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- Author / Uploaded
- Michael F. Modest

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©ElsevierInc.Allrightsreserved. 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

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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.

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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].

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[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).)

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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.

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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 → ∞).

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(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.

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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.

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[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.

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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

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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.

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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.

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[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.

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(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.

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[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.

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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

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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