal boron nitride. A number of modern applications that endeavor to enhance photon tunneling are discussed briefly in Section 24.9. If a second optically dense material is close to the first, the evanescent wave in the layer in between is reflected back toward the first interface. Interference between the two waves causes the Poynting vector to have a nonzero net component in the z-direction. However, if the gap is too wide (i.e., well more than one wavelength away), the evanescent wave reaching the second interface is too weak and net energy transfer becomes negligible. To calculate the transmissivity of the gap or intermediate film for above-critical angles of incidence we may use the thin film relations developed in Chapter 2, keeping in mind that cos θ2 may become imaginary for large incidence angles. Limiting ourselves here to three dielectrics with n1 = n3 > n2 , equations (2.131b) and (2.133) may be rewritten as t=

d t12 t21 eiβ , T = tt∗ , β = 2πw2z d = 2πn2 cos θ2 , 2 2iβ λ 1 − r21 e

(24.8)

with r21 , t12 , and t21 determined from equations (2.89) through (2.92). For θ1 < θc the interface reflection and transmission coefficients are real, and Tλ =

(1 − r221 )2 (t12 t21 )2 = , 1 − 2r221 cos 2β + r421 1 − 2r221 cos 2β + r421

θ1 < θc = sin



 n2 . n1

(24.9)

If θ1 exceeds the critical angle an evanescent wave enters Medium 2 and w2z and cos θ2 become purely imaginary. From equation (24.6) we find that the phase shift β now becomes imaginary (the exponential decay of the

890 Radiative Heat Transfer

evanescent wave), β = i (2πn2 | cos θ2 |)

d = i|β|, λ

(24.10)

and the r21 , t12 , and t21 become complex [i.e., replacing cos θ2 by i| cos θ2 | in equations (24.8) and (24.9)]. Therefore, t=

t12 t21 e−|β| , 1 + r221 e−2|β|

and Tλ = tt∗ =

(24.11)

(t12 t21 )(t∗12 t∗21 )e−2|β|  , e−2|β| + r221 r∗2 1 + r221 + r∗2 e−4|β| 21 21 

(24.12)

which, after some algebra (left as an exercise), may be reduced to ⎧ ⎪ ⎪ n1 | cos θ2 | ⎪ ⎪ , ⎪ 2 ⎪ sin 2α ⎨ n2 cos θ1 Tλ = , where tan α = ⎪ ⎪ ⎪ sin2 2α + sinh2 |β| n2 | cos θ2 | ⎪ ⎪ ⎪ , ⎩ n1 cos θ1

parallel (TM) polarization, (24.13) perpendicular (TE) polarization.

Again, equation (24.13) is valid for, both, parallel- and perpendicular-polarized light, except for the different definition of tan α (due to the different structure of r and r⊥ ). √ Example 24.1. Consider a vacuum gap surrounded by a dielectric medium with refractive index n1 = n3 = 2 = 1.4142. Determine the transmissivity for parallel-polarized light for all angles of incidence and as a function of gap width. Solution √ With n2 = 1 we have sin θc = 1/ 2, or θc = 45◦ . Writing a small computer code, using equation (24.9) for θ1 < 45◦ , and ◦ equation (24.13) for θ1 > 45 , and with tan α =

n1 | cos θ2 | , n2 cos θ1

we obtain the gap transmissivity shown in Fig. 24.3. It is observed that for small θ1 we have noticeable interference effects, but the transmissivity remains high for all gap widths (Tλ > 0.9). Wavelength of interference and magnitude increase with θ1 until, reaching Brewster’s angle (≈ 35◦ ), we have total transmission of a parallel-polarized wave (see also

FIGURE 24.3 Transmissivity of a vacuum gap surrounded by identical dielectrics (n1 = n3 = 1.4142), for parallel-polarized light.

Nanoscale Radiative Transfer Chapter | 24 891

FIGURE 24.4 Typical configuration for the generation of surface polaritons, consisting of a dielectric for incident light, and an air/metal thin layer/substrate combination.

Fig. 2.9). Beyond Brewster’s angle ρ increases rapidly, with decreasing transmissivity (but still increasing wavelength of interference). At θ1 = 45◦ we have r = −1, and an evanescent wave forms, and the larger the incident angle, the faster the strength of the evanescent wave decays across the gap. It is straightforward to verify that, at 45◦ , both equations (24.9) and (24.13), go to the same limit, i.e., Tλ (θ1 = 45◦ ) =

1 .  πd 1+ 2λ

24.5 Surface Waves (Polaritons) The interaction between electromagnetic waves and the oscillatory movement of free charges (electrons) near the surface of metallic materials is known as surface plasmons or surface plasmon polaritons. Surface plasmons are usually found in the visible to near-infrared part of the spectrum in highly conductive metals, such as gold, silver, and aluminum. They are of importance in near-field microscopy and nanophotonics [11,15–17]. In some polar dielectrics lattice vibrations (phonons) and/or oscillations of bound charges can also interact with electromagnetic waves in the mid-infrared; these are known as surface phonon polaritons, and are of interest in the tuning of emission properties [18] and nanoscale imaging [19]. In either case they result in the generation of an electromagnetic wave traveling along, and only in the immediate vicinity of both sides of an interface, i.e., a surface wave. In our brief discussion here we will mostly follow the presentation of Zhang [11]. One requirement of a surface wave, i.e., a wave decaying in both directions normal to the surface, is that there are evanescent waves on both sides of the interface. Consider the arrangement shown in Fig. 24.4, consisting of a thin layer and a thick substrate, with the thin layer bound at the top by a third medium. The thin layer may be air with a metallic substrate (Otto configuration), or a metal layer bounded by air at the bottom (Kretschmann configuration) [20]. If light is incident from the top medium, it is possible for evanescent waves to occur simultaneously in both the underlying air and metal layers, as also indicated in Fig. 24.4. A second requirement for polaritons is that the polariton dispersion relations must be satisfied, which are the poles of the Fresnel reflection coefficients, since infinite reflection coefficients are an indication of resonance. If one writes the reflection coefficients in terms of wave-vector components [4,11] as  ;   w1z w2z w1z w2z r = , (24.14a) − + ε ε2 ε ε2  . 1   1 w1z w2z w1z w2z , (24.14b) r⊥ = − + μ1 μ2 μ1 μ2 the polariton dispersion relations are defined by

892 Radiative Heat Transfer

w1z w2z + = 0, ε1 ε2 w1z w2z + = 0, μ1 μ2

for parallel-polarized light,

(24.15a)

for perpendicular-polarized light.

(24.15b)

The nature of the dispersion relations is more easily understood by first looking at the case of two dielectric media: in order to have evanescent waves we must have both w1z and w2z purely imaginary, with w1z = −i|w1z | and w2z = −i|w2z |, i.e., both with a negative sign in order to have e−2πiwr ·r = e−2πi(w1x x−w1z z) = e−2πiw1x x+2π|w1z |z (reflected wave) decay toward negative z, and e−2πiwt ·r = e−2πi(w2x x+w2z z) = e−2πiw2x x−2π|w2z |z (transmitted wave) toward positive z (see Fig. 24.4). This implies that in order to produce a surface wave with parallel-polarized incident light, the electrical permittivities of the two materials must have opposite signs. Since metals display negative permittivities over large parts of the spectrum, this condition is easily fulfilled. To produce a surface polariton with perpendicular-polarized light, on the other hand, requires a medium with negative magnetic permeability. While so-called negative index materials (NIM) exhibit both negative permittivity and permeability [21], most materials are nonmagnetic, for which surface polaritons cannot be generated with perpendicular-polarized light. Employing equation (2.31) together with m2 = ε, we may write for a general nonmagnetic medium w21 = w2x + w21z = η20 ε1 ,

(24.16a)

η20 ε2 ,

(24.16b)

w22

=

w2x

+

w22z

=

where we have made use of the fact that the tangential component of the wave vector must be continuous across the interface, w1x = w2x = wx . Using these relations the z-components may be eliminated from equation (24.15a), leading to & ε1 ε2 wx = η0 . (24.17) ε1 + ε2 This equation relates the tangential component of the wave vector to wavenumber (or frequency), and is a popular alternative statement of the polariton dispersion relation. If one of the media is vacuum or air (ε = 1), an evanescent wave exists if wx > η0 (i.e., wz has an imaginary component). Note that equation (24.17) also gives the roots to the numerator of equation (24.14a): for wx < η0 equation (24.17) describes propagating waves. Example 24.2. Determine the dispersion relation between aluminum and air, assuming that the dielectric function of Al obeys the Drude theory. Solution The Drude equation has been given by equation (3.64), when written in complex form, as εAl = 1 −

ν2p ν(ν + iγ)

;

νp = 3.07 × 1015 Hz,

γ = 3.12 × 1013 Hz,

with plasma frequency νp and damping factor γ from Fig. 3.7. With εair = 1 the tangential wave vector component may be calculated from equation (24.17). Since εAl is complex, so is wx = wx + iwx . It is common to show a dispersion relationship by plotting the real part of wx vs. frequency or wavenumber, which has been done in Fig. 24.5. The dashed line wx = η0 is called the light line. On its left wz is real in air, and a propagating wave exists. On its right, wx > η0 and the wz in air becomes imaginary, and only evanescent waves are found. It is seen that, for the evanescent waves, wx increases rapidly, √ reaching an asymptote at ν = νp / 2, when the real part of the dielectric function of Al approaches −1. For ν > νp metal becomes transparent and the real part of the dielectric function becomes positive. The solution to equation (24.17) for ν > νp corresponds to r = 0 in equation (24.14a) and shows, therefore, propagating waves.

24.6 Fluctuational Electrodynamics As indicated earlier, Maxwell’s equations do not contain a thermal radiation emission term. Such a source must be added by considering radiative transitions by elementary energy carriers (such as electrons, lattice vibrations called phonons, etc.) from a higher energy state to a lower one, accompanied by the release of a photon. Such

Nanoscale Radiative Transfer Chapter | 24 893

FIGURE 24.5 Dispersion relation for aluminum and air; top left solid line: propagating waves; dashed line: light line; bottom right solid line: evanescent waves.

a quantum-mechanical process, similar to emission from gas molecules covered in Chapter 10, must be linked to the equations describing the electromagnetic waves. This is achieved through the concept of fluctuational electrodynamics, originally developed by Rytov [1,2]. At any finite temperature above absolute zero, chaotic thermal motions take place inside any material. Charged particles of opposite sign pair up (known as dipoles), and the random motion of the dipoles induce a fluctuating electromagnetic field. Thus, in this fluctuational electrodynamics model the random thermal fluctuations generate a space- and time-dependent (but random) electric current density j (r, t) inside the medium, whose time average is zero [11]. To include the stochastic current density in the electromagnetic wave equations, several approaches are possible. The most common technique is to employ a dyadic Green’s function Ge (r, r , ν) (a 3 × 3 matrix). The induced electric and magnetic fields in the frequency domain can then be determined from  E(r, ν) = 2πiμ0 Ge (r, r , ν) · j(r , ν)dr , (24.18a) V  H(r, ν) = Gh (r, r , ν) · j(r , ν)dr , (24.18b) V

where the integral is over the volume, which contains the fluctuating dipoles, j(r , ν) is the Fourier transform of the electric current density source j (r, t) into frequency space, and μ0 is the magnetic permeability of vacuum. The dyadic Green’s function for the magnetic field is, by equation (2.13), directly related to Ge through Gh = −∇ × Ge . Physically, Ge may be interpreted as a transfer function relating the electric field at location r and frequency ν to a vector source located at r’. Mathematically, the dyadic Green’s function is found as the solution to a vector Helmholtz equation, which may be reduced to a scalar one as [4]   1  ∇∇ G0 (r, r , ν), (24.19) Ge (r, r , ν) = δ + (2πw)2 with G0 the solution to



 (2πw)2 + ∇2 G0 (r, r , ν) = −δ(r − r ),

(24.20)

where δ(r − r ) is a 3D Dirac-delta function as defined on p. 641, and w is the magnitude of the wave vector w. The time-averaged emitted energy flux may be calculated from the average Poynting vector, equation (2.41), S(r, ν) = 12 {E c × H ∗c } ,

(24.21)

894 Radiative Heat Transfer

where the angle brackets denote the ensemble average over the random fluctuations. Sticking equations (24.18) into equation (24.21) requires a two-point ensemble average of the random current density, which must be a function of local temperature. This is achieved through the fluctuation–dissipation theorem pioneered by Rytov [1], leading to E D (24.22) jm (r, ν)jn (r , ν) = 8ν 0 ε Θ(ν, T)δmn δ(r − r ), where 0 is the electrical permittivity of vacuum, ε is the imaginary part of the medium’s dielectric function, and subscripts m and n denote the x-, y-, and z-components of j. The function Θ(ν, T) is the mean energy of a Planck oscillator given by [4] hν

. (24.23) −1 A multiplicative factor of 4 is included on the right-hand side of equation (24.22), since only positive frequencies are considered in the Fourier transform for the electric current density [22]. Sticking equations (24.18) and (24.22) into equation (24.21) yields for the individual terms arising in the Poynting vector, after some manipulation, ⎫ ⎧   2 ⎪ ⎪  , ⎪ ⎪ ν ⎨ ⎬ ∗   1 , (24.24) iε E (r, ν)H (r, ν) = 8π Θ(ν, T) G (r, r , ν)G (r, r , ν)dr ⎪ ⎪ i e,im e, jm j 2 ⎪ ⎪ ⎭ ⎩ c0 V Θ(ν, T) =

ehν/kT

m

where the subscripts again denote the various x-, y-, and z-components. For example, the z-component of the Poynting vector becomes ⎫ ⎧   2 ⎪ ⎪  ,  ⎪ ⎪ ν ⎨ ⎬ ∗ ∗ ∗ ∗  1 iε . (24.25) Θ(ν, T) ⎪ Ge,xm Gh,ym − Ge,ym Gh,xm (r, r , ν)dr ⎪ Sz (r, ν) = 2 Ex H y − E y Hx = 8π ⎪ ⎪ ⎭ ⎩ c0 V m

As given, the time-averaged Poynting vector constitutes the local radiative flux caused by the surrounding electromagnetic field. A general dyadic Green’s function formalism that includes nonequilibrium van der Waals/Casimir forces between objects of arbitrary shapes, sizes, and with frequency-dependent dielectric permittivity and magnetic permeability has been presented by Narayanaswamy and Zheng [23].

24.7 Heat Transfer Between Parallel Plates Consider Medium 1 separated from Medium 2 by a small, perfectly parallel vacuum gap of width d, as shown in Fig. 24.6. To calculate the radiative flux between them we need to determine the normal component of the Poynting vector for the energy transmitted from Medium 1 across the gap, as well as the counter-flow from Medium 2 to 1. Since the problem is one-dimensional in the z-direction, it has no azimuthal (or x- and y-) dependence, making the analysis a little simpler if cylindrical coordinates are employed, i.e., we define position and wave vectors as ˆ r = rêr + zk,

ˆ w = wr êr + wz k.

(24.26)

The dyadic Green’s function for two semi-infinite media separated by a parallel gap may be determined from [4, 11,24,25]  ∞  wx    sˆ t⊥ sˆ + pˆ 1 t pˆ 2 e−2πi(w2z z−w1z z ) e−2πiwx (r−r ) dwx , Ge (r, r , z, z ; ν) = i (24.27a) 2w1z 0 where ˆ sˆ = êr × k,

pˆ i = (wx kˆ − wiz êr )/wi ,

i = 1, 2.

(24.27b)

Here t⊥ and t are the transmission coefficients from Medium 1 to Medium 2, as evaluated from Airy’s formula, equation (2.131b), and the interrelationship between wx , wiz , and wi is given by equation (24.16). Substituting this into equation (24.25) one obtains, after considerable algebra, an expression for the spectral radiative flux from Medium 1 to Medium 2:  ∞ Z12 (ν, wx )wx dwx (24.28a) qν,1→2 = 8πΘ(ν, T1 ) 0

Nanoscale Radiative Transfer Chapter | 24 895

FIGURE 24.6 Closely spaced parallel plates separated by a vacuum gap.

where   4 {w1z }  {w2z } w20 e−2iw0 d  Z12 (ν, wx ) =   2  (w0z + w1z )(w0z + w1z ) 1 − r⊥01 r⊥02 e−2iw0 d  7 6 7  6  4 ε1 w∗1z  ε2 w∗2z w20 e−2iw0 d  +  2 .  (ε1 w0z + w1z )(ε2 w0z + w1z ) 1 − r01 r02 e−2iw0 d 

(24.28b)

The Z12 (ν, wx ) may be interpreted as an exchange function, identifying the contribution of a given tangential wave vector component, wx (related to incidence angle), to the spectral flux. Observing that Z12 = Z21 , the net heat exchange between the two surfaces is readily found, after integration over all frequencies, as  qnet =

∞







qν,1→2 − qν,2→1 dν =

0

∞

 (Θ(ν, T1 ) − Θ(ν, T2 ))

0

∞

Z12 (ν, wx )wx dwx dν.

(24.29)

0

Equations (24.28) and (24.29) include contributions from both propagating and evanescent waves. We observed in Section 24.3 that we have propagating waves for wx < w0 = η0 = ν/c0 (real w0z in the vacuum layer), and evanescent waves for wx > ν/c0 (imaginary w0z ). Using the expressions for transmission coefficients developed in Section 24.4, we find (1 − r201 )(1 − r202 ) (1 − r2⊥01 )(1 − r2⊥02 ) Zprop (ν, wx ) =  2 +  2 ,  4 1 − r⊥01 r⊥02 e−2iw0 d  4 1 − r01 r02 e−2iw0 d 

wx < η0 .

(24.30a)

For the evanescent waves, the exchange function reduces to F G F G  {r⊥01 }  {r⊥02 } e−2|w0 |d  r01  r02 e−2|w0 |d Zevan (ν, wx ) =  + 2  2 ,  1 − r01 r02 e−2|w0 |d  1 − r⊥01 r⊥02 e−2|w0 |d 

wx > η0 .

(24.30b)

Clearly, similar to the evanescent transmissivity of Section 24.3, the contribution from Zevan to the flux decreases exponentially with distance between the plates. Far Field Heat Flux. As discussed in Section 2.5, as d becomes large, d  λ0 , the radiation will lose coherence, and the gap transmissivity will obey equation (2.133) (with κ = 0 for the vacuum gap). Then the exchange function reduces to, with |r|2 = ρ, Zprop,ff (ν, wx ) =

(1 − ρ⊥01 )(1 − ρ⊥02 ) (1 − ρ01 )(1 − ρ02 )    +  . 4 1 − ρ⊥01 ρ⊥02 4 1 − ρ01 ρ02

(24.31)

896 Radiative Heat Transfer

FIGURE 24.7 Radiative heat transfer coefficient, hR = qnet /(T1 − T2 ), between aluminum plates separated by a vacuum microgap of varying width and at 300.5 K and 299.5 K, respectively.

Integration over wx may be replaced by wx = (ν/c0 ) sin θ, where θ is the polar angle in vacuum, and equation (24.29) becomes, with Zevan = 0 and 1 − ρ = , ⎛ ⎞ ⎟⎟  π/2 ⎜⎜⎜⎜  ∞ ⎟⎟ 1 2π 1 ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ cos θ sin θ dθ ν2 dν. qnet,far = 2 [Θ(ν, T1 ) − Θ(ν, T2 )] (24.32) + ⎟⎟ ⎜ 1 1 1 ⎜⎜ 1 c0 0 0 ⎝ + −1 + − 1 ⎟⎠

⊥01 ⊥02

01 02 Comparison with equation (5.35) shows that these results are identical if the emissivities are assumed to be gray and diffuse. Example 24.3. Determine the radiative heat transfer coefficient, hR = qnet /(T1 − T2 ), between two plates of aluminum, separated by a vacuum gap, assuming that the dielectric function of Al obeys the Drude theory as in the previous example. The plates are isothermal and maintained at 300.5 K and 299.5 K, respectively. Determine the total radiative flux as a function of gap thickness. Distinguish contributions from propagating and evanescent waves. Solution With the dielectric function of Al given in the previous example, and with the wiz related to wx and εi by equation (24.16), the reflection coefficients in equations (24.30) may be calculated from equations (24.14). Integrating over all frequencies ν and all tangential wave vectors wx , separately 0 ≤ wx < η0 for propagating waves and η0 < wx < ∞ for evanescent waves, yields the desired total radiative flux between the two aluminum plates, as shown in Fig. 24.7 for gap widths ranging from 1 nm to 10 μm. For the far-field solution equation (24.30a) is replaced by equation (24.31) and Zevan = 0. Integration may again be over tangential wave vectors 0 ≤ wx < η0 or, alternatively, over polar angle θ. It is seen that, for gap sizes of less than about 2 μm, the heat flux is dominated by the evanescent waves. For small gap widths the propagating component approaches an asymptotic limit, which is about an order of magnitude larger than the far-field solution, but still considerably smaller than the blackbody limit of σ(T14 − T24 )/(T1 − T2 ) 6.12 W m−2 K−1 (due to the small emissivity of aluminum, see Fig. 3.7). For intermediate gap widths, both the propagating and evanescent components decrease with gap width, resulting in a decrease of the total flux (and radiative heat transfer coefficient) with gap width. At a gap width slightly larger than about 2 μm, the propagating component begins to increase, while the evanescent component continues to decay rapidly. As a result, the total flux goes through a minimum value before increasing again. At a gap width beyond about 7 μm the decrease in the evanescent component exactly matches the increase in the propagating component, resulting in a heat transfer coefficient that does not change any more with increasing gap width. This value of the radiative heat transfer coefficient, as shown in Fig. 24.7, also matches the computed far-field value.

The transition from near-field to far-field radiation has recently been carefully examined by Tsurimaki et al. [26]. As shown in Fig. 24.7, the radiative heat transfer coefficient (and total radiative flux) exhibits a dip before increasing back to the far-field value. Closer examination by Tsurimaki et al. revealed that a negative real part of

Nanoscale Radiative Transfer Chapter | 24 897

FIGURE 24.8 Spectral radiative heat fluxes between silicon carbide plates separated by a 10 nm vacuum microgap.

the dielectric function can lead to a radiative heat flux that is lower than its value in the far field. However, this is not always necessary, and depends also on the temperature. Calculations for two aluminum plates showed that a reduction of about 15% is possible—the minimum value occurring at approximately d × T = 970 μmK (T being the average temperature of the two plates), which corresponds to a gap width of approximately 3.25 μm at room temperature. Narayanaswamy and Mayo [27] observed a similar dip in the heat flux in their computed results for two plates with metal-like dielectric functions separated by a vacuum gap, and noted that the dip in the heat flux is more pronounced at low temperature. The plasma frequency of aluminum corresponds to a wavelength slightly less than 0.1 μm, while heat transfer at the example’s temperatures occurs at wavelengths between roughly 2.5 and 60 μm. Therefore, the spectral variations in heat flux essentially follow a Planck function pattern. Silicon carbide, on the other hand, has a band around 12 μm (see Fig. 3.13), giving rise to interesting spectral variations. Example 24.4. Determine the spectral radiative flux between two plates of silicon carbide, separated by a 10 nm vacuum gap, assuming that the dielectric function of SiC obeys the Lorentz model with parameters given by Fig. 3.13. The plates are again isothermal and maintained at 400 K and 300 K, respectively. Distinguish contributions from propagating and evanescent waves, as well as the influence of parallel and perpendicular polarizations, and compare with the far-field solution. Solution As noted in Chapter 3, the dielectric function of SiC is well described by the single oscillator Lorentz model of equation (3.63), with ε0 = 6.7, νpi = 4.327 × 1013 Hz, νi = 2.380 × 1013 Hz (corresponding to a wavenumber of 793 cm−1 ), and γi = 1.428 × 1011 Hz. Aside from the different dielectric function and the fixed gap width, the solution proceeds as in the previous example, but without carrying out the actual integration over frequency. Results are shown in Fig. 24.8 for the spectral region between 600 cm−1 and 2,000 cm−1 surrounding the resonance band of SiC. It is seen that the TE evanescent wave has a maximum at the resonance frequency of 793 cm−1 , before dropping by several orders of magnitude similar to the propagating waves. On the other hand, the TM evanescent wave has a maximum at 969 cm−1 (corresponding to the wavelength with near-zero reflectivity in Fig. 3.13). The far-field flux follows the behavior given in Fig. 3.13, i.e., flux decreases over wavelengths with large reflectivities.

A number of researchers have investigated near-field radiative transfer theoretically, primarily looking at different aspects of the heat flow across plane-parallel gaps [22,28–34]. In particular, Hu et al. [31] were the first to demonstrate near-field enhancement (exceeding blackbody radiation at room temperature). Other geometries that have also received attention are spheres in close contact with flat plates [24,25,35–37], with another sphere [38–41], and clusters of nanoparticles [42]. Recent research has also delved into emerging materials, such as silicongermanium alloys [43], graphene and carbon nanotubes [44–48], and doped silicon [49–52].

898 Radiative Heat Transfer

While the primary emphasis of this section has been near-field radiative transfer between two homogeneous blocks or plates, modern applications (see Section 24.9) often require consideration of blocks comprised of multiple layers of two or more materials in close proximity. Conventionally, such multilayered materials are often treated using the so-called effective medium theory, which allows calculation of the effective radiative property of the entire multilayered block (see, for example, Section 3.8 for effective property of multilayered thin films in the context of far-field radiation). Liu et al. [53] have recently laid out a set of criteria that need to be obeyed for effective medium theory to hold in the context of near-field radiation transfer between two multilayered blocks. Both multilayered blocks are comprised of alternating layers of a metal (heavily doped silicon in this particular case) and a dielectric (germanium in this particular case). It was found that for perpendicular or s-polarized radiation, effective medium theory holds for any gap width, while for parallel or p-polarized radiation, a set of criteria must be obeyed for effective medium theory to be applicable, details of which may be found in [53].

24.8 Experiments on Nanoscale Radiation It has been recognized for some time that radiative heat transfer can exceed blackbody limits at the nanoscale, and thus plays an important role in a number of applications, such as near-field microscopy, nanoelectronics thermal management, photovoltaics, etc. Correspondingly, the problem of heat transfer between closely spaced objects has been studied theoretically in some detail, as outlined in the previous sections. On the other hand, experimental verification has been limited, mostly because of the difficulties of maintaining a precise nanoscale gap between the emitter and receiver. The earliest experiments were carried out in the field of cryogenic insulation by Domoto and coworkers [54] (accompanied by some theoretical attempts [28,29]), and by Hargreaves [55,56]. At cryogenic temperatures, say below 10 K, according to Wien’s displacement law, equation (1.16), heat transfer is maximized around a wavelength of 300 μm, i.e., even plates tens of μm apart should display tunneling effects. Domoto and coworkers measured heat flow between two copper plates as close as 10 μm together, and at temperatures between 5 K and 15 K. While the measured heat transfer was only about 3% of that between blackbodies (because of copper’s small emittance), and agreement with their model was only fair, they were able to show that—contrary to far-field analysis—the heat transfer increased by a factor of 2.3 between the far field and their closest spacing of 10 μm. Hargreaves carried out similar experiments, using chromium plates with vacuum gaps down to 1.5 μm. He was able to demonstrate a factor of five heat transfer increase from far field to near field (but still considerably less than the blackbody limit). Small gaps are more easily achieved by moving a small tip close to a surface. For example, Xu et al. [57] tried to measure near-field radiative transfer by moving a 100 μm diameter indium probe of a scanning thermal microscope as close as 12 nm to a thermocouple probe, but could not detect any substantial increase in heat transfer. Kittel and colleagues [58] used a scanning tunneling microscope (STM) to measure near-field radiation between the thermocouple tip and a plate, observing the expected 1/d3 increase in heat transfer down to a gap width of 10 nm. Below that distance, there was disagreement between theory and experiment. Narayanaswamy et al. [35] measured near-field radiation with a bimetallic atomic force microscope (AFM) cantilever with a silica microsphere at its tip. The plate was heated to maintain a temperature difference with the sphere, leading to near-field radiative transfer rates in the order of nW, which was measured by monitoring the deflection of the bimetallic cantilever. Their measurements confirmed that the near-field radiation between the flat surface and the microsphere was more than two orders of magnitude larger than between blackbodies, with a 1/d-dependence. Successful measurements between parallel plates have been carried out by Hu and coworkers [31]. They employed two precise optical glass flats spaced a fixed 1.6 μm apart by using polystyrene spacer beads. Applying various temperature differences they measured heat transfer rates approximately 35% higher than the blackbody limit, and observed good agreement with theoretical predictions. Ottens et al. [59] carried out high-precision heat transfer measurements between two sapphire plates spaced a variable distance as little as 2 μm apart. They also used cryogenic temperatures to emphasize near-field effects. Figure 24.9 shows the pertinent results of their experiments, compared with theoretical results from equation (24.29), displayed in the form of a heat transfer coefficient, i.e., hR = qnet /(T1 − T2 ). Agreement between theory and experiment is good, except for a slight systematic error, which may be due to imperfect flatness of the plates, as demonstrated by the dashed lines, which correspond to near-field radiative heat transfer between two convex plates, each having a radius of curvature of 1 km. Note that the highest heat transfer coefficient measured, 8.5 W/m2 K for the ΔT = 6.8 K case, exceeds the blackbody limit of σ(T14 − T24 )/(T1 − T2 ) 6.7 W/m2 K.

Nanoscale Radiative Transfer Chapter | 24 899

FIGURE 24.9 Heat transfer coefficients between sapphire plates separated by a vacuum microgap; curves are vertically offset by 2 W/m2 K, with respective zeros indicated by the horizontal lines extending from the left axis. Solid lines = predictions from equation (24.29); dashed lines = predictions for slightly convex plates.

Many modern-day applications require measurements at higher temperature—closer to room temperature as opposed to cryogenic temperatures. Since the peak wavelength of emission shifts to smaller wavelengths at higher temperature, smaller vacuum gaps need to be created to detect near-field radiation effects. One of the most challenging aspects of direct measurement of near-field effects at or near room temperature is spacing two flat surfaces at sub-wavelength separation. Initially, this technical difficulty was addressed by placing micro/nano spacers (slender solid structures) between the plates, e.g., in [31,60,61]. However, spacers allow conduction through them, and the data have to be post-processed to exclude conduction effects, which can result in other complications and uncertainties in the measured data. Another challenge is the precise control of the spacer thickness. In the past decade, significant progress has been made to conduct direct near-field measurements across extremely small vacuum gaps without using spacers. St-Gelais et al. [62,63] measured near-field thermal radiation for vacuum gaps ranging from 42 nm to 1.5 μm under temperature differences larger than 100 K using a micro-electromechanical actuator system for controlling the gap width. Ghashami et al. [64] made near-field radiation measurements between two single crystal quartz plates placed at distances as small as 200 nm and temperature differences as large as 156 K. Their measurements showed heat fluxes that are 40 times the blackbody limit. In recent years, advancements have also been made in the area of better spectral characterization, and in isolating the various contributions to the heat flux. Wang et al. [65] designed, built, and tested an emissometer facility that is capable of directly measuring polarization-dependent emittance in the infrared up to a temperature of 800 K. A modified Fabry-Perot cavity resonator was fabricated and tested with this setup. In another study, Wang and Zhang [66] measured and analyzed the effect of magnetic polaritrons in a metallic grating structure with a dielectric spacer on a metallic film using the same setup. The emittance of the structure was measured up to a temperature of 750 K.

24.9 Applications Research on near-field radiation has exploded in recent years due to its relevance for a number of important applications in energy conversion, building technology, and electronics. These include solar photovoltaic (PV) cells [67–70], solar thermophotovoltaic (TPV) cells [71–77], radiative cooling technology [44,78], and optoelectronics [52,79,80], among others. In Section 3.6, the effect of surface roughness on radiative (optical) properties was discussed. In that discussion, it was assumed that the size of the roughness features residing on the surface

900 Radiative Heat Transfer

is much smaller than the wavelength of radiation. However, when it comes to near-field radiation, additional length scales come into play, namely the spacing between the roughness features (since there may exist a temperature difference between them), and the spacing between the emitting and receiving surfaces, i.e., the two surfaces with an imposed temperature difference. In recent years, the effect of ordered roughness has been investigated computationally using the Finite Difference Time Domain method [81] by Didari and Mengüç [82]. They studied near-field radiative transfer between two parallel silicon carbide plates one of which had a periodic array of bumps (roughness features) of various shapes—rectangles, ellipses, and triangles. The flat surface served as the receiver, while the surface with bumps served as the emitter. Results showed that the best enhancement to the radiative heat flux (over the baseline case of both surfaces being flat) was attained with rectangular bumps of a certain aspect ratio and periodicity (spacing). The study also showed that the heat flux may be reduced (as opposed to being enhanced) under certain conditions. The change in the flux—either enhancement or reduction—was attributed to the change in the local density of electromagnetic states [83] for surfaces with bumps, which makes the electromagnetic field strongly multidimensional and nonuniform across the surface, leading to additional interference effects, either constructive or destructive. The effect of random surface roughness on near-field radiative transfer has also been investigated [84] recently. With the advent of nanofabrication techniques, it is now possible to fabricate surfaces with ordered or patterned structures on them, including multiple layers of thin films, that are comparable in size to the wavelength. These nanostructured materials are often referred to as metamaterials. Development of such periodic structures, commonly referred to as gratings, is motivated by the goal to tailor the radiative property of the surface to suit a particular application. For example, for passive radiative cooling applications for buildings [78], one might desire to have selectively strong emission in the atmospheric transmission window for outgoing radiation (8–13 μm). One of the most heavily researched applications, today, of near-field radiation is solar TPV systems. Originally, TPV systems were explored as a means to utilize the “thermal” or infrared part of solar radiation, which solar cells typically underutilize. In TPV systems, concentrated sunlight is first used to heat an absorber plate. The back side of the absorber plate, which typically is designed to be an efficient emitter, then re-emits radiation (but at a temperature much lower than the sun) toward the solar PV cells that face the emitter. The efficiency of a TPV system, therefore, hinges on the efficiency of two separate processes [85]: the efficiency of sunlight absorption, and the efficiency of coupling between the emitter and the solar cell. Until recently, the latter efficiency was believed to be limited by blackbody (Planckian) radiation since the emitter and solar cells were placed at distances (typically more than several hundred μm) wherein near-field radiation is negligible. With theoretical revelation that near-field radiation fluxes can far exceed the Planckian limit, and recent experimental confirmation of the theory in prototype solar TPV systems [86], research on near-field radiative transfer in solar TPV systems has taken centerstage. A large number of strategies have been explored—mostly computationally— to tailor the near-field emission spectra of emitters and match the bandgap of the solar cells. These strategies aim to trigger one or more interactions or couplings between surface plasmon polaritons (SPPs), surface phonon polaritrons (SPhPs), magnetic polaritrons (MPs), and exploit various resonance effects, such as cavity resonance, Woody’s anomaly, and Fabry-Perot resonance, amongst others. A detailed discussion of these investigations is beyond the scope of this book. Broadly, the strategies may be classified into two categories: use of novel materials [73,76,77] and use of novel structures, such as nanowire [75], nanoholes [51], various grating shapes and configurations [71,87,88], or a combination thereof. Some studies have also considered using gratings and other nanostructures to enhance absorber efficiency [73,76,89,90].

Problems 24.1 Show that the transmissivity of a thin dielectric film, surrounded by two identical, but different dielectrics, is described by equation (24.13) for incidence angles θ1 > θc . Solve the problem separately for both TM and TE waves. 24.2 Consider an interface in the x-y-plane at z = 0 between two dielectrics (n1 , z < 0 and n2 < n1 , z > 0), and determine the z-component of the Poynting vector in Medium 2 for incidence in Medium 1 at angles exceeding the critical angle. Show that the time average of the Poynting vector is zero.

References [1] S.M. Rytov, Correlation theory of thermal fluctuations in an isotropic medium, Soviet Physics, JETP 6 (1) (1958) 130–140.

Nanoscale Radiative Transfer Chapter | 24 901

[2] S.M. Rytov, Y.A. Kravtsov, V.I. Tatarskii, Principles of Statistical Radiophysics III: Elements of Random Fields, Springer Verlag, Berlin, 1987. [3] G. Chen, Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons, Oxford University Press, New York, 2005. [4] L. Novotny, B. Hecht, Principles of Nano-Optics, Cambridge University Press, New York, 2006. [5] Z.M. Zhang, Nano/Microscale Heat Transfer, second ed., Springer, 2020. [6] S. Basu, Z.H. Zhang, C.J. Fu, Review of near-field thermal radiation and its application to energy conversion, International Journal of Energy Research 33 (2009) 1203–1232. [7] Z.H. Zhang, K. Park, Fundamentals and applications of near-field radiative energy transfer, in: Thermal-FluidsPedia, Thermal-Fluids Central, 2012, https://www.thermalfluidscentral.org/encyclopedia/index.php/Near-field_thermal_radiation. [8] X.L. Liu, L.P. Wang, Z.M. Zhang, Near-field thermal radiation: recent progress and outlook, Nanoscale and Microscale Thermophysical Engineering 19 (2015) 98–126. [9] B. Song, A. Fiorino, E. Meyhofer, P. Reddy, Near-field radiative thermal transport: from theory to experiment, AIP Advances 5 (2015) 053503. [10] S. Edalatpour, J. DeSutter, M. Francoeur, Near-field thermal electromagnetic transport: an overview, Journal of Quantitative Spectroscopy and Radiative Transfer 178 (2016) 14–21. [11] Z.M. Zhang, Nano/Microscale Heat Transfer, McGraw-Hill, New York, 2007. [12] R. Carminati, J.-J. Greffet, Near-field effect in spatial coherence of thermal sources, Physics Review Letters 82 (8) (1999) 1660–1663. [13] E.G. Cravalho, C.L. Tien, R.P. Caren, Effect of small spacings on radiative transfer between two dielectrics, ASME Journal of Heat Transfer 89 (1967) 351–358. [14] B. Zhao, Z.M. Zhang, Enhanced photon tunneling by surface plasmon–phonon polaritons in graphene/hBN heterostructures, ASME Journal of Heat Transfer 139 (2017) 022701. [15] S. Kawata (Ed.), Near-Field Optics and Surface Plasmon Polaritons, Springer, Berlin, 2001. [16] J. Tominaga, D.P. Tsai (Eds.), Optical Nanotechnologies — The Manipulation of Surface and Local Plasmons, Springer, Berlin, 2003. [17] J. Homola, S.S. Yee, G. Gauglitz, Surface plasmon resonance sensors: review, Sensors and Actuators B 54 (1999) 3–15. [18] J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, Y. Chen, Coherent emission of light by thermal sources, Nature 416 (2002) 61–64. [19] R. Hillenbrand, T. Taubner, F. Kellmann, Phonon-enhanced light–matter interaction at the nanometer scale, Nature 418 (2002) 159–162. [20] H. Raether (Ed.), Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer, Berlin, 1988. [21] R. Rupin, Surface polaritons of a left-handed medium, Physics Letters A 277 (2000) 61–64. [22] C.J. Fu, Z.H. Zhang, Nanoscale radiation heat transfer for silicon at different doping levels, International Journal of Heat and Mass Transfer 49 (2006) 1703–1718. [23] A. Narayanaswamy, Y. Zheng, A Green’s function formalism of energy and momentum transfer in fluctuational electrodynamics, Journal of Quantitative Spectroscopy and Radiative Transfer 132 (2014) 12–21. [24] J.-P. Mulet, K. Joulain, R. Carminati, J.-J. Greffet, Nanoscale radiative heat transfer between a small particle and a plane surface, Applied Physics Letters 78 (2001) 2931–2933. [25] J.-P. Mulet, K. Joulain, R. Carminati, J.-J. Greffet, Enhanced radiative heat transfer at nanometric distances, Microscale Thermophysical Engineering 6 (2002) 209–222. [26] Y. Tsurimaki, P.-O. Chapuis, J. Okajima, A. Komiya, S. Maruyama, R. Vaillon, Coherent regime and far-to-near-field transition for radiative heat transfer, Journal of Quantitative Spectroscopy and Radiative Transfer 187 (2017) 310–321. [27] A. Narayanaswamy, J. Mayo, Minimum radiative heat transfer between two metallic half-spaces due to propagating waves, Journal of Quantitative Spectroscopy and Radiative Transfer 184 (2016) 254–261. [28] E.G. Cravalho, C.L. Tien, R.P. Caren, Effect of small spacings on radiative transfer between two dielectrics, ASME Journal of Heat Transfer 89 (1967) 351–358. [29] R.F. Boehm, C.L. Tien, Small spacing analysis of radiative transfer between parallel metallic surfaces, ASME Journal of Heat Transfer 92 (1970) 412–417. [30] A. Narayanaswamy, G. Chen, Thermal radiation in 1D photonic crystals, Journal of Quantitative Spectroscopy and Radiative Transfer 93 (1–3) (2005) 175–183. [31] L. Hu, A. Narayanaswamy, X. Chen, G. Chen, Near-field thermal radiation between two closely spaced glass plates exceeding Planck’s blackbody radiation law, Applied Physics Letters 92 (2008) 133106. [32] A. Narayanaswamy, S. Shen, L. Hu, X. Chen, G. Chen, Breakdown of the Planck blackbody radiation law at nanoscale gaps, Applied Physics A 96 (2009) 357–362. [33] S. Basu, B.J. Lee, Z.M. Zhang, Near-field radiation calculated with an improved dielectric function model for doped silicon, ASME Journal of Heat Transfer 132 (2) (2010) 023302. [34] E. Rousseau, M. Laroche, J.-J. Greffet, Radiative heat transfer at nanoscale: closed-form expression for silicon at different doping levels, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (7–8) (2010) 1005–1014. [35] A. Narayanaswamy, S. Shen, G. Chen, Near-field radiative heat transfer between a sphere and a substrate, Physical Review B 72 (2008) 115303.

902 Radiative Heat Transfer

[36] E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, J.-J. Greffet, Radiative heat transfer at the nanoscale, Nature Photonics 3 (2009) 514–517. [37] S. Shen, A. Narayanaswamy, G. Chen, Surface phonon polaritons mediated energy transfer between nanoscale gaps, Nano Letters 9 (2009) 2909–2913. [38] A.I. Volokitin, B.N.J. Persson, Radiative heat transfer between nanostructures, Physical Review B 63 (2001) 205404. [39] A.I. Volokitin, B.N.J. Persson, Resonant photon tunneling enhancement of the radiative heat transfer, Physical Review B 69 (2004) 045417. [40] G. Domingues, S. Volz, K. Joulain, J.-J. Greffet, Heat transfer between two nanoparticles through near field interaction, Physical Review Letters 94 (2005) 085901. [41] A. Narayanaswamy, G. Chen, Thermal near-field radiative transfer between two spheres, Physical Review B 77 (2005) 075125. [42] J. Dong, J.M. Zhao, L.H. Liu, Near-field radiative heat transfer between clusters of dielectric nanoparticles, Journal of Quantitative Spectroscopy and Radiative Transfer 197 (2017) 114–122. [43] Z.M. Zhang, E.T. Enikov, T. Makansi, Near-field radiative transfer between heavily doped SiGe at elevated temperatures, ASME Journal of Heat Transfer 134 (2012) 092702. [44] Z.-X. Jia, Y. Shuai, H.P. Tan, Radiative flux control via graphene-based spectrum tailoring, International Journal of Heat and Mass Transfer 107 (2017) 729–735. [45] J.-Y. Chang, Y. Yang, L. Wang, Enhanced energy transfer by near-field coupling of a nanostructured metamaterial with a graphenecovered plate, Journal of Quantitative Spectroscopy and Radiative Transfer 184 (2016) 58–67. [46] A. Sisto, X. Ruan, T.S. Fisher, First principles and finite element predictions of radiative properties of nanostructure arrays: single-walled carbon nanotube arrays, ASME Journal of Heat Transfer 136 (2014) 062702. [47] R.Z. Zhang, X. Liu, Z.M. Zhang, Modeling the optical and radiative properties of vertically aligned carbon nanotubes in the infrared region, ASME Journal of Heat Transfer 137 (2015) 091009. [48] L.-Y. Zhong, Q.-M. Zhao, T.-B. Wang, T.-B. Yu, Q.-H. Liao, N. Liu, Near-field radiative heat transfer between graphene/silicon carbide multilayers, ASME Journal of Heat Transfer 140 (2018) 072701. [49] Y. Chen, Z. Zheng, The influence of the two-dimensional sinusoidal gratings on the near-field radiative heat flux between two doped silicon films, International Journal of Heat and Mass Transfer 125 (2018) 589–595. [50] M. Lim, S.S. Lee, B.J. Lee, Near-field thermal radiation between doped silicon plates at nanoscale gaps, Physical Review B 91 (2015) 195136. [51] X.L. Liu, R.Z. Zhang, Z.M. Zhang, Near-field radiative heat transfer with doped-silicon nanostructured metamaterials, International Journal of Heat and Mass Transfer 73 (2014) 389–398. [52] Y. Zhang, Y. Xuan, Spectral features of an omnidirectional narrowband emitter, ASME Journal of Heat Transfer 134 (2012) 102701. [53] X.L. Liu, T.J. Bright, Z.M. Zhang, Application conditions of effective medium theory in near-field radiative heat transfer between multilayered metamaterials, ASME Journal of Heat Transfer 136 (2014) 092703. [54] G.A. Domoto, R.F. Boehm, C.L. Tien, Experimental investigation of radiative transfer between metallic surfaces at cryogenic temperatures, ASME Journal of Heat Transfer 92 (1970) 405–411. [55] C.M. Hargreaves, Anomalous radiative transfer between closely-spaced bodies, Physics Letters A 30 (1969) 491–492. [56] C.M. Hargreaves, Radiative transfer between closely spaced bodies, Technical Report 5, Philips Research Report, 1973. [57] J.B. Xu, K. Lauger, R. Moller, K. Dransfeld, I.H. Wilson, Heat transfer between two metallic surfaces at small distances, Journal of Applied Physics 76 (1994) 7209–7216. [58] A. Kittel, W. Müller-Hirsch, J. Parisi, S. Biehs, D. Reddig, M. Holthaus, Near-field heat transfer in a scanning thermal microscope, Physical Review Letters 95 (2005) 224301. [59] R.S. Ottens, V. Quetschke, S. Wise, A.A. Alemi, R. Lundock, G. Mueller, D.H. Reitze, D.B. Tanner, B.F. Whiting, Near-field radiative heat transfer between macroscopic planar surfaces, Physical Review Letters 107 (1) (2011) 014301. [60] K. Ito, A. Miura, H. Iizuka, H. Toshiyoshi, Parallel-plate submicron gap formed by micromachined low-density pillars for near-field radiative heat transfer, Applied Physics Letters 106 (2015) 083504. [61] M.P. Bernardi, D. Milovich, M. Francoeur, Radiative heat transfer exceeding the blackbody limit between macroscale planar surfaces separated by a nanosize vacuum gap, Nature Communications 7 (2016) 12900. [62] R. St-Gelais, B. Guha, L. Zhu, S. Fan, M. Lipson, Demonstration of strong near-field radiative heat transfer between integrated nanostructures, Nano Letters 14 (2014) 6971–6975. [63] R. St-Gelais, L. Zhu, S. Fan, M. Lipson, Near-field radiative heat transfer between parallel structures in the deep subwavelength regime, Nature Nanotechnology 11 (2016) 515–519. [64] M. Ghashami, H. Geng, T. Kim, N. Iacopino, S.K. Cho, K. Park, Precision measurement of phonon-polaritonic near-field energy transfer between macroscale planar structures under large thermal gradients, Physical Review Letters 120 (2018) 175901. [65] L. Wang, S. Basu, Z. Zhang, Direct measurement of thermal emission from a Fabry-Perot cavity resonator, ASME Journal of Heat Transfer 134 (2012) 072701. [66] L.P. Wang, Z.M. Zhang, Measurement of coherent thermal emission due to magnetic polaritons in subwavelength microstructures, ASME Journal of Heat Transfer 135 (2013) 091505. [67] C. Rockstuhl, S. Fahr, F. Lederer, Absorption enhancement in solar cells by localized plasmon polaritons, Journal of Applied Physics 104 (2008) 123102.

Nanoscale Radiative Transfer Chapter | 24 903

[68] F. Beck, A. Polman, K. Catchpole, Tunable light trapping for solar cells using localized surface plasmons, Journal of Applied Physics 105 (2009) 114310. [69] W. Wang, S. Wu, K. Reinhardt, Y. Lu, S. Chen, Broadband light absorption enhancement in thin-film silicon solar cells, Nano Letters 10 (2010) 2012–2018. [70] S. Hajimirza, G. El Hitti, A. Heltzel, J.R. Howell, Specification of micro-nanoscale radiative patterns using inverse analysis for increasing solar panel efficiency, ASME Journal of Heat Transfer 134 (2012) 102702. [71] B. Zhao, L. Wang, Y. Shuai, Z.M. Zhang, Thermophotovoltaic emitters based on a two-dimensional grating/thin-film nanostructure, International Journal of Heat and Mass Transfer 67 (2013) 637–645. [72] T.J. Bright, L.P. Wang, Z.M. Zhang, Performance of near-field thermophotovoltaic cells enhanced with a backside reflector, ASME Journal of Heat Transfer 136 (2014) 062701. [73] Y. Yang, J.-Y. Chang, P. Sabbaghi, L. Wang, Performance analysis of a near-field thermophotovoltaic device with a metallodielectric selective emitter and electrical contacts for the photovoltaic cell, ASME Journal of Heat Transfer 139 (2017) 052701. [74] J.I. Watjen, X.L. Liu, B. Zhao, Z.M. Zhang, A computational simulation of using tungsten gratings in near-field thermophotovoltaic devices, ASME Journal of Heat Transfer 139 (2017) 052704. [75] J.-Y. Chang, Y. Yang, L. Wang, Tungsten nanowire based hyperbolic metamaterial emitters for near-field thermophotovoltaic applications, International Journal of Heat and Mass Transfer 87 (2015) 237–247. [76] H. Wang, J.-Y. Chang, Y. Yang, L. Wang, Performance analysis of solar thermophotovoltaic conversion enhanced by selective metamaterial absorbers and emitters, International Journal of Heat and Mass Transfer 98 (2016) 788–798. [77] N. Vongsoasup, M. Francoeur, K. Hanamura, Performance analysis of near-field thermophotovoltaic system with 2D grating tungsten radiator, International Journal of Heat and Mass Transfer 115 (2017) 326–332. [78] Z. Chen, L. Zhu, A. Raman, S. Fan, Radiative cooling to deep sub-freezing temperatures through a 24-h day-night cycle, Nature Communications 7 (2016) 13729. [79] W.-J. Wang, C.-J. Fu, W.-C. Tan, Thermal radiative properties of a SiC grating on a photonic crystal, ASME Journal of Heat Transfer 135 (2013) 091504. [80] W.-J. Wang, Y. Zhao, W.-C. Tan, C.-J. Fu, Thermal radiative properties of a two-dimensional silicon carbide grating mediated with a photonic crystal, ASME Journal of Heat Transfer 137 (2015) 091022. [81] A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, third ed., Artech House, 2005. [82] A. Didari, M.P. Mengüç, Near-field thermal emission between corrugated surfaces separated by nano-gaps, Journal of Quantitative Spectroscopy and Radiative Transfer 158 (2015) 43–51. [83] M. Francoeur, M.P. Mengüç, R. Vaillon, Local density of electromagnetic states within a nanometric gap formed between two thin films supporting surface phonon polaritons, Journal of Applied Physics 107 (2010) 034313. [84] Y. Chen, Y. Xuan, The influence of surface roughness on nanoscale radiative heat flux between two objects, Journal of Quantitative Spectroscopy and Radiative Transfer 158 (2015) 52–60. [85] Y. Wang, H. Liu, J. Zhu, Solar thermophotovoltaics: progress, challenges, and opportunities, APL Materials 7 (2019) 080906. [86] A. Fiorino, L. Zhu, D. Thompson, R. Mittapally, P. Reddy, E. Mayhofer, Nanogap near-field thermophotovoltaics, Nature Nanotechnology 13 (2018) 806–811. [87] Y. Yang, P. Sabbaghi, L. Wang, Effect of magnetic polaritons in SiC deep gratings on near-field radiative transfer, International Journal of Heat and Mass Transfer 108 (2017) 851–859. [88] K. Isobe, D. Hirashima, K. Hanamura, Spectrally enhanced near-field radiation transfer using nanometer-sized pillar array structured surfaces, International Journal of Heat and Mass Transfer 115 (2017) 467–473. [89] B.J. Lee, Y.B. Chen, S. Han, F.C. Chiu, H.J. Lee, Wavelength-selective solar thermal absorber with two-dimensional nickel gratings, ASME Journal of Heat Transfer 136 (2014) 072702. [90] X. Fang, M. Lou, H. Bao, C.Y. Zhao, Thin films with disordered nanohole patterns for solar radiation absorbers, Journal of Quantitative Spectroscopy and Radiative Transfer 158 (2015) 145–153.

Appendix A

Constants and Conversion Factors TABLE A.1 Physical constants. Speed of light in vacuum

c0

= 2.9979×108 m/s

First Planck function constant

C1

= 3.7418×10−16 W m2 = 2πhc20

Second Planck function constant

C2

= 14,388 μm K = hc 0 /k

Wien’s constant

C3

= 2897.8 μm K

Electron charge

e

= 1.6022×10−19 C

Planck’s constant

h

= 6.6261×10−34 J s

Modified Planck’s constant



= 1.0546×10−34 J s = h/2π

Boltzmann’s constant

k

= 1.3807×10−23 J/K

Electron rest mass

me

= 9.1094×10−31 kg

Neutron rest mass

mn

= 1.6749×10−27 kg

Proton rest mass

mp

= 1.6726×10−27 kg

Avogadro’s number

NA

= 6.0221×1023 molecules/mol

Solar constant (at mean RSE )

qsol

= 1367 W/m2

Radius of Earth (mean)

REarth

= 6.371×106 m

Radius of solar disk

Rsun

= 6.955×108 m

Earth–sun distance (mean)

RSE

= 1.4960×1011 m

Universal gas constant

Ru

= 8.3145 J/mol K

Effective surface T of sun

Tsun

= 5777 K

Molar volume of ideal gas

Vmol

= 22.4140 /mol = 22.4140 m3 /kmol

(at 273.15 K, 101.325 kPa) Electrical permittivity of vacuum

0

= 8.8542×10−12 C2 /N m2

Magnetic permeability of vacuum

μ0

= 4π×10−7 N s2 /C2

Stefan–Boltzmann constant

σ

= 5.6704×10−8 W/m2 K4

905

906 Constants and Conversion Factors

TABLE A.2 Conversion factors. Acceleration

1 m/s2

= 4.2520×107 ft/h2

Area

1 m2

= 1550.0 in2 = 10.764 ft2

Diffusivity

1 m2 /s

= 3.875×104 ft2 /h

Energy

1J

= 9.4787×10−4 Btu 1.6022×10−19

Force

1 eV = 1N

Heat transfer rate

1W

Heat flux

1 W/m2

Heat generation rate

1 W/m

= 1.5187×10−22 Btu = 0.22481 lb f

J

= 3.4123 Btu/h = 0.3171 Btu/h ft2

3

= 0.09665 Btu/h ft3

2

Heat transfer coefficient

1 W/m K

= 0.17612 Btu/h ft2 ◦ F

Intensity

1 W/m2 sr

= 0.3171 Btu/h ft2 sr

Kinematic viscosity

1 m2 /s

= 3.875×104 ft2 /h

Latent heat

1 J/kg

= 4.2995×10−4 Btu/lbm

Length Mass

1m 1 km 1 kg

= 39.370 in = 3.2808 ft = 0.62137 mi = 2.2046 lbm

Mass density

1 kg/m3

= 0.062428 lbm /ft3

Mass flow rate Power

1 kg/s 1W

= 7936.6 lbm /h = 3.4123 Btu/h

Pressure and stress

1 Pa = 1 N/m2

= 1.4504×10−4 lb f /in2

1.0133×105 N/m2

= 1 standard atmosphere

Specific heat

1 J/kg K

= 2.3886×10−4 Btu/lbm ◦ F

Temperature

T(K)

Temperature difference Thermal conductivity Thermal resistance

1K 1 W/m K 1 K/W

= (5/9)T(◦ R) = (5/9)(T(◦ F) + 459.67) = T(◦ C) + 273.15 = 1◦ C = (9/5)◦ R = (9/5)◦ F = 0.57782 Btu/h ft ◦ F = 0.52750 ◦ F h/Btu

Velocity and speed

1 m/s

= 3.2808 ft/s = 2.2364 mph

Viscosity (dynamic)

1 N s/m2 = 1 kg/s m

= 2419.1 lbm /ft h

Volume

1 m3

= 6.1023×104 in3 = 35.314 ft3

Volume flow rate

= 1.2713×105 ft3 /h

1 m3 /s

= 2.1189×103 ft3 /min

TABLE A.3 Conversion factors for spectral variables. Wavelength to energy

a μm = a × 103 nm

=ˆ 1.240/a eV

to frequency

a μm =

to wavenumber

a μm

=ˆ 104 /a cm−1

Energy to frequency to wavelength

a eV a eV

=ˆ 2.418×1014 a Hz =ˆ 1.240/a μm

to wavenumber

a eV

=ˆ 8.066×103 a cm−1

a cm−1

=ˆ 1.240×10−4 a eV

to frequency

a cm−1

=ˆ 2.9979×1010 a Hz

to wavelength

a cm−1

=ˆ 10+4 /a μm

a Hz

=ˆ 4.136×10−15 a eV

to wavelength

a Hz

=ˆ 2.9979×1014 /a μm

to wavenumber

a Hz

=ˆ 3.336×10−11 a cm−1

Wavenumber to energy

Frequency to energy

a × 104

Å

=ˆ 2.9979×1014 /a Hz

Appendix B

Tables for Radiative Properties of Opaque Surfaces In this appendix, tables of total normal emittances, as well as a number of total normal solar absorptances, are given. The data have been collected from several surveys [1–8] that, in turn, have assembled their data from a multitude of references dating back all the way into the 1920s. As seen from the tables, there can sometimes be considerable differences in total emittance for ostensibly the same material, as reported by different researchers. While these discrepancies are partially due to varying accuracy, the primary reason is, as outlined in Chapter 3, the fact that surface layers, surface roughness, oxidation, etc., strongly affect the emittance of materials. Therefore, it should be realized that the total normal emittance or absorptance of a given surface may, in actuality, differ considerably from these reported values. In estimating the total hemispherical emittance from total normal data, one should keep in mind that: 1. Materials with high emittance tend to behave like dielectrics, resulting in a hemispherical emittance that is 3% to 5% smaller than the normal one (cf. Fig. 3.19). 2. Materials with low emittance tend to behave like metals, resulting in hemispherical emittances that may be up to 25% larger than normal ones (cf. Fig. 3.10).

907

908 Tables for Radiative Properties of Opaque Surfaces

TABLE B.1 Total emittance and solar absorptance of selected surfaces (compiled by Edwards et al. [1]). Temperature [◦ C] Alumina, flame-sprayed Aluminum foil, as received Bright dipped

Total normal emittance

Extraterrestrial solar absorptance

−25

0.80

0.28

20

0.04

20

0.025

0.10

Aluminum, vacuum-deposited on mylar

20

0.025

0.10

Aluminum alloy 6061, as received

20

0.03

0.37

Aluminum alloy 75S-T6, weathered 65

0.16

0.54

Aluminum, hard-anodized, 6061-T6

20,000 h on a DC6 aircraft

−25

0.84

0.92

Aluminum, soft-anodized, Reflectal alloy

−25

0.79

0.23

Aluminum, 7075-T6, sandblasted with 60 mesh silicon carbide grit

20

0.30

0.55

Aluminized silicone resin paint

95

0.20

0.27

425

0.22

150

0.18

370

0.21

600

0.30

150

0.90

370

0.88

Dow Corning XP-310 Beryllium

Beryllium, anodized

Black paint, Parson’s optical black

0.77

600

0.82

−25

0.95

0.975

0.93

0.94

−25

0.89

0.95

95

0.81

Black silicone, high-heat National Lead Co. 46H47

−25 to 750

Black epoxy paint, Cat-a-lac Finch Paint and Chem. Co. 463-1-8 Black enamel paint, Rinshed-Mason Heated 1000 h at 375◦ C in air Chromium plate Heated 50 h at 600◦ C

425

0.80

95

0.12

400

0.15

35

0.15

0.78

20

0.03

0.47

35

0.16

0.91

Aluminized

−25

0.83

0.13

Silvered

−25

0.83

0.13

95

0.09

Copper, electroplated Black-oxidized in Ebonol C Glass, second surface mirror

Gold, coated on stainless steel Heated in air at 540◦ C Coated on 3 M tape Y9814

400

0.14

20

0.025

0.21

Graphite, crushed on sodium silicate

−25

0.91

0.96

Inconel X, oxidized 4 h at 1000◦ C

−25

0.71

0.90

95

0.81

425

0.79

95

0.07

Oxidized 10 h at 700◦ C Magnesium–thorium alloy Magnesium, Dow 7 coating

260

0.06

370

0.36 continued on next page

Tables for Radiative Properties of Opaque Surfaces 909

TABLE B.1 (continued) Temperature [◦ C]

Total normal emittance

Extraterrestrial solar absorptance

Mylar film, aluminized on second surface 0.0625 mm thick

20

0.37

0.17

0.025 mm thick

20

0.63

0.17

0.075 mm thick

20

0.81

0.24

Nickel, electroplated

20

0.03

0.22

110-30

35

0.05

0.85

125-30

35

0.11

0.85

Nickel, electro-oxidized on copper

Platinum-coated stainless steel Annealed in air 300 h at 375◦ C

95

0.13

400

0.15

95

0.11

425

0.13

Silica, Corning Glass 7940M Sintered, powdered, fused silica Silica, second surface mirror, aluminized Silvered

35

0.84

0.08

20

0.83

0.14

20

0.83

0.07

35

0.32

0.94

Silicon solar cell, boron-doped, no coverglass Silver, plated on nickel on stainless steel Heated 300 h at 375◦ C Silver Chromatone paint

95

0.06

400

0.08

95

0.11

425

0.13

20

0.24

95

0.27

0.20

Stainless steel Type 312, heated 300 h at 260◦ C Type 301 with Armco black oxide Type 410, heated to 700◦ C in air

425

0.32

−25

0.75

0.89

35

0.13

0.76

95

0.42

0.68

95

0.10

425

0.19

Type 303, sandblasted heavily with 80 mesh aluminum oxide grit Titanium, 75A 75A, oxidized 300 h at

450◦ C

35

0.21

425

0.25

C-110M, oxidized 100 h at 425◦ C in air

35

0.16

0.52

C-110M, oxidized 300 h at 450◦ C in air

35

0.20

0.77

Evaporated 80–100 μm, oxidized 3 h at

400◦ C

Anodized White acrylic resin paint Sherwin-Williams M49WC8-CA-10144

0.80

35

0.14

0.75

−25

0.73

0.51

95

0.92

200

0.87

White epoxy paint, Cat-a-lac Finch −25

0.88

0.25

White potassium zirconium silicate coating

20

0.89

0.13

Zinc, blackened by electrochemical treatment

35

0.12

0.89

Paint and Chemical Co. 483-1-8

910 Tables for Radiative Properties of Opaque Surfaces

TABLE B.2 Total normal emittance of various surfaces. Temperaturea [◦ C]

Total normal emittancea

A. Metals and their oxides Aluminum Highly polished plate, 98.3% pure

225–575

0.039–0.057

Commercial sheet

100

0.09

Rough polish

100

0.18

Rough plate

40

0.055–0.07

Oxidized at 600◦ C

200–600

0.11–0.19

Heavily oxidized

95–500

0.20–0.31

Aluminum oxide

275–500

0.63–0.42

500–825

0.42–0.26

40

0.216

Al-surfaced roofing Aluminum alloysb Alloy 75 ST: A, B1 , C Alloy 75 ST: Ac Alloy 75 ST: B1

c

Alloy 75 ST: Cc Alloy 24 ST: A, B1 , C

25

0.11, 0.10, 0.08

230–480

0.22–0.16

230–425

0.20–0.18

230–500

0.22–0.15

25

0.09

Alloy 24 ST: Ac

230–485

0.17–0.15

Alloy 24 ST: B1 c

230–505

0.20–0.16

Alloy 24 ST: Cc

230–460

0.16–0.13

Copper

200–600

0.18–1.19

Steel

Calorized surfaces, heated at 600◦ C 200–600

0.52–0.57

Antimony, polished

35–260

0.28–0.31

Beryllium, polished

1000–1200

0.37

75

0.34

73.2% Cu, 26.7% Zn

245–355

0.028–0.031

62.4% Cu, 36.8% Zn, 0.4% Pb, 0.3% Al

255–375

0.033–0.037

275

0.030

100

0.06

40–315

0.10

Bismuth, bright Brass Highly polished

82.9% Cu, 17.0% Zn Polished Rolled plate, natural surface

22

0.06

Rolled plate, rubbed with coarse emery

22

0.20

Dull plate

50–350

0.22

Oxidized by heating at 600◦ C

200–600

0.61–0.59

40–1100

0.08–0.36

Carefully polished electrolytic copper

80

0.018

Polished

115

0.023

Chromium, polished Copper

100

0.052

Commercial emeried, polished, pits remaining

19

0.030

Commercial, scraped shiny, not mirror-like

22

0.072

Plate heated long time, with thick oxide layer

25

0.78

Plate heated at

600◦ C

200–600

0.57

Cuprous oxide

800–1100

0.66–0.54

Molten copper

1075–1275

0.16–0.13 continued on next page

Tables for Radiative Properties of Opaque Surfaces 911

TABLE B.2 (continued) Dow

A; B1 ; C Ac c

B1

Temperaturea [◦ C]

Total normal emittancea

25

0.15, 0.15, 0.12

230–400

0.24–0.20

metal:b

Cc Germanium, polished Gold, pure, highly polished Hafnium, polished

230–425

0.16

230–405

0.21–0.18

800

0.55

225–625

0.018–0.035

1400

0.45

Inconel:b Types X and B: surface A, B2 , C

25

0.19–0.21

Type X: surface Ac

230–880

0.55–0.78

Type X: surface B2 c

230–855

0.60–0.75

Type X: surface Cc

230–900

0.62–0.73

Type B: surface Ac

230–880

0.35–0.55

Type B: surface B2

c

Type B: surface Cc

230–950

0.32–0.51

230–1000

0.35–0.40

175–225

0.052–0.064

Iron and steel (not including stainless) Metallic surfaces (or very thin oxide layer) Electrolytic iron, highly polished Steel, polished

100

0.066

Iron, polished

425–1025

0.14–0.38

Iron, roughly polished

100

0.17

Iron, freshly emeried

20

0.24

Cast iron, polished

200

0.21

Cast iron, newly turned

22

0.44

880–990

0.60–0.70

Cast iron, turned and heated Wrought iron, highly polished

40–250

0.28

Polished steel casting

770–1035

0.52–0.56

Ground sheet steel

935–1100

0.55–0.61

Smooth sheet iron

900–1040

0.55–0.60

Mild

steelb :

25

0.12, 0.15, 0.10

Mild steelb : Ac

230–1065

0.20–0.32

Mild steelb : B2 c

230–1050

0.34–0.35

230–1065

0.27–0.31

20

0.61

Mild

steelb :

A, B2 , C

Cc

Oxidized surfaces Iron plate, pickled, then rusted red Iron plate, completely rusted

20

0.69

Iron, dark gray surface

100

0.31

Rolled sheet steel

21

0.66

Oxidized iron

100

0.74

200–600

0.64–0.78

Cast iron, oxidized at 600◦ C Steel, oxidized at 600◦ C

200–600

0.79

Smooth, oxidized electrolytic iron

125–525

0.78–0.82

Iron oxide

500–1200

0.85–0.89

Rough ingot iron

925–1115

0.87–0.95

25

0.80

25

0.82 continued on next page

Sheet steel, strong, rough oxide layer Dense, shiny oxide layer

912 Tables for Radiative Properties of Opaque Surfaces

TABLE B.2 (continued) Temperaturea [◦ C]

Total normal emittancea

Cast plate, smooth

23

0.80

Cast plate, rough

23

0.82

Cast iron, rough, strongly oxidized

40–250

0.95

Wrought iron, dull oxidized

20–360

0.94

Steel plate, rough

40–370

0.94–0.97

Cast iron

1300–1400

0.29

Mild steel

1600–1800

0.28

1560–1710

0.27–0.39

1500–1650

0.42–0.53

1520–1650

0.43–0.40

Pure iron

1515–1770

0.42–0.45

Armco iron

1520–1690

0.40–0.41

125–225

0.057–0.075

Gray oxidized

25

0.28

Oxidized at 150◦ C

200

0.63

275–825

0.55–0.20

Molten surfaces

Steel, several different kinds with 0.25– 1.2% C (slightly oxidized surface) Steel

Lead Pure (99.96%), unoxidized

Magnesium Magnesium oxide Magnesium, polished Mercury

900–1705

0.20

35–260

0.07–0.13

0–100

0.09–0.12

725–2595

0.096–0.202

Molybdenum Filament Massive, polished Polished

100

0.071

35–260

0.05–0.08

540–1370

0.10–0.18

2750

0.29

Monel metalb Oxidized at 600◦ C

200–600

0.41–0.46

25

0.23, 0.17, 0.14

K Monel 5700: Ac

230–875

0.46–0.65

K Monel 5700: B2 c

230–955

0.54–0.77

230–975

0.35–0.53

K Monel 5700: A, B2 , C

K Monel 5700:

Cc

Nickel Electroplated, polished

23

0.045

225–375

0.07–0.087

Polished

100

0.072

Electroplated, not polished

20

0.11

Wire

185–1005

0.096–0.186

Plate, oxidized by heating at 600◦ C

200–600

0.37–0.48

Nickel oxide

650–1255

0.59–0.86

50–1035

0.64–0.76

100

0.059

Technically pure (98.9% Ni, + Mn), polished

Nickel alloys Chromnickel Copper–nickel, polished Nichrome wire, bright

50–1000

0.65–0.79 continued on next page

Tables for Radiative Properties of Opaque Surfaces 913

TABLE B.2 (continued) Temperaturea [◦ C]

Total normal emittancea

50–500

0.95–0.98

100

0.135

20

0.262

270–560

0.89–0.82

Pure, polished plate

225–625

0.054–0.104

Strip

925–1625

0.12–0.17

Filament

27–1225

0.036–0.192

Wire

225–1375

0.073–0.182

Polished, pure

225–625

0.020–0.032

Polished

40–370

0.022–0.031

100

0.052

Nichrome wire, oxidized Nickel–silver, polished Nickelin (18–32% Ni; 55–68% Cu; 20% Zn), gray oxidized Type ACI-HW (60% Ni; 12% Cr), smooth, black, firm adhesive oxide coat from service Platinum

Silver

Stainless

steelb

Polished

100

0.074

Type 301: A, B2 , C

25

0.21, 0.27, 0.16

230–950

0.57–0.55

230–940

0.54–0.63

Type 301: Ac Type 301: B2

c

Type 301: Cc Type 316: A, B2 , C Type 316: Ac Type 316: B2

c

Type 316: Cc Type 347: A, B2 , C Type 347: Ac Type 347: B2

c

Type 347: Cc

230–900

0.51–0.70

25

0.28, 0.28, 0.17

230–870

0.57–0.66

230–1050

0.52–0.50

230–1050

0.26–0.31

25

0.39, 0.35, 0.17

230–900

0.52–0.65

230–875

0.51–0.65

230–900

0.49–0.64

215–490

0.44–0.36

215–525

0.62–0.73

215–525

0.90–0.97

100

0.13

Type 304: (8% Cr; 18% Ni) Light silvery, rough, brown after heating After 42 h heating at

525◦ C

Type 310 (25% Cr; 20% Ni), brown, splotched, oxidized from furnace service Allegheny metal no. 4, polished Allegheny alloy no. 66, polished

100

0.11

1340–3000

0.19–0.31

275–500

0.58–0.36

500–825

0.36–0.21

Bright tinned iron

25

0.043, 0.064

Bright

50

0.06

Commercial tin-plated sheet iron

100

0.07, 0.08

Tantalum filament Thorium oxide Tin

Tungsten Filament, aged

27–3300

0.032–0.35

Filament

3300

0.39

Polished coat

100

0.066 continued on next page

914 Tables for Radiative Properties of Opaque Surfaces

TABLE B.2 (continued) Yttrium

Temperaturea [◦ C]

Total normal emittancea

1400

0.35

225–325

0.045–0.053

Zinc Commercial 99.1% pure, polished Oxidized by heating at

400◦ C

Galvanized sheet iron, fairly bright Galvanized sheet iron, gray oxidized Zinc, galvanized sheet B. Refractories, building materials, paints, and miscellaneous

400

0.11

27

0.23

25

0.28

100

0.21

Alumina (99.5–85% Al2 O3 ; 0–12% SiO2 ; 0–1% Fe2 O3 ) Effect of mean grain size

1010–1565

10 μm

0.30–0.18

50 μm

0.39–0.28

100 μm

0.50–0.40

Alumina on Inconel

540–1100

Alumina–silica (showing effect of Fe)

1010–1565

0.65–0.45

80–58% Al2 O3 ; 16–38% SiO2 ; 0.4% Fe2 O3

0.61–0.43

36–26% Al2 O3 ; 50–60% SiO2 ; 1.7% Fe2 O3

0.73–0.62

61% Al2 O3 ; 35% SiO2 ; 2.9% Fe2 O3

0.78–0.68

Asbestos Board

23

0.96

Paper

35–370

0.93–0.94

20

0.93

Brick Red, rough, but no gross irregularities Grog brick, glazed

1100

0.75

Building

1000

0.45

Fireclay

1000

0.75

White refractory

1100

0.29

1040–1405

0.526

Carbon Filament Rough plate Graphitized

100–320

0.77

320–500

0.77–0.72

100–320

0.76–0.75

320–500

0.75–0.71

Candle soot

95–270

0.952

Lampblack–waterglass coating

100–275

0.96–0.95

Thin layer on iron plate

20

0.927

Thick coat

20

0.967

Lampblack, 0.075 mm or thicker

40–370

0.945

Lampblack, rough deposit

100–500

0.84–0.78

Lampblack, other blacks

50–1000

0.96

Graphite, pressed, filed surface Carborundum (87% SiC; density 2.3 g/cm3 ) Concrete tiles

250–510

0.98

1010–1400

0.92–0.81

1000

0.63

Concrete, rough

38

0.94

Enamel, white fused, on iron

20

0.90

Glass Smooth Pyrex, lead, and soda

20 260–540

0.94 0.95–0.85 continued on next page

Tables for Radiative Properties of Opaque Surfaces 915

TABLE B.2 (continued) Temperaturea [◦ C]

Total normal emittancea

20

0.903

Gypsum, 5 mm thick on smooth or blackened plate Ice Smooth

0

0.966

Rough crystals

0

0.985

Magnesite refractory brick

1000

0.38

Marble, light gray, polished

20

0.93

White enamel varnish on rough iron plate

72

0.906

Black shiny lacquer, sprayed on iron

25

0.875

20

0.821

Paints, lacquers, varnishes

Black shiny shellac on tinned iron sheet Black matte shellac

75–145

0.91

Black or white lacquer

35–95

0.80–0.95

Flat black lacquer

35–95

0.96–0.98

100

0.92–0.96

100

0.52

Other Al paints, varying age and Al content

100

0.27–0.67

Al lacquer, varnish binder, on rough plate

20

0.39

150–315

0.35

35–150

0.87–0.97

Oil paints, 16 different, all colors Aluminum paints and lacquers 10% Al, 22% lacquer body, on rough or smooth surface

Al paint, after heating at 325◦ C Lacquer coatings, 0.025–0.37 mm thick on aluminum alloys Clear silicone vehicle coatings, 0.025–0.375 mm On mild steel

260

0.66

On stainless steels, 316, 301, 347

260

0.68, 0.75, 0.75

On Dow metal

260

0.74

On Al alloys 24 ST, 75 ST

260

0.77, 0.82

260

0.29

Aluminum paint with silicone vehicle, two coats on Inconel Paper White

35

0.95

Thin, pasted on tinned or blackened plate

20

0.92, 0.94

Roofing Plaster, rough lime Porcelain, glazed

20

0.91

10–88

0.91

20

0.92

20

0.93

Quartz Rough, fused Glass, 1.98 mm thick

280–840

0.90–0.41

Glass, 6.88 mm thick

280–840

0.93–0.47

Opaque

280–840

0.92–0.68

Rubber Hard, glossy plate

23

0.94

Soft, gray, rough (reclaimed)

25

0.86

35–260

0.83–0.90

1010–1565

0.42–0.33

1010–1565

0.62–0.46 continued on next page

Sandstone Silica (98% SiO2 ; Fe-free), grain size 10 μm 70–600 μm

916 Tables for Radiative Properties of Opaque Surfaces

TABLE B.2 (continued) Temperaturea [◦ C]

Total normal emittancea

150–650

0.83–0.96

35

0.67–0.80

Silicon carbide Slate Soot, candle

90–260

0.95

Water

0–100

0.95–0.963

Wood, sawdust

35

0.75

Oak, planed

20

0.90

Beech

70

0.94

240–500

0.92–0.80

500–830

0.80–0.52

Zirconium silicate a Temperatures

and emittances in pairs separated by dashes correspond; use linear interpolation. b Surface treatment: A, cleaned with toluene, then methanol; B , cleaned with soap and water, toluene, then methanol; B , cleaned with 2 1 abrasive soap and water, toluene, and methanol; C, polished, then cleaned with soap and water. c Results after repeated heating and cooling.

TABLE B.3 Spectral, normal emittance of metals at room temperature [9]. Wavelength [μm] Metal

0.5

0.6

1.0

3.0

5.0

10.0

Aluminum

-

-

0.08–0.27

0.03–0.12

0.03–0.08

0.02–0.04

Antimony

-

0.47

0.45

0.35

0.31

0.28

Bismuth

0.75

0.76

0.72

0.26

0.12

0.08

Cadmium

-

-

0.30

0.07

0.04

0.02

Chromium

0.45

0.44

0.43

0.30

0.19

0.08

Cobalt

-

-

0.32

0.23

0.15

0.04

Copper

0.36

0.080

0.030

0.026

0.024

0.021

Gold

0.45

0.080

0.020

0.015

0.015

0.015

Iridium

-

-

0.22

0.09

0.06

0.04

Iron

0.49

0.48

0.41

-

-

-

Lead

-

-

-

-

0.08

0.06

Magnesium

0.28

0.27

0.26

0.20

0.14

0.07

Molybdenum

-

-

0.42

0.19

0.16

0.15

Nickel

-

-

0.27

0.12

0.06

0.04

Niobium

-

0.55

0.29

0.14

0.06

0.04

Palladium

0.42

0.37

0.28

0.12

0.10

0.03

Platinum

0.40

0.36

0.24

0.11

0.06

0.05

Rhodium

0.24

0.21

0.16

0.08

0.07

0.05

Silver

0.03

0.03

0.03

0.02

0.02

0.02

Tantalum

0.62

0.55

0.22

0.08

0.07

0.06

Tellurium

-

0.51

0.50

0.47

0.43

0.22

Tin

-

-

0.46

0.32

0.24

0.14

Titanium

-

-

0.37–0.49

0.25–0.33

0.10–0.18

0.05–0.12

Tungsten

-

0.44–0.49

0.40

0.07

0.05

0.03

Vanadium

0.43–0.59

0.42–0.57

0.36–0.50

0.10–0.17

0.07–0.11

0.06–0.09

Zinc

-

0.42–0.58

0.50–0.61

0.08

0.05

0.03

Tables for Radiative Properties of Opaque Surfaces 917

TABLE B.4 Total, normal emittance of metals for elevated temperatures [9]. Temperature [◦ C] Metal

100

500

1000

1200

1400

1600

2000

Aluminum

0.038

0.064

-

-

-

-

-

Beryllium

-

-

0.55

0.87

-

-

-

Bismuth

0.06

-

-

-

-

-

-

Chromium

0.08

0.11–0.14

-

-

-

-

-

Cobalt

0.15–0.24

0.34–0.46

-

-

-

-

-

Copper

-

0.02

-

0.12m

-

-

-

Germanium

-

0.54

-

-

-

-

-

Gold

0.02

0.02

-

-

-

-

-

Hafnium

-

-

-

0.30

0.31

0.32

-

Iron

0.07

0.14

0.24

-

-

-

-

Lead

0.63

-

-

-

-

-

-

Magnesium

0.12h

-

-

-

-

-

-

Mercury

0.12

-

-

-

-

-

-

Molybdenum

0.08

0.13

0.19

0.22

0.24

0.27

-

Nickel

-

0.09–0.15

0.14–0.22

-

-

-

-

Niobium

-

-

0.12

0.14

0.16

0.18

0.21

Palladium

-

0.06

0.12

0.15

-

-

-

Platinum

-

0.086

0.14

0.16

-

-

-

Rhenium

-

-

0.22

0.25

0.27

0.29

-

Rhodium

-

0.035

0.07

0.08

0.09

-

-

Silver

0.02–0.03

0.02–0.03

-

-

-

-

-

Tantalum

0.04

0.06

0.11

0.13

0.15

0.18

0.23

Tin

0.07

-

-

-

-

-

-

Titanium

0.11

-

-

-

-

-

-

Tungsten

-

0.05

0.11

0.14

0.17

0.19

0.23

α-Uranium

-

0.33h

-

-

-

-

-

γ-Uranium

-

-

0.29–0.40h

-

-

-

-

Zinc

0.07

-

-

-

-

-

-

Zirconium Alloys

-

-

0.22

0.25

0.27

-

-

Brass

0.059

-

-

-

-

-

-

Cast iron, cleaned

0.21

-

-

-

-

0.29m

Nichrome

-

0.95

0.98

-

-

-

-

Steel, polished

0.13–0.21

0.18–0.26

0.55–0.80

-

-

-

-

0.21–0.38

0.25–0.42

0.50–0.77

-

-

-

-

cleaned h Total,

hemispherical emittance. m Value for molten state.

-

-

-

-

-

Uranium

Zirconium Alloys

Cast iron

Nichrome

m Value

for molten state.

0.35–0.40

-

Steel

0.35

-

0.37

-

0.19–0.36

-

0.48

-

0.46

0.055

0.63

Tungsten

-

Silicon

-

0.25

Titanium

-

Ruthenium

-

-

Rhodium

-

Thorium

-

Rhenium

0.29–0.31

0.40

-

-

Platinum

0.47

-

Palladium

-

-

Tantalum

-

Osmium

0.37–0.43

-

0.37

-

0.16–0.19

-

0.11

-

-

800

Silver

-

Niobium

-

Iridium

-

0.16–0.18

Gold

Molybdenum

-

Erbium

-

-

Copper

-

-

Cobalt

Manganese

-

Chromium

Iron

600

Metal

0.32–0.40

0.35

0.37

0.48

0.19–0.36

0.46–0.48

0.48

0.38

0.45

0.055

0.57

0.42

0.22

-

0.29–0.31

0.37

0.52

0.37

0.36–0.42

-

0.36

0.36

0.16–0.21

0.55

0.10

0.33–0.38

-

1000

0.30–0.40

0.35

0.37

-

0.35

0.37

0.42

0.34m

0.34m 0.45

0.42–0.47

0.47

0.38

0.42

-

0.46

0.32

0.18

0.42

0.29–0.31

0.30

0.40

0.37

0.34–0.41

-

-

0.37m

-

0.40m -

0.36

-

0.41–0.47

-

-

0.40

0.39

-

0.42–0.47

-

-

0.41

-

-

0.48m -

0.31

-

0.41

0.31

0.16

0.42

-

0.37m 0.29–0.31

0.38

0.37

0.33–0.40

0.38

0.37

0.34–0.41

0.59m -

-

-

0.35

0.43–0.48

0.48

0.38

0.44

-

0.52

0.35

0.19

-

0.29–0.31

0.34

0.44

0.37

0.35–0.42

0.59

0.35

-

0.37m

-

-

0.38m -

0.14m

-

0.12m

-

0.39

1800

0.37m

1600

-

0.32

-

0.34

0.55

0.55

0.11m

0.10m 0.13m

0.35–0.37

-

1400

0.34–0.37

-

1200

Temperature [◦ C]

TABLE B.5 Spectral, normal emittance of metals at a wavelength of 0.65 μm [9].

-

-

-

-

-

0.40–0.47

-

-

0.39

-

-

0.31

-

0.41

-

-

0.38

0.37

0.32–0.39

-

-

0.30

-

-

-

-

-

2000

-

-

-

-

-

0.38–0.46

-

-

0.38

-

-

-

-

0.40

-

-

-

0.40

0.31–0.37

-

-

-

-

-

-

-

0.39

2500

-

-

-

-

-

0.36–0.45

-

-

0.36

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

3000

918 Tables for Radiative Properties of Opaque Surfaces

0.46 0.444 0.442

1327

1727

0.36

2527

1127

0.37

2127

Zirconium

0.385

1327

Tungsten

0.490

0.36

2772

750

0.36

2118

Titanium

0.36

1537

-

1110

Rhenium

-

1200 -

-

1000

1127

-

0.260

800

0.300

2527

0.340

1245

1727

-

1200 0.335

-

1000

1327

-

0.049

985

800

-

901

-

1200 -

-

762

-

800

1.0

1000

Temperature [◦ C]

Platinum

Nickel

Molybdenum

Iron

Copper

Cobalt

Metal

-

-

-

-

-

-

-

-

-

-

0.257

0.292

0.290

0.293

0.295

-

-

-

0.316

0.291

0.294

0.294

-

-

-

0.26

0.26

0.26

1.2

-

-

-

-

-

-

0.510

-

-

-

-

0.270

0.271

0.269

0.267

-

-

-

0.298

-

-

-

-

-

-

-

-

-

1.4

0.375

-

0.422

0.30

0.292

0.28

0.500

0.32

0.30

0.29

0.227

0.250

-

-

-

0.210

0.195

0.185

0.290

-

-

-

0.037

0.079

0.031

-

-

-

1.5

TABLE B.6 Spectral, normal emittance of metals at high temperatures [9].

-

-

-

-

-

-

-

-

-

-

-

-

0.253

0.252

0.250

-

-

-

0.282

0.300

0.267

0.264

-

-

-

-

-

-

1.6

-

-

-

-

-

-

-

-

-

-

-

-

0.235

0.232

0.230

-

-

-

0.268

-

-

-

0.034

-

-

-

-

-

0.357

0.368

0.386

0.26

0.245

0.21

0.455

0.29

0.27

0.25

0.193

0.290

0.223

0.219

0.215

0.193

0.170

0.140

0.260

0.252

0.245

0.237

-

0.065

0.029

0.22

0.21

0.21

2.0

Wavelength [μm] 1.8

0.351

-

0.360

-

-

-

-

0.26

0.24

0.23

-

0.205

-

-

-

-

-

-

0.248

0.235

0.227

0.217

0.032

0.052

-

-

-

-

2.5

0.342

0.343

0.348

-

0.18

0.13

0.525

-

-

-

0.151

0.187

-

-

-

0.185

0.155

0.115

0.240

-

-

-

0.031

0.043

-

0.19

0.18

-

3.0

0.330

-

-

-

-

-

0.575

-

-

-

-

0.174

-

-

-

-

-

-

0.235

-

-

-

-

0.038

-

-

-

-

3.5

-

0.325

-

-

0.15

0.095

0.600

-

-

-

0.130

0.162

-

-

-

0.185

0.145

0.114

0.225

-

-

-

0.030

0.032

0.025

-

-

-

4.0

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.218

-

-

-

-

-

-

-

-

-

4.5

Tables for Radiative Properties of Opaque Surfaces 919

920 Tables for Radiative Properties of Opaque Surfaces

References [1] [2] [3] [4] [5] [6]

D.K. Edwards, A.F. Mills, V.E. Denny, Transfer Processes, 2nd ed., Hemisphere/McGraw-Hill, New York, 1979. H.C. Hottel, Radiant heat transmission, in: W.H. McAdams (Ed.), Heat Transmission, 3rd ed., McGraw-Hill, New York, 1954, ch. 4. H.C. Hottel, A.F. Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967. G.G. Gubareff, J.E. Janssen, R.H. Torborg, Thermal Radiation Properties Survey, Honeywell Research Center, Minneapolis, MI, 1960. W.D. Wood, H.W. Deem, C.F. Lucks, Thermal Radiative Properties, Plenum Publishing Company, New York, 1964. Y.S. Touloukian, D.P. DeWitt (Eds.), Thermal Radiative Properties: Metallic Elements and Alloys, Thermophysical Properties of Matter, vol. 7, Plenum Press, New York, 1970. [7] Y.S. Touloukian, D.P. DeWitt (Eds.), Thermal Radiative Properties: Nonmetallic Solids, Thermophysical Properties of Matter, vol. 8, Plenum Press, New York, 1972. [8] D.I. Svet, Thermal Radiation: Metals, Semiconductors, Ceramics, Partly Transparent Bodies, and Films, Plenum Publishing Company, New York, 1965. [9] W.F. Gale, T.C. Totemeier (Eds.), Smithells Metals Reference Book, 8th ed., Butterworth-Heinemann, Oxford, 2002.

Appendix C

Blackbody Emissive Power Table nλT [μm K]

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700

η/nT [cm−1 /K]

10.0000 9.0909 8.3333 7.6923 7.1429 6.6667 6.2500 5.8824 5.5556 5.2632 5.0000 4.7619 4.5455 4.3478 4.1667 4.0000 3.8462 3.7037 3.5714 3.4483 3.3333 3.2258 3.1250 3.0303 2.9412 2.8571 2.7778 2.7027 2.6316 2.5641 2.5000 2.4390 2.3810 2.3256 2.2727 2.2222 2.1739 2.1277

Ebλ /n3 T 5 [W/m2 μm K5 ]

Ebη /nT 3 [W/m2 cm−1 K3 ]

f (nλT)

0.02110 ×10−11 0.04846 0.09329 0.15724 0.23932 0.33631 0.44359 0.55603 0.66872 0.77736 0.87858 0.96994 1.04990 1.11768 1.17314 1.21659 1.24868 1.27029 1.28242 1.28612 1.28245 1.27242 1.25702 1.23711 1.21352 1.18695 1.15806 1.12739 1.09544 1.06261 1.02927 0.99571 0.96220 0.92892 0.89607 0.86376 0.83212 0.80124

0.00211 ×10−8 0.00586 0.01343 0.02657 0.04691 0.07567 0.11356 0.16069 0.21666 0.28063 0.35143 0.42774 0.50815 0.59125 0.67573 0.76037 0.84411 0.92604 1.00542 1.08162 1.15420 1.22280 1.28719 1.34722 1.40283 1.45402 1.50084 1.54340 1.58181 1.61623 1.64683 1.67380 1.69731 1.71758 1.73478 1.74912 1.76078 1.76994

0.00032 0.00091 0.00213 0.00432 0.00779 0.01285 0.01972 0.02853 0.03934 0.05210 0.06672 0.08305 0.10088 0.12002 0.14025 0.16135 0.18311 0.20535 0.22788 0.25055 0.27322 0.29576 0.31809 0.34009 0.36172 0.38290 0.40359 0.42375 0.44336 0.46240 0.48085 0.49872 0.51599 0.53267 0.54877 0.56429 0.57925 0.59366 921

922 Blackbody Emissive Power Table

nλT [μm K]

η/nT [cm−1 /K]

4800 4900 5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 6000 6200 6400 6600 6800 7000 7200 7400 7600 7800 8000 8200 8400 8600 8800 9000 9200 9400 9600 9800 10,000 10,200 10,400 10,600 10,800 11,000 11,200 11,400 11,600 11,800 12,000 12,200 12,400 12,600 12,800

2.0833 2.0408 2.0000 1.9608 1.9231 1.8868 1.8519 1.8182 1.7857 1.7544 1.7241 1.6949 1.6667 1.6129 1.5625 1.5152 1.4706 1.4286 1.3889 1.3514 1.3158 1.2821 1.2500 1.2195 1.1905 1.1628 1.1364 1.1111 1.0870 1.0638 1.0417 1.0204 1.0000 0.9804 0.9615 0.9434 0.9259 0.9091 0.8929 0.8772 0.8621 0.8475 0.8333 0.8197 0.8065 0.7937 0.7813

Ebλ /n3 T 5 [W/m2 μm K5 ]

Ebη /nT 3 [W/m2 cm−1 K3 ]

f (nλT)

0.77117 ×10−11 0.74197 0.71366 0.68628 0.65983 0.63432 0.60974 0.58608 0.56332 0.54146 0.52046 0.50030 0.48096 0.44464 0.41128 0.38066 0.35256 0.32679 0.30315 0.28146 0.26155 0.24326 0.22646 0.21101 0.19679 0.18370 0.17164 0.16051 0.15024 0.14075 0.13197 0.12384 0.11632 0.10934 0.10287 0.09685 0.09126 0.08606 0.08121 0.07670 0.07249 0.06856 0.06488 0.06145 0.05823 0.05522 0.05240

1.77678 ×10−8 1.78146 1.78416 1.78502 1.78419 1.78181 1.77800 1.77288 1.76658 1.75919 1.75081 1.74154 1.73147 1.70921 1.68460 1.65814 1.63024 1.60127 1.57152 1.54126 1.51069 1.48000 1.44933 1.41882 1.38857 1.35866 1.32916 1.30013 1.27161 1.24363 1.21622 1.18941 1.16319 1.13759 1.11260 1.08822 1.06446 1.04130 1.01874 0.99677 0.97538 0.95456 0.93430 0.91458 0.89540 0.87674 0.85858

0.60753 0.62088 0.63372 0.64606 0.65794 0.66935 0.68033 0.69087 0.70101 0.71076 0.72012 0.72913 0.73778 0.75410 0.76920 0.78316 0.79609 0.80807 0.81918 0.82949 0.83906 0.84796 0.85625 0.86396 0.87115 0.87786 0.88413 0.88999 0.89547 0.90060 0.90541 0.90992 0.91415 0.91813 0.92188 0.92540 0.92872 0.93184 0.93479 0.93758 0.94021 0.94270 0.94505 0.94728 0.94939 0.95139 0.95329

Blackbody Emissive Power Table

nλT [μm K]

η/nT [cm−1 /K]

13,000 13,200 13,400 13,600 13,800 14,000 14,200 14,400 14,600 14,800 15,000 16,000 17,000 18,000 19,000 20,000 21,000 22,000 23,000 24,000 25,000 26,000 27,000 28,000 29,000 30,000 31,000 32,000 33,000 34,000 35,000 36,000 37,000 38,000 39,000 40,000 41,000 42,000 43,000 44,000 45,000 46,000 47,000 48,000 49,000 50,000

0.7692 0.7576 0.7463 0.7353 0.7246 0.7143 0.7042 0.6944 0.6849 0.6757 0.6667 0.6250 0.5882 0.5556 0.5263 0.5000 0.4762 0.4545 0.4348 0.4167 0.4000 0.3846 0.3704 0.3571 0.3448 0.3333 0.3226 0.3125 0.3030 0.2941 0.2857 0.2778 0.2703 0.2632 0.2564 0.2500 0.2439 0.2381 0.2326 0.2273 0.2222 0.2174 0.2128 0.2083 0.2041 0.2000

Ebλ /n3 T 5 [W/m2 μm K5 ]

Ebη /nT 3 [W/m2 cm−1 K3 ]

f (nλT)

0.04976 ×10−11 0.04728 0.04494 0.04275 0.04069 0.03875 0.03693 0.03520 0.03358 0.03205 0.03060 0.02447 0.01979 0.01617 0.01334 0.01110 0.00931 0.00786 0.00669 0.00572 0.00492 0.00426 0.00370 0.00324 0.00284 0.00250 0.00221 0.00196 0.00175 0.00156 0.00140 0.00126 0.00113 0.00103 0.00093 0.00084 0.00077 0.00070 0.00064 0.00059 0.00054 0.00049 0.00046 0.00042 0.00039 0.00036

0.84092 ×10−8 0.82374 0.80702 0.79076 0.77493 0.75954 0.74456 0.72998 0.71579 0.70198 0.68853 0.62643 0.57194 0.52396 0.48155 0.44393 0.41043 0.38049 0.35364 0.32948 0.30767 0.28792 0.26999 0.25366 0.23875 0.22510 0.21258 0.20106 0.19045 0.18065 0.17158 0.16317 0.15536 0.14810 0.14132 0.13501 0.12910 0.12357 0.11839 0.11352 0.10895 0.10464 0.10059 0.09677 0.09315 0.08974

0.95509 0.95680 0.95843 0.95998 0.96145 0.96285 0.96418 0.96546 0.96667 0.96783 0.96893 0.97377 0.97765 0.98081 0.98340 0.98555 0.98735 0.98886 0.99014 0.99123 0.99217 0.99297 0.99367 0.99429 0.99482 0.99529 0.99571 0.99607 0.99640 0.99669 0.99695 0.99719 0.99740 0.99759 0.99776 0.99792 0.99806 0.99819 0.99831 0.99842 0.99851 0.99861 0.99869 0.99877 0.99884 0.99890

923

Appendix D

View Factor Catalogue In this appendix a small number of view factor relations and figures are presented. A much larger collection from a variety of references has been compiled by Howell [1,2], from which the present list has been extracted. The latest edition of this collection can be accessed on the Internet via http://www.engr.uky.edu/rtl/Catalog/. View factors for all configurations given in this appendix, as well as those between two arbitrarily orientated rectangular plates lying in perpendicular planes, as given by equations (4.41) and (4.42), can be calculated with the standalone program viewfactors (prompting for user input) or from within another program through calls to Fortran function view, both given in Appendix F. A number of commercial and noncommercial computer programs are available for the evaluation of more complicated view factors [3–13]. A list of papers and monographs that either deal with evaluation methods for view factors, or present results for specified configurations (ordered by date of publication) is also given. No attempt at completeness has been made. Note: In all expressions in which inverse trigonometric functions appear, the principal value is to be taken; i.e., for any argument ξ, π π π π − ≤ sin−1 ξ ≤ + ; 0 ≤ cos−1 ξ ≤ π; − ≤ tan−1 ξ ≤ + . 2 2 2 2 1

Differential strip element of any length z to infinitely long strip of differential width on parallel line; plane containing element does not intercept strip dFd1−d2 =

2

cos φ dφ 2

Differential planar element to differential coaxial ring parallel to the element R = r/l dFd1−d2 =

3

2R dR (1 + R2 )2

Differential planar element on and normal to ring axis to inside of differential ring X = x/r dFd1−d2 =

2X dX (X2 + 1)2

925

926 View Factor Catalogue

4

Element on surface of right-circular cylinder to coaxial differential ring on cylinder base, r2 < r1 Z = z/r1 ,

R = r2 /r1

X = 1 + Z2 + R2 dFd1−d2 =

5

2Z(X − 2R2 )R dR (X2 − 4R2 )3/2

Parallel differential strip elements in intersecting planes Y = y/x dFd1−d2 =

6

Y sin2 φ dY 2(1 + Y2 − 2Y cos φ)3/2

Strip of finite length b and of differential width, to differential strip of same length on parallel generating line B = b/r dFd1−d2 = tan−1 B

7

Differential ring element to ring element on coaxial disk R = r2 /r1 , dFd1−d2 =

8

cos φ dφ π

L = l/r1

2RL2 [L2 + R2 + 1] dR [(L2 + R2 + 1)2 − 4R2 ]3/2

Ring element on base to circumferential ring element on interior of right-circular cylinder X = x/r2 , dFd1−d2 =

R = r1 /r2

2X(X − R2 + 1) dX 2

[(X2 + R2 + 1)2 − 4R2 ]3/2

View Factor Catalogue

927

Two ring elements on the interior of right-circular cylinder

9

X = x/2r  dFd1−d2 = 1 −

10

Differential planar element to finite parallel rectangle; normal to element passes through corner of rectangle

Fd1−2 11

1 = 2π

/ √

A 1+A2

tan

−1

√

A = a/c,

B = b/c

B

B

1+A2

+ √

1+B2

tan

−1

A

0

√ 1+B2

Differential planar element to rectangle in plane 90◦ to plane of element

Fd1−2

12

 X(2X2 + 3) dX2 2(X2 + 1)3/2

X = a/b, Y = c/b   Y 1 1 1 − √ = tan−1 √ tan−1 2π Y X 2 + Y2 X 2 + Y2

Differential planar element to circular disk in plane parallel to element; normal to element passes through center of disk H = h/r Fd1−2 =

13

1 H2 + 1

Differential planar element to circular disk in plane parallel to element H = h/a,

R = r/a

Z = 1 + H + R2   Z − 2R2 1 1− √ Fd1−2 = 2 Z2 − 4R2 2

14

Differential planar element to circular disk; planes containing element and disk intersect at 90◦ ; l ≥ r H = h/l,

R = r/l

Z = 1 + H + R2   Z H Fd1−2 = − 1 √ 2 Z2 − 4R2 2

928 View Factor Catalogue

Differential planar element to right-circular cylinder of finite length and radius; normal to element passes through one end of cylinder and is perpendicular to cylinder axis

15

L = l/r,

H = h/r

X = (1 + H)2 + L2 Y = (1 − H)2 + L2

Fd1−2

L = πH

& ⎡ ⎤ & ⎢⎢ 1 X(H−1) L H−1 ⎥⎥⎥ X−2H −1 −1 −1 − tan + √ tan ⎢⎢⎣ tan √ ⎥ L Y(H+1) H+1 ⎦ XY H2 −1

16

Differential planar element to sphere; normal to center of element passes through center of sphere Fd1−2 =

17

 2 r h

Differential planar element to sphere; tangent to element passes through center of sphere

Fd1−2

18

H = h/r ⎡ ⎤ √ 1 H2 −1 ⎥⎥⎥ 1 ⎢⎢ − = ⎢⎢⎣tan−1 √ ⎥ π H2 ⎦ H2 −1

Differential planar element to sphere; element plane does not intersect sphere θ ≤ cos−1 Fd1−2 =

19

r h

 2 r cos θ h

Differential planar element to sphere L = l/r,

H≥1: −1 < H < 1 :

Fd1−2 =

H = h/r

H

(L2 + H2 )3/2 /   H H√ 2 1 Fd1−2 = cos−1 − L +H2 − 1 3/2 π (L2 + H2 ) L % √ 0 (L2 + H2 − 1)(1 − H2 ) H 2 + L2 − 1 π −1 − + − sin L2 + H 2 L 2

View Factor Catalogue

20

929

Differential element on longitudinal strip inside cylinder to inside cylinder surface Z = z/2r,

H = h/2r

(H − Z)2 + 12 Z2 + 12 − % Fd1−2 = 1 + H − √ (H − Z)2 + 1 Z2 + 1

21

Differential element on longitudinal strip on inside of rightcircular cylinder to base of cylinder Z = z/r Fd1−2 =

22

Z Z2 + 2 − √ 2 2 Z +4 2

Differential element on surface of right-circular cylinder to disk on base of cylinder, r2 < r1 (see Configuration 13) Z = z/r1 ,

Fd1−2

23

R = r2 /r1

X = 1 + Z2 + R2 / 0 X Z = −1 √ 2 X2 − 4R2

Infinite differential strip to parallel infinite plane of finite width; plane and plane containing strip intersect at arbitrary angle φ X = x/l Fd1−2 =

24

cos φ − X 1 + % 2 2 1 + X2 − 2X cos φ

Differential strip element of any length to an infinitely long strip of finite width; cross-section of A2 is arbitrary (but does not vary perpendicular to the paper); plane of dA1 does not intersect A2 Fd1−2 = 12 (sin φ2 − sin φ1 )

25

Differential strip element of any length to infinitely long parallel cylinder; r < a A = a/r, Fd1−2 =

B = b/r A A2 + B2

930 View Factor Catalogue

Differential strip element to rectangle in plane parallel to strip; strip is opposite one edge of rectangle

26

X = a/c,

Fd1−2 =

Y = b/c

  X Y 1 √ XY 1+Y2 tan−1 √ − tan−1 X + √ tan−1 √ πY 1+Y2 1+X2 1+X2 Differential strip element to rectangle in plane 90◦ to plane of strip

27

X = a/b,

Fd1−2 = 28

Y = c/b

  Y2 (X2 +Y2 +1) 1 Y 1 Y 1 −1 tan−1 + ln 2 − tan √ √ π Y 2 (Y +1)(X2 +Y2 ) X2 +Y2 X2 +Y2 Differential strip element to exterior of right-circular cylinder of finite length; strip and cylinder are parallel and of equal length; plane containing strip does not intersect cylinder S = s/r,

X = x/r,

H = h/r

A = H 2 + S2 + X 2 − 1

Fd1−2 29

S = 2 S + X2

⎡ ⎛ √ 2 2 ⎢ ⎜⎜ ⎜⎜1 − 1 ⎢⎢⎢cos−1 B − A + 4H cos−1 ⎣ ⎝ π A 2H

B = H 2 − S2 − X 2 + 1 ⎞ ⎤ ⎥⎥ A ⎟⎟⎟ B 1 B −1 ⎥ sin √ − ⎟ ⎥⎦ − √ 4H ⎠ A S2 + X2 2H S2 + X 2

Differential strip element of any length on exterior of cylinder to plane of infinite length and width Fd1−2 = 12 (1 + cos φ)

30

Differential ring element on surface of disk to coaxial sphere R1 = r1 /a, Fd1−2 = 

R2 = r2 /a R22 1 + R21

3/2

View Factor Catalogue

931

Differential ring element on interior of right-circular cylinder to circular disk at end of cylinder

31

X = x/2r X2 + 12 −X Fd1−2 = √ X2 + 1

32

Two infinitely long, directly opposed parallel plates of the same finite width

F1−2

33

H = h/w √ = F2−1 = 1 + H2 − H

Two infinitely long plates of unequal widths h and w, having one common edge, and at an angle of 90◦ to each other H = h/w F1−2 =

34

 √ 1 1 + H − 1 + H2 2

Two infinitely long plates of equal finite width w, having one common edge, forming a wedge-like groove with opening angle α F1−2 = F2−1 = 1 − sin

35

α 2

Infinitely long parallel cylinders of the same diameter X =1+ F1−2 =

36

 1 1 sin−1 + π X

s 2r √

X2 − 1 − X



Two infinite parallel cylinders of different radius R = r2 /r1 ,

F1−2 =

1 2π

8

S = s/r1 ,

C=1+R+S % % π + C2 − (R + 1)2 − C2 − (R − 1)2 + (R − 1) cos−1

R−1 R+1 − (R + 1) cos−1 C C

9

932 View Factor Catalogue

Exterior of infinitely long cylinder to unsymmetrically placed, infinitely long parallel rectangle; r ≤ a

37

B1 = b1 /a, F1−2

38

B2 = b2 /a

 1  −1 = tan B1 − tan−1 B2 2π

Identical, parallel, directly opposed rectangles X = a/c, F1−2

Y = b/c

⎧  1/2 √ ⎪ (1+X2 )(1+Y2 ) X 2 ⎪ ⎨ ln = + X 1+Y2 tan−1 √ ⎪ ⎪ 2 2 πXY ⎩ 1+X +Y 1+Y2

⎫ ⎪ √ ⎪ Y ⎬ −1 −1 −1 + Y 1+X2 tan √ − X tan X − Y tan Y⎪ ⎪ ⎭ 1+X2

39

Two finite rectangles of same length, having one common edge, and at an angle of 90◦ to each other H = h/l,

F1−2

W = w/l

⎛ √ ⎜⎜ ⎜⎜W tan−1 1 + H tan−1 1 − H2 + W 2 tan−1 √ 1 ⎜⎝ W H H2 + W 2 ⎧ ⎞ 2    H2 ⎫ W ⎪ ⎪ ⎪ H2 (1+H2 +W 2 ) 1 ⎪ ⎨ (1+W 2 )(1+H2 ) W 2 (1+W 2 +H2 ) ⎬⎟⎟⎟⎟ + ln ⎪ ⎪⎟⎠ ⎩ 1+W 2 +H2 ⎭ 4 ⎪ (1+W 2 )(W 2 +H2 ) (1+H2 )(H2 +W 2 ) ⎪

1 = πW

40

Disk to parallel coaxial disk of unequal radius R1 = r1 /a, X =1+

F1−2

41

R2 = r2 /a 1 + R22

R21 ⎫ ⎧  ⎪  2 ⎪ ⎪ ⎪ 1⎪ R ⎬ ⎨ 2 ⎪ 2 −4 X − = ⎪ X ⎪ ⎪ ⎪ 2⎪ R1 ⎪ ⎭ ⎩

Outer surface of cylinder to annular disk at end of cylinder R = r1 /r2 ,

L = l/r2

A = L + R2 − 1 2

B = L2 − R2 + 1

F1−2

& ⎡ ⎤ ⎥ (A + 2)2 1 ⎢⎢⎢ −1 A 1 A B −1 ⎥ −1 AR + − − sin R⎥⎥⎦ = −4 cos ⎢cos 8RL 2π ⎣ B 2L R2 B 2RL

View Factor Catalogue

933

Inside surface of right-circular cylinder to itself

42

F1−1

43

H = h/2r √ = 1 + H − 1 + H2

Base of right-circular cylinder to inside surface of cylinder

F1−2

44

H = h/2r √  = 2H 1 + H2 − H

Interior of finite-length, right-circular coaxial cylinder to itself R = r2 /r1 ,

H = h/r1

⎛ √ √ 2 H2 + 4R2 − H 1 ⎜⎜⎜ 2 1 −1 2 R − 1 + ⎜⎝ tan F2−2 = 1 − − R 4R π R H ⎧√ ⎫⎞ 2 2 2 2 2 ⎪⎟ H ⎪ ⎨ 4R2 + H2 −1 H +4(R −1)−2H /R −1 R − 2 ⎬⎟ sin − sin − ⎪ ⎪⎟⎟ ⎩ 2 2 2 2R H H + 4(R − 1) R ⎭⎠ 45

Interior of outer right-circular cylinder of finite length to exterior of inner right-circular coaxial cylinder R = r2 /r1 ,

H = h/r1

 / H2 − R2 + 1 1 H2 + R2 − 1 1 − 1− cos−1 2 R 4H π H + R2 − 1 % 0 (H2 +R2 +1)2 −4R2 H2 −R2 +1 H2 −R2 +1 −1 1 −1 cos − sin − 2H R(H2 +R2 −1) 2H R

F2−1 =

46

Interior of outer right-circular cylinder of finite length to annular end enclosing space between coaxial cylinders H = h/r2 , R = r1 /r2 √ X = 1 − R2 Y=

R(1 − R2 − H2 ) 1 − R2 + H2

⎧     X2  π H  −1 X 2X 1⎪ ⎨ R tan−1 − tan−1 + + sin−1 R sin (2R2 − 1) − sin−1 R + ⎪ ⎩ π H H 4 4H 2 % √  ⎫    2 2 2 2 ⎪ (1 + R + H ) − 4R π 2R2 H2 4 + H2 π ⎬ −1 −1 + sin Y + + sin 1 − − ⎪ 4H 2 4 2 4X2 + H2 ⎭

F1−2 =

934 View Factor Catalogue

47

Sphere to rectangle, r < d

F1−2

48

D1 = d/l1 , D2 = d/l2  1 1 −1 tan = 2 2 4π D1 + D2 + D21 D22

Sphere to coaxial disk

F1−2

49

R = r/a   1 1 1− √ = 2 1 + R2

Sphere to interior surface of coaxial right-circular cylinder; sphere within ends of cylinder R = r/a F1−2 = √

50

1 + R2

Sphere to coaxial cone S = s/r1 , for ω ≥ sin−1

R = r2 /r1

1 : S+1

F1−2 51

1

⎡ ⎤ ⎥⎥ 1+S +R cot ω 1 ⎢⎢⎢ ⎥⎥ = ⎢⎣1 − % ⎦ 2 (1+S +R cot ω)2 +R2

Infinite plane to row of cylinders F1−2

D D = cos−1 + 1 − s s

& 1−



D s

2

References [1] J.R. Howell, A Catalog of Radiation Configuration Factors, McGraw-Hill, New York, 1982. [2] J.R. Howell, M.P. Mengüç, Radiative transfer configuration factor catalog: a listing of relations for common geometries, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (2011) 910–912. [3] R.L. Wong, User’s manual for CNVUFAC–the General Dynamics heat transfer radiation view factor program, Technical report, University of California, Lawrence Livermore National Laboratory, 1976. [4] A.B. Shapiro, FACET–a computer view factor computer code for axisymmetric, 2D planar, and 3D geometries with shadowing, Technical report, University of California, Lawrence Livermore National Laboratory, August 1983, maintained by Nuclear Energy Agency under http://www.oecd-nea.org/tools/abstract/detail/nesc9578/.

View Factor Catalogue

935

[5] P.J. Burns, MONTE–a two-dimensional radiative exchange factor code, Technical report, Colorado State University, Fort Collins, 1983. [6] A.F. Emery, VIEW–a radiation view factor program with interactive graphics for geometry definition (version 5.5.3), Technical report, NASA Computer Software Management and Information Center, Atlanta, 1986, available from http://www.openchannelfoundation. org/projects/VIEW. [7] T. Ikushima, MCVIEW: a radiation view factor computer program or three-dimensional geometries using Monte Carlo method, Technical report, Japan Atomic Energy Research Institute (JAERI), 1986, maintained by Nuclear Energy Agency under http://www. oecd-nea.org/tools/abstract/detail/nea-1166. [8] C.L. Jensen, TRASYS-II user’s manual–thermal radiation analysis system, Technical report, Martin Marietta Aerospace Corp., Denver, 1987. [9] G.N. Walton, Algorithms for calculating radiation view factors between plane convex polygons with obstructions, in: Fundamentals and Applications of Radiation Heat Transfer, vol. HTD-72, ASME, 1987, pp. 45–52. [10] J.H. Chin, T.D. Panczak, L. Fried, Spacecraft thermal modeling, International Journal for Numerical Methods in Engineering 35 (1992) 641–653. [11] C.N. Zeeb, P.J. Burns, K. Branner, J.S. Dolaghan, User’s Manual for Mont3d – Version 2.4, Colorado State University, Fort Collins, CO, 1999. [12] G.N. Walton, Calculation of obstructed view factors by adaptive integration, Technical Report NISTIR–6925, National Institute of Standards and Technology (NIST), Gaithersburg, MD, 2002. [13] J.J. MacFarlane, VISRAD-a 3D view factor code and design tool for high-energy density physics experiments, Journal of Quantitative Spectroscopy and Radiative Transfer 81 (2003) 287–300. [14] H.B. Keene, Calculation of the energy exchange between two fully radiative coaxial circular apertures at different temperatures, Proceedings of the Royal Society LXXXVIII-A (1913) 59–60. [15] W. Nusselt, Graphische Bestimming des Winkelverhältnisses bei der Wärmestrahlung, VDI Zeitschrift 72 (1928) 673. [16] H.C. Hottel, Radiant heat transmission between surfaces separated by non-absorbing media, Transactions of ASME, Journal of Heat Transfer 53 (1931) 265–273. [17] H.C. Hottel, F.P. Broughton, Determination of true temperature and total radiation from luminous gas flames, Industrial and Engineering Chemistry 4 (1932) 166–174. [18] H.C. Hottel, J.D. Keller, Effect of reradiation on heat transmission in furnaces and through openings, Transactions of ASME, Journal of Heat Transfer 55 (1933) 39–49. [19] D.C. Hamilton, W.R. Morgan, Radiant interchange configuration factors, NACA TN 2836, 1952. [20] H.C. Hottel, Radiant heat transmission, in: W.H. McAdams (Ed.), Heat Transmission, 3rd ed., McGraw-Hill, New York, 1954, ch. 4. [21] H. Leuenberger, R.A. Pearson, Compilation of radiant shape factors for cylindrical assemblies, ASME paper no. 56-A-144, 1956. [22] M. Jakob, Heat Transfer, vol. 2, John Wiley & Sons, New York, 1957. [23] C.M. Usiskin, R. Siegel, Thermal radiation from a cylindrical enclosure with specified wall heat flux, ASME Journal of Heat Transfer 82 (1960) 369–374. [24] A.J. Buschman, C.M. Pittman, Configuration factors for exchange of radiant energy between axisymmetrical sections of cylinders, cones, and hemispheres and their bases, NASA TN D-944, 1961. [25] F.G. Cunningham, Power input to a small flat plate from a diffusely radiating sphere with application to Earth satellites, NASA TN D-710, 1961. [26] P. Moon, Scientific Basis of Illuminating Engineering, Dover Publications, New York, 1961, originally published by McGraw-Hill, New York, 1936. [27] L.D. Nichols, Surface-temperature distribution on thin-walled bodies subjected to solar radiation in interplanetary space, NASA TN D-584, 1961. [28] J.A. Plamondon, Numerical determination of radiation configuration factors for some common geometrical situations, Technical Report 32-127, California Institute of Technology, 1961. [29] W.H. Robbins, An analysis of thermal radiation heat transfer in a nuclear-rocket nozzle, NASA TN D-586, 1961. [30] J.A. Stevenson, J.C. Grafton, Radiation heat transfer analysis for space vehicles, Report SID-61-91, North American Aviation, 1961. [31] C.W. Stephens, A.M. Haire, Internal design considerations for cavity-type solar absorbers, American Rocket Society Journal 31 (7) (1961) 896–901. [32] D. Goetze, C.B. Grosch, Earth-emitted infrared radiation incident upon a satellite, Journal of the Aerospace Sciences 29 (11) (1962) 521–524. [33] P. Joerg, B.L. McFarland, Radiation effects in rocket nozzles, Report S62–245, Aerojet-General Corporation, 1962. [34] F. Kreith, Radiation Heat Transfer for Spacecraft and Solar Power Design, International Textbook Company, Scranton, PA, 1962. [35] R.L. Perry, E.P. Speck, Geometric factors for thermal radiation exchange between cows and their surroundings, Transactions of the ASAE 5 (1) (1962) 31–37. [36] W.H. Robbins, C.A. Todd, Analysis, feasibility, and wall-temperature distribution of a radiation-cooled nuclear-rocket nozzle, NASA TN D-878, 1962. [37] E.M. Sparrow, E.R.G. Eckert, Radiant interaction between fin and base surfaces, ASME Journal of Heat Transfer 84 (1) (1962) 12–18. [38] E.M. Sparrow, L.U. Albers, E.R.G. Eckert, Thermal radiation characteristics of cylindrical enclosures, ASME Journal of Heat Transfer 84 (1962) 73–81.

936 View Factor Catalogue

[39] E.M. Sparrow, V.K. Jonsson, Absorption and emission characteristics of diffuse spherical enclosures, NASA TN D-1289, 1962. [40] E.M. Sparrow, V.K. Jonsson, Absorption and emission characteristics of diffuse spherical enclosures, ASME Journal of Heat Transfer 84 (1962) 188–189. [41] E.M. Sparrow, G.B. Miller, V.K. Jonsson, Radiative effectiveness of annular-finned space radiators, including mutual irradiation between radiator elements, Journal of the Aerospace Sciences 29 (11) (1962) 1291–1299. [42] R.V. Dunkle, Configuration factors for radiant heat-transfer calculations involving people, ASME Journal of Heat Transfer 85 (1) (1963) 71–76. [43] H.C. Haller, N.O. Stockman, A note on fin-tube view factors, ASME Journal of Heat Transfer 85 (4) (1963) 380–381. [44] E.M. Sparrow, A new and simpler formulation for radiative angle factors, ASME Journal of Heat Transfer 85 (1963) 73–81. [45] E.M. Sparrow, V.K. Jonsson, Radiant emission characteristics of diffuse conical cavities, Journal of the Optical Society of America 53 (1963) 816–821. [46] E.M. Sparrow, V.K. Jonsson, Thermal radiation absorption in rectangular-groove cavities, ASME Journal of Applied Mechanics E30 (1963) 237–244. [47] E.M. Sparrow, V.K. Jonsson, Angle factors for radiant interchange between parallel-oriented tubes, ASME Journal of Heat Transfer 85 (4) (1963) 382–384. [48] J.A. Wiebelt, S.Y. Ruo, Radiant-interchange configuration factors for finite right circular cylinders to rectangular planes, International Journal of Heat and Mass Transfer 6 (2) (1963) 143–146. [49] S.J. Morizumi, Analytical determination of shape factors from a surface element to an axisymmetric surface, AIAA Journal 2 (11) (1964) 2028–2030. [50] C.J. Sotos, N.O. Stockman, Radiant interchange view factors and limits of visibility for differential cylindrical surfaces with parallel generating lines, NASA TN D-2556, 1964. [51] L.R. Jones, Diffuse radiation view factors between two spheres, ASME Journal of Heat Transfer 87 (3) (1965) 421–422. [52] K. Toups, A general computer program for the determination of radiant interchange configuration and form factors – CONFAC-I, Inc. Rept. SID-65-1043-1, North American Aviation, 1965. [53] R.G. Watts, Radiant heat transfer to Earth satellites, ASME Journal of Heat Transfer 87 (3) (1965) 369–373. [54] D.D. Bien, Configuration factors for thermal radiation from isothermal inner walls of cones and cylinders, Journal of Spacecraft and Rockets 3 (1) (1966) 155–156. [55] R.P. Bobco, Radiation from conical surfaces with nonuniform radiosity, AIAA Journal 4 (3) (1966) 544–546. [56] A. Feingold, Radiant-interchange configuration factors between various selected plane surfaces, Proceedings of the Royal Society of London 292 (1428) (1966) 51–60. [57] S.P. Kezios, W. Wulff, Radiative heat transfer through openings of variable cross sections, in: Third International Heat Transfer Conference, AIChE, vol. 5, 1966, pp. 207–218. [58] G.P. Mitalas, D.G. Stephenson, Fortran IV programs to calculate radiant interchange factors, Div. of Building Research Report DBR-25, National Research Council of Canada, 1966. [59] J.A. Wiebelt, Engineering Radiation Heat Transfer, Holt, Rinehart & Winston, New York, 1966. [60] J.S. Farnbach, Radiant interchange between spheres: accuracy of the point-source approximation, Technical Memo SC-TM-364, Sandia Laboratories, 1967. [61] H.C. Hottel, A.F. Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967. [62] C.J. Hsu, Shape factor equations for radiant heat transfer between two arbitrary sizes of rectangular planes, Canadian Journal of Chemical Engineering 45 (1) (1967) 58–60. [63] R.S. Holcomb, F.E. Lynch, Thermal radiation performance of a finned tube with a reflector, Technical Report ORNL-TM-1613, Oak Ridge National Laboratory, 1967. [64] A.L. Stasenko, Self-irradiation coefficient of a Moebius strip of given shape, Izvestiâ Akademii Nauk, SSSR, Energetika i Transport (1967) 104–107. [65] J.P. Campbell, D.G. McConnell, Radiant interchange configuration factors for spherical and conical surfaces to spheres, NASA TN D-4457, 1968. [66] E.G. Hauptmann, Angle factors between a small flat plate and a diffusely radiating sphere, AIAA Journal 6 (5) (1968) 938–939. [67] C.H. Liebert, R.R. Hibbard, Theoretical temperatures of thin-film solar cells in Earth orbit, NASA TN D-4331, 1968. [68] D.K. Edwards, Comment on ‘radiation from conical surfaces with nonuniform radiosity’, AIAA Journal 7 (8) (1969) 1656–1659. [69] N.T. Grier, Tabulations of configuration factors between and two spheres and their parts, NASA SP-3050, 1969. [70] N.T. Grier, R.D. Sommers, View factors for toroids and their parts, NASA TN D-5006, 1969. [71] J.K. Lovin, A.W. Lubkowitz, User’s manual for RAVFAC, a radiation view factor digital computer program, Lockheed Missiles and Space Rept. HREC-0154-1, Huntsville Research Park, 1969. [72] R.D. Sommers, N.T. Grier, Radiation view factors for a toroid: comparison of Eckert’s technique and direct computation, ASME Journal of Heat Transfer 91 (3) (1969) 459–461. [73] N. Wakao, K. Kato, N. Furuya, View factor between two hemispheres in contact and radiation heat-transfer coefficient in packed beds, International Journal of Heat and Mass Transfer 12 (1969) 118–120. [74] A. Feingold, K.G. Gupta, New analytical approach to the evaluation of configuration factors in radiation from spheres and infinitely long cylinders, ASME Journal of Heat Transfer 92 (1) (1970) 69–76.

View Factor Catalogue

937

[75] P.F. O’Brien, R.B. Luning, Experimental study of luminous transfer in architectural systems, Illuminating Engineering 65 (4) (1970) 193–198. [76] R.G. Rein, C.M. Sliepcevich, J.R. Welker, Radiation view factors for tilted cylinders, Journal of Fire and Flammability 1 (1970) 140–153. [77] E.M. Sparrow, R.P. Heinisch, The normal emittance of circular cylindrical cavities, Applied Optics 9 (1970) 2569–2572. [78] C.L. Sydnor, A numerical study of cavity radiometer emissivities, NASA Contractor Rept. 32-1462, Jet Propulsion Lab, 1970. [79] J.J. Bernard, J. Genot, Rayonnement thermique des surfaces de revolution, International Journal of Heat and Mass Transfer 14 (10) (1971) 1611–1619. [80] D.W. McAdam, A.K. Khatry, M. Iqbal, Configuration factors for greenhouses, Transactions of the ASAE 14 (6) (1971) 1068–1092. [81] M.E. Crawford, Configuration factor between two unequal, parallel, coaxial squares, ASME paper no. 72-WT/HT-16, November 1972. [82] B.T.F. Chung, P.S. Sumitra, Radiation shape factors from plane point sources, ASME Journal of Heat Transfer 94 (1972) 328–330. [83] E.F. Sowell, P.F. O’Brien, Efficient computation of radiant-interchange factors within an enclosure, ASME Journal of Heat Transfer 49 (3) (1972) 326–328. [84] J.O. Ballance, J. Donovan, Radiation configuration factors for annular rings and hemispherical sectors, ASME Journal of Heat Transfer 95 (2) (1973) 275–276. [85] H. Masuda, Radiant heat transfer on circular-finned cylinders, Reports of the Institute of High Speed Mechanics 27 (225) (1973) 67–89. [86] R.L. Reid, J.S. Tennant, Annular ring view factors, AIAA Journal 11 (10) (1973) 1446–1448. [87] J. Holchendler, W.F. Laverty, Configuration factors for radiant heat exchange in cavities bounded at the ends by parallel disks and having conical centerbodies, ASME Journal of Heat Transfer 96 (2) (1974) 254–257. [88] H.J. Sauer, Configuration factors for radiant energy interchange with triangular areas, ASHRAE Transactions 80 (2322) (1974) 268–279. [89] G. Alfano, A. Sarno, Normal and hemispherical thermal emittances of cylindrical cavities, ASME Journal of Heat Transfer 97 (3) (1975) 387–390. [90] F.O. Bartell, W.L. Wolfe, New approach for the design of blackbody simulators, Applied Optics 14 (2) (1975) 249–252. [91] S.N. Rea, Rapid method for determining concentric cylinder radiation view factors, AIAA Journal 13 (8) (1975) 1122–1123. [92] W. Boeke, L. Wall, Radiative exchange factors in rectangular spaces for the determination of mean radiant temperatures, Building Service Engineering 43 (1976) 244–253. [93] R.L. Cox, Radiative heat transfer in arrays of parallel cylinders, Ph.D. thesis, University of Tennessee, Knoxville, Tennessee, 1976. [94] N.H. Juul, Diffuse radiation configuration view factors between two spheres and their limits, Letters in Heat and Mass Transfer 3 (3) (1976) 205–211. [95] N.H. Juul, Investigation of approximate methods for calculation of the diffuse radiation configuration view factors between two spheres, Letters in Heat and Mass Transfer 3 (1976) 513–522. [96] C.P. Minning, Calculation of shape factors between parallel ring sectors sharing a common centerline, AIAA Journal 14 (6) (1976) 813–815. [97] C.P. Minning, Calculation of shape factors between rings and inverted cones sharing a common axis, ASME Journal of Heat Transfer 99 (3) (1977) 492–494. [98] A. Feingold, A new look at radiation configuration factors between disks, ASME Journal of Heat Transfer 100 (4) (1978) 742–744. [99] J.D. Felske, Approximate radiation shape factors between two spheres, ASME Journal of Heat Transfer 100 (3) (1978) 547–548. [100] E.M. Sparrow, R.D. Cess, Radiation Heat Transfer, Hemisphere, New York, 1978. [101] I.R. Chekhovskii, V.V. Sirotkin, V.C. Yu, V.A. Chebanov, Determination of radiative view factors for rectangles of different sizes, High Temperature (July 1979). [102] C. Garot, P. Gendre, Computation of view factors used in radiant energy exchanges in axisymmetric geometry, in: Numerical Methods in Thermal Problems; Proc. First Intl. Conf, Pineridge Press, Ltd., Swansea, Wales, 1979. [103] N.H. Juul, Diffuse radiation view factors from differential plane sources to spheres, ASME Journal of Heat Transfer 101 (3) (1979) 558–560. [104] C.P. Minning, Shape factors between coaxial annular disks separated by a solid cylinder, AIAA Journal 17 (3) (1979) 318–320. [105] C.P. Minning, Radiation shape factors between end plane and outer wall of concentric tubular enclosure, AIAA Journal 17 (12) (1979) 1406–1408. [106] I.G. Currie, W.W. Martin, Temperature calculations for shell enclosures subjected to thermal radiation, Computer Methods in Applied Mechanics and Engineering 21 (1) (1980) 75–79. [107] E. Hahne, M.K. Bassiouni, The angle factor for radiant interchange within a constant radius cylindrical enclosure, Letters in Heat and Mass Transfer 7 (1980) 303–309. [108] M.F. Modest, Solar flux incident on an orbiting surface after reflection from a planet, AIAA Journal 18 (6) (1980) 727–730. [109] W.W. Yuen, A simplified approach to shape-factor calculation between three-dimensional planar objects, ASME Journal of Heat Transfer 102 (2) (1980) 386–388. [110] B.T.F. Chung, M.H.N. Naraghi, Some exact solutions for radiation view factors from spheres, AIAA Journal 19 (8) (1981) 1077–1081. [111] U. Gross, K. Spindler, E. Hahne, Shape factor equations for radiation heat transfer between plane rectangular surfaces of arbitrary position and size with parallel boundaries, Letters in Heat and Mass Transfer 8 (1981) 219. [112] A. Ameri, J.D. Felske, Radiation configuration factors for obliquely oriented finite length circular cylinders, International Journal of Heat and Mass Transfer 33 (1) (1982) 728–736. [113] T.J. Chung, J.Y. Kim, Radiation view factors by finite elements, ASME Journal of Heat Transfer 104 (1982) 792.

938 View Factor Catalogue

[114] B.T.F. Chung, M.H.N. Naraghi, A simpler formulation for radiative view factors from spheres to a class of axisymmetric bodies, ASME Journal of Heat Transfer 104 (1982) 201. [115] N.H. Juul, View factors in radiation between two parallel oriented cylinders, ASME Journal of Heat Transfer 104 (1982) 235. [116] P.V. Kadaba, Thermal radiation view factor methods accuracy and computer-aided procedures, Contract Report NGT-01-002-099, NASA/ASEE, 1982. [117] M.H.N. Naraghi, B.T.F. Chung, Radiation configuration factors between disks and a class of axisymmetric bodies, ASME Journal of Heat Transfer 104 (1982) 426. [118] C. Buraczewski, J. Stasiek, Application of generalized Pythagoras theorem to calculation of configuration factors between surfaces of channels of revolution, International Journal of Heat and Fluid Flow 4 (3) (1983) 157–160. [119] F.W. Lipps, Geometric configuration factors for polygonal zones using Nusselt’s unit sphere, Solar Energy 30 (5) (1983) 413–419. [120] B.T.F. Chung, M.M. Kermani, M.H.N. Naraghi, A formulation of radiation view factors from conical surfaces, AIAA Journal 22 (3) (1984) 429–436. [121] B. Mahbod, R.L. Adams, Radiation view factors between axisymmetric subsurfaces within a cylinder with spherical centerbody, ASME Journal of Heat Transfer 106 (1) (1984) 244. [122] D.W. Yarbrough, C.L. Lee, Monte Carlo calculation of radiation view factors, in: F.R. Payne, et al. (Eds.), Integral Methods in Sciences and Engineering, Harper and Rowe/Hemisphere, 1984. [123] J.I. Eichberger, Calculation of geometric configuration factors in an enclosure whose boundary is given by an arbitrary polygon in the plane, Wärme- und Stoffübertragung 19 (4) (1985) 269. [124] F.U. Mathiak, Berechnung von konfigurationsfaktoren polygonal berandeter ebener gebiete (Calculation of form-factors for plane areas with polygonal boundaries), Wärme- und Stoffübertragung 19 (4) (1985) 273. [125] A.B. Shapiro, Computer implementation, accuracy and timing of radiation view factor algorithms, ASME Journal of Heat Transfer 107 (3) (1985) 730–732. [126] K.N. Shukla, D. Ghosh, Radiation configuration factors for concentric cylinder bodies in enclosure, Indian Journal of Technology 23 (1985) 244–246. [127] G.M. Maxwell, M.J. Bailey, V.W. Goldschmidt, Calculations of the radiation configuration factor using ray casting, Computer Aided Design 18 (7) (1986) 371. [128] P. Stefanizzi, Reliability of the Monte Carlo method in black body view factor determination, Termotechnica 40 (6) (1986) 29. [129] J.C.Y. Wang, S. Lin, P.M. Lee, W.L. Dai, Y.S. Lou, Radiant-interchange configuration factors inside segments of frustum enclosures of right circular cones, International Communications in Heat and Mass Transfer 13 (1986) 423–432. [130] T.L. Eddy, G.E. Nielsson, Radiation shape factors for channels with varying cross-section, ASME Journal of Heat Transfer 110 (1) (1988) 264–266. [131] J.I. Frankel, T.P. Wang, Radiative exchange between gray fins using a coupled integral equation formulation, Journal of Thermophysics and Heat Transfer 2 (4) (Oct 1988) 296–302. [132] M.F. Modest, Radiative shape factors between differential ring elements on concentric axisymmetric bodies, Journal of Thermophysics and Heat Transfer 2 (1) (1988) 86–88. [133] M.M. Mel’man, G.G. Trayanov, View factors in a system of parallel contacting cylinders, Journal of Engineering Physics 54 (4) (1988) 401. [134] M.H.N. Naraghi, Radiation view factors from differential plane sources to disks—a general formulation, Journal of Thermophysics and Heat Transfer 2 (3) (1988) 271–274. [135] M.H.N. Naraghi, Radiative view factors from spherical segments to planar surfaces, Journal of Thermophysics and Heat Transfer 2 (4) (Oct 1988) 373–375. [136] M.H.N. Naraghi, J.P. Warna, Radiation configuration factors from axisymmetric bodies to plane surfaces, International Journal of Heat and Mass Transfer 31 (7) (1988) 1537–1539. [137] M. Sabet, B.T.F. Chung, Radiation view factors from a sphere to nonintersecting planar surfaces, Journal of Thermophysics and Heat Transfer 2 (3) (1988) 286–288. [138] B.T.F. Chung, M.M. Kermani, Radiation view factors from a finite rectangular plate, ASME Journal of Heat Transfer 111 (4) (1989) 1115. [139] J. van Leersum, A method for determining a consistent set of radiation view factors from a set generated by a nonexact method, International Journal of Heat and Fluid Flow 10 (1) (1989) 83. [140] D.E. Bornside, R.A. Brown, View factor between differing-diameter, coaxial disks blocked by a coaxial cylinder, Journal of Thermophysics and Heat Transfer 4 (3) (1990) 414–416. [141] C. Saltiel, M.H.N. Naraghi, Radiative configuration factors from cylinders to coaxial axisymmetric bodies, International Journal of Heat and Mass Transfer 33 (1) (1990) 215–218. [142] J.W.C. Tseng, W. Strieder, View factors for wall to random dispersed solid bed transport, ASME Journal of Heat Transfer 112 (1990) 816–819. [143] A.F. Emery, O. Johansson, M. Lobo, A. Abrous, A comparative study of methods for computing the diffuse radiation viewfactors for complex structures, ASME Journal of Heat Transfer 113 (2) (1991) 413–422. [144] H.E. Rushmeier, D.R. Baum, D.E. Hall, Accelerating the hemi-cube algorithm for calculating radiation form factors, ASME Journal of Heat Transfer 113 (4) (1991) 1044–1047.

View Factor Catalogue

939

[145] J. Sika, Evaluation of direct-exchange areas for a cylindrical enclosure, ASME Journal of Heat Transfer 113 (4) (1991) 1040–1043. [146] A. Ambirajan, S.P. Venkateshan, Accurate determination of diffuse view factors between planar surfaces, International Journal of Heat and Mass Transfer 36 (8) (1993) 2203–2208. [147] A. Beard, D. Drysdale, P. Holborn, Configuration factor for radiation in a tunnel or partial cylinder, Fire Technology 29 (3) (1993) 281–288. [148] L.W. Byrd, View factor algebra for two arbitrary sized non-opposing parallel rectangles, ASME Journal of Heat Transfer 115 (1993) 517–518. [149] J.R. Ehlert, T.F. Smith, View factors for perpendicular and parallel, rectangular plates, Journal of Thermophysics and Heat Transfer 7 (1) (1993) 173–174. [150] A. Guelzim, J.M. Souil, J.P. Vantelon, Suitable configuration factors for radiation calculation concerning tilted flames, ASME Journal of Heat Transfer 115 (2) (May 1993) 489–491. [151] C.V.S. Murty, Evaluation of radiation reception factors in a rotary kiln using a modified Monte Carlo scheme, International Journal of Heat and Mass Transfer 36 (1) (1993) 119–132. [152] H.L. Noboa, D. O’Neal, W.D. Turner, Calculation of the shape factor from a small rectangular plane to a triangular surface perpendicular to the rectangular plane without a common edge, ASME Journal of Solar Energy Engineering 115 (1993) 117–119. [153] H. Brockmann, Analytic angle factors for the radiant interchange among the surface elements of two concentric cylinders, International Journal of Heat and Mass Transfer 37 (7) (1994) 1095–1100. [154] M. Flouros, S. Bungart, W. Leiner, Calculation of the view factors for radiant heat exchange in a new volumetric receiver with tapered ducts, ASME Journal of Solar Energy Engineering 117 (1995) 58–60. [155] K.G.T. Hollands, On the superposition rule for configuration factors, ASME Journal of Heat Transfer 117 (1) (1995) 241–244. [156] D.A. Lawson, An improved method for smoothing approximate exchange areas, International Journal of Heat and Mass Transfer 38 (16) (1995) 3109–3110. [157] R.I. Loehrke, J.S. Dolaghan, P.J. Burns, Smoothing Monte Carlo exchange factors, ASME Journal of Heat Transfer 117 (2) (1995) 524–526. [158] V.R. Rao, V.M.K. Sastri, Efficient evaluation of diffuse view factors for radiation, International Journal of Heat and Mass Transfer 39 (1996) 1281–1286. [159] C.K. Krishnaprakas, View factor between inclined rectangles, Journal of Thermophysics and Heat Transfer 11 (3) (1997) 480–482. [160] B.-W. Li, W.Q. Tao, R.X. Liu, Ray effect in ray tracing method for radiative heat transfer, International Journal of Heat and Mass Transfer 40 (14) (1997) 3419–3426. [161] A. Mavroulakis, A. Trombe, A new semianalytical algorithm for calculating diffuse plane view factors, ASME Journal of Heat Transfer 120 (1) (1998) 279–282. [162] C.P. Tso, S.P. Mahulikar, View factors between finite length rings on an interior cylindrical shell, Journal of Thermophysics and Heat Transfer 13 (3) (1999) 375–379. [163] S.S. Katte, S.P. Venkateshan, Accurate determination of view factors in axisymmetric enclosures with shadowing bodies inside, Journal of Thermophysics and Heat Transfer 14 (1) (2000) 68–76. [164] J.R. Howell, R. Siegel, M.P. Mengüç, Thermal Radiation Heat Transfer, 4th ed., Taylor and Francis-Hemisphere, Washington, 2011.

Appendix E

Exponential Integral Functions The exponential integral functions En (x) and their derivatives occur frequently in radiative heat transfer calculations; therefore, a summary of their properties as well as a brief tabulation are given here. More detailed discussions of their properties may be found in the books by Chandrasekhar [1] and Kourganoff [2], or in mathematical handbooks such as [3]. Detailed tabulations are given in [3], and formulae for their numerical evaluation are listed in [3,4]. The exponential integral of order n is defined as  ∞ dt En (x) = e−xt n , t 1

n = 0, 1, 2, . . . ,

(E.1)

or, setting μ = 1/t,  En (x) =

1

e−x/μ μn−2 dμ,

n = 0, 1, 2, . . . .

(E.2)

0

Differentiating equation (E.1), a first recurrence relationship is found as dEn (x) = −En−1 (x), dx where

 E0 (x) =

∞

n = 1, 2, . . . ,

(E.3)

e−x . x

(E.4)

e−xt dt =

1

A second recurrence is found by integrating equation (E.3), or  ∞ En (x) dx = En+1 (x), n = 0, 1, 2, . . . .

(E.5)

x

An algebraic recurrence between consecutive orders may be obtained by integrating equation (E.1) by parts, or En+1 (x) =

 1  −x e − xEn (x) , n

n = 1, 2, 3, . . . .

(E.6)

The integral of equation (E.1) may be solved in a general series expansion as [3] En (x) =

∞  (−x)n−1 (−x)m (ψn − ln x) + , (n−1)! m!(n−1−m) m=0

n = 1, 2, 3, . . . ,

(E.7a)

mn−1

where ⎧ ⎪ −γE , ⎪ ⎪ ⎪ ⎪ ⎨ n−1  ψn = ⎪ 1 ⎪ ⎪ −γ , + ⎪ ⎪ ⎩ E m

n = 1, n ≥ 2,

(E.7b)

m=1

941

942 Exponential Integral Functions

and

 γE =

∞



1 − e−t

1

 dt t

= 0.577216 . . .

(E.7c)

is known as Euler’s constant. Substituting values for n, one obtains x3 x4 x2 + − + −..., 2!2 3!3 4!4 x3 x4 x2 + − + −..., E2 (x) = 1 + x(γE − 1 + ln x) − 2!1 3!2 4!3   x2 x3 x4 1 3 E3 (x) = − x + −γE + − ln x + − + −.... 2 2 2 3!1 4!2 E1 (x) = −(γE + ln x) + x −

(E.8) (E.9) (E.10)

A function related to E1 that often occurs in radiation calculations is  1  dt  ∞   −ξ = Ein(x) = 1 − e−xt 1 − e−xe dξ t 0 0 x3 x2 + − +.... = E1 (x) + ln x + γE = x − 2!2 3!3

(E.11)

For vanishing values of x it follows from equation (E.7), or directly integrating equation (E.1), that ⎧ ⎪ +∞, ⎪ ⎪ ⎨ En (0) = ⎪ 1 ⎪ ⎪ ⎩ , n−1 Ein(0) = 0.

n = 1, (E.12a)

n ≥ 2,

(E.12b)

For large values of x, the asymptotic expansion for the exponential integrals is given by [3]   e−x n n(n+1) n(n+1)(n+2) En (x) = − + − . . . , n = 0, 1, 2, . . . . 1− + x x x2 x3

(E.13)

To estimate the relative magnitude of different orders of exponential integrals, the following inequalities are sometimes handy [3]: n−1 En (x) < En+1 (x) < En (x), n 1 1 < ex En (x) ≤ , x+n x+n−1

n = 1, 2, 3, . . . ,

(E.14)

n = 1, 2, 3, . . . .

(E.15)

Exponential Integral Functions

943

TABLE E.1 Values of exponential integral functions. x

Ein

E1

E2

E3

E4

0.00

0.000000

∞

1.000000

0.500000

0.333333

0.01

0.009975

4.037929

0.949671

0.490277

0.328382

0.02

0.019900

3.354707

0.913105

0.480968

0.323526

0.03

0.029776

2.959118

0.881672

0.471998

0.318762

0.04

0.039603

2.681263

0.853539

0.463324

0.314085

0.05

0.049382

2.467898

0.827835

0.454919

0.309494

0.06

0.059112

2.295307

0.804046

0.446761

0.304986

0.07

0.068794

2.150838

0.781835

0.438833

0.300559

0.08

0.078428

2.026941

0.760961

0.431120

0.296209

0.09

0.088015

1.918744

0.741244

0.423610

0.291935

0.10

0.097554

1.822924

0.722545

0.416291

0.287736

0.15

0.144557

1.464461

0.641039

0.382276

0.267789

0.20

0.190428

1.222650

0.574201

0.351945

0.249447

0.25

0.235204

1.044283

0.517730

0.324684

0.232543

0.30

0.278920

0.905677

0.469115

0.300042

0.216935

0.35

0.321609

0.794215

0.426713

0.277669

0.202501

0.40

0.363305

0.702380

0.389368

0.257286

0.189135

0.45

0.404039

0.625331

0.356229

0.238663

0.176743

0.50

0.443842

0.559773

0.326644

0.221604

0.165243

0.60

0.520769

0.454379

0.276184

0.191551

0.144627

0.70

0.594310

0.373769

0.234947

0.166061

0.126781

0.80

0.664669

0.310597

0.200852

0.144324

0.111290

0.90

0.732039

0.260184

0.172404

0.125703

0.097812

1.00

0.796600

0.219384

0.148496

0.109692

0.086062

1.10

0.858517

0.185991

0.128281

0.095881

0.075801

1.20

0.917946

0.158408

0.111104

0.083935

0.066824

1.30

0.975031

0.135451

0.096446

0.073576

0.058961

1.40

1.029907

0.116219

0.083890

0.064576

0.052064

1.50

1.082700

0.100020

0.073101

0.056739

0.046007

1.60

1.133528

0.086308

0.063803

0.049906

0.040682

1.70

1.182499

0.074655

0.055771

0.043937

0.035997

1.80

1.229716

0.064713

0.048815

0.038716

0.031870

1.90

1.275274

0.056204

0.042780

0.034143

0.028232

2.00

1.319263

0.048901

0.037534

0.030133

0.025023

2.50

1.518421

0.024915

0.019798

0.016295

0.013782

3.00

1.688876

0.013048

0.010642

0.008931

0.007665

4.00

1.967289

0.003779

0.003198

0.002761

0.002423

5.00

2.187802

0.001148

0.000996

0.000878

0.000783

References [1] S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960, originally published by Oxford University Press, London, 1950. [2] V. Kourganoff, Basic Methods in Transfer Problems, Dover Publications, New York, 1963. [3] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover Publications, New York, 1965. [4] W.F. Breig, A.L. Crosbie, Numerical computation of a generalized exponential integral function, Mathematics of Computation 28 (126) (1974) 575–579.

Appendix F

Computer Codes This appendix contains a listing and brief description of a number of computer programs that may be helpful to the reader of this book, and that can be downloaded from its dedicated website located at https://www.elsevier. com/books-and-journals/book-companion/9780323984065. Some of the codes are very basic and are entirely intended to aid the reader with the solution to the problems given at the end of the more basic chapters. Some of the codes were born out of research, but are basic enough to aid a graduate student with more complicated assignments or a semester project. And a few programs are so sophisticated in nature that they will be useful only to the practicing engineer conducting his or her own research. Finally, it is anticipated that the website will be kept up-to-date and augmented once in a while. Thus, there may be a few additional programs not described in this appendix. It is a fact that most engineers have done, and still do, their programming in Fortran, and the author of this book is no exception. It is also true that computer scientists and most commercial programmers do their work in C++; more importantly, the younger generation of engineers at many universities across the United States is now also learning C++. Both compiled languages have in recent years been trumped by Matlab [1], which—while an interpreted rather than compiled language—has many convenient mathematical and graphical tools. Since all the programs in this listing were written by the authors, either for research purposes or for the creation of this book, they all started their life in Fortran (older programs as Fortran77, and the later ones as Fortran90). However, as a gesture toward the C++ and Matlab communities, the most basic codes have all been converted to C++ as well as Matlab , as indicated below by the program suffixes .cpp and .m. If desired, all other programs are easily converted with freeware translators such as f2c (resulting in somewhat clumsy, but functional codes). Finally, self-contained programs that have been precompiled for Microsoft Windows have the suffix .exe. The programs are listed in order by chapter in which they first appear. More detailed descriptions, sometimes with an example, can be found on the website. Third-party codes that are also provided at the website are listed at the end.

Chapter 1 bbfn.f, bbfn.cpp, bbfn.m: planck.f, planck.cpp, planck.m, planck.exe:

Function bbfn(x) calculates the fractional blackbody emissive power, as defined by equation (1.24), where the argument is x = nλT with units of μm K. planck is a small stand-alone program that prompts the user for input (temperature and wavelength or wavenumber), then calculates the spectral blackbody emissive powers Ebλ /T5 , Ebη /T3 , and the fractional blackbody emissive power f (λT).

Chapters 2 and 3 fresnel.f, fresnel.cpp, fresnel.m:

Subroutine fresnel calculates Fresnel reflectivities from equation (2.113) for a given complex index of refraction and incidence angle.

Chapter 3 emdiel.f90, emdiel.cpp, emdiel.m:

Function emdiel calculates the unpolarized, spectral, hemispherical emissivity of an optical surface of a dielectric material from equation (3.82).

945

946 Computer Codes

emmet.f90, emmet.cpp, emmet.m:

Function emmet calculates the unpolarized, spectral, hemispherical emissivity of an optical surface of a metallic material from equation (3.77).

callemdiel.f90, callemdiel.cpp, callemdiel.m, callemdiel.exe:

Program callemdiel is a stand-alone front end for function emdiel, prompting for input (refractive index n) and returning the unpolarized, spectral, hemispherical as well as normal emissivities.

callemmet.f90, callemmet.cpp, callemmet.m, callemmet.exe:

Program callemmet is a stand-alone front end for function emmet, prompting for input (complex index of refraction n, k) and returning the unpolarized, spectral, hemispherical as well as normal emissivities.

dirreflec.f, dirreflec.cpp, dirreflec.m, dirreflec.exe:

Program dirreflec is a stand-alone front end for subroutine fresnel, returning perpendicular polarized, parallel polarized, and unpolarized reflectances.

totem.f90, totem.cpp, totem.m:

Program totem is a routine to evaluate the total, directional or hemispherical emittance or absorptance of an opaque material, based on an array of spectral data.

Chapter 4 and Appendix D view.f90, view.cpp, view.m:

A function to evaluate any of the 51 view factors given in Appendix D.

parlplates.f90, parlplates.cpp parlplates.m:

A function to evaluate the view factor between two displaced parallel plates, as given by equation (4.42).

perpplates.f90, perpplates.cpp, perpplates.m:

A function to evaluate the view factor between two displaced perpendicular plates, as given by equation (4.41).

viewfactors.f90, viewfactors.cpp, viewfactors.m, viewfactors.exe:

A stand-alone front end to functions view, parlplates, and perpplates. The user is prompted to input configuration number and arguments; the program then returns the requested view factor.

vfplanepoly.f90:

A code to calculate the view factor between two arbitrary planar triangles or quadrilaterals using the contour integral method (Section 4.10).

Chapter 5 graydiff.f90, graydiff.cpp, graydiff.m:

Subroutine graydiff provides the solution to equation (5.38) for an enclosure consisting of N gray-diffuse surfaces. For each surface the area, emittance, external irradiation, and either heat flux or temperature must be specified. In addition, the upper triangle of the view factor matrix must be provided (Fi−j ; i = 1, N; j = i, N). For closed configurations, the diagonal view factors Fi−i are not required, since they can be calculated from the summation rule. The remaining view factors are calculated from reciprocity. On output, the program provides all view factors, and temperatures and radiative heat fluxes for all surfaces.

graydiffxch.f90, graydiffxch.cpp, graydiffxch.m:

Program graydiffxch is a front end for subroutine graydiff, generating the necessary input parameters for a three-dimensional variation to Example 5.4, primarily view factors calculated by calls to function view. This program may be used as a starting point for more involved radiative exchange problems.

Computer Codes 947

Chapter 6 graydifspec.f90, graydifspec.cpp, graydifspec.m:

grspecxch.f90, grspecxch.cpp, grspecxch.m: semigray.f90, semigray.cpp, semigray.m:

semigrxch.f90, semigrxch.cpp, semigrxch.m:

Subroutine graydifspec provides the solution to equation (6.21) for an enclosure consisting of N diffusely emitting surfaces with diffuse and specular reflectance components. For each surface the area, emittance, specular reflectance, external irradiation, and either heat flux or temperature must be specified. In addition, the upper triangle of the specular view factor matrix must be provided (Fsi−j ; i = 1, N; j = i, N). Otherwise same as graydiff. Program grspecxch is a front end for subroutine graydifspec, similar to graydiffxch.

Subroutine semigray provides the solution to equations (6.41) for an enclosure consisting of N diffusely emitting surfaces with diffuse and specular reflectance components, considering two spectral ranges (one for external irradiation, one for emission). For each surface the area, emittance and specular reflectance (two values each), external irradiation, and either heat flux or temperature must be specified. Otherwise same as graydifspec. For convenience, slightly stripped versions for purely diffuse surfaces are also included, named semigraydf. Program semigrxch is a front end for subroutine semigray providing the necessary input for Example 6.9. This program may be used as a starting point for more involved radiative exchange problems. For convenience, slightly stripped versions for purely diffuse surfaces are also included, named semigrxchdf (with input for Example 6.9).

bandapp.f90, bandapp.cpp, bandapp.m:

Subroutine bandapp provides the solution to equations (6.42) for an enclosure consisting of N diffusely emitting surfaces with diffuse and specular reflectance components, considering M spectral bands. For each surface the area, emittance, specular reflectance and external irradiation (one value for each spectral band), and either heat flux or temperature must be specified. Otherwise same as graydiff. For convenience, slightly stripped versions for purely diffuse surfaces are also included, named bandappdf.

bandmxch.f90, bandmxch.cpp, bandmxch.m:

Program bandmxch is a front end for subroutine bandapp providing the necessary input for Example 6.8. This program may be used as a starting point for more involved radiative exchange problems. For convenience, slightly stripped versions for purely diffuse surfaces are also included, named bandmxchdf (also with input for Example 6.8).

Chapter 7 MCintegral.f90:

MCintegral is a little program that evaluates the integral function by the Monte Carlo method.

b a

f (x) dx for any specified

Chapter 8 ExStoSEn1D.f90:

A code to calculate the temperature distribution in a 1D slab by coupling 1D conduction to surface-to-surface radiation using explicit coupling procedure (Example 8.1).

ImStoSEn1D.f90:

A code to calculate the temperature distribution in a 1D slab by coupling 1D conduction to surface-to-surface radiation using semi-implicit coupling procedure (Example 8.2).

Chapter 10 voigt.f:

Subroutine voigt calculates the spectral absorption coefficient for a Voigt-shaped line based on the fast algorithm by Humlí˘cek [2], as a function of line intensity, and Lorentz and Doppler line widths.

nbkdistdb.f90:

Program nbkdistdb is a Fortran90 code to calculate narrow band k-distributions for a number of temperatures and a number of wavenumber ranges, for a gas mixture containing CO2 , H2 O, CH4 , and soot. The spectral absorption coefficient is calculated directly from the HITRAN or HITEMP databases.

948 Computer Codes

nbkdistsg.f90:

Program nbkdistsg is a Fortran90 code to calculate a single narrow band k-distribution from a given array of wavenumber–absorption coefficient pairs.

wbmxxx.f, wbmxxxcl.f, wbmxxxcl.exe:

Subroutines wbmxxx, where xxx stands for the different gases h20, co2, ch4, co, no, and so2, calculate for a given temperature the ratios Ψ∗ (T)/Ψ∗ (T0 ) [from equations (10.156) and (10.160)] and Φ(T)/Φ(T0 ) [from equation (10.161)], i.e., the functions shown in Figs. 10.25 through 10.27. The stand-alone programs wbmxxxcl.f are front ends for the wbmxxx.f, prompting the user for input, and printing the ratios Ψ∗ (T)/Ψ∗ (T0 ) and Φ(T)/Φ(T0 ) to the screen for all bands listed in Table 10.4.

emwbm.f, ftwbm.f, wangwbm.f:

Fortran functions to calculate the nondimensional total band absorptance A∗ from the Edwards and Menard model, Table 10.3 (emwbm), the Felske and Tien model, equation (10.168) (ftwbm), and the Wang model, equation (10.170) (wangwbm).

wbmodels.f, wbmodels.exe:

Stand-alone front end for the functions emwbm, ftwbm, and wangwbm; the nondimensional total band absorptance A∗ is printed to the screen, as calculated from three band models (Edwards and Menard, Felske and Tien, and Wang models).

wbmkvsg.f:

Fortran subroutine wbmkvsg calculates the κ∗ vs. g∗ distribution of equation (10.182).

totemiss.f:

Fortran subroutine totemiss calculates the total emissivity of an isothermal gas mixture, using Leckner’s model, equations (10.188) through (10.194).

totabsor.f:

Fortran subroutine totabsor calculates the total absorptivity of an isothermal gas mixture, using Leckner’s model, equations (10.188) through (10.194).

Leckner.f, Leckner.exe:

Stand-alone front end for totemiss and totabsor, with total emissivities and absorptivities printed to the screen.

Chapter 11 coalash.f90:

This file contains subroutine coalash (plus a front end for screen input and output) to determine nondimensionalized spectral absorption and extinction coefficients κ∗ and β∗ , as listed in Table 11.3, from the Buckius and Hwang [3] and the Mengüç and Viskanta [4] models, as functions of complex index of refraction m = n − ik and size parameter x.

mmmie.f:

Program mmmie calculates Mie coefficients (scattering coefficients an and bn , efficiencies Qsca , Qext , and Qabs , and asymmetry factor g; see Section 11.2 for definitions), and relates them to particle cloud properties (extinction coefficient β, absorption coefficient κ, scattering coefficient σs , cloud asymmetry factor g, scattering phase function Φ for specified scattering angles, and phase function expansion coefficients An , as defined in Section 11.3).

Chapter 15 P1sor.f90, P1sor.cpp:

Subroutine P1sor provides the solution to equation (15.38) with its boundary condition (15.48) for a two-dimensional (rectangular or axisymmetric cylinder) enclosure with reflecting walls and an absorbing, emitting, linear-anisotropically scattering medium. For each surface the emittance and blackbody intensities must be specified; for the medium spatial distributions of radiation properties and blackbody intensities must be input. Internal incident radiation (G) and wall flux (q) fields are calculated. Can be used for gray problems or on a spectral basis.

P1-2D.f90, P1-2D.cpp:

Program P1-2D is a front end for subroutine P1sor, setting up the problem for a gray medium with spatially constant radiative properties; it may be used as a starting point for more involved applications.

Delta.f90:

Program Delta is a stand-alone program to calculate the rotation matrix Δnmm (α, β, γ) required for the boundary conditions of higher-order PN -approximations, as given by equations (15.63) through (15.66).

Computer Codes 949

pnbcs.f90:

Program pnbcs is a stand-alone program to calculate the Legendre half-moments pm n, j and coefficients um , vm , wm , which are required for the boundary conditions of higherli li li order PN -approximations, as given by equations (15.70) through (15.71).

Chapter 18 transPN.f90:

Program transPN calculates energy from a pulsed collimated laser source transmitted through an absorbing, isotropically scattering slab as a function of time, using the P1 and P1/3 methods.

Chapter 19 wsggBrd.f90

Fortran subroutine to calculate the WSGG parameters for CO2 –H2 O–N2 mixtures with the Bordbar correlation given by equation (19.92) [28].

wsggKng.f90

Fortran subroutine to calculate the WSGG parameters for CO2 –H2 O–N2 mixtures with the Kangwanpongpan correlation given by equation (19.92) [29].

wsggsoot.f90

Fortran subroutine to calculate the WSGG parameters for soot, for gray soot as well as with the Cassol correlation [30].

wsggex1D.f90

This Fortran program (written to compute data for Example 19.8) is included here to illustrate how the WSGG routines are used for gas–soot mixtures, and may be used as a starting point for more complicated problems.

fskdist.f90:

Program fskdist is a Fortran90 code to calculate full-spectrum k-distributions for a number of Planck function temperatures and a single gas property state (temperature, partial and total pressures), for a gas mixture containing CO2 , H2 O, CH4 , and soot; weight functions a(T, T0 , g) are calculated, as well. The spectral absorption coefficient is either calculated directly from the HITRAN or HITEMP databases, or is supplied by the user.

fskdco2.f90, fskdh2o.f90:

These subroutines determine full spectrum cumulative k-distributions for CO2 and H2 O, respectively, employing the correlations of Modest and Mehta [5] and of Modest and Singh [6].

fskdco2dw.f90, fskdh2odw.f90:

Equivalent to fskdco2.f90 and fskdh2o.f90, but employing the older correlations of Denison and Webb [7,8].

kdistmix.f90:

Subroutine kdistmix finds the cumulative k-distribution for an n-component mixture from a given set of individual species cumulative k-distributions (narrow band, wide band, or full spectrum), employing the mixing scheme of Modest and Riazzi [9].

fskdistmix.f90:

This Fortran90 routine finds the full spectrum cumulative k-distribution for a CO2 –H2 O mixture, employing the correlations of Modest and Mehta [5] and Modest and Singh [6], using one of three mixing schemes described by equations (19.195) (superposition), (19.196) (multiplication), or (19.200) (uncorrelated mixture).

Chapter 20 mocacyl.f, rnarray.f:

Program mocacyl is a Monte Carlo routine for a nongray, nonisothermal, isotropically scattering medium confined inside a two-dimensional, axisymmetric cylindrical enclosure bounded by nongray, diffusely emitting and reflecting walls. Temperature and radiative properties are assumed known everywhere inside the enclosure and along the walls. Requires use of program rnarray to set up random number relationships (locations and wavenumbers of emission vs. random numbers). Calculates internal radiative heat sources ∇ · qR as well as local radiative fluxes to the walls qRw .

950 Computer Codes

FwdMCcs.f90, FwdMCck1.f90, FwdMCck2.f90:

Program FwdMCcs is a standard forward Monte Carlo code for a narrow collimated beam penetrating through a nonabsorbing, isotropically scattering slab, calculating the flux onto a small, directionally selective detector, as given in Example 20.3. FwdMCck1 and FwdMCck2 are forward Monte Carlo codes for the same problem, but also allow for absorption in the medium; FwdMCck1 uses standard ray tracing, while FwdMCck2 uses energy partitioning; see Example 20.4.

FwdMCps.f90:

Program FwdMCps is a standard forward Monte Carlo code for a radiative energy emitted by a point source penetrating through a nonabsorbing, isotropically scattering slab, calculating the flux onto a small, directionally selective detector.

RevMCcs.f90, RevMCck1.f90, RevMCck2.f90:

These programs are backward Monte Carlo implementations of the equivalent FwdMCcs, FwdMCcka1, and FwdMCcka2, as also discussed in Examples 20.3 and 20.4.

RevMCps.f90:

The backward Monte Carlo equivalent of FwdMCps.

Chapter 21 CpldP1En1D.f90:

Coupled finite-difference solution of the 1D energy and P1 equations using both explicit and semi-implicit coupling procedures (Example 21.2).

CpldP1En2D.f90:

Coupled finite-difference solution of the 2D axisymmetric energy and P1 equations using both explicit and semi-implicit coupling procedures (Example 21.4).

Software Packages MONT3D

This code, developed at Colorado State University by Burns et al. [10–14], calculates radiative exchange factors for complicated, three-dimensional geometries by the Monte Carlo method, as given by equations (7.15) and (7.21). Diffuse and specular view factors may be calculated as special cases.

VIEW3D

This code, developed at National Institute of Standards and Technology (NIST) by Walton [15], calculates radiative view factors with obstructions by adaptive integration.

RADCAL

This code, developed at NIST by Grosshandler [16,17], is a narrow band database for combustion gas properties, using tabulated values and theoretical approximations.

EM2C

This package contains a number of Fortran codes, developed at the Ecole Centrale de Paris by Soufiani and Taine [18], and updated and extended by Rivière and Soufiani [19], now based on CDSD-4000 [20] and HITEMP 2010 [21]. The codes supply atmospheric pressure statistical narrow band properties for CO2 , H2 O, CO, and CH4 , as well as narrow band k-distributions for CO2 and H2 O.

FVM2D

This Fortran77 code, developed at the University of Minnesota and Nanyang Technological University by Chai and colleagues [22–24], calculates radiative transfer in participating media using the finite-angle method of Chapter 16 for a two-dimensional, rectangular enclosure with reflecting walls and an absorbing, emitting, anisotropically scattering medium. For each surface the emittance and blackbody intensities must be specified; for the medium spatial distributions of radiation properties and blackbody intensities must be input. Internal incident radiation (G) and wall flux (q) fields are calculated. Can be used for gray problems or on a spectral basis.

H2OEmissivity.xlsx, Excel data sheets to calculate total emissivities of combustion gases CO2 [25], H2 O [26], CO2Emissivity.xlsx, and CO2 –H2 O–CO–N2 mixtures [27], with validity ranges equal to, or exceeding: MixEmissivity.xlsx temperature 300 ≤ T ≤ 3000 K, total pressure 0.1 ≤ p ≤ 40 bar, and pressure path length 0.05 ≤ pa L ≤ 1000 bar cm.

Computer Codes 951

Software Packages at Repository Some software packages developed by the first author’s group are too large for the book’s dedicated website and/or are occasionally updated, and are maintained in a repository at Marquette University, and may be downloaded from https://www.eng.mu.edu/ccl/software-data/radiation/. NBKDIR

This package contains a number of Fortran codes, developed at the Pennsylvania State University and the University of California at Merced by the primary author and his students/postdocs A. Wang, G. Pal, and J. Cai, for the assembly of full spectrum k-distributions from a narrow band k-distributions database [31,32]. At the time of printing NBKDIR contained data for five species (CO2 , H2 O, CO, CH4 , C2 H4 ), as well as nongray soot [calculated from the Chang and Charalampopoulos correlation [33] given in equation (11.119)], for temperatures up to 3000 K and pressures up to 80 bar. Spectroscopic data are taken from the HITEMP 2010 (CO2 , H2 O, CO) [21] and HITRAN 2008 (CH4 , C2 H4 ) [34].

FSK Databases

At present four different full-spectrum k-distribution databases created by Wang and coworkers are posted in the repository: the first contains distribution for 32 Gaussian quadrature points (i.e., without transformation or α = 1) for all conditions given in Table 19.4 but without soot [35]. The second and third databases include soot while calculating the stretch factor a on-the-fly, the second using standard Gaussian quadrature points [36] and in the third the quadrature points are “optimized,” i.e., transformed with α = 2 as introduced in equation (19.143) [37]. Finally, a fourth was built specifically for the FSCK-4 scheme (atmospheric pressure only) [38].

LBL Monte Carlo Database

This database contains a lookup table to determine emission wavenumbers (plus corresponding absorption coefficients) as function of random number for mixtures of H2 O, CO2 , CO, CH4 , C2 H4 , and soot, for temperatures 300–3000 K and pressures 0.1 to 80 bar, assembled by Ren and Modest [39], as outlined in Section 20.4.

FSK–PMC Database

This database contains a lookup table to determine emission pseudo-wavenumbers g0 as function of random number for mixtures of H2 O, CO2 , CO, and soot for Monte Carlo calculations with the FSCK-4 spectral model, as outlined in Section 20.4. Assembled by Wang et al. [40].

LBL Absorption Coefficient Database

This database contains a lookup table of spectral absorption coefficients, for temperatures 300–3000 K and pressures 0.1 to 80 bar, obtained from the HITEMP 2010 (CO2 , H2 O, CO) [21] and HITRAN 2008 (CH4 , C2 H4 ) [34] spectroscopic databases.

1DRTEsolv

This package contains a Fortran code that calculates the radiative transfer in a onedimensional plane-parallel medium with specified temperature field bound by two gray walls, i.e., the solution to equations (13.46) and (13.47). The code takes in the physical size of the domain, temperature field, distribution of participating media and wall properties (temperature and emittance). Three spectral models may be chosen (Planck-mean gray, LBL, FSCK-2, or user-defined). Developed by Prof. Roy and his students at Marquette University.

References [1] MathWorks MATLAB website, http://www.mathworks.com/products/matlab/. [2] J. Humlí˘cek, Optimized computation of the Voigt and complex probability functions, Journal of Quantitative Spectroscopy and Radiative Transfer 27 (1982) 437. [3] R.O. Buckius, D.C. Hwang, Radiation properties for polydispersions: application to coal, ASME Journal of Heat Transfer 102 (1980) 99–103. [4] M.P. Mengüç, R. Viskanta, On the radiative properties of polydispersions: a simplified approach, Combustion Science and Technology 44 (1985) 143–159. [5] M.F. Modest, R.S. Mehta, Full spectrum k-distribution correlations for CO2 from the CDSD-1000 spectroscopic databank, International Journal of Heat and Mass Transfer 47 (2004) 2487–2491.

952 Computer Codes

[6] M.F. Modest, V. Singh, Engineering correlations for full spectrum k-distribution of H2 O from the HITEMP spectroscopic databank, Journal of Quantitative Spectroscopy and Radiative Transfer 93 (2005) 263–271. [7] M.K. Denison, B.W. Webb, Development and application of an absorption line blackbody distribution function for CO2 , International Journal of Heat and Mass Transfer 38 (1995) 1813–1821. [8] M.K. Denison, B.W. Webb, An absorption-line blackbody distribution function for efficient calculation of total gas radiative transfer, Journal of Quantitative Spectroscopy and Radiative Transfer 50 (1993) 499–510. [9] M.F. Modest, R.J. Riazzi, Assembly of full-spectrum k-distributions from a narrow-band database; effects of mixing gases, gases and nongray absorbing particles, and mixtures with nongray scatterers in nongray enclosures, Journal of Quantitative Spectroscopy and Radiative Transfer 90 (2) (2005) 169–189. [10] P.J. Burns, MONTE–a two-dimensional radiative exchange factor code, Technical report, Colorado State University, Fort Collins, 1983. [11] J.D. Maltby, Three-dimensional simulation of radiative heat transfer by the Monte Carlo method, M.S. thesis, Colorado State University, Fort Collins, CO, 1987. [12] P.J. Burns, J.D. Maltby, Large-scale surface to surface transport for photons and electrons via Monte Carlo, Computing Systems in Engineering 1 (1) (1990) 75–99. [13] J.D. Maltby, P.J. Burns, Performance, accuracy and convergence in a three-dimensional Monte Carlo radiative heat transfer simulation, Numerical Heat Transfer – Part B: Fundamentals 16 (1991) 191–209. [14] C.N. Zeeb, P.J. Burns, K. Branner, J.S. Dolaghan, User’s Manual for Mont3d – Version 2.4, Colorado State University, Fort Collins, CO, 1999. [15] G.N. Walton, Calculation of obstructed view factors by adaptive integration, Technical Report NISTIR–6925, National Institute of Standards and Technology (NIST), Gaithersburg, MD, 2002. [16] W.L. Grosshandler, Radiative transfer in nonhomogeneous gases: a simplified approach, International Journal of Heat and Mass Transfer 23 (1980) 1447–1457. [17] W.L. Grosshandler, RADCAL: a narrow-band model for radiation calculations in a combustion environment, Technical Report NIST Technical Note 1402, National Institute of Standards and Technology, 1993. [18] A. Soufiani, J. Taine, High temperature gas radiative property parameters of statistical narrow-band model for H2 O, CO2 and CO, and correlated-k model for H2 O and CO2 , International Journal of Heat and Mass Transfer 40 (4) (1997) 987–991. [19] P. Rivière, A. Soufiani, Updated band model parameters for H2 O, CO2 , CH4 and CO radiation at high temperature, International Journal of Heat and Mass Transfer 55 (2012) 3349–3358. [20] S.A. Tashkun, V.I. Perevalov, CDSD-4000: high-resolution, high-temperature carbon dioxide spectroscopic databank, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (9) (2011) 1403–1410, available from ftp://ftp.iao.ru/pub/CDSD-4000. [21] L.S. Rothman, I.E. Gordon, R.J. Barber, H. Dothe, R.R. Gamache, A. Goldman, V.I. Perevalov, S.A. Tashkun, J. Tennyson, HITEMP, the high-temperature molecular spectroscopic database, Journal of Quantitative Spectroscopy and Radiative Transfer 111 (15) (2010) 2139–2150. [22] J.C. Chai, H.S. Lee, S.V. Patankar, Finite volume method for radiation heat transfer, Journal of Thermophysics and Heat Transfer 8 (3) (1994) 419–425. [23] J.C. Chai, H.S. Lee, S.V. Patankar, Treatment of irregular geometries using a Cartesian coordinates finite-volume radiation heat transfer procedure, Numerical Heat Transfer – Part B: Fundamentals 26 (1994) 225–235. [24] J.C. Chai, G. Parthasarathy, H.S. Lee, S.V. Patankar, Finite volume method radiative heat transfer procedure for irregular geometries, Journal of Thermophysics and Heat Transfer 9 (3) (1995) 410–415. [25] M. Alberti, R. Weber, M. Mancini, Re-creating Hottel’s emissivity charts for carbon dioxide and extending them to 40 bar pressure using HITEMP-2010 database, Combustion and Flame 162 (2015) 597–612. [26] M. Alberti, R. Weber, M. Mancini, Re-creating Hottel’s emissivity charts for water vapor and extending them to 40 bar pressure using HITEMP-2010 database, Combustion and Flame 169 (2016) 141–153. [27] M. Alberti, R. Weber, M. Mancini, Gray gas emissivities for H2 O–CO2 –CO–N2 mixtures, Journal of Quantitative Spectroscopy and Radiative Transfer 219 (2018) 274–291. [28] M.H. Bordbar, G. Wecel, T. Hyppänen, A line by line based weighted sum of gray gases model for inhomogeneous CO2 –H2 O mixture in oxy-fired combustion, Combustion and Flame 161 (2014) 2435–2445. [29] T. Kangwanpongpan, F.H.R. França, R.C. da Silva, P.S. Schneider, H.J. Krautz, New correlations for the weighted-sum-of-gray-gases model in oxy-fuel conditions based on HITEMP 2010 database, International Journal of Heat and Mass Transfer 55 (2012) 7419–7433. [30] F. Cassol, R. Brittes, F.H.R. França, O.A. Ezekoye, Application of the weighted-sum-of-gray-gases model for media composed of arbitrary concentrations of H2 O, CO2 and soot, International Journal of Heat and Mass Transfer 79 (2014) 796–806. [31] A. Wang, M.F. Modest, High-accuracy, compact database of narrow-band k-distributions for water vapor and carbon dioxide, Journal of Quantitative Spectroscopy and Radiative Transfer 93 (2005) 245–261. [32] J. Cai, M.F. Modest, Improved full-spectrum k-distribution implementation for inhomogeneous media using a narrow-band database, Journal of Quantitative Spectroscopy and Radiative Transfer 141 (2013) 65–72. [33] H. Chang, T.T. Charalampopoulos, Determination of the wavelength dependence of refractive indices of flame soot, Proceedings of the Royal Society (London) A 430 (1880) (1990) 577–591.

Computer Codes 953

[34] L.S. Rothman, I.E. Gordon, A. Barbe, D.C. Benner, P.F. Bernath, M. Birk, V. Boudon, L.R. Brown, A. Campargue, J.-P. Champion, K. Chance, L.H. Coudert, V. Dana, V.M. Devi, S. Fally, J.-M. Flaud, R.R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W.J. Lafferty, J.-Y. Mandin, S.T. Massie, S.N. Mikhailenko, C.E. Miller, N. Moazzen-Ahmadi, O.V. Naumenko, A.V. Nikitin, J. Orphal, V.I. Perevalov, A. Perrin, A. Predoi-Cross, C.P. Rinsland, M. Rotger, M. Simeckova, M.A.H. Smith, K. Sung, S.A. Tashkun, J. Tennyson, R.A. Toth, A.C. Vandaele, J.V. Auwera, The HITRAN 2008 molecular spectroscopic database, Journal of Quantitative Spectroscopy and Radiative Transfer 110 (2009) 533–572. [35] C. Wang, W. Ge, M.F. Modest, B. He, A full-spectrum k-distribution look-up table for radiative transfer in nonhomogeneous gaseous media, Journal of Quantitative Spectroscopy and Radiative Transfer 168 (2016) 46–56. [36] C. Wang, M.F. Modest, B. He, Full-spectrum k-distribution look-up table for nonhomogeneous gas–soot mixtures, Journal of Quantitative Spectroscopy and Radiative Transfer 176 (2016) 129–136. [37] C. Wang, M.F. Modest, B. He, Improvement of full-spectrum k-distribution method using quadrature transformation, International Journal of Thermal Sciences 108 (2016) 100–107. [38] C. Wang, B. He, M.F. Modest, T. Ren, Efficient full-spectrum correlated-k-distribution look-up table, Journal of Quantitative Spectroscopy and Radiative Transfer 219 (2018) 108–116. [39] T. Ren, M.F. Modest, Line-by-line random-number database for photon Monte Carlo simulations of radiation in participating media, ASME Journal of Heat Transfer 141 (2) (2019) 0227019. [40] C. Wang, M.F. Modest, B. He, Full-spectrum correlated-k-distribution look-up table for use with radiative Monte Carlo solvers, International Journal of Heat and Mass Transfer 131 (2019) 167–175.

Author Index A Abanades, S., 847, 858 Abarbanel, S. S., 281 Abbassi, M. A., 613 Abbott, E. A., 124 Abbott, G. L., 74, 76, 79, 121 Abdul-Sater, H., 848 Abe, K., 816 Abe, T., 395 Abed, A. A., 811 Abedrabbo, S., 126 Aber, J., 654 Abo, M., 398 Abraham, J., 835, 854 Abramowitz, M., 159, 259, 341, 395, 408, 446, 515, 558, 734, 943 Abrams, M., 812 Abrous, A., 159, 938 Abu-Romia, M. M., 376, 397, 399 Abulwafa, E. M., 808 Ackermann, S., 811 Adams, B. R., 593, 611 Adams, C. N., 771 Adams, R. L., 938 Adams, R. P., 882 Adams, V. H., 283 Affolter, K., 463 Afgan, N. H., 855 Afghan, N. F., 448 Agarwal, B. A., 438 Agarwal, B. M., 451 Ahluwalia, R. K., 429, 448, 559 Ai, Q., 465, 773 Aihara, T., 638 Airaksinen, V. M., 463 Åkesson, E. O., 882 Al-Ghamdi, A. S., 613 Al Hamwi, M., 125 Al-Turki, A. M., 639, 798, 809, 815 Alawi, S. M., 399 Albers, L. U., 179, 196, 233, 935 Alberti, M., 327, 374, 391, 393, 394, 398, 952 Alemi, A. A., 902 Alemi, E., 849 Alexander, R. W., 121 Alfano, G., 180, 196, 937 Alfano, O. M., 452 Alifanov, O. M., 881 Alim, M. A., 816 Allam, T. A., 679, 732

Allen, G. E., 234 Alpaydin, E., 882 Altaç, Z., 509, 511 Altenkirch, R. A., 450, 849 Altigilbers, L. L., 810 Alvarado, M. J., 398 Alwahaby, Z. T., 857 Amaya, J., 854 Ambirajan, A., 153, 159, 939 Ameri, A., 937 Amin, M., 851 Amiri, H., 612, 690, 730 Amlin, D. W., 811 An, W., 612, 883 Anders, H., 52, 58 Andersen, F. M. B., 661, 731 Andersen, K. M., 661, 731 Anderson, E. E., 465, 809, 812 Anderson, L. W., 398 Anderson, M. R., 430, 448 Anderson, R. J., 399 Andersson, K., 732, 850, 856 André, F., 342, 344, 347, 395, 396, 734, 735, 853, 885 André, S., 808–810 Andreozzi, A., 465 Anisimov, S., 654 Apelblat, Y., 449 Appel, J., 449 Apte, S. V., 855 Arambakam, R., 459, 465 Arancibia-Bulnes, C. A., 857, 858 Arduini-Schuster, M. C., 452, 465, 466, 810, 883 Argento, C., 614 Aringer, B., 394 Arking, A., 345, 396 Armaly, B. F., 216, 233, 642, 653, 809 Armante, R., 393 Armengol, J. M., 852 Armstrong, B. H., 393 Arnas, O. A., 281 Arnold, J., 395 Arnold, K. J., 881 Arnold, N., 654 Arnott, W. P., 450, 451 Arpaci, V. S., 799, 815 Arsenin, V. Y., 881 Artyukhin, E., 881

Arvo, J., 259 Ashley, E. J., 123 Ashman, P. J., 857 Askri, F., 612 Asllanaj, F., 464, 601, 613 Atalla, M. R., 494 Atalla, R., 399 Atashafrooz, M., 815 Ates, C., 850 Atkinson, W. H., 125 Atreya, A., 849 Attia, M. T., 494 Audit, E., 511, 561 Auer, L. H., 511, 654 Auwera, J. V., 392, 394, 733, 953 Ávila Marín, A. L., 844, 857 Avila, S., 772 Aydin, F., 850 Ayrancı, I., 885 Azad, F. H., 422, 424, 446, 479, 488, 493, 494, 797, 814, 815

B Babikov, Y., 395 Babrauskas, V., 855 Bachner, F. J., 123 Bader, R., 857 Badinand, T., 848 Badri, M. A., 465 Badri, P., M. A. Jolivet, 613 Baek, S. W., 592, 602, 603, 611, 613, 614, 655, 811, 814, 848–850, 854, 884 Bagley, B. G., 454, 464 Bai, L., 447 Bailey, M. J., 938 Baillis, D., 401, 425, 445, 446, 457, 458, 464–466, 883 Bakan, S., 816 Baker, B., 258 Baker, F. B., 238 Baker, L. L., 772 Balaji, C., 283, 883 Balakrishnan, A., 397, 677, 732, 813 Ball, K. S., 559, 611, 815, 816 Ballance, J. O., 937 Balridge, A. M., 124 Balsara, D., 592, 611 Baneshi, M., 123 Bang, S. Y., 122, 655 Bansal, A., 393, 395, 715, 733, 772

955

956 Author Index

Bao, H., 81, 121, 903 Barbe, A., 392–395, 729, 733, 953 Barber, R. J., 393, 394, 729, 952 Bard, S., 433, 450 Bard, Y. B., 881 Bardsley, M. E. A., 448 Barford, N. C., 239, 259 Barker, A. J., 456, 462–464, 466 Barlev, D., 856 Barlow, C. H., 449 Barlow, R. S., 399, 561, 735, 849, 850, 852, 853 Barnes, J. W., 882 Bartas, J. G., 275, 282 Bartell, F. O., 937 Bass, C. D., 464 Bass, F. G., 85, 122 Bassiouni, M. K., 937 Basu, S., 654, 901, 902 Battuello, M., 116, 126 Bauer, H.-J., 614, 852 Bäuerle, D., 654 Baum, B. A., 466 Baum, D. R., 938 Bayazitoglu, ˘ Y., 525, 538, 559, 574, 609, 772, 813 Bdéoui, F., 799, 816 Beach, H. L., 494 Beard, A., 939 Beck, F., 903 Beck, J. V., 810, 859, 881, 882 Becker, A., 449 Becker, H. A., 609, 610, 850 Becker, R., 566, 604, 609, 614 Beckett, P., 642, 653 Beckman, F. S., 882 Beckman, W. A., 97, 99, 123 Beckmann, P., 85, 87, 122, 233 Beckner, V. E., 593, 611 Beder, E. C., 454, 464 Beer, J. M., 448 Beeri, Z., 615 Beier, K., 395 Belhaj Ali, H., 612 Bell, R. J., 99, 121, 123 Bell, S. E., 121 Belosevic, S., 850 Ben-Abdallah, P., 308, 493, 643, 654 Ben-Mansour, R., 856 Ben Nasrallah, S., 612 Benedict, W. S., 393 Benner, D. C., 392, 394, 395, 733, 953 Bennethum, W. H., 125 Bennett, B. A. V., 849 Bennett, H. E., 86, 89, 122, 123 Bennett, J. M., 123 Benoist, P., 494 Bergero, S., 613, 797, 814 Bergman, T. L., 282, 611, 815 Bergquam, J. B., 809, 814 Bergstrom, R. W., 437, 451 Berland, K. H., 814 Bermejo, M., 560 Bernard, J. J., 937

Bernardi, M. P., 902 Bernath, P. F., 392, 393, 395, 729, 733, 953 Berry, M. V., 87, 122 Bertie, J. E., 449 Best, P. E., 125, 126, 884 Bethe, H. A., 392 Bevans, J. T., 126, 218, 223, 234, 399 Beyler, C. L., 855 Bharadwaj, S. P., 344, 356, 383, 385, 386, 394, 396, 466 Bhattacharjee, S., 849 Bhattacharyya, A., 453, 463 Bhushan, B., 122 Bhuvaneswari, M., 526, 559 Bi, L., 447 Bialecki, R., 281 Bianchini, G., 398 Bianco, N., 465 Biazar, M., 882 Biehs, S., 902 Bien, D. D., 936 Bilger, R. W., 849, 851 Bilgili, M., 882 Billingsley, J., 466 Biltel, A., 393 Binauld, Q., 849 Bindar, Y., 850 Binnie, J. I., 463 Birk, M., 392–395, 729, 733, 953 Birkebak, R. C., 112, 125, 200, 233 Birmingham, B. W., 196 Bittner, J., 259 Bityurin, N., 654 Bizzocchi, L., 395 Bjorge, T., 850 Blackburn, D. L., 283 Blackshear, P. L., 855 Blackwell, B., 881 Blackwell, T. M., 122 Blake, D., 398 Blanco, M., 844, 857 Blanco, M. A., 530, 560 Blättner, W. G., 771 Blokh, A. G., 448 Blomberg, M., 453, 455, 463 Blunck, D. L., 849, 851, 852, 885 Blunsdon, C. A., 615 Bobco, R. P., 216, 234, 936 Bockhorn, H., 449 Boehm, R. F., 901, 902 Boeke, W., 937 Bogomolov, S. Y., 810 Bohren, C. F., 37, 39, 41, 58, 122, 403, 410, 446, 448, 455, 461, 464 Boischot, A., 394 Boisvert, J., 857 Bokar, J., 881 Boles, M. A., 812 Bolla, M., 835, 854 Bonczyk, P. A., 884 Bond, T. C., 437, 451 Booth, K. S., 259 Bordbar, M. H., 680, 681, 732, 733, 838, 843, 856, 949, 952

Borjini, M. N., 613 Born, M., 493 Bornside, D. E., 938 Botch, M. A., 399 Botet, R., 450 Bottou, L., 882 Bouanich, J. P., 393 Boudon, V., 392, 393, 395, 729, 733, 953 Boudot, C., 611, 654 Boulanger, J., 882, 885 Boulet, C., 393 Boulet, N. F., 392 Boulet, P., 93, 123, 561, 585, 609, 610, 614, 855, 858 Bourdon, A., 393 Bouvard, D., 614 Bouzid, A., 613 Boyd, I. D., 799, 815, 851 Boyd, I. W., 453, 455, 463 Boynton, F. P., 398 Bradar, F. J., 126 Brandenberg, W. M., 77, 100, 121 Brandon, S., 559, 816 Branner, K., 159, 259, 935, 952 Branstetter, J. R., 215, 218, 233 Braren, B., 654 Breault, R. P., 125 Breene, R. G., 392 Breig, W. F., 943 Brenner, H., 514, 558 Bressloff, N. W., 615, 679, 732, 849 Brewster, M. Q., 440, 443, 445, 446, 450, 452, 690, 734 Briggs, L. L., 593, 612 Bright, T. J., 902, 903 Brittes, R., 730, 732, 852, 952 Brock, W. H., 857 Brockmann, H., 939 Brodbeck, C., 393 Brookes, S. J., 849 Brosmer, M. A., 370, 385, 397, 399 Broughton, F. P., 935 Brouillette, C. R., 126 Brown, G. W., 236, 258 Brown, L. R., 392, 394, 395, 733, 953 Brown, R. A., 938 Bryan, W. J., 882 Bryant, F. D., 451 Buchan, A. G., 613 Buckius, R. O., 87, 88, 122, 125, 236, 258, 367, 368, 398, 427, 429, 433, 434, 448, 450, 451, 658, 730, 733, 756, 761, 770, 771, 948, 951 Buckley, H., 179, 196 Buettner, D., 466, 811 Buis, A., 398 Bulat, G., 854 Bungart, S., 939 Buraczewski, C., 938 Burak, L. D., 448 Burch, D. E., 343, 393, 396 Burges, C. J. C., 882 Burgess, D., 855 Burggraf, O. R., 881

Author Index

Burka, A. L., 812 Burkhard, D. G., 209, 233 Burns, F. C., 654 Burns, P. J., 159, 254, 259, 935, 939, 950, 952 Burns, S. P., 855 Burot, D., 734, 853 Burrell, G. J., 450 Buschman, A. J., 935 Butcher, J. A., 399 Butler, C. P., 125 Bykov, A. D., 394, 735 Byrd, L. W., 939 Byun, D. Y., 614, 850 Byun, K. H., 809

C Cabrera, M. I., 443, 452 Cai, J., 349, 396, 597, 613, 694, 701, 711, 713, 718, 730, 734, 735, 771, 839, 848, 855, 951, 952 Cai, W., 879, 885 Cain, S. R., 654 Caldas, M., 429, 448 Callis, J. B., 449 Campargue, A., 392, 393, 729, 733, 953 Campbell, J. P., 936 Campo, A., 810, 816 Camy-Peyret, C., 394, 733 Candell, L. M., 259 Candler, G. V., 730 Canepa, M., 399 Cao, S., 849 Cao, W., 809 Cao, X.-Y., 609 Capitelli, M., 393 Caps, R., 463, 808, 811 Carangelo, M. D., 126 Carangelo, R. M., 126, 884 Caren, R. P., 901 Carleer, M. R., 394 Carli, B., 398 Carlson, B. G., 565, 566, 577, 579, 580, 608, 609 Carminati, R., 88, 123, 901 Carrier, G. F., 851 Carslaw, H. S., 812 Cartigny, J. D., 440, 446 Carvalho, M. G., 450, 451, 605, 614, 615, 850 Case, K. M., 494, 771 Cashwell, E. D., 236, 258 Cassano, A. E., 452 Cassol, F., 680–682, 732, 949, 952 Castro, R. O., 613 Catchpole, K., 903 Catton, I., 121 Cen, K., 771, 884 Cen, K.-F., 609, 732 Cengel, Y. A., 495 Centeno, F. R., 730, 852, 855 Cess, R. D., 28, 131, 158, 209, 233, 279, 283, 308, 365, 375, 397, 398, 783, 794, 799, 809, 813, 815, 937 Cha, H., 612

Chackerian, C., Jr., 394 Chaffa, G., 857 Chai, J.-L., 615 Chai, J. C., 398, 579, 580, 585, 593, 595, 599, 610, 611, 613, 649, 655, 733, 815, 950, 952 Chakrabarty, R. K., 436, 450 Chalhoub, E. S., 883 Challingsworth, M. J., 399 Chambers, R. L., 276, 282 Champion, J.-P., 392, 733, 953 Chan, C. S., 505 Chan, S. H., 365, 397, 495, 510, 784, 785, 788, 789, 810, 812, 814 Chance, K., 392, 394, 733, 953 Chance, K. V., 393, 394, 729 Chandrasekhar, S., 37, 58, 311, 391, 392, 490, 494, 563, 608, 668, 731, 941, 943 Chandy, A. J., 832, 853 Chang, H., 433, 434, 437, 450, 682, 726, 736, 951, 952 Chang, J.-Y., 902, 903 Chang, K. S., 395 Chang, L. C., 816 Chang, Y. P., 809 Chao, B. T., 281 Chapman, A. J., 281 Chapuis, P.-O., 901 Charalampopoulos, T. T., 433, 434, 437, 450, 682, 726, 736, 951, 952 Charbonnier, J.-M., 395 Charest, M. R. J., 563, 609 Charette, A., 612, 614, 654, 811, 859, 878, 882, 885 Charvin, P., 858 Chatterjee, P., 837, 855 Chauhan, V., 282 Chaupin, S., 465 Chauveau, S., 395 Chawla, T. C., 495, 810, 814, 816 Chazot, O., 393 Chebanov, V. A., 937 Chekhovskii, I. R., 937 Chemisana, D., 857 Chen, C. H., 559, 850 Chen, D.-L., 615, 616 Chen, G., 887, 901, 902 Chen, H. Y., 450, 451 Chen, J. C., 278, 283, 813 Chen, J. S., 89, 123 Chen, J. Y., 849 Chen, L., 396 Chen, M. F., 122 Chen, M. Y., 464 Chen, S., 903 Chen, S. H. P., 124 Chen, W., 126, 495 Chen, X., 857, 901 Chen, Y., 853, 857, 901–903 Chen, Y. B., 122, 903 Chen, Y. N., 309, 505, 510 Chen, Y. S., 613, 614 Chen, Y.-K., 730 Chen, Z., 849, 903

957

Cheng, E. H., 799, 815 Cheng, J. S., 884 Cheng, P., 514, 526, 558 Cheng, Q., 605, 615, 616, 885 Cheng, X., 283 Cheong, K. B., 593, 612 Cherkaoui, M., 675, 732, 762, 772 Chernovsky, M. K., 849 Cheung, F. B., 816 Chevrier, J., 902 Chi, D. C., 814 Chi, Y., 771, 884 Chiba, Z., 815 Chien, K. Y., 731 Chien, P. L., 884 Chin, J. H., 159, 196, 505, 510, 604, 614, 935 Chin, N., 259 Chishty, M. A., 854 Chiu, F. C., 903 Chiu, W. K. S., 465 Cho, C., 812 Cho, D. H., 812 Cho, S. K., 902 Choi, C. E., 850, 854 Choi, H. Y., 308 Choi, M. Y., 451, 452 Choi, S., 282 Chopin, T., 883 Chou, Y. S., 309 Christiansen, C., 328, 382, 393 Chu, C. M., 306, 309, 405, 446, 614 Chu, H., 730, 731, 735, 856 Chu, H. S., 526, 560, 809, 810 Chu, K. H., 397 Chuech, S. G., 851 Chueh, W. C., 858 Chui, E. H., 612, 613, 850 Chung, B. T. F., 281, 609, 633, 639, 797, 808, 812, 814, 937, 938 Chung, T. J., 937 Churchill, S. W., 306, 309, 405, 446, 505, 510, 604, 614 Churnside, J. H., 609 Chylek, P., 464 ˇ Cerný, R., 466 Clark, G. C., 446 Clark, J. A., 121 Clausen, O. W., 77, 100, 121 Clausen, S., 386, 393, 394 Clay, D. T., 399 Clement, D., 126 Clements, A. G., 735 Clerbaux, C., 394 Clodic, D., 850 Clothiaux, E. E., 733 Clough, S. A., 393 Coelho, F. R., 681, 730, 732 Coelho, P. J., 309, 398, 563, 582, 583, 592, 602, 603, 605, 609–612, 614, 615, 730, 827, 828, 830, 833, 834, 852–855 Cogley, A. C., 489, 494 Cohen, E. S., 309, 392, 617, 619, 638 Cole, J. N. S., 733 Colella, P., 848

958 Author Index

Colket, M. B., 610, 849 Colles, M. J., 463 Collin, A., 123, 610, 855 Collins, D. G., 771 Collins, M. R., 772 Collins, N. S., 124 Collins, W. D., 398 Colomer, G., 799, 816 Coltrin, M. E., 848 Comin, F., 902 Condiff, D., 495, 514, 558 Conklin, P., 609 Consalvi, J.-L., 342, 349, 396, 464, 610, 690, 711, 727, 731, 733–736, 823, 830, 832, 833, 837, 839, 842, 843, 848, 849, 852–854, 856 Cònsul, R., 816 Contento, G., 465 Coppalle, A., 853 Coquard, R., 458, 464, 465 Coray, P., 465, 466 Corbin, J. C., 449 Corcione, F. E., 450 Cortesi, U., 398 Costa, M., 282 Cotal, H., 857 Coudert, L. H., 392, 394, 733, 953 Courant, R., 179, 196 Coussement, A., 852 Cox, G., 827, 852 Cox, R. L., 937 Cravalho, E. G., 889, 901 Crawford, M. E., 282, 308, 812, 848, 937 Crepeau, J., 28 Crisp, D., 396 Crnomarkovich, N., 850 Croce, P. A., 855 Croonenbroek, T., 848 Crosbie, A. L., 423, 424, 447, 484–486, 494, 602, 603, 614, 642, 649, 652, 653, 655, 943 Cross, E. S., 450 Crouseilles, N., 309 Crow, A. J., 851 Császár, A. G., 393, 729 Cuénot, B., 854 Cui, S., 772 Cui, X., 612 Cumber, P. S., 605, 615, 677, 732, 828, 852 Cunningham, F. G., 935 Cunningham, M. M., 464 Cunnington, G. R., 446, 447, 463, 466 Cunsolo, S., 458, 465 Cunto, W., 395 Curran, H. J., 848 Currie, I. G., 937 Curry, D. M., 281 Curry, D. R., 266 Curry, R., 281 Cyr, M. A., 125

D Da Silva, C. B., 854 Da Silva, C. V., 852

Da Silva, M. L., 658, 730 Da Silva, R. C., 732, 952 D’Agostini, M. D., 734, 856 Daguse, T., 848 Dai, W. L., 938 Dalal, A., 816 Daly, B. B., 857 Dalzell, W. H., 432–434, 437, 445, 449, 451 Dana, V., 392, 394, 733, 953 Daniel, J. W., 882 Daniel, K. J., 441, 451, 615 Daniels, T. S., 398 Darabiha, N., 848 Dargaville, S., 613 Dasgupta, A., 848 Datta, A., 850 Daun, K. J., 179, 196, 259, 772, 859, 868, 877, 878, 882–885 David, C., 561, 848 Davidovits, P., 450 Davidson, G. W., 423, 424, 447 Davidson, J. H., 465, 857, 858 Davidson, N., 392 Davies, H., 86, 122 Davies, R., 447 Davison, B., 513, 514, 517–519, 558 Davisson, C., 79, 121 Dayan, A., 809, 855 de B. Alves, L. S., 810 De Bastos, R., 136, 159 De Haseth, J. A., 108, 124, 399 De Lataillade, A., 772 De Miranda, A. B., 609 De Ris, J., 849 De Ris, J. L., 855 De Silva, A. A., 125 Debye, P., 403, 446 Deem, H. W., 121, 920 Degheidy, A. R., 494 Degiovanni, A., 808, 810 Deguchi, Y., 885 Dehesa-Carrasco, U., 858 Deirmendjian, D., 403, 405, 446 Deissler, R. G., 501, 510 Deiveegan, M., 878, 883 Del Bianco, S., 398 Del Campo, L., 111, 125 Delaye, C., 324, 392 Delichatsios, M. A., 849 Delichatsios, M. M., 849 Dell’Oro, A., 430, 449 Delmas, A., 734 Demarco, R., 717, 735, 848, 852, 856 Dembele, S., 449, 614, 690, 734, 855 Denison, M. K., 367, 398, 657, 680, 683–685, 687, 694, 700, 712, 729, 732, 733, 735, 949, 952 Denny, V. E., 123, 920 Deo, R. C., 882 Depraz, S., 384, 399 Der Auwera, J. V., 393 Derby, J. J., 559, 799, 816 Deron, C., 395 DeSantis, V. J., 122

Deshmukh, K. V., 561, 854 DeSilva, C. N., 813 DeSoto, S., 797, 813 Desrayaud, G., 799, 815 DeSutter, J., 901 Devi, V. M., 392–394, 729, 733, 953 Devir, A. D., 126 DeWitt, D. P., 82, 113, 121, 282, 466, 920 Dey, C. J., 857 Diaz, L. A., 812 Didari, A., 900, 903 Dietrich, B., 466 Digonnet, H., 613 Dimenna, R. A., 122 Dixon-Lewis, G., 848 Dligatch, S., 123 Dobbins, R. A., 435, 437, 450, 451 Dobbins, R. R., 849 Doermann, D., 466, 809 Dolaghan, J. S., 159, 259, 935, 939, 952 Dollet, A., 857 Dombrovsky, L. A., 401, 425, 430, 431, 445, 447–449, 458, 464, 812, 813, 847, 855, 858 Domingues, G., 902 Domoto, G. A., 347, 396, 898, 902 Doner, N., 511, 560 Dong, J., 902 Dong, S., 733, 853 Dong, S. K., 771 Donovan, J., 937 Donovan, R. C., 281 Doornink, D. G., 448 Dorai-Raj, D. E., 609, 809 Dorigon, L. J., 732 Dorofeev, S. B., 855 Doron, B., 123 Doron, P., 857 Dörr, J., 638 Dorsey, N. E., 812 Dothe, H., 393, 729, 952 Doucen, R. L., 393 Dougherty, R. L., 649, 653 Dowrey, A. E., 399 Dozhdikov, V., 883 Dransfeld, K., 902 Drayson, S. R., 392 Drolen, B. L., 401, 402, 445, 446 Drost, M. K., 125 Drouin, B. J., 393, 729 Drude, P., 73, 121 Drummeter, L. F., 126 Drysdale, D., 939 Dsa, D. A., 884 Duan, R., 732 Duan, Y.-Y., 465, 466 Duciak, G., 732 Duffie, J. A., 97, 99, 123 Dufresne, J.-L., 732, 772 Duley, W. W., 655 Dumont, N., 773 Dunkle, R. V., 77, 83, 99, 121, 123, 126, 218, 234, 399, 661, 731, 936 Dunn, J. R., 813

Author Index

Dunn, S. T., 126 Dupoirieux, F., 734, 762, 772, 853 Dussan, B. I., 278, 283 Dutta, P., 814, 816, 884 Duval, R., 850 Dwivedi, V. H., 124 Dyer, D. F., 813

E Ebadian, M. A., 559 Ebert, H. P., 466 Ebert, J. L., 462, 466 Echegut, P., 465 Echekki, T., 834, 854 Echigo, R., 797, 798, 809, 814, 815 Eckert, E. R. G., 60, 78, 79, 112, 121, 125, 179, 196, 208, 233, 281, 282, 384, 399, 935 Edalatpour, S., 887, 901 Edamura, M., 466 Eddington, A. S., 309, 311, 391, 505, 511 Eddy, T. L., 938 Edge, P., 838, 850, 856 Edwards, D. K., 22, 29, 96, 114, 121, 123, 126, 223, 234, 306, 309, 358, 359, 361, 364, 365, 371, 381, 384, 386, 391, 397, 399, 677, 732, 771, 813, 908, 920, 936 Edwards, D. P., 394 Egan, W. G., 126 Egbert, R. B., 392 Ehlert, J. R., 939 Ehrenreich, H., 73, 121 Eichberger, J. I., 938 Eigenmann, L., 852 Einstein, T. H., 617, 627, 636, 638, 783, 797, 809, 814 El Akel, A., 885 El Ammouri, F., 643, 654 El-Asrag, H. A., 855 El-Baz, H. S., 642, 653 El Hafi, M., 772, 848, 854 El Hitti, G., 903 El Kasmi, A., 614 El Khoury, K., 882 El-Wakil, N., 566, 609 Eller, E., 638 Elliott, J. M., 813 Ellis, B., 450 Ellzey, J. L., 611, 848 Elsasser, W. M., 395 Emery, A. F., 131, 159, 935, 938 Endrud, N. E., 734, 853, 856 Enikov, E. T., 902 Epple, B., 733, 854, 856 Epstein, M., 816 Eraslan, A. H., 281 Erb, W., 126 Ertürk, H., 877, 878, 882, 883 Eryou, N. D., 811 Escudié, D., 885 Eslinger, R. G., 281 Esmaily-Moghadam, M., 815 Esquisabel, X., 125

Eswaran, V., 309, 856 Etminan, M., 398 Evans, D. D., 855 Evans, L. B., 608 Everett, C. J., 236, 258 Evseev, V., 384, 394 Eymet, V., 772 Eyre, J. A., 855 Ezekoye, O. A., 397, 559, 730, 732, 849, 882–884, 952

F Faeth, G. M., 435, 450, 451, 826, 828, 851, 852 Faghri, M., 854 Fahey, D. W., 398 Fahr, S., 902 Fairweather, M., 732 Fally, S., 392, 733, 953 Falter, C., 857 Fan, J. C. C., 123 Fan, S., 902, 903 Fang, H. S., 799, 816 Fang, X., 903 Farag, I. H., 398, 679, 732 Farbar, E., 815 Farias, T. L., 435, 436, 450, 451, 615 Farmer, J. A., 764, 773 Farmer, J. T., 615, 760, 761, 770, 772, 856 Farnbach, J. S., 936 Farnsworth, M. S., 124 Farooq, A., 392 Farouk, B., 611, 799, 814, 816 Farrell, R., 150, 159 Fateev, A., 386, 393, 394, 882 Favennec, Y., 465, 613 Faxvog, F. R., 439, 451 Fayt, A., 394 Fedorov, A. G., 877, 882 Feiner, S., 259 Feingold, A., 936, 937 Feldick, A. M., 392, 395, 658, 730, 747, 761, 770–772 Feldman, D. R., 398 Felske, J. D., 358, 363, 364, 366, 369, 397, 433, 434, 437, 450, 451, 937 Fendell, F. E., 851 Feng, G., 448 Feng, Y., 885 Feng, Y.-Y., 612 Ferguson, C., 854 Ferguson, R. E., 810 Ferkl, P., 460, 465 Fernandes, R., 810 Fernández, I., 125 Fernandez, S. F., 850 Ferriso, C. C., 384, 399 Ferziger, J. H., 282, 490, 494, 655 Fetzer, C., 857 Figueira da Silva, L. F., 852 Finkleman, D., 664, 731 Fiorino, A., 901, 903 Fippel, M., 760, 771 Fischedick, T., 466 Fischer, S. J., 852

959

Fisher, T. S., 902 Fitzgerald, S. P., 281 Fiveland, W. A., 367, 398, 563, 565, 566, 571, 574, 579, 582, 591, 592, 602, 604, 608–612, 614, 733, 848 Flamant, G., 609, 610, 815, 845, 857, 858 Flannery, B. P., 258, 734, 882 Flaud, J.-M., 392–394, 729, 733, 953 Fleck, J. A., 235, 258 Fleckl, T., 394 Fletcher, D., 395 Fletcher, E. A., 846, 856, 857 Flom, Z. H., 811 Flórez, M., 560 Flouros, M., 939 Foex, M., 124, 846, 857 Foley, J., 259 Fontes, P., 615 Ford, J. N., 113, 125 Forget, F., 393 Foster, P. J., 427, 432, 433, 448, 449, 653, 679, 732, 826, 851 Fouilloy, A., 882 Founti, M. A., 615, 850 Fournier, R., 732, 772 Fox, K., 393 Fox, R. O., 851 Fraga, G. C., 732, 828, 837, 852, 855 França, F. H. R., 681, 730, 732, 733, 760, 770, 771, 852, 855, 859, 877, 882–884, 952 Francis, J., 810, 811 Francoeur, M., 901–903 Frank, I., 881 Frank, S., 615 Frankel, J. I., 938 Frankel, S. H., 849, 853 Fransson, T., 848 Freeman, R. K., 116, 126 French, W. G., 464 Frenklach, M., 449 Fricke, J., 463, 466, 808, 811 Fried, L., 159, 196, 935 Friedman, J. N., 639, 732 Fritsch, C. A., 813 Fröberg, C. E., 182, 197, 259, 282 Fröhlich, C., 28 Froment, G. F., 633, 639, 771 Frosch, C. J., 121 Frost, W., 281 Fu, C.-J., 903 Fu, C. J., 901 Fu, K., 88, 122 Fu, Q., 345, 396, 448 Fu, X. D., 559, 731, 813 Fuentes, A., 449, 735, 849, 852, 856 Fujita, A., 855 Fujita, K., 395 Fukabori, M., 393 Fuks, I. M., 85, 122 Fukusako, S., 812 Fuller, K. A., 448 Fumeron, S., 601, 613, 654 Funai, A. I., 110, 124

960 Author Index

Fung, A. K., 122 Furmanski, ´ P., 789, 811, 812 Furtenbacher, T., 393, 729 Furuhata, T., 854 Furuya, N., 936 Fusegi, T., 816, 848 Fuss, S. P., 397 Fussell, W. B., 125, 126

G Gaffney, J. S., 464 Gaffuri, P., 848 Galarça, M., 732 Gale, W. F., 920 Gamache, R. R., 392–394, 729, 730, 733, 952, 953 Gamba, M., 883 Gan, J. N., 464 Ganesan, K., 459, 465, 847, 858 Ganz, B., 852 Gao, C., 736 Gao, H., 772 Gao, N., 612 Gao, Y.-B., 466 Garbett, E. S., 559 Gardon, R., 93, 123 Garetz, B. A., 654 Garing, J. S., 393 Garot, C., 937 Garro, M. A., 450 Garten, B., 839, 848 Gartling, D. K., 281 Gaskell, P. H., 610 Gau, C., 814 Gauglitz, G., 901 Gaumer, R. E., 123, 125 Ge, W., 538, 560, 561, 730, 731, 735, 840, 848, 856, 953 Gear, C. W., 281 Gelbard, E. M., 513, 519, 539, 558, 559 Gel’d, P. V., 466 Gendre, P., 937 Geng, H., 902 Genot, J., 937 Gentry, J. W., 450 Genzel, L., 454, 463 Gérardin, J., 123, 561 Gerencser, D. S., 281 Gero, P. J., 398 Gharebaghi, M., 850 Ghashami, M., 899, 902 Ghazvin, A. M. C., 612 Ghorbani, M. A., 882 Ghose, P., 850 Ghosh, D., 938 Ghulman, H. A., 613 Gianfrani, L., 393 Gianoulakis, S., 283 Gicquel, O., 773, 814, 852 Giedt, W. H., 381, 383, 399 Gier, J. T., 99, 123, 126, 399 Giessen, H., 449 Gillespie, G. D., 281 Gilpin, R. R., 810

Gindele, K., 126 Gissibl, T., 449 Gladen, A. C., 459, 465 Glass, D. E., 810 Glassen, L. K., 397 Glassner, A. S., 259 Glatt, L., 544, 561 Glaze, D. J., 853 Gleizes, A., 730 Glicksman, L. R., 458, 464, 811 Glouannec, P., 282 Gnoffo, P., 308 Godson, W. L., 338, 395 Goetze, D., 935 Goldbach, T., 452 Goldenstein, C. S., 885 Goldman, A., 382, 392–394, 399, 729, 733, 952, 953 Goldschmidt, V. W., 938 Goldsmith, A., 121 Goldstein, E., 126 Goldstein, R. J., 382, 399, 455, 464 Golubkin, V. N., 813 Gomart, H., 465 Gomez, R. B., 392 Gonçalves, J., 592, 611, 614 Gong, Y., 447 Gonome, H., 123 Goodwin, D. G., 429, 448 Goody, R. M., 311, 315, 338, 391, 395, 396, 734 Gordon, G. D., 124 Gordon, H. R., 771 Gordon, I., 393 Gordon, I. E., 392, 393, 395, 729, 733, 952, 953 Gordoninejad, F., 811 Gore, J. P., 399, 466, 826, 828, 849–852, 855 Gorewoda, J., 733 Goro, M., 732 Goswami, D. Y., 813 Götz, T., 560, 811 Goulard, M., 797, 814 Goulard, R., 797, 812, 814 Goulet, T., 654 Graeser, P., 733 Grafton, J. C., 935 Graham, S. C., 433, 450 Grandjean, P., 495 Grandpeix, J.-Y., 732, 772 Gratch, S., 121 Gray, D. E., 121 Green, R. M., 397 Greenspan, H., 608 Greffet, J.-J., 123, 125, 901, 902 Gregg, J. L., 196 Greif, R., 356, 397, 668, 731, 809, 814, 815 Griem, H. R., 392 Grier, N. T., 936 Griffiths, P. R., 108, 124, 399 Grimm, M., 282 Grissa, H., 612 Gritzo, L. A., 451, 452 Gronarz, T., 429, 448, 733 Grosch, C. B., 935

Grosh, R. J., 608, 780, 792, 797, 808, 812–814 Gross, K. C., 885 Gross, L. A., 399 Gross, U., 937 Grosshandler, W. L., 395, 731, 837, 852, 950, 952 Grossman, K., 345, 396 Groth, C. P. T., 609 Grove, C. I., 124 Grove, F. J., 454, 463 Grumer, J., 449 Gu, M., 731, 856 Guadarrama-Mendoza, A. J., 857 Guarnizo, G., 882 Gubareff, G. G., 121, 920 Gubba, S. R., 856 Gudat, W., 121 Guedri, K., 613 Guelzim, A., 939 Guerlet, S., 393 Guha, B., 902 Gülder, O. L., 396, 609, 732, 734, 848 Guo, H., 732, 848 Guo, K., 855 Guo, L., 447 Guo, Z., 585, 610, 611, 638, 649, 654, 655, 677, 732 Gupta, A., 561, 735, 771, 833, 834, 848, 852, 854 Gupta, K. G., 936 Gupta, R. P., 448 Gupta, S. B., 463, 466 Gurney, K., 882 Gusarov, A. V., 464 Gustafsson, M., 393

H Habermehl, M., 733 Habib, I. S., 812, 814 Habib, M. A., 856 Habibi, A., 853 Hadamard, J., 859, 881 Hadjiconstantinou, N. G., 259, 772 Hadvig, S., 731 Hagemann, H. J., 73, 121 Hagen, E., 74, 121 Haghighat, A., 772 Hagness, S., 903 Hahn, O., 810 Hahne, E., 937 Haile, S. M., 858 Haire, A. M., 935 Haji-Sheikh, A., 179, 196, 236, 258, 750, 771, 881 Hajimirza, S., 903 Hale, G. M., 126, 455, 464 Hall, D. E., 938 Hall, M. J., 397 Hall, M. L., 511, 654 Hall, R. J., 610, 849, 884 Haller, H. C., 936 Halton, J., 772 Halton, J. H., 236, 258 Hamilton, D. C., 131, 158, 935

Author Index

Hamins, A., 855 Hammersley, J. M., 236, 237, 258 Han, J.-H., 450 Han, S., 903 Han, S. D., 883, 884 Han, Y., 123 Han, Y.-H., 466 Hanamura, K., 903 Hanawa, T., 511, 561 Handa, M., 735 Handscomb, D. C., 236, 237, 258 Hangsen, L. M., 126 Hanrahan, P., 151, 159 Hansen, J. E., 422, 447 Hansen, P. C., 881 Hanson, R. K., 392 Härd, S., 439, 451 Hardoiun-Duparc, B., 852 Hardy, D. L., 465 Hargreaves, C. M., 898, 902 Harris, G. J., 394 Harris, J., 855 Harris, J. A., 123 Harris, S. J., 449 Harris, W. M., 465 Harrison, J. J., 393, 729 Hartel, W., 440, 451 Harten, A., 610 Hartmann, D. L., 124, 398 Hartmann, J.-M., 324, 327, 344, 392–396, 712, 729, 735, 808 Hartnett, J. P., 258, 309, 770, 813 Hartung-Chambers, L., 392, 730 Hartung, L., 308, 561, 658, 730 Hartz, B. A. K., 393 Harutunian, V., 877, 882 Harvazinski, M. E., 851 Harvill, L. R., 521, 559 Hasatani, M., 811 Hasegawa, S., 797, 809, 811, 814, 815 Hashempour, J., 466 Hashimoto, T., 449 Hassan, H. A., 561, 658, 730, 794, 813 Hassanzadeh, P., 600, 613 Hasse, C., 848 Hauptmann, E. G., 936 Hauser, G. M., 816 Hauser, W. C., 397 Haussener, S., 459, 464, 465, 847, 857, 858 Hauth, J. L., 810 Havran, V., 259 Havstad, M. A., 112, 125 Hawkes, E. R., 854 Haworth, D. C., 561, 734, 735, 770, 771, 829, 834, 848–854 Haya, R., 282 Hazzak, A. S., 810 He, A., 885 He, B., 734–736, 770, 856, 953 He, C., 615 He, X., 612 He, Z., 733, 853 Heaslet, M. A., 281, 474, 488, 490, 493, 494 Hebert, P., 857

Hecht, B., 887, 901 Heikal, M. R., 448, 449 Heinemann, U., 808 Heinisch, R. P., 126, 196, 259, 937 Heissler, S., 466 Heitler, W., 392 Heller, G. B., 122 Hellmann, J. R., 125, 399 Heltzel, A., 903 Hembach, R. J., 126 Hemmerdinger, L., 124, 126 Henderson-Sellers, A., 398 Hendricks, T. J., 466, 609, 883 Henline, W. D., 730 Henriquez, R., 849 Hensel, E., 881 Henson, J. C., 615 Hering, R. G., 77, 121, 122, 222, 234, 275, 282 Hernicz, R. S., 121 Hertzberg, M., 855 Herzberg, G., 316, 318, 392 Heskestad, G., 855 Hess, D., 857 Hibbard, R. R., 936 Hickman, R. S., 281 Hielscher, A. H., 884 Higano, M., 125, 638 Higenyi, J., 525, 538, 559 Highwood, E. J., 398 Hilbert, D., 179, 196 Hildebrand, F. B., 179, 196 Hilgeman, T., 126 Hill, C., 393, 729 Hillenbrand, R., 901 Hirashima, D., 903 Hirleman, E. D., 810 Hischier, I., 857 Hix, P., 281, 808 Hjärtstam, S., 856 Ho, C. H., 809, 811, 881 Ho, F. G., 233 Hodges, J. T., 393, 729 Hodkinson, J. R., 411, 446 Hoffmann, D., 28 Hogan, R. E., 281 Hogan, R. J., 733 Holborn, P., 939 Holchendler, J., 937 Holcomb, R. S., 936 Holdredge, E. S., 276, 282 Holen, J., 850 Hollands, K. G. T., 100, 124, 151, 159, 179, 196, 282, 398, 733, 939 Hollis, B. R., 393, 395, 730 Holman, J. P., 273, 282 Holthaus, M., 902 Homola, J., 901 Hong, Y. K., 884 Honnery, D., 727, 736, 853 Hook, S. J., 124 Hopkinson, R. B., 17, 28 Horak, H. G., 771 Hornbeck, R. W., 282 Horton, T. E., 209, 233

961

Hossain, M. A., 799, 816 Hossain, M. M., 885 Hostikka, S., 732, 855 Hottel, H. C., 99, 123, 143, 159, 233, 306, 309, 311, 369–372, 383, 392, 398, 402, 426, 439, 440, 445, 448, 451, 563, 608, 617, 619, 634, 636, 638, 639, 658, 661, 677, 678, 683, 731, 920, 935, 936 Hou, L., 396 Hou, M.-F., 308, 493 Hou, M. F., 483 Houchens, A. F., 122 Houde, D., 654 Houf, W. G., 642, 653, 813 Houston, R. L., 811 Howard, J. B., 449 Howard, J. N., 396 Howarth, C. R., 427, 432–434, 448, 449 Howell, J. R., 66, 100, 121, 124, 131, 158, 159, 209, 233, 235, 236, 252, 258, 259, 279, 283, 309, 466, 489, 494, 605, 609, 611, 615, 617, 638, 733, 760, 761, 770–772, 783, 809–811, 816, 839, 856, 859, 877, 878, 882–884, 903, 925, 934, 939 Howell, L. H., 593, 611, 848 Hozawa, M., 559, 816 Hrycak, P., 281 Hsia, H. M., 608 Hsia, J. J., 125 Hsieh, C. K., 123 Hsieh, T. C., 397 Hsu, C. J., 936 Hsu, K. Y., 812 Hsu, P.-F., 88, 122, 465, 611, 615, 649, 654, 655, 771, 772, 812 Hu, B., 450 Hu, L., 897, 898, 901 Hu, Z. M., 616 Huang, H., 851 Huang, J., 879, 885 Huang, Q.-X., 771, 884 Huang, W., 815 Huang, X., 879, 885 Huang, Y., 493, 559, 773 Huang, Z., 99, 124, 615, 616 Huang, Z. F., 615 Hubbard, G. L., 433, 449 Hubbard, H. A., 494 Huckaby, E. D., 856 Huffman, D. R., 37, 39, 41, 58, 122, 403, 410, 446, 455, 461, 464 Hughes, J., 259 Hughes, P. M. J., 612, 613, 850 Huh, K. Y., 600, 602, 613, 614 Hui, A. K., 393 Humlí˘cek, J., 326, 393, 947, 951 Hunger, F., 848 Hunt, A. J., 809 Hunt, G. E., 309, 642, 653 Hunt, L. A., 398 Hunter, B., 585, 610, 611 Hura, H. S., 450

962 Author Index

Husson, N., 394 Huston, V., 653 Hutchison, J. R., 677, 732, 811 Hutley, M. C., 124 Hwang, D. C., 427, 429, 434, 448, 756, 771, 948, 951 Hyde, D. J., 608 Hyppänen, T., 732, 733, 856, 952 Hyun, S. Y., 395

I Iacob, A., 282 Iacona, E., 464–466 Iacopino, N., 902 Iannetti, A. C., 855 Ibgui, L., 394 Idier, J., 881 Ignatiev, M., 883 Ihme, M., 851 Iizuka, H., 902 Ikeda, Y., 854 Ikushima, T., 159, 935 Im, H. G., 849 Im, K. H., 429, 448, 559 Imaishi, N., 559, 816 Incropera, F. P., 125, 282, 283, 451, 615, 642, 653, 810, 813 Inn, E. C. Y., 125 Inokuti, M., 121 Inoue, G., 398 Inoue, M., 398 Iqbal, M., 937 Irons, R., 850 Irrera, A., 452 Irvine, T. F., 258, 278, 281–283, 309, 770 Irvine, W. M., 455, 464, 642, 653 Ishii, K., 816 Ishimaru, A., 89, 123 Iskander, M. F., 435, 450, 451 Isobe, K., 903 Ito, K., 902 Ivarsson, A., 393 Iverson, B. D., 100, 124 ˘ 450 Ivezi´c, Z., Iwamoto, M., 809, 810 Iynkaran, K., 282

J Jackson, G. S., 394 Jacobsen, R., 451 Jacobson, A. E., 451 Jacquemart, D., 392–394, 729, 733, 953 Jaeger, J. C., 812 Jafarian, M., 857 Jagannathan, P. S., 809 Jäger, H., 394 Jakob, M., 131, 159, 186, 197, 935 Jaluria, Y., 634, 639, 799, 815 Jamaluddin, A. S., 580, 592, 610, 611 James, E. H., 615 Jang, C., 125 Jang, E. Y., 282 Janssen, J. E., 121, 126, 920 Janzen, J., 433, 450

Jasperse, J. R., 81, 122 Jaunich, M., 655 Jay-Gerin, J.-P., 654 Jayaram, K. S., 283 Jayaweera, K., 609 Jeandel, G., 464, 609, 611, 614, 654 Jeans, J. H., 6, 8, 28, 513, 514, 558 Jeffries, J. B., 392 Jellyman, P. E., 454, 463 Jendoubi, S., 593, 611 Jeng, S. M., 851 Jenkins, R. J., 125 Jensen, C. L., 159, 196, 935 Jensen, H. H., 186, 197 Jensen, M. K., 559, 731, 813 Jensen, P., 394 Jeong, C. H., 282 Jerozal, F. A., 126 Jessee, J. P., 582, 592, 602, 604, 610–612, 614, 848 Jewell, R. A., 123 Ji, H., 815 Ji, J., 385, 399, 851 Ji, Y., 885 Jia, Z.-X., 902 Jiang, J., 883 Jin, Y., 879, 885 Jing, C. Y., 448 Jinguji, T. M., 449 Jinno, H., 449 Jody, B. J., 124 Joerg, P., 935 Johansen, L. C. R., 732, 856 Johansson, O., 159, 938 Johansson, R., 427, 429, 448, 680, 732, 838, 850, 856 Johnson, T. J., 393, 729 Johnsson, F., 850, 856 Johnston, C. O., 393, 395 Jolly, A., 393, 729 Jomaas, G., 851 Jones, A. R., 434, 435, 450, 451 Jones, B. W., 125 Jones, L. R., 936 Jones, M. R., 124, 809, 877, 881 Jones, P. D., 282, 574, 609, 809, 813 Jones, R. N., 449 Jones, W. P., 851, 854 Jonsson, V. K., 209, 233, 282, 936 Jørgensen, U. G., 394 Joseph, J. H., 423, 424, 447 Joshi, Y. K., 283 Joulain, K., 901, 902 Joulain, P., 611, 849 Jourdan, G., 902 Joyeux, D., 853 Ju, Y., 732, 848, 849 Jucks, K. W., 394 Jullien, R., 450 Jury, V., 282 Juul, N. H., 937, 938

K Kääb, A., 124

Kabashnikov, V. P., 826, 852 Kadaba, P. V., 938 Kaemmerlen, A., 458, 464 Kær, S. K., 732, 856 Kahan, A., 122 Kahn, H., 236, 258 Kajiya, J. T., 253, 259 Kakinoki, K., 559, 816 Kalogirou, S., 882 Kamdem, H. T. T., 600, 611 Kamimoto, T., 885 Kaminski, C. F., 879, 885 Kaminski, D. A., 559, 668, 731, 813 Kamiuto, K., 495, 559, 809–811, 814, 815 Kanchanagom, D., 279, 283 Kang, S. H., 604, 614, 834, 854 Kang, S. J., 614 Kangwanpongpan, T., 680, 681, 732, 949, 952 Kantorovich, L. V., 495 Kanury, A. M., 450 Kapaku, R. K., 851 Kaplan, C. R., 592, 611, 822, 848, 849 Karam, N., 857 Karlekar, B. V., 281 Karman, T., 393, 729 Karni, J., 844, 845, 857 Kasper, C., 449 Kassemi, M., 633, 638, 639, 797, 814 Kassi, S., 393 Kasyutich, V. L., 885 Katika, K. M., 652, 655 Kato, A., 466 Kato, K., 936 Katsurayama, H., 395 Katta, V. R., 849 Kattawar, G. W., 405, 446, 447, 771 Katte, S. S., 939 Katz, A. J., 126 Katzoff, S., 121, 123–125 Kaufman, J. E., 28 Kaviany, M., 282, 446, 464, 815 Kawakami, S., 398 Kawata, S., 901 Kawka, P., 122 Kay, T. L., 253, 259 Kayakol, N., 592, 602, 605, 611 Kayima, K., 399 Kays, W. M., 282, 308, 812, 848 Kazakov, A., 449 Kazutaka, S. O., 125 Kedenburg, S., 449 Kee, R. J., 848 Keefe, C. D., 449 Keene, D., 464 Keene, H. B., 935 Kehtarenavaz, R. H., 813 Kelber, C. N., 608 Keller, B., 395 Keller, H. H., 276, 282 Keller, J. D., 935 Kellmann, F., 901 Kelly, J. J., 449 Keltner, Z., 385, 399 Kennedy, I. M., 449

Author Index

Kennedy, L. A., 794, 813 Kent, J. H., 727, 736, 853 Keramida, E. P., 605, 615, 850 Kerker, M., 403, 446, 451 Kermani, M. M., 938 Kersch, A., 253, 259, 282, 772 Keshock, E. G., 279, 283 Kesten, A. S., 489, 494 Keunecke, M., 857 Kez, V., 733, 836, 854, 856 Kezios, S. P., 936 Khalil, B., 393 Khan, T., 526, 559 Khatry, A. K., 937 Khersonskii, V. K., 560 Khlebtsov, N. G., 447 Kholodov, N. M., 811 Kim, C., 428, 448 Kim, D. M., 809, 814 Kim, H. K., 882–885 Kim, I. K., 580, 610 Kim, J., 125 Kim, J. S., 611, 614, 849 Kim, J. Y., 937 Kim, K., 654, 655 Kim, K. W., 884 Kim, M. Y., 613, 614, 655, 811 Kim, S. H., 600, 602, 613, 614 Kim, S. S., 559, 592, 611, 798, 814, 815 Kim, T., 902 Kim, T. H., 812 Kim, T. K., 592, 593, 609–612, 653, 677, 731, 797, 809, 811, 814 Kim, T. Y., 592, 611, 811, 814 Kim, W. S., 580, 610 Kim, Y., 282 Kind, M., 466 King, R., 857 Kinoshita, I., 811 Kinsella, K., 126, 855 Kinsey, G., 857 Kirgizbaev, D. A., 810 Kirk, D., 259 Kita, K., 398 Kittel, A., 898, 902 Klar, A., 560 Klassen, M., 855 Klausner, J. F., 857 Klein, D. E., 283, 611 Kleiner, I., 392, 393, 729, 733, 953 Kleinman, D. A., 121, 123 Klemm, R., 125 Klose, A. D., 884 Klychev, S. I., 810 Kmit, G. I., 852 Knauer, T. G., 450 Kneer, R., 448, 733, 852 Kneissl, G. J., 126 Knight, C. A., 812 Knittl, Z., 52, 55, 58 Knopken, S., 125 Kobayashi, H., 848 Kobayashi, M., 125 Kocamustafaogullari, G., 812

Koch, R., 566, 609, 614, 852 Kochanov, R. V., 393, 729 Koˇcí, V., 466 Kodama, T., 857 Koewing, J. W., 649, 652, 655 Kofink, W., 526, 559 Koh, J. C. Y., 813 Köhl, M., 126 Kollie, T. G., 124 Koltun, P. S., 811 Komiya, A., 123, 901 Komori, S., 855 Kondo, Y., 398 Kondratyev, K. Y., 345, 396 Kong, W., 734, 848, 854 Königsdorff, R., 615 Koo, H. M., 612 Kook, S., 854 Kopal, T., 259 Korder, S., 452 Korpela, S. A., 811 Kosek, J., 465 Kotan, K., 281 Kou, L., 455, 464 Kounalakis, M. E., 851 Kourganoff, V., 513, 514, 558, 941, 943 Kowalski, G. J., 810, 811 Kowsary, F., 254, 259, 884 Köylü, Ü. Ö., 435–437, 450, 451 Kramer, J. L., 281 Kramida, A. E., 395 Kratohvil, P., 451 Kratz, D. P., 398 Krautz, H. J., 732, 952 Kravtsov, Y. A., 901 Krebs, W., 614, 852 Kreider, P. B., 857 Kreith, F., 131, 158, 935 Kribus, A., 857 Kriese, J. T., 494 Krishnamoorthy, G., 838, 848, 856 Krishnan, S. S., 451 Krishnaprakas, C. K., 281, 576, 609, 797, 799, 814, 816, 939 Krook, M., 507, 511, 514, 558 Kropschot, R. H., 196 Krug, T., 399 Krylov, V. I., 495 Ku, J. C., 450, 615 Kubo, M., 816 Kubo, S., 642, 653 Kuchhal, A., 282 Kuhn, J., 443, 452, 465, 466 Kulacki, F. A., 882 Kulah, G., 850 Kulkarni, A. K., 808 Kumar, A., 560, 817, 858 Kumar, K., 446 Kumar, S., 308, 447, 450, 571, 609, 649, 650, 654, 655 Kung, S. C., 668, 731 Kunitomo, T., 449 Kunugi, M., 449 Kunz, C., 121

963

Kuo, D. C., 559, 816 Kuo, K. K., 851 Kurosaki, Y., 447, 796, 797, 813 Kurose, R., 836, 855 Kurpisz, K., 881 Kushari, A., 309, 856 Kuwahara, K., 816 Kuznetsov, E., 855 Kyuberis, A. A., 393, 729

L Labrie, D., 464 Lacis, A. A., 342, 345, 353, 396 Lacoa, U., 816 Lacome, N., 392, 733, 953 Lacona, E., 799, 816 Lacroix, D., 592, 611, 643, 654, 858 Ladenburg, R., 330, 393 Laemmerhold, M., 733 Lafferty, W. J., 392, 394, 733, 953 Lahellec, A., 732, 772 Lai, M. C., 851 Lai, Q.-Z., 465 Lalit, H. U., 852 Lallemand, M., 810, 883, 884 Lallemant, N., 362, 397 Lam, Y. C., 611, 655 Lamet, J.-M., 393 Lamouroux, J., 328, 393, 394 Landram, C. S., 216, 234 Langley, S. P., 311, 391 Langlois, S., 394 Lanza, F., 126 Lanzarotta, A., 399 Łapka, P., 789, 811, 812 Lapp, J., 858 Laraia, A. L., 393 Lari, K., 730, 815 Laroche, M., 901 Larochelle, H., 882 Larsen, E., 539, 560 Larsen, M. E., 259, 617, 638, 771, 882 Lathrop, K. D., 563, 565, 577, 579, 580, 608–610 Latimer, P., 451 Lau, A. K. C., 610 Lauger, K., 902 Launder, B. E., 851 Laurendeau, N. M., 451, 615 Lauriat, G., 799, 815 Laux, C. O., 395 Lavalle, L., 399 Laverty, W. F., 937 Lavine, A. S., 282 Lavorgna, M., 882 Lavrentieva, N. N., 394, 735 Lawler, J. E., 398 Lawson, D. A., 259, 939 Lazard, M., 809 Le Corre, S., 613 Le Dez, V., 308, 486, 493, 494, 593, 612, 654, 811 Leaf, G., 495 Lean, J., 28

964 Author Index

Leckner, B., 371, 372, 398, 732, 838, 850, 856 Lecoustre, V. R., 332, 342, 394 Lederer, F., 902 Ledin, H. S., 732 Leduc, G., 878, 882, 883 Lee, B. J., 901–903 Lee, C., 614 Lee, C. E., 563, 608 Lee, C. L., 938 Lee, E., 612, 730 Lee, E. T., 281 Lee, H. J., 122, 903 Lee, H. O., 797 Lee, H. S., 451, 565, 592, 609–611, 613, 653, 731, 732, 814, 952 Lee, J. H., 812 Lee, J. S., 811 Lee, J. W., 282 Lee, K. H., 809, 811, 882, 884 Lee, P. M., 938 Lee, P. Y. C., 367, 398, 733 Lee, S. C., 433, 434, 437, 446, 447, 450, 463, 466 Lee, S. H., 282 Lee, S. S., 902 Lee, W. J., 883 Lei, S., 560, 848 Leiner, W., 939 Lemonnier, D., 396, 593, 612, 654, 733, 734 Lentes, F. T., 810 Leonard, B. P., 582, 610 LePhat, V., 124 Lesieur, M., 851 Leuenberger, H., 131, 158, 935 Leupacher, W., 461, 466 Levenberg, K., 871, 882 Levenson, L. L., 399 Levi Di Leon, R., 395, 735 Levin, D. A., 392, 395, 730, 733, 770 Lewis, E. E., 612, 771 Li, B.-W., 566, 609, 616, 812, 939 Li, B. Q., 612 Li, F., 885 Li, G., 559, 830, 831, 853, 854 Li, H., 447, 885 Li, H. S., 566, 603, 609, 610, 614 Li, H. Y., 799, 810, 816, 881, 883, 884 Li, J., 447, 885 Li, L., 857, 858 Li, S.-N., 885 Li, T., 885 Li, T.-J., 879, 885 Li, W., 446, 526, 559 Li, X., 732 Li, Y., 465, 857 Li, Z., 885 Liakos, H. H., 615, 850 Liang, X. G., 282 Liang, Y.-C., 857 Liang, Z., 847, 858 Liao, Q.-H., 902 Lick, W., 780, 808, 809 Liddel, U., 449 Lide, D. R., 121

Liebert, C. H., 936 Lieblein, S., 275, 282 Lienhard, J. H., 392 Lii, C. C., 797, 810, 814 Lim, J., 399 Lim, M., 902 Limperis, T., 125 Lin, C. X., 559 Lin, J. D., 809 Lin, K. C., 451 Lin, Q., 89, 123 Lin, S., 938 Lin, S. H., 179, 196, 233, 279, 283 Lin, S. L., 233 Lin, S. T., 279, 283 Lin, T. H., 559, 850 Lindermeir, E., 395 Lindquist, G. H., 343, 396 Linsenbardt, T. L., 653 Lior, N., 428, 448 Liou, K. N., 345, 396, 426, 448, 514, 527, 558 Lipinski, ´ W., 464–466, 771, 812, 846, 847, 857, 858 Lipps, F. W., 938 Lipson, M., 902 Lisak, D., 393 Litkouhi, B., 639, 856 Liu, B., 465, 857 Liu, C., 447, 772 Liu, D., 771, 884, 885 Liu, F., 342, 343, 349, 396, 429, 436, 437, 447–449, 526, 559, 582, 610, 675, 677, 690, 712, 713, 727, 730–735, 823, 842, 843, 848–850, 853, 854, 856, 885 Liu, G., 885 Liu, G. R., 771 Liu, H., 857, 885, 903 Liu, H. P., 131, 159, 617, 638 Liu, H.-Y., 612 Liu, J., 600, 602, 604, 613, 614 Liu, K. V., 821, 848 Liu, L., 447, 451, 885 Liu, L. H., 308, 447, 448, 465, 483, 493, 601, 610, 612, 613, 649, 655, 810, 811, 883, 902 Liu, L. J., 655 Liu, M. B., 771 Liu, N., 902 Liu, R. X., 939 Liu, S.-B., 885 Liu, X., 902 Liu, X. L., 898, 901–903 Liu, Y., 125, 395, 734, 853 Lloyd, J. R., 816, 848 Lobo, M., 159, 938 Lobo, P., 449 Lockwood, F. C., 306, 309, 604, 611 Loehrke, R. I., 259, 939 Lomax, H., 281 London, A., 124 Long, L. L., 121 Long, M. B., 849 Longmore, J., 28

Longtin, J. P., 654 Loos, J., 393, 729 Loraud, J., 464 Lorentz, H. A., 56, 58, 323 Lorenz, L., 402, 446 Lorenz, N., 398 Loretz, M., 457, 464 Lou, C., 885 Lou, M., 903 Lou, W., 450 Lou, Y. S., 938 Lougou, B. G., 846, 857 Love, T. J., 560, 563, 608 Lovegrove, K., 856 Lovin, J. K., 936 Lowder, J. E., 364, 397 Lowe, A., 448 Loyalka, S. K., 489, 494 Lu, J. D., 592, 609, 610, 815 Lu, X., 649, 655, 771, 772 Lu, Y., 461, 466, 903 Lubkowitz, A. W., 936 Lucks, C. F., 121, 920 Ludwig, C. B., 371, 384, 398, 399 Luk’yanchuk, B., 654 Lundgren, T. S., 881 Lundock, R., 902 Luning, R. B., 937 Lunny, E. M., 451 Luo, Z., 885 Luomajarvi, M., 463 Lüthy, W., 463 Luzzi, A., 856 Lv, Z.-H., 466 Lynch, F. E., 936 Lyulin, O. M., 393, 729

M Ma, A., 617, 634, 638 Ma, B., 849 Ma, J., 732, 735, 736, 812, 849 Ma, L., 435, 447, 856 Ma, L.-X., 465 Ma, Q., 393 Ma, R. J., 816 Ma, Y., 494 Maag, G., 846, 847, 857, 858 MacDonald, J. D., 259 MacFarlane, J. J., 159, 935 Machida, T., 398 Mackay, D. B., 276, 282 Mackowski, D. W., 434, 436, 447, 450, 451 MacRobert, T. M., 493, 519, 559 Maestre, B., 810 Magi, V., 835, 854 Magin, T., 393, 395 Magin, T. E., 813 Magnotti, G., 849, 850 Magnussen, B. F., 850 Mahadeviah, A., 450 Mahbod, B., 938 Mahmoudi, Y., 815 Mahulikar, S. P., 939 Mai-Viet, T., 448

Author Index

Maillet, A. D., 809 Mainguy, S., 901 Maiorino, J. R., 495 Maire, E., 464, 465 Majumdar, A., 87, 122, 609 Makansi, T., 902 Maki, A. G., 394 Makino, T., 463, 466 Malalasekera, W. M. G., 615 Malkin, V. M., 814 Malkmus, W., 338, 395, 398 Mallery, C. F., 884 Malmuth, N. D., 281 Malpica, F., 614, 810 Maltby, J. D., 259, 952 Malyshev, A., 654 Mancaruso, E., 882 Mancini, M., 393, 394, 398, 952 Mandin, J.-Y., 392, 394, 733, 953 Mangelsdorf, H. G., 383, 392 Manickavasagam, S., 429, 435, 448, 450, 451, 881, 884 Mann, D. B., 196 Manohar, S. S., 808 Manos, P., 196 Mantell, S. C., 465 Manzello, S. L., 451 Maraseni, T., 882 Marble, F. E., 851 Marcott, C., 399 Marin, O., 367, 368, 398, 658, 730, 733 Mark, J. C., 516, 517, 558 Markham, J. R., 111, 116, 125, 126, 884 Markus, E., 855 Marley, N. A., 455, 464 Marquardt, D. W., 882 Marquez, R., 560, 730, 731, 735, 756, 771 Marquis, A. J., 854 Marschall, J., 447 Marshak, R. E., 516, 517, 558 Marston, A. J., 772 Marti, J., 460, 465 Martin, A. R., 613, 815 Martin, D. C., 99, 123 Martin, P. A., 885 Martin, W. W., 937 Martinek, J., 846, 857 Maruyama, S., 99, 123, 617, 638, 677, 732, 811, 812, 816, 901 Massie, S. T., 392–394, 729, 733, 953 Mast, J. C., 398 Mast, M., 126 Mastin, C. W., 614 Masuda, H., 125, 937 Mathiak, F. U., 938 Mathur, S. R., 585, 600, 610, 613, 806, 816, 817 Matijevic, E., 451 Matthews, L. K., 810 Maurente, A., 733, 748, 760, 770, 771 Mavroulakis, A., 939 Maxwell, G. M., 938 May, W., 855 Mayhofer, E., 903

Mayo, J., 897, 901 Mazumder, S., 153, 159, 253–255, 259, 281, 282, 553, 560, 561, 601, 603, 610, 613, 614, 668, 705, 731, 735, 806, 816, 817, 829, 852, 856, 882 McAdam, D. W., 937 McAdams, W. H., 123, 159, 392, 639, 731, 920, 935 McCall, R. P., 399 McCann, A., 394, 730 McClarren, R. G., 539, 560 McClatchey, R. A., 393 McConnell, D. G., 936 McCreight, C. R., 397 McDermott, R. J., 852 McElroy, D. L., 124 McEnally, C. S., 451 McFadden, P. W., 794, 813 McFarland, B. L., 935 McFarlane, R. A., 399 McGhee, J., 560 McGrattan, K. B., 855 McLean, W. I., 125 McLeod, D. G., 609, 809 McNicholas, H. J., 66, 121 McQueen, W. G., 855 McRae, D. S., 810 Mechi, R., 613 Medalia, A. I., 433, 450 Medvecz, P. J., 399 Medved, I., 466 Meerdink, S. K., 124 Megaridis, C. M., 435, 437, 450, 451 Mehdizadeh, A. M., 857 Mehta, R. S., 694, 695, 734, 771, 831, 832, 853, 949, 951 Meiers, R., 857 Meiers, R. A., 856 Meléndez, J., 882 Melis, S., 735, 856 Mel’man, M. M., 938 Menard, W. A., 358, 361, 364, 397 Menart, J. A., 440, 451, 609, 675, 731, 732 Mendoza, C., 395 Mengüç, M. P., 158, 159, 289, 308, 429, 435, 438, 448, 450, 451, 466, 526, 559, 560, 592, 611, 614, 735, 761, 772, 814, 821, 835, 839, 848, 854, 856, 881, 883, 884, 900, 903, 934, 939, 948, 951 Menigault, T., 857 Menon, S., 655 Merci, B., 853 Meredith, K. V., 855 Merkle, C. L., 851 Merriam, R. L., 811 Messig, D., 848 Mesyngier, C., 611, 799, 814, 816 Metcalfe, A. G., 124 Meyer, B. R., 815 Meyhofer, E., 901 Mie, G. A., 24, 29, 402, 446 Mighdoll, P., 375, 398 Mignon, P., 815

965

Mikhailenko, S. N., 392, 393, 729, 733, 953 Mikhalovsky, S. V., 448, 449 Miki, K., 733 Miliauskas, G., 811 Miller, C. E., 392, 733, 953 Miller, G. B., 282, 936 Miller, J. A., 848 Miller, W. F., Jr., 612 Miller, W. F. J., 771 Millikan, R. C., 432, 433, 449 Mills, A. F., 123, 920 Mills, D. R., 857 Milne, F. A., 505, 510 Milos, F. S., 447 Milovich, D., 902 Minkowycz, W. J., 282, 495, 610, 881 Minning, C. P., 937 Mishchenko, M. I., 425, 447, 451 Mishkin, M., 811 Mishra, S. C., 811, 812 Mital, R., 466 Mitalas, G. P., 936 Mitchell, R. P., 281 Mitcheltree, R., 308 Mitchner, M., 448 Mitra, K., 308, 609, 649, 650, 654, 655 Mitra, S. S., 122 Mittal, A., 613 Mittapally, R., 903 Mitts, S. J., 96, 123 Miura, A., 902 Miura, T., 854 Miyawaki, Y., 771 Miyazaki, Y., 809 Miyoshi, Y., 811 Mizner, G. A., 855 Mlawer, E. J., 398 Mlynczak, M. G., 398 Moazzen-Ahmadi, N., 392, 393, 729, 733, 953 Mockus, J., 882 Modak, A. T., 359, 373, 397, 398, 837, 855 Modest, M. F., 115, 122, 125, 126, 227, 234, 242, 252, 259, 326, 344, 349, 351, 354, 356, 367, 385, 392–399, 422, 424, 446, 463, 466, 479, 488, 493–495, 514, 520, 522, 527, 533, 535, 539, 544, 548, 552, 558–561, 613, 614, 627, 633, 636, 638, 646, 649, 652, 655, 658, 668, 669, 672, 673, 677–679, 686, 688, 690, 694–696, 699–701, 705, 711–713, 715–718, 721, 726, 727, 730–736, 741, 743, 745, 747–750, 756, 760, 770–772, 797, 798, 814, 815, 829–831, 834, 840, 848–850, 852–857, 882, 885, 937, 938, 949, 951–953 Moen, R. L., 846, 857 Moftakhari, A., 612 Mohamad, A. A., 850 Möller, S., 857 Moller, R., 902 Monchoux, F., 883

966 Author Index

Monda, O., 450 Moon, J. H., 282 Moon, P., 17, 28, 136, 159, 935 Moore, T. J., 809 Moore, V. S., 124 Moosmüller, H., 450 Morales, J. C., 559, 816, 882 Morel, J. E., 560, 649, 654 Morgan, W. R., 131, 158, 935 Morino, I., 398 Morizumi, S. J., 365, 397, 936 Morley, N., 126 Morlot, G., 609, 614 Morokoff, W., 772 Morozov, V. A., 881 Morton, D. P., 259, 882, 883 Moshammer, T., 282 Moskalev, A. N., 560 Moss, J. B., 615, 849, 855 Moss, J. N., 813 Moss, R. L., 653 Moss, T. S., 450 Mossi, A. C., 883 Most, J. M., 611, 849 Motte, F., 882 Mountain, R. D., 448 Mudan, K. S., 855 Mueller, G., 902 Mueller, H. F., 281 Mukhopadhyay, A., 850 Mukut, K. M., 836, 855 Mulet, J.-P., 901 Mulford, R. B., 124 Mulholland, G. W., 448, 450–452 Müller, H. S. P., 393, 729 Müller, R., 847, 858 Müller, U., 283 Müller-Hirsch, W., 902 Munipalli, R., 612 Murio, D. A., 881 Murphy, S. N. B., 851 Murray, R. L., 513, 558 Murthy, J. Y., 585, 600, 610, 613, 806, 816, 817 Murty, C. V. S., 939 Murty, V. D., 491, 495 Muthukumaran, R., 812 Myasnikova, G. I., 852 Myers, V. H., 461, 463, 466 Myhre, G., 398 Myöhänen, K., 733

N Naeimi, H., 254, 259 Nagasawa, C., 398 Nagato, Y., 816 Nagumo, Y., 809 Naidenov, V. I., 813 Nakajima, T., 854 Nakamae, E., 771 Nakamura, T., 846, 857 Nakazawa, T., 393 Nanda, A. K., 126 Nannei, E., 613, 814

Naraghi, M. H. N., 633, 636, 638, 639, 856, 937, 938 Narayana, K. B., 814, 816 Narayanaswamy, A., 152, 159, 894, 897, 898, 901, 902 Naso, V., 465 Nassab, S. A. G., 815 Natarajan, G., 816 Nathan, G. J., 846, 857 Naukkarinen, K., 463 Naumenko, O. V., 392, 393, 729, 733, 953 Negrelli, D. E., 821, 848 Neher, R. T., 126 Nehme, W., 850 Nelson, D. A., 636, 639 Nelson, J., 435, 450 Nelson, K. E., 123, 126 Nelson, R. A. J., 883 Nemer, M., 850, 882 Nemitallah, M. A., 856 Nemtchinov, V., 394 Neu, J. T., 126 Neuroth, N., 29, 93, 123, 454, 463 Newale, A. S., 852 Newnham, D. A., 394 Nguyen, V. T., 393 Nicholas, V., 282 Nichols, K. M., 399 Nichols, L. D., 281, 935 Nicolau, V. P., 466, 881 Nicolet, W. E., 395 Niederreiter, H., 772 Nielsson, G. E., 938 Nieto-Vesperinas, M., 122 Niioka, T., 848 Nikitin, A. V., 392, 393, 729, 733, 953 Nikogosyan, D. N., 654 Nilson, E. N., 281 Nilsson, O., 439, 451 Niro, F., 392 Nishimura, M., 811 Nishita, T., 771 Niu, C., 885 Nivet, M.-L., 882 Nmira, F., 690, 734–736, 831, 848, 849, 853, 854 Noble, J. J., 619, 638 Noboa, H. L., 939 Nocedal, J., 560, 881 Nogueira, F., 882 North, G., 398 Notton, G., 882 Novo, P. J., 615 Novotny, J. L., 811, 812, 848 Novotny, L., 887, 901 Nowak, A. J., 281, 881 Nudelman, S., 122 Nusselt, W., 150, 159, 935 Nüsslin, F., 760, 771 Nyland, T. W., 124

O Obernberger, I., 394 O’Brien, D. M., 772

O’Brien, P. F., 937 O’Brien, T., 771 Ochsenbein, F., 395 O’Connor, J. R., 121 Ogawa, T., 398 Ogilvy, J. A., 85, 122 Oguma, M., 883 Oinas, V., 342, 345, 353, 396 Okajima, J., 123, 901 O’Keefe, T., 95, 123 Okrent, D., 608 Okumura, M., 451 Olcott, T. M., 281 Olejniczak, J., 658, 730 Olfe, D. B., 544, 560, 561 Oliva, A., 282, 816 Oliver, C. C., 794, 813 Oliviero, M., 465 Olmstead, W. E., 281 Olsofka, F. A., 811 Olson, G. L., 511, 649, 650, 652, 654, 655 Olynick, D. R., 658, 730 Omori, T., 848 Onasch, T. B., 450 Ondruška, J., 466 O’Neal, D., 939 Ono, A., 125, 466 Onokpe, O., 828, 852 Oppenheim, A. K., 174, 196, 399 Oppenheim, U. P., 126, 382, 399 Oraevsky, A. A., 654 Oran, E. S., 611, 848 Orbegoso, E. M., 840, 852 Orchard, S. E., 440, 451 Ordal, M. A., 74, 121 Orlande, H. R. B., 881 Orloff, L., 849 Orphal, J., 392, 394, 733, 953 Orsino, S., 856 Oruma, F. O., 812 Othmer, P. W., 234 Otis, C. E., 654 Otsuki, M., 125 Ottens, R. S., 898, 902 Ou, N. R., 561, 650, 651, 653, 654, 780, 808 Ou, S. C. S., 514, 527, 558 Ozawa, T., 395, 747, 770 Ozen, G., 838, 850 Özi¸sik, M. N., 179, 196, 486, 490, 494, 495, 609, 668, 731, 797, 799, 809–812, 814, 815, 859, 881, 882

P Pabst, V. W., 399 Paeth, A. W., 259 Pagni, P. J., 433, 450 Pai, S. I., 794, 812, 813 Pain, C. C., 613 Pak, H., 452 Pal, G., 351, 397, 548, 560, 561, 711, 721, 726, 727, 730, 734–736, 771, 831, 840, 848, 951 Palchetti, L., 398 Palik, E. D., 121

Author Index

Palluotto, L., 773 Palumbo, R., 857 Pan, R., 857 Panczak, T. D., 159, 196, 935 Pandey, D. K., 489, 494 Panesi, M., 393 Panigrahi, P. K., 450 Panner, J. F., 126 Paoli, C., 882 Pareek, V. K., 855 Parent, G., 123 Parisi, J., 902 Park, C., 392, 393, 395, 658, 730 Park, C. W., 603, 614 Park, H. M., 526, 559, 592, 611, 810, 812, 883, 884 Park, J. H., 850 Park, K., 901, 902 Park, S. H., 448, 559, 798, 815 Parker, W. J., 74, 76, 79, 121 Parmananda, M., 816 Parthasarathy, G., 367, 398, 613, 733, 952 Partlow, D. P., 95, 123 Partridge, H., 394 Paschen, F., 311, 392 Pasteur, G. A., 464 Patankar, S. V., 398, 610, 613, 733, 952 Paterson, J., 395 Patra, J., 850 Paul, C., 668, 835, 849, 850 Paul, M. C., 854 Pawlak, D. T., 733 Payne, F. R., 938 Pearson, J. T., 712, 713, 718, 720, 735, 849 Pearson, R. A., 131, 158, 935 Peck, R. E., 815 Pellaud, B., 519, 559 Pember, R. B., 848 Penner, J. E., 450, 451 Penner, S. S., 311, 321, 357, 382, 392, 455, 464 Pennypecker, C. R., 426, 435, 448 Penzkofer, A., 461, 466 Pèpin, C., 654 Peraud, J. P., 259, 772 Pereira, F., 882 Pereira, P., 309 Perera, A., 856 Perevalov, V. I., 392–394, 729, 733, 735, 736, 952, 953 Pérez-Sáez, R. B., 125 Perez, P., 332, 394 Perlick, D. B., 126 Perlmutter, M., 100, 124, 179, 196, 209, 233, 235, 258, 283, 489, 494 Perrin, A., 392–394, 729, 733, 953 Perrin, M.-Y., 327, 392–395, 399, 730 Perry, R. L., 935 Persson, B. N. J., 902 Peters, N., 851 Petherbridge, P., 28 Petrasch, J., 459, 464–466, 857 Petrov, V. A., 810 Petrova, E. V., 465 Petry, A. P., 852, 855

Pettit, G. H., 654 Pezdirtz, G. F., 123 Pfefferle, L. D., 451 Phillip, H. R., 73, 121 Phillips, W. J., 342, 386, 396 Pickett, H. M., 394 Pierce, J. B. M., 855 Pierluissi, J. H., 392 Pierrot, L., 658, 690, 730 Pilon, L., 612, 652, 655 Pineda, D. I., 885 Pinheiro, I. F., 810 Pipes, L. A., 521, 559 Pipes, P. G., 126 Pitsch, H., 855 Pittman, C. M., 935 Pitz, W. J., 848 Pizzo, Y., 123 Placido, E., 465 Plamondon, J. A., 209, 216, 233, 234, 935 Planck, M., 6, 28 Plass, G. N., 311, 392, 405, 433, 434, 437, 446, 449 Plendl, J. N., 122 Pletcher, R. H., 881 Ploteau, J. P., 282 Poitou, D., 833, 854 Pokomy, R., 465 Polgar, L. G., 209, 233 Pollack, J. B., 455, 464 Pollard, A., 609, 610 Polman, A., 903 Polyansky, O. L., 393, 729 Pomraning, G. C., 308, 486, 494, 539, 560 Pontaza, J. P., 612 Pooladvand, K., 884 Poon, S. C., 227, 234, 242, 252, 259 Pope, S. B., 829, 830, 851, 853 Porter, J. M., 878, 882 Porter, R., 735, 850, 856 Porter, R. T. J., 850 Porterie, B., 123, 464 Postlethwait, M. A., 112, 125, 399 Pottas, J. J., 857, 858 Potter, J. F., 423, 447 Pourdeyhimi, B., 465 Pourkashanian, M., 735, 850, 856 Pourshaghaghy, A., 884 Poynter, R. L., 394 Prabhu, D., 730 Pranzitelli, A., 735 Predoi-Cross, A., 392, 733, 953 Press, W. H., 258, 734, 882 Price, D. J., 77, 121 Purcell, E. M., 426, 435, 448 Pyle, J., 398

Q Qi, C., 616, 885 Qi, H., 494, 612, 883, 885 Qian, B., 732 Qian, L., 766, 773 Qing, C. N., 854 Qiu, J., 465

967

Qu, W., 282 Qu, X., 885 Quenard, D., 465 Quéré, P. L., 816 Querry, M. R., 126, 455, 464 Quetschke, V., 902

R Rabinowitz, M. J., 449 Radney, J. G., 451 Radtke, G. A., 772 Raether, F., 810 Raether, H., 901 Rafferty, D. A., 448 Raimund, A., 282 Raithby, G. D., 398, 593, 595, 612, 613, 733, 850 Raj, P. K., 855 Raje, S., 655 Rajhi, M. A., 856 Ralchenko, Y., 395 Ralston, A., 882 Ramadhyani, S., 611, 849, 850 Ramamurthy, H., 592, 611, 849 Raman, A., 903 Ramanathan, K. G., 125 Ramankutty, M. A., 602, 603, 614 Ramesh, N., 283 Ramsey, J. W., 233 Rando, I., 124 Randrianalisoa, J. H., 446, 464, 465, 812 Rankin, B. A., 849–852 Rao, V. R., 283, 939 Ratzel, A. C., 809 Rauch, C., 282 Rauhala, E., 463 Ravindra, N. M., 116, 126 Ravishankar, M., 153, 159, 538, 552, 560, 561, 614, 817 Ray, P. S., 455, 464 Rayleigh, L., 6, 8, 24, 28, 29, 311, 391, 402, 411, 446 Raynaud, M., 464, 466, 808, 884 Raynor, S., 281 Razani, A., 281 Razzaque, M. M., 591, 611 Rea, S. N., 937 Reader, J., 395 Reardon, J. E., 398 Reddig, D., 902 Reddy, J. N., 491, 495, 810 Reddy, J. R., 612 Reddy, P., 901, 903 Redemann, T., 398 Rees, D. A. S., 816 Régalia, L., 393 Reiche, F., 330, 393 Reid, R. L., 937 Rein, R. G., 937 Reinhardt, K., 903 Reith, R. J., 494, 668, 731 Reitz, R. D., 835, 854 Reitze, D. H., 902 Rejeb, S. B., 834, 854

968 Author Index

Remita, H., 654 Ren, A. L., 309 Ren, T., 395, 734–736, 747–749, 756, 760, 770, 835, 850, 876, 879, 882, 885, 951, 953 Ren, Y.-T., 494, 885 Reuter, G. E. H., 121 Reviznikov, D. L., 855 Rey, M., 393, 729 Rhoby, M. R., 879, 885 Riad, H., 730 Riazzi, R. J., 351, 394, 396, 715–718, 727, 734, 949, 952 Richards, L. W., 433, 450 Richards, R. F., 677, 732, 811 Richardson, H. H., 385, 399 Richmond, J. C., 121–126 Richter, H., 449 Richter, W., 126 Ricolfi, T., 126 Riethof, T. R., 122 Rigby, F. A., 126 Rinsland, C. P., 392, 394, 733, 953 Ripoll, J.-F., 511 Rivas, D., 282 Rivera, G., 124 Riveros-Rosas, D., 857 Rivière, P., 259, 342, 349, 355, 388, 393–396, 399, 658, 681, 685, 686, 712, 713, 730, 732, 735, 772, 813, 849, 950, 952 Robbins, W. H., 935 Robert, D., 393 Robert, D. A., 124 Roberton, R. B., 810 Roberts, G. C., 435, 450 Robin, M. N., 813 Roblin, A., 394 Rochais, D., 465 Rockstuhl, C., 902 Rodat, S., 858 Roddick, R. D., 123, 126 Rodgers, W., 771 Rodrigues, L. G. P., 730 Rodrigues, P., 773 Rodríguez-Conejo, M. A., 882 Roekaerts, D., 853 Roesle, M., 465 Roessler, D. M., 439, 451 Roger, M., 309, 396, 833, 834, 854 Rogers, J. T., 851 Rohrer, W. M., 281 Rohsenow, W. M., 308 Rolon, J. C., 848 Romero, M., 846, 857 Romero-Paredes, H., 857, 858 Rosendahl, L. A., 732, 856 Rosenmann, L., 324, 392–394, 730 Rosner, D. E., 451 Rosseland, S., 500, 510 Rotger, M., 392, 393, 729, 733, 953 Rothman, L. S., 392–395, 729, 730, 733, 952, 953 Rousseau, B., 465, 613

Rousseau, E., 901, 902 Roux, J. A., 126, 563, 608, 809 Roy, S., 122, 613, 764, 773, 850, 951 Roy, S. P., 560, 561, 731, 756, 771, 773, 836, 848, 850, 855 Royne, A., 857 Ruan, J., 848 Ruan, L. M., 494, 610, 612, 653, 771, 808, 883, 885 Ruan, M., 616 Ruan, X., 81, 99, 121, 124, 902 Rubens, H., 74, 121 Rubin, R., 857 Rubini, P. A., 615, 849 Rubtsov, N. A., 812 Rudkin, R. L., 124 Rukolaine, S. A., 882, 884 Rumyantsev, A., 881 Rumynskii, A. N., 812, 813 Ruo, S. Y., 936 Rupasov, V. I., 654 Ruperti, N., Jr, 808, 884 Rupin, R., 901 Rupley, F. M., 848 Rushmeier, H. E., 938 Rusk, A. N., 464 Russell, L. D., 281 Rusu, F., 882 Ruyer, P., 561 Ryhming, I. L., 484, 494 Ryou, H. S., 282 Rytov, S. M., 887, 893, 894, 900, 901

S Sabatier, P. C., 881 Sabbaghi, P., 903 Sabella, P., 771 Sabet, M., 938 Sacadura, J.-F., 105, 124, 125, 464, 466, 566, 609, 614, 734, 808, 809, 811, 850, 881, 883, 884 Sacksteder, K., 849 Sadat, H., 486, 494, 612 Sadler, R., 124 Sadoqi, M., 654 Sagadeev, 448 Sahoo, S., 282 Said, R., 613 Saito, K., 450 Saitoh, S., 559, 815 Sakami, M., 593, 603, 612, 614, 654, 811, 884 Sakate, H., 125 Sakuma, F., 125 Sala, R., 613, 814 Salagnac, P., 282 Salah, M. B., 612 Salinger, A. G., 559, 816 Sallah, M., 494 Salpeter, E. E., 392 Saltiel, C., 613, 815, 938 Salzmann, D., 392 Samadianfard, S., 882 Sami, M., 856 Samson, R. J., 450

Sánchez, A., 566, 609 Sanchez-Gil, J. A., 122 Sani, E., 430, 449 Sankar, M., 553, 561, 601, 603, 613, 614 Sankaran, R., 561, 848 Santarelli, F., 609 Santigosa, L. R., 844, 857 Santoro, R. J., 884 Santos, R. G., 852 Sanz, J., 282 Sarma, G. S. R., 395 Sarno, A., 180, 196, 937 Sarofim, A. F., 209, 233, 369–371, 392, 398, 426, 432–434, 437, 445, 448, 449, 451, 608, 617, 619, 636, 638, 658, 683, 731, 920, 936 Sarvari, S. M. H., 612, 615, 766, 773 Sasaki, T., 121 Sasse, C., 615 Sastri, V. M. K., 939 Sathe, S. B., 815 Sato, T., 449 Sauer, H. J., 937 Sauerbrey, R., 654 Savvinova, N. A., 812 Saxena, S. C., 124 Saxon, D. S., 435, 451 Sayah, H., 850 Sazhin, S. S., 448, 449 Sazhina, E. M., 448 Scafati, F. T., 882 Scaglione, A. P., 813 Schafbauer, T., 282 Schäfer, F., 858 Scheffe, J., 811 Schenker, G. N., 395 Scherer, V., 733 Scheuerpflug, P., 811 Schiemann, M., 733 Schimmel, W. P., 811 Schlichting, H., 812 Schmidt, E., 60, 78, 79, 121, 519, 559 Schneider, G. E., 881 Schneider, P. S., 732, 952 Schneiders, L., 448 Schnell, M., 448 Schnurr, N. M., 281 Schönauer, W., 560 Schreiber, L. H., 281 Schreider, Y. A., 236, 237, 258 Schrenker, R. G., 653 Schröder, W., 448 Schroder, P., 151, 159 Schroeder, J., 394, 730 Schulz, L. G., 123 Schulze, J., 733 Schunk, L. O., 847, 858 Schuster, A., 308, 503, 510, 772 Schwander, D., 857 Schwarzschild, K., 308, 503, 510 Schweiger, H., 282 Schwenke, D. W., 394 Scoggins, J. B., 813 Scutaru, D., 393, 394, 396, 730

Author Index

Seaïd, M., 560 Seban, R. A., 809 Sediki, E., 815, 848 Seeger, P. G., 855 Seetharamu, K. N., 283 Segall, B., 121 Segarra, C. D., 282 Seiler, N., 561 Seki, N., 812 Selamet, A., 282 Selçuk, N., 159, 466, 592, 602, 605, 611, 836, 848, 850, 883, 885 Self, S. A., 125, 462, 466 Sellers, W. H., 275, 282 Semerjian, H. G., 884 Semião, V., 429, 448, 615, 850 Sen, O., 850 Senatore, G., 456, 464 Senchenko, V., 883 Seo, S. H., 593, 612 Seo, T., 731 Seong, H. L., 282 Serfaty, R., 852 Sewell, G., 560 Sexl, R. U., 233 Shackleford, W. L., 464 Shah, N. G., 306, 309, 604, 611 Shahidi-Zandi, B., 814 Shahnam, M., 856 Shamsundar, N., 259 Shan, S., 680, 732 Shang, H. M., 613, 614 Shapiro, A. B., 159, 281, 934, 938 Sharifian, A., 466 Sharpe, S. W., 393, 729 Shealy, D. L., 233 Sheffer, D., 126 Shen, S., 901, 902 Shen, T.-R., 465 Shen, Z. F., 281, 639, 732, 808, 815 Sheng, F., 883, 884 Shi, G., 773 Shi, G. Y., 366, 398 Shi, J.-W., 885 Shibata, Y., 398 Shih, T. M., 306, 309, 505, 510, 771 Shiles, E., 73, 121 Shim, K. H., 450 Shimoji, S., 216, 234 Shindini, S. A., 813 Shine, K. P., 393, 398, 729 Shippert, T. R., 398 Shirley, P., 259 Shklyar, F. R., 814 Shokri, M., 855 Shuai, Y., 771, 857, 885, 902, 903 Shukla, K. N., 938 Shurcliff, W. A., 123 Shvarev, K. M., 462, 466 Shyy, W., 613, 815 Si, M., 885 Sibulkin, M., 668, 731 Sieber, B. A., 451 Siedow, N., 810

Siegel, R., 66, 121, 131, 158, 179, 196, 209, 233, 278, 283, 614, 780, 808, 809, 935, 939 Sievers, A. J., 125 Siewert, C., 448 Siewert, C. E., 486, 490, 494, 495, 668, 731 Sifaoui, M. S., 815, 848 Sijercic, M., 850 Sika, J., 939 Sikic, I., 837, 855 Sikka, K. K., 125, 280, 283, 399, 668, 669, 673, 731 Sikorski, R. L., 464 Silva, C. B. D., 854 Silver, M., 123 Simeckova, M., 392, 733, 953 Simmons, F. S., 343, 396 Simmons, G. M., 490, 494 Singer, J. M., 449 Singh, B., 810 Singh, B. P., 282, 446, 464, 815 Singh, V., 713, 735, 949, 952 Siregar, M. R. T., 453, 455, 463 Siria, A., 902 Sirotkin, V. V., 937 Sistino, A. J., 636, 639 Sisto, A., 902 Sivathanu, Y. R., 399, 849, 851, 852 Skettrup, T., 456, 464 Skocypec, R. D., 856 Slavejkov, A. G., 734, 856 Sliepcevich, C. M., 937 Slowik, J. G., 450 Smakula, A., 463 Smalley, R., 125 Smallwood, G. J., 396, 436, 447, 449, 732, 734, 735, 848 Smart, C., 440, 451 Smiltens, J., 121 Smith, A. M., 126, 608, 794, 809, 813 Smith, D., 735, 850, 856 Smith, D. C., 563 Smith, D. Y., 121 Smith, G. B., 123 Smith, K. M., 394 Smith, M. A. H., 392, 394, 733, 953 Smith, M. G., 642, 653 Smith, M.-A. H., 393, 729 Smith, N. S. A., 849 Smith, P. J., 580, 592, 593, 610, 611, 771 Smith, T. F., 77, 96, 121–123, 222, 234, 281, 566, 609, 639, 679, 717, 732, 777, 798, 808, 809, 811, 815, 939 Smith, V. C., 392 Smooke, M. D., 849 Smurov, I., 883 Snail, K. A., 126 Snegirev, A. Y., 828, 837, 852, 855 Snoek, J., 882 Sobol, I. M., 772 Sohal, M., 279, 283 Sohn, C. H., 849 Sohn, I., 333, 395 Solomon, P. R., 125, 126, 884

969

Solovjov, V. P., 351, 352, 395, 396, 658, 685, 688, 696, 700–702, 709, 715, 718, 730, 733–735, 849 Sommer, A. J., 399 Sommerfeld, A., 558 Sommers, E. V., 276, 282 Sommers, R. D., 936 Sondheimer, E. H., 121 Song, B., 887, 901 Song, G., 850 Song, J.-L., 615 Song, K. H., 643, 654 Song, M., 611, 799, 815 Song, T. H., 125, 593, 603, 604, 612, 614, 883 Song, Y., 885 Sonntag, R. E., 731 Sørensen, G. O., 394 Sorensen, C. M., 435, 450, 451 Sotos, C. J., 936 Soucasse, L., 255, 259, 613, 763, 772, 813 Soufiani, A., 259, 311, 315, 318, 324, 342, 343, 345, 349, 355, 391–396, 399, 654, 680, 686, 688, 713, 715, 730–732, 735, 736, 772, 798, 808, 813, 815, 816, 832, 848–850, 853, 950, 952 Souil, J. M., 939 Souloukhin, R. I., 813 Sowell, E. F., 937 Sparrow, E. M., 28, 67, 87, 112, 121, 122, 125, 131, 136, 158, 159, 179, 180, 188, 196, 197, 208, 209, 233, 252, 259, 275, 276, 279, 281–283, 308, 484, 494, 610, 881, 935–937 Spearrin, R. M., 885 Specht, E., 398 Speck, E. P., 935 Speranza, A. C., 126 Sphaier, L. A., 810 Spindler, K., 937 Spinrad, R. W., 443, 452 Spitzer, W. G., 80, 121, 123 Spizzichino, A., 85, 122, 233 Sposovin, A. V., 855 Spuckler, C. M., 614, 809 St. Clair, C. R., 881 St-Gelais, R., 899, 902 Stair, F., 125 Stamnes, K., 571, 609 Stanmore, B. R., 615, 850 Starikova, E., 393, 729 Stasenko, A. L., 936 Stasiek, J., 938 Stefanizzi, P., 938 Stegun, I. A., 159, 259, 341, 395, 408, 446, 515, 558, 734, 943 Steinfeld, A., 465, 466, 771, 811, 846, 847, 857, 858 Stelzner, B., 848 Stephens, C. W., 935 Stephenson, D. G., 936 Stephenson, P., 850 Stetson, A. R., 124 Stevens, D., 552, 560, 638 Stevenson, J. A., 935

970 Author Index

Stevenson, W. H., 809 Stewart, I. M., 392, 448 Stewart, J. V., 125 Stewart, S. M., 8, 28 Stierwalt, D. L., 461, 466 Stockman, N. O., 281, 936 Stolberg-Rohr, T., 393 Stolz, G., 881 Stone, J. M., 31, 43, 58 Story, G. M., 399 Stouffer, S. D., 849 Streed, E. R., 123 Streetman, B. G., 453, 463 Strieder, W., 233, 281, 938 Stroeve, P., 856 Ströhle, J., 398, 733, 854, 856 Stubican, V. S., 122 Stull, V. R., 433, 434, 437, 449 Su, B., 526, 559 Su, K. C., 123 Subramaniam, S., 526, 559, 883 Subramanian, S. V., 813 Sugawara, M., 812 Sugioka, K., 816 Sugiyama, S., 811 Sullivan, R., 123 Sumitra, P. S., 937 Sun, B., 855 Sun, C., 465 Sun, F.-X., 613, 772 Sun, H.-F., 600, 613, 772 Sun, J., 612, 885 Sun, W., 426, 448 Sun, X., 771 Sun, Y., 309, 720, 736, 810, 811, 816, 844, 856 Sun, Y.-S., 812 Sun, Z., 99, 124 Sundararajan, T., 283 Sunden, ´ B., 854 Sung, K., 259, 392, 393, 729, 733, 953 Suo-Anttila, J., 451, 452 Sutton, G., 882 Sutton, K. A., 393, 395 Sutton, W. H., 560 Svet, D. I., 121, 920 Swanson, A. D., 464 Swathi, P. S., 446, 466, 526, 559 Swithenbank, J., 429, 448, 559 Syamlal, M., 771 Sydnor, C. L., 937 Szalmas, L., 772 Szel, J. V., 196 Szeles, D. M., 125 Szema, K. Y., 813

T Tabaczynski, R. J., 794, 813 Tabanfar, S., 646, 649, 652, 655, 815 Tabor, H., 99, 123 Taflove, A., 903 Tafreshi, H. V., 465 Tainaka, K., 885 Taine, J., 311, 315, 318, 324, 331, 342, 343, 345, 349, 355, 391–396, 458, 463,

464, 466, 654, 680, 686, 713, 715, 730, 731, 734–736, 772, 797–799, 808, 813–816, 832, 850, 853, 950, 952 Taitel, Y., 813 Tajouri, A., 882 Takacs, M., 811 Takara, E. E., 617, 634, 638, 811, 856 Takashima, N., 730 Takasuka, E., 125 Takita, K., 732 Tamanovich, V., 883 Tamehiro, H., 815 Tan, E., 851 Tan, H. P., 465, 493, 494, 610, 612, 613, 643, 653, 733, 766, 771, 773, 789, 808, 810–812, 846, 853, 857, 883, 885, 902 Tan, J. Y., 308, 448, 465, 483, 493, 494, 612 Tan, T., 857 Tan, W.-C., 903 Tan, Y., 393, 729 Tan, Z. M., 605, 615, 649, 655, 812, 856 Tanaka, M., 393 Tancrez, M., 458, 463, 464 Tanev, S. e. a., 448 Tang, K., 88, 122, 125, 394 Tang, K. C., 690, 734 Tang, S., 823, 849 Tanner, D. B., 399, 902 Tanno, K., 885 Tanno, S., 854 Tao, W. Q., 939 Tashkun, S. A., 392–394, 729, 733, 735, 736, 952, 953 Tatarskii, V. I., 901 Tauber, M. E., 730 Taubner, T., 901 Tauc, J., 464 Taussky, O., 237, 258 Taylor, P. B., 732 Taylor, R. E., 124 Taylor, T. B., 679 Teffo, J.-L., 394, 735 Tekkalmaz, M., 511 Tello, M. J., 125 Tencer, J., 611 Tenkolsky, S. A., 258, 734 Tennant, J. S., 937 Tennyson, J., 392–394, 729, 733, 952, 953 Terashima, K., 125 Terrapon, V. E., 851 Tessé, L., 690, 734, 762, 772, 832, 853 Teukolsky, S. A., 882 Tezuka, S., 772 The Opacity Project Team, 395 Thekaekara, M. P., 28, 124 Thellier, F., 883 Thibault, F., 393 Thirumalaisamy, R., 816 Thomas, A., 526, 559 Thomas, X., 393 Thommen, D., 858 Thömmes, G., 560

Thompson, D., 903 Thompson, J. F., 614 Thomson, J. A. L., 398 Thorsen, R. S., 279, 283 Thorsos, E., 122 Thring, M. W., 433, 449 Thunman, H., 732 Thurgood, C. P., 566, 609 Thynell, S. T., 486, 494, 668, 731, 808, 811, 883, 884 Tian, H., 616 Tian, J., 816 Tien, C. L., 87, 122, 216, 233, 275, 282, 309, 311, 312, 315, 358, 363–366, 369, 370, 375, 376, 381, 383, 385, 391, 392, 397, 399, 401, 402, 427, 433, 434, 437, 440, 445, 446, 448–451, 609, 654, 780, 781, 808, 809, 855, 901, 902 Tikhonov, A. N., 859, 881 Timans, P. J., 454, 463 Timchenko, V., 812 Timofeyev, V. N., 797, 814 Tindall, J. W., 259 Tinet, E., 526, 559 Tipping, R. H., 393, 394, 733 Tishkovets, V. P., 465 Tiwari, S. N., 364, 397, 813 Todd, C. A., 935 Todd, D. C., 608 Todd, J., 237, 258 Tokizaki, E., 125 Tolchenov, R. N., 394 Tominaga, J., 901 Tong, F. M., 126 Tong, T. W., 446, 466, 526, 559, 653, 808, 810, 811, 815, 856 Tong, Z., 124 Toon, G. C., 393, 729 Toor, J. S., 210, 222, 226, 227, 233, 234, 242, 259 Torborg, R. H., 121, 126, 920 Torn, M. S., 398 Torpey, H., 458 Torpey, M., 464 Torrance, K. E., 67, 87, 112, 121, 122, 125 Toshiyoshi, H., 902 Tosi, M. P., 464 Totemeier, T. C., 920 Toth, R. A., 392, 394, 733, 953 Touloukian, Y. S., 82, 113, 121, 920 Toups, K., 936 Townsend, M. A., 281 Tran, H., 393, 729 Traugott, S. C., 664, 731 Travis, L. D., 447 Trayanov, G. G., 938 Trelles, J. P., 613 Tremante, A., 614, 810 Trimis, D., 848

Author Index

Triolo, J. J., 126 Tripp, C. P., 399 Trivedi, A., 654 Trombe, A., 939 Trombe, F., 100, 124, 846, 857 Trovalet, L., 561 Truelove, J. S., 563, 565, 566, 591, 592, 608, 609, 679, 680, 732 Tsai, D. P., 901 Tsai, D. S., 209, 233 Tsai, J. H., 809 Tsai, J. R., 574, 609, 811 Tsao, C. K., 794, 813 Tsay, S.-C., 609 Tseng, C. C., 458, 464 Tseng, C. J., 810 Tseng, J. W. C., 938 Tso, C. P., 939 Tsukada, T., 559, 799, 816 Tsurimaki, Y., 896, 901 Tualle, J.-M., 526, 559 Tucakovic, D., 850 Tuchscher, J. S., 397 Tulloch, B. B., 884 Tuntomo, A., 430, 448, 780, 808 Tuomi, T., 463 Turner, W. D., 939 Turns, S. R., 734, 849, 856 Twynstra, M. G., 885 Tyuterev, V. G., 393, 729

U Uchino, O., 398 Ugnan, A., 448 Usiskin, C. M., 179, 196, 494, 935

V Vachon, R. I., 813 Vader, D. T., 112, 125, 280, 283 Vaglieco, B. M., 450, 882 Vaillon, R., 342, 344, 395, 396, 885, 901, 903 Vajta, T. F., 123 Valadés-Pelayo, P. J., 857 Van Dam, A., 259 Van de Hulst, H. C., 37, 58, 402, 403, 413–415, 422, 445, 447, 653 Van Eyk, P. J., 857 Van Leer, B., 610 Van Leersum, J., 259, 938 Van Wylen, G. H., 731 Vandaele, A. C., 392, 733, 953 Vander Auwera, J., 393, 729 Vanderwood, P. C., 392 Vantelon, J. P., 939 Varanasi, P., 394 Variot, B., 609, 815 Varshalovich, D. A., 529, 560 Vasalos, I. A., 445, 451, 608 Velho, H. F. C., 883 Velusamy, K., 283 Venkateshan, S. P., 153, 159, 283, 883, 939 Venstrom, L., 857 Vercammen, H. A. J., 633, 639, 771 Verma, A. K., 309, 856

Verma, N. N., 282 Versteeg, H. K., 605, 615 Vetterling, W. T., 258, 734, 882 Vicquelin, R., 773, 814, 852 Videen, G., 447, 448 Vidu, R., 856 Vielmo, H. A., 770, 883 Vieweg, M., 449 Villafán-Vidales, H. I., 847, 857, 858 Vineeth, S., 282 Viskanta, R., 125, 227, 234, 242, 259, 283, 289, 308, 427, 429, 448, 464–466, 484–486, 494, 526, 560, 592, 611, 614, 664, 731, 780, 792, 793, 797, 799, 808–816, 821, 848–850, 854, 882, 948, 951 Visona, S. P., 615, 850 Vitkin, E. I., 883 Vo, C., 464 Vogel, E. M., 464 Volokitin, A. I., 902 Volz, S., 902 Von Zedtwitz, P., 771 Vongsoasup, N., 903 Vossier, A., 857 Voyant, C., 882 Vyushin, V. D., 448

W Wagner, G., 393, 394, 729 Wagner, J. C., 772 Wakao, N., 936 Wakatsuki, K., 394, 855 Wall, L., 937 Wall, T. F., 427, 448 Walsh, D. J., 121, 123 Walters, D. V., 236, 258, 761, 770, 771 Walther, V. A., 617, 636, 638 Walton, G. N., 159, 935, 950, 952 Wang, A., 326, 349, 367, 392, 396, 561, 712, 713, 718, 721, 734, 745, 748, 750, 760, 770, 771, 831, 832, 848, 853, 951, 952 Wang, B., 857, 858 Wang, B.-X., 465, 466, 614 Wang, B. Y., 559 Wang, C., 696, 703, 705, 708, 709, 711, 718, 720, 734–736, 748, 770, 840, 856, 951, 953 Wang, C.-A., 460, 465, 483, 494, 612 Wang, C.-H., 612, 773, 812 Wang, C. C., 430, 448 Wang, C. J., 486, 494 Wang, D., 885 Wang, F., 771, 857, 884 Wang, F.-Q., 857, 885 Wang, G., 615, 849 Wang, G. X., 609, 812 Wang, H., 449, 853, 903 Wang, H. Y., 611, 849 Wang, J., 493, 559, 884 Wang, J. C. Y., 938 Wang, L., 561, 585, 610, 721, 734, 736, 748, 770, 840, 848, 849, 856, 902, 903

971

Wang, L. P., 899, 901–903 Wang, L. S., 365, 397, 781, 808, 809 Wang, L. W., 809 Wang, N. S., 814 Wang, Q., 732 Wang, S., 816, 885 Wang, S.-M., 616 Wang, T.-B., 902 Wang, T. P., 938 Wang, T. S., 614 Wang, W., 903 Wang, W.-J., 903 Wang, W. C., 358, 364, 366, 397, 398 Wang, X.-D., 465, 466 Wang, Y., 609, 855, 903 Wang, Z., 615, 616, 732, 772, 885 Wang, Z.-C., 615 Ward, C. A., 121 Warming, R. F., 474, 488, 490, 493, 494 Warna, J. P., 938 Warnatx, J., 848 Warsi, Z. U. A., 614 Waslander, S. L., 884 Wassel, A. T., 677, 732, 813 Watanabe, H., 855 Waterman, P. C., 425, 447 Waterman, T. E., 121 Watjen, J. I., 903 Watts, R. G., 936 Wattson, R. B., 393, 394, 730 Wayne, F. D., 855 Wcisło, P., 393, 729 Weast, R. C., 58, 121 Webb, B. W., 351, 352, 395, 396, 657, 658, 680, 683–685, 687, 694, 700, 701, 710–712, 715, 718, 729, 730, 732–735, 799, 810, 816, 849, 949, 952 Weber, R., 362, 393, 394, 397, 398, 952 Wecel, G., 732, 856, 952 Weeks, J. R., 79, 121 Wei, C., 885 Wei, L.-Y., 494 Weidler, P. G., 466 Weimar, A. W., 846, 857 Weinberg, F. J., 851 Weinberger, H., 123 Weinberger, K. Q., 882 Weiner, M. M., 242, 259, 397 Weinman, J. A., 447 Welker, J. R., 937 Wells, M. B., 771 Welty, J. R., 125 Wen, J. X., 449, 614, 734, 855 Wen, S., 885 Wendlandt, B. C. H., 810 Wersborg, B. L., 449 Werther, J., 603, 614 West, R., 396 Westbrook, C. K., 848 Westlye, F. R., 328, 393, 855 Weston, K. C., 810 Wheeler, V. M., 857, 858 White, F. M., 21, 28

972 Author Index

White, S. M., 447, 466 Whiting, B. F., 902 Whiting, E., 395 Whitton, M. C., 159 Wibberley, L. J., 448 Wiebelt, J. A., 936 Wiedenhoefer, J. F., 835, 854 Wien, W., 6, 8, 9, 28 Wilcox, L., 612 Wild, M. W., 813 Wilf, H. S., 882 Wilkins, J. E., 281 Willey, R. R., 126 Williams, A., 735, 850, 856 Williams, D., 396, 464 Williams, D. R., 266 Williams, F. A., 851 Williams, G. C., 449 Williams, M. L., 811 Williams, S. D., 281 Wilson, B., 463 Wilson, I. H., 902 Wilzewski, J., 393, 729 Wiscombe, W. J., 405, 446, 447, 609 Wise, S., 902 Witte, L. C., 884 Wittig, S., 614, 852 Woertz, B. B., 99, 123 Wolfe, E., 493 Wolfe, W. L., 125, 937 Wöll, C., 466 Wong, B. T., 761, 772 Wong, R. L., 158, 934 Wood, B. E., 126 Wood, R., 449 Wood, W. D., 89, 121, 920 Woodruff, T. O., 456, 464 Woodward, D. H., 440, 451 Wooten, F., 461, 466 Worsnop, D. R., 450 Wray, A. A., 393 Wriedt, T., 425, 447 Wright, M., 730 Wright, S. J., 560, 881 Wu, B., 823, 837, 849, 850, 855 Wu, C.-Y., 308, 493 Wu, C. Y., 483, 486, 494, 526, 544, 559–561, 615, 649–651, 653, 654, 780, 808, 877, 881 Wu, J., 732 Wu, J. W., 526, 560, 809 Wu, K., 885 Wu, P. K., 855 Wu, S., 903 Wu, S. H., 615, 649, 654, 877, 881 Wu, S. T., 810 Wu, T., 855 Wu, X., 812 Wu, Y., 760, 771, 834, 854 Wu, Z., 447 Wulff, W., 936 Wylie, C. R., 58, 123, 135, 159, 233, 446, 808 Wyss, P., 465

X Xia, X. L., 465, 613, 772, 808, 857 Xing, H., 857 Xiong, Y., 885 Xu, C., 885 Xu, C.-L., 616 Xu, J. B., 898, 902 Xu, X., 643, 654, 853 Xu, Y.-L., 426, 447 Xuan, Y., 123, 902, 903

Y Yadav, R., 309, 856 Yamada, J., 447 Yamada, Y., 440, 443, 446, 452, 654, 816 Yamaguchi, S., 848 Yan, J.-H., 771, 884 Yan, W. M., 799, 816 Yan, Y. Y., 771 Yang, B., 450 Yang, C. Y., 883 Yang, G., 99, 124 Yang, J., 514, 535, 544, 558, 560, 613, 734, 770, 772 Yang, K. T., 812, 816, 848 Yang, P., 426, 447, 448 Yang, R., 851 Yang, W. M., 816 Yang, X., 733, 831, 853 Yang, Y., 122, 902, 903 Yao, C., 609, 808, 812 Yao, Q., 609 Yarbrough, D. W., 938 Yaroslavsky, A. N., 452 Yaroslavsky, I. V., 443, 452 Ye, B., 736 Yee, S. K., 426, 448 Yee, S. S., 901 Yen, S. H., 125 Yener, Y., 495, 797, 814 Yi, H.-L., 494, 612, 773, 812 Yin, C., 680, 732, 838, 856 Yin, J. Y., 447 Yin, Y. S., 616 Yin, Z., 634, 639, 799, 815 Yoldas, B. E., 95, 123 Yon, J., 449 Yong, S., 846, 857 Yoo, D. H., 883 Yoon, H., 857 Yoon, T. Y., 592, 611, 810, 884 Yoshi, S., 449 Yoshida, A., 466 Yoshikawa, T., 835, 854 Yoshino, K., 394 You, R., 451 Young, S. J., 336, 393, 396 Yousefian, F., 883, 884 Yu, C., 857 Yu, E., 283 Yu, G., 885 Yu, H.-T., 466 Yu, K., 125 Yu, Q. Z., 493, 883

Yu, S., 393, 729 Yu, T.-B., 902 Yu, V. C., 937 Yu, Y., 736 Yuan, J., 615, 850 Yuan, P., 883 Yuan, Y., 857, 885 Yücel, A., 811, 813 Yuen, W. W., 617, 634, 638, 809, 811, 856, 937 Yung, Y. L., 311, 315, 391, 734 Yutevich, I. B., 813

Z Zabaras, N., 884 Zachariah, M. R., 451 Zak, E. J., 393, 729 Zakharova, N. T., 447 Zakhidov, R. A., 810 Zamuner, B., 734, 772, 853 Zamuraev, V. P., 793, 813 Zaneveld, J. R. V., 452 Zangmeister, C. D., 451 Zara, J., 259 Zargarabadi, M. R., 849 Zaworski, J. R., 125 Zeeb, C. N., 159, 254, 259, 935, 952 Zeghondy, B., 459, 463, 464, 466 Zeippen, C. J., 395 Zeng, S. Q., 783, 809 Zerlaut, G. A., 124 Z’Graggen, A., 858 Zhang, B., 616, 885 Zhang, B. X., 281 Zhang, F., 398 Zhang, H., 354, 395, 397, 398, 658, 686, 690, 705, 712, 721, 730, 733, 734 Zhang, H. C., 613 Zhang, J., 124 Zhang, J. Q., 884 Zhang, K., 111, 125 Zhang, Q., 885 Zhang, Q. J., 816 Zhang, R. Z., 902 Zhang, S.-D., 465 Zhang, W., 616, 885 Zhang, X., 309, 616, 810, 811, 816, 844, 856 Zhang, X.-L., 885 Zhang, X.-R., 466 Zhang, Y., 494, 612, 773, 902 Zhang, Y. F., 814 Zhang, Z., 115, 126, 463, 466, 559, 772, 849, 902 Zhang, Z. H., 901 Zhang, Z. M., 55, 58, 88, 122, 887, 889, 891, 899, 901–903 Zhao, B., 889, 901, 903 Zhao, C., 736, 885 Zhao, C. Y., 99, 124, 903 Zhao, J., 851 Zhao, J.-J., 465, 466 Zhao, J. M., 297, 308, 493, 612, 902 Zhao, Q.-M., 902 Zhao, W., 882 Zhao, X. Y., 823, 849, 850, 855

Author Index

Zhao, Y., 124, 125, 435, 447, 903 Zhdanovich, O., 883 Zhen, B., 613 Zheng, B., 559 Zheng, C. G., 883, 884 Zheng, S., 616, 885 Zheng, Y., 399, 852, 894, 901 Zheng, Z., 902 Zhong, L.-Y., 902 Zhong, M., 851

Zhou, H., 615, 616, 730, 731, 735, 856, 885 Zhou, H. C., 605, 615, 883, 884 Zhou, P.-C., 494 Zhou, W., 812 Zhou, Y., 122, 720, 732, 736 Zhou, Z., 732 Zhu, J., 903 Zhu, J. Y., 451, 452 Zhu, K.-Y., 483, 493, 526, 559 Zhu, L., 902, 903

Zhu, Q. Z., 122 Zhu, T., 612 Zhu, X., 732 Ziering, M. B., 209, 233 Zimberg, M. J., 849 Zivanovic, T., 850 Zoby, E. V., 730 Zweifel, P. F., 494, 668, 731

973

Index A Abel’s transformation, 878 Absorbance, 286 Absorbing medium, 48 Absorptance, 20, 62–65 solar, 98, 908 spectral, directional, 63 spectral, hemispherical, 64, 69 total, directional, 64, 65 total, hemispherical, 65 Absorption, 2, 22, 312 collision-based, 741 energy partitioning, 252, 741, 756 gray, diffuse, 127 in a participating medium, 286 multiphoton, 643 negative, 312, 314 pathlength method, 741 saturable, 643 suppression, 741 Absorption band, 56, 57, 81, 83 Absorption coefficient, xxii, 22, 36, 82 band-integrated, xxii correlated, 353, 687 database, 712 density-based, 286, 314, 319, 684 effective, 314, 664, 665 for a particle cloud, 407, 409 for coal particles, 427 for Rayleigh scattering, 412 line-integrated, xxi, 323 linear, 286, 314, 683 mean, 374–376 modified Planck-mean, 375 narrow band average, 336, 337 Planck-mean, 374, 408, 427, 428, 437, 665, 752 pressure-based, 286, 314, 319 Rosseland-mean, 376, 408, 427, 428, 437, 665 scaled, 354, 685, 687 soot, 432, 437 spectral, 335 of carbon dioxide, 320, 333 of clear ice, 455, 456 of clear water, 455, 456 of halides, 454 of ionic crystals, 453, 454 of lithium fluoride, 456 of nitrogen, 321

of silicon, 455 of window glass, 23, 24, 454 true, 314 Absorption cross-section, 403 molar, 683 molecular, 683 Absorption Distribution Function model, 685, 798 Absorption edge, 453, 456 Absorption efficiency factor, 403 for absorbing spheres, 406 for fuel sprays, 430 for specularly reflecting spheres, 416 for water sprays, 431 Absorption suppression, 761 Absorptive index, xx, 6, 35, 82, 83, 287, 295 Absorptivity, xxii, see also Absorptance, 286, 327 of a gas layer, 22 of a thick slab, 92 of an isothermal medium, 678 of carbon dioxide, 23 spectral of a participating medium, 286, 674 total of a gas, 370, 675 of a gas–particulate suspension, 676 of a participating medium, 674 of an isothermal medium, 677 Acetylene, 442, 822, 823 Acrylic paint, 90 ADA, see Advanced differential approximation ADF method, 685, 721, 798, 799 Advanced differential approximation, 548–552 Aerogel, 463, 780 Aggregate fractal, 425 soot, 425, 426 Air mass, 5 Air plasma, 320 Airy’s formulae, 55 ALBDF, 685, 693 database, 718 Albedo, scattering, xxiii, 24, 25, 290 Alumina, 89, 111 Aluminum, 73, 89, 111, 406, 892, 893, 896 Ammonia, 376 Amorphous solid, 85, 454

Amplitude function, 403, 404, 411 for diffraction, 415 Angle azimuthal, xxiii, 11 Brewster’s, 46 critical, 47, 296 divergence, 439 of incidence, 44, 46, 68 of refraction, 44 opening, 100 phase, 33, 53, 461 polar, xxii, 11, 14 polarizing, 46 scattering, xxii, 405 solid, xxiii, 10–12 zenith, 5 Angle factor, see also View factor, 128 Angular frequency, xxiii, 3 Anomalous diffraction, 414 Anomalous skin effect, 76, 79 Apparent emittance, 173, 180, 212 Ash particle, 427 Asymmetric top, 315 Asymmetry factor, 405, 422, 424 for a particle cloud, 408, 409 for coal particles, 427 Atmosphere, 377 absorption coefficient, 377 transmissivity, 377 Atmospheric greenhouse effect, 377, 378 Atomic force microscope, 898 Attenuation by absorption, 286 by scattering, 287 Attenuation vector, 33 Average emission temperature, 705 Azimuth, 38 Azimuthal angle, xxiii, 11

B Babinet’s principle, 415 Band absorption, 81, 83 electron energy, 56 electronic–vibration–rotation, 320 fundamental, 318 molecular vibration, 455 overtone, 318 Reststrahlen, 81, 83, 85, 453, 455 symmetric, 358 vibration–rotation, 312, 315, 318

975

976 Index

with a head, 319, 358 Band absorptance, xix, 341, 357, 358, 669, 677 for nonisothermal gas, 365 Band absorptance correlation, 364 Band approximation, 217, 218, 226, 227, 846 Band center, 318 Band gap, 56, 453 Band intensity, 319, 357 Band model narrow band, 335–344 wide band, 356–369 Band origin, 318, 319 Band overlap, 369 Band strength parameter, xxii, 358, 362, 370 Band width, 668 effective, xix, 357 parameter, 358 Band wing, 673 bandapp, 219, 232, 947 bandappdf, 218 bandmxch, 219, 232, 947 bandmxchdf, 218 Bandpass filter, 81 Basis function, 860 Bayesian optimization, 876 Bayesian statistics, 878 bbfn, 10, 945 Beam channeling, 210 Beam splitter, 109 Beer’s law, 22, 286 BFGS scheme, 869 Bidirectional reflection function, 200, 220, 223 for magnesium oxide, 67, 87 spectral, 66, 86 total, 69 Binary spatial partitioning algorithm, 253 Biot number, xix, 272 Black carbon, 431 Black chrome, 98 Black nickel, 98 Black surface, 4 Black-walled enclosure, 4, 161, 162 Blackbody, 5 manufacture of, 180 reference, 462 Blackbody cavity, 173 Blackbody cavity source, 106 Blackbody emissive power, xix, 6, 14, 921–923 fraction of, xx, 9, 921–923 total, 9 Blackbody intensity, xx, 13, 14, 313 Bleaching, 643 Blending, 750, 820, 839, 842 Boltzmann number, xix, 792 Boltzmann’s constant, xx, 6 Boltzmann’s distribution law, 313 Boltzmann, Ludwig Erhard, 9 Bouguer–Lambert–Beer Law, 286 Bound electron transition, 73 Boundary layer, 790–794 Bounding box algorithm, 253

Box model, 357, 667–673 Brass, oxidized, 200 BRDF, see Bidirectional reflection function Bremsstrahlung, 312 Brewster’s angle, 46, 461, 890 Brewster, Sir David, 46 Broadening collision, 323 Doppler, 323–325 line, 22, 23, 312, 323 Lorentz, 323 natural line, 323 Stark, 323, 324 Voigt, 326 Bundle, energy, 236, 737, 738, 740, 741 Bundle, photon, 236 Buoyant flame, 837 Butanol, 430

C Calcium carbonate, 847 callemdiel, 946 callemmet, 946 Candela, 17 Carbon foam, 457, 463 particle, 845 Carbon capture, 824 Carbon dioxide, 319, 322, 332, 359, 360, 362, 367, 370–372, 380, 386, 679, 680, 685, 712 Planck-mean absorption coefficient, 375 total emissivity, 372 Carbon monoxide, 359, 360, 362, 363, 370, 376 Carbon particle, 429, 847 Case’s normal-mode expansion technique, 490 Causal relationship, 461 Cavity conical, 209 cylindrical, 172, 179, 209 hemispherical, 184 spherical, 209 CCD camera, 879 CDSD database, 332, 349, 712, 713, 725 Cell cold-window, 383 hot-window, 382 nozzle seal, 384 Cement, 847 Central limit theorem, 239 Ceramics reticulated porous, 459, 463, 845 Cerium dioxide, 847 Cesium, 74 Chamotte, 845 Char, 426 Charge density, xxiii, 32 Chemiluminescence, 879 CHEMKIN, 822 Chopper, 110, 381 Chrome-oxide coating, 98 Chromium, 898

Closing condition, 507 Cluster T-matrix method, 425 CO2Emissivity, 374, 950 Coal gasification, 847 Coal particle properties, 427–430 coalash, 948 Coating, 51, 93–95 antireflective, 51 chrome-oxide, 98 for glass, 99 nickel-oxide, 98 pigmented, 99 reflectivity, 94 surface, 91 thermal barrier, 99 Cobalt, 74 Coherence, 887 Coherence length, 887 Coherence time, 887 Cold medium approximation, 292 Collimated irradiation, 64, 210, 211, 566, 641–653 Collision broadening, 323 Collisional interference, 323 Color center, 453 Colors of the sky, 411 Combustion, 821–844 Complex index of refraction, xx, 35, 69, 72, 402 of ash, 427 of coal, 427 of metals, 74 of semiconductors, 74 of various fuels, 430 of various soots, 434 Composition PDF method, 830 Composition variable, 826, 829 Computational fluid dynamics, 824 Computer codes, 945–951 Concentrator, compound parabolic, 845 Condition number, 863, 865 Conduction, 1, 472, 750 Conduction-to-radiation parameter, xxi, 274, 776, 792, 795 Conductivity dc-, 74, 76 electrical, xxiii, 32, 35, 55, 73, 74, 76 radiative, xx, 500, 776 thermal, xx, 1 Cone of rays, 594 Configuration factor, see also View factor, 128 Conjugate gradient method, 872–874, 878 Conjugation coefficient, 872 Conservation of energy, 2 overall, 303–305 radiative, 300, 470 Contour integration, 132 planar polygons, 151 Control function, 254 Control temperature, 255 Control variates, 761 Convection, 1, 472, 750

Index 977

free, 799 in boundary layers, 790–794 Convection-to-conduction parameter, xix Convection-to-radiation parameter, xix, 792 Copper, 73, 74, 89, 898 Correlated k-distribution, 334 global, 335 narrow band, 353 wide band, 335, 366–369 Correlation length, xxiii, 87 Cosine law, 14 Couette flow, 797 Coupling radiation and conduction/convection, 261–271 Coupling procedure, 781, 783, 796, 799–806 explicit, 264, 777, 779 fully implicit, 806 semi-implicit, 265, 270, 777, 805 CpldP1En1D, 783, 950 CpldP1En2D, 806, 950 Critical angle, 47, 296, 888, 889 Cross-section for absorption, 403 for extinction, 403 for scattering, 403 mass absorption, 432, 437 supplementary, 684 Crossed-strings method, 132, 143, 144, 163 Crossover wavelength, 79 Crystal lattice, 56, 80 Cumulative k-distribution, 346, 367, 692, 698 Current density, 893 Curtis–Godson approximation, 343 Cutoff wavelength, 97 Cylinders, concentric at radiative equilibrium, 490, 500, 506, 525 discrete ordinates method, 575, 576 without participating medium, 177, 203, 207, 621 Cylindrical fiber absorption and scattering by, 420 Cylindrical medium, 487–490 discrete ordinates method, 575, 576

D Damping parameter, 871 Darkening, 643 Database absorption coefficient, 712 CDSD, 332, 349, 712, 713, 725 EM2C, 342, 349, 385, 680, 690, 694, 713 HITEMP, 327, 332, 349, 355, 367, 368, 375, 657, 658, 680, 681, 686, 691, 694, 707–709, 712, 713, 716, 725, 951 HITRAN, 319, 322, 327, 331, 332, 339, 342, 343, 349, 375, 385, 657, 686, 712, 713, 951 k-distribution, 713 NBKDIR, 349, 713 NEQAIR, 332 NIST, 332 RADCAL, 342

SPECAIR, 332 SPRADIAN, 332 Dc-conductivity, 74, 76 Decane, 430 Degeneracy, xx, 312, 315, 361 Degrees of freedom, of a molecule, 315 Deissler’s jump boundary conditions, 501 Delta, 948 Delta–Eddington approximation, 422, 429 Density, xxiii charge, xxiii, 32 optical, 47 partial, of absorbing gas, 329 Density path length, 329 Detectivity, xix, 109 Detector, 108, 109, 462 fiber-optic, 441 photon or quantum, 108 pyroelectric, 381 thermal, 108 Deviational energy, 255 Deviational Monte Carlo, 254, 763 Diamond scheme, 579 Dielectric film, 54 Dielectric function, xxii, 55–57, 72, 73, 894 Dielectric layer, 89 Dielectric medium, 35, 44, 54, 71 Differential approximation, see also P1 -approximation, 505, 514–780 advanced, 548–552 hybrid, 552 improved, 552 modified, 544–548 ordinary, 547 Diffraction, 23 by a particle, 401 from large spheres, 414, 415 Diffraction peak, 410 Diffraction theory, 86 Diffuse emission, 14, 127 Diffuse emitter, 60 Diffuse irradiation, 68, 69 Diffuse reflectance, 67 Diffuse reflector, 68 Diffuse view factor, see also View factor, 128 Diffusion approximation, 306, 458, 498–502, 792, 845, 847, 848 Diffusion flame laminar, 822–824 Dipole, 893 Dipole element, 321 Dipole moment, 316, 455 Dirac-delta function, xxii, 346, 421, 423, 641 Direct exchange area, 552 Direct Exchange Monte Carlo, 761 Direct numerical simulation, 824 Direction of incidence, 68 of propagation, 287 specular, 67, 221, 294 Direction cosine, xx, xxii, xxiii, 132, 133, 470, 507 Direction vector, xxi, 10, 35, 132, 469, 740 dirreflec, 946

Discrete dipole approximation, 425, 426, 435 Discrete ordinates method, 306, 505, 553, 563–608, 846 anisotropic scattering, 585 boundedness-preserving scheme, 582 Cartesian coordinates, 576 comparison with FAM, 601 diamond scheme, 580 even-parity, 603 finite element method, 593 graded media, 593 higher-order scheme, 582 meshless method, 593 modified, 602, 603 natural element method, 593 non-Cartesian, 592 ray effect, 601 related methods, 604–606 spatial discretization, 579 step scheme, 580 three-dimensional, 592 two-dimensional, 583 Discrete transfer method, 306, 592, 602, 604 Dispersion, 55, 433 anomalous, 57 normal, 57 Dispersion exponent, 432 Dispersion relation, 891, 893 Dispersion theory, 69–72 Dissipation function, xxiii, 303 Distribution function cumulative, 237 Gaussian, 410 particle, xxi, 408, 427, 429 Divergence angle, 439 Dopant, 453 Doppler broadening, 323–325 Doppler effect, 325, 439 Doppler shift, 325 DRESOR method, 605 Drude theory, 73–75, 892, 896 Duct flow, 797 Dyadic Green’s function, xx, 893, 894

E Eddington approximation, 306 Eddington factor, 507 Effective medium theory, 898 Efficacy, luminous, 17, 18 Efficiency factor, 403 absorption, 403 for absorbing spheres, 406 for fuel sprays, 430 for specularly reflecting spheres, 416 for water sprays, 431 extinction, 403 for dielectric spheres, 406 for long cylinders, 421 for water droplets, 410 Rayleigh scattering, 411 scattering, 403 for absorbing spheres, 406 for fuel sprays, 430 for long cylinders, 421

978 Index

for specularly reflecting spheres, 416 for water sprays, 431 Efficiency, luminous, 17, 18 Eigenfrequency, 317 Einstein coefficients, xix, 312–315, 321 Electric field, xix, 32 Electrical conductivity, xxiii, 32, 35, 55, 73, 74, 76 Electrical conductor, 35, 56, 57, 72 Electrical network analogy, 174–177, 209, 210 Electrical permittivity, xxii, 32, 55, 892, 894 Electrical resistivity, 74 Electromagnetic energy, 35 Electromagnetic wave, 1, 2, 31 Electromagnetic wave spectrum, 2, 3 Electromagnetic wave theory, 2, 31–58, 69–72 Electron, 892 bound, 56, 57 free, 32, 56, 57, 73, 454 Electron energy, 56, 57 Electron volt, 3 Electronic transition, 83, 453, 455 Electronic–vibration–rotation band, 320 Electrostatic approximation, 436 Ellipsometric parameter, 38, 40 Ellipsometric technique, 461, 462 Ellipticity, 38 Elsasser model, 336, 337 EM2C database, 342, 349, 385, 680, 690, 694, 713, 950 emdiel, 84, 945 emdielr, 84 Emission, 1 blackbody, 6 diffuse, 14, 60, 61, 127 from a gas volume, 369 from a volume element, 287 from any isothermal volume, 299 gray, diffuse, 127 luminous, 426 spontaneous, 312, 315 stimulated, 312, 314, 322 Emission coefficient, 287, 315 Emission measurement, 110–112 Emissive power, xix, 5–10 apparent, 212 blackbody, xix, 6, 14, 921–923 blackbody spectrum, 7, 8 directional, 14 effective, 166 maximum, 8 spectral, 5, 61 spectral, directional, 60 total, 5, 9 weighted, 635 Emissivity, xxii, see also Emittance narrow band, 336 of a nonhomogeneous layer, 353 of an isothermal medium, 287, 678 spectral, 335 of a participating medium, 674

of an isothermal layer, 288, 659 spectral, directional of nonconductors, 84 spectral, hemispherical, 77 of nickel, 77 of nonconductors, 83 spectrally averaged, 337 total of a gas, 369–374 of an isothermal layer, 663 of carbon dioxide, 372 of water vapor, 371, 372 total, directional, 78 Emittance, 21, 60–62 apparent, 173, 180, 212 hemispherical, 21 infrared, 98 of selected materials, 21 spectral, directional, 60, 65 spectral, hemispherical, 60, 61 of tungsten, 80 spectral, normal of aluminum, 89, 90 of zirconium carbide, 86 total, directional, 60, 61, 65 of several metals, 60 of several nonmetals, 60 total, hemispherical, 61, 65 of a metal, 79 total, normal, 76 of polished metals, 76 tables, 908 Emitted energy, 12 emmet, 77, 946 emwbm, 948 Enclosure, 128 black-walled, 4, 161, 162, 553, 554, 673 closed, 128, 165 idealized, 127, 128, 161, 205, 209 isothermal, 205 long, 143 open, 165, 208 Energy deviational, 255 electromagnetic, 35 internal, xxi, 21, 303, 311 of a photon, 3 solar, 2 Energy balance Earth, 102, 379 Energy bundle, 236, 737, 738, 740, 741 path, 241 Energy conservation equation, 303 Energy density, radiation, 298 Energy level electronic, 21 molecular, 311, 317 rotational, 21, 312, 315, 316 vibrational, 21, 80, 312, 315, 317, 319 Energy partitioning, 252, 741, 756, 761 Enthalpy, 820 Environment, large, isothermal, 171 Epoxy coating, 90, 99

Equation of transfer, see Radiative transfer equation (RTE) Equilibrium radiation, 313 Error, statistical, 235 Ethane, 332, 342 Ethanol, 430 Ethene, 442 Ethylene, 332, 342, 822, 832 Ethylene glycol, 430 Euler angles, xxii, 529, 530 Euler’s constant, xxii, 341, 942 Evanescent wave, 887–889 Even-parity formulation, 603 Exchange area direct, xx, xxi, 617, 618, 622, 623, 628 total, xx, xxi, 618–622, 624–627, 629–631 Exchange factor, xx, 240, 241, 617, 618, 737 Exchange function, 895 Exponential integral, xix, 341, 472, 941–943 Exponential kernel approximation, see Kernel approximation ExStoSEn1D, 265, 947 Extinction, 287, 403 Extinction coefficient, xxii, 24, 287, 438 for a particle cloud, 407, 409 for coal particles, 427 modified, 585 Planck-mean, 408, 427, 428, 437 Rosseland-mean, 408, 427, 428, 437, 458, 500, 664 Extinction efficiency factor, 403 for dielectric spheres, 406 for long cylinders, 421 for water droplets, 410 Extinction paradox, 414

F False scattering, 586 FAM, see Finite angle method Favre averaging, 825 FDF method, 833 FDTD method, 425, 426 Fiberglass, 463 Fibers, scattering by, 421, 463 Fictitious gas technique, 355, 685, 721 Figure of merit, 755, 760, 765 Film dielectric, 54 metallic, 177 nonmetallic, 98 porous, 95 slightly absorbing, 54 thick, 54 thin, 51, 460 Filter bandpass, 81 optical, 107 Filtered density function, 833 Fin efficiency, 274, 275 Fin radiator, 273 Finite angle method, 306, 553, 593–602 angular grid selection, 597 body-fitted, 602 boundary conditions, 596

Index 979

comparison with DOM, 601 FTn discretization, 600 Generalized Source Finite Volume Method, 606 graded media, 601 multi-dimensional, 594 Finite volume method, 306, see also Finite angle method Fire pool fire, 837 Flame axisymmetric, 555 laminar diffusion, 821 luminous, 832, 833 nonluminous, 832 Flame D, Sandia, 555, 708, 830–832 Fluctuation–dissipation theorem, 894 Fluctuational electrodynamics, 887, 892 Fluidized bed, 420 Fluidized bed combustion, 836 Fluidized bed, solar, 845 Flux heat, xxi, 1 luminous, xxi, 18 momentum, 16 Flux method, 604 Fly ash properties, 427–430 FN -method, 490 Foam carbon, 457 closed cell, 458 open cell, 457 Foam insulation, 463 Forced collisions, 761 Fourier’s law, 1, 26, 304 Fractal aggregate, 425, 435 Fractal prefactor, xx, 435, 436 Fractal surface, 87 Fredholm integral equation, 179, 474 Free electron, 32, 73, 454 Freezing, 784–789 Frequency, xxiii, 3 angular, xxiii, 3 of radiation, 2 plasma, 57, 73 resonance, 57 fresnel, 46, 50, 83, 945 Fresnel’s relation, 46, 50, 71, 72, 76, 82, 83, 295 Fresnel, Augustin-Jean, 46 FSCK method, 555, 697, 725, 748 FSCK Monte Carlo, 748 FSK method, 686, 690–727 fskdco2, 694, 713, 949 fskdco2dw, 713, 949 fskdh2o, 713, 949 fskdh2odw, 713, 949 fskdist, 694, 712, 949 fskdistmix, 949 FSSK method, 704, 725 FTIR spectrometer, 107, 381 ftwbm, 364, 948 Fuel Spray properties, 430, 431 Fuel sprays, 750

Full spectrum k-distribution, 690–727 Function estimation, 859 Fundamental band, 318 Furnace, 462 high-temperature, 383 sealed-chamber, 462 FVM, see Finite angle method FVM2D, 950 FwdMCcs, 768 FwdMCxx, 950

G Galerkin method, 490 Gamma distribution, 408 Gamma rays, 90 Gas emission from, 369 mixture, 342, 349, 714 sum of gray gases, 634, 677–683 total absorptivity, 370 total emissivity, 369 Gas layer isothermal, 288, 330 nonisothermal, 343 Gas properties atmosphere, 377 carbon dioxide, 377 greenhouse gas, 377 water vapor, 377 Gas–particulate mixture, 665, 676 Gauss’ theorem, 42 Gauss-Newton method, 861 Gaussian distribution function, 410 Genetic algorithms, 878 Geometric mean, 694 Geometric optics, 25, 51, 88, 91, 403, 414 Geometric path length, 329 Glass, 83, 500 multiple panes, 95 single pane, 91 soda–lime, 93 Global model, 335, 686 Global warming, 2, 311, 679 Globar light source, 106, 440 Godson approximation, 337 Gold, 74 Goody model, 338, 745 Graded index radiative transfer equation, 297 Graded medium, 740, 766 PN -approximation, 526 DOM, 593 DTM, 605 FAM, 601 radiative equilibrium, 480–483 ray trajectory, 482, 483 Graphite pyrolytic, 682 Gray gas weights, 684, 699 Gray medium, 300, 305 Gray source, 65 Gray surface, 62 Gray, diffuse surface, 62, 161, 166, 294 graydiff, 173, 946

graydiffxch, 173, 188, 194, 946 graydifspec, 206, 230, 947 Greenhouse effect, 2, 92, 377 Greenhouse gas, 377 Groove right-angled, 164, 188 V-corrugated, 100, 101, 211, 222 grspecxch, 206, 230, 947

H H2OEmissivity, 374, 950 Hagen–Rubens relation, 74, 76 Half-width, line, 323, 325 Halide, 456 Harmonic oscillator, 56, 72, 317, 318, 321 Heat conduction, 303 Heat flux, xxi, 1, 2 at a surface, 204 average, 162 directional, 14 outgoing, 204 prescribed, 276 radiative, 12, 14–16, 63, 298, 299 reflected, 67 Heat flux vector, 303 Heat of fusion, 786 Heat rate, xxi, 1 Heat rejector, radiative, 97, 98 Heat source, 303, 305 radiative, 304 Heat transfer coefficient, xx, 1, 898 convective, 175, 276 radiative, 272, 776 Heaviside’s unit step function, xx, 346, 650, 668, 767 Helmholtz equation, 521, 893 Hemisphere, 10 Hemispherical cavity, 184 Hemispherical volume, 659 Hencken burner, 876 Henyey–Greenstein phase function, 422, 423 Heptane, 430 Hessian matrix, 868 Hexane, 430 HITEMP database, 327, 332, 349, 355, 367, 368, 375, 657, 658, 680, 681, 686, 691, 694, 695, 707–709, 712, 713, 716, 725, 951 HITRAN database, 319, 322, 327, 331, 332, 339, 342, 343, 349, 375, 385, 657, 686, 712, 713, 951 Hohlraum, 106, 111 Hole, cylindrical, 172, 179 Hot band, 322 Hot line, 322, 332, 355, 375 Hottel, Hoyte Clark, 143 Hybrid differential approximation, 552 Hybrid methods, 306 Hyperspectral imaging, 879

I Ice, 455

980 Index

IDA, see Improved differential approximation Ideal spectrum, 687 Ill-conditioned problem, 861 Ill-posed problem, 859 Illumination, 18, 212 Image, 201 Imaging hyperspectral, 879 Importance sampling, 240, 251, 761 Improved differential approximation, 552 ImStoSEn1D, 267, 947 In-scattering, 26, 288 Incidence angle, 44, 68 Incidence direction, 68 Incident radiation, xx, 292, 298 for a plane-parallel medium, 470 Index of refraction, complex, see Complex index of refraction Induced emission, see Emission, stimulated Infrared emittance, 98 Infrared radiation, 4, 311, 453 Inside sphere method, 132 Insulation, foam, 463 Insulator, 56 Integral equation for outgoing intensity, 220 for radiosity, 167, 204 for specular reflections, 208 Fredholm, 179 Integrating sphere, 439, 443 Intensity, xx, 12–14 blackbody, xx, 13, 14, 313 in vacuum, 19, 219 outgoing, 220 reflected, 67, 220 weighted, 679 Interaction radiation and conduction, 273–276, 775–784 radiation and convection, 276–279, 790–799 radiation and laminar combustion, 821–823 radiation and melting/freezing, 784–789 radiation and turbulence, 824–834 radiation and turbulent combustion, 824–838 Interface moving, 787 optically smooth, 44, 69 plane, 41, 43 Interface condition, 42, 43, 48, 787 Interface reflectivity, 46, 47, 49, 51 Interference structure, 410 Interference, wave, 51, 54, 93–95 Internal energy, xxi, 21, 303, 311 Invariance, principle of, 490 Inverse Bremsstrahlung, 312 Inverse heat transfer, 859 Inverse radiation, 859–879 Ionic crystal, 453, 454 Iron, 74 Irradiation, xx, 128, 161, 167, 553

collimated, 64, 210, 211, 566, 641–653 diffuse, 64, 68, 69 diffuse and gray, 65 directional, 62 external, 161, 162, 204 gray, 64 laser, 199, 641, 642 polarized, 83, 199 solar, 5, 642 spectral, directional, 63 spectral, hemispherical, 68 total, 65 Isotropic medium, 32 Isotropic scattering, 289, 293, 305, 405, 423 Isotropic surface, 61

J Jacobian, 861 Jeans, Sir James Hopwood, 6 Jet diffusion flame, 555 nonluminous, 830 Jump boundary condition, 502

K k-distribution, xix, 334, 366–369, 798 cumulative, xx, 346, 367, 692, 698 database, 718 databases, 713 for mixtures, 714 global, 335, 690–727 narrow band, 344–356, 688–690 Planck function weighted, 691, 696 wide band, 335 K-moments, 342, 344 Kalman filtering, 878 kdistmix, 949 Kernel, 179, 507 Kernel approximation, 179, 182, 306, 507–509 Kirchhoff approximation, 88 Kirchhoff’s law, 4, 13, 21, 72, 221 for absorptance, 63 for bidirectional reflection, 66 Kirchhoff, Gustav Robert, 4 Kramers–Kronig relation, 461 Kronecker’s delta, xxii, 165, 205, 860

L Ladenburg–Reiche function, 330 Lambert, 17 Lambert surface, 60 Lambert’s law, 14, 60 Lambert, Johann Heinrich, 14 Landweber inverse method, 879 Laplace transform, xx Laplace’s equation, 522 Large eddy simulation, 824 Laser, 780 pulsed, 641, 649–652 Laser irradiation, 199, 641, 642 Laser light source, 106 Latex particles, 439, 443 Lattice defect, 76, 79, 80, 82, 98, 453 Lattice vibration, 56, 85, 892

Lattice, crystal, 56, 80 Law of reciprocity for bidirectional reflection function, 66 for diagonally opposed pairs, 142 for direct exchange areas, 618, 623 for exchange factors, 241 for specular view factors, 201 for total exchange areas, 619, 624 for view factors, 129, 130 Layer dielectric, 89 of alumina, 89 of silica, 89 opaque, 82 oxide, 89 surface, 76, 81, 89 thick dielectric, 93 thin, 89 LBL, see Line-by-line calculations Least squares norm, 860 Leckner, 373, 948 Legendre polynomials, xxi, 404, 405, 469 associated, 469, 514, 519 orthogonality of, 515 polyadic, 514 Leibniz’s rule, 99, 183 Lens, 109 Levenberg–Marquardt method, 871, 872, 878 Light, 2, 17 polarized, 461 Light field imaging, 879 Light guide, 227 Light line, 892 Light source, 105, 381 blackbody cavity, 106 globar, 106, 440 laser, 106 Nernst glower, 106 Lighting, 17, 213 Lime, 847 Line absorption–emission, 22 collision-broadened, 323 Doppler-broadened, 323 hot, 355 isolated, 323 Lorentz, 323 no overlap, 337 rotational, 312 spectral, 311, 315, 323 strong, 337, 339 Voigt-broadened, 323 weak, 337, 339 Line broadening, 22, 23, 312, 323 Line half-width, xxii, 323, 325 Line intensity, 314, 321 Line mixing, 323, 327 Line overlap parameter, xxii, 337, 362, 363 Line shape, 323 Line shape function, xxiii, 314, 323 line mixing, 327 Line spacing, xxii, 336, 337 Line strength, xxi, 314, 323, 324

Index 981

Line strength parameter, xxii, 321, 330, 337 Line structure effects, 745 Line width, equivalent, xxii, 330, 338, 339 nonhomogeneous path, 343 Line-by-line calculations, 238, 334, 657, 681, 682, 685, 693, 695, 707–709, 745 Line-by-line Monte Carlo, 745 Linear-anisotropic phase function, 423 Linear-anisotropic scattering, 305, 423 Liquid high-temperature, 462 semitransparent, 455 Lithium, 74 Lithium fluoride, 456 Lorentz broadening, 323 Lorentz model, 73, 453, 897 single oscillator, 80, 897 Lorentz, Hendrik Anton, 56 Lorenz, Ludvig, 402 Lorenz–Mie scattering, see Mie scattering Lumen, 17, 212 Luminance, xx, 17, 18 Luminous efficacy, xx, 17, 18 Luminous efficiency, xxii, 17, 18 Luminous emission, 426 Luminous flame, 832, 833 Luminous flux, xxi, 18 Lumped mass approximation, 272 Lux, 17

M M1 -approximation, 507, 543 Machine learning, 875 Magnesium oxide, 81, 85–87, 115 bidirectional reflection function, 67 Magnetic field, xx, 32 Magnetic permeability, xxiii, 32, 36, 45, 892 Malkmus model, 338, 341, 347, 686 Manganese sulfide, 53 Mark’s boundary condition, 517 Marshak’s boundary condition, 517, 529–535, 539, 546 Mass fractal dimension, xix, 435, 436 Mass fraction, xxii Maxwell’s equations, 31, 32, 403 Maxwell, James Clerk, 31 MCintegral, 947 MDA, see Modified differential approximation Mean beam length, xx, 658–664, 669 definition, 659, 660 for an isothermal gas layer, 660 for optically thin media, 660, 661 geometric, 661 spectrally averaged, 661–664 Mean free path for a photon, 2, 286 for absorption, 286 for collision, 2 for scattering, 287 Measurement absolute, 439 emission, 110–112 gas properties, 381–386

hyperspectral, 879 multi-projection, 878 multiple-scattering, 440 reflection, 112–116 relative, 439 scattering, 439 semitransparent media, 460–463 transmission, 381 Medium absorbing, 48 cold, 292 conducting, 69 cylindrical, see Cylindrical medium dielectric, 44, 54, 71 gray, 300, 305 isotropic, 32 nonabsorbing, 37 nongray, 305, 634–636, 657–727 nonhomogeneous, 343, 353 nonmagnetic, 35, 36, 55 nonparticipating, 19, 127 nonscattering, 291 opaque, 59, 72 optically thick, 474 participating, 19, 285 plane-parallel, see Plane-parallel medium scattering, 293 semitransparent, 82, 90, 285, 453–463, 784–789 spherical, see Spherical medium transparent, 474 Melting, 784–789 Metaheuristics, 865, 874–878 Metallic foam, 457 Methane, 332, 359, 360, 362, 363, 376, 555, 709, 821, 822, 845, 847 Method of steepest descent, 872 Microgravity, 823 Mie scattering, 24, 403, 479 equivalent-sphere, 435 Mie scattering coefficient, 404 Mie theory, 847 Mie, Gustav, 402 Milne–Eddington approximation, 505–507 Minimization, 872 Mirror, 13, 67, 109, 115, 199 platinum, 462 spherical, 462 MixEmissivity, 374, 391, 950 Mixture gas, 342, 714 gas–particulate, 665, 676 mmmie, 407, 409, 948 mocacyl, 949 Modified differential approximation, 544–548 Molar density, 684 Mole fraction, xxii Moment method, 306, 505–507, 513 Momentum, of photons, 16 Monochromatic radiation, 5 Monochromator, 107, 381, 462 MONT3D, 241, 950

Monte Carlo method, 132, 235–258, 306, 553, 737–770, 820, 845–847 absorption reciprocal method, 762 bidirectional reciprocal method, 762 convergence, 238 deviational, 254, 763 direct exchange, 761 emission reciprocal method, 762 optimized reciprocal method, 762 Quasi-random, 764–766 reciprocity, 762 results for a V-groove, 211, 222 results for a gas slab, 667 shift, 763 MSFSK method, 721, 725, 727 Mueller matrix, 41 Mullite, 459, 463 Multi-layer perceptron (MLP), 875 Multiphase combustion, 836 Multiphoton absorption, 643 Multisphere Mie solution, generalized, 425, 426 Mushy zone, 784, 786

N Nanoscale radiation, 887–899 Narrow band k-distribution, 344–356, 688–690 Narrow band model, 334–344, 743 Narrow band model Monte Carlo, 743 Narrow band parameter, 336 Natural line broadening, 323 NBCK method, 690 NBKDIR database, 349, 713, 725, 951 nbkdistdb, 347, 947 nbkdistsg, 347, 948 Negative index materials, 892 Nephelometer, 439, 442 NEQAIR database, 332 Nernst glower, 106 Net radiation method, 166, 223–227, 847 Neural network, 875, 878 convolutional, 879 Neutron transport theory, 513, 563 Newton’s direction, 868 Newton’s method, 868 Nickel, 74 Nickel-oxide coating, 98 NIST database, 332 Nitric oxide, 360, 376 Nitrous oxide, 359, 376 Nonane, 430 Nonconductor, of electricity, 56 Nonequilibrium radiation, 290, 313, 330 Nongray medium, 305, 634–636, 657–727 Nongray stretching factor, 694, 696, 698, 699, 702, 704, 714, 720, 722 Nongray surface, 215 Nonluminous flame, 832 Number density, 409 molecular, 314, 323 Numerical quadrature, 179, 181, 564 Nusselt number, xxi, 796, 797, 805

982 Index

O Objective function, xx, 859, 860, 864, 870, 872, 873, 876 Obstruction, visual, 144 Off-specular peak, 87 Opacity Project, 332 Opaque, 4 Opaque medium, 59, 72 Opaque surface, 4, 10, 20, 21, 59 OpenFOAM, 538 Opening angle, 100 Optical constants, 55–58 Optical coordinate, xxiii, 290 Optical density, 47 Optical depth, 467 Optical filter, 107 Optical path length, xxii, 329 Optical thickness, xxiii, 286, 467 for absorption, 286 for extinction, 287 for scattering, 287 narrow band, 337 of a spectral line, 330 Optically thick approximation, 498–502 Optically thin approximation, 306, 497, 498 Optics collection, 463 geometric, 25, 51, 91, 403, 414 thin film, 51 Optimization, 878 gradient-based, 865, 867–874 Oscillator double, 81 harmonic, 56, 72, 317, 318, 321 isolated, 72 single, 80, 897 OTFA, 827, 830 Out-scattering, 25, 287 Overlap band, 369 line, 336 Overlap parameter, xxii, 666 (for MSFSK), 722 Overtone band, 318 Oxide film, 89, 90 Oxy-fuel, 679, 683, 690, 824, 837

P P1-2D, 948 P1 -approximation, 507, 514, 519–526, 845 for box model, 672, 673 semigray, 665 transient, 650 with collimated irradiation, 646–649 with conduction, 780 with convection, 795, 803 P1/3 -approximation, 650 P1sor, 523, 948 P3 -approximation, 514, 526 Packed bed, 847 Palladium, 74 Parallel plates, 174, 177, 179, 180, 202, 206, 894, 895, 898 Parameter vector, xxi, 860, 873

parlplates, 143, 158, 946 Particle ash, 427 coal, 427 large, 414 model, 750 soot, 411 spherical, 401 stochastic, 750 Particle beds, 420 Particle distribution function, 408, 427, 429 Particle size parameter, see Size parameter Particle suspension, 439, 440, 676, 750 Partition function, 321 Path length density, 329 geometric, xxi, 329 optical, xxii, 329 pressure, 329 Pathlength method, 761 PDF method, 750, 826, 829, 830, 832 Peak backward-scattering, 410, 420, 422, 424 diffraction, 410 forward-scattering, 407, 410, 414 off-specular, 87 specular, 67 Pellet-reflection technique, 433 Pencil of rays, 14 Permeability, magnetic, xxiii, 32, 36, 45, 892 Permittivity complex, xxii, 33, 42 electrical, xxii, 32, 55, 892, 894 relative, 55 perpplates, 143, 158, 946 Phase angle, 33, 53, 461 of polarization, xxii, 39 Phase function, xxiii, 24, 288, 403, 405 approximate, 422–425 for a particle cloud, 408, 409 for absorbing particles, 409 for diffraction, 415 for diffusely reflecting spheres, 420 for large spheres, 417 for Rayleigh scattering, 412 for Rayleigh–Gans scattering, 414 for single sphere, 407 for specularly reflecting spheres, 417 Henyey–Greenstein, 422, 423 isotropic, 423 linear-anisotropic, 423 Phase velocity, 33, 35, 413 Phenomenological coefficient, 32, 35, 55 Phonon, 56, 891, 892 Photoacoustic, 439 Photolysis, 643 Photometer, scattering, 439 Photon, 1–3 Photon bundle, 236 multi-spectral, 760 Photon detector, 108 Photon energy, 3 Photon gas, 520 Photon momentum, 16

Photon pressure, 16 Photon tunneling, 889 Photon–phonon interaction, 80 Photovoltaics, 844 Pigmented coating, 99 planck, 10, 945 Planck function, xx, 14 Planck number, 274 Planck oscillator, xxii, 894 Planck’s constant, xx, 3 modified, 316 Planck’s law, 6, 8, 313 Planck, Max, 6 Planck-mean absorption coefficient, 374–376, 665, 748, 752 for coal particles, 427, 428 for particles, 408 for soot, 437 modified, 375 Planck-mean extinction coefficient for coal particles, 427, 428 for particles, 408 for soot, 437 Planck-mean temperature, 705 Plane of equal amplitude, 33, 43 of equal phase, 33, 43 of incidence, 44, 87 Plane wave, 33, 38, 43 Plane-parallel medium approximate methods, 497–510 at radiative equilibrium, 473–475, 477, 479, 669–673 discrete ordinates method, 566–571 exact formulation, 467–471 graded index, 480–483 isothermal, nongray gas, 663 isothermal, nonscattering, 538, 626 nonscattering, 471–478 optically thick, 498–502 optically thin, 497, 498 scattering, 478, 479 specified temperature field, 472, 473, 475, 476, 478 Plasma, 325 Plasma frequency, 57, 73 Platinum, 74, 76, 77, 215 Plexiglass, 93 PMMA, 93 PN -approximation, 513–555 Simplified, 513, 539–543 pnbcs, 949 Point collocation method, 491 Poiseuille flow, 794, 796, 797, 802 Polar angle, xxii, 11, 14 Polaritons, 891 Polarization, 37–41, 643 circular, 37, 40 degree of, 40 elliptical, 37 linear, 37, 40 parallel, 39, 71, 97 perpendicular, 39, 71, 97 plane, 37

Index 983

state of, 37, 199 Polarization ellipse, xix, xxii, 37 Polarization phase angle, xxii, 39 Polarized light, 461 Polarizer, 439 Polarizing angle, 46 Pollutants, 822 Polystyrene, 458 Porous film, 95 Position vector, 132, 136 Potassium, 74 Poynting vector, xxi, 35, 37, 38, 403, 888, 893, 894 Poynting, John Henry, 36 Prévost’s law, 5 Prandtl number, xxi, 792 Prefactor, fractal, 435, 436 Pressure, xxi correction chart, 371 correction factor, 371 effective, 361, 371 partial, 324 partial, of absorbing gas, 329, 370 photon, 16 radiation, xxi, 16, 17, 304 solar, 17 Pressure path length, 329 Principle of invariance, 490 Probability, 829 Probability density function, xx, 239, 750, 826, 829, 832 Probability distribution, 236 Profilometer, 86 Property, radiative, see Radiative properties Propylene, 342 Pseudorandom number, 237 Pulverized coal, 750 Pyroelectric detector, 108 Pyrometer, 28

Q QMC, see Quasi-Monte Carlo Quadrature, numerical, 179, 181, 564 Quantum detector, 108 Quantum mechanics, 2, 3 Quantum number rotational, xx, 316 vibrational, xxii, 317, 361 Quartz, 7, 111, 454, 846 Quasi-Monte Carlo, 764–766 Quasi-Newton method, 868–871, 873

R RADCAL database, 342, 950 Radiation background, 462 external, 211 from isolated lines, 329 midinfrared, 4 monochromatic, 5 oxy-fuel, 837 sky, 211 transient, 649–652 ultraviolet, 4, 73, 89, 93

visible, 2, 3, 17, 18 Radiation energy density, xxi, 298 Radiation pressure, xxi, 16, 17, 304 Radiation shield, 177–179, 210 Radiation tunneling, 889 Radiation–turbulence interaction, 824–834 Radiative combination, 312 Radiative conductivity, xx, 500, 776 Radiative equilibrium, 304, 599 between concentric cylinders, 490, 525 between concentric spheres, 484–487 in a gray medium, 749 in a nongray gas slab, 666, 669–673 in a nongray medium, 676, 749 in a nonscattering slab, 473–475, 477, 478 in a scattering slab, 479, 568, 569, 597, 598 Radiative forcing, 381 Radiative heat flux, 12, 14–16, 63, 298, 299 divergence of, 299–301 for a cylindrical medium, 488 for a plane-parallel medium, 470 for a spherical medium, 485 Radiative heat transfer, 1 Radiative heat transfer coefficient, 272, 776 Radiative intensity, see Intensity Radiative properties, 2, 26 climate change, 102, 377, 379 definitions for surfaces, 60–69 directional, 21 Earth, 102 hemispherical, 21 of coal particles, 427–430 of fly ash, 427–430 of Fuel Sprays, 430, 431 of gases, 21–23, 311–391 of materials, 2 of metals, 72–79 of nonconductors, 80–85 of particles, 23–25, 401–445 of plexiglass, 93 of PMMA, 93 of selective absorbers, 99 of semitransparent media, 23, 453–463 of semitransparent sheets, 90–97 of soot, 431–438 of window glass, 91 spectral, 21 summary for surfaces, 70 temperature dependence, 76, 79, 85, 631 total, 21 Radiative resistance, xxi, 174, 175, 209 Radiative source, xxi, 25, 291, 467 for anisotropic scattering, 469 for isotropic scattering, 468 for linear-anisotropic scattering, 469 modified, 585 time-averaged, 826 Radiative surface properties Earth, 103 Radiative transfer equation (RTE), 25, 26, 285–298, 329, 467 boundary conditions, 293 graded index, 297 integral formulation, 302, 303

solution methods, 305 Radiative transport theory, 26 Radiosity, xx, 166, 202, 204 artificial, 619 for a semitransparent wall, 212 spectral, 166 volume zone, 628 Radiosity equation, 167, 169, 179, 204 Raman effect, 401 Random number, xxi, 235, 237 Random number generators, 237 Random number relation for absorption, 740, 744 for absorption and reflection, 248 for direction of emission, 247, 740 for point of emission, 243, 738 for scattering, 741 for wavelength of emission, 247 for wavenumber of emission, 739, 743 inversion, 251 Rank correlated, 688 Ray effect, 588, 599, 602 Ray model, 750, 751 Ray tracing, 91, 95, 202, 249, 250 acceleration, 252 binary spatial partitioning algorithm, 253 bounding box algorithm, 253 uniform spatial division algorithm, 254 volume-by-volume advancement algorithm, 254 Rayleigh scattering, 24, 405, 411–413, 435, 642 Rayleigh, John William Strutt, Lord, 6 Rayleigh–Debye–Gans scattering, 435 Rayleigh–Gans scattering, 405, 413, 414 Rayleigh–Jeans distribution, 8 Reaction mechanism, 821 Reactive flow, 830 Reciprocity, see Law of reciprocity Reciprocity Monte Carlo, 762 Reflectance, 20, 66–69 bidirectional, 66 components of, 200 diffuse, 67 of silicon carbide, 84 rock, 102 snow, 102 spectral, directional of platinum, 77 spectral, directional–hemispherical, 67 spectral, hemispherical, 68, 69 spectral, hemispherical–directional, 68 spectral, normal of magnesium oxide, 85 of silicon, 81 total, directional–hemispherical, 69 total, hemispherical, 69 total, hemispherical–directional, 69 total, normal, 76 vegetation, 102 water, 102 Reflection, 23, 41–55, 401 by a slab, 51–55

984 Index

by a thin film, 51–54 from large spheres, 414 gray, diffuse, 127 irregular, 210 specular, 43, 67, 199 Reflection coefficient, xxi, 45, 47, 50, 461, 891 for a thin film, 52 Reflection function bidirectional, 200, 220, 223 spectral, 66 total, 69 Reflection measurement, 112–116 Reflection technique, 460 Reflectivity, xxiii, see also Reflectance, 46 coating, 94 for polarized light, 76 interface, 46, 47, 49, 51 of a dielectric thin film, 54 of a slab, 463 of a thick slab, 54, 91 of a thin film, 53, 54 of aluminum, 51 spectral, directional, 76 spectral, directional, polarized of glass, 83 spectral, normal, 80, 82 of aluminum, 73 of an In2 O3 film on glass, 95 of copper, 73 of magnesium oxide, 81 of metals, 73 of silicon carbide, 80 of silver, 73, 75 Reflectometer heated cavity, 114 integrating mirror, 115 integrating sphere, 114 Reflector diffuse, 68 perfect, 68 specular, 67, 69 Refraction, 23, 401 in large spheres, 414 Refraction angle, 44 Refractive index, xxi, 2, 35, 296 for semitransparent materials, 82, 83 of air, 2 of vacuum, 2 varying, 643 Refractive index function, xix, 432 Regularization, 859, 865–867 parameter, xxiii, 866, 872 Relaxation factor, 278 Relaxation time, 74 Remote sensing, 706 Resistance, radiative, xxi, 174, 175, 209 Resistivity, electrical, 74 Resonance frequency, 57 Reststrahlen band, 81, 83, 85, 453, 455 Reticulated porous ceramics, 459, 463, 500, 845 RevMCcs, 768 RevMCxx, 950 Reynolds number, xxi, 792

Rigid rotator, 316, 318, 321 Ripple, 410 RMS error, 765 rnarray, 949 Rosseland approximation, 500, 845, 847, 848 Rosseland-mean absorption coefficient, 376, 665 for coal particles, 427, 428 for particles, 408 for soot, 437 Rosseland-mean extinction coefficient, 458, 500, 664 for coal particles, 427, 428 for particles, 408 for soot, 437 Rotation matrix, xxii, 530 Rotational energy level, 312, 315, 316 Rotational quantum number, xx, 316 Rotator, rigid, 316, 318, 321 Roughness root-mean-square, xxiii, 85–87 surface, 67, 85–89 RPC, see Reticulated porous ceramics RTE (Radiative transfer equation), see Radiative transfer equation (RTE)

S Sandia Flame D, 555, 708, 830–832 Sapphire, 111, 382, 846, 898 Saturable absorption, 643 Sauter mean diameter, 428, 431 Scaled k-distribution global, 704 narrow band, 354 Scaled SLW method, 704 Scaling approximation, 354, 683 Scaling function, xxi, 354, 687 Scanning tunneling microscope, 898 Scattering, 23, 24, 305 anisotropic, 585 attenuation by, 287 augmentation by, 288, 289 by fibers, 421, 463 by nonspherical particles, 434 dependent, 401 elastic, 401 false, 586 independent, 401 inelastic or Raman, 401 isotropic, 289, 293, 305, 405, 423 linear-anisotropic, 305, 423 multiple, 439 Rayleigh, 405, 411–413, 642 Rayleigh–Gans, 405, 413, 414 single, 439 Scattering albedo, xxiii, 24, 25, 290 Scattering angle, xxii, 405 Scattering coefficient, xxiii, 24, 287 for a particle cloud, 407, 408 soot, 437 Scattering cross-section, 403 Scattering efficiency factor, 403 for absorbing spheres, 406

for fuel sprays, 430 for long cylinders, 421 for specularly reflecting spheres, 416 for water sprays, 431 Scattering measurement, 439 Scattering peak, 407, 410, 414, 420, 422, 424 Scattering phase function, see Phase function Scattering photometer, 439 Scattering regimes, 402 Schrödinger’s wave equation, 316, 317 Schuster–Schwarzschild approximation, 306, 503–505, 569, 604 Search direction, 868, 872, 873 Search step size, 868, 872, 873 Selection rule, 316, 317 Selective surface, 97, 98, 199, 208, 214 Self-broadening coefficient, xix Self-correlation Planck function, 830 temperature, 827 Semiconductor, 56, 80, 81 semigray, 219, 232, 947 Semigray approximation, 215, 216, 226, 664–667 semigraydf, 216 semigrxch, 219, 232, 947 semigrxchdf, 216 Semitransparent, 4 Semitransparent liquid, 455 Semitransparent medium, 82, 453–463, 784–789 Semitransparent sheet, 72, 211–214 Semitransparent surface, 211–214 Semitransparent wall, 204, 295 Semitransparent window, 211–214, 846 Sensitivity matrix, xxii, 861, 864, 873 Shadowing, 87 Shape factor, see also View factor, 128 Sheet, semitransparent, 72, 211–214 Shield, radiation, 177–179, 210 Signal velocity, 35, 649 Silica, 89, 845 Silicon, 81, 111, 878 absorption coefficient, 455 phosphorus-doped, 453 Silicon carbide, 80, 84, 85, 89, 845, 897 Silver, 73, 74 Simplified PN -approximation, 539–543 Simulated annealing, 875, 878 Single scattering albedo, see Scattering albedo Singular value, 862, 864, 865 Singular value decomposition, 862 truncated, 865 Six-flux method, 306, 505, 604 Size parameter, xxii, 24, 402 SKN method, 509 Sky radiation, 211 Skylight, 212 Slab, see Plane-parallel medium reflection by, 51–55 transmission through, 51–55 Slab absorptivity, xix, 92

Index 985

Slab reflectivity, xxi, 91, 92 spectral of plexiglass, 94 of PMMA, 94 spectral, normal for several glass panes, 92 of soda–lime glass, 93 Slab transmissivity, xxi, 92 spectral of PMMA, 94 spectral, normal for several glass panes, 92 of soda–lime glass, 93 Slag, 463 SLW method, 683, 693–695, 699, 799 SN -approximation, 563–606 Snell’s law, 44, 295, 888 generalized, 50, 71 Soda–lime glass, 93 Sodium, 74 Solar absorptance, 98 Solar cell, 108 Solar collector, 97, 98 Solar concentration ratio, 844 Solar constant, xxi, 16 Solar energy, 2 Solar furnace, 846 Solar irradiation, 5, 642, 844 concentrated, 844–848 Solar pressure, 17 Solar reactor, 844 Solar receiver, 844 Solar sail, 16 Solar temperature, 7, 10, 16, 18 Solar transmittance, 99 Solid amorphous, 85, 454 high-temperature, 461 semitransparent, 453 Solid angle, xxiii, 10–12 infinitesimal, 11 overhang, 600 pixelation, 600 total, 11 Solidification, 784–789 Soot, 411, 426, 679, 680, 726 aggregate, 425, 426, 435 cylindrical, 434 size distribution, 437 Soot model, 832 Soot properties, 431–438, 441 Soot radiation, 822 Source, radiative, see Radiative source Space radiator, 273 SPECAIR database, 332 Special surfaces, 97–102 Species concentration, 820 Spectral line, 311, 315, 323 strength, 321 Spectral models, 333 Spectral range, 215 Spectral variable, xxii, 6, 285, 316 Spectral window, 23, 635, 670, 678 Spectrometer, 107, 438, 440, 461

Spectroscopic database, 331 Spectrum electromagnetic wave, 2, 3 of the sun, 5 vibration–rotation band, 319 Specular direction, 67, 221, 294 Specular peak, 67 Specular reflection, 43, 67, 199 paths, 200 peak, 199 Specular reflector, 67, 69 Specular view factor, xx, 128, 200–203 Speed of light, xix, 2, 19 in vacuum, 2, 34 Sphere integrating, 439, 443 large, diffusely reflecting, 418 large, opaque, 414, 847 large, specularly reflecting, 415 near-dielectric, 413 Spheres, concentric at radiative equilibrium, 486 discrete ordinates method, 571–574 without participating medium, 163, 170, 177, 203, 207 Spherical harmonics, xxii, 507, 514 Spherical harmonics method, 306, 513–555 Spherical medium, 483–487 discrete ordinates method, 571–574 isothermal, 291, 298, 301 isothermal, nongray gas, 663 Spherical top, 315 Spline, 491 SPRADIAN database, 332 Spray combustion, 836 Stabilization, 859 Standard deviation, 238, 764 Standard error, 238 Stanton number, xxi, 277 Stark broadening, 323, 324 Stark effect, 325 Statistical error, 235 Statistical model, 336, 338, 339 general, 358 rough surface, 88 Statistical sampling, 235 Statistical uncertainty, 860 Steepest descent, 868, 873 Stefan number, xxi, 788 Stefan, Josef, 9 Stefan–Boltzmann constant, xxiii, 9 Step size, 868 Stepwise-gray model, 667–673 Steradian, 11 Stimulated emission, 322 Stochastic particle, 750 Stokes’ parameter, xx, 39–41 for polarization, xxi, xxii Stokes’ theorem, 42, 135, 136 Stretching factor, 693 Stretching factor, nongray, 694, 696, 698, 699, 702, 704, 714, 720, 722 Successive approximation, method of, 179, 180

Sulfur dioxide, 359, 360, 376 Summation relation for exchange factors, 241 for specular view factors, 202, 205 for view factors, 131 Sun, see Solar Surface artificial, 128, 164, 165 black, 4 concave, 131 convex, 131, 144, 171 curved, 201, 208, 209 cylindrical, 188 directionally nonideal, 219–226 flat, 131 fractal, 87 gray, 62 gray, diffuse, 62, 161, 166, 294 grooved, 101 ideal, 127 isotropic, 61 nongray, 215 nonideal, 295 opaque, 4, 10, 20, 21, 59 optically smooth, 44, 67, 75, 199 polished, 75 rough, 210 selective, 97, 98, 199, 208 semitransparent, 211–214 solar collector, 97 specularly reflecting, 200, 294 V-grooved, 100, 101, 211, 222 vector description, 242 Surface coating, 91 Surface damage, 89, 90 Surface integration, 132 Surface layer, 76, 81, 89 Surface modification, 89 Surface normal, xxi, 11, 129, 132, 136 Surface phonon polaritons, 891 Surface plasmons, 891 Surface polaritons, 891 Surface preparation, 73 Surface radiosity, 166 Surface roughness, 67, 85–89 Surface waves, 891 Suspension, particle, 439, 440, 676, 750 Switching function, 578, 585 Symmetric top, 315 Symmetry number, 322

T T-matrix method, cluster, 425 Tables: apparent emittance for cylindrical cavities, 180 associate Legendre polynomial half-moments pm , 532 n,j blackbody emissive powers, 921 comparison of different Monte Carlo implementations, 769 conversion factors, 906 discrete ordinates (one-dimensional), 567

986 Index

discrete ordinates (three-dimensional), 566 Drude parameters for metals, 74 exponential integrals, 943 full spectrum k-distributions, 719 mean beam lengths, 662 narrow band correlations, 340 optical properties of coal and ash, 427 physical constants, 905 radiative equilibrium between concentric cylinders, 490 between concentric spheres, 486 in a plane-parallel medium, 475 radiative heat flux from an isothermal cylinder, 489 radiative properties of coal particles, 429 spectral, normal emittances of metals, 918, 919 Stokes’ parameters for polarization, 40 total emissivity correlation for CO2 , 372 total emissivity correlation for H2 O, 372 total, normal emittances, 908, 910 total, normal emittances of metals, 917 total, normal solar absorptances, 908 view factor catalogue, 925 weighted-sum-of-gray-gases coefficients, 680 wide band model correlation, 359 wide band model parameters, 360 TE wave, 39 Temperature average emission, 705 bulk, 276 control, 255 Planck-mean, 705 solar, 7, 10, 16, 18 Temperature dependence of radiative properties, 79, 85 Temperature discontinuity, 474 Temperature measurement of gases, 381 Tempered averaging, 820, 839 Thermal barrier coating, 99 Thermal conductivity, xx, 1 Thermal detector, 108 Thermal radiation, 1, 3, 26 Thermal runaway, 382 Thermopile, 108 Thick film, 54 Thin eddy approximation, 827, 830 Thin film, 51, 460 reflectivity, 54 transmission through, 51–54 Thin film optics, 51 Thin layer, 89 Tikhonov regularization, 860, 865, 866, 872, 873, 878 Titanium dioxide, 90, 443 TM wave, 39 Toluene, 430 Tomography, 458, 878 multi-projection, 879 totabsor, 373, 948 totem, 946

totemiss, 373, 374, 664, 948 Transient radiation, 649–652 Transition bound electron, 73 bound–bound, 22, 311 bound–free, 22, 311, 319 electronic, 83, 453, 455 forbidden, 319 free–bound, 22 free–free, 22, 311, 320 interband, 56 vibrational, 80, 81 Translucent, 4, 150 Transmission, 41–55 through a slab, 51–55 Transmission coefficient, xxi, 45 for a thin film, 52 Transmission measurement, 381, 441 Transmission method, 460, 462 Transmissivity, xxiii, see also Transmittance, 46, 82 full spectrum, 715 narrow band, 345 of a dielectric thin film, 54 of a fictitious gas, 355 of a gas layer, 22 of a material layer, 23 of a nonhomogeneous layer, 343, 353 of a slab, 463 of a thick slab, 54, 92 of a thin film, 53 of a thin gap, 890 of an In2 O3 film on glass, 95 of multiple glass sheets, 97 of window glass, 92 spectral, 336 vacuum gap, 890 Transmittance, 20, 21 solar, 99 Transparent, 4 Transparent medium, 474 transPN, 651, 652, 949 Transverse electric, 39 Transverse magnetic, 39 TRI, see Turbulence–radiation interaction Truncated singular value decomposition, 860, 878 Tunneling, of radiation, 889 Turbulence interaction with radiation, 824–834 Turbulence model, 826 Turbulence moment, 826 Turbulence–radiation interaction, 824–834 Turbulent combustion buoyant flame, 837 engines, 834 fluidized bed, 836 gas turbine, 834 multiphase, 836 oxy-fuel, 837 pool fire, 837 sprays, 836 Turbulent diffusivity, 825 Turbulent flames

confined, 834 nonpremixed, 829 Two-flux approximation, 306, 503, 504, 604, 845

U Ultraviolet radiation, 4, 73, 89, 93, 311, 455 Uncertainty, statistical, 860 Uniform spatial division algorithm, 254 Unit sphere method, 132 Unit tensor, 137, 520, 620, 860 Unit vector, 132 direction, 11 for direction, xxi, 10, 35, 249, 469 surface normal, xxi, 11, 243 surface tangent, 243

V

V-groove, 100, 101, 211, 222 Vacuum, 127 Van Royen, Willebrord van Snel, 44 Variance, 239, 860 Variational calculus, 179 Velocity, xxii, 303 mean, 276 phase, 33, 35, 413 signal, 35, 649 vfplanepoly, 153, 946 Vibration ellipse, 37 Vibration, lattice, 56, 85 Vibration–rotation band, 22, 312, 315, 318 spectrum, 319 Vibrational energy level, 80, 312, 315, 317, 319 Vibrational quantum number, 317, 361 Vibrational transition, 80, 81 view, 131, 143, 158, 925, 946 View factor, xx, 127 by area integration, 132 by contour integration, 135, 151 by crossed-strings method, 143, 144 by inside sphere method, 148 by unit sphere method, 150 catalogue, 925–934 definition of, 128 diffuse, 128 evaluation methods, 131 specular, xx, 128, 200–203 View factor algebra, 132, 140 VIEW3D, 950 viewfactors, 131, 158, 925, 946 Visible radiation, 2, 3, 17, 18, 455 voigt, 326, 947 Voigt broadening, 326 Voigt profile, 326 Volume fraction, 407, 408, 412 of particles, 402 Volume-by-volume advancement algorithm, 254

W Wall, semitransparent, 204, 295 wangwbm, 364, 948 Water, 455

Index 987

Water droplets, 422–424 Water vapor, 359, 360, 362, 367, 370–372, 679, 680, 685, 712, 717 Planck-mean absorption coefficient, 375 total emissivity, 371, 372 Wave homogeneous, 33 inhomogeneous, 33 plane, 33, 38, 43 Wave equation, Schrödinger’s, 316 Wave interference, 51, 54, 93–95, 99 Wave vector, xxii, 33, 888 for transmission, 48 Wavefront, 43 Wavelength, xxii, 3 crossover, 79 cutoff, 97 Wavenumber, xxii, 3, 33 reordered, 692

wbmkvsg, 368, 948 wbmodels, 364, 948 wbmxxx, 362, 948 Weight factor (for WSGG), 678, 680, 683 Weight function, 752 Weight function (for FSK), 693, 708, 722 Weighted sum of gray gases, 634, 677–683, 799 Weighting matrix, xxii Wide band model, 334, 356–369 exponential, 358 for isothermal media, 677 Wien’s displacement law, 8, 898 Wien’s distribution, 9, 314 Wien’s law, 9 Wien, Wilhelm, 6 Wigner-D functions, xix, 530 Window, 91, 462 semitransparent, 211–214, 846

spectral, 635, 670, 678 Window glass absorption coefficient, 454 WSGG, see Weighted sum of gray gases wsggBrd, 949 wsggKng, 949 wsggsoot, 949

Y YIX method, 605

Z Zenith angle, 5 Zinc oxide, 847 Zinc selenide, 383 Zirconia, 845 Zirconium carbide, 85 Zonal method, 306, 617–638