Thermal radiation heat transfer [Seventh ed.] 9780429327308, 0429327307, 9781000257816, 1000257819, 9781000257823, 1000257827, 9781000257830, 1000257835

Citation preview

Thermal Radiation Heat Transfer

Thermal Radiation Heat Transfer Seventh Edition

John R. Howell, M. Pinar Mengüç, Kyle Daun, and Robert Siegel

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.

Seventh edition published 2021 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2021 Taylor & Francis Group, LLC Sixth edition published by CRC Press 2015 CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected]. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

Library of Congress Cataloging‑in‑Publication Data Names: Howell, John R., author. | Mengüç, M. Pinar, author. | Daun, Kyle J., author. | Siegel, Robert, 1927- author. Title: Thermal radiation heat transfer / John R. Howell, M. Pinar Mengüc, Kyle Daun, and Robert Siegel. Description: Seventh edition. | Boca Raton : CRC Press, 2021. | Revised edition of: Thermal radiation heat transfer / John R. Howell, M. Pinar Mengüç, Robert Siegel. Sixth edition. 2015. | Includes bibliographical references and index. Identifiers: LCCN 2020026407 (print) | LCCN 2020026408 (ebook) | ISBN 9780367347079 (hardback) | ISBN 9780429327308 (ebook) Subjects: LCSH: Heat--Radiation and absorption. | Heat--Transmission. | Materials--Thermal properties. Classification: LCC QC331 .S55 2021 (print) | LCC QC331 (ebook) | DDC 621.402/2--dc23 LC record available at https://lccn.loc.gov/2020026407 LC ebook record available at https://lccn.loc.gov/2020026408 ISBN: 978-0-367-34707-9 (hbk) ISBN: 978-0-429-32730-8 (ebk) Typeset in Times by Deanta Global Publishing Services, Chennai, India Visit the eResources: www.routledge.com/9780367347079

Contents Preface.............................................................................................................................................xix Acknowledgments......................................................................................................................... xxiii Authors............................................................................................................................................xxv List of Symbols.............................................................................................................................xxvii Chapter 1 Introduction to Radiative Transfer................................................................................1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Thermal Radiation and the Natural World......................................................... 3 Thermal Radiation in Engineering..................................................................... 4 Thermal Radiation and Thermodynamics.........................................................5 Nature of the Governing Equations.................................................................... 6 Electromagnetic Waves vs Photons....................................................................8 Radiative Energy Exchange and Radiative Intensity....................................... 11 Solid Angle....................................................................................................... 12 Spectral Intensity.............................................................................................. 12 Characteristics of Blackbody Emission............................................................ 14 1.9.1 Perfect Emitter.................................................................................... 15 1.9.2 Radiation Isotropy of a Black Enclosure............................................. 15 1.9.3 Perfect Emitter for Each Direction and Wavelength........................... 16 1.9.4 Total Radiation into a Vacuum............................................................ 16 1.9.5 Blackbody Intensity and Its Directional Independence...................... 16 1.9.6 Blackbody Emissive Power: Cosine Law Dependence....................... 18 1.9.7 Hemispherical Spectral Emissive Power of a Blackbody................... 18 1.9.8 Planck’s Law: Spectral Distribution of Emissive Power..................... 19 1.9.9 Approximations to the Blackbody Spectral Distribution....................24 1.9.9.1 Wien’s Formula....................................................................24 1.9.9.2 Rayleigh–Jeans Formula......................................................24 1.9.10 Wien’s Displacement Law...................................................................25 1.9.11 Total Blackbody Intensity and Emissive Power..................................26 1.9.12 Blackbody Radiation within a Spectral Band..................................... 27 1.9.13 Summary of Blackbody Properties..................................................... 31 1.10 Radiative Energy along a Line-of-Sight........................................................... 35 1.10.1 Radiative Energy Loss due to Absorption and Scattering.................. 35 1.10.2 Mean Penetration Distance................................................................. 38 1.10.3 Optical Thickness................................................................................ 38 1.10.4 Radiative Energy Gain Due to Emission............................................ 39 1.10.5 Radiative Energy Density and Radiation Pressure.............................40 1.10.6 Radiative Energy Gain due to In-Scattering....................................... 41 1.11 Radiative Transfer Equation............................................................................. 42 1.12 Radiative Transfer in Enclosures with Nonparticipating Media......................44 1.13 Probabilistic Interpretation............................................................................... 47 1.14 Concluding Remarks........................................................................................ 49 Homework Problems................................................................................................... 49 Chapter 2 Radiative Properties at Interfaces............................................................................... 53 2.1 Introduction...................................................................................................... 53 v

vi

Contents

2.2 Emissivity......................................................................................................... 56 2.2.1 Directional Spectral Emissivity, ελ(θ,φ,T).......................................... 57 2.2.2 Directional Total Emissivity, ε(θ,φ,T)................................................. 59 2.2.3 Hemispherical Spectral Emissivity, ελ(T)...........................................60 2.2.4 Hemispherical Total Emissivity, ε(T).................................................. 61 2.3 Absorptivity......................................................................................................64 2.3.1 Directional Spectral Absorptivity, αλ(θi,φi,T)..................................... 65 2.3.2 Kirchhoff’s Law..................................................................................66 2.3.3 Directional Total Absorptivity, α(θi,φi,T)........................................... 67 2.3.4 Kirchhoff’s Law for Directional Total Properties............................... 68 2.3.5 Hemispherical Spectral Absorptivity, αλ(T)........................................ 69 2.3.6 Hemispherical Total Absorptivity, α(T).............................................. 70 2.3.7 Diffuse-Gray Surface.......................................................................... 72 2.4 Reflectivity........................................................................................................ 73 2.4.1 Spectral Reflectivities.......................................................................... 73 2.4.1.1 Bidirectional Spectral Reflectivity, ρλ(θr,ϕr,θi,ϕi)................ 73 2.4.1.2 Reciprocity for Bidirectional Spectral Reflectivity............. 74 2.4.1.3 Directional Spectral Reflectivities, ρλ(θr,ϕr), ρλ(θi,ϕi).......... 76 2.4.1.4 Reciprocity for Directional Spectral Reflectivity................ 77 2.4.1.5 Hemispherical Spectral Reflectivity, ρλ............................... 77 2.4.1.6 Limiting Cases for Spectral Surfaces.................................. 78 2.4.2 Total Reflectivities............................................................................... 79 2.4.2.1 Bidirectional Total Reflectivity, ρ(θr,ϕr,θi,ϕi)......................80 2.4.2.2 Reciprocity for Bidirectional Total Reflectivity..................80 2.4.2.3 Directional-Hemispherical Total Reflectivity, ρ(θr,ϕr), ρ(θi,ϕi)....................................................................80 2.4.2.4 Reciprocity for Directional Total Reflectivity..................... 81 2.4.2.5 Hemispherical Total Reflectivity, ρ..................................... 81 2.4.3 Summary of Restrictions on Reflectivity Reciprocity Relations........ 82 2.5 Transmissivity at an Interface.......................................................................... 82 2.5.1 Spectral Transmissivities..................................................................... 82 2.5.1.1 Bidirectional Spectral Transmissivity, τλ(θt,ϕt,θi,ϕi)........... 82 2.5.1.2 Directional Spectral Transmissivities, τλ(θt,ϕt), τλ(θi,ϕi)..... 83 2.5.1.3 Hemispherical Spectral Transmissivity, τλ..........................84 2.5.2 Total Transmissivities..........................................................................84 2.5.2.1 Bidirectional Total Transmissivity, τ(θi,ϕi,θt,ϕt)..................84 2.5.2.2 Directional Total Transmissivities, τ(θi,ϕi).........................84 2.5.2.3 Hemispherical-Directional Total Transmissivity, τ(θt,ϕt), τ(θi,ϕi)..................................................................... 85 2.5.2.4 Hemispherical Total Transmissivity, τ................................ 86 2.6 Relations among Reflectivity, Absorptivity, Emissivity, and Transmissivity.......86 Homework Problems...................................................................................................90 Chapter 3 Radiative Properties of Opaque Materials.................................................................. 95 3.1 Introduction...................................................................................................... 95 3.2 Electromagnetic Wave Theory Predictions......................................................96 3.2.1 Dielectric Materials.............................................................................96 3.2.1.1 Reflection and Refraction at the Interface between Two Perfect Dielectrics (No Wave Attenuation, k → 0)......96 3.2.1.2 Reflectivity...........................................................................97

vii

Contents

3.2.1.3 Emissivity............................................................................99 Radiative Properties of Conductors.................................................. 101 3.2.2.1 Electromagnetic Relations for Incidence on an Absorbing Medium............................................................ 101 3.2.2.2 Reflectivity and Emissivity Relations for Metals (Large k)............................................................................. 103 3.2.2.3 Relations between Radiative Emission and Electrical Properties.......................................................... 107 3.2.3 Extensions to the Theory of Radiative Properties............................ 112 3.3 Measurements on Real Surfaces.................................................................... 114 3.3.1 Heterogeneity and Surface Coatings................................................. 114 3.3.2 Surface Roughness Effects................................................................ 119 3.3.3 Variation of Radiative Properties with Surface Temperature........... 131 3.3.3.1 Non-metals......................................................................... 131 3.3.3.2 Metals................................................................................ 131 3.3.4 Properties of Liquid Metals............................................................... 137 3.3.5 Properties of Semiconductors and Superconductors......................... 138 3.4 Selective Surfaces for Solar Applications...................................................... 139 3.4.1 Characteristics of Solar Radiation.................................................... 139 3.4.2 Modification of Surface Spectral Characteristics............................. 140 3.4.3 Modification of Surface Directional Characteristics........................ 146 3.5 Concluding Remarks...................................................................................... 147 Homework Problems................................................................................................. 148 3.2.2

Chapter 4 Configuration Factors for Diffuse Surfaces with Uniform Radiosity....................... 153 4.1

4.2

4.3

Radiative Transfer Equation for Surfaces Separated by a Transparent Medium.................................................................................... 153 4.1.1 Enclosures with Diffuse Surfaces..................................................... 154 4.1.2 Enclosures with Directional (Nondiffuse) and Spectral (Nongray) Surfaces............................................................................ 155 Geometric Configuration Factors between Two Surfaces.............................. 155 4.2.1 Configuration Factor for Energy Exchange between Diffuse Differential Areas.............................................................................. 156 4.2.1.1 Reciprocity for Differential Element Configuration Factors........................................................ 157 4.2.1.2 Sample Configuration Factors between Differential Elements......................................................... 157 4.2.2 Configuration Factor between a Differential Area Element and a Finite Area...................................................................................... 160 4.2.2.1 Reciprocity Relation for Configuration Factors between Differential and Finite Areas.............................. 161 4.2.2.2 Configuration Factors between Differential and Finite Areas....................................................................... 161 4.2.3 Configuration Factor and Reciprocity for Two Finite Areas............. 164 Methods for Determining Configuration Factors........................................... 165 4.3.1 Configuration Factor Algebra............................................................ 165 4.3.1.1 Configuration Factors Determined by Symmetry............. 170 4.3.2 Configuration Factor Relations in Enclosures................................... 173 4.3.3 Techniques for Evaluating Configuration Factors............................. 175 4.3.3.1 Hottel’s Crossed-String Method........................................ 175

viii

Contents

4.3.3.2 Contour Integration............................................................ 178 4.3.3.3 Differentiation of Known Factors...................................... 185 4.3.4 Unit-Sphere and Hemicube Methods................................................ 188 4.3.5 Direct Numerical Integration............................................................ 189 4.3.6 Computer Programs for the Evaluation of Configuration Factors.........190 4.4 Constraints on Configuration Factor Accuracy.............................................. 190 4.5 Compilation of Known Configuration Factors and Their References: Appendix C and Web Catalog........................................................................ 192 Homework Problems................................................................................................. 192 Chapter 5 Radiation Exchange in Enclosures Composed of Black and/or Diffuse-Gray Surfaces..................................................................................................................... 201 5.1 Introduction.................................................................................................... 201 5.2 Radiative Transfer for Black Surfaces............................................................202 5.2.1 Transfer between Black Surfaces Using Configuration Factors........204 5.2.2 Radiation Exchange in a Black Enclosure.........................................205 5.3 Radiation Among Finite Diffuse-Gray Areas................................................208 5.3.1 Net-Radiation Method for Enclosures...............................................208 5.3.1.1 System of Equations Relating Surface Heating Rate Q and Surface Temperature T................................... 214 5.3.1.2 Solution Method in Terms of Radiosity J.......................... 219 5.3.2 Enclosure Analysis in Terms of Energy Absorbed at Surface.......... 221 5.3.3 Enclosure Analysis by the Use of Transfer Factors........................... 222 5.3.4 Matrix Inversion for Enclosure Equations........................................ 223 5.4 Radiation Analysis Using Infinitesimal Areas............................................... 228 5.4.1 Generalized Net-Radiation Method Using Infinitesimal Areas........ 228 5.4.1.1 Relations between Surface Temperature T and Surface Heat Flux q........................................................... 230 5.4.1.2 Solution Method in Terms of Outgoing Radiative Flux J................................................................................. 231 5.4.1.3 Special Case When Imposed Heating q Is Specified for All Surfaces.................................................................. 232 5.4.2 Boundary Conditions Specifying Inverse Problems.........................240 5.5 Computer Programs for Enclosure Analysis..................................................240 Homework Problems................................................................................................. 241 Chapter 6 Exchange of Thermal Radiation among Nondiffuse Nongray Surfaces................... 251 6.1 Introduction.................................................................................................... 251 6.2 Enclosure Theory for Diffuse Nongray Surfaces........................................... 252 6.2.1 Parallel-Plate Geometry.................................................................... 253 6.2.2 Spectral and Finite Spectral Band Relations for an Enclosure......... 256 6.2.3 Semigray Approximations................................................................. 258 6.3 Directional-Gray Surfaces.............................................................................. 258 6.4 Surfaces with Directionally and Spectrally Dependent Properties................ 265 6.5 Radiation Exchange in Enclosures with Specularly Reflecting Surfaces....... 271 6.5.1 Representative Cases with Simple Geometries................................. 271 6.5.2 Ray Tracing and Image Formation.................................................... 275 6.5.3 Radiative Transfer among Simple Specular Surfaces for Diffuse Energy Leaving a Surface.................................................... 277

ix

Contents

6.5.4

Configuration-Factor Reciprocity for Specular Surfaces; Specular Exchange Factors............................................................... 281 6.6 Net-Radiation Method in Enclosures Having Both Specular and Diffuse Reflecting Surfaces............................................................................ 287 6.6.1 Enclosures with Planar Surfaces....................................................... 287 6.6.2 Curved Specular Reflecting Surfaces............................................... 293 6.7 Multiple Radiation Shields............................................................................. 297 6.8 Concluding Remarks......................................................................................300 Homework Problems.................................................................................................302 Chapter 7 Radiation Combined with Conduction and Convection at Boundaries.................... 313 7.1 Introduction.................................................................................................... 313 7.2 Energy Relations and Boundary Conditions.................................................. 314 7.2.1 General Relations.............................................................................. 314 7.2.2 Uncoupled and Coupled Energy Transfer Modes............................. 316 7.2.3 Control Volume Approach for One- or Two-Dimensional Conduction along Thin Walls............................................................ 317 7.3 Radiation Transfer with Conduction Boundary Conditions........................... 318 7.3.1 Thin Fins with 1D or 2D Conduction............................................... 318 7.3.1.1 1D Energy Flow................................................................. 318 7.3.1.2 2D Energy Flow................................................................. 324 7.3.2 Multidimensional and Transient Heat Conduction with Radiation................................................................................... 325 7.4 Radiation with Convection and Conduction................................................... 326 7.4.1 Thin Radiating Fins with Convection............................................... 327 7.4.2 Channel Flows................................................................................... 329 7.4.3 Natural Convection with Radiation................................................... 333 7.5 Numerical Solution Methods.......................................................................... 336 7.5.1 Numerical Integration Methods for Use with Enclosure Equations........................................................................................... 337 7.5.2 Numerical Formulations for Combined-Mode Energy Transfer....... 337 7.5.2.1 Finite-Difference Formulation........................................... 339 7.5.2.2 Finite Element Method Formulation................................. 343 7.5.3 Numerical Solution Techniques........................................................348 7.5.4 Verification, Validation, and Uncertainty Quantification................. 349 7.5.4.1 Verification........................................................................ 349 7.5.4.2 Validation........................................................................... 350 7.5.4.3 Uncertainty Quantification................................................ 350 7.6 Concluding Remarks...................................................................................... 350 Homework Problems................................................................................................. 351 Chapter 8 Electromagnetic Wave Theory.................................................................................. 363 8.1 Introduction.................................................................................................... 363 8.2 The Electromagnetic Wave Equations...........................................................364 8.3 Wave Propagation in a Medium..................................................................... 365 8.3.1 EM Wave Propagation in Perfect Dielectric Media.......................... 366 8.3.2 Wave Propagation in Isotropic Media with Finite Electrical Conductivity...................................................................................... 369 8.3.3 Energy of an EM Wave..................................................................... 371 8.3.4 Polarization....................................................................................... 372

x

Contents

8.4

Laws of Reflection and Refraction................................................................. 375 8.4.1 Reflection and Refraction at the Interface between Perfect Dielectrics............................................................................. 376 8.4.2 Reflection and Refraction at the Interface of a Perfect Dielectric and a Conducting Medium............................................... 379 8.5 Classical Models for Optical Constants......................................................... 381 8.5.1 Lorentz Model (Non-conductors)...................................................... 383 8.5.2 Drude Model (Conductors)................................................................ 385 8.6 EM Wave Theory and the Radiative Transfer Equation................................ 387 Homework Problems................................................................................................. 387 Chapter 9 Properties of Participating Media............................................................................. 389 9.1 Introduction.................................................................................................... 389 9.2 Propagation of Radiation in Absorbing Media............................................... 391 9.3 Spectral Lines and Bands for Gas Absorption and Emission........................ 392 9.3.1 Physical Mechanisms........................................................................ 392 9.3.2 Local Thermodynamic Equilibrium (LTE)...................................... 396 9.3.3 Spectral Line Broadening................................................................. 397 9.3.3.1 Natural Broadening........................................................... 398 9.3.3.2 Doppler Broadening........................................................... 399 9.3.3.3 Collision Broadening and Narrowing................................ 399 9.3.3.4 Stark Broadening...............................................................400 9.3.4 Absorption or Emission by a Single Spectral Line........................... 401 9.3.4.1 Property Definitions along a Path in a Uniform Absorbing and Emitting Medium...................................... 401 9.3.4.2 Weak Lines........................................................................402 9.3.4.3 Lorentz Lines.....................................................................402 9.3.5 Band Absorption...............................................................................403 9.3.5.1 Band Structure...................................................................403 9.3.5.2 Types of Band Models.......................................................405 9.3.5.3 Databases for the Line Absorption Properties of Molecular Gases................................................................407 9.4 Band Models and Correlations for Gas Absorption and Emission................408 9.4.1 Narrow-Band Models........................................................................408 9.4.1.1 Elsasser Model...................................................................408 9.4.1.2 Goody Model..................................................................... 410 9.4.1.3 Malkmus Model................................................................. 410 9.4.2 Wide Band Models............................................................................ 410 9.4.3 Probability Density Function-Based Band Correlations................... 410 9.4.3.1 k-Distribution Method....................................................... 411 9.4.3.2 Correlated-k Assumption................................................... 413 9.4.3.3 Full Spectrum k-Distribution Methods.............................. 416 9.4.3.4 Effect of Temperature and Concentration Gradients in the Medium................................................... 418 9.4.4 Weighted Sum of Gray Gases........................................................... 419 9.5 Gas Total Emittance Correlations.................................................................. 425 9.6 Absorption Coefficient Neglecting Induced Emission................................... 430 9.7 Definitions and Use of Mean Absorption Coefficients................................... 431 9.7.1 Use of Mean Absorption Coefficients............................................... 431 9.7.2 Definitions of Mean Absorption Coefficients................................... 431

xi

Contents

9.7.3

Approximate Solutions of the Radiative Transfer Equation Using Mean Absorption Coefficients................................................ 432 9.8 Radiative Properties of Translucent Liquids and Solids................................ 433 Homework Problems................................................................................................. 438 Chapter 10 Absorption and Scattering by Particles and Agglomerates....................................... 441 10.1 Overview........................................................................................................ 441 10.2 Absorption and Scattering: Definitions.......................................................... 443 10.2.1 Background....................................................................................... 443 10.2.2 Absorption and Scattering Coefficients, Cross-Sections, Efficiencies........................................................................................444 10.2.3 Scattering Phase Function.................................................................446 10.3 Scattering by Spherical Particles....................................................................449 10.3.1 Scattering by a Large Specularly Reflecting Sphere......................... 450 10.3.2 Reflection from a Large Diffuse Sphere........................................... 453 10.3.3 Large Ideal Dielectric Sphere with n ≈ 1.......................................... 454 10.3.4 Diffraction by a Large Sphere........................................................... 455 10.3.5 Geometric Optics Approximation..................................................... 456 10.4 Scattering by Small Particles......................................................................... 458 10.4.1 Rayleigh Scattering by Small Spheres.............................................. 458 10.4.2 Cross-Section for Rayleigh Scattering..............................................460 10.4.3 Phase Function for Rayleigh Scattering............................................ 462 10.5 The Lorenz–Mie Theory for Spherical Particles........................................... 462 10.5.1 Formulation for Homogeneous and Stratified Spherical Particles............................................................................................. 463 10.5.2 Cross-Sections for Specific Cases..................................................... 467 10.6 Approximate Anisotropic Scattering Phase Functions..................................469 10.6.1 Forward Scattering Phase Function..................................................469 10.6.2 Linear-Anisotropic Phase Function...................................................469 10.6.3 Delta-Eddington Phase Function...................................................... 470 10.6.4 Henyey–Greenstein Phase Function.................................................. 470 10.7 Prediction of Properties for Irregularly Shaped Particles.............................. 471 10.7.1 Integral and Differential Formulations............................................. 472 10.7.2 T-Matrix Approach............................................................................ 472 10.7.3 Discrete Dipole Approximation........................................................ 474 10.7.4 Finite-Element Method..................................................................... 479 10.7.5 Finite-Difference Time-Domain Method.......................................... 479 10.8 Dependent Absorption and Scattering........................................................... 481 Homework Problems................................................................................................. 486 Chapter 11 Fundamental Radiative Transfer Relations and Approximate Solution Methods........ 489 11.1 Introduction.................................................................................................... 489 11.2 Energy Equation and Boundary Conditions for a Participating Medium......... 490 11.3 Radiative Transfer and Source-Function Equations....................................... 492 11.3.1 Radiative Transfer Equation.............................................................. 492 11.3.2 Source-Function Equation................................................................. 494 11.4 Radiative Flux and Its Divergence within a Medium..................................... 498 11.4.1 Radiative Flux Vector........................................................................ 498

xii

Contents

11.4.2 Divergence of Radiative Flux without Scattering (Absorption Alone)............................................................................ 501 11.4.3 Divergence of Radiative Flux Including Scattering.......................... 503 11.5 Summary of Relations for Radiative Transfer in Absorbing, Emitting, and Scattering Media.....................................................................504 11.5.1 Energy Equation................................................................................504 11.5.2 Radiative Energy Source................................................................... 505 11.5.3 Source Function................................................................................. 505 11.5.4 Radiative Transfer Equation..............................................................506 11.5.5 Relations for a Gray Medium............................................................506 11.6 Treatment of Radiation Transfer in Non-LTE Media..................................... 507 11.7 Net-Radiation Method for Enclosures Filled with an Isothermal Medium of Uniform Composition..................................................................508 11.7.1 Definitions of Spectral Geometric-Mean Transmission and Absorption Factors............................................................................ 511 11.7.2 Matrix of Enclosure Theory Equations............................................. 511 11.7.3 Energy Balance on a Medium........................................................... 512 11.7.4 Spectral Band Equations for an Enclosure........................................ 514 11.7.5 Gray Medium in a Gray Enclosure................................................... 515 11.8 Evaluation of Spectral Geometric-Mean Transmittance and Absorptance Factors....................................................................................... 517 11.9 Mean Beam Length Approximation for Spectral Radiation from an Entire Volume of a Medium to All or Part of Its Boundary...................... 518 11.9.1 Mean Beam Length for a Medium between Parallel Plates Radiating to Area on Plate................................................................ 519 11.9.2 Mean Beam Length for the Sphere of a Medium Radiating to Any Area on Its Boundary............................................................ 519 11.9.3 Radiation from the Entire Medium Volume to Its Entire Boundary for Optically Thin Media................................................. 519 11.9.4 Correction to Mean Beam Length When a Medium Is Not Optically Thin................................................................................... 520 11.10 Exchange of Total Radiation in an Enclosure by Use of Mean Beam Length.................................................................................................. 521 11.10.1 Total Radiation from the Entire Medium Volume to All or Part of Its Boundary.......................................................................... 521 11.10.2 Exchange between the Entire Medium Volume and the Emitting Boundary............................................................................ 524 11.11 Optically Thin and Cold Media..................................................................... 525 11.11.1 Nearly Transparent Medium............................................................. 525 11.11.2 Optically Thin Media with Cold Boundaries or Small Incident Radiation: The Emission Approximation........................... 527 11.11.3 Cold Medium with Weak Scattering................................................. 528 Homework Problems................................................................................................. 529 Chapter 12 Participating Media in Simple Geometries............................................................... 535 12.1 Introduction.................................................................................................... 535 12.2 Radiative Intensity, Flux, Flux Divergence, and Source Function in a Plane Layer..................................................................................................... 535 12.2.1 Radiative Transfer Equation and Radiative Intensity for a Plane Layer..................................................................................... 535

xiii

Contents

12.2.2 12.2.3 12.2.4 12.2.5 12.2.6

Local Radiative Flux in a Plane Layer.............................................. 537 Divergence of the Radiative Flux: Radiative Energy Source............ 538 Equation for the Source Function in a Plane Layer.......................... 539 Relations for Isotropic Scattering......................................................540 Diffuse Boundary Fluxes for a Plane Layer with Isotropic Scattering.......................................................................................... 541 12.3 Gray Plane Layer of Absorbing and Emitting Medium with Isotropic Scattering........................................................................................ 542 12.4 Gray Plane Layer in Radiative Equilibrium...................................................546 12.4.1 Energy Equation................................................................................546 12.4.2 Absorbing Gray Medium in Radiative Equilibrium with Isotropic Scattering........................................................................... 547 12.4.3 Isotropically Scattering Medium with Zero Absorption................... 547 12.4.4 Gray Medium with dqR /dx = 0 between Opaque Diffuse-Gray Boundaries........................................................................................ 548 12.4.5 Solution for Gray Medium with dqr/dx = 0 between Black or Diffuse-Gray Boundaries at Specified Temperatures....................... 549 12.4.5.1 Gray Medium between Black Boundaries......................... 549 12.4.5.2 Gray Medium between Diffuse-Gray Boundaries............ 551 12.4.5.3 Extended Solution for Optically Thin Medium between Gray Boundaries.................................................. 553 12.5 Multidimensional Radiation in a Participating Gray Medium with Isotropic Scattering........................................................................................ 555 12.5.1 Radiation Transfer Relations in Three Dimensions.......................... 555 12.5.2 Two-Dimensional Transfer in an Infinitely Long Right Rectangular Prism............................................................................. 557 12.5.3 One-Dimensional Transfer in a Cylindrical Region......................... 560 12.5.4 Additional Information on Non-planar and Multidimensional Geometries........................................................................................564 Homework Problems................................................................................................. 565 Chapter 13 Numerical Solution Methods for Radiative Transfer in Participating Media........... 569 13.1 Introduction.................................................................................................... 569 13.2 Series Expansion and Moment Methods........................................................ 570 13.2.1 Optically Thick Media: Radiative Diffusion.................................... 571 13.2.1.1 Simplified Derivation of the Radiative Diffusion Approximation................................................................... 572 13.2.1.2 General Radiation–Diffusion Relations in a Medium....... 574 13.2.2 Moment-Based Methods................................................................... 583 13.2.2.1 Milne–Eddington (Differential) Approximation............... 584 13.2.2.2 General Spherical Harmonics (PN) Method....................... 587 13.2.2.3 Simplified PN (SPN) Method............................................... 597 13.2.2.4 M N Method........................................................................600 13.3 Discrete Ordinates (SN) Method.....................................................................602 13.3.1 Two-Flux Method: The Schuster–Schwarzschild Approximation......... 603 13.3.2 Radiative Transfer Equation with SN Method...................................606 13.3.3 Boundary Conditions for the SN Method..........................................607 13.3.4 Control Volume Method for SN Numerical Solution.........................608 13.3.4.1 Relations for 2D Rectangular Coordinates........................609 13.3.4.2 Relations for 3D Rectangular Coordinates........................ 611

xiv

Contents

13.3.5 Ordinate and Weighting Pairs........................................................... 614 13.3.6 Results Using Discrete Ordinates...................................................... 615 13.4 Other Methods That Depend on Angular Discretization............................... 617 13.4.1 Discrete Transfer Method.................................................................. 617 13.4.2 Finite Volume Method...................................................................... 618 13.4.3 Boundary Element Method............................................................... 619 13.5 Zonal Method................................................................................................. 619 13.5.1 Exchange Area Relations.................................................................. 619 13.5.2 Zonal Formulation for Radiative Equilibrium.................................. 622 13.5.3 Developments for the Zone Method.................................................. 624 13.5.3.1 Smoothing of Exchange Area Sets.................................... 624 13.5.3.2 Other Formulations of the Zone Method........................... 624 13.5.3.3 Numerical Results from the Zone Method........................ 625 13.6 Additional Solution Methods.......................................................................... 628 13.6.1 Reduction of the Integral Order........................................................ 628 13.6.2 Spectral Methods............................................................................... 628 13.6.3 Finite Element Method (FEM) for Radiative Equilibrium............... 629 13.6.4 Lattice-Boltzmann Method............................................................... 632 13.6.5 Additional Information on Numerical Methods............................... 632 13.7 Comparison of Results for the Methods......................................................... 633 13.8 Benchmark Solutions for Computational Verification................................... 634 Homework Problems................................................................................................. 635 Chapter 14 The Monte Carlo Method.......................................................................................... 641 14.1 Introduction.................................................................................................... 641 14.2 Fundamentals of the Monte Carlo method..................................................... 641 14.2.1 Monte Carlo as an Integration Tool................................................... 642 14.2.2 Sampling from a Probability Density................................................644 14.2.3 Markov-Chain Monte Carlo..............................................................648 14.2.4 Uncertainty Quantification................................................................649 14.3 Monte Carlo for Surface-to-Surface Radiation Exchange............................. 650 14.3.1 Radiation between Black Surfaces.................................................... 650 14.3.2 Radiation between Nonblack Surfaces.............................................. 655 14.3.3 Wavelength-Dependent Properties.................................................... 658 14.3.4 Ray-Tracing.......................................................................................660 14.4 Monte Carlo for Enclosures Containing Participating Media........................ 661 14.4.1 Relations for Absorption and Scattering within the Medium........... 661 14.4.2 Net Radiative Energy Transfer between Volume and Surface Elements............................................................................... 663 14.4.3 Treatment of Wavelength-Dependent Properties.............................. 665 14.5 Variance Reduction Methods......................................................................... 667 14.5.1 Energy Partitioning........................................................................... 669 14.5.2 Importance Sampling........................................................................ 670 14.5.3 Reciprocity........................................................................................ 670 14.5.4 Quasi-Monte Carlo............................................................................ 672 14.6 Backwards (Reverse) Monte Carlo................................................................. 674 14.7 The Null-Collision Technique........................................................................ 677 14.8 Sensitivities and Nonlinear Monte Carlo....................................................... 677 14.9 Summary and Outlook................................................................................... 679 Homework Problems................................................................................................. 679

Contents

xv

Chapter 15 Conjugate Heat Transfer in Participating Media....................................................... 683 15.1 Introduction.................................................................................................... 683 15.2 Radiation Combined with Conduction...........................................................684 15.2.1 Energy Balance................................................................................. 685 15.2.2 Plane Layer with Conduction and Radiation..................................... 685 15.2.2.1 Absorbing–Emitting Medium without Scattering............. 685 15.2.2.2 Absorbing–Emitting Medium with Scattering.................. 689 15.2.3 Rectangular Region with Conduction and Radiation........................ 691 15.2.4 PN Method for Radiation Combined with Conduction...................... 693 15.2.5 Approximations for Combined Radiation and Conduction............... 696 15.2.5.1 Addition of Energy Transfer by Radiation and Conduction.................................................................. 696 15.2.5.2 Diffusion Method for Combined Radiation and Conduction......................................................................... 698 15.3 Transient Solutions Including Conduction..................................................... 704 15.4 Combined Radiation, Conduction, and Convection in a Boundary Layer..........707 15.4.1 Optically Thin Thermal Boundary Layer......................................... 707 15.4.2 Optically Thick Thermal Boundary Layer....................................... 708 15.5 Numerical Solution Methods for Combined Radiation, Conduction, and Convection in Participating Media.......................................................... 709 15.5.1 Finite-Difference Methods................................................................ 711 15.5.1.1 Energy Equation for Combined Radiation and Conduction......................................................................... 711 15.5.1.2 Radiation and Conduction in a Plane Layer...................... 712 15.5.1.3 Radiation and Conduction in a 2D Rectangular Region....... 714 15.5.1.4 Boundary Conditions for Numerical Solutions................. 719 15.5.2 Finite-Element Method..................................................................... 720 15.5.2.1 FEM for Radiation with Conduction and/or Convection... 721 15.5.2.2 Results from Finite-Element Analyses.............................. 722 15.5.3 Pitfalls of Iterative Methods.............................................................. 724 15.6 Results for Combined Radiation, Convection, and Conduction Heat Transfer........................................................................................................... 724 15.6.1 Forced Convection Channel Flows................................................... 725 15.6.2 Natural Convection Flow, Radiative Energy Transfer, and Stability...................................................................................... 730 15.6.3 Radiative Transfer in Porous Media and Packed Beds..................... 734 15.6.4 Radiation Interactions with Turbulence............................................ 734 15.6.5 Additional Topics with Combined Radiation, Conduction, and Convection......................................................................................... 734 15.7 Inverse Multimode Problems......................................................................... 735 15.8 Verification, Validation, and Uncertainty Quantification.............................. 735 Homework Problems................................................................................................. 736 Chapter 16 Near-Field Thermal Radiation.................................................................................. 741 16.1 Introduction.................................................................................................... 741 16.2 Electromagnetic Treatment of Thermal Radiation and Basic Concepts........ 746 16.2.1 Near-Field versus Far-Field Thermal Radiation................................ 746 16.2.2 Electromagnetic Description of Near-Field Thermal Radiation....... 747 16.2.3 Near-Field Radiative Energy Flux..................................................... 750

xvi

Contents

16.2.4 Density of Electromagnetic States.................................................... 750 16.2.5 Spatial and Temporal Coherence of Thermal Radiation................... 751 16.3 Evanescent and Surface Waves...................................................................... 751 16.3.1 Evanescent Waves............................................................................. 752 16.3.2 Surface Waves................................................................................... 752 16.4 Near-Field Radiative Energy Flux Calculations............................................. 756 16.4.1 Near-Field Radiative Energy Transfer in 1D Layered Media........... 756 16.4.2 Near-Field Radiative Energy Transfer between Two Bulk Materials Separated by a Vacuum Gap............................................. 759 16.5 Computational Studies of Near-Field Thermal Radiation.............................. 763 16.6 Experimental Studies of Near-Field Thermal Radiation................................ 766 16.6.1 Earlier Work...................................................................................... 766 16.6.2 Experimental Determination of Near-Field Radiative Transfer Coefficient......................................................................................... 768 16.6.3 Near-Field Effects on Radiative Properties and Metamaterials........ 770 16.7 Concluding Remarks...................................................................................... 773 Homework Problems................................................................................................. 774 Chapter 17 Radiative Effects in Translucent Solids, Windows, and Coatings............................ 777 17.1 Introduction.................................................................................................... 777 17.2 Transmission, Absorption, and Reflection for Windows................................ 778 17.2.1 Single Partially Transmitting Layer with Thickness D ≫ λ (No Wave Interference Effects)............................................. 779 17.2.1.1 Ray-Tracing Method.......................................................... 780 17.2.1.2 Net-Radiation Method....................................................... 780 17.2.2 Multiple Parallel Windows................................................................ 782 17.2.3 Transmission through Multiple Parallel Glass Plates....................... 784 17.2.4 Interaction of Transmitting Plates with Absorbing Plate.................. 785 17.3 Enclosure Analysis for Partially Transparent Windows................................. 787 17.4 Effects of Coatings or Thin Films on Surfaces.............................................. 790 17.4.1 Coating without Wave Interference Effects...................................... 790 17.4.1.1 Nonabsorbing Dielectric Coating on Nonabsorbing Dielectric Substrate............................................................ 790 17.4.1.2 Absorbing Coating on Metal Substrate............................. 791 17.4.2 Thin Film with Wave Interference Effects........................................ 792 17.4.2.1 Dielectric Thin Film on Dielectric Substrate.................... 792 17.4.2.2 Absorbing Thin Film on a Metal Substrate....................... 796 17.4.3 Films with Partial Coherence............................................................ 797 17.5 Refractive Index Effects on Radiation in a Participating Medium................ 797 17.5.1 Effect of Refractive Index on Intensity Crossing an Interface.......... 797 17.5.2 Effect of Angle for Total Reflection.................................................. 799 17.5.3 Effects of Boundary Conditions for Radiation Analysis in a Plane Layer.....................................................................................800 17.5.3.1 Layer with Nondiffuse or Specular Surfaces....................800 17.5.3.2 Diffuse Surfaces................................................................804 17.5.4 Emission from a Translucent Layer (n > 1) at Uniform Temperature with Specular or Diffuse Boundaries.......................... 805 17.6 Multiple Participating Layers with Heat Conduction.....................................807 17.6.1 Formulation for Multiple Participating Plane Layers........................808 17.6.2 Translucent Layer on a Metal Wall................................................... 810

Contents

xvii

17.6.3 Composite of Two Translucent Layers.............................................. 812 17.6.3.1 Temperature Distribution Relations from the Energy Equation................................................................ 813 17.6.3.2 Relations for Radiative Energy Flux................................. 814 17.6.3.3 Equation for the Source Function...................................... 816 17.6.3.4 Solution Procedure and Typical Results............................ 816 17.7 Light Pipes and Fiber Optics.......................................................................... 819 17.8 Final Remarks................................................................................................ 821 Homework Problems................................................................................................. 821 Chapter 18 Inverse Problems in Radiative Transfer.................................................................... 825 18.1 Introduction.................................................................................................... 825 18.2 Inverse Analysis and Ill-Posed Problems....................................................... 826 18.3 Mathematical Properties of Inverse Problems............................................... 827 18.3.1 Linear Inverse Problems................................................................... 827 18.3.2 Nonlinear Inverse Problems.............................................................. 833 18.4 Solution Methods for Inverse Problems......................................................... 836 18.4.1 Linear Regularization....................................................................... 836 18.4.2 Nonlinear Programming................................................................... 839 18.4.3 Metaheuristics...................................................................................840 18.4.4 Bayesian Methods............................................................................. 843 18.4.5 Artificial Neural Networks................................................................ 847 18.4.6 Is Least-Squares Minimization an Inverse Technique?....................848 18.5 Inverse Design Problems................................................................................848 18.5.1 Linear Inverse Design Problems....................................................... 849 18.5.2 Nonlinear Inverse Design Problems.................................................. 852 18.6 Parameter Estimation Problems..................................................................... 853 18.6.1 Problems Involving 1D Participating Media..................................... 854 18.6.2 Tomography....................................................................................... 854 18.6.3 Light-Scattering by Droplets and Particles....................................... 858 18.6.4 Inverse Crimes.................................................................................. 859 18.7 Summary........................................................................................................ 859 Homework Problems.................................................................................................860 Chapter 19 Applications of Radiation Energy Transfer............................................................... 865 19.1 Introduction.................................................................................................... 865 19.2 Solar Energy................................................................................................... 865 19.2.1 Solar Energy Conversion................................................................... 865 19.2.1.1 Central Solar Power Systems............................................. 865 19.2.1.2 Photovoltaic (PV) Conversion...........................................866 19.2.1.3 Solar Enhanced HVAC Systems........................................ 868 19.2.2 Radiation Transfer, Architecture, and Visual Comfort..................... 870 19.2.3 Astronomy, Astrophysics, and Atmospheric Radiation.................... 871 19.2.4 Solar Energy, Global Warming, and Climate Change...................... 872 19.3 Radiation from Combustion........................................................................... 872 19.3.1 Radiation from Non-Sooting Flames................................................ 873 19.3.2 Radiation from Sooting Flames........................................................ 873 19.3.3 Modeling of Furnaces and Burners................................................... 875 19.4 Radiation in Porous Media and Packed Beds................................................. 876

xviii

Contents

19.4.1 CSP Absorbers Using Porous Media................................................. 876 19.4.2 Porous Media Burners....................................................................... 876 19.5 Aerospace Applications.................................................................................. 876 19.5.1 Spacecraft Thermal Control.............................................................. 877 19.5.2 Radiation Energy Transfer in Rocket Nozzles.................................. 879 19.6 Advanced Manufacturing and Materials Processing..................................... 879 19.6.1 Radiation Transfer in Process Industries.......................................... 880 19.7 Biomedical Applications of Thermal Radiation............................................. 881 19.8 Conclusions..................................................................................................... 882 Homework Problems................................................................................................. 882 Appendix A: Conversion Factors, Radiation Constants, and Blackbody Functions.............. 887 Appendix B: Radiative Properties............................................................................................... 893 Appendix C: Catalog of Selected Configuration Factors.......................................................... 901 Appendix D: Exponential Integral Relations and Two-Dimensional Radiation Functions.......... 907 On-Line Appendices (at www.ThermalRadiation.net) A: Wide-Band Models B: Derivation of Geometric Mean Beam Length Relations C: Exponential Kernel Approximation D: Curtis-Godson Approximation E: The YIX Method F: Charts for CO2, H2O and CO Emittance G: Radiative Transfer in Porous and Dispersed Media H: Benchmark Solutions for Verification of Radiation Solutions I: Integration and Numerical Solution Methods J: Radiation Cooling K: Radiation from Flames L: Commercial Codes M: References to Reviews and Historical Papers N: Short Biographies and a History of Thermal Radiation O: Timeline of Important Events in Radiation P: Additional Homework Problems Q: Proposed One-Semester Syllabus References...................................................................................................................................... 913 Index...............................................................................................................................................997

Preface This Seventh Edition of Thermal Radiation Heat Transfer represents a major advancement on the Sixth Edition and revises the structure for easier reading. Engineering applications of radiative transfer remain the focus of the book and are updated throughout the text. Chapters added to the Fifth Edition, and augmented in the Sixth Edition (Inverse Methods; Electromagnetic Theory; Scattering and Absorption by Particles; and Near-Field Radiative Transfer) are retained, updated, and expanded. The book has also been updated with 350 added references to contemporary, stateof-the-art research, making the book a comprehensive and up-to-date reference for engineers, researchers, and students. There is a new emphasis throughout the book that connects the Second Law of Thermodynamics to radiative transfer, including its relationship to blackbody emission and energy transfer between surface and volume elements. The Second Law connects to a probabilistic interpretation for radiative properties, which also lays a foundation for Monte Carlo techniques, which have been consolidated and expanded in Chapter 14. Chapter 19 is added to examine contemporary applications of radiation. The text consists of three main sections. The first section (Chapters 1 through 3) describes the characteristics of blackbody radiation as well as the radiative properties of opaque materials as predicted by electromagnetic theory and obtained by measurements. The second section (Chapters 4 through 7) covers radiative exchange in enclosures containing transparent media, although heat conduction and convection at and through boundaries is discussed. The remaining chapters (Chapters 8 through 18) deal with electromagnetic theory, the radiative properties of gases and particles, and radiative transfer in the presence of participating materials. Chapters 14 (Monte Carlo Methods) and 18 (Inverse Problems in Radiative Transfer) are expanded and updated to include contemporary developments in these rapidly expanding fields. Older material has been removed from the main body of the textbook and incorporated as online appendices to aid in readability. These remain available at www.ThermalRadiation.net. Chapter 1 covers fundamentals of radiation, including blackbody radiation, radiative properties, and provides an introduction to the propagation of radiation in participating media and the radiative transfer equation. Chapters 2 and 3 define the radiative properties of surfaces, and compare measured properties to predictions from electromagnetic theory. A detailed discussion of electromagnetic wave theory and the behavior of waves at interfaces is deferred to Chapter 8, which has been completely revised, to aid in the flow of discussion. Chapter 8 also provides a theoretical foundation for later chapters on advanced topics concerning radiative transfer through participating media, scattering, and nanoscale heat transfer. The analysis of radiative exchange among multiple surfaces separated by transparent media is developed in Chapters 4 through 7. This discussion includes surfaces with black, gray, spectral, diffuse, and directional (including specular) properties. Radiative transfer between diffuse surfaces is calculated using configuration factors, which are discussed in Chapter 4. A short appendix (Appendix C) provides convenient analytical expressions for configuration factors that are useful for homework problems and basic geometries. (A greatly extended catalog of factors is posted at www. ThermalRadiation.net/indexCat.html, which constitutes one of the most complete and exhaustive sources of configuration factors available anywhere.) The catalog allows built-in numerical computation of values for many factors. Sections on numerical techniques are also updated, with new material on the emerging use of graphical program units (GPUs) for parallel processing in radiative transfer solutions. Chapter 9 introduces the radiative properties of gases, liquids, and nonopaque solids, including contemporary methods for calculating the spectral and total properties of gases. An updated section on the weighted sum of gray gases and the k-distribution methods is included, along with xix

xx

Preface

an introduction to problems where non-local thermodynamic equilibrium is present. The radiative properties of scattering media are discussed in Chapter 10, including absorption and scattering by particles and agglomerates of particles, as well as recent analytical advancements in this field. The development of radiative transfer in participating media is contained in the updated Chapters 11 through 13, with an emphasis on gaseous media. This material emphasizes the close relationship among the many differential methods (diffusion, two-flux, PN, SPN, MN, and others). Monte Carlo techniques are consolidated into Chapter 14, which has been extensively expanded and updated from the previous edition. This chapter includes a more rigorous probabilistic interpretation of the Monte Carlo method, as well as recent innovations to Monte Carlo algorithms that significantly reduce variance, improve computational efficiency, and expand the range of problems that can be tackled using this approach. Chapter 15 deals with multimode problems where radiation in participating media is important. Chapter 16 outlines the important developing area of near-field radiation effects within matter. Applications to radiation interactions with nanopatterned surfaces are included. Chapter 17 considers participating media that have a refractive index larger than unity, such as glass and multiple layers of glass, where reflections and interface effects need to be included. This material is extended to discuss the near-field radiative interactions in multiple nano-layers. Chapter 18 describes inverse problems arising in radiative transfer, with a focus on inverse design and parameter estimation. This chapter has been significantly expanded to include a pedagogical discussion of the mathematical properties of inverse problems, the algorithms used to solve these problems, and a survey of historical and recent applications of inverse analysis in radiative transfer. Chapter 19 is new in this edition; it outlines contemporary applications of thermal radiation transfer to areas such as climate change, spacecraft atmospheric re-entry, advanced manufacturing, solar absorption within solar photovoltaic cells, radiative cooling, applications in architecture, central solar power systems, among others. This textbook has long emphasized a historical perspective of radiative transfer, including the pivotal roles played by individual scientists and engineers and the broader historical context of their contributions. Understanding how this discipline evolved, and its impacts on other areas of science and engineering (e.g., quantum mechanics, astrophysics, electromagnetism, scientific computing), is key to developing a holistic understanding of radiative transfer and the current state-of-the-art. In this edition, each chapter begins with biographies of pioneers in the area of radiative heat transfer. Some of the analytical and semianalytical techniques presented in previous editions have become less important due to advancements in computer hardware and access to high performance computing. Material on the exponential wide-band properties of gases, geometric-mean beam lengths, and certain approximations to solving the radiative transfer equation (exponential kernel approximation, Curtis–Godson approximation, YIX method) are now less used in practice. Since many widely used software packages such as MATLAB® have built-in and optimized routines for numerical integration, material on numerical integration methods has also been moved to the online appendix. Descriptions of radiation in porous media and radiative cooling technology have also been moved to the dedicated site www.ThermalRadiation.net/book.html, where it is available for download. Teachers using this book as a text should be aware that a full solution manual is available from the publisher. In addition, continually updated Errata, the appendix material moved from earlier editions, and the full catalog of configuration factors are available at www.ThermalRadiation.net/ book.html. This website also provides a history of the early development of the theory and practice of thermal radiation energy. This includes biographies of more pioneers in radiative transfer and a timeline of major events in radiation. Further, a list of important reviews and seminal references is provided for students and researchers wishing to expand their knowledge in specific areas. A detailed syllabus keyed to book sections is provided in the online Appendix Q of the book at www.ThermalRadiation.net. The syllabus is based on a 15-week semester with two 90-minute classes per week, with 27 lecture sessions and three sessions left for examinations, class evaluation, etc. This one-semester syllabus by necessity ignores many important subjects such as the emerging

Preface

xxi

areas of micro-nanoscale effects and inverse problems. The skipped material could be incorporated into a second course, particularly one that concentrates on radiative transfer in participating media. This Seventh Edition continues the tradition of providing both a comprehensive textbook for those interested in the study of thermal radiative transfer and serving as a source of important references to the literature for researchers. The authors wish to recognize Prof. Hakan Ertürk and his students at Boğaziçi University, Istanbul, for coordinating the editing and revision of the homework solution manual for this edition. We further acknowledge the contributions of Mathieu Francoeur to the original version of Chapter 16 in the Fifth Edition of the book. Since then, several of our students (Azadeh Didari in particular) have contributed to the description of near-field effects. John R. Howell M. Pinar Mengüç Kyle J. Daun MATLAB® is a registered trademark of The MathWorks, Inc. For product information, please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 USA Tel: 508-647-7000 Fax: 508-647-7001 E-mail: [email protected] Web: www.mathworks.com

Acknowledgments We sadly note the passing of Robert Siegel in September 2017. Bob was a founding inspiration for this text and was a major contributor through the first five editions before choosing to enjoy a wellearned retirement. He will always be missed in the radiation and heat transfer communities. To my wife Susan and to my children and grandchildren who have kept me lively enough to continue to contribute to this Seventh Edition. Also, to all those students and colleagues who have made this effort enjoyable and gave a feeling of being useful through interactions and interchanges with them. Jack Howell To my parents and to my children who gave me the pleasure of life all along, and to all my mentors, students, and colleagues who helped me to gather each and every piece of knowledge and wisdom gathered here. M. Pinar Mengüç To my wife, Caryn, and to my parents Jim and Lynda for introducing me to science, and to Scott, Terry, Jack, and Greg, for helping me along the path. Kyle Daun And, from all of us, our sincere thanks to Jonathan Plant and Kyra Lindholm who ushered this work through the contracting, editing, printing, and distribution process along with the CRC/Taylor & Francis team who made the final version possible.

xxiii

Authors John R. (Jack) Howell received his academic degrees from Case Western Reserve University (then Case Institute of Technology), Cleveland, Ohio. He began his engineering career as a researcher at NASA Lewis (now Glenn) Research Center (1961–1968) and then took academic positions at the University of Houston (1978–1988) and the University of Texas at Austin, where he remained until his retirement in 2012. He is presently Ernest Cockrell, Jr., Memorial Chair Emeritus. He pioneered the use of the Monte Carlo method for the analysis of radiative heat transfer in complex systems that contain absorbing, emitting, and scattering media. Jack has concentrated his research on computational techniques for radiative transfer and combined-mode problems for over 60 years. Recently, he has adapted inverse solution techniques to combined-mode problems and to radiation at the nanoscale. His awards in heat transfer include the ASME/AIChE Max Jakob Medal (1998), the ASME Heat Transfer Memorial Award (1991), the AIAA Thermophysics Award (1990), the Poynting Award from JQSRT, (2013), and the 2018 International Center for Heat and Mass Transfer Luikov Medal. He is an elected member of the US National Academy of Engineering (2005) and a foreign member of the Russian Academy of Sciences (1999). He is an Honorary Fellow of the American Society of Mechanical Engineering (ASME). M. Pinar Mengüç completed his BSc and MS in mechanical engineering from the Middle East Technical University (METU) in Ankara, Turkey. He earned his PhD in mechanical engineering from Purdue University in 1985. He joined the faculty at the University of Kentucky the same year. and was promoted to the ranks of associate and full professor in 1988 and 1993, respectively. In 2008, he was named an Engineering Alumni Association professor. Later that year he joined Özyeğin University in Istanbul as the founding head of the Mechanical Engineering Department. In 2009 he established the Center for Energy, Environment, and Economy (CEEE) which he still directs. He is currently one of the three editors-in-chief of the Journal of Quantitative Spectroscopy and Radiative Transfer and an editor of Physics Open. He has organized several conferences, including five international symposia on radiative transfer as the chair or co-chair. His research expertise includes radiative transfer in multidimensional geometries, light scattering-based particle characterization and diagnostic systems, nanoscale transport phenomena, near-field radiative transfer, applied optics, and sustainable energy and its applications to buildings and cities. He has guided more than 60 MS and PhD students and postdoctoral fellows both in the United States and in Turkey and has six assigned and two pending patents. He is the co-author of more than 150 journal papers, more than 200 conference papers, two books and several book chapters. He was the co-owner of a start-up company on particle characterization and served as the director of the Nanoscale Engineering Certificate Program at the University of Kentucky. He has been the Turkish delegate to the European Framework programs (FP-7 and Horizon 2020) on energy-related topics and a Turkish delegate to the United Nations Framework Convention on Climate Change (COP). He is a fellow of both the ASME and ICHMT and received the ASME Heat Transfer Memorial Award in 2018. He was elected to the Science Academy, Turkey in 2017, and currently serves on its executive committee. Kyle J. Daun received his BSc from the University of Manitoba (1997), his MASc from the University of Waterloo (1999), and his PhD from the University of Texas at Austin (2003). From 2004 to 2007 he worked at the National Research Council Canada as a Natural Sciences and Engineering Research Council of Canada postdoctoral fellow and later as a research officer. While at NRC he investigated radiation heat transfer in solid oxide fuel cells and helped develop and improve laser-based combustion diagnostics. In 2007 he returned to the University of Waterloo, where he is now a Professor xxv

xxvi

Authors

in the Department of Mechanical and Mechatronics Engineering. Professor Daun’s research interests include inverse problems in radiative transfer, laser-based combustion diagnostics, nanoscale transport phenomena, and heat transfer in manufacturing. He has published over 250 contributions, including 90 refereed journal papers. Professor Daun is a fellow of the Humboldt Foundation, a DFG Mercator Fellow, and a Fellow of the American Society of Mechanical Engineering. In 2010 he received the JQSRT Ray Viskanta Young Scientist Award. Robert Siegel received his ScD in mechanical engineering from Massachusetts Institute of Technology in 1953. For two years, he worked at General Electric Co. in the Heat Transfer Consulting Office analyzing the heat transfer characteristics of the Seawolf submarine nuclear reactor. He joined NASA in 1955 and was a senior research scientist at the Lewis/Glenn Research Center where he researched in heat transfer for more than 50 years until he retired in 1999. Apart from his work in thermal radiation, he studied reduced gravity effects on heat transfer, and developed the first counterweighted drop tower in the late 1950s. His pioneering work on reduced gravity effects, particularly its effect on nucleate and film boiling, led to the design of the Glenn Zero Gravity Research Facility, built in 1966 and now a National Historic Test Facility. Dr. Siegel was inducted into the inaugural class of the NASA Glenn Research Center Hall of Fame in 2015 as one of the NASA “Giants of Heat Transfer.” He wrote some 150 technical papers and gave graduate heat transfer courses as an adjunct professor at three universities. He was a fellow of the American Society of Mechanical Engineering (ASME) and the American Institute of Aeronautics and Astronautics (AIAA). He received the ASME Heat Transfer Memorial Award in 1970, the NASA Exceptional Scientific Achievement Award in 1986, both the Space Act Award and the AIAA Thermophysics Award in 1993, and the ASME/AIChE Max Jacob Memorial Award in 1996. Bob died in September 2017.

List of Symbols This is a consolidated list of symbols for the entire text. Some symbols that are only used in a local development are defined where they are used and are not included here. The symbols used in radiative transfer have evolved from many different disciplines where radiation is important. This has led to the same quantity being defined by a variety of symbols and to multiple quantities designated with the same symbol. The symbols listed here are typical of those used for engineering heat transfer and follow, where possible, those adopted formally by the major heat transfer journals (Howell 1999). The study of radiative transfer combined with conduction and convection involves many types of applications and hence requires definitions for many different quantities and parameters. There is an insufficient number of convenient symbols that can be used, so some symbols must be used for multiple quantities. Attention has been devoted to making the definition clear from the context of its use. Some typical units have been indicated. Some care must be observed as quantities may have multiple units, such as a spectral bandwidth that can be in terms of wavelength, wave number, or frequency; some of these are designated by (mu), meaning multiple units. A length could, for example, be in nm, μm, cm, m, or other units, so that only a typical unit is shown. Some quantities are nondimensional; these are designated by (nd). Quantity in reflectivity relations (nd); spacing between surfaces, m; thickness, m; coefficient in phase velocity of electromagnetic wave (nd) a0 Autocorrelation distance of surface roughness, m akj Matrix elements (mu or nd) a Matrix of elements akj a−1 Inverse matrix (mu or nd) A Surface area, m2; absorptance of a translucent plane layer (nd) A, B, C, D Field amplitude coefficients A R Aspect ratio of rectangle (nd) m Al Coefficients in spherical harmonics expansion Aij Equivalent spectral line width (mu, wavelength, wave number, frequency) Al , A Equivalent spectral bandwidth (mu) B Spacing, m; width of a base, m; a dimension, m; coefficient in phase velocity of electromagnetic wave (nd); quantity in reflectivity relations (nd) B Pressure broadening parameter (nd); length dimension, m B Magnetic induction vector, Wb/m2 c Speed of electromagnetic radiation propagation in medium other than a vacuum, m/s c0 Speed of electromagnetic radiation propagation in vacuum, m/s c, cp, cv Specific heat, J/(kg⋅K) C A coefficient or constant (mu or nd); clearance between particles, m; particle volume fraction (nd) C1 Constant in Planck’s spectral energy distribution (Table A.4), W ⋅ μm4/(m2⋅sr) C2 Constant in Planck’s spectral energy distribution (Table A.4), μm ⋅ K C3 Constant in Wien’s displacement law (Table A.4), μm ⋅ K Ci Concentrations of component i in a mixture (nd) Ckj Elements of matrix C (mu or nd) CCO2 , CH2O Pressure-correction coefficients (nd) d Number of diffuse surfaces (nd); a dimension, m dA* Differential element on the same surface area as dA, m2 a

xxvii

xxviii

D Df Dp D e E En Ew Eff E f(ξ) F F0→λ F g g(kη) gg , gs g G G G h hv H H Iλ I i, j, k i Iˆ Im j J J Jr k kB K Kij l l, m, n l1, l2, l3 lm L

List of Symbols

Thickness of a layer or plate, m; a dimension, m; diameter of tube or hole, m; diameter of atom or molecule, m; number of dimensions (nd) Fractal dimension (nd) Particle diameter, m Electric displacement, C/m2 Energy level of a quantized state or photon, J Emissive power (usually with a subscript), W/m2; amplitude of electric intensity wave, N/C; overall emittance of a translucent layer (nd); the quantity (1−ε)/ε, (nd) Exponential integral (Appendix D) (nd) Weighted error (nd) Absorption efficiency for a grooved directional absorber (nd) Electric field vector, V/m Frequency distribution of events occurring at ξ (nd) Configuration factor (nd); objective function in optimization (mu); separation variable in Equation (12.52) Fraction of total blackbody intensity or emissive power in spectral region 0 to λ (nd) Transfer factors in enclosure (nd) Gravitational acceleration, m/s2 Cumulative distribution function in k-distribution method, Equation (9.47) Gas–gas and gas–surface direct exchange areas, m2 Weyl component of the dyadic Green’s function, m Incident radiative flux onto a surface, W/m2 Green’s function Dyadic Green’s function, 1/m Planck’s constant, J⋅s (Table A.1); height dimension, m; convective heat transfer coefficient, W/(m2⋅K); enthalpy, J/kg Volumetric heat transfer coefficient, W/(m3⋅K) Wave amplitude of magnetic intensity, C/(m⋅s); convection-radiation parameter (nd) Magnetic field vector, C/(m⋅s) Spectral radiation intensity, W/(m2⋅μm⋅sr) Radiation intensity, W/(m2⋅sr) Unit vectors in x, y, z coordinate directions (nd) Number of increments (nd) Source function, W/m2 Imaginary part Emission coefficient, Equation (1.59) Radiosity; outgoing radiative flux from a surface; auxiliary variational function (mu or nd); number of increments (nd) Current density vector, A/m2 Random current density vector, A/m2 Thermal conductivity, W/(m⋅K); absorption index for electromagnetic radiation, m−1; wave vector (= k′ + ik″) rad/m Boltzmann constant (Table A.1), J/K Kernel of integral equation (mu or nd) Finite element function defined in Equation (13.165) A length, m, or a dimensionless length (nd) Direction cosines for normal direction used in contour integration method (nd) Direction cosines for rectangular coordinates designated as x1, x2, x3 (nd) Mean penetration distance, m Length dimension, m

List of Symbols

Le Le, 0 L M Mkj n n n N NCR Nx, Ny Nu p pˆ P P, Q, R P0 Pe Pl m Pr q  q qc ql qr qr Q Qa Q s QoI r re rij r R Re sλ s sˆ sjgg s j sk S S  S Sc

xxix

Mean beam length of gas volume, m Mean beam length for the limit of very small absorption, m Ladenberg–Reiche function (nd) Mass of a molecule or atom; molecular weight Minor of matrix element akj (mu or nd) Index of refraction of a lossless material c0/c (nd); ratio n2 /n1 in a few equations (nd); index in summation (nd); sample index (nd); number of a surface (nd); ordinate directions in Sn approximation (nd); normal direction (nd) Complex refractive index n−ik (nd) Unit normal vector (nd) Number of surfaces in an enclosure (nd); number of sample bundles per unit time, s−1; number of particles per unit volume, m−3; density of electromagnetic states, s/(m3⋅rad) Conduction-radiation parameter (Stark number) k k /4sT 3 (nd) Number of x and y grid points (nd) Nusselt number hD/k (nd) Partial pressure of gas in mixture, atm; probability density function TM-polarized unit vector Perimeter, m; probability density function (nd); total pressure of gas, atm; probability Functions used in contour integration, m−1 Pressure of 1 atm Effective broadening pressure (nd) Associated Legendre polynomials (nd) Prandtl number cpμf/k (nd) Energy flux, energy per unit area and per unit time, W/m2 Internal energy generation per unit volume, W/m3 Energy per unit area per unit time resulting from heat conduction, W/m2 Radiative energy flux in spectral band l (mu) Net radiant energy per unit area per unit time leaving a surface element, W/m2 Radiative flux vector, W/m2 Energy per unit time, W; ray origin point Absorption efficiency factor (nd) Scattering efficiency factor (nd) Quantity of interest Radial coordinate, m; radius, m Electrical resistivity, N⋅m2⋅s/C2 = Ω ⋅ m Fresnel’s reflection coefficient at interface i–j Position vector (mu or nd) Radius, m; overall reflectance of a translucent plane layer or a group of multiple layers (nd); random number in range 0 to 1 (nd) Reynolds number Dumρf /μf (nd); real part Scattering cross-section, m2 Unit vector in S direction (nd) TE-polarized unit vector Surface–gas direct exchange area in zonal method Surface–surface direct exchange area in zonal method Coordinate along the path of radiation, m; distance between two locations or areas, m; surface, m2; number of sample energy bundles per unit time, s−1 Poynting vector, W/m2; energy per unit area and time, W/m2 Dimensionless internal energy source (nd) Collisional line intensity (mu)

xxx

Sdv Sij Sk - j Sn Sr St t tij t t(S) t j -k T Tl Tw1, Tw2 u uk u , um u, v U, V U U(x, y) Uλ V Vγ w W x, y, z X X, Y, Z Yl m Z

List of Symbols

Number of energy bundles absorbed per unit time in volume dV (nd) Spectral line intensity (mu) Geometric-mean beam length from Ak to Aj, m Two-dimensional radiation integral functions (Appendix D) (nd); singular values from matrix decomposition Dimensionless radiative heat source (nd) Stanton number, Nu/(Re⋅Pr) (nd) Time, s Fresnel’s transmission coefficient at interface i–j Dimensionless time (nd) Transmittance of a medium (nd) Geometric-mean transmittance (nd) Absolute temperature, K; overall transmittance of a plane layer or a group of multiple layers (nd) Mean transmission in a spectral band (nd) Temperatures of walls 1 and 2, K Fluid velocity, m/s; the variable Xα/ω (nd); energy density, J/m3 Spectral band parameter hcηk /kT (nd) Mean fluid velocity, m/s Velocity, m/s Orthogonal matrices resulting from singular value decomposition Total number of unknowns for an enclosure (nd); radiant energy density, J/m3 Approximate solution in finite-element method (mu) Spectral radiant energy density, J/(m3⋅µm) Volume, m3; voltage signal, V = N/C Volume of element γ in zoning method, m3 Width, m; energy carried by the sample Monte Carlo bundle, J; weighting factors (nd) Weighting function in finite-element method (nd); width dimension, m Coordinates in Cartesian system, m Coordinate, m, or dimensionless coordinate (nd); mass path length, g/m2 Optical or dimensionless coordinates (nd) Normalized spherical harmonics (nd) Height of surface roughness, m

Greek Symbols α α(S) α, α0 α, β, γ α, δ, γ a j-k β βi βR γ γ2

Absorptivity (nd); thermal diffusivity, m2/s; coefficient in soot scattering correlations Absorptance of a medium (nd) Regularization parameter in Tikhonov regularization Direction cosines (nd) Angles measured from normal direction in contour integration method, rad Geometric-mean absorptance, m−1 Extinction or attenuation coefficient κ + σS, m−1; angle in x-y plane, rad; coefficient of volume expansion, K−1; the parameter πγc/δ (nd) Coefficients in shape function in the finite-element method (nd) Rosseland mean attenuation coefficient, m−1 Electrical permittivity, C2/(N ⋅ m2); polynomial coefficients (nd); half-width of a spectral line (mu) Variance in a statistical solution (nd)

List of Symbols

Γ δ δkj δ(r′−r″) Δ Δε Δφ ε ε(S) ϵ εd εh ερ γ ζ η θ θo Θ ϑ κ κe κi κP κR κλ λ λm μ μf ν ξ ξ, η ξC ρ ρe ρij ρf ρM ρS ρ*, ρ0 σ σe σs

xxxi

Number of gas elements (nd); factors in Gebhart’s method (fraction of energy leaving one surface that is absorbed by another) (nd); separation variable in Equation (12.53); function in integral equation (Equation (12.63)), W/m2 Propagation angle in medium, rad; boundary layer thickness, m; average spacing between lines in absorption band (mu); penetration distance of evanescent waves, m Kronecker delta; = 1 when j = k; = 0 when j ≠ k Dirac delta function Distance above a radiating body Correction for spectral overlap (nd) Intermediate function in alternating direction implicit method (mu) Emissivity of a surface (nd) Emittance of a medium (nd) Dielectric constant, γ/γ0 (can be complex) Deformation parameter in T-matrix method Eddy diffusivity for turbulent flow, m2/s Porosity (nd) Electrical permittivity Arbitrary direction (nd); the quantity C2/λT (nd) Fin efficiency (nd); Blasius similarity variable (nd); wave number, 1/λ, m−1 Polar (cone) angle measured from the surface normal, rad Scattering angle, rad Separation variable in Equations (12.52) and (12.53); mean energy of a Planck oscillator, J Dimensionless temperature T/Tref (nd) or T(σ/qmax)1/4 Absorption coefficient, m−1 Effective mean absorption coefficient, m−1 Incident mean absorption coefficient; absorption coefficients in weighted sum of gray gases emittance model, m−1 Planck mean absorption coefficient, m−1 Rosseland mean absorption coefficient, m−1 Spectral absorption coefficient, m−1 Wavelength, m Wavelength in a medium other than vacuum, m Magnetic permeability, N/A2; dimensionless fin conduction parameter (nd); the quantity cos θ (nd); the quantity SSc/δ (mu) Fluid viscosity, kg/m•s Frequency, c0/λ0 = c0/λ = c/λm, s−1 = Hz Length coordinate, m; parameter πD/λ for scattering (nd); parameter SSij/2πγc for equivalent line width Dimensionless coordinates (nd) Clearance parameter for particle separation criteria, πC/λ Reflectivity (nd); gas density, kg/m3 Electric charge density, C/m3 Reflectivity at interface i–j Density of a fluid, kg/m3 Density of a material, kg/m3 Specular reflectivity, (nd) Distances between points, m or (nd) Stefan–Boltzmann constant, Equation (1.34) and Table A.4, W/(m2⋅K4) Electrical conductivity, (Ω·m)-1; C2/(N·m2·s) Scattering coefficient, m−1

xxxii

σ0 τ τD ϕ Φ Φ d χ χe ψ ψ1 ψ(3) ψb ω Ω Ωi F

List of Symbols

Root-mean-square height of surface roughness, m Roughness correlation length, m; optical thickness (nd); transmittance (Chapter 17) (nd) Optical thickness for path length D (nd) Circumferential (azimuthal) angle, rad; dimensionless function, Equation (12.71) (nd) Scattering phase function (nd); shape function in finite-element method (nd); function in integral equation (mu or nd) Viscous dissipation function, J/(kg⋅m2) Angle of refraction, rad Susceptibility (nd) Dimensionless heat flux (nd); stream function Temperature jump coefficient (nd) Pentagamma function (nd) Dimensionless energy flux for black walls (nd) Albedo for scattering (nd); angular frequency, rad/s; width of spectral band (mu); band width parameter, cm−1 Solid angle, sr Incident solid angle, sr Transfer factor, Equations (5.34)–(5.36)

Subscripts α α0, α1, … abs A b b, black bi-d c cond c CO2 D d, dif d1, d2 d-h D e eq evan E f fc fd F g h H

Absorbed; absorption; absorber; apparent value Coefficients Absorbed Property of surface A Base surface; at the base of a fin; bottom Blackbody condition Bidirectional Evaluated at cutoff wavelength; corrected values; collision broadening; at a collector (absorber) plate; cylinder; cross-section Conduction Coating Carbon dioxide Disk Diffuse Evaluated at differential elements dA1, dA2 Directional hemispherical Doppler broadening Emitted or emitting; entering; environment; element of area; energy input; electrical; effective value At thermal equilibrium Evanescent wave Electric Fluid Free convection Fully developed Final Gas Hemispherical Magnetic

List of Symbols

H2O i i,j I j, k l L LO m m, m′ m, n mc mh mP max min M n nd N, S, E, W o p prop P r rad ref R s sol sub S t TE TM TO u w x, y, z η λ Δλ λ1→λ2 λT ν ω 0 1, 2 ⊥ || ∩

xxxiii

Water vapor Incident; inner; incoming Energy states Initial Property of surface Aj or Ak Spectral band index; layer index Long wavelength region Longitudinal optical Mean value; in a medium; maximum value; metal Outgoing and incoming angular directions Number of identical semitransparent plates in a system Metal on cold side Metal on hot side Evaluated at midpoint Corresponding to maximum energy; maximum value; maximum refraction angle Minimum Maximum value; material Normal direction; natural broadening Nondiffuse Directions in Figures 13.13 and 13.14 Outer; outgoing; evaluated in vacuum Projected; particle Propagating wave Planck mean value; perimeter; point in discrete ordinates method Reflected; reduced temperature; reservoir; radiative Radiation Reference value Radiator; radiating source; Rosseland mean value; radiative Surface of a sphere; sun; scattering; surroundings; source; solid; specular Solar Substrate Short-wavelength region Transmitted; top Transverse electric Transverse magnetic Transverse optical Uniform conditions Wall; window Components in x, y, z directions Wave number dependent Wavelength (spectrally) dependent For a wavelength band Δλ In wavelength region from λ1 to λ2 Evaluated at λT Frequency dependent Angular frequency dependent In vacuum Surface or medium 1 or 2 Perpendicular component Parallel component Hemisphere of solid angles

xxxiv

List of Symbols

Superscripts i n o s (0), (1), (2) + − ~ *

Inside of an interface nth time interval Inlet value; outside of an interface Specular exchange factor Zeroth-, first-, or second-order term; designation for moments Along directions having positive cos θ Along directions having negative cos θ (Overbar) averaged over all incident or outgoing directions; mean value; complex value Dimensionless quantity Complex conjugate

1

Introduction to Radiative Transfer Wilhelm Carl Werner Otto Fritz Franz (Willy) Wien (1864–1928) studied mathematics and physics at the Universities of Göttingen and Berlin. In 1886, he completed his doctorate with a thesis on diffraction and how materials impact the color of refracted light. In 1893, he announced the Law of Displacement, which states that the product of wavelength and absolute temperature for a blackbody is constant. In 1896, he proposed a formula which described the spectral composition of radiation from an ideal body, which he called a blackbody. This work earned Wien the 1911 Nobel Prize in Physics, and later impelled Max Planck to propose quantum effects to bring Wien’s distribution into agreement with experimental measurements. Max Planck (1858–1947) arguably laid the basis for quantum mechanics and was one of the forerunners of modern physics. He originally developed his blackbody spectral distribution based on the observation that the denominator in classically derived distributions such as that of Wien needed to be slightly smaller to fit the experimental data. His attempts to explain the theoretical basis of his proposed spectral energy equation led him to hypothesize the existance of quantized energy levels, a concept that was at odds with all of classical physics and thermodynamics. He was forced to accept Boltzmann’s statistial interpretation of the Second Law of Thermodynamics, as opposed to the more classical deterministic view.

Energy radiates from all types of matter under all conditions and at all times. The emission of thermal radiation arises from random fluctuations in the quantized internal energy states of the emitting matter. Temperature is a measure of the internal energy level of matter, and the nature of the fluctuations can be related to an object’s temperature. Once the energy is radiated, it propagates as an electromagnetic (EM) wave. If the waves encounter matter, they may partially lose their energy and increase the internal energy of the receiving matter. This is called absorption. The amount of emitted and absorbed radiation are functions of the physical and chemical properties of the material as well as its internal energy level, as quantified by its temperature. An EM wave can also undergo scattering as it propagates through a heterogeneous medium, e.g., a porous ceramic, a layer of freshly fallen snow or even the molecules in a gas. As waves encounter these scattering centers, they are reflected, refracted, or diffracted, or any combination of these phenomena, which redirects the wave without increasing the internal energy of the scatterer. The propagation and scattering of EM waves, including all effects of reflection, refraction, transmission, polarization and coherence, are governed by the Maxwell equations, as we discuss in Chapters 8, 10, and 16. Since all matter emits and absorbs radiation under all conditions, there is always radiative transfer of energy, even within an isothermal system. If two objects are at different temperatures, there will be net radiation energy transfer between them, even if there is no matter between the objects. An obvious example is radiation emitted by the sun, which travels through the vacuum of space and is partially absorbed and scattered upon entering the earth’s atmosphere. A significant part of this radiation reaches the earth’s surface, where again, some is absorbed and some is reflected (Figure 1.1). These three distinct physical mechanisms – emission, absorption, and scattering – are all spectral in nature; that is, they all depend on the wavelength or frequency of the EM wave. 1

2

Thermal Radiation Heat Transfer

Global warming, for example, is driven by spectral variations in the radiation emitted by the sun, the radiation emitted by the earth, and the fact that the atmosphere absorbs and scatters radiation differently at different wavelengths. (See Section 19.2.4 for further discussion of this.) The strength and spectral distribution of emitted energy is correlated with the internal energy state of the emitter. The spectral distribution of emitted radiation is connected to the temperature of the emitting matter through the distribution derived by Max Planck in 1901 (Section 1.9.8). For large objects containing many atoms and molecules, the internal energy of this matter can be described using the principles of statistical thermodynamics. Statistical averaging allows for the definition of average thermodynamic and thermophysical properties, including internal energy and temperature. We can use relatively simple laws derived within a classical framework, such as Planck’s distribution and Kirchhoff’s law, to explain these phenomena. Each emitting object exchanges radiant energy with the others that it can “see.” A fraction of the incident energy that it receives from other bodies is absorbed and converted into its internal energy. As discussed earlier, the quantitative calculation of the exchange of EM energy between matter at different energy levels constitutes radiative transfer – the focus of this book. We start our discussion with a section on the importance and fundamentals of radiation transfer, and then provide a largely empirical discussion that ties the traditional radiation exchange formulation in enclosures with the radiative transfer equation (RTE) in participating media. In this context, the word “participating” refers to a medium such as the earth’s atmosphere (Figure 1.1) that alters the propagating radiative energy because of absorption, emission, or scattering mechanisms. We introduce the fundamentals of blackbody radiation, define radiation intensity, and outline the phenomena of emission, absorption, and propagation, including reflection, refraction, and scattering. Throughout the book, both the fundamentals and applications of radiative transfer are emphasized. Several practical engineering problems are discussed, ranging from combustion chambers to solar radiators to nanoscale devices. Chapter 19 is devoted to such applications. Radiation can be emitted through many mechanisms. For thermal radiation, emission is caused by spontaneous fluctuations in quantized energy levels, and the average magnitude of these levels is evidenced by the temperature of matter. This phenomenon is also called incandescence. Radiation emission also occurs due to non-thermal phenomena. For example, some chemical reactions can stimulate emission. This phenomenon is called chemiluminescence, e.g., the blue color of a natural gas flame, and bioluminescence (fireflies and foxfire). These do not constitute thermal radiation, since the transitions are caused by chemical and biochemical reactions and not

FIGURE 1.1  Atmospheric scattering effects on radiation from the sun.

Introduction to Radiative Transfer

3

induced by random thermal transitions or the temperature of the emitter. Neither does electroluminescence, which is induced in some materials by electrical inputs (e.g., light-emitting diodes, or LEDs). These other phenomena are not part of thermal radiation transfer and will only be mentioned in a few cases of special interest.

1.1 THERMAL RADIATION AND THE NATURAL WORLD It is difficult to overstate the importance of thermal radiation to the natural phenomena that shape our daily lives, starting with the sun. The sun converts hydrogen fuel into helium and thermal radiation. The immense gravitational body forces acting toward the center of the sun are balanced by the radiation pressure (Section 1.1.3) acting outward; when these forces become out-of-balance, stars can collapse under their own gravitational force. Radiation is the only mechanism for transporting thermal energy through a vacuum, and thus is the means by which energy is conveyed from the sun to the planets in the solar system. Solar radiation evaporates water from the oceans, irrigates the land masses, and induces currents in the oceans. Climatic zones and seasonal variations are caused by the differences in the solar angle of incidence at different latitudes and as the earth tilts on its axis. Solar irradiation is converted to chemical energy via photosynthesis, which is the basis for almost all plant and animal life on earth. The temperature of the earth, which allows the chemical reactions that support organic life, is determined by a balance between absorbed solar radiation and emitted radiation. This balance is changing due to increased absorption of solar irradiation because of anthropogenic pollutants such as CO2, soot particles, and agglomerates. On a clear day the sky is blue due to the scattering of solar irradiation by nitrogen and oxygen molecules in the air, as well as by very small dust and pollen particles. Most animals on earth see by virtue of reflected solar radiation, which, not coincidentally, has a peak emission intensity aligned with the center of the visible spectrum (approximately 550 nm, green light). Interestingly, Figure 1.2 shows that the visible spectrum also corresponds to the wavelengths at which water

FIGURE 1.2  Representative data for the absorption coefficient of pure water collected from various sources. (Redrawn from Irvine, W.M. and Pollack, J.B., “Infrared optical properties of water and ice spheres”. Icarus, 8, 324, 1968).

4

Thermal Radiation Heat Transfer

absorbs the least amount of EM radiation, and therefore is most transparent; this fact is critical for aquatic species who possess eyes, although creatures who reside in the ocean depths rely on other types of sensory organs. Note that the short-wavelength (blue) part of the visible spectrum is weakly absorbed compared with the longer-wavelength (red) part, which accounts for the blue shade common to underwater scenes.

1.2 THERMAL RADIATION IN ENGINEERING Thermal radiation is also important in many engineering applications, particularly at high temperatures. For conduction and convection, energy transfer between two objects depends approximately on the temperature difference between them. For natural (free) convection, or when variable property effects are included, the energy transferred can be a function of the difference between T1n and T2n, where the power n may be slightly larger than 1 and can reach 2. Thermal radiation transfer that takes place between two distant bodies depends on the difference between the fourth power of their absolute temperatures (n = 4). If the temperature-dependent material properties are accounted for in the calculations, the radiative flux can be proportional to an even higher power n of absolute temperatures, and consequently the importance of radiation transfer is significantly enhanced at high temperatures. Because of this, radiation contributes substantially to energy transfer in furnaces, combustion chambers, fires, rocket plumes, spacecraft atmospheric entry, high-temperature heat exchangers, and during explosions of chemicals. Infrared processing furnaces are used to anneal metals, dry paint, and cure composites. Thermal radiation is also used to monitor and control the temperature of these processes via sensing by infrared cameras and pyrometers. Radiation heat transfer is also important in the boilers and furnaces used by the petrochemical industry for hydrocarbon distillation and cracking. Thermal radiation plays an especially important role in power generation. At the time of writing, approximately 80% of the world’s energy is generated via the combustion of fossil fuels, and radiation heat transfer within boilers and combustion chambers is a key factor in both thermal efficiency and emitted pollutants. Understanding the directional and spectral principles of radiative transfer is key to the design and operation of efficient fossil-fuel burning combustion systems, which means less energy use and minimum effects on the environment. As a case in point, natural gas boilers may use porous ceramic heaters, which preheat the reactants by conduction and radiative transfer through the semitransparent porous ceramic. Increasingly, combustion-based devices are being displaced by more environmentally sustainable solar energy systems. Understanding the spectral and directional nature of solar radiation is crucial in optimizing the design of these systems. It is a commonly held misconception that, since radiation dominates heat transfer at high temperatures, it must be unimportant at lower temperatures. This is far from true. For example, when designing heating, ventilation, and air conditioning (HVAC) systems for buildings, engineers must consider both convection and radiation between a person and their surroundings, which are often similar in magnitude. Furthermore, HVAC systems function to counteract undesirable seasonal heating of buildings in the summer and cooling in the winter, which is becoming more pronounced with climate change. Because of their large energy consumption, HVAC systems themselves are one of the growing sources of greenhouse gas emission. While architecturally appealing, windows consume about 4% of the total world generated energy, because HVAC systems must counteract both transmitted solar radiation in the summer and emission in the winter. Accordingly, thermal radiation is a crucial element in developing more efficient building energy systems and making more energy-efficient architectural choices. At even lower temperatures, radiation is a key element of spacecraft thermal control and cryogenic insulation systems. These systems are often under vacuum, so radiation is the only mode of energy transfer. Spacecraft undergo extreme variation in radiative heating and cooling from the sun, the earth, and deep space. A well-known example that highlights the importance of spacecraft thermal control is Skylab, a space station that orbited the earth between 1973 and 1979, although

5

Introduction to Radiative Transfer

it operated for only 24 weeks after its launch in May 1973. During its launch, Skylab lost a crucial radiation shield intended to protect the space station from solar irradiation, and a replacement made from Mylar was urgently provided to the space station 11 days after the incident. Waste heat from onboard electronics must also be radiated to space through specially designed radiant emitter panels. Deep-space probes (e.g., Voyager 1 and 2, Pioneer 10 and 11) also incorporate radiant emitters into radioisotope-based thermoelectric generator systems that are required to power the onboard electronics because solar irradiation is too weak at the outer reaches of the solar system to provide power from solar cells. The above examples represent only a small subset of the many engineering systems that feature thermal radiation heat transfer. More details are given in Chapter 19.

1.3 THERMAL RADIATION AND THERMODYNAMICS The laws that govern thermal radiation largely descend from the First Law of Thermodynamics (conservation of energy) and the Second Law of Thermodynamics. The latter can be stated in several ways, but broadly concerns the increase in entropy principle, or, alternatively, the direction of time. Application of the First Law is usually quite straightforward. Energy must be conserved across interfaces, i.e. applying the First Law over a control surface, which envelops no mass, shows that the radiation incident upon a surface must exactly balance the absorbed, reflected, and, in the case of semitransparent surfaces, transmitted radiation. By extending the control volume to include mass, it is possible to relate radiative transfer to other modes of heat transfer. The net rate at which radiation is absorbed by a surface can be linked to the rate at which the temperature of a surface rises, or the rate at which heat is conducted through the surface. In the case of a participating medium, the difference between absorbed and emitted radiation at any given point becomes a volumetric radiative energy source term in a coupled radiation-advection-diffusion equation. Concepts based on the Second Law are just as important, if more nuanced. In its most fundamental form, energy transfer can be envisioned as the way by which systems reach thermal equilibrium with their surroundings. When a system is at thermal equilibrium with its surroundings, it is in a state of uniform thermal potential, and there can be no net, spontaneous energy transfer across the system boundaries without some work input. This is the Clausius statement of the Second Law of Thermodynamics, which is used for many proofs and discussions throughout this text. The thermal potential of a system is defined by its temperature, which is a macroscopic averaging of the internal energy of the individual microscopic states (microstates) of matter, e.g., the random motion of individual atoms and molecules induced by temperature, over a given volume and time. The connection between macroscopic temperature and microscopic energy states requires the assumption of some probability density function (PDF) pi of their energy states. This is often a Boltzmann distribution,

æ E pi µ exp ç - i è kBT

ö ÷ (1.1) ø

where Ei is the energy of state i and k B = 1.381 × 10 −23 J/K is Boltzmann’s constant (cf. Table A.1). The Boltzmann distribution arises from the state of local thermodynamic equilibrium (LTE) that occurs when the constituent atoms and molecules have undergone a sufficient number of random interactions so that their energy states have reached a maximum state of disorder subject to the constraints of an average value (i.e. the temperature). Moreover, at LTE, energy is shared, on average, equally between different types of microstates; this is the concept of equipartition. Because of this requirement, LTE cannot be assumed for very fast processes, such as femto- or attosecond laser beam heating, or in regions with extremely large gradients in properties such as near-shock layers, as these conditions do not allow the microstates sufficient opportunity to interact with each

6

Thermal Radiation Heat Transfer

other. In addition, it is not possible to assume LTE when an object is so small that the constituent number of atoms or molecules does not allow a statistically meaningful averaging. For most practical engineering systems, however, the LTE assumption is valid as it pertains to thermal radiation, since the time scales needed for a system to reach local radiative equilibrium are much faster than those typical of the macroscopic processes, e.g., temperature change. Another Second Law concept that impacts radiative transfer is microscopic reversibility. Loosely stated, microscopic reversibility requires a process that has been reduced to its simplest and deterministic form to be reversible. This is because irreversibility arises from random processes, and, in the absence of randomness, the process must be reversible. For example, suppose a gas molecule undergoes adiabatic scattering (reflection) at a perfectly smooth surface; its incident direction and energy can be exactly determined from its scattered direction and energy by applying conservation of linear momentum. Because this process is deterministic, it must also be reversible, so, were the scattered molecule to “reverse course,” it would recreate the incident trajectory. Were the scattering influenced by the random thermal motion of the surface, it would only be possible to calculate a distribution of probable incident trajectories, and the interaction would be irreversible.

1.4 NATURE OF THE GOVERNING EQUATIONS While the equations governing conductive, convective, and radiative transfer are all derived from the First and Second Laws of Thermodynamics, the equations that describe radiation are distinct from those that govern conduction and convection, reflecting the fundamental differences in the underlying physics. In the case of a material having a thermal conductivity k, the conduction heat flux in direction S is given by Fourier’s Law, qcond (S ) = -k



dT (1.2) dS

where qcond has units of W/m2. (Although this law was derived empirically, it is a consequence of the Second Law of Thermodynamics and follows from the increase in entropy principle.) Within a purely conductive medium, this equation can be applied to each surface of the infinitesimal cube shown in Figure 1.3a to derive an energy balance that quantifies the time-dependent energy change in cubical element dV due to a volumetric energy source term q ̇ (which can result from internal electrical heating, nuclear or chemical reactions, or in this book, usually by net thermal radiation sources or sinks) is

rcP

¶T  . (1.3) dV + Ñ(kÑT )dV = qdV ¶t

When thermal radiation is present, q ̇ is a volumetric energy source or sink and may correspond to the rate of radiative energy either deposited in (absorbed, positive) or rejected by (emitted, negative) the elemental cube. As we discuss later in this book, in this scenario q ̇ is obtained from the solution of the radiative transfer equation (RTE). At steady state, Equation (1.3) simplifies to

Ñ(kÑT ) = q. (1.4)

With appropriate boundary conditions, Equations (1.3) and (1.4) can be solved for the temperature distribution in a conducting medium. Key observations are that: (i) conduction requires an intervening medium; and (ii) the local energy flux depends only on the local temperature distribution, and therefore the equations governing the temperature distribution are differential (and, at steady state, elliptical). When these equations are discretized into matrix form, they have a sparse “banded” structure that is convenient to solve. A similar analysis can be made for convection, again

7

Introduction to Radiative Transfer

FIGURE 1.3  Terms for conduction and radiation energy balances: (a) Energy conduction terms for a volume element in a solid, and (b) radiation terms for an enclosure filled with radiating material.

demonstrating that the energy balance depends only on the conditions in the immediate vicinity of the location being considered. In contrast, radiative energy can be transmitted between distant area and volume elements without requiring an intervening medium. As an example, consider a heated enclosure with surface A and volume V, filled with radiating material such as a hot gas, a cloud of hot particles, or molten glass, as shown in Figure 1.3b. If qAdA is the rate of radiant energy arriving at dA1 from a surface element dA and the energy rate qVdV arrives at dA1 from a volume element dV, then the rate of radiative energy from the entire enclosure arriving at dA1 is

ò

ò

Qrad (dA1 ) = qA dA + qV dV . (1.5) A

V

Equation (1.5) highlights that, in contrast to the equations that govern conduction and convection, which are differential and elliptical in space, those governing radiation are integral in nature. Equation (1.5) does not explicitly account for the radiation that is absorbed between the emitting element and dA1; when this is done, the governing equations become integrodifferential. Discretizing integral-type equations produces a full matrix, reflecting the fact that the incident radiation on any point on the surface is influenced by the radiation leaving every other point on the surface, and, in the case of a participating medium, every point in the volume. The directional nature of radiation makes numerical solution procedures computationally expensive compared with similarly sized conduction and convection problems.

8

Thermal Radiation Heat Transfer

In addition to the complexity caused by the directional nature of radiation, we must also consider its spectral aspect. Radiation transfer calculations must be performed at each wavelength or frequency, and the total radiative energy is then determined by integrating overall wavelengths or frequencies of the EM spectrum. This is complicated by the fact that radiative properties of surfaces, particles, and gases usually vary substantially over the wavelength (or frequency) spectrum, necessitating a finely resolved spectral quadrature. The boundary conditions may be specified in terms of total energies, but detailed spectral energy flows at the boundaries are usually unknown. This means that solutions must be obtained iteratively to satisfy the total energy flow conditions at the boundaries. In addition to mathematical complexities, uncertainty in the radiative properties also hampers the accuracy of radiative transfer calculations. This shortcoming is usually a major bottleneck and underscores the difficulty of performing radiation analysis in engineering practice. Radiative properties for solids and surfaces depend on many variables such as surface roughness and degree of polish, purity of the material, and thickness of any thin film (e.g., oxide) or paint coating on the surface. The coating thickness, its material properties, and its temperature may affect the radiation absorbed by the material or radiation leaving the surface, and, moreover, in many processes, the surface state may change with time. Unfortunately, most available experimental data for surface radiative properties do not include the necessary information on the coatings, making their use guesswork at best. For radiating gases, radiative properties display even more complicated and spectrally irregular behavior. These properties are functions not only of the constituent molecules, but also of gas pressure and temperature. In addition, the atmospheric or combustion gases that attenuate radiation are usually laden with various particles, including soot, dust, pollen, fly ash, or char particles. The particles absorb and scatter radiation depending on their relative size with respect to incoming radiation and show different characteristics according to their size and shape distributions and material properties. Even though some of these physical characteristics can be approximated for engineering applications, their relative importance needs to be established carefully for each problem to be solved. Even if all properties are precisely known, the solution of governing integrodifferential equations of radiation transfer in a spectrally absorbing, emitting, and scattering medium is not trivial and requires systematic study. In this text, we try to shed some light on these complex problems.

1.5 ELECTROMAGNETIC WAVES VS PHOTONS Radiative energy propagation is often considered from one of two viewpoints: The flow of discrete quanta of energy, or the propagation of electromagnetic waves. The conflict between these viewpoints originates with the well-known and bitter dispute between Isaac Newton and Christiaan Huygens in the 17th century. Newton favored the interpretation of light as discrete quanta (called “corpuscles”), while Huygens saw light as the propagation of longitudinal waves, much like soundwaves. Natural philosophers were divided along these lines of thinking throughout the 19th century, with a gradual shift toward the wave interpretation based on observations of wave interference, diffraction, polarization, and the changing speed of light in different media. In 1865 James Clerk Maxwell successfully synthesized earlier work by Faraday, Ørsted, Coulomb, Ampere, Gauss, and others and described the propagation of electromagnetic waves using a set of 20 equations. These equations were then distilled by Oliver Heaviside (who also introduced the properties of electrical permittivity and magnetic permeability) into the four equations we know as “Maxwell’s equations” today. The wave–photon debate was reignited with Hertz’s observation of the photoelectric effect in 1885, which appeared to be incompatible with the strictly wave interpretation of light. Max Planck struggled to explain his heuristically derived distribution of emitted energy from an ideal surface. He found that assuming discrete energy states, rather than a continuum of states, was a necessary

9

Introduction to Radiative Transfer

ingredient and introduced the at the time controversial idea of quantized energy states in 1901. Einstein suggested the existence of “light quantum” (Lichtquanten), which both explained the photoelectric effect and allowed Planck to correctly interpret the basis of the spectral distribution of blackbody radiation, which is often regarded as the moment of the birth of quantum theory. The equivalence of these two competing perspectives was finally postulated by de Broglie in 1924, who showed that all matter could be interpreted as either particles or waves. The photon concept introduces a simplification that allows the treatment of radiation in simple terms and is particularly convenient for understanding processes involving molecular gases. Nevertheless, it fails to explain some of the fundamental physics predicted from full-fledged quantum mechanics that includes polarization, coherence, and tunneling; these phenomena are important in a range of key applications, including those pertaining to radiation at the nanoscale. There is a growing awareness that the conventional interpretation of “photons” in the Einstein sense is a grossly oversimplified and inaccurate model, and a more sophisticated approach (based on quantum electrodynamics, QED) is needed to describe the true nature of radiation. Moreover, concepts like “photons” and “photon bundles,” which are, in reality, particle-based mathematical constructs for solving the governing radiative transfer equations (and are thus sometimes called “RTE photons”), are often conflated with “QED photons” that are more like “quantized disturbances in the local electromagnetic field,” as opposed to massless particles propagating at the speed of light (Mishchenko 2014a). To avoid confusion between these concepts, there is a growing trend of referring to RTE photons as something else, e.g., energy “packets” or “bundles.” Nevertheless, we will adopt the traditional concept and nomenclature of photons, since they simplify the treatment of many practical engineering problems, and provide convenient interpretations of key phenomena, with the understanding that these interpretations can only be taken so far. The concept of “QED photons” is discussed in Chapter 16. The classical EM wave view of radiation, in most cases, provides conservation equations like those obtained from quantum mechanics. Within the framework of wave theory, EM waves are comprised of coupled oscillations in the electric and magnetic fields perpendicular to the direction of wave propagation. (In other words, EM waves are transverse waves, as opposed to the longitudinal waves envisioned by Huygens.) The Maxwell equations describe this wave propagation in a simple, beautifully symmetric way. EM waves propagate with the speed of light in vacuum; indeed, light itself is an EM wave within the narrow visible range of the spectrum. The light speed c0 in vacuum (as designated by the subscript “0”) is 2.9979 × 108 m/s. This speed is related to the speed of light in matter through the refractive index n, where n = c0/c. The refractive index, which is the real part of the complex index of refraction, n = n - ik, is greater than unity for almost all natural materials within the visible and the infrared spectra. For example, for glass, it is about 1.5 in the visible spectrum; for water, it is about 1.33 and for most gases, it is close to unity. The imaginary component k is the absorption index and is a measure of how an EM wave is attenuated during propagation through an absorbing medium. The real part of the index of refraction can sometimes be less than unity within a narrow wavelength interval, which is the indication of strong absorption. This does not mean that speed of the propagating waves would be greater than c0. The propagation velocity is determined using a group velocity (Section 8.5.1). Any property used for radiative transfer calculations must be spectral in nature, specified at a given wavelength, λ, or frequency, n. These quantities are related to the wave speed by

c = ln (1.6)

while the energy of the wave is given by

E = hn = hc / l (1.7)

where h = 6.626 × 10 −34 J·s is Planck’s constant (Table A.1 in Appendix A).

10

Thermal Radiation Heat Transfer

As the EM radiation propagates from one medium to another, its wavelength changes as a function of the index of refraction, as n = λ0/λ = (c0/n)/(c/n). The frequency of the oscillations, ν, however, remains the same in each medium; for that reason, the use of frequency (in units of s−1 or Hz) to describe the spectral behavior of matter is common practice. When wavelength is used, it is customary to specify the wavelength in a vacuum, λ0 = c0/n. The wavenumber may also be used to represent spectral effects, especially in quantitative spectroscopy applications, and is defined as η0 = 1/λ0. Finally, in some applications, it is convenient to work in units of angular frequency, ω = 2πn, having units of rad/s. In this case, Equation (1.7) is rewritten as E = w where  = h / 2p is the reduced Planck’s constant. Wavelength is usually given in units of micrometers (μm) or nanometers (nm), where 1 μm is 10 −6 m or 10 −4 cm and 1 nm is 10 −9 m. Sometimes, the units are expressed in Angstroms (Å), where 1 μm = 104 Å or 1 nm = 10 Å. Wavenumber is customarily given in units of cm−1; the wavelength spectrum can be converted to wave number by using the multiplier 10,000/λ where λ in μm yields η in cm−1. Spectral units of electron volts (eV) are sometimes used, where eV = hn = hc/λ. The conversion hence yields 1 eV = 1.602 × 10 −19 J. Additional conversion factors and units for radiative transfer applications are provided in Tables A.2 and A.3 in Appendix A. The EM spectrum is shown in Figure 1.4. Radio and television wavelengths may be as long as thousands of meters. At the other end of the spectrum, the wavelengths of cosmic rays can be as small as 10 −14 m. Within this vast range spanning more than 18 orders of magnitude, visible light is confined to 400–740 nm (i.e. 0.4–0.7 μm). However, this can differ for individuals, as some individuals (and other species) may “see” beyond this range.* For most practical purposes, thermal radiation is concerned with a wavelength range of 100 nm to 100 μm. Different physical mechanisms produce EM waves at different frequencies. Some of these mechanisms include atoms or molecules undergoing transitions from one energy state to another lower energy state. These transitions may occur spontaneously or be induced by the presence of an external radiation field. Detailed discussions can be found in the physics literature; they are outside the scope of the present text.

FIGURE 1.4  Spectrum of EM radiation (wavelength in vacuum). * What most people perceive as “red light” corresponds to the narrow wavelength range of roughly 620–740 nm; green light is roughly 510–530 nm and violet is 400–430 nm.

Introduction to Radiative Transfer

11

1.6 RADIATIVE ENERGY EXCHANGE AND RADIATIVE INTENSITY Before further discussing the fundamentals of radiative heat transfer, it is instructive to provide an overview of a few physical problems where radiative exchange is important. A common objective in radiative transfer calculations is to determine the amount of energy leaving one surface and reaching another after traveling through an intervening space. If the intervening space is a vacuum, the problem is simply one of geometry. On the other hand, if the space is filled with a participating medium, such as the mixture of water vapor and carbon dioxide common to combustion applications, the medium may selectively absorb radiation at certain wavelengths, which attenuates the radiation as it travels along its path. These same gases also emit at infrared wavelengths, which augments the radiation along the path. The extent of augmentation depends on the local temperature and gas concentration, which can vary along the path. Particles in the medium (e.g., soot, char) may also absorb and emit radiation. In addition, they may scatter the radiation in a different direction. A complete radiative transfer prediction requires the analyses of all these interactions, which are highlighted in Figure 1.5. These interactions will be used as the basis of our discussion throughout this chapter. Emitted radiation that leaves a surface is always spectral and directional in nature, meaning that it may change both as a function of wavelength and direction (as in the case of surface 1, in Figure 1.5). If the medium is participating, the radiative energy will change along the path from surface 1 to surface 2. The small cylindrical volume element shown in Figure 1.5 will act as a control volume for conservation of radiative energy, which is expressed only in a specific direction Ω as indicated. Once the radiative energy is incident on the second surface, it will be either absorbed (hence, the energy will be converted to thermal energy in the body) or reflected to travel in another direction, for example, toward surface 3. For example, the arrow in Figure 1.5 could represent the propagation of a laser beam through a semitransparent medium, or it may represent emission from surface 1. Alternatively, we can consider the emitting body as the sun; then, the problem is the same as the propagation of solar rays through the earth’s atmosphere, which includes absorbing and scattering gases and particles. In this case, the temperature of the air is relatively low, so the radiative energy emitted by the atmosphere is small compared to the incident solar radiation or that reflected by the earth. As a more practical engineering problem, we can visualize a three-dimensional (3D) configuration with several surfaces. This can simply be a kitchen oven where the air inside does not reduce the strength of the radiation propagating between the surfaces. The problem, however, is still complicated, as the exchange of energy from one surface element to another needs to be carefully accounted for in order

FIGURE 1.5  General schematic for radiative transfer discussion.

12

Thermal Radiation Heat Transfer

to determine the energy balance everywhere. For a combustion chamber, the flame or hot gases or particles inside will absorb, emit, and scatter the energy incident on them. In this case, energy may be emanating from any point in the medium or on any surface; therefore, radiation propagating in any direction carries the radiative fingerprints of all particles, gas molecules, and surfaces in the enclosure. Radiative transfer needs to be determined after accounting for all directional effects, making the governing equation a complicated one. It is obvious that in order to quantitatively analyze any of these problems, we need to properly define and quantify the radiative properties, including those that govern emission and absorption of radiation by surfaces, as well as absorption, emission, and scattering in the media. For this, we start with the concepts of solid angle and radiation intensity, which are fundamental to all radiation analyses. After that, we consider emission and absorption by an idealized surface, called a “blackbody.” Following that, we discuss absorption, emission, and scattering in the medium, which are defined using absorption and scattering coefficients and the scattering phase function. These effects are combined into the radiative transfer equation (RTE) which describes the conservation of radiative energy along a line-of-sight. The RTE is the backbone of the discussion provided throughout this book.

1.7 SOLID ANGLE To develop an understanding of radiative energy propagating from one surface to another, consider the spectral radiative energy propagating along a direction S and incident on a small control volume dV at S(x,y,z). This energy is confined to a small conical region, which is called the solid angle, Ω having units of steradians. The solid angle is defined in 3D space and is analogous to a planar angle. The planar angle is the ratio of the arc length to the distance from the apex to the base, while the solid angle is the ratio of the projected area dAn to the square of the chord length r from the apex to the area, dW =



dAn . (1.8) r2

It is often convenient to work in spherical coordinates with an origin centered on elemental surface area dA, as shown in Figure 1.6. Direction is defined by the zenith and azimuthal angles, θ and ϕ, respectively, where θ is measured from the direction normal to dA. The angular position for ϕ = 0 is arbitrary but is usually measured from the x-axis. The dAn is a differential area that is normal to a vector in the θ, ϕ direction. An infinitesimal solid angle dΩ can be related to infinitesimal increments dθ and dϕ. As shown in Figure 1.6, the area element on the hemisphere is given by dAn = r2sinθdθdϕ, so that dW =



dAn = sinqdqdf (1.9) r2

and p/2 2p



ò dW = ò ò sin qdqdf = 2p (1.10) Ç

q=0 f=0

where the symbol “∩” denotes a hemispherical domain. Thus, a complete hemisphere subtends a solid angle of 2π steradians when viewed from its base.

1.8 SPECTRAL INTENSITY We can use this definition of a solid angle, along with the coordinate system shown in Figure 1.6, to quantify the spectral and directional nature of the radiation emitted by dA, and radiation incident

13

Introduction to Radiative Transfer

FIGURE 1.6  Hemisphere showing solid angle relations. Schematic for the definition of solid angle.

upon dA. The spectral intensity leaving a surface in the Ωo = (θo, ϕο) direction and at a wavelength λ is defined by

I l ,o ( W o ) º

dQl,o (1.11) dA cos qo dldWo

where dΩo is an infinitesimal solid angle centered on Ωο, dAcosθo is the projected area of dA in the Ωo direction, dλ is an infinitesimal wavelength interval centered on λ and dQλ,o is the rate at which radiation leaves dA into dΩο over dλ. These quantities give spectral intensity units of W/(m2·μm·sr). The intensity leaving a surface consists of emitted intensity, as well as the reflected portion of the spectral intensity incident upon the surface. The incident intensity is given by

I l ,i ( W i ) =

dQl,i (1.12) dA cos qi dldW i

where dQλ,i is the rate that radiation is incident upon dA within an infinitesimal solid angle dΩi centered on the Ωi = (θi, ϕi) direction, and for wavelengths contained within an infinitesimal interval dλ centered on λ. While these equations define spectral intensity emitted by and incident upon a surface, it is also possible to define spectral intensity for an arbitrary location and direction within a volume, as described later in this chapter. At first glance, it may be confusing as to why intensity is defined relative to the surface area projected in the direction of radiation propagation, as opposed to simply the surface area. The reasons for this will become clear later. It is also important to appreciate the meaning of the infinitesimal quantities: In a probabilistic sense, the actual probability of a photon being emitted at any given wavelength λ is zero, just as the probability of encountering a person who is exactly 2.000000… m tall is zero, while the probability of finding someone who has a height between 2 m +/− a small amount is small, but finite. Accordingly, the infinitesimal quantity Iλ,o·dA·cosθ·dλ·dΩ·dt is proportional to the infinitesimal, but nonzero probability that a photon will be emitted by the surface element having an energy contained between dλ, within the solid angle dΩo, and over a time interval dt. We will revisit this probabilistic interpretation throughout the text.

14

Thermal Radiation Heat Transfer

Equations (1.11) and (1.12) define the spectral intensity leaving or incident upon a surface at a given wavelength, λ. It is also possible to consider the total intensity by integrating the spectral quantify for all wavelengths. For example, the total intensity leaving the surface in direction Ωi is given by ¥



ò

I i ( W i ) = I l,i ( W i ) dl. (1.13) 0

1.9 CHARACTERISTICS OF BLACKBODY EMISSION The key aspect that distinguishes thermal radiation from other types of radiant emission is that it is caused by matter undergoing random, spontaneous transitions in quantized energy levels due to its finite temperature. In the context of statistical thermodynamics, these fluctuations can be explained by the Second Law; both the rate of fluctuations and the size of the “jumps” between states increases with temperature. What, precisely, is the relationship between emitted intensity, wavelength, and temperature? The answer to this question had a profound impact on the history of modern physics. In principle, emission from a given body is a function of the material properties, the temperature, and direction:

I le = f (T , l, q, f) (1.14)

but it is not clear at this point exactly how these parameters influence radiant emission. The quest to clarify this relationship mirrors the concept of the idealized thermodynamic cycle. Motivated by the quest to improve the efficiency of steam engines, in 1824 Sadi Carnot laid the foundations for the Second Law of Thermodynamics by asking the question “What is the maximum amount of work that can be extracted from a source of heat?” The answer is a cycle comprised of reversible processes, which is independent of the working fluid. An analogous question in thermal radiation could be “What is the greatest rate at which a surface can emit radiation for a given wavelength and temperature?” The concept of such an ideal body, which is the so-called blackbody, is a thermodynamic limiting case, as is the Carnot cycle. In contrast to real surfaces, we can also show that the blackbody absorbs all incident radiation, and that its properties are independent of direction. As an ideal emitter and absorber, it serves as a hypothetical standard with which the properties of real surfaces can be compared. (We defer the discussion of the radiative properties of real surfaces, and how they depart from the idealized case of a blackbody, to Chapters 2 and 3). The blackbody, conceived by Gustav Kirchhoff in 1860, derives its name from observing that good absorbers of incident visible light appear black to the eye. However, except for the visible region, the eye is not a good indicator of absorption capability over the entire wavelength range corresponding to thermal radiation. For example, a surface coated with white oil-based paint is a very good absorber for infrared radiation, although it is a poor absorber for the shorter wavelengths of visible light, as evidenced by its white appearance. Only a few materials, such as carbon black, carborundum, platinum black, gold black, some specialty formulated black paints on absorbing substrates, and some nanostructured surfaces, approach the blackbody in their ability to absorb radiant energy. Vantablack (derived from “Vertically Aligned Nanotube Array black”), consists of a “forest” of carbon nanotubes grown through chemical vapor deposition, and absorbs 99.96% of incident visible radiation, very closely approaching blackbody characteristics in the visible spectrum. Other nanostructured near-black surfaces have also been proposed (Zhao and Zhang 2017; Zhao, Yu, and Zhang 2019). We further elucidate the important characteristics of a blackbody below.

Introduction to Radiative Transfer

15

1.9.1 Perfect Emitter Consider a blackbody object at a uniform temperature contained within a perfectly insulated, evacuated enclosure of arbitrary shape whose walls are also blackbodies at a uniform temperature, but one that is different from that of the object (Figure 1.7). After some time, the blackbody object and enclosure will reach thermal equilibrium. In equilibrium, the blackbody must emit as much energy as it absorbs, otherwise there would be net energy transfer between two objects at the same temperature, which violates the Clausius statement of the Second Law of Thermodynamics. Because the blackbody is, by definition, absorbing the maximum possible radiation incident from the enclosure at each wavelength and from each direction, it must also be emitting the maximum total amount of radiation. This deduction becomes clear by considering any less-than-perfect absorber, which must consequently emit less energy than the blackbody to remain in equilibrium. This example highlights a commonly held misconception about thermal radiation. It is often assumed that the Second Law prohibits radiative transfer between objects at thermal equilibrium. This is not the case; the Second Law requires that there be no net energy transfer between objects at thermal equilibrium. Since all objects emit and absorb thermal radiation, there must always be radiative transfer between objects, even those in a state of thermal equilibrium. The fact that a body emits radiation even when in thermal equilibrium with its surroundings is called Prevost’s law.

1.9.2 Radiation Isotropy of a Black Enclosure Now, consider the isothermal enclosure with black walls and an arbitrary shape in Figure 1.7; this time, we move the blackbody to another position and rotate it to another orientation. The blackbody object must still be at the same temperature because the whole enclosure remains isothermal. Consequently, the blackbody must emit the same amount of radiation as before. To be in equilibrium, the body must still receive the same amount of radiation from the enclosure walls. Thus, the total radiation received by the blackbody is independent of the orientation or position throughout

FIGURE 1.7  Enclosure geometry for derivation of blackbody properties.

16

Thermal Radiation Heat Transfer

the enclosure. Hence, the radiation traveling through any point within the enclosure is independent of the position or direction. This means that the radiation filling the black enclosure is isotropic.

1.9.3 Perfect Emitter for Each Direction and Wavelength Consider an area element on the surface of the black isothermal enclosure and an elemental blackbody within the enclosure. Some of the radiation emitted by the surface element strikes the elemental body at a random direction. All this radiation, by definition, is absorbed by the blackbody. To maintain thermal equilibrium and isotropic radiation throughout the enclosure, the radiation emitted back into the incident direction must be equal to that received. Since the body is absorbing the maximum radiation from any direction, it must be emitting the maximum in any direction. Because the radiation filling the black enclosure is isotropic, the radiation received and hence emitted in any direction by the enclosed black surface, per unit projected area normal to that direction, must be the same as that in any other direction. By considering the detailed balance of absorbed and emitted energy in each small wavelength interval, we also reach the conclusion that the blackbody is a perfect emitter and absorber at every wavelength.

1.9.4 Total Radiation into a Vacuum If the enclosure temperature is altered, the enclosed blackbody temperature also changes and becomes equal to the new temperature of the enclosure. The absorbed and emitted energy by the blackbody are again equal, although the new equilibrium temperature differs from the original temperature of the enclosure. Because a blackbody absorbs, and hence emits, the maximum amounts of radiation corresponding to its temperature, the characteristics of the surroundings do not affect its emissive behavior. Hence, the total radiant energy emitted by a blackbody in a vacuum is a function only of its temperature. As mentioned earlier, the Second Law of Thermodynamics forbids spontaneous net energy transfer from a cooler to a hotter surface. If the radiant energy emitted by a blackbody increased with decreasing temperature, we could build a device to violate this law. Without considering the proof in detail, we observe that radiant energy emitted by a blackbody must increase with temperature. The total radiant energy emitted by a blackbody is therefore proportional to a monotonically increasing function of temperature.

1.9.5 Blackbody Intensity and Its Directional Independence As we defined previously, the intensity is the fundamental quantity to describe radiation propagating in any direction. The spectral intensity of a blackbody is denoted as Iλb, where the subscripts denote, respectively, that a spectral quantity is being considered and that the properties are for a blackbody. The emitted spectral intensity from a surface is defined as the radiative energy emitted per unit time per unit infinitesimal wavelength interval around the wavelength λ, per unit elemental projected surface area normal to (θ, ϕ) direction and into the elemental solid angle centered on the direction (θ, ϕ). The directional independence of Iλb can be shown by considering a spherical isothermal blackbody enclosure of radius R with a blackbody element dA at its center (Figure 1.8a). The enclosure and the central element are in thermal equilibrium, and therefore, all radiation throughout the enclosure is isotropic. Consider radiation in a wavelength interval dλ emitted by an element dAs on the enclosure surface and traveling toward the central element dA (Figure 1.8b). The emitted energy in this direction per unit solid angle and time is written Iλb,n dAs dλ. The normal spectral intensity of a blackbody is used because the energy is emitted normal to the black wall element dAs of the spherical enclosure. The amount of energy per unit time that strikes

17

Introduction to Radiative Transfer

FIGURE 1.8  Energy exchange between element of enclosure surface and element within enclosure. (a) Black element dA within black spherical enclosure, (b) energy transfer from dA to dAp, and (c) energy transfer from dAp to dAs.

dA depends on the solid angle that dA occupies when viewed from dAs. This solid angle is the projected area of dA normal to the (θ, ϕ) direction, dAp = dAcosθ, divided by R2. Then, the energy absorbed by dA is Iλb,n(θ, ϕ)dAs(dAcosθ/R2)dλ. The energy emitted by dA in the (θ, ϕ) direction and incident on dAs in solid angle dAs/R2 (Figure 1.8c) must equal that absorbed from dAs, or the equilibrium would be disturbed. Hence,



æ dA I lb (q, f)dAp ç 2s è R

ö æ dAs ÷ dl = I lb (q, f)dA cos q ç R 2 ø è

ö ÷ dl ø

æ dA cos q ö = I lb,n (q, f)dAs ç ÷ dl 2 è R ø

(1.15)

which shows that

I lb (q, f) = I lb,n = I lb . (1.16)

The intensity of radiation from a blackbody, Iλb, as defined based on its projected area, is independent of the direction of emission. Neither the subscript “n” nor the (θ, ϕ) notation is needed for the intensity emanating from a blackbody. Since the blackbody is always a perfect absorber and emitter, its properties are independent of its surroundings. Hence, these results are independent of both assumptions used in the derivation of a spherical enclosure and thermodynamic equilibrium with the surroundings. The isotropic nature of blackbody intensity means that the amount of radiation emitted by a blackbody that is intercepted by another surface depends only on the solid angle subtended by the blackbody as viewed by the intercepting surface, and thus it is its projected area, and not the true area of the surface, that matters in radiative exchange between surfaces. For example, the radiation intercepted by an observer from a blackbody trapezoid would be the same as the radiation intercepted by a tilted rectangle that looks like a trapezoid to the observer. Indeed, the observer would have no way of distinguishing between these two scenarios. Some exceptions exist for most of the blackbody “laws.” They are of minor importance in almost any practical engineering situation, but need to be considered when conditions do not favor the

18

Thermal Radiation Heat Transfer

assumption of local thermodynamic equilibrium, e.g., extremely rapid transients or extreme temperature gradients (e.g., near shock layers). For example, if the transient period is of the order of the timescale of the process that governs the emission of radiation from the body in question, then the emission properties of the body may lag the absorption properties. In such a case, the concepts of temperature used in the derivation of the blackbody laws no longer hold rigorously. The treatment of such non-local thermodynamic equilibrium (non-LTE) problems is generally outside the scope of this work. (Some general comments are in Sections 9.2.2 and 11.6).

1.9.6 Blackbody Emissive Power: Cosine Law Dependence Recall that intensity is defined relative to the surface area projected in the direction of propagation. Hence, the spectral emission from dA per unit time and unit surface area passing through the element on the hemisphere shown in Figure 1.8 is given by use of Equation (1.9) as

dQle ( q, f ) = I lb cos qdAdWdl

(1.17) = I lb cos qdA sin qdqdfdl.

The directional spectral emissive power of a surface is defined as the rate of spectral emission in each direction, per solid angle, dΩ, per wavelength interval, dλ, and per unit area, dA. This gives

Elb ( q, f ) =

dQle ( q, f ) dAdWdl

= I lb cos q = Elb ( q ) . (1.18)

Since blackbody intensity is isotropic, this means that the blackbody emissive power is a maximum if the surface is normal to a receiving surface or a detector. Equation (1.18) is known as Lambert’s cosine law, proposed by Johann Heinrich Lambert in 1760. Surfaces having a directional emissive power that follow this relation are called Lambertian, diffuse or cosine-law surfaces. A blackbody, because it is always diffuse, serves as a standard for comparison with the directional properties of real surfaces that do not follow the cosine law. Real surfaces emit less energy than the ideal blackbody, which will be discussed further in Chapters 2 and 3. For many practical engineering problems, the radiative emission from a surface can be considered isotropic, and does not have (θ, ϕ) dependence. With the advances in micro- and nanomanufacturing techniques, however, novel structures are currently built to obtain highly directional emission (Greffet et al. 2002). We discuss such developments in Chapters 16 and 17. In Figure 1.9, both the blackbody intensity and blackbody energy are depicted as a function of zenith angle θ, although I λb is independent of θ. In the case of nonblack surfaces, emitted energy may depend on both the zenith and azimuthal angles and can be expressed more properly as Eλ(θ, ϕ).

1.9.7 Hemispherical Spectral Emissive Power of a Blackbody Equation (1.18) provides the radiant flux emitted by a blackbody in a given direction, per unit surface area, per unit solid angle and per unit wavelength interval centered on λ. The hemispherical spectral emissive power, Eλb, is the energy emitted by a black surface per unit time, per unit area and per unit wavelength interval around λ, into all directions. This quantity is found by integrating Equation (1.17) over all solid angles of the hemisphere to give

El b =

dQleÇ = I lb dAdl

2p p/2

q dqdf = pI  ò ò cos q sin

f=0 q=0

dW

lb

(1.19)

19

Introduction to Radiative Transfer

FIGURE 1.9  Cosine law: The angular distribution of blackbody intensity (independent of θ) and blackbody directional emissive power (cos θ function).

which is a remarkably simple relation: The blackbody hemispherical emissive power is π times the blackbody intensity. This relation for a blackbody will prove useful in the chapters that follow.

1.9.8 Planck’s Law: Spectral Distribution of Emissive Power So far, we have inferred the three blackbody characteristics from basic thermodynamic reasoning, but this is insufficient to obtain the spectral nature of blackbody emission. Indeed, this was Planck’s impetus in exploring a more exact theoretical expression, which eventually became the foundation of quantum theory. In Section 1.9, we review the history of the blackbody formulation by Planck (1901), which is built upon Boltzmann’s (1884) derivation of Equation (1.1) using statistical thermodynamics and the concept of thermal equilibrium.* We now adopt the result of Planck’s blackbody formulation to express the spectral distribution of hemispherical emissive power and radiant intensity in vacuum. This expression is based on wavelength and the absolute temperature of the blackbody

Elb (T ) =

2phc02 é æ hc0 n2 l 5 êexp ç è nkBlT ë

ö ù ÷ - 1ú ø û

=

2pC1 . (1.20) é æ C2 ö ù 2 5 n l êexp ç ÷ - 1ú è nlT ø û ë

Here, λ is the wavelength in the medium of refractive index n. This is the Planck spectral distribution of emissive power. The blackbody spectral intensity is given by

I lb (T ) =

Elb (T ) = p

2C1 é æ C n2 l 5 êexp ç 2 è nlT ë

ö ù ÷ - 1ú ø û

. (1.21)

For most engineering applications, radiative emission is into air or other gases, for which the index of refraction is near unity. Note that Planck’s blackbody function, which provides quantitative values for the emission, is expressed in terms of two universal constants: Planck’s constant, h, which defines the smallest energy quantum in terms of wavelength, and Boltzmann’s constant, k B, which relates temperature to the average energy of the microstates. Two auxiliary radiation constants are defined as C1 = hc02 and C2 = hc0/k B. The values of these constants are given in Table A.4 for two * A short history of important events in radiative transfer, biographies of contributors and a timeline are in the online Appendices M and N for this text at www.ThermalRadiation.net.

20

Thermal Radiation Heat Transfer

common unit systems. Planck initially derived Equations (1.20) and (1.21) for the restricted case of blackbody emission into a vacuum, using the speed of light in a vacuum c0 which appears in both constants C1 and C2 (Planck 1913). He assumed correctly that air and other gaseous media had a simple refractive index near unity, so little error was introduced in most cases by using this assumption. Correcting for emission into substances with a non-unity refractive index (which will be shown to be important in some cases, such as radiative transfer in glass) requires that the speed of light in the medium c = c0/n be substituted into the relations, resulting in the forms given. Example 1.1 A black surface at temperature 1000°C is radiating into a vacuum. At a wavelength of 6 μm, what are the emitted intensity, the directional spectral emissive power of the blackbody at an angle of 60° from the surface normal, and the hemispherical spectral emissive power? From Equation (1.21), Ilb (6 mm) =



2 ´ 0.59552 ´ 108 W × mm4 /m2

(

14 ,388 / [6´1273]

65 mm5 e

)

- 1 sr

= 2746 W/(m2 × mm × sr).

From Equation (1.18), the directional spectral emissive power is Elb (6 mm, 60° ) = 2746 cos 60° = 1373 W/(m2 × mm × sr).



The hemispherical spectral emissive power is Eλb(6 μm) = πIλb(6 μm) = 8627 W/(m2·μm).

Example 1.2 To a good approximation, the sun can be considered to emit as a blackbody at 5780 K. Determine the sun’s intensity at the center of the visible spectrum. From Figure 1.4, the wavelength of interest is 0.55 μm. Then, from Equation (1.21),



Ilb ( 0.55 mm) =

(

2 ´ 0.59552 ´ 108 W × mm4 /m2

(

)

0.55 mm éëe 5

5

(14.388 / 0.55´5780 )

)

- 1ùû ( sr )

= 0.256 ´ 108 éë W/(m2 × mm × sr)ùû .

Alternative forms of Equation (1.20) are employed if frequency or wave number is used rather than wavelength. Working in terms of frequency (which was how Planck’s distribution was first derived) is advantageous when spectral radiation travels between media with different refractive indices, since frequency remains constant, while wavelength changes because of the change in propagation speed. We can rewrite Equation (1.20) in terms of frequency after noting that the wavelength in the medium is related to the wavelength in the vacuum and the frequency through nλ = λ0 = c0/ν, and hence dλ0 = −(c0/ν2) dν. Then, the emissive power in the wavelength interval dλ0 becomes El b ( T ) d l 0 =

2pn2C1 2pn2C1 dl0 dn dl0 = 5 é ù dn é ù æ ö C æ ö n C c æ 2 5 0 ö 2 l 0 êexp ç ÷ - 1ú ç n ÷ êexp ç c T ÷ - 1ú è l 0T ø û è ø ë è 0 ø û ë

2pn2C1n3 d n = Enb ( T ) d n. = æ C2 n ö ù 4é c0 êexp ç ÷ - 1ú è c0T ø û ë

(1.22)

21

Introduction to Radiative Transfer

The quantity Eνb(T) is the emissive power per unit frequency interval about ν. Likewise, in terms of wavenumber,

Ehb ( T ) dh0 =

2pn2C1h30 dl 0 dh0 . (1.23) d h = 0 5 é æ C2 h0 ö ù æ 1 ö é æ C2 h0 ö ù d h0 êexp ç T ÷ - 1ú ç ÷ êexp ç ÷ - 1ú è ø û ë è T ø û è h0 ø ë 2pn2C1

As before, the blackbody spectral intensity is given by Iνb = Eνb/π and Iηb = Eηb/π. The spectral nature of Planck’s law is shown in Figure 1.10, where the hemispherical spectral emissive power, Equation (1.20), is plotted as a function of wavelength for several temperatures. We have already shown that the total radiated energy, which is the spectral energy integrated over all wavelengths, increases with temperature. Figure 1.10 shows that the spectral emissive power also increases monotonically with temperature at any given wavelength. Also, the maximum spectral emissive power shifts toward smaller wavelengths as temperature increases. The energy emitted at shorter wavelengths increases more rapidly with temperature than the energy at long wavelengths. The connection between Eλb and temperature can be conceptualized if we remember that thermal radiation originates from random fluctuations in the quantized energy levels of matter, which coincides with the emission or absorption of a photon due to the First Law of Thermodynamics (i.e. the conservation of radiative energy when no other modes of energy transfer are present). The transition probability between two energy states, i and j, depends on three things: The population of each state, the difference in energy between the states, and the temperature. States are populated according to a Boltzmann distribution, Equation (1.1), so higher temperatures permit a broader distribution of energy states and occupation of higher energy states. Therefore, transitions involving larger energy changes become possible, which corresponds to a shift toward emission at higher frequencies/lower wavelengths through Ei→j = Ei − Ej = hn. The transition probability also increases with temperature, meaning that the transitions become more frequent; this accounts for the increase in spectral intensity or spectral emissive power. Planck’s distribution also connects the visible appearance of a blackbody to its temperature. For a blackbody at 555 K, for example, only a small amount of energy is emitted within the visible region, which would not be detectable by the eye. Red light first becomes visible to the unaided eye from a heated object in darkened surroundings at the Draper point of 798 K (Draper 1847). At higher temperatures, additional wavelengths in the visible light spectral range appear. At a sufficiently high temperature, the visible emitted light becomes white as it is composed of a mixture of radiation at

FIGURE 1.10  Hemispherical spectral emissive power, Eλb, of a blackbody at several temperatures.

22

Thermal Radiation Heat Transfer

all the visible wavelengths. The visible brightness of an object also increases substantially with its increased temperature due to the overall increase in intensity. It is not a coincidence that our eyes are most sensitive to the peak spectral emissive power of the sun, which resembles that of a blackbody at 5780 K. This blackbody-like spectrum arises due to a bremsstrahlung emission in which thermally excited free electrons in the sun’s photosphere decelerate around positively charged nuclei and must emit photons to conserve energy. Although human vision is well-tuned to the solar spectrum, this is not true of all life on Earth. Turtles have eyes sensitive to infrared but not to blue; bees have eyes sensitive to ultraviolet but not red. If we find life in other solar systems having a sun with an effective temperature different from ours, it will be interesting to discover what wavelength range encompasses the “visible spectrum” if beings there possess sight. While we perceive the sun as emitting near-white light, and stars that are hotter or colder than our sun as blue or red, respectively, we will never encounter a green star because the wavelength corresponding to green light lies in the middle of the visible spectrum. Incandescent lightbulbs also emit radiation according to a blackbody spectrum. Most tungsten filament lamps are limited to a temperature of 3000 K due to vaporization of the filament at higher temperatures, and they therefore emit a large fraction of their energy in the infrared (centered at about 1 μm). In compact fluorescent bulbs, an electric current is passed through a vapor of argon and mercury, which stimulates high-energy ultraviolet emission corresponding to a transition in electronic energy states. The ultraviolet radiation irradiates the phosphor-coated inner bulb surface, which produces fluorescence. Light-emitting diodes (LEDs), on the other hand, work according to electroluminescence. A phosphor coating over an LED source is used to tune the emitted spectrum to give adjusted “white” spectra that correspond to various incandescent temperature spectra. Both bulbs are more efficient than incandescent bulbs because they avoid emission in the infrared spectrum. Neither LEDs nor compact fluorescent bulbs emit significant amounts of thermal radiation, which arises due to temperatureinduced quantum transitions in matter. Because of their low efficiency, incandescent light bulbs are routinely replaced by compact fluorescent bulbs and LEDs in most applications. The unique relationship between temperature and blackbody emissive power is the basis of pyrometry. The first pyrometer was developed by Josiah Wedgwood to correlate the firing temperature of pots against their color in the kiln in the mid-18th century, although blacksmiths were using this principle for millennia before that to determine the optimal temperature at which metal should be removed from the fire and forged, based on its color. In the modern world, pyrometers are ubiquitous, and are used in almost every situation in which it is impractical to make a contact temperature measurement. It is important to have easy-to-use expressions to calculate the energy available in different wavelength intervals. Noting the similarity of the curves shown in Figure 1.10, it is possible to collapse these into a single curve by dividing Equation (1.20) by T5 to obtain

El b ( T ) T

5

=

pI lb ( T ) T5

=

n ( lT ) 2

5

2pC1 é æ C2 êexp ç nlT è ë

ö ù ÷ - 1ú ø û

(1.24)

which is a univariate function of λT. The plot in Figure 1.11 replaces the multiple curves in Figure 1.10. Illustrative numerical values are shown in Table A.5 for n = 1. Likewise, Equation (1.23) can also be placed in a more universal form in terms of the variable η/T (=1/λT) 3



æhö 2C1 ç ÷ I hb Ehb ( T ) èT ø = = . (1.25) 3 3 T pT æ C2 h ö ù 2 é n êexp ç ÷ - 1ú è nT ø û ë

This function is shown in Figure 1.12.

Introduction to Radiative Transfer

FIGURE 1.11  Spectral distribution of blackbody hemispherical emissive power as a function of λT.

FIGURE 1.12  Spectral distribution of blackbody intensity as a function of η/T.

23

24

Thermal Radiation Heat Transfer

1.9.9 Approximations to the Blackbody Spectral Distribution Some approximate forms of Planck’s distribution are occasionally useful in derivations because of their relative simplicity. Care must be taken to use them only in ranges where their accuracy is acceptable. 1.9.9.1 Wien’s Formula If the term exp(C2/nλT) is much larger than unity, Equation (1.24) reduces to

El b ( T ) T

5

=

2pC1 5 æ C ö n2 ( lT ) exp ç 2 ÷ è nlT ø

(1.26)

which is known as Wien’s approximation. In 1896 Wien conjectured the exp(a/λT) relationship, where a is a constant, based on the Maxwell–Boltzmann distribution for the velocities of gas molecules, which in turn descends from Equation (1.1). This is the earliest attempt to provide a theoretical basis for the spectral distribution of blackbody emission. Wien’s approximation is accurate to within 1% for λT less than 3000 μm·K. It is still widely used due to its convenient functional form. In two-wavelength pyrometry, for example, the emitted radiation from an object measured at two wavelengths is taken to be proportional to Iλb(T) at each wavelength. The ratio of Equation (1.26) at the two-measurement wavelength can be rearranged into an explicit function of temperature, which is not possible using Equation (1.20) due to the “−1” in the denominator. 1.9.9.2 Rayleigh–Jeans Formula Another approximation is found by expanding the exponential in the denominator of Equation (1.24) in a series (for n = 1) to obtain

2 3 é C ù 1æC ö 1æC ö eC2 /lT - 1 = ê1 + 2 + ç 2 ÷ + ç 2 ÷ + ú - 1. (1.27) êë lT 2! è lT ø 3! è lT ø úû

For λT that is much larger than C2, the term eC2 /lT - 1 can be approximated by the single term C2/λT, and Equation (1.24) becomes

El b ( T ) T5

=

2pC1

C2 × ( lT )

4

. (1.28)

This is the Rayleigh–Jeans formula. It was first postulated by Lord Rayleigh (1900), and later improved upon by Rayleigh (1905a,b) and James Jeans (1905). Like Wien’s distribution, the Rayleigh–Jeans formula is also derived from statistical thermodynamics, but it follows a more rigorous route based on the concept of equipartition. Equipartition is the principle that at thermodynamic equilibrium the internal energy of a system is evenly distributed across the different types of microstates. In a molecular gas the microstates consist of the individual gas molecules, while in the case of radiation Rayleigh and Jeans envisioned the microstates as individual frequencies. The distribution is accurate to within 1% for λT greater than 7.78 × 105 μm·K. At lower wavelengths (higher energies) the error increases due to the so-called ultraviolet catastrophe, named because the Rayleigh–Jeans formula fails to predict the blackbody spectral emissive power at wavelengths shorter than the visible spectrum. This failure arises from the fact that, in contrast to the finite number of molecules in a molecular gas, the number of frequency microstates as envisioned by Rayleigh and Jeans is unbounded; if the total energy of the system is E, the equipartition principle states that each of these infinite frequencies is assigned an energy of E/∞. Planck fixed this problem in 1901 by supposing that energy can only be emitted in discrete quanta as an integer multiple of hν, which

25

Introduction to Radiative Transfer

corresponds to finite wavelengths. As noted earlier, while he initially envisioned this as a “kluge” to make the equation work, it was shortly shown by Einstein and Bose that this was consistent with other phenomena, including the photoelectric effect.

1.9.10 Wien’s Displacement Law A useful quantity is the wavelength λmax at which the blackbody intensity Iλb(T) is a maximum for a given temperature. This maximum shifts toward shorter wavelengths as the temperature increases (Figure 1.10). The value of λmaxT is at the peak of the distribution curve in Figure 1.11. It is obtained by differentiating Equation (1.24) with respect to λT and setting the left side equal to zero. This gives the transcendental equation (assuming n = 1): l maxT =



1 C2 . (1.29) 5 1 - e -C2 /( lmaxT )

The solution to this equation for λmaxT is a constant l maxT = C3 = 2897.7729 mm × K (1.30)



which is one form of Wien’s displacement law. Values of the constant C3 in other units are listed in Table A.4. Equation (1.30) indicates that with increased temperature, the maximum emissive power and the intensity shift to a shorter wavelength in inverse proportion to T. Wien proposed a form of Equation (1.30) in 1893, which predates Planck’s distribution (1901), and even Wien’s own approximation to Eλb (1896). Wien derived the displacement law from a conceptual argument centered on how radiation contained within a volume would change as it slowly expanded or contracted. The argument exploited the concept of radiation pressure exerted on a surface, the Doppler principle, and the Stefan–Boltzmann law to obtain the displacement law. Substituting Wien’s displacement law, Equation (1.30), into Equation (1.21) gives the maximum intensity as I lmax, b =

Elmax, b 2C = 5 C /C1 T5 p C3 e 2 3 - 1

(

)

(1.31)

ö 5 æ W 5 = 4.09570 ´ 10 -12 ç 2 ÷ T = C 4T 5 × × m m m K × sr è ø where C4 = 4.09570 × 10 −12 W/(m2·μm ·K5 ·sr) and n = 1. In Table A.4, C4 is also listed in other units. Equation (1.31) shows that the maximum blackbody intensity increases with temperature to the fifth power. Example 1.3 What temperature is needed for a blackbody to radiate its maximum intensity at the center of the visible spectrum? Figure 1.4 shows that the visible spectrum spans the range λ = 0.4–0.7 μm with its center at 0.55 μm. From Equation (1.30),

T=

2897.8 mm × K C3 = = 5269 K. lmax 0.55 mm

This compares with the observed effective radiating surface temperature of the sun of 5780 K.

26

Thermal Radiation Heat Transfer

Example 1.4 At what wavelength do you observe the maximum emission from a blackbody at room temperature? Using a room temperature of 21°C (294 K), Wien’s displacement law gives lmax =



C3 2897.8 mm × K = = 9.86 mm T 294 K

which is in the middle of the near-infrared region. About 25% of radiative emission at room temperature is at wavelengths smaller than this value.

1.9.11 Total Blackbody Intensity and Emissive Power The previous discussion provided the energy per unit wavelength interval that a blackbody radiates into a vacuum at each wavelength. Now the total intensity is determined, where a “total” quantity includes radiation for all wavelengths. The intensity emitted in the wavelength interval dλ is Iλb(T)dλ. Integrating over all wavelengths gives the total blackbody intensity: ¥

ò

I b (T ) = I lb (T )dl. (1.32)



0

This is evaluated by substituting Planck’s distribution from Equation (1.21) with n = 1 and transforming variables using ζ = C2/λT. Equation (1.32) then becomes ¥



Ib =

2C1

ò ( l e 0

5

C2 lT

)

-1

2C T 4 dl = 14 C2

¥

ò 0

2C1T 4 p4 s 4 z3 d z = = T (1.33) ez - 1 C24 15 p

where the constant σ is

sº

2C1p5 = 5.670367 ´10 -8 W/ (m 2 × K 4 ). (1.34) 15C24

The hemispherical total emissive power of a blackbody radiating into vacuum is then ¥



ò

¥

ò

Eb = Elb dl = pI lb dl = pI b = sT 4 (1.35) 0

0

which is the emitted total blackbody energy flux (W/m2). Equation (1.35) is the Stefan–Boltzmann law, and σ is the Stefan–Boltzmann constant. This Stefan–Boltzmann law was postulated by Josef Stefan (1879) based on empirical observations and confirmed theoretically by Ludwig Boltzmann in 1884. The astute observer will notice that Boltzmann theoretically verified Equation (1.35) well before Planck derived his distribution in 1901. Boltzmann did this by using the concept of radiation pressure within an isentropically expanding volume along with the theory of Carnot, much as Wien used similar principles (including the Stefan–Boltzmann law) to derive his displacement law without detailed knowledge of how Eλb varies with wavelength. For a more complete history with short biographies of important contributors and a timeline of events, see Appendices M and N on the online site www.ThermalRadiation.net.

27

Introduction to Radiative Transfer

Example 1.5 Energy emitted in the normal direction from a blackbody surface is found to have a total radiation per unit solid angle and per unit surface area of 10,000 W/(m2·sr). What is the surface temperature? The hemispherical total emissive power is related to the total intensity in the normal direction by Eb = πIb. Hence, from Equation (1.33), T = ( pIb /s)1/ 4 = (10, 000p / 5.67040 ´ 10-8 )1/ 4 = 862.7 K .

Example 1.6 A black surface is radiating with a hemispherical total emissive power of 20 kW/m2. What is its surface temperature? At what wavelength is its maximum spectral intensity? From the Stefan–Boltzmann law, the temperature of the blackbody is T = (Eb/σ)1/4 = (20,000/5.67 040) × 10 −8)1/4 = 770.6 K. Then, from Wien’s displacement law, λmax = C3/T = 2898/770.6 = 3.76 μm.

Example 1.7 An electric flat-plate heater that is square with a 0.1 m edge length is radiating 100 W from its face. If the heater can be considered black, what is its temperature? Using the Stefan–Boltzmann law, 1/ 4



æ Q ö T =ç ÷ è As ø

100 W é ù =ê -8 2 2 4 ú ( 0 . 1 m ) 5 . 67040 10 W/m K ´ × ´ ë û

1/ 4

= 648 K.

1.9.12 Blackbody Radiation within a Spectral Band The hemispherical total emissive power of a blackbody radiating into a vacuum is given by the Stefan–Boltzmann law, Equation (1.35). In calculating radiative exchange between two objects, it is often necessary to determine the fraction of the total emissive power emitted in a wavelength band, as illustrated in Figure 1.13. This fraction is designated by Fl1T ®l2T and is given by the ratio l2

Fl1T ®l2T =

ò ò

l = l1 ¥ l =0

=

1 s

El b ( T ) d l El b ( T ) d l

( l T )2

=

ò

l2

l = l1

El b ( T ) d l sT 4 (1.36)

2pC1

ò( ) ( lT ) éëexp(C /lT ) - 1ùû d ( lT ). 5

2

lT = lT 1

The wavelength band integral in Equation (1.36) can be expressed as the difference between two integrals, each with a lower limit λ = 0, so that l1 l2 ù 1 êé Fl1T ®l2T = 4 Elb dl - Elb dl ú = F0®l2T - F0®l1T . (1.37) sT ê ú 0 ë0 û The fraction of the emissive power for any wavelength band can therefore be found by having F0®l1T as a function of λT. The F0®l1T function is illustrated by Figure 1.14a, where it equals the light gray area divided by the total area (shaded) under the curve. For a blackbody, the hemispherical emissive power is related to intensity. Equation (1.19) and the Fl1T ®l2T function give the fraction of the total intensity within the wavelength interval λ1 to λ2.

ò

ò

28

Thermal Radiation Heat Transfer

FIGURE 1.13  Emitted energy in wavelength band.

Expressing the F function in terms of a single variable λT simplifies the calculations significantly (Figure 1.14b). This is done by rewriting Equation (1.37) as

Fl1T ®l2T

1é = ê sê ë

l 2T

El b d ( lT ) T5

ò 0

l1T

ò 0

ù El b ú d l T = F0®l2T - F0®l1T . (1.38) ( ) T5 ú û

As shown by Equation (1.24), Eλb/T5 is a function only of λT, so the integrands in Equations (1.37) and (1.38) depend only on the λT variable. A convenient series form for F0→λT is found by using the substitution ζ = C2/λT to obtain, from Equation (1.33),

F0®lT =

2pC1 sT 4

l

dl

ò ( l e 5

0

C2 lT

)

-1

=

2pC1 sC24

¥

ò z

z3 dz. (1.39) e -1 z

Using the definition of σ and the value of the definite integrals in Equation (1.33), this reduces to

F0®lT

15 = 1- 4 p

z

ò 0

z3 dz. (1.40) ez - 1

By using the series expansion (ez - 1)-1 = e - z + e -2 z + e -3z +  and then integrating by parts, the F0→λT becomes (Chang and Rhee 1984):

F0®lT =

15 p4

¥

é

- mz

å êêë em m =1

æ 3 3z 2 6z 6 öù + 2 + 3 ÷ ú (1.41) çz + m m m ø úû è

Introduction to Radiative Transfer

29

FIGURE 1.14  Physical representation of F factor, where F0→λ1 or F0→λ1T is the ratio of light gray to total shaded area, (a) in terms of curve for specific temperature (the entire area under curve is σΤ4) and (b) in terms of universal curve (the entire area under universal curve is σ).

where ζ = C2/λT. The series in Equation (1.41) converges very rapidly, and using the first three terms of the summation gives good results over most of the range of F0→λT. As λT becomes large (small ζ), a larger number of terms is required. The series is useful in computer solutions. Illustrative F0→λT values from Equation (1.41) are in Table A.5. A plot of F0→λT as function of λT is in Figure 1.15. When working with wave number or frequency, the fraction F0®h/T = F0®v / c0T = F0®1/ lT is needed. The inverse relation between wave number and wavelength gives F0→η/T = 1−F0→λT. The use of the F0→λT function is illustrated in the examples that follow. Example 1.8 A blackbody is radiating at a temperature of 3000 K. An experimenter wants to measure total radiant emission by using a radiation detector. The detector absorbs all radiation in the range of λ = 0.8–5 μm but detects no energy outside that range. What percentage correction does the experimenter need to apply to the energy measurement? If the sensitivity of the detector could be extended in the range of λ by 0.5 μm at only one end of its sensitivity range, which end should be extended?

30

Thermal Radiation Heat Transfer

FIGURE 1.15  Fractional blackbody emissive power in the range 0 to λT. Using λ1T = 0.8 × 3000 = 2400 μm·K and λ2T = 5 × 3000 = 15,000 μm·K in Equation (1.37) gives the fraction of energy outside the sensitivity range as F0®l1T + Fl2T ®¥ = F0®l1T + (1 - F0®l2T )  = (0.14 026 + 1 – 0.96893) = 0.17133, or a correction of 17.1% of the total incident energy. Extending the sensitive range to the longer-wavelength side of the measurement interval adds little accuracy because of the small slope of the curve of F against λT in that region, so extending to shorter wavelengths would provide the greatest increase in the detected energy. The quantitative results to demonstrate this are found from the factors F0 → 0.3 × 3000 = 0.00009 and F0 →5.5 × 3000 = 0.97581. The correction factors after extending the short- and long-wavelength boundaries of the interval are then 0.00009 + (1 – 0.96893) = 0.03116 and 0.14026 + (1 − 0.97581) = 0.16445.

Example 1.9 The experimenter of the previous example has designed a radiant energy detector that can be made sensitive over only a 1 μm range of wavelength. They want to measure the total emissive power of two blackbodies, one at 2500 K and the other at 5000 K and plans to adjust the 1 μm interval to give a 0.5 μm sensitive band on each side of the peak blackbody emissive power. For which blackbody would you expect to detect the greatest percentage of the total emissive power? What will the percentage be in each case? Wien’s displacement law tells us that the peak emissive power exists at λmax = 2897.8/T μm in each case. For the higher temperature, a wavelength interval of 1 μm will give a wider spread of λT values around the peak of λmaxT on the normalized blackbody curve (Figure 1.13), so the measurement should be more accurate for the 5000 K case. For the 5000 K blackbody, λmax = 0.5796 μm, and λ1T = (0.5796 − 0.5000) × 5000 = 398 μm·K. Similarly, λ2T = 1.0796 × 5000 = 5398 μm·K. The percentage of detected emissive power is then 100(F0 → 5398 − F0→398) = 100 × (0.6801 − 0) = 68.0%. A similar calculation for the 2500 K blackbody shows that 48.6% of the emissive power is detected.

Example 1.10 A lightbulb filament is at 3000 K. If the filament radiates as a blackbody, what fraction of its emission is in the visible region? The visible region is within λ = 0.4→0.7 μm. The desired fraction is then

F0®2100 mm×K - F0®1200 mm×K = 0.0831 - 0.0021 = 0.0810.

This very low efficiency (8%) of converting energy into visible light is a compelling reason for replacing incandescent bulbs with more efficient lighting.

31

Introduction to Radiative Transfer

Some commonly used values of the F0→λT and F0→η/T are listed in Table 1.1 for quick reference. It is interesting that very nearly one-fourth of the total emissive power lies in the wavelength range below the peak of Planck spectral distribution at any temperature.

1.9.13 Summary of Blackbody Properties It has been shown that the blackbody possesses certain fundamental properties that make it a useful standard for quantifying the properties of real radiating bodies. These properties, listed here for convenience, are the following:

1. The blackbody is the maximum absorber and emitter of radiant energy at all wavelengths and in all directions. 2. The total (including all wavelengths) radiant intensity and hemispherical total emissive power of a blackbody into a medium with constant index of refraction n are given by the Stefan–Boltzmann law:

pI b = Eb = n2 sT 4 .



3. The blackbody spectral and total intensities are independent of the direction so that emission of energy into a direction θ away from the surface normal direction is proportional to the projected area of the emitting element, dA cosθ. This is known as Lambert’s cosine law. 4. The blackbody spectral distribution of intensity for emission into medium with refractive index n is given by Planck’s distribution









El b ( T ) p

= I lb ( T ) =

2C1 . æ C2 ö ù 2 5é n l êexp ç ÷ - 1ú è nlT ø û ë

5. The wavelength at which the maximum spectral intensity of radiation into a medium with refractive index n for a blackbody occurs is given by Wien’s displacement law: l max,n =

l max,0 C3 = . n nT

For radiation into a vacuum or air, n will equal 1, so λmax,vacuum = λmax,0. The definitions for blackbody properties introduced in this chapter are summarized in Table 1.2. The formulas for the quantities are given in terms of either the spectral intensity Ιλb, which is computed from Planck’s law, or the surface temperature T.

Thermal Radiation Heat Transfer

(Continued)

32

33

(Continued)

Introduction to Radiative Transfer

34

Thermal Radiation Heat Transfer

35

Introduction to Radiative Transfer

1.10 RADIATIVE ENERGY ALONG A LINE-OF-SIGHT Now, let us return to our discussion of radiative transfer as depicted in Figure 1.5. Here, we discuss the fate of radiative transfer in a medium as it propagates after emission from a surface. We begin by extending our definition of radiant intensity, defined for a surface element in Equations (1.11) and (1.12), to any point in a volume. If we consider radiative energy propagating at a rate dQλ(S,Ω,t) in direction Ω along path S, we can define the radiant intensity in a volume as

I l (S, W, t ) =

dQl (S, W, t ) (1.42) dAdl dW

where dA is a surface element having its unit normal vector aligned with S. As it propagates along the direction Ω over a path interval dS, radiative energy loses some of its strength due to scattering and absorption. On the other hand, it gains additional strength because of emission and in-scattering of radiation incident from other directions into the direction Ω. Along this direction and path, we can tally the change in radiative energy, which provides a conservation equation of radiative energy within a small directional cone along a given direction, for a small wavelength interval around a fixed wavelength, and within an infinitesimal time duration at a given time (see Figure 1.16) This formulation leads us to the radiative transfer equation (RTE), expressed in terms of Iλ. Before discussing the RTE further, we need to introduce the definitions of absorption, scattering, and emission coefficients. Once we obtain the conservation of radiative energy along a line of sight, it can be written over a finite length of the medium, from one surface to another. After these fundamental expressions are outlined, we are poised to model and solve the expressions for engineering calculations.

1.10.1 Radiative Energy Loss due to Absorption and Scattering Consider spectral radiation of intensity Iλ incident on a volume element of length dS along the direction of propagation S. As radiation passes through dS, its intensity is reduced by absorption and scattering. This reduction is also proportional to the magnitude of the local intensity, since the matter contained along dS absorbs and scatters a fixed fraction of Iλ incident on the volume. The decrease in radiation intensity can be expressed as

dI l ( S, W ) = -bl ( S ) I l ( S, W ) dS (1.43)

FIGURE 1.16  Radiation intensity propagating through a volume element.

36

Thermal Radiation Heat Transfer

where βλ is called the spectral extinction coefficient of the medium and includes losses due to both absorption and scattering. The extinction coefficient is a physical property, has units of reciprocal length, and depends on the local properties of the medium, including temperature T, pressure P, composition of the material (e.g., the concentration of absorbers/scatters, Ci), and wavelength of the incident radiation. Therefore, it depends on local parameters, as indicated by the dependency on S, and in explicit form, it is expressed as βλ(S) = βλ(T,P,Ci). If βλ in each medium is constant, it can be interpreted as the inverse of the mean penetration distance (often called the mean free path) of radiation, as will be shown later. The extinction coefficient consists of two parts: The absorption coefficient κλ and the scattering coefficient σs,λ, bl = kl + ss,l . (1.44)



The subscript s on the scattering coefficient is included to avoid confusion with the Stefan– Boltzmann constant. All these parameters have units of reciprocal length and are called linear, or volumetric, coefficients. Absorption describes how the radiative energy is converted to the internal energy of the matter. It is one of two mechanisms coupling the radiative energy propagation with the thermodynamic state of the matter. The other is radiative emission, which is proportional to the internal energy of matter and to its temperature. The absorption coefficient describes the coupling between the internal energy states of matter and an EM wave, and provides a pathway through which matter can move to thermal equilibrium with the local electromagnetic fields. Scattering, on the other hand, causes a change in the direction of radiation propagating along S. During a scattering event, no radiative energy is converted to thermal energy (i.e. scattering in radiation is assumed to be perfectly elastic, except in some very special cases). However, scattering changes the balance of the energy propagating in each direction. In a nonparticipating medium, βλ is zero, and there would be no change in intensity along a path. In a translucent, or participating, medium, however, the initial radiative intensity changes as it propagates along a line of sight. In order to determine how the energy decreases, we can integrate Equation (1.43) over a finite path length S to obtain Il ( S )



ò

Il = Il (0 )

dI l =Il

S

ò b ( S )dS (1.45) l

*

*

S *= 0

where Iλ(0) is the intensity incident on the control volume at S = 0, and S* is a dummy variable of integration. Carrying out the integration yields the intensity at location S for a given direction Ω ù é S I l ( S ) = I l ( 0 ) exp ê - bl (S * )dS * ú . (1.46) ê ú ë 0 û Equation (1.46) is known as Bouguer’s law, Beer’s law, Bouguer–Lambert law, Beer–Lambert law, and various permutations and combinations of these names. This law shows that the intensity of spectral radiation is attenuated exponentially as it passes through an absorbing–scattering medium. It has extensive use in many fields, from astrophysics to spectroscopy. Historically, it was first proposed by Pierre Bouguer (bōo’gâr) in 1729, and later cited by Johann Heinrich Lambert in 1760. August Beer recognized the relationship between the attenuation of light and the concentration of the medium (i.e. the connection between βλ and Ci) in 1852. We will call Equation (1.46) Bouguer’s law in recognition of his priority, and to avoid confusion with Lambert’s cosine distribution, Equation (1.18).

ò

37

Introduction to Radiative Transfer

For a homogeneous medium, Equation (1.46) simplifies to

I l ( S ) = I l ( 0 ) exp(-bl S ). (1.47)

If the medium is only absorbing, then the intensity will be converted to internal energy of the matter along the line-of-sight of the beam without being redirected to other directions due to scattering. If σsλ = 0, then βλ = κλ and Equation (1.43) becomes dI l = - kl (S )I l (1.48) dS



which again is in the direction Ω. This expression can be integrated to determine the intensity leaving the boundary after it travels over a distance S:

ù é S I l ( S ) = I l ( 0 ) exp ê - kl (S * )dS * ú (1.49) ê ú ë 0 û

ò

and, if κλ is uniform within the medium,

I l ( S ) = I l ( 0 ) exp(- kl S ). (1.50)

We can obtain similar expressions for a purely scattering medium, by substituting βλ = σsλ into Equations (1.43), (1.46), and (1.47). The decrease in intensity due to the scattering of radiative energy out of the direction in propagation is often called “out-scattering” to distinguish from the augmentation of intensity caused by “in-scattering” from other directions, which is discussed later. In many fields, for example, in spectroscopy, Bouguer’s law can be written using mass extinction, mass absorption, and mass scattering coefficients. In this case,

bl, m = kl, m + ss, l, m =

bl kl ss, l = + (1.51) r r r

where ρ is the local density of the absorbing and scattering medium. Mass coefficients have units of area per unit mass and are analogous to the concept of cross-sections in molecular physics. Because the extinction coefficient βλ increases as the density of the absorbing or scattering species is changed, the βλ,m = βλ/ρ is more uniform with density changes than βλ. In the case of molecular gases, it is also commonplace to see pressure coefficients, in which the extinction, absorption, and scattering coefficients are defined per unit pressure. This exploits Dalton’s law of partial pressure, which connects the partial pressure of a gas to the molar concentration, and thus the number of molecules over dS. Bouguer’s law only accounts for the attenuation in intensity along S due to absorption and scattering. This implicitly assumes: (i) Emission from the medium is negligible, which is reasonable only if the incident intensity Iλ(0) is much larger than Iλb within the medium and the path length is sufficiently short that no multiple scattering takes place; and (ii) we neglect the in-scattering of radiation from other directions into the S direction. This can include multiple scattering, in which some of the energy is scattered out of the beam path along S and then finds its way back into the same direction after multiple scattering events. Accounting for these effects requires a more thorough analysis of the conservation of radiative energy and leads to the integrodifferential RTE, as discussed later.

38

Thermal Radiation Heat Transfer

1.10.2 Mean Penetration Distance We can use the previous analysis to find out how much radiation would penetrate a medium. Starting from Equation (1.43), the fraction of the radiation attenuated in the small volume element from S to S + dS is expressed as

é S ù Il ( S ) dI l (S ) dS = bl ( S ) exp ê - bl S * dS * ú dS. (1.52) = bl ( S ) Il ( 0 ) I l (0) ú ê ë 0 û

ò ( )

A mean penetration distance of the radiation, lm, is obtained by multiplying the fraction absorbed at S by the distance S and integrating over all path lengths from S = 0 to ∞: é S ù Sbl (S )exp ê - bl (S * ) dS * ú dS. (1.53) ê ú S =0 û ë 0 ¥



lm =

ò

ò

If βλ is uniform along the path, carrying out the integration yields ¥



lm = bl

1

ò S exp(-b S)dS = b l

S =0

(1.54)

l

demonstrating that the average penetration distance is the reciprocal of βλ. This is a very useful expression for the interpretation of the experimental measurements in many applications. Similar penetration depths can be obtained for absorbing-only or scattering-only media by replacing the extinction coefficient βλ with either the absorption coefficient κλ or the scattering coefficient σs,λ.

1.10.3 Optical Thickness The exponential factor in Equation (1.46) is often written as a useful dimensionless quantity, τλ(S), called the optical thickness or opacity along the path length S: S



ò

tl ( S ) = bl (S * ) dS * (1.55) 0

and Equation (1.47) becomes

I l ( S ) = I l ( 0 ) exp éë -tl ( S ) ùû (1.56)

which is an integral function accounting for all the values of βλ between 0 and S. Because βλ is a function of the local parameters P, T, and Ci, the optical thickness is also a function of these variables along the path between 0 and S. For a medium with uniform properties, Equation (1.55) is reduced to τλ(S) = βλ·S. The optical thickness indicates how strongly a medium attenuates radiation at a given wavelength. If τλ(S) ≪ 1, the medium is called optically thin, whereas if τλ ≫ 1, the medium is optically thick. With decreasing optical thickness, the participating medium approaches a nonparticipating one; on the other hand, with increasing τλ, less and less energy is transferred and the medium eventually becomes opaque. The optical thickness can also be given as the number of mean penetration distances within the path length, lm:

39

Introduction to Radiative Transfer



tl ( S ) =

S . (1.57) lm

The notation for the optical thickness τλ should not be confused with the surface transmissivity defined in Chapter 2, which uses the same symbol τ.

1.10.4 Radiative Energy Gain Due to Emission As radiative energy propagates in a medium along a given direction, its strength increases due to volumetric emission from every point along its path. If we denote the strength of the emission at a given location along S as jλ(S,θ,ϕ), then the incremental increase in radiative energy due to emission is

dI l (q, f) = jl (S, q, f) dS. (1.58)

Here jλ(S,θ,ϕ) is any function representing the physics of the emission mechanism (e.g., incandescence, chemiluminescence, electroluminescence) and has units of W/m3·μm·sr. In some radiation literature, jλ is called the emission coefficient. Under LTE, the thermal radiation strength is directly related to how much energy is absorbed by the medium and is correlated with the Planck blackbody function:

jl ( S, q, f ) = kl I lb (S ). (1.59)

This relationship can be intuited by remembering that κλ defines the coupling between the electromagnetic wave and the internal energy states of the matter, and under local thermal equilibrium conditions at which the local EM field is a blackbody at temperature T, the rate at which energy exits the wave and enters the internal modes of matter must exactly balance the rate at which energy is emitted by the matter into the EM field. For a more rigorous analysis, consider an elemental volume dV with a spectral absorption coefficient κλ, which can be a function of local temperature, pressure, and species concentration (T,P,Ci). Let dV be at the center of a large black hollow sphere of radius R at uniform temperature T (Figure 1.17). Assume the space between dV and the sphere boundary is filled with nonparticipating (transparent) material. Then, the spectral intensity incident at the location of dAs on dV, from element dA on the enclosure surface (at S = 0), is

I l ( 0 ) = I l ( S = 0 ) = I lb ( S = 0 ) = I lb (0). (1.60)

The change of this intensity in dV as a result of absorption is

- I l (0)kl dS = - I lb (0)kl dS. (1.61)

The energy absorbed by the differential subvolume dV = dSdAs is Iλb(S = 0) κλdSdAsdλdΩ, where dΩ = dA/R2 is the solid angle subtended by dA when viewed from dV and dAs and dA is the projected area normal to Iλ(0). The energy emitted by dA and absorbed by all dV is found by integrating over dV (over all dAs and dA elements) to obtain κλIλbdVdλdΩ. To account for all energy incident upon dV from the entire spherical enclosure, integration over all solid angles gives 4πκλIλbdVdλ. To maintain equilibrium in the enclosure, dV at location S must emit energy equal to that absorbed. Hence, the spectral emission by an isothermal volume element is

4pkl I lb ( S ) dVdl = 4 kl Elb ( S ) dVdl . (1.62)

40

Thermal Radiation Heat Transfer

FIGURE 1.17  Emission from a volume element.

This is a very important result for emission within a volume. The shape of dV is arbitrary, but small enough that energy emitted within dV escapes before any reabsorption occurs within dV. The emitting material must be in thermodynamic equilibrium with respect to its internal energy. This is usually reasonable given the very short timescales of the atomic-level interactions through which a Boltzmann distribution, Equation (1.1), is attained, and is only violated for very rapid processes (e.g., femtosecond laser pulse heating of material.) For all conditions discussed here, the spontaneous thermal emission is uniform over all directions (isotropic), so the spectral intensity emitted by a volume element into any direction, dIλ,e, is obtained by dividing by 4πdλ and the cross-sectional area to give

dI l,e ( S ) = kl ( S ) I lb ( S ) dS = jl ( S, q, j ) dS. (1.63)

If the refractive index of the medium is n ≠ 1, the spectral volumetric emission is modified by a factor of nλ2:

dI l,e ( S ) = nl2 ( S ) kl ( S ) I lb ( S ) dS. (1.64)

1.10.5 Radiative Energy Density and Radiation Pressure Radiation traversing a volume element dV produces a radiative energy density within the element. The radiative energy per unit volume at a particular frequency, the spectral radiant energy density, Uλ, is given by integrating the intensity passing through an element at location r to give

U l (l, r ) =

1 c

4p

ò I ( r, S ) d W = l

W =0

where I l (r, S ) is the local mean spectral intensity.

4p I l (r ) (1.65) c

41

Introduction to Radiative Transfer

When photons are emitted from a surface, the corresponding reaction force imposes a pressure on a surface in the opposite direction. The pressure on a body from a surrounding isotropic blackbody radiation field is (Feynman 1963): P=



U 4p 4 = I b = Eb . (1.66) 3 3c 3c

Because of the speed of light c in the denominator of Equation (1.66), radiative pressures are usually small in comparison with pressures caused by molecular interactions. Nevertheless, photon pressure is important in some applications: Most notably, the photon pressure arising from the anisotropic radiation field within the sun results in a net outwards force that balances gravitational forces acting inwards. Also, photon pressure forms the basis of solar sails, which show promise as a technology to propel space probes and nanosatellites. In 2010 the Japanese Aerospace eXploration Agency (JAXA) launched IKAROS, the first space probe to successfully use solar sails to navigate interplanetary space. From a theoretical standpoint, of course, photon pressure was used to derive the Stefan–Boltzmann law, Equation (1.35), and Wien’s displacement law, Equation (1.30), using concepts fundamental to the Second Law of Thermodynamics. Both relationships predated Planck’s distribution by many years.

1.10.6 Radiative Energy Gain due to In-Scattering If radiative energy is scattered away from the original direction of radiative energy, it constitutes a “loss” that is expressed by an “out-scattering” term in the RTE. If the scatterers (particles, molecules, impurities, voids, or any inhomogeneities) in the small volume element direct the energy coming from other angles into the direction of the original beam, then it is a “gain” for the radiative energy propagating in the direction S. This is considered the “in-scattering” term. Consider the radiation within solid angle dΩi that is incident on an element dA. The portion of the incident intensity scattered away within the increment dS is dIλ,s(S) = −σs,λIλdS. More specifically, the term dIλ,s(S) is the spectral energy scattered within path length dS per unit incident solid angle and area normal to the incident direction. The drop in intensity along dS due to scattering manifests as an increase in intensity in a direction defined by azimuthal angles Ωs = (θs, ϕs) measured relative to the direction Ωi = (θi,ϕi) of the incident intensity. The scattered intensity in the Ωs direction is defined as the energy scattered in that direction per unit solid angle of the scattered direction and per unit normal area and solid angle of the incident radiation:

dI l¢ ,s ( S ) = dI l¢ ,s ( Ws ) =

Spectral energy scattered in direction Ws = ( qs , fs ) dWs dAdW i dl

. (1.67)

The “prime” notation is used to distinguish the increase of intensity into direction Ωs, dIs′, due to scattering, caused by the scattering of intensity out of Ωi, dIs. Now we introduce the concept of the scattering phase function, Φ, which is a measure of the amount of radiative energy propagating in incident direction Ωi(θi,ϕi) that is redirected into Ωs(θs, ϕs) and can be expressed as

F éëW i ( qi , fi ) , Ws ( qs , fs ) ùû = F ( W i , Ws ) = F éë( qi , fi ) , ( qs , fs ) ùû (1.68)

in units of sr−1. The angular profile of Φ(Ωi, Ωs) is directly related to the size (relative to the wavelength λ), shape, and material properties of the scatterer and varies with the wavelength of the incident radiation. In Chapter 10, we discuss the scattering phase function and show how it can be derived for different scatterers.

42

Thermal Radiation Heat Transfer

Using the definition of the phase function, the directional magnitude of dIλs′(Ωs) is related to the entire out-scattered intensity dIλ,s(Ωi) scattered away from the incident radiation as

dI l¢, s ( Ws ) =

s 1 Fl ( W i , Ws ) dI ls = s, l F l ( W i , Ws ) I l ( S ) dS. (1.69) 4p 4p

The spectral energy scattered into all dΩ, per unit dλ, dΩi, and dA is 4p

dI l, s =



ò dI ¢

l, s

( Ws) dWs. (1.70)

Ws= 0

Using Equation (1.70) to eliminate dIλ,s in Equation (1.69) yields the expression for the phase function as

F ( Ws ) =

4p dI l¢, s ( Ws ) dI l, s

=

dI l¢, s ( Ws )

(1/4p ) ò

4p

Ws* = 0

. (1.71)

( )

dI l¢, s Ws* dWs*

Thus Φ(Ωs) is the scattered intensity in a direction Ωs, divided by the mean intensity scattered in all directions. Integrating Equation (1.71) over all possible directions allows the normalization of Φ such that

1 4p

4p

ò

1 F l ( Ws ) d Ws = 4p

Ws = 0

2p p

òòF ( W ) sinq dq df = 1. (1.72) l

s

s

s

s

0 0

Finally, the increase of intensity in each direction Ω due to out-scattering from all other directions is given by

s dI l, in-scat ( W ) = s, l 4p

4p

ò F (W , W) I l

i

l

( S, W i ) dS. (1.73)

W i =0

If the scattered intensity is isotropic, that is, the same for all scattered directions, Φ is equal to unity. Isotropic scattering is a convenient mathematical approximation, which significantly simplifies the radiative transfer calculations. However, it should be used with care, as no single scatterers are truly isotropic in nature; in reality, because of the way in which the EM wave interacts with the scatterers (e.g., particle, gas molecule, etc.), the intensity will be scattered preferentially in some directions over others. On the other hand, multiple scattering systems (i.e. where many scatters occur before a boundary is reached) can often be adequately approximated by isotropic scattering. Examples of multiple scattering systems include sand, flour, granulated sugar, porous ceramics, and some types of clouds.

1.11 RADIATIVE TRANSFER EQUATION So far, we have discussed the change in radiative energy along a path in absorbing and scattering media. We have introduced the concepts of radiation intensity; absorption, scattering, and extinction coefficients; and the scattering phase function. We have also provided a formulation for the attenuation of the radiative energy of an energy beam (like a laser beam) along the direction of its propagation. This theoretical background is enough to describe many practical problems and the working

43

Introduction to Radiative Transfer

principles of several diagnostic tools where a single beam is incident on a medium. Equipped with this formulation, we can now determine the amount of radiative energy reaching a surface element or a detector at the observation location. If the medium is scattering or if the incident radiation arrives from several different sources (like several laser beams arriving from different directions), then the recorded signal at the observation point is the combined effect of all these energy sources. Similarly, if radiant energy is emitted by the walls of the enclosure, then each emitting surface contributes to the detected signal, and each volume element dV redirects energy incident toward the detector, regardless of the original direction of radiative energy. In addition, each beam arrives at other surface elements after being scattered by the absorbing and emitting particles and gases in the medium or after reflection by other surfaces. The combined effect of these multiple beams needs to be tallied for calculating the net radiative exchange. In addition, the emission process effectively “cools” each surface element or volume element in the medium. These complex mechanisms are interrelated and take place practically simultaneously, as the radiative energy propagates with the speed of light. To be able to express this series of events, we write the conservation of radiative energy along a line of sight for a given wavelength. This conservation equation constitutes the radiative transfer equation (RTE), which is a fundamental relation of radiation heat transfer. In deriving the RTE, we consider a small volume element dV with macroscopic parameters such as absorption, scattering, and extinction coefficients, κλ, σs,λ, and βλ, and the scattering phase function Φ(Ωi, Ωs). Also, we consider emission by the matter within each small volume element. Now, we can write conservation of radiative energy within a small volume element dV for a small path increment dS along S within a small solid angle dΩ around the direction (θ, ϕ) for a small time interval dt at time t and within a small wavelength interval dλ around a given wavelength λ as Change in radiative energy = gain due to emission - loss due to absorption





- loss due to outscattering + gain due to inscattering. This conservation relation over infinitesimal path dS is expressed mathematically as



I l ( S + dS, W, t + dt ) - I l ( S, W, t ) = kl I lb ( S, t ) dS - kl I l ( S, W, t ) dS - ss,l I l ( S, W, t ) dS +

s s ,l 4p

ò

(1.74)

I l ( S, W i , t ) F l ( W i , W ) dW i dS.

Wi = 4 p

We can recast this expression into a differential form: ¶I l ( S, W, t )

c¶t

+

¶I l ( S, W, t ) ¶S

= kl I lb ( S, t ) - kl I l ( S, W, t ) - ss,l I l ( S, W, t ) s + s ,l 4p

ò

I l ( S , Wi , t ) F l ( W i , W ) d W i

(1.75)

Wi = 4 p

where ∂ represents the partial differential with respect to time or space. This energy balance is the radiative transfer equation (RTE). Note that in Equation (1.75), the first term is important only for ultrafast phenomena; however, for most engineering applications, it is neglected due to the large value of the speed of light c appearing in the denominator.

44

Thermal Radiation Heat Transfer

1.12 RADIATIVE TRANSFER IN ENCLOSURES WITH NONPARTICIPATING MEDIA The RTE expresses the general conservation of radiative energy along a line of sight in an absorbing, emitting, and scattering medium. The solution of this integrodifferential equation is by no means trivial, especially in 3D geometries. It is the stepping-stone to determining the contribution of radiation to an overall energy conservation relation. However, it is not necessary to have its full-fledged solution for every engineering problem. It is indeed possible to introduce certain physical approximations for a given application, which reduces this equation to much simpler forms. Among all complications, it is the in-scattering integral that couples the intensity in one direction to the intensity in all other directions that is most difficult to resolve. It should be noted that scattering is a natural phenomenon, and physically it is always present, unless there is pure vacuum between the surfaces exchanging radiation. All gases scatter light; the most obvious example of this is the blue color of the sky, caused by scattering by air molecules (N2, O2, NO2, and others). However, gas scattering is orders-of-magnitude smaller than scattering by dust and other particles. We show in Chapter 10 that both phenomena are often much smaller than absorption, especially at the infrared wavelengths important to thermal radiation. If scattering can be neglected, the RTE is reduced to a partial differential equation. Applications abound for non-scattering media, ranging from kitchen ovens to satellite energy management systems, and therefore constitute an important class of problems. In many cases the RTE can be further simplified by assuming the medium is nonparticipating. Since all matter absorbs and emits thermal radiation, at first glance it may seem that this assumption only applies to a vacuum. In reality, however, the absorption and emission of common monatomic (e.g., Ar) and nonpolar diatomic (e.g., N2, O2) gases is almost always negligible, and other participating species like CO2 and H2O may only have a strong influence if they are present in significant amounts (e.g., in flue gases) or over very long pathlengths (e.g., the earth’s atmosphere.) The energy transfer in this simplified (non-scattering and nonabsorbing medium) case is limited to that among the surfaces, each of which emits, mostly uniformly, in all directions. Each surface also reflects the energy received from other surfaces. Emission characteristics of surfaces and discussion of reflection (whether it is diffuse, i.e. with uniform intensity in all directions, or specular, i.e. mirrorlike, or a combination) are covered in Chapters 2 and 3. We start the analysis with the radiative intensity Iλ(θ,ϕ) leaving surface element dA1. To analyze the radiative exchange between two finite surfaces, we need to carry out integration over the entire area of each surface. For this, consider radiative energy leaving a small area element dA1 and traveling in a nonparticipating medium. Assume that this energy is incident on a second small area element dA2 on finite area A2, at distance S12 from dA1. The projected areas are formed by taking the area that the energy is passing through and projecting it normal to the direction of the radiation; therefore, dA1cosθ1 and dA2cosθ2 are the normal components of the infinitesimal areas along direction S12 as shown in Figure 1.18a. The elemental solid angle is centered about the direction of the radiant path and has its origin at dA. Using the definition of spectral intensity Iλ,1 as the rate of energy passing through dA1 per unit projected area per unit solid angle and per unit wavelength interval, the energy dQλ,1 from dA passing through dA1 in the direction of S12 is

æ dAcosq ö dQ l,1 = I l,1 dA1cosq1dW21dl = I l,1 dA1cosq1 ç ÷ dl. (1.76) 2 è S12 ø

If dA1 is placed at a farther distance along the same direction, the rate of energy passing through would be smaller, as in Figure 1.18b. Let’s assume another small area element dA2 = dA1 is located at a distance S2. The energy rate at this new position is Iλ,2dA1cosθ1dΩ2dλ = Iλ,2dA1cosθ1(dA2cosθ2/ S22)dλ. If all areas are perpendicular to the direction of propagation, then all cosθ terms equal unity, and the ratio of energy rates for distances S1 and S2 is given as Iλ,1S12/Iλ,2S22.

Introduction to Radiative Transfer

45

FIGURE 1.18  Demonstration that intensity is invariant along a path in a transparent medium. (a) Black element dA within black spherical enclosure, (b) energy transfer from dA to dA1, (c) energy transfer from dA1 to dA, and (d) solid angle subtended by dA1 on sphere surface.

Now, consider a differential source emitting energy equally in all directions. Assume we draw two concentric spheres around this source, as in Figure 1.18c. If dQλ,sdλ is the entire spectral energy leaving the differential source, the energy flux crossing the inner sphere is dQλ,sdλ/4πS12, and that crossing the outer sphere is dQλ,sdλ/4πS22. The ratio of the energies passing through the two elements dA1 at S1 and dA2 = dA1 at S2 is then [(dQλ,sdλ/4πS12)dA2]/[(dQλ,sdλ/4πS22)dA1] = S22/S12. This is indeed expected, as the radiative energy density decreases with the square of the distance from the origin (the inverse square law). However, in the previous paragraph, this energy ratio was derived as Iλ,1S12/Iλ,2S22. For these two expressions to be equal, the intensities must remain constant along the line of sight:

I l,1 = I l,2 (1.77)

Thus, the intensity in each direction in a nonattenuating and nonemitting (nonparticipating) medium is independent of position along that direction, including that at the starting point, that is, Iλ,s originating at the surface dAs. This demonstrates again the invariance of intensity along a ray transecting a nonparticipating medium. The invariance of intensity provides a convenient way to specify

46

Thermal Radiation Heat Transfer

the magnitude of attenuation or emission when they are present in a medium, as their effects are directly shown by the change of intensity with distance along a path. If the medium is absorbing, scattering, and/or emitting, the intensity will differ at any two locations along a ray. This is another key advantage of defining the radiative field using intensity. Next, consider a cold absorbing but non-scattering medium, which is more complicated, but allows us to start our discussion straight from the RTE, Equation (1.75). An example of such a medium would be cold flue gases, or a cold unaglommerated soot-laden aerosol (for which scattering can often be neglected), bounded by hot surfaces. Since the medium is not emitting or scattering, the equation will not have any gain terms due to emission or in-scattering, and the only loss is due to absorption along the path S. The governing equation is a first-order differential equation, as given by Equation (1.48), and solution is relatively simple. It can be obtained by integrating the equation from S = S1 = 0 on dA1 to S = S2 on dA2. Using the boundary condition at S = S1 = 0, which is I l ( S = 0, q, f ) = I l,dA1 ( q, f ) = I l,emission,dA1 ( q, f ) + I l,reflection,,dA1 ( q, f ) , we obtain ù é S2 I l,dA2 ( S2 , q, f ) = I l,dA1 ( S1 , q, f ) exp ê - kl (S )dS ú . (1.78) ê ú ë S =S1 û

ò



Equation (1.78) is further simplified to I l,dA2 ( S2 , q, f ) = I l,dA1 ( S1 , q, f ) exp éë - kl (S2 - S1 ) ùû for a medium with spatially uniform absorption coefficient. This is a restatement of Equation (1.50). In these equations, S1 and S2 correspond to the central locations of infinitesimal area elements dA1 and dA2 on the directly opposed surfaces A1 and A2, respectively. Equation (1.78) is valid only along a single line connecting S1 to S2. If we want to determine the radiative energy leaving from a finite area and reaching another one, then we need to integrate Equation (1.78) over each of these areas. These area integrations are equivalent to those carried over solid angles subtended by each area. First, consider infinitesimally small area elements; spectral radiative energy leaving dA1 toward dA2 is given as dQl,1 = I l,1 cos q1dA1dldW2 = I l,1 cos q1dA1dl

where dW2 =

cos q2 dA2

( S2 - S1 )

2

cos q2 dA2 (S2 - S1 )2

.

The radiative energy arriving to dA2 is then found from dQl,2

é S2 ù = I l,1 cos q1dA1 exp ê - kl (S )dS ú dldW2 ê ú ë S =S1 û (1.79) S é 2 ù cos q2 dA2 exp ê - kl (S )dS ú dl. = I l,1 cos q1dA1 ê ú (S2 - S1 )2 ë S =S1 û

ò

ò

To write a similar expression for finite surfaces, Equation (1.79) is integrated over both surfaces A1 and A2.

Ql,1®2 =

òò

A1, A2

dQ2 =

òò

A1, A2

I l,1 cos q1

é S2 ù cos q2 ê - kl (S )dS ú dl dA1dA2 . (1.80) exp 2 ê ú (S2 - S1 ) ë S =S1 û

ò

47

Introduction to Radiative Transfer

If the medium absorption approaches zero, that is, if the medium becomes optically thin and eventually completely nonabsorbing, the exponential term approaches unity. If radiation leaving surface A1 is directionally uniform, then the intensity term can be taken outside the integral sign, yielding

Ql,1®2 = I l,1dl

cos q2

òò cos q ( S - S ) dA dA = E 1

A1, A2

1

2

2

2

1

l ,1

dl

òò

A1 , A2

cos q1 cos q2 p ( S2 - S1 )

2

dA1dA2 . (1.81)

The double integral in this equation is a purely geometrical factor and can be evaluated independently of the spectral surface properties:

Ql1®2 = El,1 F1-2 A1dl (1.82)

where

A1F1-2 =

cos q1 cos q2

òò

p ( S2 - S1 )

A1 , A2

2

dA1dA2 . (1.83)

Here, F1−2 is called a configuration factor, view factor, or shape factor and is treated extensively in later chapters. This formulation can be expanded to absorbing and scattering media. If the medium properties are uniform, then the attenuation of radiation along a path connecting one surface to another one would be included in the configuration factor analysis with little difficulty. We will return to these discussions later.

1.13 PROBABILISTIC INTERPRETATION The above description of blackbody emission, absorption, emission, out-scattering, and in-scattering from a participating medium, and radiation propagating along a line-of-sight, largely focuses on the EM wave interpretation, which is deterministic. As outlined in Section 1.5, these phenomena can also be interpreted in the context of photons, although one should keep in mind these “RTE photons” are mostly conceptual constructs that facilitate the solution of the equations governing radiative transfer, and are only distantly related to true “QED” photons. The photonic interpretation also lays the foundation for Monte Carlo solutions (Chapter 14) to the equations governing radiative transfer, which are particularly advantageous for complicated problems in which a deterministic analysis may be difficult or impossible to carry out. Central to this concept is the idea that photons are indivisible quanta of energy, so emission, absorption, and scattering events are random in nature and occur with a defined probability. In the case of emission, the probability of a photon emitted by blackbody having an energy between hn and h(n+dn) is given by

Pn ®n + dn ( n ) =

En b ( T ) d n ¥

òE

nb

= p ( n ) dn (1.84)

(T ) dn

0

where p(n) is the probability density function (PDF) in units of seconds. An equivalent function p(λ) can be derived in terms of wavelength, and the blackbody fraction, given by Equation (1.39), is simply the corresponding cumulative distribution function (CDF)

1 P0®l ( l ) = CDF ( l ) = sT 4

¥

òE 0

bl

( l ) dl = F0-l ( lT ) . (1.85)

48

Thermal Radiation Heat Transfer

It is also possible to define PDFs and CDFs for the direction of emission using Lambert’s cosine law. From Equation (1.17) we can show that for a diffuse surface

Pq®q+ dq ( q ) =

cos ( q ) sin ( q ) dq

ò

p2

0

cos ( q ) sin ( q ) dq

= 2 cos ( q ) sin ( q ) dq = p ( q ) dq. (1.86)

Since all azimuthal angles ϕ are equally probable, Pf®f+ df ( f ) =



1 df = p ( f ) df (1.87) 2p

such that p(ϕ) = 1/2π is a constant over its domain f Î[0, 2p]. Likewise, the absorption and scattering coefficients and the scattering phase function define probabilities for how an individual photon may interact with matter encountered along its trajectory. The quantity κλ(S)dS defines the probability that a photon having an energy hc0/λ will be absorbed by matter as it travels between S and S + dS, and under conditions of local thermodynamic equilibrium this probability must equal the probability of matter emitting a photon having the same energy over the same pathlength. Likewise, the probability of a photon being scattered into another direction is σs,λ(S)dS, and the joint probability that a photon will be absorbed or scattered over dS is βλ(S)dS. In this context, κλ, σs,λ, and βλ are PDFs, while the corresponding CDFs define the probability that a photon will be absorbed or scattered over a finite distance. For example, Pno abs sca,l ( S ) =



0®S

ù é S = exp ê - bl S * dS * ú = exp éë -tl ( S ) ùû (1.88) Il ( 0 ) ê ú ë 0 û

Il ( S )

ò ( )

is the probability that a photon having an energy between hc/(λ+dλ) and hc/λ will not be absorbed or scattered after traveling a distance S. As τλ(S) becomes large, the probability approaches zero. The complementary probability that a photon will be absorbed or scattered over an optical thickness τλ is Pext ,l ( S ) = 1 - Pno abs sca,l ( S ) = 1 - exp éë -tl ( S ) ùû = CDFbl (1.89)



0®S

0®S

which is the cumulative density function for extinction. The mean penetration distance, introduced in Section 1.10.2, is the average, or expected, distance that a photon will travel before it is either absorbed or scattered. In this context, Equation (1.53) is interpreted as follows: The probability that a photon will be absorbed or scattered having traveled a distance in between S and S + dS is given by



Pext ,l

S ® S + dS

( S ) = - éê Pno abs sca,l ( S + dS ) - Pno abs sca,l ( S )ùú = ë

0®S

0®S

û

d ( CDFbl ) d tl dS d tl dS (1.90)

= exp éë -tl ( S ) ùû bl ( S ) dS. The expected value for this probability is then found from ¥



ò

lm = SPext ,l 0

S ® S + dS

(S ) =

¥

ò Sb (S)exp éë-t ( S )ùû dS. (1.91) l

S =0

l

49

Introduction to Radiative Transfer

Finally, the scattering phase function defines the conditional probability of a photon scattering into a given direction. If an incident photon within a solid angle dΩi is scattered, the probability of it scattering into a second solid angle dΩs is

(

)

P Ws W i =



F l ( Ws , W i ) 4p

dWs (1.92)

so that the scattering phase function is a PDF having units of sr−1.

1.14 CONCLUDING REMARKS In this chapter, we discussed the importance, definitions and equations of thermal radiative transfer. We derived the fundamental RTE and discussed its simplified forms for different medium properties. We also presented an intuitive relationship between the RTE formulations and the configuration factor analyses which is used extensively for nonparticipating media calculations. The historical development of the blackbody relations in this chapter differed from the sequence in which they are presented here. A detailed discussion of the history of the blackbody, biographies of the important contributors, and a timeline of events in developing the theory of the blackbody are given in Appendices M and N on the online site www.ThermalRadiation.net.

HOMEWORK PROBLEMS (Additional problems are available in the online Appendix P at www.ThermalRadiation.net) 1.1

What ranges of wave numbers and frequencies delineate the visible spectrum (0.4–0.7 μm)? What are the wave number and frequency values at the spectral boundary (25 μm) between the near- and the far-infrared regions? Answers: 2.5 × 106 to 1.4286 × 106 m−1; 4.2827 × 1014 to 7.4948 × 1014 s−1; 4 × 104 m−1; 1.1992 × 1013 s−1 1.2 Radiation propagating within a medium is found to have a wavelength within the medium of 1.633 μm and a speed of 2.600 × 108 m/s. (a) What is the refractive index of the medium? (b) What is the wavelength of this radiation if it propagates into a vacuum? Answers: (a) 1.153; (b) 1.883 μm 1.3 A material has an index of refraction n(x) that varies with position x within its thickness. Obtain an expression in terms of c0 and n(x) for the transit time for radiation to pass through a thickness L. If n(x) = ni(1 + kx), where ni and k are constants, what is the relation for transit time? How does the wave number (relative to that in a vacuum) vary with position within the medium? Answer: t =

ni æ kL2 ö çL + ÷ ; ni (1 + kx ) c0 è 2 ø

Derive Equation (1.30) by analytically finding the maximum of the Eλb/T5 versus λT relation (Equation (1.24)). 1.5 A blackbody is at a temperature of 1350 K and is in air. (a) What is the spectral intensity emitted in a direction normal to the black surface at λ = 4.00 μm? (b) What is the spectral intensity emitted at θ = 50° with respect to the normal of the black surface at λ = 4.00 μm? 1.4

50

Thermal Radiation Heat Transfer

(c) What is the directional spectral emissive power from the black surface at θ = 50° and λ = 4.00 μm? (d) At what λ is the maximum spectral intensity emitted from this blackbody, and what is the value of this intensity? (e) What is the hemispherical total emissive power of the blackbody? Answers:  (a) 8706 W/(m2·μm·sr); (b) 8706 W/(m2·μm·sr); (c) 5596 W/(m2·μm·sr); (d) 2.1465 μm, 18365 W/(m2·μm·sr); (e) 188.34 kW/m2. 1.6 For a blackbody at 2450 K that is in air, find (a) The maximum emitted spectral intensity (kW/m2·μm·sr) (b) The hemispherical total emissive power (kW/m2) (c) The emissive power in the spectral range between λ0 = 1.5 and 7 μm (d) The ratio of spectral intensity at λ0 = 1.5 μm to that at λ0 = 7 μm Answers: (a) 361.54 kW/(m2·μm·sr); (b) 2043.1 kW/m2; (c) 1142.8 kW/m2; (d) 59.17 1.7

Determine the fractions of blackbody energy that lie below and above the peak of the blackbody curve. Answers: 25% and 75% 1.8 The surface of the sun has an effective blackbody radiating temperature of 5780 K. (a) What percentage of the solar radiant emission lies in the visible range λ = 0.4–0.7 μm? (b) What percentage is in the ultraviolet range? (c) At what wavelength and frequency is the maximum energy per unit wavelength emitted? (d) What is the maximum value of the solar hemispherical spectral emissive power? Answers: (a) 36.7%; (b) 12.2%; (c) 0.5013 μm, 5.98 × 1014 Hz; (d) 8.301 × 107 W/(m2·μm) 1.9 A blackbody has a hemispherical spectral emissive power of 0.0390 W/(m2·μm) at a wavelength of 85 μm. What is the wavelength for the maximum emissive power of this blackbody? Answer: 19.71 μm 1.10 Blackbody radiation is leaving a small hole in a furnace at 1350 K (see the Figure). What fraction of the radiation is intercepted by the annular disk? What fraction passes through the hole in the disk?

Answers: 0.1000; 0.1000

Introduction to Radiative Transfer

1.11 A solid copper sphere 2.5 cm in diameter has a thin black coating. Initially, the sphere is at 750 K, and it is then placed in a vacuum with very cold surroundings. How long will it take for the sphere to cool to 300 K? Because of the high thermal conductivity of copper, it is assumed that the temperature within the sphere is uniform at any instant during the cooling process. (Properties of copper: density, ρ = 8950 kg/m3; specific heat, c = 383 J/kg·K). Answer: 0.807 h 1.12 Spectral radiation at λ = 2.500 μm and with intensity 8.00 kW/(m2·μm·sr) enters a gas and travels through the gas along a path length of 21.5 cm. The gas is at uniform temperature 1200 K and has an absorption coefficient κ 2.500 μm = 0.625 m−1. What is the intensity of the radiation at the end of the path? Neglect scattering, but include emission by the gas. Answer: 8.27 kW/(m2·μm·sr) 1.13 A science fiction writer uses a “slow glass” window as a plot device. The window material has such a high value of constant simple refractive index n in the visible range that light takes 5.0 minutes to traverse a 1 m thick window. Thus, the observer on one side sees a murder scene 5 minutes after it occurs on the other. He rushes outside, but the perpetrator and corpse have been gone for 5 minutes. Chaos ensues. What must the slow glass refractive index be for this to be possible? Answer: n = 9.00 × 1010 (For another approach, see Hau et al. 1999, Hau 2001) 1.14 If the window in Problem 1.13 is 1 m thick and is made of crown glass (refractive index n = 1.50 in the visible), what will be the time delay between light entering and exiting the window in the normal direction? Answer: Δt = 5 ns

51

2

Radiative Properties at Interfaces Johann Heinrich Lambert (1728–1777) was a self-taught mathematician, astronomer, logician, and philosopher. Aside from his work in radiation, he offered the first proof that π was an irrational number. In 1758, he published his first book, describing the exponential decay of light, followed in 1760 by his more complete book Photometrie that describes the exponential decay as well as the cosine dependence of the emission from a diffuse surface. Gustav Robert Kirchhoff (1824–1887) had a broad influence on physics and engineering. He proposed in 1859, and provided a proof in 1861 that, in simple terms, “For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.” This was shown to apply for spectral, directional and total properties. He first described the ideal radiation emitter in 1862 and called it the schwarzer Körper (blackbody) because, as the perfect emitter, it must also be the perfect absorber and thus a zero reflector that would appear black to the eye.

2.1 INTRODUCTION The radiative behavior of a blackbody was presented in Chapter 1. The ideal blackbody is a thermodynamic limiting case against which the performance of real radiating bodies can be compared. The radiative behavior of a real body depends on many factors such as composition, surface finish, temperature, radiation wavelength, angle at which radiation is either emitted or intercepted and spectral distribution of the incident radiation. We need to define spectral, directional, or averaged emissive, absorptive, and reflective properties to describe the radiative behavior of real materials relative to blackbody behavior. This chapter introduces definitions of the surface radiative properties of materials. A complete understanding of radiative transfer relies on many parameters to be defined clearly, but the introduction of many surface properties and their dependence on wavelength, angle, and temperature may be overwhelming at first. For this reason, the reader is encouraged to scan the information first, and then use the details as necessary for a given application. This reference chapter is the foundation for the property definitions used for radiation/surface interactions discussed in the rest of the text. The most general definitions governing emission, absorption, reflection, and transmission are presented first. Consider Figure 2.1, where an infinitesimal area element dA is part of an object at temperature T and is exchanging radiation with other objects. The location of dA can be specified in any coordinate system; for convenience, consider a Cartesian system so the temperature of dA can be specified as T(x,y,z). The emitted energy from dA is thus an implicit function of location through its dependence on T(x,y,z) and is also almost always a function of wavelength and direction of emission. The ability of the surface to absorb and reflect radiation depends not only on the temperature of the surface, T(x,y,z), but on the directional and spectral distribution of the incident radiation, which, in turn, depend on the source of this radiation. A surface irradiated by a blackbody source at 6000 K will have a maximum spectral incident radiation (irradiation) near 0.5 μm (Wien’s law, 53

54

Thermal Radiation Heat Transfer

FIGURE 2.1  Description of radiation interactions at a surface.

Equation (1.30)), while a source at 300 K will have a spectral peak near 10 μm. The fraction of irradiation absorbed by dA will depend on whether its absorption ability is highest near 0.5 or 10 μm. The rates at which irradiation is reflected from, and transmitted through, dA at a given wavelength and direction also depend on the spectral characteristics of the source, in addition to the spectral, directional, and temperature-dependent properties of dA. In general, then, reflected and transmitted radiation are functions of the source spectral characteristics, the direction of the source, the direction of the reflected or transmitted radiation and the wavelength considered. Time dependence may be important in ultrafast processes such as femto- or picosecond laser applications, where the incident energy may change the temperature and/or properties of the surface, and the assumption of local thermodynamic equilibrium (LTE) may be invalid, although we will assume LTE applies in all of our discussions. Generally, emission properties depend on (λ,θe,ϕe,T), absorption will depend on (λ,θi,ϕi,T), reflection on (λ,θi,ϕi,θr,ϕr,T), and transmission on (λ,θi,ϕi,θt,ϕt,T). To simplify the notation, the time dependence is dropped, and the wavelength dependence will be shown through the presence or absence of the subscript λ. Experimental data in the literature that provide both detailed directional and spectral data are scarce. Because of the complexities in making these measurements (e.g., see Shafey et al. 1982, Makino 2002, Makino and Wakabayashi 2003), tabulated property values are often averaged over all directions, all wavelengths, or both. The definitions reveal relations among averaged properties that enable maximum use of available property information; for example, absorptivity data can be obtained from emissivity data if certain restrictions are observed. Table 2.1 lists each property, its symbolic notation, the equation number of its definition and the figure showing a physical interpretation. Emissivity, ε, and absorptivity, α, are surface attributes. For an opaque material, the portion of incident radiation that is not reflected is transmitted through the surface and then absorbed within material that extends below the surface. This layer can be quite thin (tens of nanometers) for a material with high internal absorptance, such as a metal, or it can be a few millimeters or more for a less strongly internally absorbing dielectric material. Emission comes from within a material. The amount of energy leaving the surface depends on what portion of the energy emitted by each

Radiative Properties at Interfaces

55

internal volume element reaches the surface and can then be transmitted through the surface. As discussed in Section 17.5, some of the energy reaching the surface can be internally reflected from the surface back into the body, depending on the refractive index of the body relative to that of the external medium. Thus, the emissive ability of a body depends on the refractive index of its surrounding medium. With rare exceptions, emissivity measurements are made for emission into air or vacuum, so the tabulated values of spectral or total emissivity ελ or ε are for an external medium with n ≈ 1. If the medium adjacent to the surface has n > 1, the ελ or αλ, and hence ε and α, can be different from the tabulated values. In this chapter, the properties are discussed for the surface of a medium that has enough thickness to attenuate and absorb all radiation that passes through the surface; thus, the medium is opaque. The radiant energy transmitted through the surface of the medium is then equal to the radiant energy absorbed by the medium. Although the ability of a surface to transmit radiation is

56

Thermal Radiation Heat Transfer

not important for radiative transfer among opaque surfaces, the definitions describing transmission across a surface or interface are provided for reference when nonopaque materials are treated. Layers of material that are not opaque, and therefore transmit a portion of the incident radiation, emit in a way that depends on their thickness. It is common practice in most fields of science to assign the -ivity ending to intensive properties such as electrical resistivity, thermal conductivity, or thermal diffusivity. The -ivity ending is used throughout this book for radiative properties of opaque materials, whether for ideal surfaces or for properties with a given surface condition. The -ance ending is for an extensive property, such as the emittance of a partially transmitting isothermal layer of water or glass, where the emittance depends on layer thickness. The -ance ending is also often found in literature dealing with the experimental determination of surface properties. The term emittance is also used in some references to describe what we have called emissive power. Over the years, suggestions have been made to standardize radiation nomenclature. The US National Institute of Standards and Technology (NIST) has published a nomenclature useful in the fields of illumination and measurement (Nicodemus et al. 1977). That nomenclature is close to the set used in this text, and we follow the notation recommended by the editors of the major heat transfer journals (Howell 1999). Because radiative properties involve many independent variables, a concise but accurate notation is required. The notation used here is an extension of that in Chapter 1. A functional notation is used to give explicitly the variables upon which a quantity depends. For example, ελ(θ,ϕ,T) shows that the spectral emissivity is directional and can depend on four variables. The λ subscript specifies that the quantity is spectral and is convenient when the functional notation is omitted. A total quantity does not have a λ subscript. Certain quantities depend on two directions and have four angles in their functional notation. A hemispherical-directional or directional-hemispherical quantity, for example, has only two angles. Bidirectional properties require four explicit angles. In later chapters, the functional notation may be abbreviated or omitted to simplify the equation format when the functional dependencies are evident. Additional notation is needed for Q, the energy rate for a finite area, to keep consistent mathematical forms for energy balances. The Qλdλ is used to designate spectral energy in the interval dλ; this is consistent with blackbody spectral energy where terms such as Eλbdλ are used in Chapter 1. In dQλ(θ,ϕ)dλ, the Q has a derivative to indicate that the spectral energy is also of differential order in solid angle. The total energy dQ(θ,ϕ) is a differential quantity with respect to solid angle. If the energy is for a differential area, the order of the derivative is correspondingly increased. The notation

ò dW Ç

signifies integration over the hemispherical solid angle where dΩ = sin

θdθdϕ. Hence, for any function f(θ,ϕ),

2p

p/2

f= 0

q= 0

ò ò

f (q, f) sin q d qd f º

ò

Ç

f (q, f)d W .

Each of the four main sections of this chapter deals with a different property: emissivity, absorptivity, reflectivity, and transmissivity. In each section, the most fundamental property is presented first; for example, Section 2.2 begins with the directional spectral emissivity. Then, averaged quantities are obtained by integration. Section 2.3 on absorptivity also contains various forms of Kirchhoff’s law that relate absorptivity to emissivity. Section 2.4 on reflectivity and Section 2.5 on transmissivity include reciprocity relations.

2.2 EMISSIVITY Emissivity specifies the rate at which thermal radiation is emitted from a real surface as compared with emission from a blackbody at the same temperature. It can be spectral, and in general depends on surface temperature and direction. In detailed radiative exchange calculations, data for spectraldirectional emissivity at the correct surface temperature are needed. Spectral-directional emissivity is also needed to carry out pyrometry, the process through which the surface temperature is

57

Radiative Properties at Interfaces

inferred from the radiation emitted by the surface. Such detailed data for particular materials are scarce. Most available data are for a limited spectral range and are mostly for the normal direction. Because of this, averaged emissivity values are often used. Averages may be over direction, wavelength, or both. Values averaged with respect to all wavelengths are termed total quantities; averages with respect to all directions are termed hemispherical quantities. In this section, the basic definition of the directional spectral emissivity is given. This emissivity is then averaged with respect to wavelength, direction, or both wavelength and direction simultaneously. Usually the total hemispherical quantity is implied unless otherwise specified: for example, if we simply refer to a property as “emissivity” without a modifier, we are likely referring to hemispherical total emissivity. Similarly, “spectral emissivity” usually implies hemispherical spectral emissivity. This may not be the case, however, and it is always good to avoid ambiguity.

2.2.1 Directional Spectral Emissivity, ελ(θ,φ,T) Consider the geometry for emitted radiation in Figure 2.2a. As discussed in Chapter 1, the radiation intensity is the energy per unit time emitted in direction (θ,ϕ) per unit of projected area dAp normal to this direction, per unit solid angle, and per unit wavelength interval. Basing the intensity on the projected area has the advantage that for a black surface the intensity has the same value in all directions. Unlike the intensity from a blackbody, the intensity emitted from a real body does depend on direction. The energy leaving a real surface dA of temperature T per unit time in the wavelength interval dλ and within the solid angle dΩ = sinθdθdϕ is then given by

d 2Ql (q, f, T )d l = I l ( q, f, T ) dA cos q d W d l. (2.1)

For a blackbody, the intensity Iλb(T) is independent of direction. The T notation is introduced here to clarify when quantities are temperature-dependent, so the blackbody intensity is Iλb(T). It is important to note that the T-dependence may imply a spatial variation (x,y,z in Cartesian coordinates) in the case of nonisothermal surfaces. The energy leaving a black area element per unit time within dλ and dΩ is

d 2Ql,b (q, f, T )d l = I lb ( T ) dA cos qd Wd l. (2.2)

The directional spectral emissivity is then defined as the ratio of the rate at which a real surface radiates energy in each direction and at a given wavelength compared to a blackbody: Directional spectral emissivity º el ( q, f, T )

º

d 2Ql ( q, f, T ) d l

d 2Ql,b ( q, f, T ) d l

=

I l ( q, f, T ) (2.3) . I lb ( T )

This is the most fundamental emissivity expression because it includes dependencies on wavelength, direction, and surface temperature. Surface temperature influences Iλ in two ways: via the Boltzmann distribution of energy states in the surface, giving rise to Planck’s distribution, and the bulk electromagnetic properties of the surface material, which are discussed in more detail in Chapter 3. The former effect is “cancelled out” by the blackbody intensity in the denominator of Equation (2.3), so the temperature influences ελ only through the electromagnetic properties of the surface material.

58

Thermal Radiation Heat Transfer

FIGURE 2.2  Pictorial descriptions of directional and hemispherical radiation properties. For spectral properties, the subscript λ is added the property definitions: (a) directional emissivity ε(θ,ϕ,T); (b) hemispherical emissivity ε(T); (c) directional absorptivity α(θi,ϕi,T); (d) hemispherical absorptivity α(T); (e) bidirectional reflectivity ρ(θr,ϕr,θi,ϕi,T); (f) directional-hemispherical reflectivity ρ(θi,ϕi,T); (g) hemispherical-directional reflectivity ρ(θ,ϕ,T); (h) hemispherical reflectivity ρ(T).

Example 2.1 A surface heated to 1000 K in vacuum has a directional spectral emissivity of 0.70 at a wavelength of 5 µm. The emissivity is independent of azimuthal angle φ. What is the emitted spectral intensity in this direction? From Equation (2.3), Iλ(5μm,60°,1000 K) = ελ(5μm,60°,1000K) Iλb (5µm,1000K). Using Equation (1.21) for Iλb(5 µm, 1000 K), Il ( 5 mm, 60°,1000 K ) = 0.70 ´

l

5

(e

2C1 C 2 / lT

)

-1

= 0.70 ´

(

2 ´ 0.59552 ´ 108

(

)

55 e14387.75/ 5000 - 1

)

= 0.70 ´ 2272.61 = 1591 W/ m2 × mm × sr .

59

Radiative Properties at Interfaces

From the directional spectral emissivity in Equation (2.3), an averaged emissivity can be derived by two approaches: averaging over all wavelengths, or averaging over all directions.

2.2.2 Directional Total Emissivity, ε(θ,ɸ,T) The radiation emitted at all wavelengths into direction (θ,ϕ) is found by integrating the directional spectral intensity to obtain the directional total intensity (the term total denotes that radiation at all

ò

¥

wavelengths is included): I (q, f, T ) º I l (q, f, T )d l . Using Equation (1.33), the total intensity is l =0 ¥ sT 4 I b (T ) = I lb (T )d l = . p l =0 The directional total emissivity is the ratio of I(θ,ϕ,T) for the real surface to I b(T) emitted by a blackbody at the same temperature; that is

ò

Directional total emissivity º e ( q, f, T ) º =

p

¥

ò

l =0

I ( q, f, T ) I b (T )

I l ( q, f, T ) d l sT 4

(2.4) .

The Iλ(θ,ϕ,T) in the numerator can be replaced in terms of ελ(θ,ϕ,T) by using Equation (2.3) to give directional total emissivity in terms of the directional spectral emissivity

e ( q, f, T ) =



p

ò

¥

l =0

el ( q, f, T ) I lb ( T ) d l sT 4

=

ò

¥

l =0

el ( q, f, T ) Elb ( T ) d l sT 4

, (2.5)

Thus, if the wavelength dependence of ελ(θ,ϕ,T) is known, ε(θ,ϕ,T) can be found from Equation (2.5). The total directional emissivity can be calculated accurately only if the spectral-directional emissivity is available. Total emissivity is an integrated average weighted by the blackbody intensity Iλb(T). Therefore, it strongly correlates with the ελ values within the spectral range where Iλb(T) peaks, which varies with the temperature of the emitting surface, T. Note that the temperature dependence of the directional total emissivity arises from two sources: the spectral distribution of the emitted intensity, which affects the weighting of the spectral emissivity, ελ, through Equation (2.5), and the temperature dependence of ελ itself, which is influenced via the electromagnetic properties of the surface as described in Chapters 3 and 8. Example 2.2 For a surface at 700 K, the ελ(θ,ϕ,T) can be approximated by 0.8 for λ = 0–5 μm, and ελ(θ,ϕ,T) = 0.4 for λ > 5 μm. What is the value of ε(θ,ϕ,T)? From Equation (2.5), 5T



ò

e ( q, f, T ) = 0.8 0

Elb (T ) sT

5



ò

d ( l T ) + 0 .4 5T

Elb (T ) sT 5

d ( lT ) .

From Equation (1.38) and by using Equation (1.41),

e ( q, f, T ) = 0.8F0 -3500 + 0.4F3500 -¥ = 0.8 ´ 0.38291 + 0.4 ´ 0.61709 = 0.553.



60

Thermal Radiation Heat Transfer Because 61.7% of the emitted blackbody radiation at 700 K is in the region of 5 μm, the result is heavily weighted toward the emissivity value of 0.4.

2.2.3 Hemispherical Spectral Emissivity, ελ(T) Now return to Equation (2.3) and consider the average obtained by integrating the directional spectral quantities over all directions of a hemispherical envelope centered over dA (Figure 2.2b). The spectral radiation emitted by a unit surface area into all directions of the hemisphere is the hemispherical spectral emissive power, which is found by integrating the spectral energy per unit solid angle over all solid angles. This is analogous to Equation (1.19) for a blackbody and is given by (see integration notation in Section 2.1) 2 p p/2



El (T ) º

ò ò I (q, f, T ) cos q sin qdqdf =ò I (q, f, T ) cos qdW. l

l

f= 0 q= 0

Ç

FIGURE 2.2  Pictorial descriptions of directional and hemispherical radiation properties. For spectral properties, the subscript λ is added the property definitions: (i) bidirectional transmissivity τ(θi,ϕi,θt,ϕt,Τ); (j) directional-hemispherical transmissivity τ(θi,ϕi,Τ); (k) hemispherical-directional transmissivity τ(θt,ϕt,Τ); and (l) hemispherical transmissivity τ(Τ).

61

Radiative Properties at Interfaces

Using Equation (2.3), this becomes

ò

El ( T ) = I lb ( T ) el ( q, f, T ) cos qd W. (2.6)



Ç

For a blackbody, the hemispherical spectral emissive power is Eλb(T) = πIλb(T). The ratio of actual to blackbody emission from the surface is then the hemispherical spectral emissivity Hemispherical spectral emissivity º el ( T ) º

El ( T )

El b ( T )

1 el ( q, f, T ) cos qd W. = p

(2.7)

ò Ç

2.2.4 Hemispherical Total Emissivity, ε(T) Hemispherical total emissivity represents the average of the spectral directional emissivity over both wavelength and direction. To derive the hemispherical total emissivity, consider that from a unit area, the spectral emissive power in any direction is derived from Equation (2.3) as ελ(θ,ϕ,T)Iλb(T) cosθ. This is integrated over all λ and all directions to give the hemispherical total emissive power. Dividing by σT4, the hemispherical emissive power of a blackbody, results in the hemispherical total emissivity: E (T ) Hemispherical º e (T ) º Eb ( T ) total emissivity =

òò Ç

¥

l =0

[el ( q, f, T ) I lb ( T ) d l]cos qd W sT 4

(2.8) .

Using Equation (2.5), this becomes: e (T ) =



1 e ( q, f, T ) cos qd W. (2.9) p

ò Ç

If the order of integrations in Equation (2.8) is interchanged and Equation (2.7) is then used, a third form is obtained:



e (T ) =

p

ò

¥

l =0

e l ( T ) I lb ( T ) d l sT 4

=

ò

¥

l =0

e l ( T ) El b ( T ) d l sT 4

. (2.10)

To interpret Equation (2.10) physically, consider Figure 2.3. The spectral profile of ελ(T) is provided at surface temperature T = 1000 K in Figure 2.3a. The solid curve in Figure 2.3b is the hemispherical spectral emissive power for a blackbody at T. The area under the solid curve is σT4, which is the denominator in Equation (2.10) and equals the radiation emitted per unit area by a blackbody including all wavelengths and directions. The dashed curve in Figure 2.3b is the product ελ(T)Eλb(T), and the area under this curve is the numerator of Equation (2.10), which is the emission from the real surface. Hence, ε(T) is the ratio of the area under the dashed curve to that under the solid curve.

62

Thermal Radiation Heat Transfer

FIGURE 2.3  Physical interpretation of hemispherical spectral and total emissivities. (a) Measured emissivity values at T = 1000 K; (b) interpretation of emissivity as ratio of actual emissive power (dashed curve) to blackbody emissive power (solid curve).

At each λ, the ελ(T) is the ordinate of the dashed curve divided by the ordinate of the solid curve. In Figure 2.3b, the hemispherical spectral emissivity at 7.5 μm is ελ (λ = 7.5 μm, T = 1000 K) = b/a (the ratio of the lower curve value to those of the upper curve in Figure 2.3b). Example 2.3 A surface at 1000 K has an ε(θ,φ,TA) that is independent of φ, but depends on θ as in Figure 2.4. What are the hemispherical total emissivity and the hemispherical total emissive power? The ε(θ, 1000 K) is approximated by the function 0.85cosθ. Then, from Equation (2.9), the total hemispherical emissivity is

ò

e ( T ) = 0.85 cos2 qd W = = 1.70

3

p/ 2

1 p

2p p / 2

ò ò 0.85 cos

 =0 q =0

cos q = 0.567. 3 0

2

q sin qd qd f

63

Radiative Properties at Interfaces

FIGURE 2.4  Directional total emissivity at 1000 K for Example 2.3.

FIGURE 2.5  Hemispherical spectral emissivity for Example 2.4. Surface temperature T = 1000 K. The hemispherical total emissive power is then

E (T ) = e (T ) sT 4 = 0.567 ´ 5.6704 ´ 10 8 ´ 1, 0004 = 32,150 W/m2 .

Generally, ε(θ,T) cannot be approximated well by a convenient analytical function, and numerical integration of Equation (2.9) is necessary.

Example 2.4 The ελ(TA) for a surface at TA = 1000 K is approximated by the solid line as shown in Figure 2.5. What are the hemispherical total emissivity and the hemispherical total emissive power of the surface? From Equation (2.10), e (T ) =

1 sT 4 1 + s



ò

el (T ) Elb (T ) d l =

l =0 6000

ò

1 s

2000

ò 0 .1

lT = 0

Elb (T ) 1 0 .4 d ( lT ) + s T5

lT = 2000

Elb (T ) d ( lT ) T5



ò

Elb (T ) 0 .2 d ( lT ) . T5

lT = 6000



64

Thermal Radiation Heat Transfer From Equations (1.38) and (1.41), this is evaluated as e (TA ) = 0.1F0®2000 + 0.4 (F0®6000 F0®2000 ) + 0.2 (1F0®6000 ) = 0.3275.



It is often easier and/or more accurate to numerically integrate Equation (2.10) using the actual ελ(T). The hemispherical total emissive power is E (T ) = e (T ) sT 4 = 0.3275 ´ 5.67040 ´ 10-8 ´ 10004 = 18.57 kW/m2 .



Example 2.5 The ελ(T) for a surface at T = 650 K is approximated as shown in Figure 2.6. What is the hemispherical total emissivity? Numerical integration is used for the portion where 3.5 ≤ λ ≤ 9.5 µm. This part of the spectral emissivity is given by ε(T)= 1.27917 − 0.10833 λ. The λT bounds are 3.5 × 650 = 2275 µm · K and 9.5 × 650 = 6175 µm · K. Then, as in Example 2.4, and by the use of Equations (1.32), (1.34), (1.37), and (1.41),



e (650 K ) = 0.90F0®2275 +

1 s6504

9 .5

2pC1

ò (1.279170.10833l ) l (e 5

3 .5

C2 / 650 l

)

1

dl



+ 0.25 (1F0®6175 ) . Evaluating the integral numerically yields

e (650 K ) = 0.90 ´ 0.11514 + 0.40141 + 0.25 (1 - 0.75214 ) = 0.5670.

2.3 ABSORPTIVITY Absorptivity is defined as the fraction of energy incident on a body that is absorbed by the body. The incident radiation depends on the radiative conditions (spectral intensity) at the source of the incident energy. The spectral distribution of incident radiation is independent of the temperature or physical nature of the absorbing surface unless radiation emitted from the surface is partially reflected from

FIGURE 2.6  Spectral emissivity for Example 2.5.

65

Radiative Properties at Interfaces

the source or surroundings back to the surface. Compared with emissivity, the absorptivity has additional complexities, because directional and spectral characteristics of the incident radiation must be included along with the absorbing surface temperature. It is desirable to have relations between emissivity and absorptivity so that measured values of one will allow the other to be calculated. These relations are developed in this section, along with the absorptivity definitions.

2.3.1 Directional Spectral Absorptivity, αλ(θi,φi,T) Figure 2.2c illustrates the energy incident on a surface element dA from the (θi,ϕi) direction. The line from dA in the direction (θi,ϕi) passes normally through an area element dAi on the surface of a hemisphere of radius R centered over dA. The incident spectral intensity passing through dAi is Iλ,i(θi,ϕi). This is the energy per unit area of the hemisphere, per unit solid angle dΩi, per unit time, and per unit wavelength interval. The energy within the incident solid angle dΩi strikes the area dA of the absorbing surface. Hence, the energy per unit time incident from the direction (θi,ϕi) in the wavelength interval dλ is

d 2Ql,i ( qi , fi ) d l = I l,i ( qi , fi ) dAi d Wd l = I l,i ( qi , fi ) dAi

dA cos qi d l (2.11) R2

where dA cos θi /R2 is the solid angle dΩ subtended by dA when viewed from dAi as in Figure 2.7a. Equation (2.11) can also be expressed in terms of the solid angle dΩi in Figure 2.7b. This is the solid angle subtended by dAi when viewed from dA. Solid angle dΩi has its vertex at dA and hence is the convenient solid angle to use when integrating to obtain energy incident on dA from more than one direction. For a nonabsorbing medium in the region above the surface, as is being considered here, the incident intensity does not change along the path from dAi to dA (see Section 1.9.5). For these reasons, in the figures that follow, the energy incident on dA from dAi will be pictured as that arriving in dΩi, as shown in Figure 2.7b, rather than as the energy leaving dA in dΩ, as in Figure 2.7a. To place Equation (2.11) in terms of dΩi, note that

d WdAi cos qi = dAi

dA cos qi dAi = 2 dA cos qi = d W i dA cos qi . (2.12) R2 R

Equation (2.11) is then

d 2Ql,i ( qi , fi ) d l = I l,i ( qi , fi ) d W i dA cos qi d l. (2.13)

FIGURE 2.7  Equivalent ways of showing energy from dAi that is incident upon dA. (a) Incidence within solid angle dΩi having origin at dAi; (b) incidence within solid angle dΩ having origin at dA.

66

Thermal Radiation Heat Transfer

The fraction of the incident energy d2Qλ,i(θi,ϕi,T)dλ that is absorbed is defined as the directional spectral absorptivity αλ(θi,ϕi,T). The amount of the incident energy that is absorbed is d2Qλ,a(θi,ϕi,T)dλ. Then, the ratio is formed Directional spectral absorptivity º a l (qi , fi , T )

=

(2.14) d 2Ql,a (qi , fi , T )d l . I l,i (qi , fi )dA cos qi d W i d l

In addition to depending on the wavelength and direction of the incident radiation, the spectral absorptivity is also a function of the absorbing surface temperature, as it affects the electromagnetic properties of the surface, in the same way that temperature influences ελ. The directional spectral absorptivity also has a probabilistic interpretation – it is the finite probability that an incident photon* contained within a solid angle dΩi centered on (θi,ϕi) and having an incident energy corresponding to a wavelength between λ and λ + dλ will be absorbed by a surface. We can write this as

(

)

a l (qi , fi , T ) = P abs qi , fi , l , T . (2.15)



The “x|y” notation highlights that this is a conditional probability, i.e. the probability that a photon will be absorbed is conditional on it having a given incident direction and wavelength, and the surface having a defined temperature, T.

2.3.2 Kirchhoff’s Law Kirchhoff’s law relates the emitting and absorbing abilities of a body and is derived from the Second Law of Thermodynamics. Kirchhoff’s law has necessary conditions imposed on it, depending on whether spectral, total, directional, or hemispherical quantities are being considered. From Equations (2.1) and (2.3), the energy emitted per unit time by an element dA in a wavelength interval dλ and solid angle dΩ is

d 2Ql,e d l = el ( q, f, T ) I lb ( T ) dA cos qd Wd l. (2.16)

If the element dA at temperature T is placed in an isothermal black enclosure also at temperature T, then the intensity of the energy incident on dA from the direction (θi,ϕi) (recalling the isotropy of intensity in a black enclosure) is Iλb(T). Since both element dA and the emitting element on the black enclosure surface are at the same temperature, T, the Clausius statement of the Second Law requires the net radiation heat transfer at that wavelength to be zero. Setting these terms as equal results in





d 2Ql,e d l = el ( q, f, T ) I lb ( T ) dA cos qd Wd l = d 2Ql,a d l = a l ( qi = q, f i = f, T ) I lb ( T ) dA cos ( qi = q ) d Wd l

(2.17)

el ( q, f, T ) = a l ( qi = q, f i = f, T ) . (2.18)

This relation between the properties of the material holds without restriction. This is the most general form of Kirchhoff’s law. Were this condition not satisfied, one could conceive an arrangement * While we conceptually speak of “a photon” interacting with the surface, it is important to keep in mind the limitations of the “RTE photon” interpretation discussed in Chapter 1. For more discussion, see Section 1.5 and Mishchenko 2014a.

67

Radiative Properties at Interfaces

of surfaces that would permit net energy transfer between two objects at the same temperature, which would violate the Second Law of Thermodynamics. While Equation (2.18) was formulated by considering radiative transfer between an object within a black-walled enclosure at the same temperature, these conditions are not required in order for Kirchhoff’s law to apply. However, the relationship between ελ(θ,ϕ,T) and αλ(θ,ϕ,T) must generally apply for the Clausius statement to be satisfied under radiative equilibrium. This approach exemplifies the methodology for applying the Second Law of Thermodynamics to radiative transfer: variables and properties must always be related in a way such that the Second Law is satisfied under all possible scenarios. This methodology will be applied throughout the textbook. The equality between αλ(θ,ϕ,T) and ελ(θ,ϕ,T) implies that, if the former is a probability, then the latter must also be a probability. But a probability of what? As described in Chapter 1, local thermodynamic equilibrium normally applies within surfaces, and therefore the energy levels of photons within the surface obey a probability density proportional to Planck’s distribution at the surface temperature. A fraction of the radiative transfer equation (RTE) photons emitted by matter within a thin layer beneath the surface cross the surface boundary and issue into the surrounding medium (e.g., a vacuum or the air). The spectral directional emissivity can then be interpreted as the probability that, upon reaching the interface, a photon will cross the surface boundary and be emitted within a solid angle dΩi centered on (θi,ϕi)

(

)

el (q, f, T ) = P emit q, f, l , T . (2.19)

In this context, Kirchhoff’s law can be explained using the concept of microscopic reversibility. In thermodynamics, processes are irreversible due to some sort of inherent randomness. If a process is reduced to its simplest possible form, it must be time-reversible. Accordingly, the trajectory of a photon* from its origin within the surface to its emission into the surrounding medium must be reversible, hence the probability of emission must equal the probability of absorption. As discussed in Chapters 3 and 8, radiation is polarized, having two wave components vibrating at right angles to each other and the propagation direction. For the special case of blackbody radiation, the two components are equal. Equation (2.18) actually holds only for each component of polarization, and for Equation (2.18) to be valid as written for all incident energy, the incident radiation must have equal components of polarization. Strictly speaking, Equation (2.18) also only applies when the surfaces are at local thermodynamic equilibrium, under which conditions the populations of energy states that govern the emission and absorption processes are closely approximated by their equilibrium distributions at the local temperature. As discussed in Chapter 1, this condition is almost always satisfied, except for highly nonequilibrium processes such as those that involve ultrafast lasers.

2.3.3 Directional Total Absorptivity, α(θi,φi,T) The directional total absorptivity is the energy including all wavelengths that is absorbed from a given direction, divided by the energy incident from that direction. The total energy incident from the given direction is obtained by integrating the spectral incident energy (Equation (2.13)) over all wavelengths to obtain ¥



d 2Qi (qi , fi ) = dA cos qi d W i

òI

l ,i

(qi , fi )d l. (2.20)

l =0

* Again, it is important to remember the limitations of the RTE photon. Microscopic reversibility comes from the timereversal invariance of Maxwell’s equations, which holds in the absence of an external magnetic field. Consequently, our conceptual “RTE photons” satisfy detailed balancing.

68

Thermal Radiation Heat Transfer

The radiation absorbed is determined from Equation (2.20) by integrating over all λ, ¥



d Qa (qi , fi , Ti ) = dA cos qi d W i 2

ò a (q , f , T ) I l

i

i

l ,i

(qi , fi )d l. (2.21)

l =0

The ratio is then formed: Directional total absorptivity º a(qi , fi , T ) =

¥

ò =

l =0

d 2Qa (qi , fi , T ) d 2Qi (qi , fi )

a l ( q i , f i , T ) I l ,i ( q i , f i ) d l

ò

¥

l =0

(2.22) .

I l,i (qi , fi )d l

Using Kirchhoff’s law, Equation (2.18), an alternative form is



a ( qi , fi , Ti ) =

ò

¥

l =0

e l ( q i , f i , T ) I l ,i ( q i , f i ) d l

ò

¥

l =0

I l ,i ( q i , f i ) d l

. (2.23)

The directional total absorptivity can also be interpreted as the probability that a photon incident from a solid angle dΩi centered on (θi,ϕi) will be absorbed for any wavelength. This probability must account for the probability density of incident wavelengths, ¥

òI

p ( l ) = Il ( l )



l

( l ) dl (2.24)

0

which, in many cases, is proportional to Planck’s distribution, as described in Section 1.13. Equation (2.22) can then be rewritten in terms of probabilities as

(

)

¥

ò (

)

a(qi , fi , T ) = P abs qi , fi , T = P abs qi , fi , l, T p ( l ) d l. (2.25)     0

al ( qi ,fi ,T )

In the language of probability, the integral in Equation (2.25) is the marginalization over wavelength, and we can say that the wavelength dependence has been “marginalized out.”

2.3.4 Kirchhoff’s Law for Directional Total Properties The general form of Kirchhoff’s law, Equation (2.18), shows that the directional-spectral properties ελ(θ,ϕ,T) and αλ(θi,ϕi,T) are equal. Now examine this equality for the directional total quantities. This can be accomplished by comparing a special case of Equation (2.23) with Equation (2.5). If the incident intensity originates from a blackbody source at the same temperature as the surface, it follows from the Second Law of Thermodynamics that the directional total emissivity must equal the directional total emissivity in each direction. In this scenario



a ( qi , fi , T ) =

ò

¥

l =0

e l ( q = q i , f = f i , T ) I lb ( T ) d l

ò

¥

l =0

(

I lb ( T ) d l = sT 4 /p

)

= e ( q = qi , f = fi , T ) . (2.26)

69

Radiative Properties at Interfaces

Hence, when ελ(θi,ϕi,T) and αλ(θi,ϕi,T) are dependent on wavelength, there is the equality α(θi,ϕi,T) = ε(θi,ϕi,T) only when the spectral distribution of incident and blackbody intensities are identical. In practice, this happens most often when a small object is at radiative equilibrium with large, isothermal surroundings, in which case the incident intensity field corresponds to that of a blackbody at the temperature of the surroundings. Otherwise, α(θi,ϕi,T) = ε(θi,ϕi,T) only if ελ and αλ are independent of wavelength, i.e. a directional-gray surface.

2.3.5 Hemispherical Spectral Absorptivity, αλ(T) Now, consider energy in a wavelength interval dλ. The hemispherical spectral absorptivity is the fraction of the spectral energy that is absorbed from the spectral energy that is incident from all directions of a surrounding hemisphere (Figure 2.2d). The spectral energy from an element dAi on the hemisphere that is intercepted by a surface element dA is given by Equation (2.13). The incident energy on dA from all directions of the hemisphere is then given by the integral

dQl,i d l = dAd l

òI Ç

l ,i

(qi , fi ) cos qi d W i . (2.27a)

The amount absorbed is given by integrating Equation (2.14) over the hemisphere:

dQl,a (T )d l = dAd l

ò a (q , f , T ) I Ç

l

i

i

l ,i

(qi , fi ) cos qi d W i . (2.27b)

The ratio of these quantities gives Hemispherical spectral absorptivity º a l (T ) =

dQl,a (T )d l dQl,i d l

ò a (q , f , T )I (q , f ) cos q dW = ò I (q , f ) cos q dW



Ç

l

i

Ç

l ,i

i

l ,i

i

i

i

i

i

i

i

(2.28)

i

or, in terms of emissivity, by using Kirchhoff’s law, Equation (2.18),



al (T ) =

òe Ç

l

( q = qi , f = fi , T ) I l,i ( qi , fi ) cos qi dWi

ò

Ç

I l,i ( qi , fi ) cos qi d W i

. (2.29)

The hemispherical spectral absorptivity and emissivity can now be compared by looking at Equations (2.29) and (2.7). For the general case where αλ and ελ are functions of (λ,θ,ϕ,T), αλ(T) = ελ(T) only if Iλi(θi,ϕi) is independent of θi and ϕi, that is, if the incident spectral intensity is uniform from all directions. If this is so, the Iλi can be canceled in Equation (2.29) and the denominator becomes π, and Equation (2.29) then compares with Equation (2.7). Continuing with the probabilistic interpretation, the hemispherical spectral absorptivity can be viewed as a marginalization of the absorption probability at a given wavelength over all incident directions, so Equation (2.28) can be rewritten as

ò (

)

a l (T ) = P abs qi , fi , l, T p ( W i ) d W i (2.30)    Ç

al ( qi ,fi ,T )

70

Thermal Radiation Heat Transfer

where p(Ωi) is the probability density for incident angles, p ( Wi ) =



I l,i ( qi , fi ) cos qi

ò

Ç

I l,i ( qi , fi ) cos qi d W i

. (2.31)

Since the incident azimuthal and polar angles are independent, this probability density can be written as the product of probability densities p(θi) and p(ϕi) (see, e.g., Walpole et al. 2016) leading to 2p p 2



a l (T ) =

abs q , f , l, T ) p ( q ) p ( f ) d q d f . (2.32) ò ò P(   i

0

i

i

i

i

i

al ( qi ,fi ,T )

0

If the surface irradiation is diffuse (e.g., from isothermal surroundings) then p(θi) = 2cos(θi)sin(θi) and p(ϕi) = 1/(2π) following Equations (1.86) and (1.87). This is not generally the case, however; for example, terrestrial solar irradiation typically has a lobular distribution with a peak centered on the direction of the sun and a width caused by atmospheric scattering. For the case where αλ(θ,ϕ,T) = ελ(θ,ϕ,T) = αλ(T) = ελ(T), that is, the directional spectral properties are independent of angle, the hemispherical spectral properties are related by αλ(T) = ελ(T) for any angular variation of incident spectral intensity. Such a surface is termed a diffuse spectral surface.

2.3.6 Hemispherical Total Absorptivity, α(T) The hemispherical total absorptivity represents the fraction of energy absorbed that is incident from all directions of the enclosing hemisphere and for all wavelengths, as shown in Figure 2.2d. The total incident energy that is intercepted by a surface element dA is determined by integrating Equation (2.13) over all λ and all (θi,ϕi) of the hemisphere, which results in

é dQi = dA ê ë Ç

ù I l,i (qi , fi )d l ú cos qi d W i . (2.33) l =0 û

òò

¥

Similarly, by integrating Equation (2.14), the total amount of energy absorbed is

ù é¥ dQa (T ) = dA ê a l (qi , fi , T )I l,i (qi , fi )d l ú cos qi d W i . (2.34) ê ú Ç ë l =0 û

ò ò

The ratio of absorbed to incident energy provides the definition Hemispherical total absorptivity º a(T ) =

é ê Ç = ë

dQa (T ) dQi

ù a l (qi , fi , T )I l,i (qi , fi )d l ú cos qi d W i (2.35) l =0 û . é ¥ ù I l,i (qi , fi )d l ú cos qi d W i ê Ç ë l =0 û

ò ò

¥

ò ò

Following the probabilistic interpretation, the hemispherical total absorptivity comes from marginalizing the photon absorption probability over both incident wavelength and direction,

71

Radiative Properties at Interfaces ¥ 2p p 2



a(T ) =

abs q , f , l, T ) p ( l ) p ( q ) p ( f ) d q d f d l (2.36) ò ò ò P(   i

0 0

i

i

i

i

i

al ( qi ,fi ,T )

0

which is the probability that an incident photon will be absorbed for any incident wavelength and direction. Note that the marginalization has made α(T) independent of θi, ϕi, and λ. Equation (2.35) can also be written in terms of emissivity by using Kirchhoff’s law, Equation (2.18),



é ê Ç a (T ) = ë

ù el ( q = qi , f = fi , T ) I l,i ( qi , fi ) d l ú cos qi d W i l =0 û . (2.37) é ¥ ù I l,i ( qi , fi ) d l ú cos qi d W i ê ë l =0 û Ç

ò ò

¥

òò

Equation (2.37) can be compared with Equation (2.8) to determine the conditions when the hemispherical total absorptivity and emissivity are equal. From Equation (2.8),



e (T ) =

òò Ç

¥

l =0

[el ( q, f, T ) I lb ( T ) d l]cos qd W sT 4

. (2.38)

The comparison reveals that, for the general case when ελ(θ,ϕ,T) and αλ(θi,ϕi,T) vary with both wavelength and angle, α(T) = ε(T) only when the incident intensity is independent of the incident angle and has the same spectral form as that emitted by a blackbody with temperature equal to the surface temperature T; in other words, Iλ,i (θ,ϕ) = C(θ,ϕ) Ιλb(Τ). In general, this only happens when an object is at radiative equilibrium with its surroundings, in which case there can be no net energy transfer, and thus Kirchhoff’s law, in this form, is rarely useful for analyzing radiative transfer. (The only other scenario in which α(T) = ε(T) is when these properties are independent of wavelength, i.e., gray, as discussed below.) Some more restrictive cases are summarized in Table 2.2. A special case of hemispherical total absorptivity is assigned in Homework Problem 2.1 at the end of this chapter. If there is a uniform incident intensity from a gray source at Ti, and if ελ(T) is

72

Thermal Radiation Heat Transfer

independent of T, then the hemispherical total absorptivity for the incident radiation is equal to the hemispherical total emissivity of the material evaluated at the source temperature Ti, α(T) = ε(Τi). This relation also applies if the spectral directional absorptivity/emissivity is independent of direction (i.e. a diffuse spectral surface). Example 2.6 The hemispherical spectral emissivity of a surface at 300 K is approximated by a step function: ελ(T) = 0.8 for 0 ≤ λ ≤ 3 µm, and 0.2 for λ > 3 µm. What is the hemispherical total absorptivity for diffuse incident radiation from a black source at Ti = 1000 K? What is it for diffuse incident solar radiation? For diffuse incident radiation, αλ(T) = ελ(T), and for a black source at Ti, Equation (2.37) becomes



a (T ) =

ò

¥

l =0

el (T ) Ilb,i (Ti ) d l

ò

¥

l =0

Ilb,i (Ti ) d l

=

ò

¥

l =0

el (T ) Elb,i (Ti ) d l sTi 4

.

Expressing α(T) in terms of the two wavelength regions over which ελ(T) is constant gives αλ(T) =  0.8F0→3Ti + 0.2(1 − F0→3Ti). From Equation (1.41), F0→3 μm ⋅ 1000 K = 0.27323 so that

a(T ) = 0.8 F0®3000 + 0.2 (1 - F0®3000 ) = 0.364.

For incident solar radiation, Ti = 5780 K, F0→3 μm ⋅ 5780 K = 0.97880 so that

a(T ) = 0.8 F0®17.340 + 0.2 (1 - F0®17,340 ) = 0.787.

Hence, for a surface with this type of spectral emissivity variation, there is a considerable increase in α(T) by the shift of the incident-energy spectrum toward shorter wavelengths as the source temperature is raised.

2.3.7 Diffuse-Gray Surface As discussed in Chapters 4 and 5, a common assumption in enclosure calculations is that surfaces are diffuse-gray. Diffuse signifies that the directional emissivity and directional absorptivity do not depend on direction. Hence, for emission, the emitted intensity is uniform over all directions, as for a blackbody. The term gray signifies that the spectral emissivity and absorptivity do not depend on wavelength. They can, however, depend on temperature. Thus, at each surface temperature, for any wavelength the emitted spectral radiation is a fixed fraction of blackbody spectral radiation. The diffuse-gray surface therefore absorbs a fixed fraction of incident radiation from any direction and at any wavelength. It emits radiation that is a fixed fraction of blackbody radiation for all directions and all wavelengths (this is the motivation for the term “gray”). The directionalspectral absorptivity and emissivity are then independent of λ, θ, ϕ so that αλ(θi,ϕi,T) = α(T) and ελ(θ,ϕ,T) = ε(T), and from Kirchhoff’s law, Equation (2.18), α(T) = ε(T). In Equations (2.8), (2.35), and (2.37), since the αλ(θi,ϕi,T) and ελ(θ,ϕ,T) are not functions of either direction or wavelength, they can be taken out of the integrals. Then, for a diffuse-gray surface, the directional-spectral and hemispherical-total values of absorptivity and emissivity are all equal, and the hemispherical total absorptivity is independent of the nature of the incident radiation. This provides a great simplification for radiation transfer analysis, but rarely does this reflect the true character of the surface properties. The diffuse-gray assumption is frequently confused with Kirchhoff’s law, but the latter only leads to α(T) = ε(T) under conditions of radiative equilibrium, which do not involve net radiative transfer between surfaces. These are entirely separate concepts.

73

Radiative Properties at Interfaces

For diffuse-gray and other surface characteristics, the restrictions for applying Kirchhoff’s law are summarized in Table 2.2. Example 2.7 The surface in Example 2.5 at T = 650 K is subject to incident radiation from a diffuse-gray source at Ti = 925 K. What is the hemispherical-total absorptivity? For a diffuse-gray source at Ti, Iλ,i(θi,φi) = CIλb(Ti) where C is a constant. From Table 2.2, since there is no angular dependence, αλ(T) = ελ(T). Then from Equation (2.37), é





ò êò e a (T ) = ë é ò êëò l =0

l

ù

(T ) CIlb (Ti ) d l ú cos qi d Wi û

ù CIlb (Ti ) d l ú cos qi d Wi l =0 û 

=

1 sTi 4



òe

l

(T ) Elb (Ti ) d l.

l =0

As in Example 2.5, using λTi values of 3.5 × 925 = 3238 μm · K and 9.5 × 925 = 8788 μm · K,



a = 0.90 F0®3238

1 + s9254

9 .5

ò (1.27917 - 0.10833l) l (e 5

3 .5

2pC

1 C2 / 925l

- 1)

dl



+ 0.25(1 - F0®8788 ). Numerical integration yields

a = 0.90 ´ 0.32650 + 0.37872 + 0.25 (1- 0.88377 ) = 0.7016.

2.4 REFLECTIVITY The reflective properties of a surface are more complicated to specify than the emissivity or absorptivity. This is because reflected energy depends not only on the angle at which the incident energy impinges on the surfaces, but also on the direction being considered for the reflected energy. Reflectivities are also functions of the reflective surface temperature T. For clarity, we omit explicitly showing this temperature-dependence, but it should be understood as for emissivity and absorptivity. The important reflectivity quantities are now defined.

2.4.1 Spectral Reflectivities 2.4.1.1 Bidirectional Spectral Reflectivity, ρλ(θr,ϕr,θi,ϕi) Consider spectral radiation incident on a surface from direction (θi,ϕi), as in Figure 2.2e. Part of this energy is reflected into the (θr,ϕr) direction and accounts for a portion (but not necessarily all) of the reflected intensity in the (θr,ϕr) direction. The subscript “r” denotes quantities evaluated at the reflected angle. The entire Iλ,r(θr,ϕr) is the result of summing the reflected intensities produced by the incident intensities Iλ,i(θi,ϕi) from all incident directions (θi,ϕi) of the hemisphere surrounding the surface element. The contribution to Iλ,r(θr,ϕr) by the incident energy from only one direction (θi,ϕi) is designated as Iλ,r(θr,ϕr,θi,ϕi). The energy from direction (θi,ϕi) intercepted by dA per unit area and wavelength is, from Equation (2.13),

d 2Ql,i (qi , fi )dl = I l,i (qi , fi ) cos qi dW i . (2.39) dAdl

74

Thermal Radiation Heat Transfer

The bidirectional spectral reflectivity is a ratio expressing the contribution Iλ,i(θi,ϕi)cosθidΩi makes to the reflected spectral intensity in the (θr,ϕr) direction: Bidirectional spectral reflectivity º rl (qr , fr , qi , fi )

=

I l,r (qr , fr , qi , fi ) (2.40) . I l,i (qi , fi ) cos qi dW i

The reflectivity also depends on surface temperature, but the T notation is omitted for simplicity. The ratio in Equation (2.40) is a reflected intensity divided by the intercepted energy arriving within solid angle dΩi. Having cosθidΩi in the denominator means that when ρλ(θr,ϕr,θi,ϕi)Iλ,i(θi,ϕi) cosθidΩi is integrated over all incidence angles (θi,ϕi) to provide the reflected intensity Iλ,r(θr,ϕr), the reflected intensity is properly weighted by the amount of energy incident from each direction. Moreover, since Iλ,r(θr,ϕr,θi,ϕi) is generally one differential order smaller than Iλ,i(θi,ϕi), the dΩi in the denominator prevents ρλ(θr,ϕr,θi,ϕi) from being a differential quantity. This also gives the bidirectional spectral reflectivity units of sr−1, which is unusual for a radiative property. While emissivity and absorptivity are probabilities of RTE photon emission and absorption, and thus are bound between zero and one, the bidirectional reflectivity is the product of the probability that a photon incident from the (θi,ϕi) direction having a wavelength λ will be reflected and the probability density function that defines the direction of reflection. In other words,

(

) (

)

rl (qr , fr , qi , fi ) cos qr d W r = p qr , fr ref , qi , fi , l P ref qi , fi , l dW r . (2.41)   probability density for probability of reflection direction on reflected direction into any direction

The probability that the photon will be reflected exactly into the (θr,ϕr) direction is infinitely small (≈0), but, by rearranging Equation (2.41), the probability that a photon will be reflected into a solid angle dΩr centered on the (θr,ϕr) direction is

(

)

P ref qi , fi , qr , fr , l = rl ( qi , fi , qr , fr ) cos qr d W r . (2.42)

The presence of the cosθr is important, since, in the case of a diffuse surface, the probability of reflection depends on cosθr through the projected area in the reflected direction. Therefore, a diffuse surface has a bidirectional reflectance that is independent of the reflectance angle, even though the probability of reflection depends on cosθr. Equation (2.40) leads to some convenient reciprocity relations. For a diffuse reflection, the incident energy from each (θi,ϕi) contributes equally to the reflected intensity for all (θr,ϕr). A mirror-like (specular) reflection is a special case and is discussed separately. The measurement and application of the bidirectional reflectivity are discussed by Zaworski et al. (1996a, b). 2.4.1.2 Reciprocity for Bidirectional Spectral Reflectivity The ρλ(θr,ϕr,θi,ϕi) is symmetric with regard to reflection and incidence angles; the reflectivity for energy incident at (θi,ϕi) and reflected into (θr,ϕr) is equal to that for energy incident at (θr,ϕr) and reflected into (θi,ϕi). In the context of the Second Law of Thermodynamics, this is demonstrated by considering a nonblack element dA2 in an isothermal black enclosure, as in Figure 2.8. For the isothermal condition, the net energy exchange between black elements dA1 and dA3 must be zero. The energy exchange is by two paths. The direct exchange along the dashed line is between black elements at the same temperature and hence is zero. If the net exchange along this path is zero and net exchange including all paths between dAl and dA3 is zero, then net exchange along the path having reflection from dA2 must be zero. For the energy traveling along the reflected path,

75

Radiative Properties at Interfaces

FIGURE 2.8  Enclosure used to examine reciprocity of bidirectional spectral reflectivity.

d 3Ql,1-2-3 d l = d 3Ql,3-2-1d l. (2.43)



The energy reflected from dA2 that reaches dA3 is

d 3Ql,1-2-3 d l = I l,r (qr , fr , qi , fi ) cos qr dA2 (dA3 cos q3 /S22 )d l, (2.44)

or, using Equation (2.40),



æ dA cos q ö d 3Ql,1-2-3 d l = rl (qr , fr , qi , fi )I l,1 (T ) cos qi ç 1 2 1 ÷ S1 è ø æ dA cos q ö cos qr dA2 ç 3 2 3 ÷ d l. S2 è ø

(2.45)

Similarly,



æ dA cos q ö d 3Ql,3-2-1d l = rl (qi , fi , qr , fr )I l,3 (T ) cos qr ç 3 2 3 ÷ S2 è ø æ dA cos q ö cos qi dA2 ç 1 2 1 ÷ d l. S1 è ø

(2.46)

Then from Equation (2.43) and because Iλ,1(T) = Iλ,3(T) = Iλb(T), there is the reciprocity relation

rl (qr , fr , qi , fi ) = rl (qi , fi , qr , fr ). (2.47)

The probabilistic interpretation based on RTE photons is particularly useful when considering microscopic reversibility, which requires any trajectory traveled by an individual photon to be

76

Thermal Radiation Heat Transfer

reversible.* Since the probability density function for the direction of incident photons is proportional to cosθi dΩi, we can write

r¢¢l ( l, qi , fi , qr , fr ) cos qr dW r cos qi dW i = r¢¢l ( l, qr , fr , qi , fi ) cos qi dWi cos qr dWr (2.48)       P ( ref qi ,fi ,qr ,fr ,l ) P ( ref qi ,fi ,qr ,fr ,l )

which results in Equation (2.47). There is some debate as to whether detailed balancing provides adequate proof of bidirectional reciprocity (Clarke and Parry 1985, Snyder et al. 1998). It is presently believed that reciprocity is valid for conventional reflecting materials, but there may be exceptions for materials with very unusual optical characteristics, or those that are in the presence of a strong magnetic field. An analysis of the bidirectional reflectivity distribution was made by Greffet and Nieto-Vesperinas (1998) using electromagnetic theory (basic electromagnetic theory relations are derived in Chapter 8 and applied in Chapter 3). Zhang et al. (2020) investigate Kirchhoff’s law for macroscopic objects that obey Helmholtz reciprocity. The results developed from these basic principles rather than from thermodynamic arguments indicate that reciprocity and Kirchhoff’s law are indeed valid. The analysis accounts for penetration of electromagnetic waves into the medium beyond the reflecting surface, the scattering of these waves within the medium, and their re-emergence through the surface as part of the reflected component. 2.4.1.3 Directional Spectral Reflectivities, ρλ(θr,ϕr), ρλ(θi,ϕi) Multiplying Iλ,r(θr,ϕr,θi,ϕi) by dλ cosθrdA dΩr and integrating over all θr and ϕr gives the energy per unit time reflected into the entire hemisphere due to an incident intensity from a single direction (θi,ϕi):

ò

d 2Ql,r (qi , fi )d l = d ldA I l,r (qr , fr , qi , fi ) cos qr d W r . Ç

From Equation (2.40), this equals

ò

d 2Ql,r (qi , fi )dl = dldAI l,i (qi , fi ) cos qi dW i rl (qr , fr , qi , fi ) cos qr dW r . (2.49) Ç

The directional-hemispherical spectral reflectivity is defined as the energy reflected into all solid angles divided by the energy incident from direction (θi,ϕi) (Figure 2.2f). This gives Equation (2.49) divided by the incident energy from Equation (2.39):



Directional-hemispherical spectral reflectivity d 2 Q (q , f )d l º rl (qi , fi ) = 2 l,r i i ( in terms of bidirectional spectral reflectivity ) d Ql,i (qi , fi )d l

ò

(

= rl (qr , fr , qi , fi ) cos qr d W r = P ref qi , fi , l

)

(2.50)

Ç

which amounts to marginalizing the bidirectional spectral reflectivity over all possible reflection directions to obtain the probability that a photon will be reflected in any direction conditional on the incident (θi,ϕi) direction. Likewise, by integrating Equation (2.40) over all incident directions (θi,ϕi):

ò

I l,r (qr , fr ) = rl (qr , fr , qi , fi ) I l,i (qi , fi ) cos qi dW i . (2.51) Ç

* Again, this is due to the time reversal invariance of Maxwell’s equations.

77

Radiative Properties at Interfaces

The hemispherical-directional spectral reflectivity is then defined as the reflected intensity into the (θr,ϕr) direction divided by the mean incident intensity: Hemispherical-directional spectral reflectivity º r (q , f ) ( in terms of bidirectional spectral reflectivity ) l r r

ò r (q , f , q , f )I (q , f ) cos q dW = (1 / p ) ò I (q , f ) cos q dW Ç

l

r

i

r

Ç

i

l ,i

l ,i

i

i

i

i

i

i

(2.52) i

(

)

= P ref qr , fr , l .

i

Equation (2.52) is also a marginalization of the bidirectional spectral reflectivity, but this time over incident directions, and ρλ(θr,ϕr) is the probability that a photon will be reflected from all incident directions conditional on the (θr,ϕr) direction, i.e. it is “known” that the photon reflected into (θr,ϕr). 2.4.1.4 Reciprocity for Directional Spectral Reflectivity When the incident intensity is uniform over all incident directions (θi,ϕi), Equation (2.52) reduces to



Hemispherical-directional spectral reflectivity º r l (q r , f r ) ( for uniform incident intensity )

ò

(2.53)

= rl (qr , fr , qi , fi ) cos qi dW i . Ç

Comparing Equations (2.50) and (2.52), and noting Equation (2.47), the reciprocal relation for ρλ is obtained (restricted to uniform incident intensity):

rl (qi , fi ) = rl (qr , fr ) (2.54)

where (θr,ϕr) and (θi,ϕi) are the same angles, that is, θi = θr and ϕi = ϕr. This means that the reflectivity of a material irradiated at a given angle of incidence (θi,ϕi), as measured by the energy collected over the entire hemisphere of reflection, is equal to the reflectivity for uniform irradiation from the hemisphere as measured by collecting the reflected energy at a single angle of reflection (θr,ϕr) when (θr,ϕr) are the same angles as (θi,ϕi). This relation is used for the design of hemispherical reflectometers for measuring radiative properties (Brandenberg 1963). 2.4.1.5 Hemispherical Spectral Reflectivity, ρλ If the incident spectral radiation arrives from all angles of the hemisphere (Figure 2.2h), then all the radiation intercepted by the area element dA of the surface is given by Equation (2.26) as

ò

dQl,i dl = dAdl I l,i (qi , fi ) cos qi dW i . Ç

The amount of dQλ,idλ that is reflected is, by integration using Equation (2.52), to include all incident directions

ò

dQl,r dl = dAdl rl (qi , fi ) I l,i (qi , fi ) cos qi dW i . Ç

The fraction of dQλ,idλ that is reflected provides the definition.

78

Thermal Radiation Heat Transfer

Hemispherical spectral reflectivity dQl,r dl º rl = dQl,i dl ( in terms of directional--hemispherical spectral reflectivity )

ò r (q , f )I (q , f ) cos q dW = ò I (q , f ) cos q dW Ç

l

i

Ç

i

l ,i

l ,i

i

i

i

i

i

i

(2.55) i

(

)

= P ref l .

i

In a probabilistic sense, the hemispherical spectral reflectivity comes from marginalizing the bidirectional spectral reflectivity over both incident and reflected directions, and defines the probability of a photon incident from any direction being reflected into any other direction, conditional on the photon having a wavelength λ. 2.4.1.6 Limiting Cases for Spectral Surfaces In the following, we use a subscript notation for clarification: “d-h” denotes directional-hemispherical, and “bi-d” is for bidirectional. 2.4.1.6.1 Diffuse Surfaces For a diffuse surface, the incident energy from the direction (θi,ϕi) that is reflected produces a reflected intensity that is uniform over all (θr,ϕr) directions. When an irradiated diffuse surface is viewed, the surface is equally bright from all viewing directions, regardless of the distribution of incident intensity. The bidirectional spectral reflectivity is independent of (θr,ϕr), and Equation (2.52) simplifies to

ò

rl,d -h = rl,bi-d cos qr dW r = prl,bi-d (qi , fi ). Ç

However, note that for an opaque surface, ρλ,d–h(θi,ϕi) = 1 − αλ, where for a diffuse surface αλ is not a function of incidence angle (θi,ϕi). Hence, the directional-hemispherical reflectivity must be independent of angle of incidence, so that for a diffuse surface,

rl,d -h = prl,bi-d (2.56)

for any incidence angle. The π arises because ρλ,d−h accounts for energy reflected into all (θr,ϕr) directions, while ρλ,bi−d accounts for the reflected intensity into only one direction. This is analogous to the relation between blackbody hemispherical emissive power and intensity, Eλb = πIλb. Equation (2.51) provides the intensity in the (θr,ϕr) direction when the incident radiation is distributed over (θi,ϕi) values. If the surface is diffuse, and if the incident intensity is uniform for all incident angles, Equation (2.51) reduces to

ò

I l,r = rl,bi-d I l,i cos qi dW i = prl,bi-d I l,i . (2.57) Ç

But, using Equation (2.56),

I l,r = rl,d -h I l,i = (1 - a l ) I l,i (2.58)

so that the reflected intensity in any direction for a diffuse surface is the uniform incident intensity multiplied by either the directional-hemispherical reflectivity or the hemispherical reflectivity. For angularly nonuniform irradiation on a diffuse surface, the incident intensity can be replaced by

79

Radiative Properties at Interfaces

dQλ,i/π, where dQλ,i is the spectral energy per unit wavelength and time intercepted by the surface element dA from all angular directions of the hemisphere. 2.4.1.6.2 Specular Surfaces Mirror-like, or specular, surfaces obey well-known Fresnel laws of reflection. For incident radiation from a single direction, a specular reflector, by definition, provides reflected radiation at the same magnitude of the angle from the surface normal as for the incident intensity, and in the same plane as the incident intensity and the surface normal. Hence qr = qi ; fr = fi + p. (2.59)



At all other angles, the bidirectional reflectivity of a specular surface is zero; then

rl (l, qr , fr , q, f)specular = rl (qr = qi , fr = fi + p, qi , fi ) º rl,bi-d (qi , fi ) (2.60)

and the bidirectional reflectivity is a function of only the incident direction, as this direction also provides the reflected direction. For the intensity of radiation reflected from a specular surface into the solid angle around (θr, ϕr), Equation (2.51) becomes, for an arbitrary directional distribution of incident intensity,

ò

I l,r ( qr , fr ) = rl,bi-d ( qi , fi ) I l,i ( qi , fi ) cos qi dW i . Ç

The integrand has a nonzero value only in the small solid angle around (θi,ϕi) because of the properties of ρλ,bi-d(θi,ϕi) for a specular surface. When examining the radiation reflected from a specular surface in a given direction, only that radiation at the incident direction (θi,ϕi) defined by Equation (2.48) need be considered as contributing to the reflected intensity. Hence

I l,r ( qr = qi , fr = fi + p ) = rl,bi-d ( qi , fi ) I l,i ( qi , fi ) cos qi dW i . (2.61)

If a surface tends to be specular, the use of the bidirectional reflectivity can become less practical than for a situation with a more diffuse reflection behavior. For a specular reflection, the reflected intensity can be of the same order as the incident intensity; hence, the bidirectional reflectivity becomes very large because of the dΩi on the right side of Equation (2.61). In a statistical sense, the probability density that defines (θr,ϕr) in Equation (2.41) approaches a delta function. A reflectivity that equals the well-behaved quantity ρλ,bi-d(θi,ϕi)cosθidΩi in Equation (2.61) is more useful and is now defined as ρλ,s(θi,ϕi). From Equation (2.61), it is this quantity that equals the ratio of reflected to incident intensities. Using reciprocity, we can then define a convenient specular reflectivity as

I l ,r ( qr = qi , fr = fi + p ) I l ,i ( qi , fi )

= rl,s ( qi , fi ) = rl,s ( qr , fr ) . (2.62)

The angular relations of Equation (2.59) apply so that all the reflected radiation is in one direction.

2.4.2 Total Reflectivities The previous reflectivity expressions for spectral radiation can be marginalized over all wavelengths to obtain corresponding total reflectivities.

80

Thermal Radiation Heat Transfer

2.4.2.1 Bidirectional Total Reflectivity, ρ(θr,ϕr,θi,ϕi) This gives the contribution made by the total energy incident from direction (θi,ϕi) to the reflected total intensity into the direction (θr,ϕr). By analogy with Equation (2.40), Bidirectional total reflectivity º r ( qr , fr , qi , fi )

ò

=



¥

l =0

I l ,r ( qr , fr , qi , fi ) d l

cos qi dW i =

ò

¥

l =0

I l ,i ( qi , fi ) d l

I r ( qr , fr , qi , fi )

I i ( qi , fi ) cos qi dW i

(2.63)

.

Using Equation (2.40), another form is Bidirectional total reflectivity

( in terms of bidirectional spectral reflectivity ) º r ( qr , fr , qi , fi ) =

ò

¥

l =0

(2.64)

rl ( qr , fr , q i , fi ) I l , i ( qi , fi ) d l

ò

¥

l =0

I l ,i ( qi , fi ) d l

.

2.4.2.2 Reciprocity for Bidirectional Total Reflectivity Rewriting Equation (2.64) for energy incident from direction (θr,ϕr) and reflected into direction (θi,ϕi) gives



r ( qi , fi , qr , fr ) =

ò

¥

l =0

rl ( qi , fi , qr , fr ) I l ,i ( qr , fr ) d l

ò

¥

l =0

I l,i ( qr , fr ) dl

. (2.65)

Comparing Equations (2.64) and (2.65) shows that

r ( qi , fi , qr , fr ) = r ( qr , fr , qi , fi ) (2.66)

if the spectral distribution of incident intensity is the same for all directions or, less restrictively, if Iλ,i(θi,ϕi) = CIλ,i(θr,ϕr). 2.4.2.3 Directional-Hemispherical Total Reflectivity, ρ(θr,ϕr), ρ(θi,ϕi) The directional-hemispherical total reflectivity is the fraction of the total energy incident from a single direction that is reflected into all angular directions, or, in other words, the probability that a photon having any wavelength will be reflected in any direction, conditional on an incident direction. The spectral energy from a given direction (θi,ϕi) that is intercepted by the surface is Iλ,i(θi,ϕi) cosθidΩidλdA. The reflected portion is ρλ(θi,ϕi)Iλ,i(θi,ϕi)cosθidΩidλdA. Integrating over all wavelengths and forming a ratio gives

81

Radiative Properties at Interfaces

Directional-hemispherical total reflectivity º r(qi , fi ) (in terms of diirectional-hemispherical spectral reflectivity)

ò =

¥

l =0

r l (q i , f i ) I l , i (q i , f i )d l

ò

¥

l =0

(2.67)

(

)

= P ref qi , fi .

I l , i (q i , f i )d l

Another directional total reflectivity specifies the fraction of radiation reflected into the angle (θr,ϕr) direction when there is diffuse irradiation. The total radiation intensity reflected into the (θr,ϕr) direction when the incident intensity is uniform over all incident directions is ¥



ò r (q , f ) I

I r (q r , f r ) =

l

r

r

l ,i

dl

l =0

where ρλ(θr,ϕr) is in Equation (2.52). The total reflectivity is defined as the reflected intensity divided by the uniform incident intensity: Hemispherical-directional total reflectivity I r (qr , fr ) º Ii ( for diffuse irrradiation )

ò =

¥

l =0

r l (q r , f r ) I l , i d l

ò

¥

l =0

(2.68) .

I l ,i d l

2.4.2.4 Reciprocity for Directional Total Reflectivity Compare Equations (2.67) and (2.68), bearing in mind that the latter is restricted to uniform incident intensity. With this restriction, Equation (2.54) gives r(qr , fr ) = r(qi , fi ) (2.69)



where (θr,ϕr) and (θi,ϕi) are the same angles, that is, θi = θr and ϕi = ϕr. In addition, there is a fixed spectral distribution of the incident intensity such that Iλ,i in Equation (2.67) is related to that in Equation (2.68) by Iλ,i(θi,ϕi) = CIλi. 2.4.2.5 Hemispherical Total Reflectivity, ρ If the incident total radiation arrives from all angles of the hemisphere, the total radiation intercepted by a unit area at the surface is given by Equation (2.26). The amount of this radiation that is reflected is

dQr = dA

ò r(q , f )I (q , f ) cos q dW . i

Ç

i

i

i

i

i

i

The fraction of all the incident energy that is reflected into all directions of reflection is then



Hemispherical total reflectivity dQr ºr= dQi (in terms of directional-heemispherical total reflectivity) dA = dQi

ò r(q , f )I (q , f ) cos q dW = P (ref ) i

Ç

i

i

i

i

i

i

(2.70)

82

Thermal Radiation Heat Transfer

which is the probability that an incident photon will be reflected, marginalized over all wavelengths and all incident and reflected directions. Another form is found by using the incident hemispherical spectral energy. The amount reflected is ρλdQλ,idλ where ρλ is the hemispherical spectral reflectivity from Equation (2.55). Integrating yields



Hemispherical total reflectivity ºr= ( in terms of hemispherical spectral reflectivity )

ò

¥

0

rl dQl,i dl

dQi =

ò

¥

0

. (2.71)

dQl,i dl

2.4.3 Summary of Restrictions on Reflectivity Reciprocity Relations Table 2.3 summarizes the restrictions necessary for applying the reflectivity reciprocity relations.

2.5 TRANSMISSIVITY AT AN INTERFACE As for reflectivity, the transmissivity of radiation across a single interface depends on two sets of angles: the angle of incidence on the interface, and the direction at which the radiation is transmitted after crossing the surface, and also on the wavelength of the radiation. The properties defined in this section pertain only to the fraction of incident radiation crossing the single interface. For many real configurations, radiation that is transmitted across the interface may return to the interface by reflection from another nearby surface, as is the case in a pane of glass, or by scattering from material in the region of the interface. These situations are considered in Chapter 17. For opaque surfaces, the transmissivity across the surface of the material is not germane to radiative transfer calculations, but the definitions are included here for later reference. Given that the transmissivity is defined here for a single interface, it follows that the radiation incident on a surface is either reflected from the interface or transmitted through the surface, so that the transmissivity is directly related to the reflectivity. Because the definitions in this section are derived in parallel to those for reflectivity, the derivations are given more tersely.

2.5.1 Spectral Transmissivities 2.5.1.1 Bidirectional Spectral Transmissivity, τλ(θt,ϕt,θi,ϕi) The ratio of intensity that is transmitted across an interface into the direction (θt,ϕt) to the incident spectral energy on the interface from an incident direction (θi ϕi) (Figure 2.2i) is thus defined by the relation

83

Radiative Properties at Interfaces

Bidirectional spectral transmissivity º tl ( qt , ft , qi , fi )

I l,t ( qt , ft , qi , fi ) (2.72)

=

I l,i ( qi , fi ) cos qi dW i

which, as for the other properties, depends on the surface temperature T, and that notation is again omitted for simplicity. Like the bidirectional spectral reflectivity, the bidirectional spectral transmissivity also has units of sr−1, and has a similar meaning as Equation (2.41), i.e.

(

) (

)

tl (qt , ft , qi , fi ) cos qt d W t = p qt , ft trans, qi , fi , l P tranns qi , fi , l dW t . (2.73)     probability density for direction of transmitted photon

probability of transmission into any direction

2.5.1.2 Directional Spectral Transmissivities, τλ(θt,ϕt), τλ(θi,ϕi) The directional hemispherical spectral transmissivity τλ(θi,ϕi) is the fraction of the incident spectral energy that is transmitted across the interface in all directions (Figure 2.2j). By energy conservation, it is related to the directional-hemispherical spectral reflectivity, so that Directional-hemispherical spectral transmissivity

òI º t (q , f ) = l

i

Ç

i

( qt , ft , qi , fi ) cos qt dWt I l,i ( l, qi , fi ) cos qi dW i l ,t

(2.74) = 1 - rl (qi , fi ).

Using Equation (2.72), this becomes

tl ( q i , f i ) =

I l,i ( qi , fi ) cos qi dW i

ò t ( q , f , q , f ) cos q dW Ç

l

t

t

i

i

t

t

I l,i ( qi , fi ) cos qi dW i



(2.75)

ò

= tl ( qt , ft , qi , fi ) cos qt dW t . Ç

Following Equations (2.51) and (2.52), the hemispherical-directional spectral transmissivity τλ(θt,ϕt) is the spectral intensity transmitted into the (θt,ϕt) direction over the integrated average spectral intensity incident on the surface (Figure 2.2k), or Hemispherical-directional spectral transmissivity º tl (qt , ft )

=

I l ,t ( qt , ft ) æ1ö çp÷ è ø

òI Ç

l ,i

(2.76)

( qi , fi ) cos qi dWi

or, using Equation (2.74), Hemispherical-directional spectral transmissivity º tl ( q t , f t ) (in terms of thhe bidirectional spectral transmissivity)

ò t ( q , f , q , f ) I ( q , f ) cos q dW . = æ1ö ç p ÷ ò I ( q , f ) cos q dW è ø Ç

l

t

t

i

Ç

l ,i

l ,i

i

i

i

i

i

i

i

i

i

(2.77)

84

Thermal Radiation Heat Transfer

For uniform incident spectral intensity from all directions, Equation (2.76) reduces to



Hemispherical-directional spectral transmissivity º tl ( q t , f t ) ( for uniforrm incident intensity )

ò

= tl ( qt , ft , qi , fi ) cos qi dW i .

(2.78)

Ç

Comparing Equation (2.78) with Equation (2.77) shows that, for uniform incident radiation, tl ( qt , ft ) = tl ( qi , fi ) = 1 - rl (q, f). (2.79)



2.5.1.3 Hemispherical Spectral Transmissivity, τλ The fraction of spectral energy incident on the surface from all directions that is transmitted across the surface into all directions (Figure 2.2l) is the hemispherical spectral transmissivity. It is related through energy conservation to the hemispherical spectral reflectivity: Hemispherical spectral transmissivity

º tl =

òI òI Ç

Ç

l ,t

( qt , ft ) cos qt dWt

l ,i

( qi , fi ) cos qi dWi

= 1 - rl

(2.80)

or, using Equation (2.74), Hemispherical-spectral transmissivity ºt ( in terms of directionaal-hemispherical transmissivity ) l (2.81)



ò t ( q , f ) I ( q , f ) cos q dW . = ò I ( l, q , f ) cos q dW Ç

l

i

Ç

i

l ,i

l ,i

i

i

i

i

i

i

i

i

2.5.2 Total Transmissivities Total transmissivities are obtained by direct analogy with the spectral property definitions. 2.5.2.1 Bidirectional Total Transmissivity, τ(θi,ϕi,θt,ϕt) The ratio of total intensity that is transmitted across an interface into the direction (θt,ϕt) to the incident total energy on the interface from direction (θi,ϕi) (Figure 2.2i) is defined by Bidirectional total transmissivity º t ( qt , ft , qi , fi )

=

I t ( qt , ft , qi , fi )

I i ( qi , fi ) cos qi dW i

(2.82) .

2.5.2.2 Directional Total Transmissivities, τ(θi,ϕi) The directional-hemispherical total transmissivity is the fraction of the incident total energy that is transmitted across the interface in all directions (Figure 2.2j). By energy conservation, it is related to the directional-hemispherical total reflectivity, so that

85

Radiative Properties at Interfaces

Directional-hemispherical total transmissivity

º t ( qi , fi ) =

ò I ( q , f , q , f ) cos q dW Ç

t

t

t

i

i

t

t

I i ( qi , fi ) cos qi d W i

(2.83) = 1 - r(qi , fi ).

Using Equation (2.82), this can be expressed as Directional-hemispherical total transmissivity º t ( qi , fi ) ( in terms of bidirectional transmissivity )



=

I i ( qi , fi ) cos qi dW i

ò t ( q , f , q , f ) cos q dW t

Ç

t

i

i

t

t

(2.84)

I i ( qi , fi ) cos qi d W i

ò

= t ( qt , ft , qi , fi ) cos qt dW t . Ç

2.5.2.3 Hemispherical-Directional Total Transmissivity, τ(θt,ϕt), τ(θi,ϕi) The hemispherical-directional total transmissivity τ(θt,ϕt) is the total intensity transmitted into the (θt,ϕt) direction over the integrated average total intensity incident on the surface (Figure 2.2k), or Hemispherical-directional total transmissivity

º t(qt , ft ) =

I t ( qt , ft ) æ1ö ç p ÷ I i ( qi , fi ) cos qi dW i è ø

(2.85)

ò

or, using Equation (2.82), Hemispherical-directional total transmissivity º t ( qt , ft ) (in terms of the bidirectional total transmissivity)

t ( q , f , q , f ) I ( q , f ) cos q d W (2.86) ò . = æ1ö I q , f cos q d W ) ç p÷ò ( è ø Ç

t

t

i

Ç

i

i

i

i

i

i

i

i

i

i

i

For uniform incident total intensity from all directions, Equation (2.86) reduces to



Hemispherical-directional total transmissivity º t ( qt , ft ) ( for uniform incident intensity )

ò

= t ( qt , ft , qi , fi ) cos qi dW i .

(2.87)

Ç

Comparing Equation (2.87) with Equation (2.84) shows that, for uniform incident radiation and because of the reciprocity of τ(θt,ϕt,θi,ϕi),

t ( qt , ft ) = t ( qi , fi ) . (2.88)

86

Thermal Radiation Heat Transfer

2.5.2.4 Hemispherical Total Transmissivity, τ The fraction of total energy incident on the surface from all directions that is transmitted across the surface into all directions (Figure 2.2l) is the hemispherical total transmissivity:



Hemispherical total transmissivity º t =

ò I ( q , f ) cos q dW (2.89) ò I ( q , f ) cos q dW Ç

Ç

t

t

t

t

t

i

i

i

i

i

or, using Equation (2.82), Hemispherical total transmissivity ºt ( in terms of directional-hhemispherical transmissivity )

ò t ( q , f ) I ( q , f ) cos q dW . = ò I ( q , f ) cos q dW Ç

i

Ç

i

i

i

i

i

i

i

i

i

(2.90)

i

i

For uniform incident intensity, this reduces to

Hemispherical total transmissivity

( for uniform incident inteensity )

ºt=

1 t ( qi , fi ) cos qi dW i . (2.91) p

ò Ç

2.6 RELATIONS AMONG REFLECTIVITY, ABSORPTIVITY, EMISSIVITY, AND TRANSMISSIVITY From the definitions of absorptivity, reflectivity, and transmissivity as fractions of incident energy absorbed, reflected, or transmitted, it is evident that some relations exist among these properties through the First Law of Thermodynamics. For an opaque material, all energy that is transmitted across a surface is absorbed by the medium beneath the surface, so it follows that transmissivity and absorptivity are equal. This will not be true when translucent materials of finite thickness are considered. With the use of Kirchhoff’s law and its restrictions (Table 2.2), further relations between emissivity and reflectivity can be obtained. These relationships are summarized in Table 2.4. The spectral energy d2Qλ,i(θi,ϕi)dλ incident per unit time on an element dA of an opaque body from within a solid angle dΩi at (θi,ϕi) is either absorbed or reflected, so that

d 2Ql,a (qi , fi )dl d 2Ql,r (qi , fi )dl + = 1. (2.92) d 2Ql,i (qi , fi )dl d 2Ql,i (qi , fi )dl

87

Radiative Properties at Interfaces

Because the radiation is incident from the angles (θi,ϕi), the two energy ratios in Equation (2.92) are the directional spectral absorptivity, Equation (2.14), (or the directional spectral transmissivity, Equation (2.74)) and the directional hemispherical spectral reflectivity, Equation (2.50). Substituting gives

a l ( qi , fi ) + rl ( qi , fi ) = tl ( qi , fi ) + rl ( qi , fi ) = 1. (2.93)

Equivalently, these parameters define the probability of photons being absorbed and reflected, or transmitted and reflected, conditional on an incident direction (θi,ϕi) and wavelength λ. Since photons must either cross the surface interface (absorption or transmission) or be reflected by the interface, the Law of Total Probability requires that the probability of all outcomes must sum to unity, which is consistent with Equation (2.93). Kirchhoff’s law, Equation (2.18), can be applied without restriction to give

el ( qi , fi ) + rl ( qi , fi ) = tl ( qi , fi ) + rl ( qi , fi ) = 1. (2.94)

When the total energy arriving at dA in dΩi from the given direction (θi,ϕi) is considered, Equation (2.92) becomes

d 2Qa (qi , fi ) d 2Qr (qi , fi ) + = 1. (2.95) d 2Qi (qi , fi ) d 2Qi (qi , fi )

Substituting Equation (2.22) or (2.83) and Equation (2.67) results in

a ( qi , fi ) + r ( qi , fi ) = t ( qi , fi ) + r ( qi , fi ) = 1. (2.96)

Again, this relation is also enforced by the Law of Total Probabilities, since the quantities in Equation (2.96) come from marginalizing the probabilities in Equation (2.93) over all wavelengths. The absorptivity (and transmissivity) are directional-total values, and the reflectivity is directional-hemispherical total. Kirchhoff’s law for directional total properties (Section 2.3.4) is applied to give

e ( qi , fi ) + r ( qi , fi ) = t ( qi , fi ) + r ( qi , fi ) = 1 (2.97)

under the restriction that the incident radiation obeys the relation Iλ,i(θi,ϕi) = C(θi,ϕi)Iλb(T), which usually only happens under conditions of radiative equilibrium, or if the surface is directional-gray. If the incident spectral intensity is arriving at dA from all directions over the hemisphere, then Equation (2.95) gives

d 2Ql,a dl d 2Ql,r dl + = 1. (2.98) d 2Ql,i dl d 2Ql,i dl

Equation (2.98), using the property definitions, is

a l + rl = tl + rl = 1 (2.99)

where the properties are hemispherical spectral values from Equations (2.28), (2.55), and (2.80). Once again, this is consistent with the Law of Total Probabilities. Substitution of the hemispherical spectral emissivity ελ for αλ in this relation is only valid if the incident intensity is independent of incident angle (diffuse irradiation), or if the ελ and αλ do not depend on the angle (see Table 2.2). Under these restrictions, Equation (2.99) becomes

el + rl = tl + rl = 1. (2.100)

88

Thermal Radiation Heat Transfer

If the energy incident on dA is integrated over all wavelengths and directions, then Equation (2.95) becomes dQa dQr + = 1 (2.101) dQi dQi

or

a + r = t + r = 1. (2.102)



Note that, while ελ(θi,ϕi) = αλ(θi,ϕi) without restriction, and ελ = αλ if either the surface or the irradiation is diffuse, the conditions under which total emissivity can be equated with total absorptivity are far more restrictive. Specifically, this requires: (i) Ii,λ(T) = CIbλ(T), which usually implies radiative equilibrium between a surface and its surroundings and therefore no net radiative transfer, or (ii) for gray surfaces, an idealization commonly invoked to simplify analysis, but one that rarely reflects reality. Table 2.4 may be used to develop further the relationship of surface transmissivity to the other radiative properties of opaque surfaces (see Table 2.3). Example 2.8 Radiation from the sun is incident on a surface in orbit above the Earth’s atmosphere. The surface is at 1000 K, and the directional total emissivity is given in Figure 2.4. If the solar energy is incident at an angle of 25° from the normal to the surface, what is the reflected energy flux? From Figure 2.4, ε(25°, 1000 K) = 0.8. Table 2.2 shows that for directional total properties α(25°, 1000 K) = ε(25°, 1000 K) only when the incident spectrum is proportional to that emitted by a blackbody at 1000 K. This is not the case here, since the solar spectrum is like that of a blackbody at 5780 K. Hence α(25°, 1000 K) ≠ 0.8, and without α(25°, 1000 K) we cannot determine ρ(25°, 1000 K). The emissivity data given are insufficient to solve the problem. Spectral values are needed.

Example 2.9 A surface at T = 500 K has a spectral emissivity in the normal direction that is approximated as shown in Figure 2.9. The surface is maintained at 500 K by cooling water and is enclosed by a black hemisphere heated to Ti = 1500 K. What is the reflected radiant intensity in the direction normal to the surface? From Equation (2.94), ρλ(θi = 0°) = 1 − ελ(θ = 0°), which is the reflectivity into the hemisphere for radiation arriving from the normal direction. From reciprocity, for uniform incident intensity over the hemisphere, ρλ(θr = 0°) = ρλ(θ = 0°). Hence, the reflectivity into the normal direction resulting from the incident radiation from the hemisphere is (by use of Figure 2.9):

rl (0 £ l < 2, qr = 0°, T ) = 0.7;

rl ( 2 £ l £ 5, qr = 0°, T ) = 0.2;

rl (5 £ l £ ¥, qr = 0°, T ) = 0.5

The incident intensity is Iλ,i(Ti) = Iλb(1500 K). The reflected intensity is ¥



Ir (qr = 0°) =

òI

lb

0

From Equations (1.38) and (1.41), this becomes

(Ti ) rl (qr = 0°, T ) d l.



89

Radiative Properties at Interfaces

FIGURE 2.9  Directional spectral emissivity in normal direction for Example 2.9.



Ir (qr = 0°) =

sTi 4 (0.7F0 ® 2Ti + 0.2F2Ti ®5Ti + 0.5F5Ti ®¥ ) p

=

5.67040 ´ 10 -8 (1500)4 [0.7(0.27323) p



+ 0.2(0.83437 - 0.27323) + 0.5(1- 0.83437)ùû = 35.3 kW/(m2 × sr).

Example 2.10 A large flat plate at T = 925 K has a hemispherical spectral emissivity that can be approximated by a straight line decreasing from 0.850 to 0 as λ increases from 0 to 10.5 μm. The flat plate faces a second large plate that has a reflectivity of 0.35 for 0  6. Such large n2/n1 values are uncommon for dielectrics, so the curve is not extended to smaller εn. The hemispherical emissivity for dielectrics is compared to the normal value in Figure 3.3b.

Radiative Properties of Opaque Materials

101

FIGURE 3.3  Predicted emissivities of dielectric materials for emission into a medium with a refractive index n1 from a medium with a refractive index n2 where n2 ≥ n1. (a) Normal emissivity as a function of n2/n1, and (b) comparison of hemispherical and normal emissivity.

For large n2/n1, the εn values are relatively low, and with increasing n2/n1, the curves in Figure 3.2 depart more from a circular form. Figure 3.3b reveals that the flattening of the curves of Figure 3.2 in the region near the normal causes the hemispherical emissivity to exceed the normal value at large n2/n1 (ελ,n ≈ 0.5−0.6). For n2/n1 near 1 (ελ,n near 1), the hemispherical ελ is lower than ελ,n because of poor emission at large θ. Example 3.3 A dielectric medium has n = 1.41. What is its hemispherical emissivity into air at the wavelength at which n was measured? From Equation (3.13), for n1/n2 = n = 1.41, ελ = 0.92. Alternatively, from Equation (3.14), the spectral normal ελ,n = 1−(0.41/2.41)2 = 0.97, and from Figure 3.2b, ελ/ελ,n = 0.95; then the spectral hemispherical emissivity is ελ = 0.97 × 0.95 = 0.92.

3.2.2 Radiative Properties of Conductors 3.2.2.1 Electromagnetic Relations for Incidence on an Absorbing Medium In the preceding section, we discussed perfect dielectrics where there is no attenuation (k = 0) of radiation as it travels within the material. In other words, we assumed that the material is perfectly

102

Thermal Radiation Heat Transfer

transparent except for interface reflections. For real dielectrics, there is attenuation, so k is nonzero, although it is small. The propagation of an EM wave within an attenuating medium is governed by the same relations as a nonattenuating medium if the refractive index n is replaced by the complex index, n = n - ik , although this leads to some complexities in interpretation, particularly at the interface between media. For example, in the case of an EM wave within a dielectric medium (n1) incident upon a conducting medium ( n2 = n2 - ik2 ), Equation (3.5) becomes sin c n1 n1 . (3.15) = = sin qi n2 n2 - ik2



Consequently, sin χ is complex, and χ can no longer be interpreted as the angle of refraction for propagation into the material because the transmitted wave is heterogeneous (cf. Section 8.4.2). Also, except for normal incidence, n2 is not directly related to the wave propagation velocity. The equations for the reflected components of incident radiation are also more complicated than the case of two dielectrics. One of the earlier analyses was provided by Hering and Smith (1968) who gave the following form for incidence from air or vacuum (n1 = 1, k1 = 0, n2 = n, k2 = k):



rl,^ ( qi ) =

rl, ( qi ) =

( + (n

) )a - n b

(3.16)

) )a - n g

(3.17)

(nb - cosqi )2 + n2 + k 2 a - n2b2 (nb + cosqi )

2

2

( + (n

+k

2

2 2

(ng - a /cosqi )2 + n2 + k 2 a - n2 g 2 (ng + a /cosqi )

2

2

+k

2

2 2

where



æ sin 2 q ö 4n2 a 2 = ç 1 + 2 i2 ÷ - 2 2 è n +k ø n +k b2 =

n2 + k 2 2n2

æ sin 2 qi ö (3.18) ç 2 2 ÷ èn +k ø

æ n2 - k 2 sin 2 qi ö - 2 + a ÷ (3.19) ç 2 2 2 n +k èn +k ø

and 1/ 2



g=

ö 2nk æ n2 + k 2 n2 - k 2 b+ 2 a - b ÷ . (3.20) ç 2 2 n +k n + k 2 è n2 ø

The reflectance at normal incidence reduces to 2 ( n - 1) + k 2 . (3.21) é n - ik - 1 ù = rl,n = rl, ( 0 ) = rl,^ ( 0 ) = ê 2 ú ë n - ik + 1 û ( n + 1) + k 2 2



The directional emissivity can be found from Kirchhoff’s law, ελ(θ) = l − [ρλ,⊥(θ) + ρλ,∥(θ)]/2, which can then be integrated over the hemisphere to obtain the hemispherical emissivity, el =

1

ò e ( q) d(sin q) , 0

l

2

as a function of n and k. One set of results, for n ≥ 1, is plotted in Figure 3.4 as a function of n and k/n, which shows how the hemispherical emittance decreases as both n and k increase.

103

Radiative Properties of Opaque Materials

FIGURE 3.4  Exact and approximate hemispherical emissivity results (n ≥ 1.0). (From Hering, R.G. and Smith, T.F., IJHMT, 11(10), 1567, 1968).

These results apply for radiation incident from air or a vacuum, with n1 ≈ 1. Harpole (1980) presented an analysis for the more complex case of oblique incidence from an adjacent absorbing material. 3.2.2.2 Reflectivity and Emissivity Relations for Metals (Large k) Usually conductors (such as metals) are internally highly absorbing; this means that the extinction coefficient k is sufficiently large to simplify the above equations into more convenient results. For large k, the sin2θ terms in Equations (3.18) and (3.19) can be neglected relative to n2 + k2. Then α ≈ β ≈ γ ≈ 1 and Equations (3.16) and (3.17) yield

rl, ( qi ) =

(n cosqi - 1)2 + ( k cosqi )

2

(n cosqi + 1)2 + ( k cosqi )

2

(3.22)

and

rl , ^ ( qi ) =

(n - cos qi )2 + k 2 . (3.23) (n + cos qi )2 + k 2

For unpolarized incident radiation, ρ(θi) = [ρ∥(θi)+ ρ⊥(θi)]/2, and for the normal direction, ρn reduces to Equation (3.21). Emission is unpolarized, so the directional emissivity is given by Equation (3.12). In the normal direction (θi = 0), this becomes

el,n = 1 - rl,n =

4n . (3.24) (n + 1)2 + k 2

These emissivity relations are demonstrated in Figure 3.5 for a pure smooth platinum surface at a wavelength of 2 μm, and in Figure 3.6 for a titanium surface at multiple wavelengths. For metals,

104

Thermal Radiation Heat Transfer

FIGURE 3.5  Directional-spectral emissivity of platinum at λ = 2 μm. Refractive indices are from Lide (2008).

FIGURE 3.6  Directional-spectral emissivity of titanium at 306 ± 14 K. Surface ground to 0.4 μm rms roughness. Symbols denote data from Edwards, D. K., and Catton, I.: Radiation characteristics of rough and oxidized metals, in S. Gratch (ed.) Advances in Thermophysical Properties at Extreme Temperatures and Pressures, pp. 189–199, ASME, New York, 1964. Dashed curves are predictions from EM theory using optical constants in Palik, E. D. (ed.) Handbook of Optical Constants of Solids, Elsevier, New York, 1998.

Radiative Properties of Opaque Materials

105

the emissivity is essentially constant for θi  ~5 μm

n=k=

l 0m0 co 0.003l 0 = 4pre re

( l 0 in mm,

re in W × cm ) (3.26)

where μ0 is the magnetic permeability, which for most metals can be taken to be the vacuum value. This is the Hagen–Rubens equation (Hagen and Rubens 1900). Predictions of n and k from this equation can be greatly in error, particularly at short wavelengths, as shown in Table 3.2. The restrictions of this equation to wavelengths greater than approximately 5 μm should be carefully noted. With the conditions in Equation (3.26) that n = k, Equation (3.24) reduces to e l , n = 1 - rl , n =



4n (3.27) 2n2 + 2n + 1

for a material with refractive index n radiating in the normal direction into air or vacuum: Substituting Equation (3.26) into Equation (3.27) and expanding into a series gives the Hagen–Rubens emissivity relation: 1/ 2



el,n

2 æ re ö = ç ÷ 0.003 è l 0 ø

-

2 re + . (3.28) 0.003 l 0

Because the index of refraction of metals as predicted from Equation (3.26) is generally large at longer wavelengths so that (re/λo) is small, that is, for λ > ~5 μm (see Table 3.2), one or two terms of the series are usually adequate. A two-term approximation is made by adjusting the coefficients of the second term for the remaining terms in the series to yield 1/ 2



el,n

ær ö = 36.5 ç e ÷ è l0 ø

- 464

re . (3.29) l0

Data for polished nickel are shown in Figure 3.8, and the extrapolation to long wavelengths by Equations (3.29) is reasonable. The predictions of normal spectral reflectivity at long wavelengths (ρλ,n = 1−ελ,n), as presented in Table 3.2, are much better than the predictions of the optical constants. The angular behavior of the spectral emissivity can be obtained by substituting the refractive index, Equation (3.26), into Fresnel’s relations, Equations (3.22) and (3.23), and this result into Kirchhoff’s law, Equation (3.12), to yield the spectral directional emissivity, ελ(θ). Figure 3.6 shows

108

Thermal Radiation Heat Transfer

FIGURE 3.8  Comparison of measured values with theoretical predictions for normal spectral emissivity of polished nickel. Hagen–Rubens predictions are calculated using electrical resistivity from Chu, T. K., and Ho, C. Y., Electrical Resistivity of Chromium, Cobalt, Iron, and Nickel (CINDAS Report 60), Thermophysical Properties Research Centre, 1982.

that the directional spectral emissivity of titanium tends to decay with increasing wavelength, in accordance with Hagen–Rubens theory. The normal spectral emissivity in Equation (3.28) can be integrated with respect to wavelength

ò

¥

to yield a normal total emissivity, e n (T ) = p el,n ( T ) I lb ( T ) dl /sT 4 . Note that Equation (3.28) 0 is accurate only for λ0 > ~5 μm, so in performing the integration starting from a small value of λ (e.g., in the visible spectrum) assumes that the energy radiated at wavelengths shorter than 5 μm is assumed to be small compared with that at wavelengths longer than 5 μm. Then, substituting into the integral, the first term of Equations (3.28) and (1.20) for Iλb provides æ re 2 çç 1 0 ç 0.003l 0 ) 2 è e n (T ) » p

ò



=

¥

4pC1 (Tre )1/ 2 (0.003)1/ 2 sC24.5

¥

ò 0

ö ÷ 2C é 5 C2 / l0T - 1) ù dl 0 û ÷ 1 ë l 0 (e ÷ ø (3.30) sT 4

z 3.5 dz ez - 1

where ζ = C2/λ0 T, as was used in conjunction with Equation (1.39). The integration is carried out by use of gamma functions to yield

en (T ) »

4pC1 (Tre )1/ 2 (12.27 ) = 0.575(reT )1/ 2 . (3.31) (0.003)1/ 2 sC24.5

If additional terms in the series Equation (3.28) are retained, the recommended three-term approximation for total normal emissivity of a metal is obtained (Parker and Abbott 1964)

109

Radiative Properties of Opaque Materials



e n ( T ) = 0.578 (reT )1/ 2 - 0.178 reT + 0.584(reT )3 / 2

(T in K,

re in W × cm ) . (3.32)

For pure metals, re near room temperature is approximately described by

re » re,273

T . (3.33) 273

where re,273 is the electrical resistivity at 273 K. Substituting Equation (3.33) into Equation (3.31) gives the approximate result

e n ( T ) » 0.0348T re,273 . (3.34)

This indicates that the total emissivity of pure metals increases directly with absolute temperature. This expression was originally derived by Aschkinass (1905). In some cases, it holds to unexpectedly high temperatures where considerable emitted radiation is in the short-wavelength region (for platinum, to near 1800 K), but often it applies only to temperatures below about 550 K. This is illustrated in Figure 3.9 for platinum and tungsten (data from Lide 2008). In Figure 3.10, the normal emissivity for various polished surfaces of pure metals is compared with predictions based on Equation (3.34). The experimental data are obtained from three standard compilations (Hottel 1954, Eckert and Drake 1959, Lide 2008). Likewise, the spectral directional emissivity found from the refractive index, Equation (3.26), and the Fresnel relations, Equations (3.22) and (3.23) can be integrated over all directions to obtain total hemispherical quantities. The following approximate equations for the hemispherical total emissivity fit the results in two ranges:

e ( T ) = 0.751 reT - 0.396reT

0 < reT < 0.2 (3.35)

FIGURE 3.9  Temperature dependence of normal total emissivity of polished metals.

110

Thermal Radiation Heat Transfer

FIGURE 3.10  Comparison of data with calculated normal total emissivity for polished metals at 373 K.

e ( T ) = 0.698 reT - 0.266reT



0.2 < reT < 0.5 (3.36)

where T and re are in units of K and Ω-cm, respectively. The resistivity re depends approximately on T to the first power so that the first term of each of these equations provides a T proportionality, indicating that the temperature dependence for energy emission by metals is higher than the blackbody dependence of T4. A different hemispherical total emissivity expression suggested by Parker and Abbott (1964) is e ( T ) = 0.766 reT - éë0.309 - 0.0889 ln ( reT ) ùû reT - 0.0175(reT )3 / 2 . (3.37)



Figures 3.11 and 3.12 compare predictions for normal total emissivity, Equation (3.32), and hemispherical total emissivity, Equation (3.37), with experimental measurements. Example 3.4 A polished platinum surface is maintained at T = 250 K. Energy is incident upon the surface from a black enclosure at Ti = 500 K that surrounds the surface. What is the hemispherical-directional total reflectivity into the direction normal to the surface? The directional-hemispherical total reflectivity for the normal direction can be found from

rn (T = 250 K ) = 1 - a n (T = 250 K )

where αn(T = 250 K) is the normal total absorptivity of a surface at 250 K for incident black radiation at 500 K. ¥



a n (T = 250 K ) =

òa 0

l ,n

(T = 250 K)Il ,b ( 500 K ) d l

ò

¥ 0

Il ,b ( 500 K ) d l

.

111

Radiative Properties of Opaque Materials

FIGURE 3.11  Normal total emissivity of various metals compared with theory. (From Parker, W.J. and Abbott, G.L., Theoretical and experimental studies of the total emittance of metals, Symposium on Thermal Radiation of Solids, NASA SP-55, pp. 11–28, 1964).

FIGURE 3.12  Hemispherical total emissivity of various metals compared with theory. (From Parker, W.J. and Abbott, G.L., Theoretical and experimental studies of the total emittance of metals, Symposium on Thermal Radiation of Solids, NASA SP-55, pp. 11–28, 1964). For spectral quantities, αλ,n(T = 250 K) = ελ,n(T = 250 K). From Equation (3.28) the near-linear variation of re with temperature provides the approximate emissivity variation ελ,n(T) ∝ T1/2. Then ελ,n (T = 250 K) = ελ,n(T = 500 K)(250/500)1/2, and we obtain

a n (T = 250 K ) =

1/ 2

¥

òe 0

l ,n

(T = 250 K )Il ,b ( 500 K ) d l ¥

òI



0

=

l ,b

en (T = 500 K ) 2

( 500 K ) d l



112

Thermal Radiation Heat Transfer

where the last equality is obtained by examining the emissivity definition, Equation (2.5). The normal total emissivity of platinum at 500 K is given by Equation (3.34), as plotted in Figure 3.9, as en (T = 500 K ) = 0.348 re ,273T = 0.348 re ,293 = 0.348 10 ´ 10

-6

273 T 293



273 ´ 500 = 0.053. 293

Note that Equation (3.34) is to be used only when temperatures are such that most of the emitted radiation is at wavelengths greater than 5 μm. From the blackbody functions, for T = 500 K, about 10% of the energy is below λ = 5 μm, so a small error is introduced. The reciprocity relation of Equation (2.58) for uniform incident intensity can now be employed to give the desired result for the hemispherical-directional total reflectivity: rn (T = 250 K ) = 1 - a n (T = 250 K )

» 1-

1 0.053 en (T = 500 K ) = 1 = 0.963. 2 2

3.2.3 Extensions to the Theory of Radiative Properties Accurate EM predictions of surface properties requires precise knowledge of the refractive index or dielectric function of the medium. This motivates development of dispersion models that account for how these quantities vary with respect to wavelength. A brief discussion concerning dispersion models is presented below; a more complete derivation is included in Chapter 8. The key difference between metals and dielectrics concerns the nature of the charged particles through which the EM wave interacts with the medium. Charges within dielectrics (e.g., lattice ions or valence electrons) are localized and can be modeled as damped harmonic oscillators that vibrate due to the columbic forces imposed by the local EM field. This model, called the Lorentz model, is highly successful for predicting the variation of the complex dielectric function with respect to the angular frequency of the wave, ω = 2πν = 2πc0/λ

ε = ε0 +

å j

w2p,j . (3.38) w20, j - w2 - iz j w

In Equation (3.38), ωo,j is the resonant frequency, ωp,j is the plasma frequency, and ζj is the damping coefficient for the jth oscillator. These parameters are normally found by nonlinear regression to experimental data. The dielectric function components can then be transformed into n and k via Equation (3.2) if so desired. Figure 3.13 shows the reflectance of MgO predicted using Equation (3.38), which can be modeled using two oscillators (Jasperse et  al. 1966). The abrupt variations of spectral reflectivity with respect to wavelength are representative of many dielectrics, and this behavior can be exploited in many engineering applications, e.g., optics and in designing selective surfaces (cf. Section 3.4). In the case of metals, the EM wave interacts principally (but not exclusively) with conduction band electrons. The classical Hagen–Rubens relations captures this behavior only at long wavelengths, λ >~ 5 μm, as outlined by Table 3.2, which shows much better agreement between the predicted and measured ελ,n value at 10 μm compared to 0.589 μm. This is for two main reasons. First, EM theory presumes that the surface is optically smooth (i.e. the wavelength is much larger than the RMS roughness), which is more likely to be the case at longer wavelengths. Second, Hagen– Rubens theory assumes that electrical conductivity is independent of wavelength; while this is true at long wavelengths, at shorter wavelengths/higher frequencies, electron mobility depends on the

Radiative Properties of Opaque Materials

113

FIGURE 3.13  Normal spectral reflectivity of MgO at room temperature. Lorentz parameters and experimental data from Jasperse, J. R., Kahan, A., Plendl, J. N.: Temperature dependence of infrared dispersion in ionic crystals LiF and MgO, Phys. Rev., 146(2), 526, 1966.

oscillatory motion of electrons induced by the EM wave. This effect is captured by Drude theory, which models the metal as a “sea” of free conduction band electrons that scatter from a background of ions and neutral atoms. The complex dielectric constant is then derived from the damped harmonic motion of electrons:

ε = 1-

w2p . (3.39) w2 + izw

The Drude parameters in Equation (3.39) are the plasma frequency, ωp, and the damping coefficient, ζ, which in principle can be calculated from the number density of electrons, Np, (found from the density and valence of the metal) and DC conductivity, σDC = 1/re, according to Equations (8.117) and (8.118). Aluminum is particularly well-approximated using Drude theory over the visible and near-infrared wavelengths. Figure 3.14 compares the measured spectral reflectance of polished aluminum and deposited aluminum film with predictions from Hagen–Rubens theory, Equation (3.29), and Drude theory. The Hagen–Rubens prediction is computed using the resistivity for aluminum in Table 3.2, while the Drude parameters of ωp = 2.48 × 1015 rad/s and ζ = 5.88 × 1012 rad/s are also found from re and electron number density; the electron number density is found using the bulk density of aluminum, assuming each atom contributes three conduction band electrons. Drude theory assumes that the EM field interacts exclusively with conduction band electrons; in many metals, however, transitions between bound states (e.g., interband transitions) strongly affect the radiative properties. In some cases Drude theory can still be applied to model the radiative properties of the metal by treating ωp and τ as parameters to be fitted to reference reflectance data, as opposed to being calculated from the density and DC resistivity of the bulk metal. Ordal et  al. (1985) followed this procedure for 14 metals in the infrared and far-infrared. In this scenario Equation (3.39) is simply a convenient means of parameterization; therefore, good agreement between modeled and measured radiative properties should not be construed to mean that Drude theory is an accurate representation of the true electron dynamics within metals in all cases.

114

Thermal Radiation Heat Transfer

FIGURE 3.14  Normal spectral reflectivity of Ag at room temperature. Drude parameters are calculated using density and DC resistivity. Data from Touloukian, Y.S. and Ho, C.Y. (eds.): Thermophysical Properties of Matter, TRPC Data Services; Volume 7, Thermal Radiative Properties: Metallic Elements and Alloys, 1970.

Alternatively, Lorentz–Drude theory can be used to account for transitions between bound states, e.g., the vibration of free electrons. At still shorter wavelengths (e.g., visible and UV) the role of bound electrons and electron band structure must be considered. The theory for radiative properties of materials has been improved significantly over the years by explicitly modeling these effects. The contributions of Davisson and Weeks (1924), Foote (1915), Schmidt and Eckert (1935), and Parker and Abbot (1964) extended the emissivity relations for metals to shorter wavelengths and higher temperatures, and Mott and Zener (1934) derived predictions for metal emissivity at very short wavelengths on the basis of quantummechanical relations. Kunitomo (1984) provides accurate predictions of high- and low-temperature properties of metals and alloys by including the effect of bound electrons on optical properties. Chen and Ge (2000) introduced a damping frequency term into the Drude model of free-electron oscillations, giving a more accurate prediction of the complex dielectric constant. This modifies the relations between the complex refractive index and the complex dielectric constant.

3.3 MEASUREMENTS ON REAL SURFACES The results discussed in the previous section apply to pure materials having perfectly smooth and uncontaminated interfaces. While there are situations in which these conditions are approximately satisfied, e.g., lenses and optical components, the surfaces encountered in engineering practice more often deviate from this ideal scenario, and their radiative properties will differ significantly from EM predictions.

3.3.1 Heterogeneity and Surface Coatings In many practical situations the surface may be heterogeneous, which profoundly affects the radiative properties. As an example, in the visible wavelengths, solid ice behaves as a dielectric having a

Radiative Properties of Opaque Materials

115

refractive index of approximately 1.3, so it is nearly transparent at normal incidence (e.g., on sidewalks, much to the chagrin of pedestrians) with a normal reflectance of 0.02 according to Equation (3.14). On the other hand, freshly fallen snow has a reflectivity of approximately 0.85 due to multiple scattering caused by the ice crystals, each of which acts as scattering centers, directing the majority of incident irradiation through multiple scattering events back out of the surface. This phenomenon, sometimes called the “snow effect” (Hollands 2004), is shown schematically in Figure 3.15. Multiple random scattering within the layer dramatically increases the reflectivity and also makes the surface more diffuse-like. The radiative properties of snow are pivotal to climate change, through the overall global radiation budget (i.e. fraction of solar energy reflected vs absorbed) and local changes in climate patterns, which has motivated recent theoretical research and experimental measurements. Ball et al. (2015) measured the bidirectional reflectance function of snow-covered tundra. The snow effect also applies to paper, fabric, chalk, ceramics, and some pigments and coatings. Figure 3.16 plots the hemispherical normal spectral reflectivity of three paint coatings on steel. White paint often consists of ceramic (e.g., TiO2) nanoparticles in a silicone or polymer binder and has a high reflectivity at short wavelengths because of the snow effect. Black paint, however, which is often made of absorbing particles like carbon black, has a relatively low reflectivity over the entire wavelength region. Using aluminum powder in a silicone binder produces a reflectivity more representative of a metallic surface. Coatings based on TiO2 nanoparticles can be engineered to enhance the solar reflectivity of surfaces (Baneshi et al. 2010), thereby reducing cooling requirements of buildings during summer months. (Spectrally selective surfaces are discussed in Section 3.4 and in Chapter 19). Fuel storage tanks and aircraft are also often treated using white, TiO2-based paints to prevent them from heating up due to solar radiation. On the other hand, the absorptance of surfaces can be enhanced by embedding highly absorbing particles in a dielectric matrix, e.g., for solar energy applications (Wang et al. 2018). Hadley and Kirchstetter (2012) consider how black carbon particles reduce the high reflectance of snow and ice, along with the implications for climate change. Scattering and absorption of radiation by particles are discussed in more detail in Chapter 10. Even in the absence of heterogeneities, applying EM theory to dielectrics is complicated by the fact that, for a dielectric to be considered opaque, a specimen must have sufficient thickness to absorb essentially all the radiation that enters through boundaries; otherwise, radiation transmitted through and reflected from all boundaries must be taken into account. Often, then, it is necessary to conceive the dielectric as a coating on a substrate. For a surface to behave completely as the coating material, the coating thickness must be large enough so that no significant radiation is transmitted through it. Otherwise, some incident radiation can be reflected from the substrate and transmitted

FIGURE 3.15  Heterogeneities in a dielectric can dramatically increase the reflectivity and make the surface more diffuse.

116

Thermal Radiation Heat Transfer

FIGURE 3.16  Spectral reflectance of a TiO2 based pigment. Data from Touloukian, Y.S. and Ho, C.Y. (eds.), Thermophysical Properties of Matter, TRPC Data Services; Volume 9 Thermal Radiative Properties: Coatings, 1972b.

back through the coating to reappear as measured reflected energy, in which case the surface properties would depend on both the coating material and thickness and the substrate. One of the earlier studies on this subject was conducted by Liebert (1965), who examined the spectral emissivity of zinc oxide coatings on various substrates. In Figure 3.17, the effect of coating thickness is shown on the spectral emissivity of a composite of zinc oxide on a substrate that has an approximately constant normal spectral emissivity. The effect of increasing coating thickness becomes small in the thickness range of 0.2–0.4 mm, indicating that the emissivity approaches that of zinc oxide alone. Figure 3.18 shows the spectral reflectance of a 14.4 μm TiO2 paint film on various substrates. It is clear that the substrate may have a considerable impact on the spectral reflectivity. For a black substrate, as the wavelength decreases and becomes small relative to the film thickness, the reflectivity increases and becomes more like that of the white paint film. The most common surface impurities on metal surfaces are thin layers deposited either by absorption, such as water vapor, or by chemical reaction, such as a thin oxide layer. Because dielectrics generally have high emissivities, an oxide layer or other nonmetallic contaminant usually increases the emissivity of an otherwise ideal metallic surface. Figure 3.19 shows the directional-spectral emissivity of titanium at θ = 25° (Edwards and Catton 1964, Edwards and Bayard de Volo 1965). The data points are for unoxidized metal, and the solid line is the ideal emissivity predicted by EM theory. The dashed curve is the observed emissivity when the surface is contaminated with a 0.06 µm thick oxide layer, which nearly doubles ελ compared to pure titanium over much of the wavelength range. Similarly, Figure 3.20 shows how oxidation affects the normal spectral emissivity of Inconel X over visible and near-infrared wavelengths (Wood 1964). Finally, Figure 3.21 shows how the normal total emissivity of copper and the hemispherical total emissivity of aluminum change with oxide layer thickness. The copper oxide layer was produced by heating copper in air, while the aluminum oxide layer was produced through anodization. The emissivity of both metals is initially very low (typical for a highly conducting metal) but increases as the oxide coating becomes thicker. The surface roughness of the oxide layer may also evolve as it grows, as discussed in Section 3.3.2.

Radiative Properties of Opaque Materials

117

FIGURE 3.17  Spectral emissivity of zinc oxide coatings on an oxidized stainless-steel substrate. (From Liebert, C. H., Spectral emittance of aluminum oxide and zinc oxide on opaque substrates, NASA TN D-3115, 1965).

FIGURE 3.18  Effect of substrate reflectivity characteristics on the hemispherical-spectral reflectivity of a TiO2 paint film for normal incidence. Film thickness, 14.4 μm; volume concentration of pigment, 0.017. (From Shafey, H.M. et al., AIAA J., 20(12), 1747, 1982).

118

Thermal Radiation Heat Transfer

FIGURE 3.19  Effect of oxide layer on directional-spectral emissivity of titanium. Emission angle, θ = 25°; surface lapped to 0.05 mm RMS; temperature 294 K.

FIGURE 3.20  Effect of oxidation on normal spectral emissivity of Inconel X. (Data from Wood, W. D., et al., Thermal Radiative Properties, Plenum Press, New York, 1964).

In some cases, an oxide film can induce oscillatory patterns in ελ or ρλ due to the phase shift caused by the oxide layer, leading to constructive and destructive interferences between the incident and reflected EM wave. Figure 3.22 shows the spectral emissivity of an iron surface heated in air as a function of time (del Campo et al. 2008.) The rate at which the peaks shift is connected to the oxide layer growth, which is used to infer the oxidation reaction kinetics. Makino et al. (1988) reported similar results for oxide layers on chromium. This effect is also responsible for the attractive colors of anodized metals, which can be controlled through careful engineering of the oxide layer thickness. Thin film interference effects are discussed further in Chapter 17.

Radiative Properties of Opaque Materials

119

FIGURE 3.21  Effect of oxide layer thickness on the normal total emissivity of copper and hemispherical total emissivity of aluminum. Data for copper: Brannon, R. R., and Goldstein, R. J., ASME J. Heat Trans., 92(2) 257, 1970. Data for aluminum: Touloukian, Y.S. and Ho, C.Y. (eds.), Thermophysical Properties of Matter, TRPC Data Services, Volume 9, Thermal Radiative Properties: Coatings, Touloukian, Y.S., DeWitt, D.P., and Hernicz, R.S. (1972b).

FIGURE 3.22  Thin film interference effect due to oxide layer growth. From del Campo, L. Pérez-Sáez, R. B., Tello, M. J.: Iron oxidation kinetics study by using infrared spectral emissivity measurements below 570°C, Corros. Sci. 50, 194, 2008.

3.3.2 Surface Roughness Effects Surface topography has a profound and varied impact on the radiative properties of metals and nonmetals, depending on the bulk refractive index of the surface and the size of the surface features relative to wavelength. While surfaces may appear smooth macroscopically, microscopically they consist of asperities that may be induced by the manufacturing process, mechanical damage, or

120

Thermal Radiation Heat Transfer

a chemical reaction (e.g., oxidation). In some cases, microscale roughness may be imparted with the specific intent of altering the radiative properties of a surface. If the process that caused the roughness is random, it is often reasonable to approximate the roughness as a sequence of normally distributed peaks, where the asperity heights z follow a normal distribution

p(z) =

1 so

æ z2 exp ç - 2 2p è 2s o

ö ÷ , (3.40) ø

where σo is the root-mean-square (rms) roughness, (sometimes written Rq)

so =

( z - zm )

2

, (3.41)

the mean surface height is zm, and represents the average, as shown in Figure 3.23. The rms roughness is readily measured using a contact profilometer, consisting of a stylus or needle that is dragged across a surface. The rms roughness is then used to derive the “optical roughness,” σo/λ. At small values of optical roughness, the surface may appear specular or “mirror-like,” while at large values the reflection may be more diffuse in nature. For this reason, many surfaces that appear diffuse at visible wavelengths can often be approximated as specular in the infrared. A good demonstration of this phenomenon involves looking at a classroom whiteboard with an infrared camera; the board reflects visible light diffusely, while a clearly defined image of the reflected body heat from the camera operator is visible in the infrared because the white board is optically smooth at these wavelengths. Generally, when σo/λ is greater than about unity, multiple reflections occur in the cavities between roughness elements; this increases the trapping of incident radiation, thereby increasing the observed surface absorptivity and consequently the emissivity. Graphite has one of the highest emissivities of commonplace materials due in part to its porous structure, and consequently it is often used to construct surfaces having radiative properties that mimic those of a blackbody.

FIGURE 3.23  Definition of RMS roughness. Both surfaces shown here have similar RMS roughness, but would have different radiative properties. Moreover, roughness exists at different scales, which interact with an EM wave in different ways depending on the wavelength.

Radiative Properties of Opaque Materials

121

VantaBlackTM enhances this effect via a field of closely packed and vertically aligned carbon nanotubes, giving it an even higher emissivity than that of graphite (Theocharous et al. 2006). Surface roughness may be caused by the manufacturing process, or it may be due to chemical reactions like oxidation. Figure 3.24 shows the normal spectral emissivity of three types of aluminum surface: commercial aluminum, mechanically polished aluminum and a surface formed by ultrahigh vacuum deposition of aluminum onto a quartz surface. The spectral emissivity is largest for the roughest sample and is smallest for the smoothest surfaces. The influence of roughness is most pronounced at the short wavelengths, where the optical roughness is the largest. Williams et al. (1963) showed that the hemispherical-spectral emissivity of an aluminum surface coated with lead sulfide is a strong function of the crystalline structure of the lead sulfide, which may appear as cubes or dendrites. In a more recent study, Yu et al. (2019) attribute variations in spectral emissivity of copper surfaces oxidized in air at various temperatures and durations to the differing surface morphology of the oxides, as shown in Figure 3.25. Shi et  al. (2015) measured the spectral emissivity of steel sheets coated with an Al-Si layer, which prevents oxidation and decarburization of the steel during furnace heating. The hemispherical-spectral emissivity is plotted in Figure 3.26 at various stages of furnace heating. The radiative properties of the as-received aluminum coating are typical of a metallic surface having a roughness in the order of 1–2 μm and is similar to the normal spectral emissivity of commercial aluminum in Figure 3.24. The coating melts at approximately 575°C, giving the surface a very low emissivity, characteristic of a smooth, highly conducting metal. At higher temperatures the coating reacts with iron from the substrate steel to form an intermetallic layer having radiative properties more representative of an insulator. The emissivity continues to increase with heating due to the increasing surface roughness. This effect is highlighted by examining the evolving spectral emissivity of samples heated in air and argon. The argon-heated samples are much smoother, and consequently have a lower spectral emissivity compared to the ones heated in air, as shown in Figure 3.27.

FIGURE 3.24  Normal spectral emissivity of aluminum surfaces having different surface finishes. Data from Touloukian, Y.S. and Ho, C.Y. (eds.), Thermophysical Properties of Matter, TRPC Data Services; Volume 7, Thermal Radiative Properties: Metallic Elements and Alloys, 1970.

122

Thermal Radiation Heat Transfer

FIGURE 3.25  Normal spectral emissivity of copper surfaces heated in air for 1 hr at different soak temperatures. Scanning electron micrographs show the corresponding surface topographies. From Yu, K., Zhang, H., Liu, Y., Liu, Y.: Study of normal spectral emissivity of copper during thermal oxidation at different temperatures and heating times, IJHMT, 129, 1066, 2019.

FIGURE 3.26  Spectral emissivity of Al-Si coated 22MnB5 steel. (Data from ArcelorMittal).

Radiative Properties of Opaque Materials

123

FIGURE 3.27  Influence of atmosphere on surface morphology and spectral emissivity of Al-Si coated 22MnB5 steel during two-step heating. The samples are initially heated at 610°C for 5 minutes and then 900°C for varying durations. Solid curves show in-situ normal spectral emissivity measurements, while symbols show ex-situ hemispherical spectral emissivity. Arithmetic roughness (Ra) was measured with a contact profilometer. From Shi, C. J., Daun, K. J., and Wells, M. A.: Spectral emissivity characteristics of the Usibor® 1500P steel during austenitization in argon and air atmospheres, IJHMT, 91, 818, 2015.

While spectral emissivity generally increases with surface roughness, this is not always the case. Cox (1965) showed that for dielectric materials with σoβλ, a ratio of roughness to mean penetration distance in the material of about 0.05 (the mean penetration distance is l = 1/βλ), the emissivity may be less than a smooth surface. This can be considered as an extension of the “snow effect” described in the preceding section. Surface roughness also affects the directional nature of emission and reflection. Figure 3.28 shows the ratio of the reflectivity in the specular direction of a rough nickel surface compared to that of a polished specimen, which increases as σo/λ becomes large. Figure 3.29 shows the bidirectional spectral reflectivity of an aluminum surface having an optical roughness of σ0/λ of 2.6 in the incidence plane (cf. Figure 3.1.) Detailed bidirectional reflectivity measurements for magnesium oxide ceramic with optical roughness σo/λ varying from 0.46 to 11.6 are listed by Torrance and Sparrow (1966). As the roughness and incidence angle of the incoming radiation are increased, off-specular peaks are obtained. Bidirectional-spectral reflectivity measurements are reported by DeSilva and Jones (1987) for six materials used for solar energy absorption. Peaks in the specular direction were found. Drolen (1992) provides bidirectional reflectivity measurements for 12 materials commonly used for spacecraft thermal control, such as white paint, black Kapton, and aluminized Kapton. The “specularity” given is the fraction of the directional-hemispherical reflectivity contained within the specular solid angle. Specularity values can be used in surface property models that assume a combination of diffuse and specular reflectivities (cf. Sections 6.5 and 6.6). Most measurements reported by Drolen are at a wavelength of λ = 0.488 μm, but some white paints were measured at four discrete wavelengths covering the range 0.488 ≤ λ ≤ 10.63 μm. Zaworski et al. (1996a) present detailed bidirectional reflectance data for a variety of surfaces, including Krylon white paint.

124

Thermal Radiation Heat Transfer

FIGURE 3.28  Effect of roughness on bidirectional reflectivity in the specular direction for ground nickel specimens. Mechanical roughness for polished specimen, 0.0015 μm. (From Birkebak, R. C., and Eckert, E. R. G., JHT; 87(1) 1965).

FIGURE 3.29  Bidirectional reflectivity in the incidence plane for various incidence angles; 2024–T4 aluminum (coated); σo = 1.3 μm; wavelengths of incident radiation, λ = 0.5 μm. (From Torrance, K. E., and Sparrow, E. M., JOSA, 57(9), 1105, 1967).

Figure 3.30 shows that the painted surface is quite diffuse until incident angles of 60° or greater are reached. Combined, the results suggest that the bidirectional reflectivity can be modeled as the superposition of a diffuse reflector and a specular reflector. This approximation has merit in some cases and simplifies the radiant exchange calculations compared with using exact directional properties (Sparrow and Liu 1965, Sarofim and Hottel 1966). In other cases, however, this approximation would fail. A particularly interesting example concerns the lunar surface, which is highly reflective due to its roughness and porosity. When the moon is full in phase, the sun, earth, and moon are nearly coincident and the moon is uniformly

Radiative Properties of Opaque Materials

125

FIGURE 3.30  Bidirectional reflectivity of Krylon white paint. Plots are of ln[ρλ(λ,θi,ϕi, θr, ϕr,)] at λ = 0.6328 μm. (From Zaworski, J. R., et al., IJHMT, 39(6) 1149, 1996a).

irradiated with nearly collimated solar radiation, as shown in Figure 3.31. Were the lunar surface diffuse, as one may expect for a rough surface, the reflected light would follow a Lambertian cosine distribution, and the moon would appear brightest at its center and darker near its edges. On the contrary, however, we see the moon as a white disk of nearly uniform intensity. The reason for this observation can be traced back to the bidirectional reflectivity measurements plotted in Figure 3.32 that show a strong back-scattering component, located 180° away from the direction of specular reflectance (Orlova 1956). Consequently, the bidirectional reflectance component increases approximately proportional to 1/cosθi (shown by the dashed line in Figure 3.32). This compensates for the reduced energy incident per unit area at large angles. Further discussions of the radiative properties of the lunar surface are given by Saari and Shorthill (1967), Harrison (1969), Birkebak (1974), and Sailsbury et al. (1997). Ohtake et al. (2013) describe recent measurements carried out from a lunar orbiter. While optical roughness provides a general indication of the importance of surface topography over a particular spectrum and can be used to interpret trends in the data, it is inadequate to make quantitative predictions of radiative properties. It does not account for the horizontal spacing of the roughness, for example, nor does it indicate the distribution of the roughness size around the rms value (which may not be normally distributed), nor information on the average slope of the sides of the roughness peaks. Both surfaces shown in Figure 3.23 may have the same rms roughness, but the slopes and spacing between the asperities are quite different, which would lead to dissimilar

126

Thermal Radiation Heat Transfer

FIGURE 3.31  Reflected energy at full moon. (Not to scale).

FIGURE 3.32  Bidirectional total reflectivity in the incidence plane for mountainous regions of the lunar surface. After Orlova, N. S. Astron. Z. 33(1), 93, 1956).

radiation properties. A further complication is that surface topography exists at multiple scales, and often has a self-similar, “fractal like” quality (Majumdar and Tien 1990a, 1990b) that is not captured by the rms roughness measured by contact profilometry. The influence of additional geometric scales that characterize surface roughness is also addressed by Yang and Buckius (1995). An important example concerns the radiative properties of metals, which are needed to carry out multiwavelength pyrometry during thermal processing. The spectral emissivity of these alloys is known to evolve due to oxide nodules that appear as the steel is heated within a reducing atmosphere. However, contact profilometry measurements made on samples extracted from a furnace at

127

Radiative Properties of Opaque Materials

intermediate heating times did not correlate with the changing emissivity; this is because the rms roughness measured using the contact profilometer was dominated by large-scale rolling artifacts that did not influence the radiative properties of the specimen (Ham et al. 2018). Additional parameters and more sophisticated measurement techniques are therefore needed to derive quantitative estimates of radiative properties. Some contact profilometers provide the average surface slope, or additional “skewness” and “kurtosis” parameters to describe the shape of the asperities. The local radius of curvature and spacing between asperities are also important. For randomly rough surfaces this latter quantity can be quantified by the Gaussian correlation function

C (L) =

1 so2

(z ( x) - z )(z ( x + L) - z ) m

m

2 - L t = e ( ) (3.42)

where τ is the correlation length, which is the width of a Gaussian correlation function over which the roughness correlation decreases by a factor of e. Optical profilometers use the principle of interferometry to construct a “2D” picture of roughness. Many of these instruments provide a “power spectrum” of the surface topography, showing the prominence of different length scales in accordance with the fractal-like nature of surface roughness. Still higher resolution can be obtained through atomic force microscopy (AFM), which have horizontal and vertical resolution on the order of ~30 nm and 0.1 nm, respectively. Example optical and AFM profilograms are shown in Figure 3.33. Detailed surface topography information can sometimes be used to derive theoretical or numerical models for radiative properties of surfaces. Wen and Mudawar (2006) present a taxonomy of models used to predict the spectral emissivity of metallic surfaces, shown in Figure 3.34. These surfaces can broadly be categorized as perfectly smooth; slightly rough; moderately rough; and very rough. A schematic of the bidirectional reflectances that correspond to this case is shown in Figure 3.35. The radiative properties of perfectly smooth surfaces can be modeled directly using the relations in Section 3.2.2. For slightly rougher surfaces, small surface asperities act as “scattering centers” for the incident wave, and the observed radiative properties are due to the superposition of the scattered waves in the far field. Davies (1954) derived an analytical model using the Kirchhoff approximation,

FIGURE 3.33  Optical profilogram, atomic force micrograph, and scanning electron micrograph of DP980 steel alloy heated in a reducing atmosphere. (The box outline on the optical profilogram does not coincide with the AFM region, but emphasizes scale.) Liu, K., and Daun, K. J.: Interpreting the spectral reflectance of advanced high strength steels using the Davies’ model, JQSRT, 242, 106796, 2020.

128

Thermal Radiation Heat Transfer

FIGURE 3.34  Types of models used to predict spectral emissivity of metallic surfaces. Wen, C.-D and Mudawar, I.: Modeling the effects of surface roughness on the emissivity of aluminum alloys, IJHMT, 49, 4279, 2006.

FIGURE 3.35  Transition from specular reflectance to diffuse scattering. Surfaces are: (a) smooth; (b) slightly rough; (c) moderately rough; (d) very rough. Beckman, P., and Spizzichino, A.: The Scattering of Electromagnetic Waves from Rough Surfaces, Pergamon Press, New York, 1963.

in which the Fresnel reflection equations are applied to the tangent plane of the surface roughness at each point of incidence; this may be reasonable, provided that the local radius of curvature is less than the wavelength, and the correlation length is sufficiently large. The material is further to be a perfect electrical conductor so that, from EM theory, the extinction coefficient is infinite. This provides perfect reflection, and consequently, the theory gives the directional distribution of the energy that is reflected distribution about the specular peak as shown in Figure 3.35b, as opposed to the reflectivity. According to this theory the reflectance in the specular direction is given by

2 é æ 4pso ö ù rl = exp ê - ç cos qi ÷ ú (3.43) ø úû êë è l

which diminishes as the surface becomes optically rough and the lobular scattering pattern in Figure 3.35b becomes broader. This treatment can be extended to surfaces having finite conductivity by introducing a coefficient ρλ,s to the right-hand side of Equation (3.43) that equals the reflectance of a perfectly smooth surface made of the same material (e.g., from Fresnel’s relations). Davies treatment is inaccurate at near-grazing angles because shadowing by the roughness element s is neglected. Modeling the radiative properties of a moderately rough surface is considerably more complex. Porteus (1963) extended Davies approach by removing the restrictions on the relation between σ0

Radiative Properties of Opaque Materials

129

and λ and including more parameters for specifying the surface roughness characteristics. Some success was obtained in predicting the roughness characteristics of prepared samples from measured reflectivity data. Measurements were mainly at normal incidence, and the neglect of shadowing makes the results doubtful at near-grazing angles. Beckmann and Spizzichino (1963) also extend the applicability of Davies’ theory by accounting for the autocorrelation distance of the roughness, τ. The method provides better data correlation than the earlier analyses and captures the more diffuse reflectance that one would expect for large values of optical roughness. A critical evaluation of Davies and Beckmann analyses is given by Houchens and Hering (1967). Data for aluminum have been well-correlated in Smith and Hering (1970) by use of Beckmann’s theory. The emissivity of a surface with random submicron roughness elements is discussed by Carminati et  al. (1998) and Ghmari et  al. (2004). Through scaling arguments, it is shown that interference effects between radiation emitted by closely spaced elements can be neglected, which is not the case for reflected energy. Based on this observation, predictions of directional-spectral emissivity from surfaces with random submicron roughness elements can be made through numerical simulation. Because the model initially neglects shadowing and blocking effects, the predictions are restricted to rms height to wavelength ratios of σo/λ  15 μm) the surface is “slightly rough” and the specular reflectance approaches the hemispherical reflectance, indicating that the surface behaves in a specular manner and ρλ can be modeled

FIGURE 3.36  Spectral hemispherical reflectance of DP980 steel alloy annealed in a reducing atmosphere. Liu, K., and Daun, K. J.: Interpreting the spectral reflectance of advanced high strength steels using the Davies’ model, JQSRT, 242, 106796, 2020.

130

Thermal Radiation Heat Transfer

using Equation (3.43). At shorter wavelengths the surface becomes moderately rough, as indicated by the drop in the specular reflectivity and an increase in the diffuse reflectivity. At still shorter wavelengths (λ  1.27 μm. In particular, since the Hagen–Rubens theory predicts that ελ ∝ re1/2 and the resistivity of many metals varies linearly with temperature, cf. Equation (3.33), it follows that ελ should also vary with the square root of temperature (in K). This should also approximately hold for hemispherical values, which is shown to be the case in Figure 3.43 at λ = 2 μm, although at this wavelength one may not expect an accurate quantitative prediction using Hagen–Rubens theory. Figure 3.43 also illustrates a characteristic of many metals, as discussed by Sadykov (1965). At short wavelengths (in the case of tungsten, λ > 1.27 μm), the temperature effect is reversed and the spectral emissivity decreases with increased temperature. The emissivity curves all cross at the same point, called the “X point.” Sell et al. (1964) provide a theoretical explanation of the X-point based on how the electron density-of-states varies with temperature. Other X points for different metals include iron, 1.0 μm; nickel, 1.5 μm; copper, 1.7 μm; and platinum, 0.7 μm. Thermionic energy conversion devices use high-temperature alloys of tungsten and rhenium compounded with thoria and hafnium carbide. Their normal spectral emittances were measured by Tsao et al. (1992), and results were given for temperatures from 1500 to 2500 K. Most of the results are at λ = 0.535 μm, and the normal spectral emittances were found to decrease linearly with increasing temperature. This is the expected trend, since the wavelength is smaller than at the X point and hence spectral emissivity should decreases as temperature increases.

Radiative Properties of Opaque Materials

135

FIGURE 3.43  Effect of wavelength and surface temperature on hemispherical-spectral emissivity of tungsten. (From De Vos, J.C., Physica, 20, 690, 1954).

FIGURE 3.44  Effect of temperature on hemispherical total emissivity of several metals. (From Touloukian, Y.S. and Ho, C.Y. (eds.): Thermophysical Properties of Matter, TRPC Data Services; Volume 7, Thermal Radiative Properties: Metallic Elements and Alloys, Touloukian, Y.S. and DeWitt, D.P., 1970).

The increase of spectral emissivity with decreasing wavelength for metals in the IR region (wavelengths longer than in the visible region) accounts for the increase in total emissivity with temperature. With increasing temperature, the peak of the blackbody radiation distribution (Figure 1.10) moves toward shorter wavelengths. Consequently, as the surface temperature increases, proportionately more radiation is emitted in the region of higher spectral emissivity, which increases the total emissivity. Some examples are plotted in Figure 3.44; in the case of tungsten, experimental

136

Thermal Radiation Heat Transfer

measurements are compared with Equation (3.37), derived by Parker and Abbott (1964) from Hagen–Rubens theory. The radiative properties of surfaces at extremely low temperatures provide key fundamental insights into the electronic structure of materials, and are also important in many practical engineering applications, for example, vacuum cryogenic insulation systems where the dominant heat transfer mechanism is through radiative transfer. Were electrical resistivity directly proportional to temperature, Hagen–Rubens theory would indicate that emissivities should become quite small at low T and large λ. Likewise, Drude theory, which reduces to Hagen–Rubens relation at long wavelengths, predicts ε values that decrease much more rapidly with temperature than has been observed. Experimental measurements summarized by Toscano and Cravalho (1976) for copper, silver, and gold indicate that ε does not decrease to such small values. The anomalous skin-effect model with diffuse electron reflections was found to predict the emissivity most accurately. Figure 3.45 shows results of the model. Similar results for gold films are reported by Tien and Cunnington (1973). Frolec et al. (2019) provide a database of measured total hemispherical absorptivity and emissivity data in the cryogenic range from 20 K to between 120 and 320 K for 58 polished metals, metal foils, and metals with various surface finishes. They show agreement with previous measurements, and also show the expected increases in emissivity when polished surfaces are abraded to various degrees. It is fitting to conclude this section with two final remarks: First, while EM theory may predict how the emissivity of a metal surface varies with changes in wavelength and temperature, quantitative agreement between these values and experimental measurements should only be expected for polished specimens, and otherwise should be considered as a lower bound due to surface roughness.

FIGURE 3.45  Effect of low temperatures on hemispherical total emissivity of copper. (Results from Toscano, W.M. and Cravalho, E.G., JHT, 98(3), 438, 1976).

Radiative Properties of Opaque Materials

137

Second, a temperature-induced change in surface morphology of a metal, e.g., oxidation, often has a far greater influence on the radiative properties than one would expect via Hagen–Rubens theory, again keeping in mind that this theory is limited to longer wavelengths.

3.3.4 Properties of Liquid Metals Liquid metals tend to be more reflective compared to their solid counterparts, since their surface is perfectly smooth. Among numerous studies that have investigated the radiative properties of liquid metals, of particular note is a series of papers by Krishnan et al., who employed a containerless technique through magnetic levitation to avoid chemical reactions between the sample and gas or crucible that can occur at extreme temperatures. These studies include Krishnan et al. (1990a) (copper, silver, gold, nickel, palladium, platinum, and zirconium), Krishnan et al. (1990b), (liquid silicon, aluminum, titanium, and niobium), Krishnan and Nordine (1993) (aluminum), and Krishnan et al. (1997) (nickel and iron). A subset of these results is plotted in Figure 3.46, showing that ελn is nearly independent of temperature. Moscowitz et al. (1972) reported the spectral emissivity values at λ = 0.645 μm of three rare earth liquid metals, which were similarly independent of temperature. This can be attributed to the fact that in solid metals the electrical resistance increases with temperature largely due to temperature-induced lattice vibrations, which scatter the electrons, while liquid metals have no atomic lattice. Figure 3.47 shows how the normal spectral reflectance of several liquid metals varies with respect to wavelength. Drude theory, discussed in Section 3.2.3, is often successful for liquid metals since the electronic band structure caused by the periodicity of the crystal lattice disappears upon melting, and in many cases the radiative properties at infrared wavelengths can be interpreted solely in terms of intraband electronic transitions involving conduction band electrons (Faber 1972). Several studies show very good agreement between the dielectric constants calculated from ellipsometry measurements on molten metal and those predicted from Drude theory using the published density and DC conductivity, e.g., Miller (1969) and Comins (1972). (See Figure 8.10). In other cases where interband transitions are important, e.g., liquid iron and nickel (Krishnan et al. 1997), Drude theory cannot be applied. Havstad et al. (1993) reported the normal spectral emissivity of molten uranium, another transition metal, between 0.4–10 μm and at temperatures

FIGURE 3.46  Normal spectral emissivity of molten copper as a function of temperature at four different wavelengths. From Krishnan, S., Hansen, G. P., Hauge, R. H., Margrave, J. L.: Emissivities and optical constants of electromagnetically levitated liquid metals as functions of temperature and wavelength, In Materials Chemistry at High Temperatures: Volume 1 Characterization, J. W. Hastie (ed), Springer, 1990.

138

Thermal Radiation Heat Transfer

FIGURE 3.47  Normal spectral emissivity of liquid uranium between 1410 K and 1630 K. (From Havstad, M.A. et al., JHT, 115, 1013, 1993).

between 1410–1630 K. Both direct emittance and ellipsometry measurements were carried out. At longer wavelengths ελn is found to diminish with respect to wavelength, as shown in Figure 3.47. This is consistent with the general trend expected from Hagen–Rubens theory, although the refractive index inferred from ellipsometry could not be reproduced using Equation (3.26) due to interband transitions that are unaccounted for in this model. Havstad and Qiu (1996) used the electrical resistivity and Hall coefficients for liquid metals to predict the spectral normal emissivity of liquid cerium/copper alloys. Agreement with published data is good, and extension to copper-, gold-, silver-, and aluminum-rare Earth alloys is also expected to be good from similarities in property behavior for these alloys (important in lasercladding technologies).

3.3.5 Properties of Semiconductors and Superconductors Semiconductors are a particularly important case since the spectral emissivity must be known in order to measure the temperature of silicon wafers during rapid thermal processing. When the photon energy is greater than the band gap energy of silicon (~1.14 eV, or ~1.09 μm) the radiative properties are dominated by electron transitions between the valence and conduction bands. At longer wavelengths, however, the EM field interacts primarily with free carriers in the valence band (holes) and the conduction band (electrons). Semiconductors are often “doped” with impurities to tune their electronic properties, and, as one may expect, this has a pronounced effect on the radiative properties as shown in Figure 3.48 for silicon. Radiative properties depend both on the carrier density (holes and electrons) as well as the scattering, which is dominated by the presence of impurities. Several researchers have attempted to model how doping may influence the radiative property through a modified Drude theory that accounts for both holes and electrons (e.g., Basu et al. 2010.) Liebert and Thomas (1967) demonstrated that the radiative properties of doped silicon could be modeled as that of a metal with high resistivity using the Hagen–Rubens relations. While silicon and germanium are semiconductors in solid state, in liquid form they are metals because the valence band electrons that are localized by covalent bonds are released and form a conduction band upon melting. Consequently, liquid silicon and germanium have a high normal reflectance in the visible wavelengths, in contrast to the solid state (Lampert et al. 1981), as shown

Radiative Properties of Opaque Materials

139

FIGURE 3.48  Spectral normal reflectance of pure and doped silicon. (Adapted from Touloukian, Y.S. and Ho, C.Y. (eds.), Thermophysical Properties of Matter, TRPC Data Services; Volume 8, Thermal Radiative Properties: Nonmetals).

in Figure 3.49. A number of Drude models for liquid Si and Ge have been derived either by fitting ωp, and ζ from Equation (3.39) to ellipsometry measurements or from the density and DC resistivity of the molten semiconductor, as described in Chapter 8 (Shvarev et al. 1977, Li and Fauchet 1987, Jellison and Lowndes 1987, Kawamura et al. 2005).

3.4 SELECTIVE SURFACES FOR SOLAR APPLICATIONS It is often desirable to tailor the radiative properties of surfaces to increase or decrease their natural ability to absorb, emit, or reflect radiant energy. This is particularly the case in solar-related applications: In some instances the goal is to maximize the fraction of solar energy absorbed at short wavelengths and minimize emission losses at longer wavelengths, e.g., solar collectors, while in other cases the fraction of reflected sunlight should be maximized to reduce the cooling load of buildings in the summer. (This is often called “radiative cooling.”) While so-called spectrally selective surfaces have long been used for these purposes, recent advancements in nanotechnology allow engineers to precisely “tune” surface properties according to the requirements of the specific application, leading to significant performance improvements. To understand the performance of these types of surfaces, we first review the key characteristics of solar radiation.

3.4.1 Characteristics of Solar Radiation The key attributes that describe solar irradiation are the solar constant and the solar temperature. The solar constant describes the average total solar irradiation incident on a surface normal to the sun at a distance equal to the Earth’s mean radius from the sun; it accounts for the solid angle subtended by the sun as viewed by the earth, but excludes attenuation of solar energy by the atmosphere. The accepted value by many standards organizations including the American Society for Standards and Measurement (ASTM) is 1366 W/m2, although the National Oceanographic and Atmospheric

140

Thermal Radiation Heat Transfer

FIGURE 3.49  Spectral normal reflectance of polished solid and liquid silicon. Solid silicon data is from Green and Keevers (1995). Liquid silicon measurements are from Shvarev et  al. (1975) and Jellison and Lowndes (1987); Drude model is from Kawamura et al. (2005).

Administration (NOAA) uses a value of 1376 W/m2. The value fluctuates slightly with time due to changes in the solar energy output. Larger changes are due to the eccentricity of the earth’s orbit about the sun, causing variations from 1322 W/m2 at aphelion to 1412 W/m2 at perihelion. In this book we use a value of 1366 W/m2 unless specified otherwise. The solar temperature defines the spectral distribution of solar irradiation, which is close to that of a blackbody at a temperature of 5780 K. The radiation reaching the earth’s surface comes from the plasma contained in the sun’s photosphere and originates from bremsstrahlung radiation (“braking radiation”) emitted as excited electrons in the plasma decelerate around the ions and neutrals. The photosphere has a much lower temperature than the sun’s interior (millions of degrees) where a thermonuclear reaction is occurring.

3.4.2 Modification of Surface Spectral Characteristics For surfaces that collect solar energy, such as in solar distillation units, solar furnaces, or solar collectors for energy conversion, it is desirable to maximize the energy absorbed while minimizing emission losses. In solar thermionic or thermoelectric devices, the best performance is obtained by maintaining the solar-irradiated surface at its highest possible equilibrium temperature. For photovoltaic (PV) solar cells, absorption should be maximized within the bandgap of the PV material to maximize electrical output, but absorption should be minimized in the IR spectrum to minimize heating of the cells and minimize cooling requirements (Hajimirza et al. 2011, 2012, Hajimirza and Howell 2012, 2013a, b, 2014a, b.). For situations where a surface is to be kept cool while exposed to the sun, it is desired to have maximum reflection of solar energy with maximum radiative emission from the surface. For solar energy collection, a black surface maximizes the absorption of incident solar energy; unfortunately, it also maximizes the emissive losses. However, if a surface could be manufactured

Radiative Properties of Opaque Materials

141

FIGURE 3.50  Characteristics of some spectrally selective surfaces.

that had a large absorptivity in the spectral region of short wavelengths about the peak solar energy, yet small in the spectral region of longer wavelengths where the peak surface emission would occur, it might be possible to absorb nearly as well as a blackbody while emitting very little energy. Such surfaces are called “spectrally selective.” One method of manufacture is to coat a thin, nonmetallic layer onto a polished metallic substrate. As discussed in Section 3.3.1, the thin coating is essentially transparent at long wavelengths, and the surface is highly reflective due to the substrate metal. At short wavelengths, however, the radiation characteristics approach those of the nonmetallic coating, so the spectral emissivity and absorptivity are relatively large. Some examples of this behavior are in Figure 3.50 (Shaffer 1958, Hibbard 1961, Long 1965). An ideal solar-selective surface maximizes the amount of absorbed solar irradiation while minimizing losses due to emission. A good surface should therefore have a spectral absorptivity near unity over short wavelengths where the incident solar energy has a large intensity and a low spectral emissivity close to zero at longer wavelengths. The wavelength at which this transition occurs, as shown in Figure 3.50, is the cutoff wavelength. Example 3.5 An ideal selective surface is exposed to a normal radiation flux equal to the average solar constant Gsolar = 1366 W/m2. The only means of heat transfer to or from the exposed surface is by radiation. Determine the maximum equilibrium temperature Teq corresponding to a cutoff wavelength of λc = 1 μm. The solar energy can be assumed to have a spectral distribution proportional to that of a blackbody at 5780 K. Because the only means of heat transfer is by radiation, the radiant energy absorbed must equal that emitted from the exposed side. For an ideal selective absorber, the hemispherical emissivity and absorptivity are

el = a l = 1; 0 £ l £ l c el = a l = 0; l c < l < ¥.

142

Thermal Radiation Heat Transfer

The energy absorbed by the surface per unit time is Qa = a lF0®lc (TR )Gsolar A = (1)F0®lc (TR )Gsolar A



where F0®lc (TR ) is the fraction of blackbody energy in the range of the wavelengths between zero and the cutoff value, for a radiating temperature TR. In the case, TR is the effective solar radiating temperature 5780 K. Similarly, the energy emitted by the selective surface is Qe = elF0®lc (Teq )sTeq 4 A = (1)F0®lc (Teq )sTeq 4 A.

Equating Qe and Qa gives

Teq 4F0®lc (Teq ) =

qsolF0®lc (TR ) . s

For the chosen value for λc, all terms on the right are known, and we can solve for Teq by trial and error. The equilibrium temperature for λc = 1 μm is 1337 K. Here are the values of Teq for various λc:

For a blackbody surface (λc → ∞), the equilibrium temperature is 394 K; this is the equilibrium temperature of the surface of a black object in space at the earth’s orbit when exposed to normally incident solar radiation and with all other surfaces of the object perfectly insulated. The same equilibrium temperature is reached by a gray body, since a gray emissivity cancels out of the energy-balance equation. As λc is decreased, the Teq continues to increase even though less energy is absorbed; this is because it also becomes relatively more difficult to emit energy.

A performance parameter for a solar-selective surface is the ratio of its directional total absorptivity α(θi, ϕi, T) for incident solar energy to its hemispherical total emissivity ε(T). The ratio α(θi, ϕi, T)/ε(T) for the condition of incident solar energy is a measure of the theoretical maximum temperature that an otherwise insulated surface can attain when exposed to solar radiation. In general, the energy absorbed per unit time by a surface element dA



dQa ( qi , fi , T ) = a ( qi , fi , T ) I solar ( qi , fi ) dWsolar dAcosqi = a ( qi , fi , T ) Gsolar ( qi , fi ) dAcosqi

(3.44)

where Isolar is the incident (total) solar intensity, dΩsolar is the solid angle subtended by the sun as viewed by dA, and Gsolar is the corresponding directional energy flux. The total energy emitted per unit time by the surface element is

dQe = e(T )sT 4 dA. (3.45)

143

Radiative Properties of Opaque Materials

If there are no other modes of energy transfer (e.g., conduction or convection losses), the emitted and absorbed energies are equated to give sTeq4 a(q i , f i , T ) = (3.46) e(Teq ) qsol (qi , fi )cosfi



where Teq is the equilibrium temperature that is achieved. Thus, the ratio α(θi,ϕi,T)/ε(T) for T ≈ Teq is a measure of the equilibrium temperature of the element. Note that the temperature at which the properties α and ε are selected should be the equilibrium temperature that the body attains. For the collection and utilization of solar energy on Earth or in outer space applications, it is common to have normal solar incidence so that α = αn and cosθi = 1. Naturally, a high αn/ε is desired. For the relatively low temperatures of solar collection in ground-based systems without solar concentrators, selective paints on an aluminum substrate have αn = 0.92 and ε = 0.10 (Moore 1985). Low-emittance metallic flakes also mixed in a binder with high-absorptance metallic oxides to yield coatings with αn = 0.88 and ε = 0.40. For ground-based solar collectors, convection heat transfer must be included in the energy balances, and solar collectors are often designed to minimize these losses. To attain high equilibrium temperatures for space power systems, polished metals attain αn/ε of 5–7, and specially manufactured surfaces have αn/ε approaching 20. Coatings with αn/ε ≈ 13 and stability at temperatures up to about 900 K in air are reported by Craighead et al. (1979). Space power systems usually have a concentrator such as a parabolic mirror. This increases the collection area relative to the area for emission and thus effectively increases the absorption-to-emission ratio even further. Economical and durable paints are desired for application to large solar collection areas. Many of these ideas are discussed in reviews by Granqvist (2003) and Wijewardane and Goswami (2012). Example 3.6 The properties of a real SiO–Al selective surface are approximated by the long dashed curve in Figure 3.46 (it is assumed that this curve can be extrapolated toward λ = 0 and λ = ∞). What is the equilibrium temperature of the surface for normally incident solar radiation at Earth orbit when energy transfer is only by radiation? What is αn/ε for the surface? Describe the spectra of the absorbed and emitted energy at the surface. Assume normal and hemispherical emissivities are equal. As in the derivation of Equation (3.46), equate the absorbed and emitted energies. The emissivity has nonzero constant values on both sides of the cutoff wavelength, so that ¥

Qa = A

òa

Gl ,solar d l

l ,n

l =0



= éëe0®lc F0®lc (TR ) + e lc ®¥ Flc ®¥ (TR ) ùû Gsolar A = a nGsolar A ¥

Qe = A

òe E

l lb



(Teq ) d l

l =0

= éëe0®lc F0®lc (TR ) + elc ®¥ Flc ®¥ (Teq ) ùû sTeq4 A = esTeq4 A where TR is the temperature of the radiating source. Equating Qe and Qa gives

{0.95F

0 ®lc



{

(TR ) + 0.05 éë1- F0®l (TR )ùû} Gsolar c

}

= 0.95F0®lc (Teq ) + 0.05 éë1 - F0®lc (Teq ) ùû sTeq4 .



144

Thermal Radiation Heat Transfer

Solving by trial and error, for λc = 15 μm, we obtain Teq = 790 K. The small difference in the properties of an ideal selective surface produces a significant change in Teq, which in the previous example was 1048 K for an ideal selective surface with the same λc. The spectral curve of incident solar energy is given by Eλ,i(TR) ∝ Eb,λ(TR). It has the shape of the blackbody curve at the solar temperature, TR = 5780 K, but it is reduced in magnitude so that the integral of Eλ over all λ is equal to Gsolar, the total incident solar energy per unit area at Earth orbit. Multiplying this curve by the spectral absorptivity of the selective surface gives the spectrum of the absorbed energy. The spectrum of emitted energy is that of a blackbody at 790 K multiplied by the spectral emissivity of the selective surface. The integrated energies under the spectral curves of absorbed and emitted energy are equal.

The energy equation solved in Example 3.6 is a two-spectral-band approximation to the following more general energy-balance equation for a diffuse surface: ¥

ò



a l ( Teq ) Gl,solar ( l ) dl =

¥

ò e (T l

eq

) Elb (Teq ) dl.

l =0

l =0

The Gλ,solardλ can have any spectral distribution, and by Kirchhoff’s law αλ(Teq) = ελ(Teq) for a diffuse surface. A more general situation is in Figure 3.51, where the absorption of incident energy from the θi direction depends on the directional-spectral absorptivity αλ(θi,ϕi,Teq). The emission from the surface depends on its hemispherical-spectral emissivity ελ(Teq). The qe is the heat flux supplied to the surface by any other means, such as convection, electrical heating, or radiation to its lower side. The heat balance then becomes ¥



cosqi

òa

l =0

l

( qi , fi , Teq ) Gl,solar dl + qe =

¥

ò e (T l

eq

) Elb (Teq ) dl. (3.47)

l =0

Equation (3.47) is readily solved by using integration subroutines and root solvers. This analysis can be used for temperature control of space vehicles, as shown by Furukawa (1992), who determined the solar and Earth radiation fluxes incident on the different surfaces of an orbiting vehicle.

FIGURE 3.51  Radiative energy incident on a selective surface.

145

Radiative Properties of Opaque Materials

Example 3.7 A spectrally selective surface is to be used as a solar energy absorber. The surface that has the same properties as given in Example 3.6 is to be maintained at T = 394 K by extracting energy for a power-generating cycle. If the absorber is in orbit around the sun at the same radius as the Earth and is normal to the solar direction, how much energy will a square meter of the surface provide? How does this energy compare with that provided by a black surface at the same temperature? Emitted energy and reflected solar energy by Earth are neglected. The energy extracted from the surface is the difference between the absorbed and emitted radiation. The absorbed energy flux is calculated as in Example 3.6, where TR = 5780 K, ¥



qa =

òe

l

(T ) Gl ,solar (TR ) d l = {0.95F0®l (TR ) + 0.05 éë1- F0®l (TR )ùû} Gsolar c

c

l =0



= éë0.95 ( 0.880 ) + 0.05 (1- 0.880 ) ùû 1366 = 1142.1 W/m . 2

The emitted flux is ¥



qe =

òe E

l lb

(T ) d l = {0.95F0®l (T ) + 0.05 éë1- F0®l (T )ùû} sT 4 c

c

l =0



95 ´ (~ 0 ) + 0.05 (~ 1) ùû 5.6704 ´ 10-8 ´ 3944 = 70.7 W/m2 . = éë0.9 Therefore, the energy that can be used for power generation is 1142.1 − 70.7 = 1071.4 W/m2. For a blackbody or gray body, the equilibrium temperature is 394 K as obtained from Example 3.6 so, for a black or gray absorber, no useful energy can be removed for the stated conditions.

Spectrally selective surfaces can also be useful where it is desirable to cool an object exposed to incident radiation from a high-temperature source. Common situations are objects exposed to the sun, such as a hydrocarbon storage tank, a cryogenic fuel tank in space, or the roof of a building. Equation (3.46) shows that the smaller the value of α/ε that can be reached, the lower will be the equilibrium temperature. For a cryogenic storage tank exposed to solar flux in the vacuum of outer space, αn/ε should be as small as possible in order to reduce losses by heating the stored cryogen. In practice, values of α(θi)/ε in the range 0.20–0.25 can be obtained for normal incidence (cosθi = 1). A highly reflecting coating such as a polished metal can also be used for some applications. This would reflect much of the incident energy but would be poor for radiating away energy that was absorbed or generated within an enclosure, such as by electronic equipment. This behavior is important for the energy balance in the vacuum of outer space and may not provide a low α/ε, but this may not be important when there is appreciable convective cooling that dominates over radiant emission. Some metals may not work well because they have a tendency toward lower reflectivity at the shorter-wavelengths characteristic of the incident solar energy, e.g., aluminum. For some applications, spectrally selective materials are used. As discussed in Section 3.3.1, white paint is another example of spectrally selective surface. For thermal control in outer space, different spectrally selective surfaces have been defined. Among them, the optical solar reflector (OSR) is a mirror composed of a glass layer silvered on the back side. The glass, being transparent in the short-wavelength region, λ  λc, plot the curve of equilibrium temperature vs λc for the range 2 £ l c £ 10 mm .

150

Thermal Radiation Heat Transfer

3.10 For a free convective heat transfer coefficient of 6.0 W/m2 °C, an air temperature of 0°C, and an incident solar flux on the surface of 900 W/m2, calculate the equilibrium temperature of the earth’s surface when it is (a) covered with fine fresh snow. (b) covered with plowed soil. Discuss how this might impact global warming in snow-covered regions as the average global air temperature increases. Answers: a) 18.4°C; b) 87.1°C. 3.11 A diffuse spectral coating has the spectral emissivity approximated as below. The coating is placed on one face of a thin sheet of metal. The sheet is placed in an orbit around the sun where the solar flux is 1350 W/m2. The other face of the sheet is coated with a diffuse gray coating of hemispherical total emissivity ε = 0.530. What is the temperature (K) of the sheet if: (a) The side with the spectral coating is facing normal to the sun? (b) The gray side is facing normal to the sun? (c) What is the normal-hemispherical total reflectivity of the diffuse-spectral coating when exposed to solar radiation? Take the effective solar radiating temperature to be 5780 K. Note any necessary additional assumptions.

Answers: (a) 424.4 K; (b) 376.2 K; (c) 0.143. 3.12 A spinning spherical satellite 0.85 m in diameter is in earth orbit and is receiving solar radiation of 1353 W/m2. As a consequence of the rotation, the sphere surface temperature is assumed uniform. The sphere exterior is an SiO-Al selective surface as in Example 3.6, with a cutoff wavelength of 2.00 μm. The surface properties do not depend on the angle. Neglect energy emitted by the earth, and solar energy reflected from the earth. (a) What is the equilibrium surface temperature for heat transfer only by radiation? How does the temperature depend on the sphere diameter? (b) It is desired to maintain the sphere surface at 750 K. At what rate must energy be supplied to the entire sphere to accomplish this? Answers: 569 K; 1820 W. 3.13 A cylindrical concentrator (reflector) is long in the direction normal to the cross-section shown, so that end effects may be neglected. The concentrator is gray and reflects 94% of the incident solar energy onto the central tube. The central tube receiving the energy is assumed to be at uniform temperature and is coated with a material that has the spectral properties shown. The properties are independent of direction. Emitted energy from the tube that is reflected by the concentrator may be neglected, as may emission from the concentrator. The surrounding environment is at low temperature.

Radiative Properties of Opaque Materials

151

(a) If the heat exchange is only by radiation, compute the temperature of the central tube. (b) If the tube is cooled to 395 K by passing a coolant through its interior, how much energy must be removed by the coolant per meter of tube length?

Answers: (a) 816.3 K; (b) 1239 W/meter of length. 3.14 For a small diameter light-pipe with refractive index n2 = 1.4400, plot the reflectivity of the light-pipe end vs incident angle θ1 for radiation incident on the end. Also, plot the angle of refraction χ vs angle of incidence, and determine the range of incident angles θ1 that will have angles θ2 above the critical angle when the radiation encounters the cylindrical light-pipe wall.

4

Configuration Factors for Diffuse Surfaces with Uniform Radiosity Hoyt Clarke Hottel (1903–1998) was a Professor at MIT from 1928 until his death and became an Emeritus Professor in 1968. He developed gas emissivity charts for the important combustion gases, the crossed-string method for determining configuration factors in 2D geometries, and in 1927, papers established the engineering basis for treating radiation in furnaces, including the zone method. Hottel also contributed to the fields of combustion and solar energy.

4.1 RADIATIVE TRANSFER EQUATION FOR SURFACES SEPARATED BY A TRANSPARENT MEDIUM Radiation interchange among surface areas must be quantified to allow calculation of heat transfer rates for effective and efficient processes, building energy loads for interior thermal comfort, illumination engineering for visual comfort, and applied optics for the development of diagnostic tools. Studies have been conducted for many years, as evidenced by the publication dates of d’Aguillon (1613) and Charle (1888). Since 1960, the study of radiative transfer has been given impetus by technological advances that involve systems in which thermal radiation is very important. Examples include radiant heating of materials, curing, surface modification systems, satellite temperature control, devices for collection and utilization of solar energy, advanced engines with increased operating temperatures, reduction of energy leakage into cryogenic fuel storage tanks, space station power systems, thermal control during spacecraft entry into planetary atmospheres, heat islands in cities, atmospheric–ocean–land coupling to understand climate change, and many others. The general radiative transfer equation describes the propagation of intensity along a path and was introduced in Sections 1.10 and 1.11. For the important cases when the medium separating two surfaces is transparent (i.e. there is no attenuation of intensity along a path by scattering or absorption, and the medium along the path does not emit), then Equation (1.77) shows that the intensity leaving a radiating surface is invariant along a path, as shown in Figure 4.1. Under certain assumptions, the geometric relations between surfaces 1 and 2 can be separated from the radiative intensity, allowing the description of the geometric configuration of surface 2 relative to surface 1 to be treated independently of the thermal states of the two surfaces. The factors that contain only the geometric relations are called configuration factors (Equation (1.83)) The geometric configuration factors derived in this chapter are an important component for analyzing radiation exchange. Before considering them, some introductory comments are in order about “enclosure” theory that motivates the use of configuration factors. In this chapter and Chapters 5 through 7, the theory is developed for radiation exchange in enclosures that are evacuated or contain radiatively nonparticipating media. First, we must define what is meant by an enclosure. Any surface can be considered as being surrounded by an envelope of other solid surfaces or open areas. This envelope is the enclosure for the surface; thus, an enclosure accounts for all directions surrounding a surface. By considering the radiation from the surface to all parts of the enclosure, and the radiation arriving at the surface from all parts of the enclosure, it is assured that all radiative 153

154

Thermal Radiation Heat Transfer

FIGURE 4.1  Intensity leaving surface 1 and transferring energy to surface 2.

contributions are included. A convenient enclosure is usually evident from the physical configuration. The opening into a large empty environment can often be treated as a bounding area of zero reflectivity. The opening may also act as a radiation source that can be diffuse or directional when radiation is entering from the surrounding environment. In Chapters 1 through 3, we defined the radiative properties of solid opaque surfaces. For some materials, the properties vary substantially with wavelength, surface temperature, and direction. For radiation computations within enclosures, the geometric aspects of the exchange introduce another challenge in addition to the surface property variations. For simple geometries, it may be possible to account in detail for property variations without the analysis becoming complex. As the geometry becomes more involved, it may be necessary to invoke more idealizations of the surface properties so that solutions can be obtained with reasonable effort, if the resulting decrease in accuracy is acceptable. The rapid increase in computational speed may eliminate the need for such idealizations and assumptions, but the simplified treatment allows rapid approximate designs and calculations. The development here begins with simple situations; successive complexities are then added to build more comprehensive treatments in Chapters 5 through 7.

4.1.1 Enclosures with Diffuse Surfaces Consider an enclosure where each surface is black and isothermal. For black surfaces, there is no reflected radiation, and all emitted energy is diffuse (the intensity leaving a surface is independent of direction). The local energy balance at a surface involves the enclosure geometry, which governs how much radiation leaving a surface will reach another surface. For a black enclosure, the geometric effects are expressed in terms of the diffuse configuration factors developed in this chapter. A configuration factor is the fraction of uniform diffuse radiation leaving a surface that directly reaches another surface. The computation of configuration factors involves analytical or numerical integration over the solid angles by which surfaces can view each other. Some examples are given to demonstrate analytical integrations. Factors are usually evaluated numerically, and numerical routines have been incorporated

Configuration Factors for Diffuse Surfaces

155

into many of the computer programs developed for thermal analysis. Because these integrations can be tedious, it is often helpful to use relations that exist between configuration factors. This may make it possible to obtain a desired factor from factors that are already known. These relations, along with various shortcut methods that can be used to obtain configuration factors, are given in this chapter. The next step in treating more complex enclosures is to have gray rather than black surfaces that emit and reflect diffusely. The enclosure surfaces must be specified such that it can be assumed that both emitted and reflected energies are uniform over each surface. The configuration factors can then be used for radiation leaving a surface by both emission and reflection. For gray surfaces, reflections among surfaces must be accounted for in the analysis; we discuss this in Chapter 5.

4.1.2 Enclosures with Directional (Nondiffuse) and Spectral (Nongray) Surfaces Sometimes the approximations of black or diffuse-gray surfaces are not sufficiently accurate, and directional and/or spectral effects must be included. The necessity of treating spectral effects was noticed quite early in the field of radiative transfer. Sir William Herschel (1800) published “Investigation of the Powers of the Prismatic Colours to Heat and Illuminate Objects; with Remarks, that prove the Different Refrangibility of Radiant Heat, to which is added, an Inquiry into the Method of Viewing the Sun Advantageously, with Telescopes of Large Apertures and High Magnifying Powers,” which includes the following statement: In a variety of experiments I have occasionally made, relating to the method of viewing the sun, with large telescopes, to the best advantage, I used various combinations of differently coloured darkening glasses. What appeared remarkable was, that when I used some of them, I felt a sensation of heat, though I had but little light; while others gave me much light, with scarce any sensation of heat. Now, as in these different combinations, the sun’s image was also differently coloured, it occurred to me, that the prismatic rays might have the power of heating bodies very unequally distributed among them.

This paper was the first in which the infrared region of the spectrum was mentioned, and the energy radiated as “heat” shown to be of wavelengths different than those for “light.” The quotation shows an early awareness that in some instances spectral effects must be included in a radiative analysis. The performance of spectrally selective surfaces, such as those for satellite temperature control and solar collectors, can be understood only by considering wavelength variations of the surface properties. If the spectral surfaces are diffuse, the configuration factors developed here remain applicable for enclosure analysis. In some instances, surface properties have significant directional characteristics. In Chapter 3, directionally dependent surface properties were examined. They sometimes differ considerably from the diffuse approximation. Consequently, diffuse configuration factors cannot be used when surfaces emit or reflect in a significantly directional (nondiffuse) manner. A mirror is a special directional surface. However, emission from this type of surface is often approximated as being diffuse; hence, the emitted energy is treated by using diffuse configuration factors. Reflected energy is then followed within the enclosure by using the directional (mirror-like) characteristic that the angles of reflection and incidence are equal in magnitude.

4.2 GEOMETRIC CONFIGURATION FACTORS BETWEEN TWO SURFACES When calculating radiative transfer among surfaces, geometric relations are needed to determine how the surfaces view each other. In this section, a geometric configuration factor is developed to account for geometric effects. Such factors allow computation of radiative transfer in many systems by referring to formulas or tabulated values for the geometric relations between two surfaces. This simplifies a time-consuming portion of the analysis. Radiative exchange in systems with directional properties is analyzed in Chapter 6; first, we discuss the factors for uniformly distributed diffuse energy leaving a surface.

156

Thermal Radiation Heat Transfer

4.2.1 Configuration Factor for Energy Exchange between Diffuse Differential Areas The radiative transfer from a diffuse differential area element to another area element is used to derive relations for transfer between finite areas. Consider the two differential area elements in Figure 4.1. The dA1 and dA2 are at T1 and T2, are arbitrarily oriented, and have their normal at angles θ1 and θ2 to the line of length S between them. If I1 is the total intensity leaving dA1, the total energy per unit time leaving dA1 and incident on dA2 is d 2Qd1-d 2 = I1dA1 cos q1d W1 (4.1)



where dΩ1 is the solid angle subtended by dA2 when viewed from dA1, and the dash in the subscript of Q means “to.” This equation follows directly from the definition of I1 as the total energy leaving surface 1 per unit time, per unit area projected normal to S, and per unit solid angle. Because I1 is diffuse, it is independent of the angle at which it leaves dA1. It may consist of both diffusely emitted and diffusely reflected portions. The d2Q is a second-order differential as it depends on two differential quantities: dA1 and dΩ1. Equation (4.1) can also be written for radiant energy in a wavelength interval dλ:

d 2Ql,d1- d 2 d l = I l,1d l dA1 cos q1d W1. (4.2)

The total radiant energy is then found by integrating over all wavelengths: ¥



d Qd1-d 2 = dA1 cos q1d W1 2

òI

l ,1

d l. (4.3)

l =0

For a diffuse surface, Iλ does not depend on direction. Because the geometric factors are independent of λ, they can be removed from under the integral sign, and the integration over λ is independent of geometry. Thus, the results that follow for diffuse geometric configuration factors involving finite areas apply for both spectral and total quantities. For simplicity in notation, the remaining development is carried out for total quantities. The solid angle dΩ1 is related to the projected area of dA2 and the distance between the differential elements by

d W1 =

dA2 cos q2 . (4.4) S2

Substituting this into Equation (4.1) gives the following relation for the total energy per unit time leaving dA1 that is incident upon dA2:

d 2Qd1- d 2 =

I1dA1 cos q1dA2 cos q2 . (4.5) S2

An analogous derivation for the radiation leaving a diffuse dA2 that arrives at dA1 results in

d 2Qd 2-d1 =

I 2 dA2 cos q2 dA1 cos q1 . (4.6) S2

157

Configuration Factors for Diffuse Surfaces

Note that both Equations (4.5) and (4.6) contain the same geometric factors. The fraction of energy leaving diffuse surface element dA1 that arrives at element dA2 is defined as the geometric configuration factor dFd1–d2. The dash in the subscript means “to” (it is not a minus sign). Using Equation (4.5), dF can be expressed as dFd1-d 2 =



I1 cos q1 cos q2 dA1dA2 /S 2 cos q1 cos q2 = dA2 (4.7) pI1dA1 pS 2

where πI1dA1 is the total diffuse energy leaving dA1 within the entire hemispherical solid angle over dA1. Equation (4.7) shows that dFd1–d2 depends only on the size of dA2 and its orientation with respect to dA1. By substituting Equation (4.4), Equation (4.7) can also be written as dFd1- d 2 =



cos q1d W1 . (4.8) p

Consequently, elements dA2 have the same configuration factor if they subtend the same solid angle dΩ1 when viewed from dA1 and are positioned along a path at angle θ1 with respect to the normal of dA1. The factor dFd1–d2 has various names, being called the view, angle, shape, interchange, exchange, or configuration factor. The notation used here has subscripts designating the areas involved and a derivative consistent with the mathematical description. For the subscript notation, d1, d2, etc. indicate differential area elements, while 1, 2, etc. indicate finite areas. Thus, dFd1–d2 is a factor between two differential elements, and dF1–d2 is from finite area A1 to differential area dA2. The derivative dF indicates that the factor is for energy to a differential element; this keeps the mathematical form of equations such as Equation (4.7) consistent by having a differential quantity on both sides. The F denotes a factor to a finite area; thus, Fd1–2 is from differential element dA1 to finite area A2. 4.2.1.1 Reciprocity for Differential Element Configuration Factors The configuration factor for energy from dA2 to dA1 can be found using a derivation similar to that for Equation (4.7). dFd 2 - d1 =



cos q1 cos q2 dA1. (4.9) pS 2

Multiplying Equation (4.7) by dA1 and Equation (4.9) by dA2 gives the reciprocity relation between dFd1–d2 and dFd2–d1 dFd1- d 2 dA1 = dFd 2 - d1dA2 =



cos q1 cos q2 dA1dA2 . (4.10) pS 2

4.2.1.2 Sample Configuration Factors between Differential Elements The derivation of configuration factors in terms of system geometry parameters is illustrated by a few examples. Example 4.1 In Figure 4.2, two elemental areas are shown that are located on strips that have parallel generating lines. Derive an expression for the configuration factor between dA1 and dA 2. The angle β is in the y–z plane.

158

Thermal Radiation Heat Transfer

FIGURE 4.2  Geometry for configuration factor between elements on strips formed by parallel generating lines. The y–z plane is normal to the generating lines. The distance S = (l 2 + x 2 )1/ 2 and cos q1 = (l cos b) /S = (l cos b) /(l 2 + x 2 )1/ 2 . The angle β is in the y–z plane normal to the two strips. The solid angle subtended by dA 2, when viewed from dA1, is d W1 =

Projected area of dA2 S2

=

(Projected width of dA2 )(Projected length of dA2 ) S2

=

(ld b)(dx cos y) ld bdx l = . S2 S2 S

Substituting into Equation (4.8) gives

dFd1-d 2 =

l 3 cos bdbdx cos q1d W1 l cos b 1 l 2dbdx = 2 = 2 1/ 2 2 2 3/ 2 p (l + x ) p (l + x ) p(l 2 + x 2 )2

which is the desired configuration factor in terms of convenient parameters that specify the geometry.

Example 4.2 Find the configuration factor between an elemental area and an infinitely long strip of differential width as in Figure 4.3, where the generating lines of dA1 and dAstrip,2 are parallel. Example 4.1 gave the configuration factor between element dA1 and element dA 2 of length dx. To find the factor when dA 2 becomes an infinite strip, as in Figure 4.3, integrate over all x to obtain

Configuration Factors for Diffuse Surfaces

159

FIGURE 4.3  Geometry for configuration factor between elemental area and infinitely long strip of differential width; area and strip are on parallel generating lines.



dFd1- strip,2 =

l 3 cos bdb p

¥

ò

-¥

l 3 cos bdb p 1 dx = = d (sin b) 2 2 2 3 2 p 2l (l + x )

where β is in the y–z plane normal to the strips. Figure 4.3 also shows that since dA1 lies on an infinite strip dAstrip,1 with elements parallel to dAstrip,2, the dFd1–strip,2 is valid for dA1 at any location along dAstrip,1. Then, since any element dA1 on dAstrip,1 has the same fraction of its energy reaching dAstrip,2, it follows that the fraction of energy from the entire infinite dAstrip,1 that reaches dAstrip,2 is the same as the fraction for each element dA1. Thus, the configuration factor between two infinitely long strips of differential width and having parallel generating lines must also be the same as for element dA1 to dAstrip,2, or (1/2)d(sinβ).

Example 4.3 Consider an infinitely long wedge-shaped groove as shown in cross-section in Figure 4.4. Determine the configuration factor between the differential strips dx and dξ in terms of x, ξ, and α. From Example 4.2, the configuration factor is

dFdx - d x =

1 1 d (sin b) = cos bd b. 2 2

From the construction in Figure 4.4b, cos β = (ξ sin α)/L. The dβ is the angle subtended by the projection of dξ normal to L, that is,

db =

d x cos(a + b) d x x sin a = . L L L

160

Thermal Radiation Heat Transfer

FIGURE 4.4  Configuration factor between two strips on sides of wedge groove: (a) Wedge-shaped groove geometry, and (b) auxiliary construction. From the law of cosines, L2 = x2 + ξ2 − 2xξ cos α. Then

dFdx - d x =

xx sin2 a 1 1 xx sin2 a 1 dx = cos bd b = d x. 3 2 2 2 2 L 2 ( x + x - 2xx cos a)3 / 2

4.2.2 Configuration Factor between a Differential Area Element and a Finite Area Consider an isothermal diffuse element dA1 with uniform emissivity at temperature T1 exchanging energy with a finite area A2 that is isothermal at temperature T2 as in Figure 4.5. The relations for exchange between two differential elements must be extended to a finite A2. Figure 4.5 shows that θ2 is different for different positions on A2 and that θ1 and S will also vary as different differential elements on A2 are viewed from dA1. Two configuration factors need to be considered. The Fd1–2 is from the differential area dA1 to the finite area A2, and dF2–d1 is from A2 to dA1. Each of these is obtained by evaluating the fraction of

FIGURE 4.5  Radiant interchange between differential element and finite area.

161

Configuration Factors for Diffuse Surfaces

energy leaving one diffusely emitting and reflecting area that reaches the second area. The radiation leaving dA1 is πI1 dA1. The energy reaching dA2 on A2 is πI1(cos θ1 cos θ2 /πS2)dA1 dA2. Then, integrating over A2 to obtain all of the energy reaching A2 and dividing by the energy leaving dA1 result in



Fd1- 2

ò =

I1 cos q1 (cos q2 dA1 /S 2 )dA2

A2

pI1dA1

=

ò

A2

cos q1 cos q2 dA2 . (4.11) pS 2

From Equation (4.7), the quantity inside the integral of Equation (4.11) is dFd1–d2, so Fd1–2 can also be written as Fd1- 2 =



ò dF

d 1- d 2

. (4.12)

A2

This gives the fraction of the energy reaching A2 as the sum of the fractions reaching all the elements of A2. For obtaining the configuration factor from the finite area A2 to the element dA1, the energy leaving A2 is

ò

A2

ò

A2

pI 2 dA2 . The energy reaching dA1 from A2 is by integrating Equation (4.6) over A2,

I 2 (cos q1 cos q2 /S 2 ) dA2 . The configuration factor dF2–d1 is then

dF2 - d1 =



dA1

ò

A2

I 2 (cos q1 cos q2 /S 2 ) dA2

ò

A2

=

pI 2 dA2

dA1 A2

ò

A2

cos q1 cos q2 dA2 . (4.13) pS 2

The last integral on the right was obtained by imposing the condition that A2 has uniform emitted plus reflected intensity I2 over its entire area. From Equation (4.7), the quantity under the integral sign in Equation (4.13) is dFd1–d2, so the alternative form is obtained: dF2 - d1 =



dA1 A2

ò dF

d1- d 2

. (4.14)

A2

4.2.2.1 Reciprocity Relation for Configuration Factors between Differential and Finite Areas By use of Equation (4.12) to eliminate the integral in Equation (4.14), we get the reciprocity relation: A2 dF2 - d1 = dA1Fd1- 2 . (4.15)



4.2.2.2 Configuration Factors between Differential and Finite Areas Certain geometries have configuration factors with closed-form algebraic expressions, while others require numerical integration of Equation (4.11). Two examples illustrate how factors that have algebraic forms are obtained. Example 4.4 An element dA1 is perpendicular to a circular disk A 2 with outer radius r as in Figure 4.6a. Find an equation for the configuration factor Fd1–2 in terms of h, l, and r.

162

Thermal Radiation Heat Transfer

FIGURE 4.6  Geometry for radiative exchange between differential area and circular disk: (a) Geometry of problem, (b) auxiliary construction for determining cos θ1 and cos θ2, and (c) auxiliary construction for determining S. The first step is to find expressions for the quantities inside the integral of Equation (4.11) in terms of known quantities so that the integration can be carried out. The element dA 2 = ρdρ dϕ. Because the integral in Equation (4.11) must be carried out over all ρ and ϕ, the other quantities in the integral must be put in terms of these variables. Figure 4.6b is drawn to evaluate cos θ1 and cos θ2, which are cos θ1 = (l + ρ cos ϕ)/S and cos θ2 = h/S. Figure 4.6c shows that S2 = h2 + B2, where B2 can be evaluated by using the law of cosines on triangle a0b. This gives

B2 = l 2 + r2 - 2l r cos( p - f) = l 2 + r2 + 2l r cos f.

Substituting into Equation (4.11) gives Fd1- 2 =

ò

A2

h = p

cos q1 cos q2 dA2 = pS 2 r

2p

òò

r =0 f =0

ò

A2

h(l + r cos f) rd rd f pS 4

r(l + r cos f) d fd r. (h2 + l 2 + r2 + 2rl cos f)2

This integration is carried out using the symmetry of the configuration and is nondimensionalized to give, after considerable manipulation, Fd1-2 =

2h p

H = 2

r

p

ò ò (h

r =0 f =0

2

r(l + r cos f) d fd r + r2 + l 2 + 2rl cos f)2

ì ü H2 + R2 + 1 - 1ý í 2 2 2 2 1/ 2 [( H + R + 1 ) 4 R ] î þ



163

Configuration Factors for Diffuse Surfaces where H = h/l R = r/l.

To avoid the complicated double integration, the analysis can be carried out by contour integration, as in Section 4.3.3.

Example 4.5 An infinitely long 2D wedge has an opening angle α. Derive an expression for the configuration factor from one wall of the wedge to a strip element of width dx on the other wall at x as in Figure 4.7a. Such configurations approximate the geometries of long fins and ribs used in radiators for devices in outer space. Use the configuration factor between two infinitely long strip elements having parallel generating lines from Example 4.2. Note that β is measured clockwise from the normal of dx; Equation (4.5) then gives l



Fdx -1 =

ò

b¢

0

dFdx - d x =

x =0

ò

b =- p / 2

d (sin b) +

1

1

ò 2 d(sin b) = 2 (1+ sin b¢). 0

The function sinβ′ is found by the auxiliary construction in Figure 4.7b to be sinβ′ = B/C =  (l cos a - x ) /( x 2 + l 2 - 2xl cos a)1/ 2 . Then Fdx -1 =



1 l cos a - x . + 2 2( x 2 + l 2 - 2xl cos a)1/ 2

The problem requires dFl–dx. Using the reciprocal relation Equation (4.15) gives

dFl - dx =

dx cos a - X ù é1 Fdx - l = dX ê + . 2 1/ 2 ú l ë 2 2( X + 1- 2X cos a) û

where X = x/l. The only parameters are the wedge opening angle and the dimensionless position of dx from the vertex.

FIGURE 4.7  Configuration factor between one wall and strip on other wall of infinitely long wedge cavity: (a) Wedge-cavity geometry, and (b) auxiliary construction to determine sin β′.

164

Thermal Radiation Heat Transfer

4.2.3 Configuration Factor and Reciprocity for Two Finite Areas Consider the configuration factor for radiation with uniform intensity leaving the diffuse surface A1 and reaching A2, as shown in Figure 4.8. The energy leaving A1 is πI1A1. The radiation leaving an element dA1 that reaches dA2 was given previously as πIl(cosθ1cosθ2/πS2)dA1dA2. This is integrated over both A1 and A2 to give the energy leaving A1 that reaches A2. The configuration factor is then (since I1 is constant)



F1- 2

òò = A1

A2

pI1 (cos q1 cos q2 /pS 2 )dA2 dA1 pI1 A1

=

1 A1

òò

A1 A2

cos q1 cos q2 dA2 dA1. (4.16) pS 2

This expression can be written in terms of configuration factors with differential areas:

F1- 2 =

1 A1

ò ò dF

d1- d 2

A1 A2

dA1 =

1 A1

òF

d1- 2

dA1. (4.17)

A1

A similar derivation to Equation (4.16) gives the configuration factor from A2 to A1 as

F2 -1 =

1 A2

òò

A1 A2

cos q1 cos q2 dA2 dA1. (4.18) pS 2

The reciprocity relation for configuration factors between finite areas is found from the identical double integrals in the previous equations:

A1F1- 2 = A2 F2 -1. (4.19)

Further relations among configuration factors are found by using Equation (4.17) in conjunction with the reciprocity relations Equations (4.15) and (4.19):

FIGURE 4.8  Geometry for energy exchange between finite areas.

165

Configuration Factors for Diffuse Surfaces

F2 -1 =



A1 A 1 F1- 2 = 1 A2 A1 A2

òF

d1- 2

dA1 =

A1

1 A2

ò dF

A =

2 - d1 2

A1

ò dF

2 - d1

. (4.20)

A1

Example 4.6 Two plates of the same finite width and of infinite length are joined along one edge at angle α, as in Figure 4.7. Using the same nondimensional parameters as in Example 4.5, derive the configuration factor between the plates. Example 4.5 gives the configuration factor dFl–dx between one plate and an infinite strip on the other plate. Substituting into Equation (4.20) gives l



Fl -l* =

*

ò

x =0

l

dFl -dx =

é1

ò êë 2 + 2(X

2

0

cos a - X ù dX + 1 - 2X cos a)1/ 2 úû

where the width of the side in Figure 4.7 having element dx is designated as l*, X = x/l, and l* = l. Upon integration, it yields 1/ 2



æ 1 - cos a ö Fl -l * = 1 - ç ÷ 2 è ø

= 1 - sin

a . 2

For the present case with equal plate widths, the only parameter is α; hence, the configuration factor is the same for plates of any equal width.

Table 4.1 summarizes the integral definitions of the configuration factors and the configuration factor reciprocity relations.

4.3 METHODS FOR DETERMINING CONFIGURATION FACTORS 4.3.1 Configuration Factor Algebra Configuration factor algebra is the manipulation of various relations among configuration factors to derive new factors from those already known. The algebra is based on the reciprocal relations, the definition that the F factor is the fraction of energy that is intercepted, and energy conservation for a complete enclosure.

166

Thermal Radiation Heat Transfer

FIGURE 4.9  Energy exchange between two finite areas with one area subdivided: F1–3 + F1–4 = F1–2.

Consider an arbitrary isothermal area A1 in Figure 4.9 exchanging energy with a second area A2. The F1–2 is the fraction of diffuse energy leaving A1 that is incident on A2. If A2 is divided into A3 and A4, the fractions of the energy leaving A1 that are incident on A3 and A4 must add to F1–2:

F1-2 = F1-(3+4 ) = F1-3 + F1-4 (4.21)



F1-3 = F1-2 - F1-4 . (4.22)

The reciprocity relation, Equation (4.19), gives F3-1 =



A1 A F1-3 = 1 ( F1- 2 - F1- 4 ). (4.23) A3 A3

In Equations (4.21) through (4.23), the dashes in the subscript as before mean “to” and are not to be confused with negative signs. Thus, F1–2 means from area A1 to area A2. Combined areas are grouped with parentheses so that (3 + 4) means the combination of A3 and A4. Thus, the notation F1–(3 + 4) means from A1 to the combined areas of A3 and A4. The reciprocity relation of Equation (4.23) is a powerful computational tool for obtaining new configuration factors from those available. The following examples demonstrate how this is done. Example 4.7 An elemental area dA1 is perpendicular to a ring of outer radius ro and inner radius ri as in Figure 4.10. Derive an expression for the configuration factor Fd1–ring.

167

Configuration Factors for Diffuse Surfaces

FIGURE 4.10  Energy exchange between elemental area and finite ring. From Example 4.4, the configuration factor between dA1 and the entire disk of area A 2 with outer radius ro is

Fd1-2



H = 2

ü ì ïï ïï H 2 + Ro2 + 1 1 ý í 1/ 2 ï ï é H 2 + Ro2 + 1 2 - 4Ro2 ù úû ïþ ïî êë

(

)

where H = h/l, Ro = ro/l. The configuration factor to the inner disk of area A3 with radius ri has the same form, with Ri = ri/l substituted for Ro. Using configuration factor algebra, the desired configuration factor from dA1 to the ring A 2 – A3 is Fd1–2 − Fd1–3, so that



Fd1- ring

ì ü ïï H ïï H 2 + Ro2 + 1 H 2 + Ri2 + 1 = í . 1/ 2 ý 1/ 2 2 2 ïé 2 é H 2 + R 2 + 1 2 - 4R 2 ù ï H + Ro2 + 1 - 4Ro2 ù i i êë ïî ëê ûú ûú ïþ

(

)

(

)

Example 4.8 Suppose the configuration factor is known between two parallel disks of arbitrary size whose centers lie on the same axis. From this, derive the configuration factor between the two rings A 2 and A3 of Figure 4.11. Give the result in terms of known disk-to-disk factors from disk areas on the lower surface to disk areas on the upper surface. The factor desired is F2–3. From configuration factor algebra, F2–3 is equal to F2–3 = F2–(3 + 4) − F2–4. The factor F2–(3 + 4) can be found from the reciprocal relation A 2 F2–(3 + 4) = (A3 + A4)F(3 + 4)–2. Applying configuration factor algebra to the right-hand side results in

168

Thermal Radiation Heat Transfer

FIGURE 4.11  Energy exchange between parallel ring areas having a common axis. A2F2-(3+ 4) = ( A3 + A4 )[F(3+ 4) -(1+ 2) - F(3+ 4) -1]



= ( A3 + A4 )F(3+ 4) -(1+ 2) - ( A3 + A4 )F(3+ 4) -1.



Applying reciprocity to the right side gives A2F2-(3+ 4) = ( A1 + A2 )F(1+ 2)-(3+ 4) - A1F1-(3+ 4)



where the F factors on the right are both disk-to-disk factors from the lower surface to the upper. Next, the factor F2–4 is to be determined. Applying reciprocity relations and configuration factor algebra,

F2- 4 =

A A4 1 F4 - 2 = 4 [F4 -(1+ 2) - F4 -1] = [( A1 + A2 )F(1+ 2) - 4 - AF 1 1- 4 ]. A2 A2 A2

Substituting the relations for F2–4 and F2–(3 + 4) into the first equation gives

F2-3 =

A A1 + A2 [F(1+ 2)-(3+ 4) - F(1+ 2)-4 ] - 1 [F1-(3+ 4) - F1-4 ] A2 A2

and all configuration factors on the right-hand side are for exchange between two disks from the lower surface to the upper surface.

Because small differences in large numbers can occur in obtaining an F factor by using configuration factor algebra, as might occur on the right side of the last equation of the preceding example, care must be taken that enough significant figures are retained. Feingold (1966) gives one example

169

Configuration Factors for Diffuse Surfaces

in which an error of 0.05% in a known factor causes an error of 57% in another factor computed from it by using configuration factor algebra. Example 4.9 The internal surface of a hollow circular cylinder of radius R is radiating to a disk A1 of radius r, as in Figure 4.12. Express the configuration factor from the internal cylindrical side A3, to the disk in terms of disk-to-disk factors for the case of r ≤ R. From any position on A1, the solid angle subtended when viewing A3 is the difference between the solid angle when viewing A2 or dΩ2, and that when viewing A4 or dΩ4. This gives the Fd1–3 factor from an area element dA1 on A1 to area A3 as Fd1–3 = Fd1–2 – Fd1–4. By integrating over A1 and using Equation (4.17), this can be written for the entire A1 as Fd1–3 = Fd1–2 – Fd1–4. The factors on the right are between parallel disks. The final result for F from the internal cylindrical side A3 to the disk A1 is

F3-1 =

A1 (F1- 2 - F1- 4 ) A3

(r £ R).

From symmetry, the configuration factor from A1 to any sector As of A1 is (As/A1)F3–1.

In formulating relations between configuration factors, it is sometimes useful to use energy quantities rather than fractions of energy leaving a surface that reach another surface. For example, in Figure 4.9, the energy leaving A2 that arrives at A1 is proportional to A2F2–1 and is equivalent to the sums of the energies from A3 and A4 that arrive at A1. Thus,

( A3 + A4 )F(3+ 4 ) -1 = A3 F3-1 + A4 F4 -1. (4.24)

FIGURE 4.12  Geometry for a cylindrical cavity radiating to circular disk A1 for r ≤ R.

3+ 4 )

170

Thermal Radiation Heat Transfer

This can also be proved by using reciprocity relations: ( A3 + A4 )F(3+4 )-1 = A1 F1-(3+4 ) = A1 F1-3 + A1 F1-4 = A3 F3= A1 F1-3 + A1 F1-4 = A3 F3-1 + A4 F4-1. 4.3.1.1 Configuration Factors Determined by Symmetry A reciprocity relation can be derived from the symmetry of a geometry. Consider the opposing areas in Figure 4.13a. From symmetry, A2 = A4 and F2–3 = F4–l, so A2F2–3 = A4F4–1. From reciprocity, A4F4–1 = A1F1–4. Hence, the expression

A2 F2-3 = A1 F1-4 (4.25)

relates the diagonal directions shown by the arrows. Similarly, the symmetry of Figure 4.13b yields

A2 F2-7 = A3 F3-6 . (4.26)

Figure 4.14a shows four areas on two perpendicular rectangles having a common edge. Since these areas are of unequal size, there is no apparent symmetry relation. However, a valid relation is

A1 F1-2 = A3 F3-4 . (4.27)

To prove this, begin with the basic definition, Equation (4.16), A1 F1-2 =

1 p

òò

A1 A2

cos q1 cos q2 dA2 dA1. S2

FIGURE 4.13  Geometry to determine reciprocity relations between opposing rectangles: (a) Two pairs of opposing rectangles, A1F1–4 = A2F2–3, and (b) four pairs of opposing rectangles, A2F2–7 = A3F3–6.

171

Configuration Factors for Diffuse Surfaces

FIGURE 4.14  Geometry for application of reciprocity relations among diagonally opposite pairs of rectangles on two perpendicular planes having a common edge: (a) Representation of reciprocity, A1F1 2 = A3F3–4, (b) construction for F1–2, and (c) construction for F3–4.

From Figure 4.14b, S 2 = ( x2 - x1 )2 + y12 + z22 , cos q1 = z2 /S, and cos q2 = y1 /S so that

A1F1- 2 =

1 p

c

a

c+d

b

ò ò ò ò

y1z2

2 2 2 é ù x1 = 0 y1 = 0 x2 = c z2 = 0 ë( x2 - x1 ) + y1 + z2 û

2

dz2 dx2 dy1dx1. (4.28)

Similarly, Figure 4.14c reveals that A3 F3- 4 =

1 p

1 = p

òò

cos q3 cos q4 dA4 dA3 S2

c+d

a

A3 A4

c

b

ò ò ò ò

(4.29) y3 z4

2 2 2 ù é x3 = c y3 = 0 x4 = 0 z4 = 0 ë( x3 - x4 ) + y3 + z4 û

2

dz4 dx4 dy3dx3 .

172

Thermal Radiation Heat Transfer

By interchanging the dummy integration variables x1, y1, x2, and z2 for x4, y3, x3, and z4, the integrals in Equations (4.28) and (4.29) are identical, thus proving Equation (4.27). Lebedev (1988) extends these diagonal relations to nonplanar surfaces: Two infinitely long parallel cylindrical areas of finite width and two coaxial surfaces of rotation. Example 4.10 If the configuration factor is known for two perpendicular rectangles with a common edge, as in Figure 4.15a, derive F1–6 for Figure 4.15b. If F1–2 and F1–4 are known and F3–1 is desired, then first consider the geometry in Figure 4.15c and derive F7–6 as follows: F(5+ 6) -(7 + 8) = F(5+ 6) -7 + F(5+ 6) - 8 =

A8 A7 F8 -(5+ 6) F7 -(5+ 6) + A5 + A6 A5 + A6

A8 A7 = (F7 - 5 + F7 -6 ) + (F8 - 5 + F8 -6 ). A5 + A6 A5 + A6



Substitute A7F7–6 for A8F8–5 and solve the resulting relation for F7–6 to obtain

F7-6 =

1 [( A5 + A6 )F(5+6)-(7+8) - A7F7-5 - A8F8-6 ] . 2A7

FIGURE 4.15  Orientation of areas for Example 4.10: (a) Perpendicular rectangles with one common edge, (b) geometry for F1–6, and (c) auxiliary geometry.

173

Configuration Factors for Diffuse Surfaces

FIGURE 4.16  Geometry for an isothermal enclosure composed of black surfaces. Now, in Figure 4.15b F1-6 =



A A A6 F6-1 = 6 F6-(1+3) - 6 F6-3 . A1 A1 A1

The factors F6–(1 + 3) and F6–3 are of the same type as F7–6, so F1–6 can finally be written as F1-6 =

A6 A1

ì 1 [( A1 + A2 + A3 + A4 )F(1+ 2+ 3+ 4)-(5+6) - A6F6 -( 2+ 4) - A5F5-(1+ 3) ] í î 2A6 1 [( A3 + A4 )F(3+ 4)-(5+6) - A6F6 - 4 - A5F5-3 ]üý . 2A6 þ



All the F factors on the right side are for two rectangles having one common edge, as in Figure 4.15a.

4.3.2 Configuration Factor Relations in Enclosures So far, only configuration factors between two surfaces have been considered, although subdivision of one or both surfaces into smaller areas has been examined. Consider the very important situation where the configuration factors are among surfaces that form a complete enclosure. For an enclosure of N surfaces, such as in Figure 4.16 where N = 8 is an example, the entire radiative energy leaving any surface k inside the enclosure must all be incident on surfaces of the enclosure. Thus, all the fractions of energy leaving one surface and reaching the surfaces of the enclosure must total to one: N

Fk -1 + Fk - 2 + Fk -3 +  + Fk - k +  + Fk - N =



åF

k- j

= 1. (4.30)

j =1

The factor Fk–k is included because when Ak is concave, it will intercept a portion of its own outgoing radiative energy. Example 4.11 Two diffuse isothermal concentric spheres are exchanging energy. Find all the configuration factors for this geometry if the surface area of the inner sphere is A1 and the surface area of the outer sphere is A 2.

174

Thermal Radiation Heat Transfer

All energy leaving A1 is incident upon A 2, so F1–2 = 1. Using the reciprocal relation gives F2–1 =  A1F1–2/A 2 = A1/A 2. From Equation (4.30), F2–1 + F2–2 = 1, giving F2–2 = 1 – F2–1 = (A 2 − A1)/A 2.

Example 4.12 An isothermal cavity of internal area A1 has a plane opening of area A 2. Derive an expression for the configuration factor of the internal surface of the cavity to itself. Assume that a black plane surface A 2 replaces the cavity opening; this has no effect on the F1–1 factor, which depends only on geometry for diffuse surfaces. Then F2–1 = 1 and F1–2 =  A 2F2–1/A1 = A 2/A1, which is the configuration factor from the entire internal area to the opening. Since A1 and A 2 form an enclosure, F1–1 = 1 – F1–2 = (A1 – A 2)/A1, which is the desired F factor.

Example 4.13 A long enclosure of triangular cross-section consists of three plane areas, each of finite width and infinite length, thus forming a hollow infinitely long triangular prism. Derive an expression for the configuration factor between any two of the areas in terms of their widths L1, L2, and L3. For area 1, F1–2 + F1–3 = 1, since F1–1 = 0. Using similar relations for each area and multiplying through by the respective areas results in

AF 1 1- 2 + AF 1 1- 3 = A1;

A2F2-1 + A2F2- 3 = A2 ;

A3F3-1 + A3- 2 = A3.

By applying reciprocal relations to some terms, these equations become

A1F1-2 + A1F1-3 = A1;

A1F1-2 + A2F2-3 = A2 ;

A1F1-3 + A2F2-3 = A3

thus giving three equations for the three unknown F factors. Solving gives

F1- 2 =

A1 + A2 - A3 L1 + L2 - L3 = . 2A1 2L1

When L1 = L2, this reduces to the factor between infinitely long adjoint plates of equal width separated by an angle α as in Example 4.6:

F1- 2 =

L /2 2L1 - L3 a = 1- 3 = 1- sin . L1 2L1 2

Examine the set of three simultaneous equations yielding the final result in Example 4.13 more closely. The first equation involves two unknowns, F1–2 and F1–3; the second has one additional unknown F2–3; and the third has no additional unknowns. Generalizing the procedure from a threesurface enclosure to any N-sided enclosure of plane or convex surfaces shows that, of N simultaneous equations, the first involves N–1 unknowns, the second N–2 unknowns, and so forth. The total number of unknowns U is then

U = ( N - 1) + ( N - 2) + 1 =

N ( N - 1) . (4.31) 2

Thus, [N(N – 1)/2] – N = N(N – 3)/2 factors must be provided. For a four-sided enclosure made up of planar or convex areas, we can write four equations relating 4(4 − 1)/2 = 6 unknown configuration factors. Specifying any two factors allows calculation of the rest by solving the four simultaneous equations. If a surface k can view itself, the factor F k–k must be included in each of the equations. Analyzing this situation shows that an N-sided enclosure provides N equations in N(N + 1)/2 unknowns. Thus, [N(N + 1)/2] − N = N(N − 1)/2 factors must be specified. For a four-sided enclosure, four equations

175

Configuration Factors for Diffuse Surfaces

involving ten unknown F factors can be written. The specification of six factors is required, then the simultaneous relations can be solved for the remaining four factors. If only M surfaces can view themselves, then [N(N − 3)/2] + M factors must be specified. For some geometries, the use of symmetry can reduce the number of factors that are required. Use of these relations is discussed further in Section 4.4.

4.3.3 Techniques for Evaluating Configuration Factors Evaluating configuration factors Fd1–2 and F1–2 requires integration over the finite areas involved. Various mathematical methods are useful for evaluating configuration factors when analytical integration is too cumbersome or is not possible. A few methods that are especially valuable are discussed next. 4.3.3.1 Hottel’s Crossed-String Method Consider configurations such as long grooves in which all surfaces are assumed to extend infinitely along one coordinate. Such surfaces are generated by moving a line such that it is always parallel to its original position. A typical configuration is shown in cross-section in Figure 4.17. Suppose that the configuration factor is needed between A1 and A2 when some blockage of radiant transfer occurs because of other surfaces A3 and A4. To obtain F1–2, first consider that A1 may be concave. In this case, draw the dashed line agf across A1. Then draw in the dashed lines cf and abc to complete the enclosure abcfga, which has three sides that are either convex or planar. The relation found in Example 4.13 for enclosures of this type can be written as

Aagf Fagf - abc =

Aagf + Aabc - Acf . (4.32) 2

FIGURE 4.17  Geometry for determining configuration factors using Hottel’s crossed-string method.

176

Thermal Radiation Heat Transfer

For the three-sided enclosure adefga, similar reasoning gives Aagf Fagf - def =



Aagf + Adef - Aad . (4.33) 2

Further, note that Fagf -abc + Fagf -cd + Fagf -def = 1. (4.34)



Observing that Fagf -cd = Fagf -2 and substituting Equations (4.32) and (4.33) into Equation (4.34) results in Aagf Fagf -2 = Aagf (1 - Fagf -abc - Fagf -def ) =



Acf + Aad - Aabc - Adef . (4.35) 2

Now F2–agf = F2–1 since Aagf and A1 subtend the same solid angle when viewed from A2. Then, with the additional use of reciprocity, the left side of Equation (4.35) is Aagf Fagf - 2 = A2 F2 - agf = A2 F2 -1 = A1F1- 2 . (4.36)



Substituting Equation (4.36) into Equation (4.35) results in A1F1- 2 =



Acf + Aad - Aabc - Adef . (4.37) 2

If the dashed lines in Figure 4.17 are imagined as being lengths of string stretched tightly between the outer edges of the surfaces, then the term on the right of Equation (4.37) is interpreted as onehalf the total quantity formed by the sum of the lengths of the crossed strings connecting the outer edges of A1 and A2 minus the sum of the lengths of the uncrossed strings. This is a very useful way for determining configuration factors, although it is limited to two-dimensional (2D) geometries. It was first described by Hottel (1954). Example 4.14 Two infinitely long semicylindrical surfaces of radius R are separated by a minimum distance D as in Figure 4.18a. Derive the configuration factor F1–2. The length of crossed string abcde is denoted as L1 and that of uncrossed string ef as L2. From the symmetry of the geometry, Equation (4.37) gives F1-2 =



2L1 - 2L2 L1 - L2 . = 2pR pR

Then, L2 = D + 2R. The L1 is twice the length cde. The segment of L1, from c to d, is found from right triangle 0cd to be

2 éæ D ù ö L1,c - d = êç + R ÷ - R 2 ú ø êëè 2 úû

1/ 2

é æD öù = êD ç + R ÷ ú 4 è øû ë

1/ 2

.

The segment of L1 from d to e is L1, d–e = Rφ where, from triangle 0cd, φ = sin−1[R/(D/2 + R)]. Then F1- 2 =

L1 - L2 2(L1,c - d + L1,d - e ) - L2 = pR pR

[4D(D / 4 + R)]1/ 2 + 2R sin-1[R /(D / 2 + R)] - D - 2R = . pR



177

Configuration Factors for Diffuse Surfaces

FIGURE 4.18  Examples of applying the crossed-string method: (a) Configuration factor between infinitely long semicylindrical surfaces, and (b) partially blocked view between parallel strips. Letting X = 1 + D/2R gives F1- 2 =



2é 2 1 ù ( X - 1)1/ 2 + sin-1 - X ú . (4.14.1) ê pë X û

Example 4.15 The view between two infinitely long parallel strips of width a is partially blocked by strips of width b as in Figure 4.18b. Obtain the factor F1–2. The length of each crossed string is (a2 + c2)1/2, and the length of each uncrossed string is 2[b2 + (c/2)2]1/2. From the crossed-string method the configuration factor is then 2

F1- 2 =



2

2

a2 + c 2 - 2 b2 + (c / 2)2 æcö æ 2b ö æ c ö = 1+ ç ÷ - ç ÷ +ç ÷ . a a è ø è a ø èaø

As expected, F1–2 → 0 as b is extended inward so that b → a/2.

Example 4.16 The view between two infinitely long parallel strips of width c and spaced c apart is partially blocked by a thin plate of width c/2 as in Figure 4.19. Obtain the configuration factor F1–2. Following the idea of a partially blocked view as in Figure 4.19, area A1 can view A 2 either to the right or to the left of the center plate. Figure 4.19a shows the strings drawn to the right side by considering the region to the left of the center plate to be obstructed. Figure 4.19b shows the remainder of the view by drawing the strings as if the region to the right of the center plate were obstructed. The final configuration factor is the sum of the results from the two parts. For each

178

Thermal Radiation Heat Transfer

FIGURE 4.19  Partially blocked view between two parallel strips: (a) View through the right side of vertical plate, and (b) view through the left side. part, the crossed strings are of equal length, so for clarity, only one is shown. For part (a), the lengths of the crossed and uncrossed strings are, respectively, 2 2c and c + c / 2 + 2 2(c / 4). Similarly, for part (b), the lengths are 4c [(1/4)2 + (3/4)2]1/2 =  10c and c + c / 2 + 10c / 2. Then, from Equation (4.37), cF1- 2 =



c öù 1 é 1é c öù æ 3c æ 3c 2 2c - ç + 2 ÷ ú + ê 10c - ç + 10 ÷ ú . 2 2 øû 2 2 2 2 êë è è øû ë

This simplifies to F1- 2 =



1 (3 2 + 10 - 6) = 0.35123. 4

When the center plate is absent, the crossed-string method yields F1-2 = 2 - 1 = 0.41421, so the center plate has provided about 15% blockage.

4.3.3.2 Contour Integration Another useful tool for evaluating configuration factors is to apply Stokes’ theorem to reduce the multiple integration over a surface area to a single integration around the boundary of the area. This method was developed by Moon (1961), de Bastos (1961), and Sparrow (1963a), and later outlined by Sparrow and Cess (1978). Consider a surface area A as in Figure 4.20 with its boundary designated as C (where C is piecewise continuous). An arbitrary point on the area is at position x, y, z. At this point the normal to A is constructed, and the angles between this normal and the x, y, and z axes are α, γ, and δ. Let P, Q, and R be any twice-differentiable functions of x, y, and z. Stokes’ theorem in 3D provides the following relation between an integral of P, Q, and R around the boundary C of the area and an integral over the surface A of the area:

ò (Pdx + Qdy + Rdz)

C

éæ ¶R ¶Q ö ù æ ¶Q ¶P ö æ ¶P ¶R ö = êç cos g + ç ÷ cos d ú dA. ÷ cos a + ç ÷ è ¶z ¶x ø è ¶x ¶y ø ëè ¶y ¶z ø û

ò A

(4.38)

Configuration Factors for Diffuse Surfaces

179

FIGURE 4.20  Geometry for quantities used in Stokes’ theorem.

This relation is now applied to express area integrals in finding configuration factors in terms of integrals around the boundaries of the areas. 4.3.3.2.1 Configuration Factor between a Differential and a Finite Area The integrand in the configuration factor Fd1–2 is (cos θ1 cos θ2/πS2)dA2. The two cosines can be written (Figure 4.21):

cos q1 =

x2 - x1 y -y z -z cos a1 + 2 1 cos g1 + 2 1 cos d1. (4.39) S S S



cos q2 =

x1 - x2 y -y z -z cos a 2 + 1 2 cos g 2 + 1 2 cos d2 . (4.40) S S S

This follows from the relation that for two vectors V1 and V2 having direction cosines (l1, m1, n1) and (l2, m2, n2), the cosine of the angle between the vectors is l1l2 + m1m2 + n1n2. Substituting Equations (4.39) and (4.40) into the integral relation for Fd1–2 gives Fd1- 2 =

ò

cos q1 cos q2 dA2 pS 2

1 p

ò

A2



=

A2

( x2 - x1 ) cos a1 + ( y2 - y1 ) cos g1 + ( z2 - z1 ) cos d1 S4

(4.41)

´ éë( x1 - x2 ) cos a 2 + ( y1 - y2 ) cos g 2 + ( z1 - z2 ) cos d2 ùû dA2 . Now, let

l = cos a; m = cos g; n = cos d (4.42)

180

Thermal Radiation Heat Transfer

FIGURE 4.21  Geometry for contour integration.

and f =



( x2 - x1 )l1 + ( y2 - y1 )m1 + ( z2 - z1 )n1 . (4.43) pS 4

Equation (4.41) becomes

Fd1- 2 =

ò éë( x - x ) fl + ( y - y ) fm + (z - z ) fn ùû dA . (4.44) 1

2

2

1

2

2

1

2

2

2

A2

Comparison of Equation (4.44) with the right side of Equation (4.38) shows that Stokes’ theorem can be applied if

¶R ¶Q = ( x1 - x2 ) f (4.45) ¶y2 ¶z2



¶P ¶R = ( y1 - y2 ) f (4.46) ¶z2 ¶x2



¶Q ¶P = ( z1 - z2 ) f . (4.47) ¶x2 ¶y2

Useful solutions to these three equations are (Sparrow 1963a) of the form

P=

-m1 ( z2 - z1 ) + n1 ( y2 - y1 ) (4.48) 2pS 2

181

Configuration Factors for Diffuse Surfaces

Q=



R=



l1 ( z2 - z1 ) - n1 ( x2 - x1 ) (4.49) 2pS 2

-l1 ( y2 - y1 ) + m1 ( x2 - x1 ) . (4.50) 2pS 2

Equation (4.38) is used to express Fd1–2 in Equation (4.44) as a contour integral; that is, Fd1- 2 =



ò (Pdx + Qdy + Rdz ). (4.51) 2

1

2

C2

Substitute P, Q, and R from Equations (4.48), (4.49), and (4.50), and the result is rearranged to obtain Fd1- 2 =



l1 2p

ò

( z2 - z1 )dy2 - ( y2 - y1 )dz2 S2

+

m1 2p

ò

( x2 - x1 )dz2 - ( z2 - z1 )dx2 (4.52) S2

n1 2p

ò

( y2 - y1 )dx2 - ( x2 - x1 )dy2 . S2

+

C2

C2

C2

The double integration over area A2 for Fd1–2 has been replaced by a set of three-line integrals. Sparrow (1963a) discusses the superposition properties of Equation (4.44) that allow additions of the configuration factors of elements aligned parallel to the x, y, and z axes to obtain the factors for arbitrary orientation. A further derivation of the superposition procedure is in Hollands (1995), based on radiation flux vector rather than using contour integration. The numerical integration of the configuration factor for parallel directly opposed squares is discussed by Shapiro (1985). The contour integral method was found to be advantageous in terms of both accuracy and computing time. Example 4.17 Determine the configuration factor Fd1–2 from an element dA1 to a right triangle as in Figure 4.22. The normal to dA1 is perpendicular to both the x and y axes and is thus parallel to z. The direction cosines for dA1 are then cos α1 = l1 = 0, cos γ1 = m1 = 0, and cos δ1 = n1 = 1, and Equation (4.52) becomes

Fd1- 2 =

1 2p

ò C2

(y 2 - y1)dx2 - ( x2 - x1)dy 2 . S2

Since dA1 is situated at the origin of the coordinate system, x1 = y1 = 0 and Fd1–2 further reduces to

Fd1- 2 =

1 2p

ò C2

y 2dx2 - x2dy 2 . S2

The distance S between dA1 and any point (x2, y2, z2) on A 2 is

S 2 = x22 + y 22 + z22 = x22 + y 22 + d 2.

182

Thermal Radiation Heat Transfer

FIGURE 4.22  Geometry to determine the configuration factor between a plane-area element and a right triangle in a parallel plane. The contour integration for Fd1–2 must now be carried out around the three sides of the right triangle. To keep the sign of Fd1–2 positive, the integration is performed by traveling around the boundary lines I, II, and III in a particular direction. The correct direction is that of a person walking around the boundary with their head in the direction of the normal n2 and always keeping A 2 to their left. Along boundary line I, x2 = 0, dx2 = 0, and 0 ≤ y2 ≤ a. On boundary II, y2 = a, dy2 = 0, and 0 ≤ x2 ≤ b. On boundary III, the integration is from ξ = 0 to c, where ξ is a coordinate along the hypotenuse of the triangle, so that x2 = (c − ξ) sin φ, y2 = (c − ξ) cos φ, and dx2 = −sin φ dξ, dy2 = −cos φ dξ. Substituting into the integral for Fd1–2 gives 2pFd1- 2 =

ò

I,II,III

y 2dx2 - x2dy 2 x22 + y 22 + d 2 b



ò

=0+

x2 = 0 b

=

òx 0

2 2

adx2 + x + a2 + d 2 2 2

c

ò

x =0

-(cc - x) cos j sin jd x + (c - x) sin j cos j d x (c - x)2 sin2 j + (c - x)2 cos2 j + d 2

adx2 X X tan j tan-1 = . (1+ X 2 )1/ 2 + a2 + d 2 2p(1+ X 2 )1/ 2

Where: X = a/d, tan φ = b/a.

4.3.3.2.2 Configuration Factor between Finite Areas For configuration factors between two finite areas, substituting Equation (4.52) into Equation (4.17) gives

183

Configuration Factors for Diffuse Surfaces

A1 F1-2 = A2 F2-1 =

òF

d 1-2

dA1

A1

=

1 2p

ù é ê ( y2 - y1 )n1 - ( z2 - z1 )m1 dA1 ú dx2 ê ú S2 û ë A1

ò ò C2

1 + 2p 1 2p

+

é ù ê ( z2 - z1 )l1 - ( x2 - x1 )n1 dA1 ú dy2 ê ú S2 ë A1 û

(4.53)

ò ò C2

ù é ê ( x2 - x1 )m1 - ( y2 - y1 )l1 dA1 ú dz2 2 ê ú S ë A1 û

ò ò C2

where the integrals have been rearranged and dx2, dy2, and dz2 have been factored out since they are independent of the area integration over A1. Stokes’ theorem is applied in turn to each of the three area integrals. Compare the first of these integrals

ò



A1

( y2 - y1 )n1 - ( z2 - z1 )m1 dA1 S2

with the area integral in Stokes’ theorem, Equation (4.38). This gives the identities

¶R ¶Q = 0; ¶y1 ¶z1

¶P ¶R -( z2 - z1 ) ; = S2 ¶z1 ¶x1

¶Q ¶P y2 - y1 . = ¶x1 ¶y1 S2

A solution to this set of partial differential equations (Sparrow and Cess 1978) is P = 1n S, Q = 0, and R = 0. By use of Equation (4.38) to convert it into a surface integral, the area integral becomes

ò



A1

( y2 - y1 )n1 - ( z2 - z1 )m1 dA1 = S2

ò ln Sdx . 1

C1

By applying Stokes’ theorem in a similar fashion to the other two integrals in Equation (4.53), that equation becomes A1 F1-2 =

1 2p

æ

ò çè ò

C1

C2

1 + 2p

æ ç è

1 ö æ ln Sdx1 ÷ dx2 + ç p 2 ø è C

ò ò

C1

2

ö ln S dy1 ÷ dy2 ø

ö ln Sdz1 ÷ dz2 C1 ø

ò ò C2

or more compactly as

F1- 2 =

1 2pA1

ò ò ( ln Sdx dx + ln Sdy dy + ln Sdz dz ). (4.54) 2

1

2

1

2

1

C1 C2

Thus, the integrations over two areas, which would involve integrating over four variables, are replaced by integrations over the two surface boundaries. This allows considerable computational savings when numerical evaluations must be carried out. It also may make analytical integration possible.

184

Thermal Radiation Heat Transfer

Example 4.18 Using the contour integration method, formulate the configuration factor for the parallel rectangles in Figure 4.23. Note that on both surfaces dz is zero. First, integrate Equation (4.54) around the boundary C2. The value of S to be used in Equation (4.54) is measured from an arbitrary point (x1, y1, 0) on A1 to a point on the portion of the boundary C2 being considered. This gives

F1-2

1 = 2pab

ì b 1/ 2 ï 2 2 2 í ln éë x1 + (y 2 - y1) + c ùû dy 2 C1 ï î y 2 =0

ò ò

0

+

ò

ln éë( a - x1)2 + (y 2 - y1)2 + c 2 ùû

ü ï dy 2 ý dy1 ïþ

1/ 2

y 2 =b

ì a 1/ 2 1 ï 2 2 2 + í ln éë( x2 - x1) + (b - y1) + c ùû dx2 2pab ï C1 î x2 =0



ò ò

0

+

ò

ln éë( x2 - x1)2 + y12 + c 2 ùû

1/ 2

x2 = a

ü ï dx2 ý dx1. ïþ

Carrying the integration out over C1 gives, in this case, eight integrals. The first four, corresponding to the first two integrals of the previous equation, are b

2pabF1-2 =

b

ò ò ln éëa

2

+ (y 2 - y1)2 + c 2 ùû

1/ 2

dy 2dy1

y1=0 y 2 =0 0

+

b

ò ò ln éë(y

2

- y1)2 + c 2 ùû

2

- y1)2 + c 2 ùû

1/ 2

dy 2dy1

y1=b y 2 =0 b

+

0

ò ò ln éë(y

1/ 2

dy 2dy1

y1=0 y 2 =0 0

+



0

ò ò ln éëa

2

+ (y 2 - y1)2 + c 2 ùû

1/ 2

dy 2dy1

y1=b y 2 =b

+ ( 4 integral terms in x ) b

=

b

é a2 + (y 2 - y1)2 + c 2 ù ln ê ú dy 2dy1 (y 2 - y1) + c 2 û ë y1=0 y 2 =0

ò ò a

+

a

é ( x - x1)2 + b 2 + c 2 ù ln ê 2 ú dx2dx1 2 2 ë ( x2 - x1) + c û x1=0 x2 =0

ò ò

and the configuration factor is now given by the sum of two double integrals. These can be integrated analytically, and the result is factor 4 in Appendix C.

185

Configuration Factors for Diffuse Surfaces

FIGURE 4.23  Contour integration to determine configuration factor between two parallel rectangles.

When the contour integration method is applied to geometries in which two surfaces share a common edge, the contour integrals become indeterminate. This difficulty is overcome for two quadrilaterals by a partial analytical integration, as developed in Mitalas and Stephenson (1966) and Shapiro (1983). 4.3.3.3 Differentiation of Known Factors An extension of configuration factor algebra is to obtain configuration factors to differential areas by differentiating known factors to finite areas. This technique is useful in certain cases, as is demonstrated by the following example. Example 4.19 As part of the determination of radiative exchange in a square channel whose temperature varies longitudinally, the configuration factor dFd1–d2 is needed between an element dA1 at one corner of the channel end and a differential length of wall section dA 2 as in Figure 4.24a. Configuration factor algebra plus differentiation can be used to find the required factor. Referring to Figure 4.24b, the fraction of energy leaving dA1 that reaches dA2 is the difference between the fractions reaching the squares A3 and A4, so the factor dFd1–d2 is the difference between Fd1–3 and Fd1–4. Then

dFd1-d 2 = Fd1-3 - Fd1-4 = -

¶F DFd1- Dx = - d1- dx. ¶x Dx Dx ®0

Thus, if the configuration factor Fd1–◽ between a corner element and a square in a parallel plane were known, the derivative of this factor with respect to the separation distance could be used to obtain the required factor. From Example 4.17, the configuration factor between a corner element and a parallel isosceles right triangle is given by setting tan φ = 1 in the expression for a general right triangle. This yields for the present case, where the distance d = x

Fd1- D =

a a tan-1 2 . 2 1/ 2 2p( a + x ) ( a + x 2 )1/ 2 2

186

Thermal Radiation Heat Transfer

FIGURE 4.24  Geometry for the derivation of the configuration factor between differential length of square channel and element at corner of channel end: (a) Configuration factor between dA1 and differential length of channel wall dA2, and (b) configuration factor between dA1 and squares A3 and A4. Inspection shows that, by symmetry, the factor between a corner element and a square is twice the factor Fd1−Δ. The dFd1–d2 is then dFd1- d 2 =

adx ¶ é a ¶Fd1- 1 ù dx = tan-1 2 ¶x p ¶x êë ( a2 + x 2 )1/ 2 (a + x 2 )1/ 2 úû é -1 a a( a2 + x 2 )1/ 2 ù + ê tan ú 2 2 1/ 2 (a + x ) x 2 + 2a2 û ë

=

axdx p( a + x 2 )3 / 2

=

é -1 (1+ X 2 )1/ 2 ù XdX 1 tan + ê ú; p(1+ X 2 )3 / 2 ë (1+ X 2 )1/ 2 2+ X2 û

2

X=

x . a

More generally, start with F1–2 for two parallel areas A1 and A 2 that are cross-sections of a cylindrical channel of arbitrary cross-section (Figure 4.25a). This factor depends on the spacing |x2−x1| between the two areas and includes blockage due to the channel wall; that is, it is the factor by which A 2 is viewed from A1 with the channel wall present. Note that for simple geometries such as a circular tube or rectangular channel, the wall blockage is zero. The factor between A1 and dA 2 in Figure 4.25b is then given by

dF1- d 2 = -

¶F1- 2 dx2. (4.19.1) ¶x2

Configuration Factors for Diffuse Surfaces

187

FIGURE 4.25  Geometry for derivation of the configuration factors for differential areas starting from the factors for finite areas: (a) Two finite areas, F1–2; (b) finite to differential area, dF1–d2 = −(∂F1–2/∂x2)dx2; and (c) two differential areas, dFd1–d2 = −(A1/dA1)(∂2F1–2/∂x1∂x2)dx2dx1. as in Example 4.19. Equation (4.19.1) is now used to obtain dFd1–d2, the configuration factor between the two differential area elements dA1 and dA 2 in Figure 4.25c. By reciprocity, Fd2–1 = (−A1/dA 2)(∂F1–2/∂x2)dx2. Then in a fashion similar to the derivation of Equation (4.19.1), dFd2–d1 = (∂Fd2–1/∂x1)dx1. Substituting Fd2–1 results in

dFd 2- d1 = -

A1 ¶ 2F1- 2 dx2dx1. (4.19.2) dA2 ¶x1¶x2

dFd1- d 2 = -

A1 ¶ 2F1- 2 dx2dx1. (4.19.3) dA1 ¶x1¶x2

or after using reciprocity,

Hence, by two differentiations, the factor dFd1–d2 can be found from F1–2 for the cylindrical configuration.

188

Thermal Radiation Heat Transfer

4.3.4 Unit-Sphere and Hemicube Methods Experimental determination of configuration factors is possible by the unit-sphere method introduced by Nusselt (1928). If a hemisphere of unit radius r = 1 is constructed over the area element dΑ1 in Figure 4.26, the configuration factor from dA1 to an area A2 is, by Equation (4.11),

Fd1- 2 =

1 p

ò cos q

1

A2

cos q2 dA2 1 = p S2

ò cos q dW . 1

1

A2

dA cos q2 dA2 The projection of dA2 onto the surface of the hemisphere is dΩ1, because d W1 = 2 s = dAs = . r S2 1 cos q1dAs . However, dAs cos θ1 is the projection of dAs onto the base The Fd1–2 is then Fd1-2 = p As of the hemisphere. It follows that integrating cos θ1 dAs gives the projection Ab of As onto the base of the hemisphere or

ò



Fd1- 2 =

1 p

Ab

ò cos q dA = p . (4.55) 1

s

As

The relation in Equation (4.55) forms the basis of several graphical and experimental methods for determining configuration factors. In one method, a spherical sector mirrored on the outside is placed over the area element dA1. A photograph taken by a camera placed directly above the sector and normal to dA1 then shows the projection of A2, which is Ab. The measurement of Ab on the photograph then provides Fd1–2 from Fd1-2 = Ab /pre2 , where re is the radius of the experimental mirrored sector. A means of optical projection was given by Farrell (1976), and numerical techniques were

FIGURE 4.26  Geometry of unit-sphere method for obtaining configuration factors.

Configuration Factors for Diffuse Surfaces

189

described in Lipps (1983) and Rushmeier et al. (1991). A semianalytical computational algorithm based on this method was developed by Mavroulakis and Trombe (1998) and may be useful in complex enclosures where two surfaces can share part of a contour. The hemicube approach to finding configuration factors is a modified unit-sphere method to meet speed requirements for applications in computer graphics. The method uses the fact that the configuration factor is the same from any object that subtends the same solid angle to the viewer, regardless of the actual shape and orientation of the object. Projection onto the face of one-half of a cube centered over the receiving element dA1 rather than onto a hemisphere, as shown in Figure 4.27, is more convenient for some purposes. In practice, the five surfaces of the hemicube are divided into small square elements (pixels), and the configuration factors from the element dA1 to each pixel are precomputed using an element-pixel configuration factor. This can be approximated by Equation (4.7) or, more exactly, by using factors for square finite small areas. The “patch” B projected by A onto the surface of the hemicube is found by available efficient techniques (Cohen and Greenberg 1985, Cohen et al. 1988, Watt and Watt 1992), and the configuration factor is found by summing the factors for the pixels contained within the patch. The projection on the unit hemisphere as used in the unit-sphere method is patch C. The precomputation of factors makes this summation very fast, and the method easily handles blocking and shading by intermediate elements (Kramer et al. 2015).

4.3.5 Direct Numerical Integration It is common to determine configuration factors between a differential area and a finite area, or between two finite areas, by direct numerical integration of Equation (4.11) or Equation (4.16). An example for a differential-to-finite area geometry is carried out in the online Appendix G, Example G.1, at www.ThermalRadiation.net. Such integrations require care to incorporate the effects of shading or blocking by intervening surfaces, and it can be quite computationally intensive

FIGURE 4.27  Geometry for hemicube analysis and relation to unit sphere.

190

Thermal Radiation Heat Transfer

to determine all the factors in a complex system with many surfaces. Methods are available for increasing computational speed and/or accuracy, including adaptive gridding to reduce the number of grid nodes while retaining accuracy (Campbell and Fussell 1990, Saltiel and Kolibal 1993) and partitioning of space by various search algorithms to find and minimize the number of equal-intensity areas that must be summed in the numerical integration (Thibault and Naylor 1987, Cohen et al. 1988, Buckalew and Fussell 1989). A numerical integration method is developed in Krishnaprakas (1998b) by dividing the surfaces into triangular elements and using Gaussian integration. A triangular grid can fit some surface shapes better than a rectangular grid. Contour integration carried out with Gaussian integration is used in Rao and Sastri (1996), and a procedure is derived that is similar to a finite-element line integral method. Using numerical integration, configuration factors can be calculated between surfaces having common or curved edges. The discrete ordinates method, described in Chapter 13, was developed to determine radiative transfer in enclosures filled with a medium that absorbs, emits, and scatters radiation. The method involves following radiation along paths between surfaces of an enclosure with the radiative effects of the intervening medium included. By omitting the radiative interaction with the intervening medium, the discrete ordinates method can also be used to evaluate configuration factors, as considered in Sanchez and Smith (1992) and Byun and Smith (1997). The radiation exchange for a 3D rectangular box compared very well with those computed from the known formulas for exchange between rectangles. In addition, the effect of an obstruction in the view between the enclosure surfaces in the form of a rectangular solid was introduced and investigated. Another approach uses a technique common in computer graphics and visualization (Daun and Hollands 2001) based on use of nonuniform rational B-splines to transform a parametrically defined 2D surface into the kernel of the integral equation describing radiative exchange. Some representative 2D radiative exchange solutions are obtained, and the method is shown to be fast and accurate. Mazumder and Ravishankar (2012) provide a general solution for factors between arbitrary polygons. Hajji et al. (2015) compare Monte Carlo, the crossed-string method, and analytical solutions for the case of a strip to an in-plane semicylinder, although there appears to be an error in the analytical solution. Gupta et al. (2017) provide a review where different methods are compared. Jiang et al. (2020) use a modified numerical method to find the configuration factors among arbitrary arrays of finite and infinite parallel cylinders. Muneer and Ivanova (2020) compare four routines based on finite element approaches using uniform and nonuniform grids, in some cases coupled with a Monte Carlo algorithm, and apply the method to various nonhomogeneous surfaces.

4.3.6 Computer Programs for the Evaluation of Configuration Factors Commercial and freeware computer programs are available that use one or more of the methods outlined in this chapter for numerical calculation of configuration factors. These are discussed in Section L.1 of the online book appendices listed at www.ThermalRadiation.net. These codes provide a means to generate configuration factors for complex geometries and are invaluable for radiative analyses in such systems.

4.4 CONSTRAINTS ON CONFIGURATION FACTOR ACCURACY To obtain accurate results when configuration factors are used for calculating radiative heat transfer in an enclosure, the configuration factors must satisfy their physical constraints. One constraint is that factors for any pair of surfaces satisfy reciprocity. A second constraint provides that the sum of all the factors from a surface to all surrounding surfaces in the enclosure, including the view to itself, must equal 1 for energy to be conserved; this is called the “closure” constraint. These constraints may not be accurately satisfied by factors individually evaluated in complex enclosures with many surfaces. Factors may become difficult to evaluate to high precision when enclosures have complex shapes with obstructions and partial views of surfaces. Individual factors are often

191

Configuration Factors for Diffuse Surfaces

computed by numerical integrations that introduce approximations, or by statistical methods such as Monte Carlo that may be converged to only a certain degree of accuracy. Vujiĉić et al. (2006a, 2006b) calculated the sensitivity to grid size and sample number for Monte Carlo configuration factor calculation for finite areas for the cases of directly opposed rectangles, hinged rectangles, and concentric parallel disks of equal diameter. Koptelov et al. (2012, 2014) present criteria for choosing the best integration method and number of nodes for numerically computing configuration factors. The latter paper includes criteria for a priori choice of number of nodes to achieve a specified level of accuracy. It was shown in Section 4.3.2 that if a certain number of configuration factors are calculated in an enclosure, the physical constraints can be used to obtain the remaining factors by solving a set of simultaneous equations. Sowell and O’Brien (1972) point out that in an enclosure with many surfaces, the configuration factors specified or computed easily may not lend themselves to subsequent accurate calculation of the remaining factors. Reciprocity or the energy conservation (“closure”) relation, Equation (4.30), may be difficult to apply. They present a computer scheme using matrix algebra that allows calculations of all remaining factors in an N-surface planar or convex-surface enclosure once the required minimum number of configuration factors is specified. To improve the accuracy of computations, some methods are discussed in Larsen and Howell (1986) for smoothing sets of “direct exchange areas” for enclosures, using the constraints imposed by reciprocity and energy conservation. These methods are directly applicable to configuration factors and can be used when some inaccurate or ill-defined configuration factors are present in a set. Another presentation of the compatibility conditions for enclosure configuration factors is given by van Leersum (1989). This guarantees overall energy conservation for the enclosure. An uncertainty analysis in Taylor et al. (1995) shows that strict enforcement of the reciprocity and closure constraints yields an order-of-magnitude reduction in the uncertainty in heat flux results that arises from uncertainties in the areas and configuration factors. Methods for adjusting the factors to provide enforcement are provided in Taylor and Luck (1995). A least-squares smoothing method is recommended, similar to that in Larsen and Howell and in Vercammen and Froment (1980) for enclosures filled with a gas that is not transparent. If some of the factors are known to have better accuracy, weighting factors wij can be used to give them more importance in the least-squares smoothing method; otherwise, the weighting factors are all equal to one. The least-squares optimization for an enclosure with N surfaces is done by minimizing the weighted error, given by Ew, as N



Ew =

N

åå w (F ij

i =1

- Fi - j ) (4.56) 2

c ,i - j

j =1

where the Fi–j are the configuration factors that have been calculated with approximations, and the Fc,i–j are the corrected configuration factors that result from the least-squares smoothing procedure. During the minimization procedure for Ew, constraints are imposed of reciprocity and closure:

Ai Fc,i - j = A j Fc, j -i

i = 1,… , N - 1,

j = i + 1,… , N (4.57)

N



åF

c,i - j

= 1 i = 1,… , N (4.58)

j =1



Fc,i - j ³ 0 i = 1,… , N ,

j = 1,… , N . (4.59)

192

Thermal Radiation Heat Transfer

The corrected Fc,i–j are then used in the enclosure calculations. This provides significantly improved results compared with using configuration factors that have not been optimized to guarantee that the physical constraints are satisfied. Other approaches are presented by Clarksean and Solbrig (1994), Lawson (1995), and Loehrke et al. (1995). A method that enforces both reciprocity and conservation while also enforcing a nonnegativity constraint on all factors using least-squares minimization is given by Daun et al. (2005).

4.5 COMPILATION OF KNOWN CONFIGURATION FACTORS AND THEIR REFERENCES: APPENDIX C AND WEB CATALOG Many configuration factors for specific geometries have been derived in analytical form or tabulated, and they are spread throughout the literature. Some factors that have convenient analytical forms are given in Appendix C for use in examples and homework problems. An open website giving extensive information on configuration factors is housed at www.ThermalRadiation.net/indexCat.html. The website provides references, algebraic relations, tables, and graphical results for some 325 configurations. A review of the site organization and contents is given by Howell and Mengüç (2011). The catalog also provides calculation capability to find numerical values for many of the factors.

HOMEWORK PROBLEMS (Additional problems available in the online Appendix P at www.ThermalRadiation.net) 4.1

Derive the configuration factor Fd1–2 between a differential area centered above a disk and a finite disk of unit radius parallel to the dA1.

Answer: 1/(H2 + 1). 4.2 Find the configuration factor Fd1–2 from a planar element to a coaxial parallel rectangle as shown below; the dA1 is above the center of the rectangle. Use any method except contour integration.

Configuration Factors for Diffuse Surfaces

Fd1-2 = Answer:

193

ù -1 é ù X Y 2 ïì é tan ê íê 2 1/ 2 ú 2 1/ 2 ú p ïî ë (1 + X ) û ë (1 + X ) û é ù -1 é ù X Y tan ê , +ê 2 1/ 2 ú 2 1/ 2 ú ( 1 + Y ) 1 ( ) Y + ë û ë û

where X = L/2H Y = W/2H. 4.3 Find the configuration factor F1–2 for the geometries shown below. Give a numerical value using any method: (a) Two plates infinitely long normal to the cross-section shown

(b) Two plates of finite length as shown

Answers: (a) 0.126; (b) 0.0417.

194

4.4

Thermal Radiation Heat Transfer

Find the configuration factor F1–2 for the geometries shown below. Give a numerical value using any method: (a) Disk to concentric ring

(b) Configuration factor F1–2 between the interior conical surface A1 of a frustum of a right circular cone and a hole in its top

(c) Configuration factor from infinitely long rectangular shape 1 shown in cross-section to infinitely long plane 2

Answers: (a) 0.2973; (b) 0.02743; (c) 0.4164.

Configuration Factors for Diffuse Surfaces

4.5

Derive a formula for F2–2 in terms of α. The A1 is inside the cone, and A2 is part of a sphere.

Answer: F2-2 = 4.6

195

aö 1æ 1 + sin ÷ . 2 çè 2ø

Find the configuration factor between the two infinitely long parallel plates shown below in cross-section. Use (a) The crossed-string method (b) Configuration factor algebra with factors from Appendix C (c) Integration of differential strip-to-differential strip factors

Answer: 0.3066. 4.7 For the cylindrical enclosure of diameter d and length l = 3.3d shown below, find all configuration factors among the surfaces.

Answer: F2–2 = 0.852.

196

4.8

Thermal Radiation Heat Transfer

Using the factor for an “infinitely long enclosure formed of three plane areas” derived in Example 4.13, find the factor from one side of an infinitely long enclosure to its base. The enclosure cross-section is an isosceles triangle with apex angle α.

Answer: F1-3 = sin 4.9

a . 2

A four-sided enclosure is formed of three mutually perpendicular isosceles triangles of short side S and hypotenuse H and an equilateral triangle of side H. Find the configuration factor F1–2 between two perpendicular isosceles triangles sharing a common short side.

Answer: 0.2113. 4.10 Using the crossed-string method, derive the configuration factor F1–2 between the infinitely long plate and the parallel cylinder shown below in cross-section. Compare your result with that given for configuration 11 in Appendix C.

Configuration Factors for Diffuse Surfaces

197

é r ù é -1 æ b ö -1 æ a ö ù Answer: F1-2 = ê ú ê tan ç c ÷ - tan ç c ÷ ú . ( ) b a è ø è øû ë ûë 4.11 For the 2D geometry shown, the view between A1 and A2 is partially blocked by two identical cylinders. Determine the configuration factor F1–2.

Answer: 0.667. 4.12 An infinitely long enclosure is shown below in cross-section. The outer surfaces form a square in cross-section, and the outer surfaces are parallel to the inner circular coaxial cylinder, so the geometry is 2D. Find the configuration factors F2–1, F2–3, and F2–4.

Answers: F2–1 = π/8; F2–3 = 0.2482; F2–4 = 0.1109.

198

Thermal Radiation Heat Transfer

4.13 (a)  Derive the configuration factor between a sphere of radius R and a coaxial disk of radius r. (Hint: Think of the disk as being a cut through a spherical envelope concentric around the sphere).

(b) What is the F from a sphere to a sector of a disk?

(c) What is the dF from a sphere to a portion of a ring of differential width?

Configuration Factors for Diffuse Surfaces

199

Answers:

(a )

ù ù 1é a a é a a ar 1- 2 ; ( b) 1- 2 ; (c ) dr. ê ê 2 1/ 2 ú 2 1/ 2 ú 2 2 ë (a + r ) û 4p ë (a + r ) û 4p (a + r 2 )3/ 2

4.14 Consider the interior of a black cubical enclosure. Determine the configuration factors between (a) two adjacent walls, and (b) two opposite walls. A sphere of diameter equal to one-half the length of a side of the cube is placed at the center of the cube. Determine the configuration factors between (c) the sphere and one wall of the enclosure, (d) one wall of the enclosure and the sphere, and (e) the enclosure and itself. Answers: (a) 0.20004; (b) 0.19982; (c) 1/6; (d) 0.1309; (e) 0.8691. 4.15 By use of the contour integration method in Section 4.3.3.2, obtain the result in Example 4.4 for the configuration factor from an elemental area to a perpendicular circular disk. Answer: Fd1-2 =

ù Hé H 2 + R2 + 1 ê - 1ú . 2 ê ( H 2 + R 2 + 1)2 - 4 R 2 úû ë

5

Radiation Exchange in Enclosures Composed of Black and/or Diffuse-Gray Surfaces Leopoldo Nobili (1784–1835) (left) and Macedonio Melloni (1798–1854) (right) developed a thermopile-based radiometer read by a galvanometer, and investigated radiation from various sources. They showed (1831) that different surfaces emitted differing amounts of radiation at the same temperature, and that the radiometer reacted similarly to light sources and heated surfaces.

5.1 INTRODUCTION We begin this chapter with a discussion of the analysis of radiation exchange in an enclosure where all surfaces are black, so that no reflections need to be considered. These ideal surfaces emit diffusely, so the intensity leaving each surface is independent of direction. For emission from an isothermal surface, the configuration factors discussed in Chapter 4 can then be used to calculate how much radiation will reach another surface. The relation for exchange between two surfaces is applied to multiple surfaces, each at a different uniform temperature, in an enclosure of black surfaces. In the next step toward accounting for real surface properties, the surfaces of the enclosure are assumed to be diffuse and gray (some surfaces can be black), so the directional spectral emissivity and absorptivity of each surface do not depend on direction or wavelength, but can depend on surface temperature. At any surface temperature T, the hemispherical total absorptivity and emissivity are equal and depend only on T, α(T) = ε(T). Even though this behavior is approached by only a limited number of real materials, the diffuse-gray approximation is often made to greatly simplify enclosure theory. A specific comment is warranted as to what is meant by the individual “surfaces” or “areas” of an enclosure. Usually, the geometry tends to divide the enclosure into surface areas, such as the individual sides of a rectangular chamber. It also may be necessary to specify surface areas by their heating or cooling conditions; for example, if one side of an enclosure has portions at two different temperatures, that side would be divided into two areas so this difference in boundary condition could be included. An area may also be subdivided by surface characteristics, such as separate smooth and rough portions with different emissivities. Surface areas in the radiation analysis are then defined as each portion of the enclosure boundary for which an energy balance is formed; these portions are selected based on geometry, imposed heating or temperature conditions, or surface characteristics. A further consideration is solution accuracy. If too few areas are designated, the accuracy may be poor because significant nonuniformity in reflected flux over an area is not accounted for in the analysis. Dividing a surface into too many smaller areas may require excessive computation time (although this is becoming less and less of a concern). Thus, engineering judgment is required in selecting the size and shape of the enclosure areas and their number. The surface areas of the enclosure can have various imposed thermal boundary conditions. A surface can be at a specified temperature, have a specified energy input, or be perfectly insulated 201

202

Thermal Radiation Heat Transfer

from external energy transfer. The present analysis requires that each surface area for the enclosure analysis must be at a uniform temperature. If the heating conditions are such that the temperature would vary markedly over an area, the area should be subdivided into smaller, more nearly isothermal portions; if necessary, these portions can be of differential size. From this isothermal area requirement, the emitted energy is uniform over each surface. A gray surface reflects only a portion of the incident energy. Two assumptions are made for reflected energy: (1) It is diffuse, so the reflected intensity at each position is uniform over all directions, and (2) it is uniform over each surface area. If the reflected energy is expected to vary over an area, the area should be subdivided so that the variation is not significant over each surface area considered for the analysis. With these restrictions reasonably met, the reflected energy for each area is assumed to be diffuse and uniform, as is the emitted energy. Hence, the reflected and emitted energies can be combined into a single diffuse energy flux leaving the area. The geometric configuration factors can then be used for the enclosure analysis. The derivation of the F factors was based on a diffuse-uniform intensity leaving the surface; this diffuse-uniform condition must be valid for the sum of both emitted and reflected energies. Radiative energy balances are not limited to steady-state conditions, but can be directly applied when there are transient temperature changes. The net energy flux q in the enclosure theory is the instantaneous net radiative loss from the location being considered on the boundary. For example, if a solid body is cooling only by radiation, q provides the boundary condition for the transient heat conduction solution for the temperature distribution within the solid. In some instances, an analysis assuming diffuse-gray surface areas may not yield good results. For example, if the temperatures of the individual areas of the enclosure differ considerably from each other, then an area will be emitting predominantly in the range of wavelengths characteristic of its temperature, while receiving energy predominantly in different wavelength regions. If the spectral emissivity varies with wavelength, the fact that the incident radiation has a different spectral distribution from the emitted energy makes the gray assumption invalid; that is, ε(T) ≠ α(T). When polished (specular) surfaces are present, the diffuse reflection assumption is invalid, and the directional paths of the reflected energy should be considered. These complications are treated later. In summary, for the analysis that follows, the enclosure boundary must be subdivided into areas so that over each such area the following restrictions are met:

1. All surface is opaque 2. The temperature is uniform 3. The surface properties are uniform 4. The ελ, αλ, and ρλ are independent of wavelength and direction so that ε(T) = α(T) =  1 − ρ(T), where ρ is the reflectivity 5. All energy is emitted and reflected diffusely 6. The incident, and hence reflected energy flux, is uniform over each individual area.

5.2 RADIATIVE TRANSFER FOR BLACK SURFACES Equations (4.5) and (4.6) can be written for black surfaces to give the total energy per unit time leaving dA1 that is incident upon dA2 as Ib,1 dA1 cosθ1 dA2 cosθ2/S2 and the total radiation leaving dA2 that arrives at dA1 is Ib,2 dA2 cosθ2 dA1 cosθ1/S2. For a black receiving element, all incident energy is absorbed. From Equation (1.35), the blackbody total intensity is related to the blackbody hemispherical total emissive power by Ib = Eb/π = σT 4/π, so the net energy per unit time d2Qd1↔d2 transferred from black element dA1 to black element dA2 along path S is (Figure 5.1)

(

d 2Qd1« d 2 = s T14 - T24

) cos qpScos q 1

2

2

dA1dA2 . (5.1)

203

Radiation Exchange in Enclosures

FIGURE 5.1  Radiative exchange between two black isothermal elements.

Example 5.1 The sun emits energy at a rate that can be approximated by a blackbody at Ts = 5780 K. A blackbody area element in orbit around the sun at the mean radius of the earth’s orbit, 1.49 × 1011 m, is oriented normal to the line connecting the centers of the area element and the sun. If the sun’s radius is RS = 6.95 × 108 m, what solar flux is incident upon the element? To the element in orbit, the sun appears as a diffuse isothermal black disk element of area

(

dA1 = pRS2 = p 6.95 ´ 108

)

2

= 1.52 ´ 1018 m2.

From the derivation of Equation (5.1), since

(

) (

)

θ1 = θ2 = 0, the incident energy flux on element dA2 in orbit is sTs4 /p dA1 cos q1 cos q2 /S 2  = 

(

)(

) (

)

(

)

2 sT /p dA1 /S = 5.6704 ´ 10 ´ 5780 /p é1.52 ´ 1018 / 1.49 ´ 1011 ù  = 1379 W/m2. This value êë úû is consistent with measured values of the solar constant, 1353 to 1394 W/m2, and the accepted standard value of 1366 W/m2 (Section 3.4.1). An alternative procedure is to utilize the fact that radiant energy leaves the sun in a spherically symmetric fashion. The energy radiated from the solar sphere is sTs4 4pRS2 , and the area of a sphere surrounding the sun and having a radius equal to the earth’s orbit is 4πS2. Hence, the flux 4 s

2

-8

4

received at the earth’s orbit is sTs4 4pRS2 / 4pS 2 = sTs4 (RS /S ) , as obtained before. 2

Example 5.2 As shown in Figure 5.2, a black square with side 0.25 cm is at T1 = 1100 K and is near a tube 0.30 cm in diameter. The tube opening acts as a black surface at 700 K. What is the net radiative transfer from A1 to A 2 along the connecting path S? For this geometry, the A1 and A 2 have small dimensions compared with the length S between them. If Equation (5.1) is used, the cos θ1, cos θ2, and S do not vary significantly for positions along A1 and A 2. Hence, accurate transfer results can be obtained without integrating to obtain an F factor. Then, for Equation (5.1), cos θ1 is found from the sides of the right triangle A 2-0-A1 as

204

Thermal Radiation Heat Transfer

FIGURE 5.2  Radiative transfer between a small square area and a small circular tube opening. cos θ1 = 5/(82 + 52)1/2 = 5/891/2. The other factors in the energy exchange equation are given, so the net radiative exchange is

(

s T14 - T24

) cos qpScos q 1

2

2

A1A2

(

= 5.6704 ´ 10 -8 11004 - 700 4

) 895

1/ 2

cos 20° 0.252 p0.30 2 4 4 p 89 / 10 4 10 4 ´ 10

(

)

= 5.462 ´ 10 -5 W.

5.2.1 Transfer between Black Surfaces Using Configuration Factors Using Equation (4.7), the geometric factors in Equation (5.1) for black area elements can be written in terms of configuration factors to provide a shorter form:

(

)

(

)

d 2Qd1« d 2 = s T14 - T24 dFd1- d 2 dA1 = s T14 - T24 dFd 2 - d1dA2 . (5.2)

Similarly, for radiative transfer between a black differential element and a black finite area,

dQd1«2 = dQd1®2 - dQ2®d1 = sT14 dA1 Fd1-2 - sT24 A2 dF2-d1 (5.3)

or, using Equation (4.15), the net transfer is

(

)

(

)

dQd1« 2 = s T14 - T24 dA1 Fd1- 2 = s T14 - T24 A2 dF2 - d1. (5.4)

For two black surfaces with finite areas

Q1«2 = Q1®2 - Q2®1 = sT14 A1 F1-2 - sT24 A2 F2-1 (5.5)

205

Radiation Exchange in Enclosures

or, using the reciprocity relation, Equation (4.20),

(

)

(

)

Q1« 2 = s T14 - T24 A1 F1- 2 = s T14 - T24 A2 F2 -1. (5.6)

5.2.2 Radiation Exchange in a Black Enclosure The derivations so far are for enclosures with black surfaces, which is seldom the case in practical applications. In addition, a surface may not be isothermal, in which case the surfaces must be subdivided into smaller ones that meet the isothermal approximation. The analysis here is for many black surfaces in an enclosure. An energy balance is formed on a typical enclosure surface Ak as in Figure 5.3. The Qk is the combined energy supplied to Ak by all sources other than by radiation inside the enclosure; this maintains Ak at Tk while Ak exchanges radiation with the enclosure surfaces. The Qk could be composed of convection heat transfer to the inside of the wall, and/or conduction heat transfer through the wall from an energy source outside, and/or an energy source in the wall itself such as an electrical heater. For wall cooling, such as by cooling channels within the wall, the contribution to Qk is negative. The emission from Ak into the enclosure is sTk4 Ak . The radiant energy received by Ak from another surface Aj is sT j4 A j Fj -k . The energy balance is then N



Qk = sTk4 Ak -

å sT A F 4 j

j

j -k

(5.7)

j =1

where the summation includes energy arriving from all surfaces inside the enclosure including Ak if Ak is concave. Equation (5.7) can be written in alternative forms. Applying reciprocity to the terms in the summation results in

æ Qk = Ak ç sTk4 ç è

N

å j =1

ö sT j4 Fk - j ÷ . (5.8) ÷ ø

FIGURE 5.3  Enclosure composed of N black isothermal surface areas (shown in cross-section for simplicity).

206

Thermal Radiation Heat Transfer

For a complete enclosure, from Equation (4.30), å j =1 Fk - j = 1, so that N

æ Qk = Ak ç sTk4 ç è



N

åF

k- j

N

-s

j =1

åT F

4 j k- j

j =1

ö ÷ = sAk ÷ ø

N

å (T

4 k

)

- T j4 Fk - j . (5.9)

j =1

This is in the form of a sum of the net radiative energy transferred from Ak to each surface within the enclosure. Example 5.3 The three-sided black enclosure of Example 4.13 has its black surfaces maintained at T1, T2, and T3. To maintain each temperature, determine the amount of energy that must be supplied to each surface by means other than radiation inside the enclosure. This energy equals the net radiative loss from each surface by radiative exchange within the enclosure. Equation (5.9) is written for each surface as

(

)

(

)

(

)

(

)

(

)

(

)

Q1 = A1 éF1- 2s T14 - T24 + F1- 3s T14 - T34 ù ë û

Q2 = A2 éF2-1s T24 - T14 + F2- 3s T24 - T34 ù ë û Q3 = A3 éF3-1s T34 - T14 + F3- 2s T34 - T24 ù . ë û

The configuration factors are in Example 4.13. Thus, all factors on the right sides are known, and the Qk values can be computed. A check on the numerical results is that, from overall energy conservation, the net Qk added to the entire enclosure,

å

N

k=1

Qk , must be zero to maintain steady

temperatures. This is also shown by adding all the terms on the right sides of the three equations and using reciprocity, such as A 2F2–1 = A1F1–2.

Example 5.4 The enclosure of Example 4.13 has two sides maintained at T1 and T2. The third side is insulated on the outside, and there is only radiative transfer on the inside so that Q3 = 0. Determine Q1, Q2, and T3. Equation (5.9) is written for each surface as

(

)

(

)

(

)

(

)

(

)

(

)

Q1 = A1 éF1- 2s T14 - T24 + F1- 3s T14 - T34 ù ë û

Q2 = A2 éF2-1s T24 - T14 + F2- 3s T24 - T34 ù ë û 0 = A3 éF3-1s T34 - T14 + F3- 2s T34 - T24 ù . ë û

The final equation is solved for T34. This is inserted into the first two equations to obtain Q1 and Q2.

Example 5.5 A very long black heated tube A1 of length L is enclosed by a concentric black split cylinder as in Figure 5.4. The diameter of the split cylinder is twice that of the heated tube, and one-half as much energy flux is to be removed from the upper area A3 as from the lower area A 2. If T1 = 1700 K, and

207

Radiation Exchange in Enclosures

FIGURE 5.4  Radiant energy exchange in split-circular cylinder configuration, L ≫ R2. (a) Geometry of enclosure, and (b) auxiliary construction to determine F2–2. a heat flux Q1/A1 = 3 × 105 W/m2 is supplied to the heated tube, what are the values of T2, T3, Q2, and Q3? Neglect the effect of the tube ends. Writing Equation (5.9) for each surface gives

(

)

(

)

(

)

(

)

(

)

(

)

Q1 = A1 éF1-2s T14 - T24 + F1-3s T14 - T34 ù ë û

Q2 = A2 éF2-1s T24 - T14 + F2-3s T24 - T34 ù ë û Q3 = A3 éF3-1s T34 - T14 + F3-2s T34 - T24 ù . ë û

From the geometry, A1 /A3 = A1 /A2 = pD1L / 21 pD2L = 1, since D2 = 2D1. From an energy balance, Q1 + Q2 + Q3 = 0 and, since A1 = A 2 = A3, (Q1/A1) + (Q2/A 2) + (Q3/A3) = 0. From the statement of the problem, Q3/A3 = Q2/2A 2, and this yields

Q2 2 Q1 == -2.0 ´ 105 W/m2 , 3 A1 A2

1 Q1 Q3 == -1.0 ´ 105 W/m2 . 3 A1 A3

From the symmetry and configuration-factor algebra, F1–2 = F1–3 = 1/2, F2–1 = F3–1 = A1 F1–3/A3 = 1/2, and F2–3 = F3–2. To determine F2–3, it is known that F2–1 + F2–2 + F2–3 = 1. Using F2–1 = 1/2 gives F2–3 = (1/2) − F2–2. In the auxiliary construction of Figure 5.4b, F2–2 = 1 − F2–E. The effective area AE has been drawn in to leave unchanged the view of surface 2 to itself, and to simplify the geometry so that the crossedstring method can be used to determine F2–E. The uncrossed strings from a to a′ and b to b′ have zero length. The crossed strings extend from a to b′ and a′ to b, and each has length 2 3R1 + pR1 / 3 . Then,

(

)

from Section 4.3.1 and the fact that A2 = A1 = 2pR1, F2-E = 2 3R1 + pR1 / 3 / 2pR1 = It then follows that F2-3 = (1/ 2) - F2-2 = (1/ 2) - (1 - F2-E ) = mation, the energy exchange equations become 3 ´ 105 =

(

)

)

(

)

-2.0 ´ 105 =

s 4 T2 - 17004 + 0.2 2180s T24 - T34 2

-1.0 ´ 105 =

s 4 T3 - 17004 + 0.2180s T34 - T24 . 2

( (

) )

)

3 /p + (1/6 ) .

3 /p - (1/ 3) = 0.2180. Using this infor-

s s 17004 - T24 + 17004 - T34 2 2

(

(

( (

) )

Adding the second and third equations results in the first, so only two equations are independent. Solving the first and second equations gives T2 = 1207 K and T3 = 1415 K.

208

Thermal Radiation Heat Transfer

5.3 RADIATION AMONG FINITE DIFFUSE-GRAY AREAS 5.3.1 Net-Radiation Method for Enclosures Consider an enclosure of N discrete internal surface areas as in Figure 5.5. To analyze the radiation exchange between the surface areas within the enclosure, two common problems are to find: (1) The required energy supplied to a surface to be determined when the surface temperature is specified, and (2) the temperature that a surface will achieve to be found when a known heat input is imposed. Some more general boundary conditions are considered later. A complex radiative exchange occurs inside the enclosure as radiation leaves a surface, travels to other surfaces, is partially reflected, and is then re-reflected many times within the enclosure, with partial absorption at each contact with a surface. It is complicated to follow the radiation as it undergoes this process; fortunately, this is not always necessary. A convenient analysis can be formulated by using the net-radiation method. This method was first devised by Hottel and later developed in a different manner by Poljak (1935). An alternative approach was given by Gebhart (1961, 1971). All the methods are basically equivalent, as demonstrated by Sparrow (1963b) and shown here in an example. The Hottel/Poljak approach, which is usually convenient, is now given; the other formulations are then presented more briefly. Consider the kth inside surface area Ak of the enclosure in Figures 5.5 and 5.6. The Gk and Jk are the rates of incoming and outgoing radiant energy per unit area of Ak. The quantity qk is the energy flux supplied to Ak by some means other than the radiation inside the enclosure, to make up for the net radiative gain or loss and thereby maintain the specified inside surface temperature. For example, if Ak is the inside surface of a wall of finite thickness, Qk could be the heat conducted through the wall from the outside to Ak. An energy balance for Ak provides the relation

Qk = qk Ak = ( J k - Gk ) Ak . (5.10)

FIGURE 5.5  Enclosure composed of N discrete surface areas with typical surfaces j and k (shown in crosssection for simplicity).

209

Radiation Exchange in Enclosures

FIGURE 5.6  Energy quantities incident upon and leaving typical surface inside enclosure.

A second equation results from the energy flux leaving the surface being composed of emitted plus reflected energy. This gives, noting αk = εk for a gray surface,

J k = e k sTk4 + rk Gk = e k sTk4 + (1 - a k ) Gk = e k sTk4 + (1 - e k ) Gk (5.11)

where ρk = 1 − αk = 1 − εk has been used for opaque gray surfaces. The term radiosity is often used for Jk. The incident flux or irradiation, Gk, is derived from the portions of radiant energy leaving the surfaces inside the enclosure that arrive at the kth surface. If the kth surface can view itself (is concave), a portion of its outgoing flux will contribute directly to its incident flux. The incident energy is then equal to

Ak Gk = A1 J1 F1-k + A2 J 2 F2-k +  + A j J j Fj -k +  + Ak J k Fk -k +  + AN J N FN -k . (5.12)

From configuration-factor reciprocity, Equation (4.19),

A1 F1-k = Ak Fk -1; A2 F2-k = Ak Fk -2 ;  AN FN -k = Ak Fk - N . (5.13)

Then Equation (5.12) can be written so the only area appearing is Ak:

Ak Gk = Ak J1 Fk -1 + Ak J 2 Fk -2 +  + Ak J j Fk - j +  + Ak J k Fk -k +  + Ak J N Fk - N (5.14)

so that the incident flux is N



Gk =

åJ F

j k- j

. (5.15)

j =1

Equations (5.10), (5.11), and (5.15) are simultaneous relations between qk, T k, Gk, and Jk for each surface, and, through Equation (5.15), to the other surfaces. One solution procedure is to note that

210

Thermal Radiation Heat Transfer

Equations (5.11) and (5.15) provide two different expressions for Gk. These are each substituted into Equation (5.10) to eliminate Gk and provide two energy balance equations for qk in terms of T k and Jk: Qk e = qk = k sTk4 - J k (5.16) Ak 1 - ek

(



Qk = qk = J k Ak



N

N

å

)

J j Fk - j =

j =1

å(J

k

- J j ) Fk - j . (5.17)

j =1

The qk can be regarded as either the energy flux supplied to surface k by means other than internal radiation (such as by convection or conduction to Ak) or as the net radiative loss from Ak by radiation exchange inside the enclosure. Equation (5.16) or (5.17) is the balance between net radiative energy loss and energy supplied by means other than the radiation inside the enclosure. Equations (5.16) and (5.17) are written for each of the N surfaces in the enclosure. This provides 2N equations in 2N unknowns. The Jk are N of the unknowns, and the remaining unknowns consist of q and T, depending on what boundary quantities are specified. Later, the Jk are eliminated to give N equations relating the N unknown q and T. Other forms of Equation (5.16) are J k = sTk4 -



1 - ek qk ek

or sTk4 =

1 - ek qk + J k . (5.18) ek

Equations (5.16) and (5.18) were obtained by eliminating Gk from Equations (5.10) and (5.11). If Jk is eliminated instead, then the results are qk = e k sTk4 - e k Gk



or Gk = sTk4 -

qk . (5.19) ek

Equation (5.19) shows that the net energy leaving the surface by radiation is the emitted energy minus the absorbed incident energy εkGk. Hence, if qk and Tk4 are available from a solution, the incident energy that is absorbed by a gray surface can be found from a k Gk = e k Gk = e k sTk4 - qk . (5.20)



The absorbed energy added to the energy supplied by other means is equal to the emitted energy, a k Gk + qk = e k sTk4 . Before continuing with the development, examples are given to illustrate using Equations (5.16) and (5.17) as simultaneous equations for each enclosure surface. Example 5.6 Derive the expression for the net radiative heat exchange between two infinite parallel flat plates in terms of their temperatures T1 and T2 (Figure 5.7). Since for infinite plates all radiation leaving one plate will arrive at the other plate, the F1–2 =  F2–1 = 1. Equations (5.16) and (5.17) for each plate are then

q1 =

e1 sT14 - J1 , 1 - e1

(

)

q1 = J1 - J2 (5.6.1)

211

Radiation Exchange in Enclosures

FIGURE 5.7  Heat fluxes for radiant interchange between infinite parallel flat plates.

q2 =



e2 sT24 - J2 , q2 = J2 - J1. (5.6.2) 1- e 2

(

)

Comparing Equations (5.6.1) and (5.6.2), q1 = −q2, so the energy added to surface 1 is removed from surface 2. The q1 is thus the net energy transferred from 1 to 2, as requested in the problem statement. Equations (5.6.1) and (5.6.2) yield

J1 = sT14 -

1- e1 q1, e1

J2 = sT24 -

1- e 2 1- e 2 q2 = sT24 + q1. e2 e2

These are substituted into Equation (5.6.1), and the result solved for q1:

q1 = -q2 =

(

s T14 - T24

)

1/e1 (T1 ) + 1/ e 2 (T2 ) - 1

. (5.6.3)

The functional notation ε(T) is used to emphasize that ε1 and ε2 can be functions of temperature. Since T1 and T2 are specified, ε1 and ε2 are substituted at their proper temperatures, and q1 is directly calculated.

Example 5.7 For the parallel-plate geometry of the previous example, what temperature will surface 1 reach for a given energy flux input q1 while T2 is held at a specified value? Equation (5.6.3) applies and, when solved for T1, gives 1/ 4



ìï q é 1 üï ù 1 - 1ú + T24 ý T1 = í 1 ê + s e T e 2 (T2 ) úû îï êë 1 ( 1 ) þï

. (5.7.1)

Since ε1(T1) is a function of T1, which is unknown, an iterative solution is necessary. A trial T1 is selected, and ε1 is chosen at this value. Equation (5.6.3) is solved for T1, and this value is used to select ε1 for the next approximation. The process is continued until ε1(T1) and T1 do not change with further iterations.

212

Thermal Radiation Heat Transfer

Example 5.8 Derive an expression for the net-radiation exchange between two uniform temperature concentric diffuse-gray spheres as in Figure 5.8. This situation is more complicated than for infinite parallel plates, as the two surfaces have unequal areas and surface 2 can partially view itself. The configuration factors were derived in Example 4.11 as F1–2 = 1, F2–1 = A1/A 2, and F2–2 = 1 − (A1/A 2). Equations (5.16) and (5.17) are written for each of the two sphere surfaces as

Q1 = A1

(

Q2 = A2





e1 sT14 - J1 , 1 - e1

)

Q1 = A1 ( J1 - J2 ) (5.8.1)

e2 sT24 - J2 (5.8.2) 1- e2

(

)

é A ö ù A æ Q2 = A2 ê J2 - 1 J1 - ç 1- 1 ÷ J2 ú = A1 ( J2 - J1 ) . (5.8.3) A A 2 2 ø è ë û

Comparing Equations (5.8.1) and (5.8.3) reveals that Q1 = −Q2, as expected from an overall energy balance. The four equations (Equations (5.8.1), (5.8.2), and (5.8.3)) can be solved for the four unknowns J1, J2, Q1, and Q2. This yields the net energy transfer (supplied to surface 1 and removed at surface 2):

Q1 =

(

A1s T14 - T24

)

1/e1 (T1 ) + ( A1 /A2 ) éë1/e 2 (T2 ) - 1ùû

. (5.8.4)

When the spheres in Example 5.8 are not concentric, all the radiation leaving surface 1 is still incident on surface 2. The configuration factor F1–2 is again 1 and, with the use of the same assumptions, the analysis follows as before, leading to Equation (5.8.4). However, when sphere 1 is relatively small (say, one-half the diameter of sphere 2) and the eccentricity is large, the geometry is so different from the concentric case that using Equation (5.8.4) seems intuitively incorrect. The error in using Equation (5.8.4) is that it was derived on the basis that q, G, and J are uniform over each of A1 and A 2. These conditions are exactly met only for the concentric case. For the eccentric case, the A1 and A 2 need to be subdivided to improve accuracy.

FIGURE 5.8  Energy quantities for radiant interchange between two concentric spheres.

213

Radiation Exchange in Enclosures

Example 5.9 A gray isothermal body with area A1 and temperature T1 and without any concave indentations is completely enclosed by a much larger gray isothermal enclosure having area A 2. How much energy is being transferred by radiation from A1 to A 2? The area A1 cannot see any part of itself, and A1 is not near A 2. Since A1 is completely enclosed and F1–1 = 0, the configuration factors and analysis are the same as in Example 5.8, which results in Q1 given by Equation (5.8.4). This is valid, as A1 is specified as being rather centrally located within A 2, and hence the heat fluxes tend to be uniform over A1. For the present situation, A1 ≪ A 2, and Equation (5.8.4) (unless ε2 is very small) reduces to

(

)

Q1 = A1 e1 (T1 ) s T14 - T24 . (5.9.1)

Note that this result is independent of the emissivity ε2 of the enclosure (the enclosure acts like a black cavity unless ε2 is very small so that A 2 is highly reflective).

Example 5.10 Consider a long enclosure made up of three surfaces as in Figure 5.9. The enclosure has a uniform cross-section and is long enough that its ends can be neglected in the radiative energy balances. How much energy must be supplied to each surface (equal to the net radiative energy loss from each surface resulting from exchange within the enclosure) to maintain the surfaces at temperatures T1, T2, and T3? Write Equations (5.16) and (5.17) for each surface:

Q1 e = 1 sT14 - J1 (5.10.1) A1 1 - e1



Q1 = J1 - F1-1J1 - F1-2 J2 - F1-3 J3 (5.10.2) A1



Q2 e = 2 sT24 - J2 (5.10.3) A2 1 - e 2

(

(

)

)

FIGURE 5.9  Long enclosure composed of three surfaces (ends neglected); cross-section is uniform.

214

Thermal Radiation Heat Transfer



Q2 = J2 - F2-1J1 - F2-2 J2 - F2-3 J3 (5.10.4) A2



Q3 e = 3 sT34 - J3 (5.10.5) A3 1 - e3



Q3 = J3 - F3-1J1 - F3- 2 J2 - F3- 3 J3. (5.10.6) A3

(

)

The first of each of these three pairs is solved for J, and the J is substituted into the second equation of each pair to obtain Q1 æ 1 1 - e1 ö Q2 1 - e 2 Q3 1 - e3 F1-3 F1-2 ç - F1-1 ÷e3 A3 e2 e1 ø A2 A1 è e1 = (1 - F1-1 ) sT14 - F1-2sT24 - F1-3sT34



(

)

(

= F1-2s T14 - T24 + F1-3s T14 - T34 -

(5.10.7)

)

Q1 1 - e1 Q2 æ 1 1 - e 2 ö Q3 1 - e3 F2-3 F2-1 + ç - F2-2 ÷e3 A2 è e 2 e 2 ø A3 A1 e1 = -F2-1sT14 + (1 - F2-2 )sT24 - F2-3sT34



(

)

(

= F2-1s T24 - T14 + F2-3s T24 - T34

(5.10.8)

)

Q1 1- e1 Q2 1- e 2 Q3 æ 1 1- e 3 ö F3- 2 F3-1 + ç - F3- 3 e ÷ A3 è e3 e2 A2 e1 A1 3 ø (5.10.9) = -F3-1sT14 - F3- 2sT24 + (1- F3- 3 )sT34

(

)

(

)

= F3-1s T34 - T14 + F3- 2s T34 - T24 . Since the T values are known, the ε can be specified from surface-property data at their appropriate T values and the three simultaneous equations solved for the Q supplied to each surface. Note that the results are approximations, because the radiosity leaving each surface is not uniform, as assumed by using Equations (5.16) and (5.17). This is because the reflected flux is not uniform as a result of the enclosure geometry. Greater accuracy is obtained by dividing each of the three sides into additional surface areas to yield a larger set of simultaneous equations.

5.3.1.1 System of Equations Relating Surface Heating Rate Q and Surface Temperature T The form of Equations (5.10.7) through (5.10.9) shows that the Q and T for an enclosure of N surfaces can be related by a system of N equations. It is evident from Equations (5.10.7) through (5.10.9) that the general form for the kth surface is N



å j =1

æ dkj 1 - e j ö Qj = - Fk - j ç ÷ e e j ø Aj è j

N

å (d

kj

- Fk - j ) sT = 4 j

j =1

N

åF s (T k- j

4 k

)

- T j4 . (5.21)

j =1

Corresponding to each surface, k takes on one of the values 1, 2, … N, and δkj is the Kronecker delta, defined as

ìï1 when k = j dkj = í . îï0 when k ¹ j

215

Radiation Exchange in Enclosures

When the surface temperatures are specified, the right side of Equation (5.21) is known and there are N simultaneous equations for the unknown radiation energy rate Q. In general, the energy inputs to some of the surfaces may be specified and the temperatures of these surfaces are to be determined. There is still a total of N unknown Q and T, and Equation (5.21) provides the necessary number of relations. If the values of ε depend on temperature, it is necessary initially to guess the unknown T, then the ε values can be chosen and the system of equations solved. The T values are used to select new ε, and the process repeated until the T and ε values no longer change. Again, note that the results may be approximate because the uniform radiosity assumption is not perfectly fulfilled over each finite area. Division into smaller areas is required to improve accuracy. As a practical example, the solution of an enclosure with mixed boundary conditions to simulate heat-generating electronic modules can be found in Arimilli and Ketkar (1988). At least one enclosure surface must have a specified temperature as a boundary condition. If all surfaces have a specified radiative energy flux (even if they meet the energy conservation requirement that å k qk Ak = 0 ), then the left-hand side of Equation (5.21) is a known constant, and the result is that the surface temperatures become indeterminate. Example 5.11 Consider an enclosure of three sides as in Figure 5.9. Side 1 is maintained at T1, side 2 is uniformly heated with flux q2, and the third side is perfectly insulated on the outside. What are the equations to determine Q1, T2, and T3? The conditions give Q2/A 2 = q2 and Q3 = 0. Then Equation (5.21) yields three equations, where the unknowns have been gathered on the left sides: Q1 æ 1 1 - e1 ö 1- e2 4 4 4 (5.11.1) ç - F1-1 ÷ + F1-2sT2 + F1-3sT3 = (1 - F1-1 ) sT1 + q2F1-2 A1 è e1 e1 ø e2





-

Q1 1 - e1 1- e2 ö æ 1 F2-1 - (1 - F2-2 ) sT24 + F2-3sT34 = -F2-1sT14 - q2 ç - F2-2 ÷ (5.11.2) A1 e1 e2 ø è e2



-

Q1 1- e1 1- e 2 F3-1 . (5.11.3) + F3- 2sT24 - (1- F3- 3 ) sT34 = -F3-1sT14 + q2F3- 2 A1 e1 e2

Note that for this simple situation, the solution could be shortened by using overall energy conservation to give Q1 = −Q2. However, it is generally a good idea to solve directly for all the unknowns and use the overall energy balance

å

N

k =1

Qk = 0 as a check.

Example 5.12 A hollow cylinder is heated on the outside in such a way that the cylinder is maintained at a uniform temperature. The outside of the cylinder is otherwise insulated so the energy must be transferred by radiation from the inside of the cylinder through the cylinder ends. The system is in a vacuum, so radiation is the only mode of energy transfer. As shown in Figure 5.10, there is a disk centered on the cylinder axis and facing normal to the ends of the cylinder. The disk is exposed on both sides, so the side facing away from the cylinder radiates to the surroundings at Te. The other side receives radiation from the inside of the cylinder. Provide equations to compute the disk temperature, Td, if the inside of the cylinder is at a uniform temperature Tc.

216

Thermal Radiation Heat Transfer

FIGURE 5.10  Hollow cylinder and disk configuration for Example 5.12. The heat loss from the surface of the disk facing away from the cylinder can be considered as radiation into a large environment at Te. Hence, from Equation (5.9.1), the net energy flux leaving this surface is qd,e = ed,e s(Td4 - Te4 ). The exchange with the cylinder is analyzed as a three-surface enclosure consisting of the inside of the cylinder, the side of the disk facing the cylinder, and the open area between the disk and the cylinder (a frustum of a cone) combined with the open area at the left side of the cylinder (this is a first approximation without subdividing the cylinder and disk areas to achieve greater accuracy). In the configuration-factor designations, the subscript e designates the combination of the two open areas at Te. Since the energy lost to the environment is not asked for, only two equations are written: One for the inside of the cylinder and one for the

(

)

surface of the disk facing the cylinder. The qd,c = -ed,e s Td4 - Te4 is the radiative heat flux supplied to the left surface of the disk. From Equation (5.21), the two equations are

æ 1 1- e c ö 1- ed,c qc ç - Fc - c = Fc - ds Tc4 - Td4 + Fc - e s Tc4 - Te4 (5.12.1) ÷ - qd,cFc - d e ec ø d ,c è ec



(

-qcFd - c

)

(

)

1- e c 1 + qd = Fd - c s Td4 - Tc4 + Fd - e s Td4 - Te4 . (5.12.2) ec ed,c

(

)

(

)

The qc is eliminated from Equations (5.12.1) and (5.12.2); this yields an analytical relation for Td.

Example 5.13 As an example with a more complex geometry, consider the long rectangular enclosure shown in cross-section in Figure 5.11 (the end walls are neglected). There is a thin baffle along a diagonal, dividing the interior into two triangular enclosures. Heat fluxes are specified along two sides, and the other two sides are each cooled to a specified uniform temperature. It is required to find the temperatures T1 and T2 of the heated walls and the heat fluxes q5 and q6 that must be removed through the cooled walls to maintain their temperatures. For simplicity, the surfaces are not subdivided into smaller areas. Subdivision improves the results (see Example 5.14). Since T3, T4, q3, and q4 are also unknown, there are eight unknowns, so eight simultaneous equations are required. Two are the constraints at the baffle wall that yield T3 = T4 and q3 = −q4. The other six equations are found from Equation (5.21) and are written here in abbreviated form using the notation Fmn ≡ Fm−n, ϑ ≡ T4, En ≡ (1−εn)/εn:

217

Radiation Exchange in Enclosures

FIGURE 5.11  Rectangular enclosure divided into two triangular enclosures for Example 5.13.

( q1 / e1) - q2E2F12 - q3E3F13 = s ( J1 - F12J2 - F13J3 ) -q1E1F21 + ( q2 /e 2 ) - q3E2F23 = s ( -F21J1 + J2 - F23J3 ) -q1E1F31 - q2E2F32 + ( q3 /e3 ) = s ( -F31J1 + F32J2 + J3 )



( q5 /e5 ) - q6E6F56 - q4E4F54 = s ( J5 - F56J6 - F54J4 )



-q5E5F65 + ( q6 /e6 ) - q4E4F64 = s ( -F65J5 + J6 - F64J4 ) -q5E5F45 - q6E6F46 + ( q4 / e 4 ) = s ( -F45J5 - F46J6 + J4 ) . For each three-sided enclosure, the configuration factors are found from Example 4.13. Typical values are F1–2 = F5–6 = 0.33632 and F1–3 = F5–4 = 0.66369. A matrix solver can be used to substitute the values and solve the equations to obtain T1 = 817.0 K, T2 = 821.7 K, T3 = 709.1 K, q5 = −559.0 W/m2, and q6 = −2890.1 W/m2. Most of the energy is leaving through the more strongly cooled side A6. The effect of various parameters can easily be examined. If ε1 = ε2 are changed, the only results affected are T1 and T2. This is because the same total energy must pass through surface 4, so T4 remains the same in order to transfer the energy to sides 5 and 6. For ε1 = ε2 = 1.0, the T1 and T2 are slightly decreased to T1 = 812.9 K and T2 = 815.8 K. When ε1 = ε2 = 0.50 these temperatures increase to T1 = 824.9 K and T2 = 833.1 K. The results should be checked by determining whether

å

N

k =0

qk Ak = 0. In this case (0.55 × 1500.0) + (0.75 × 2200.0) − (0.55​ × 55​9.0)–​(0.75​ × 28​90.1)​  = 

−0​.025 ​W/m of length, which is within the round-off accuracy.

218

Thermal Radiation Heat Transfer

Example 5.14 A triangular enclosure has long sides normal to the cross-section shown in Figure 5.12. The triangular end walls can be neglected in the radiative exchange. Specified quantities are the temperatures of two sides and the heat flux added to the third. The q1, q2, and T3 are required. The solution is obtained first with side 3 being a single area. Then, this side is divided into two equal parts to refine the calculation. Equation (5.21) is used for analysis. From Example 4.13, F1−2 = (A1 + A 2−A 3)/2A1 = 0.2929 = F2−1. Then, F1−3 = 1−F1−2 = 0.7071 = F2−3. From symmetry, F3–1 = F3–2 = 0.5. The self-view factors are zero. The first three equations from Example 5.13 are used. If ε1 = ε2 = 0.80 and ε3 = 0.90, then the results are q1 = −6346 W/m2, q2 = 1820 W/m2, and T3 = 649.1 K. Thus, energy must be added to maintain the high temperature of A 2. The energy added to A 2 combined with that from A 3 flows out of A1 at a lower temperature. Now consider what happens if the emissivities are reduced by half, ε1 = ε2 = 0.40 and ε3 = 0.45. The energy supplied to A 3 has greater difficulty in being transferred to the other walls, and the temperature rises to T3 = 733.9 K. Since T3 is now larger than both T1 and T2, energy is transferred out from both A1 and A 2: q1 = −4101 W/m2 and q2 = −424.6 W/m2. To refine the calculation, A 3 is divided into two equal parts A4 and A 5. From the geometry, F1–4 = 0.5 = F1–5 + F1–2 = F2–5. With F1–2 known, this gives F1–5 = 0.2071 = F2–4. As in the previous examples, four equations are written from Equation (5.21). With T1, T2, q4, and q5 known, these are solved for q1, q2, T4, and T5. For ε1 = ε2 = 0.80 and ε3 = 0.90, this gives q1 = −6049 W/m2, q2 = 1524 W/m2, T4 = 626.3 K, and T5 = 669.7 K. This reveals the temperature variation along A 3 as compared with the uniform value obtained in the first part of this example. The q2 is somewhat smaller because A 2 is adjacent to the higher-temperature portion of A 3. By further subdividing A 3, its temperature distribution would be obtained. For ε1 = ε2 = 0.40 and ε3 = 0.45, T4 = 726.8 K and T5 = 740.8 K, compared with T3 = 733.9 K in the previous calculation. The q are changed somewhat to q1 = −4038 W/m2 and q2 = −487.2 W/m2. The q values in this example have all been verified to satisfy overall energy conservation (remembering to multiply the energy fluxes by the respective surface areas for the energy balances – the results given have been rounded off).

FIGURE 5.12  Triangular enclosure for Example 5.14. Areas A4 and A5 are each one-half of A3.

219

Radiation Exchange in Enclosures

5.3.1.2 Solution Method in Terms of Radiosity J An alternative approach for computing the radiative exchange is to solve for J for each surface and then compute q and T. When the surface is viewed with a radiation detector, it is J that is intercepted, that is, the sum of emitted and reflected radiation. Hence, it is sometimes desirable to determine the J as primary quantities. In the previous formulation, the J can be found from the q and T by using Equation (5.18). When the surface temperatures are all specified, the simultaneous equations for J are obtained by eliminating Qk from Equations (5.17) and (5.18). This gives either of the following equations for the kth surface: N

N

å

J k - (1 - e k )



J j Fk - j =

j =1

kj

- (1 - e k )Fk - j ùû J j = e k sTk4 (5.22)

j =1

(1 - e k ) Jk + ek



å éëd N

å(J

k

- J j ) Fk - j = sTk4 . (5.23)

j =1

To illustrate, for a system of three surfaces, Equation (5.22) becomes

éë1 - (1 - e1 )F1-1 ùû J1 - (1 - e1 )F1-2 J 2 - (1 - e1 )F1-3 J 3 = e1sT14 (5.24)



-(1 - e2 )F2-1 J1 + éë1 - (1 - e2 )F2-2 ùû J 2 - (1 - e2 )F2-3 J 3 = e2 sT24 (5.25)



-(1 - e3 )F3-1J1 - (1 - e3 )F3- 2 J 2 + éë1 - (1 - e3 )F3-3 ùû J 3 = e3sT34 . (5.26)

With the T given, the J can be found. Then, if desired, Equation (5.16) can be used to compute q for each surface. When q is specified for some surfaces and T for others, Equation (5.22) is used for the surfaces with known T in conjunction with Equation (5.17) for the surfaces with known q to obtain simultaneous equations for the unknown J. Once J is obtained for a surface, it can be combined with the given q (or T) and Equation (5.16) used to determine the unknown T (or q). In general, if an enclosure has surfaces 1, 2, …, m with specified T, and the remaining surfaces m + 1, m + 2, …, N with specified q, then the system of equations for the J is, from Equations (5.17) and (5.22), N

å éëd



kj

- (1 - e k )Fk - j ùû J j = e k sTk4 1 £ k £ m (5.27)

j =1

N

å éëd



j =1

kj

- Fk - j ùûJ j =

Qk Ak

m+1 £ k £ N . (5.28)

For a black surface with Tk specified, Equation (5.27) gives J k = sTk4 , so the Jk is known and the number of simultaneous equations is reduced by one. The following example demonstrates obtaining a solution first by using Equation (5.21) and then by using Equations (5.27) and (5.28). Example 5.15 A frustum of a cone has its base uniformly heated as in Figure 5.13. The top is maintained at 550 K while the side is perfectly insulated on the outside. Surfaces 1 and 2 are diffuse-gray, while surface

220

Thermal Radiation Heat Transfer

FIGURE 5.13  Enclosure used in Example 5.15. 3 is black. For energy transfer only by radiation, what is the temperature of side 1? How important is the value of ε2? By using the configuration factor for two parallel disks (factor 10 in Appendix C), F3–1 = 0.3249. Then, F3–2 = 1 − F3–1 = 0.6751. From reciprocity, A1F1–3 = A3F3–1 and A 2F2–3 = A3F3–2, so that F1–3 =  0.1444 and F2–3 = 0.1310. Then, F1–2 = 1 − F1–3 = 0.8556. From A1F1–2 = A 2F2–1, F2–1 = 0.3735. Finally, F2–2 = 1 − F2–1 − F2–3 = 0.4955. From Equation (5.21), and by noting that Q2 = 0 and 1 − ε3 = 0, the three equations are





3000 4 = s éT14 - 0.8556T24 - 0.1444 ( 550 ) ù . ë û 0 .6 -3000 ( 0.3735)

1 - 0 .6 4 = s é -0.3735T14 + (1 - 0.4955) T24 - 0.1310 ( 550 ) ù ë û 0 .6

-3000 ( 0.3249 )

1- 0.6 Q3 4 + = s é -0.3249T14 - 0.6751T24 + ( 550 ) ù . û ë A3 0 .6

These can be solved for T1, T2, and Q3. (For this particular example, Q3 can also be obtained from overall energy conservation, Q3 = −Q1.) The result is T1 = 721.6 K (T2 = 667.4 K). Since Q2 = 0, all the terms involving ε2 are zero, so ε2 does not appear in the simultaneous equations, and the emissivity of the insulated surface is not important. Physically, this results from the fact that, for no convection or conduction, all absorbed energy must be re-emitted and hence J2 = G2 independent of ε2. This example is now solved using Equations (5.27) and (5.28). Because T3 is specified and A3 is black, Equation (5.27) gives J3 = sT34 . Equation (5.28) is used at A1 and A 2, since q1 and q2 are specified:



J1 - F1- 2 J2 = F1- 3sT34 + q1 -F2-1J1 + (1- F2- 2 ) J2 = F2- 3sT34 .



221

Radiation Exchange in Enclosures

This yields J1 = 13,370 W/m2 and J2 = 11,250 W/m2. From Equation (5.18), sT14 = J1 + [(1 - e1)/e1 ] q1, and similarly for sT24 . This gives the same temperatures as in the first part of this example.

Example 5.16 The second half of Example 5.14 is now calculated by using Equations (5.27) and (5.28). A1 and A 2 have specified T, while A4 and A5 have specified q. Then, Equations (5.27) and (5.28) yields J1 - (1- e1) [F1- 2 J2 + F1- 4 J4 + F1- 5 J5 ] = e1sT14 J2 - (1- e 2 ) [F2-1J1 + F2- 4 J4 + F2- 5 J5 ] = e 2sT24 -F4 -1J1 - F4 - 2 J2 + J4 - F4 - 5 J5 = q4 -F5-1J1 - F5- 2 J2 - F5- 4 J4 + J5 = q5.



The solution gives J in W/m2: J1 = 3277, J2 = 9741, J4 = 8370, and J5 = 11,048. Then, from Equation (5.16), q1 = éëe1 / (1 - e1 ) ùû sT14 - J1 , and similarly for q2. From Equation (5.18), sT44 = éëe 4 / (1 - e 4 ) ùû q4 + J4, and similarly for T5. This provides the same results as for Example 5.14.

(

)

5.3.2 Enclosure Analysis in Terms of Energy Absorbed at Surface An alternative procedure to the net-radiation method is to examine the energy absorbed at a surface. In the net-radiation method, the energy absorbed at each surface is found from Equation (5.20) as αG = εG for a gray surface. A somewhat different viewpoint is briefly presented here; however, additional discussion may be found in Gebhart (1961, 1976). This formulation yields coefficients that provide the fraction of energy emitted by a surface that is absorbed at another surface after reaching that surface by all possible paths. These coefficients are useful in formulating some thermal analyses. For a typical surface Ak, the net energy loss is the emission from the surface minus the energy that is absorbed by the surface from all incident radiation sources. Let Γjk be the fraction of the emission from Aj that reaches Ak and is absorbed. The Γjk includes all the paths for reaching Ak, that is, the direct path and paths by means of one or multiple reflections. Thus, A j e j sT j4 G jk is the amount of energy emitted by Aj that is absorbed by Ak. An energy balance on Ak then gives N



Qk = Ak e k sTk4 -

åA e sT G j j

4 j

jk

. (5.29)

j =1

The Γkk would generally not be zero, since even for a plane or convex surface, some of the emission from the surface is returned to itself by reflection from other surfaces. Equation (5.29) is written for each surface; this relates each Q to the surface temperatures in the enclosure. The Γ factors must now be found. The Γjk is the fraction of energy emitted by Aj that reaches Ak and is absorbed. The total emitted energy from Aj is A j e j sT j4 . The portion traveling directly to Ak and then absorbed is A j e j sT j4 Fj -k e k , where for a gray surface αk = εk. All other radiation from Aj arriving at Ak will first undergo one reflection. The emission from Aj that arrives at a typical surface An and is then reflected is A j e j sT j4 Fj -n rn . The fraction Γnk reaches Ak and is absorbed. Then all the energy absorbed at Ak that originated by emission from Aj is

(

A j e j sT j4 Fj - k e k + A j e j sT j4 Fj -1r1G1k + A j e j sT j4 Fj - 2r2G 2 k + 

)

+ A j e j sT j4 Fj - k rk G kk +  + A j e j sT j4 Fj - N r N G NK .



222

Thermal Radiation Heat Transfer

Dividing this energy by emission from Aj gives the fraction

G jk = Fj - k e k + Fj -1r1G1k + Fj - 2r2G 2 k + ××× . + Fj - k rk G kk + ××× + Fj - N r N G Nk

This is rearranged to the form

- Fj -1r1G1k - Fj - 2r2G 2 k - ××× + G jk - ××× - Fj - N r N G Nk = Fj - k e k .

By letting j take on all values from 1 to N, the set of equations is

(1 - F1-1r1 ) G1k - F1-2r2G2 k - F1-3r3G3k - ××× - F1- N rN G Nk = F1- k ek - F2 -1r1G1k + (1 - F2 - 2r2 ) G 2 k - F2 -3r3G3k - ××× - F2 - N r N G Nk = F2 - k e k

- F3-1r1G1k - F3- 2r2G 2 k + (1 - F3-3r3 ) G3k - ××× - F3- N r N G Nk = F3- k e k ×

×

×

×

×

×

×

×

(5.30)

- FN -1r1G1k - FN - 2r2G 2 k - FN -3r3G3k - ××× + (1 - FN - N r N ) G Nk = FN - k e k . Equations (5.30) are solved simultaneously for Γ1k, Γ2k, …, ΓNk. This is done for each k where 1 ≤ k ≤ N, and the solution is simplified by the fact that the coefficients of the Γ do not depend on k. The amount of calculation can be reduced by using some relations between the Γ values. Since all emission by a surface is absorbed by all of the surfaces, the sum of the fractions absorbed must be unity: N

åG



kj

= 1. (5.31)

j =1

Gebhart also showed that there is the reciprocity relation e k Ak G kj = e j A j G jk . (5.32)



Equations (5.31) and (5.32) can be substituted into Equation (5.29) to yield Qk = Ak e k sTk4

N

N

N



å

G kj -

j =1

å

Ak e k G kj sT j4 = Ak e k

j =1

åG s (T kj

4 k

)

- T j4 . (5.33)

j =1

Example 5.17 demonstrates using these relations.

5.3.3 Enclosure Analysis by the Use of Transfer Factors The transfer factors F are defined by stating that Q has the form N



Qk = Ak

åF s (T kj

j =1

4 k

- T j4

) (1 £ k £ N ) . (5.34)

223

Radiation Exchange in Enclosures

It is evident by comparing with Equation (5.33) that F kj = e k G kj . (5.35)

Hence, from Equations (5.31) and (5.32),

N

åF



= e k . (5.36)

kj

j =1

Ak F kj = A jF jk . (5.37)



The F can obviously be calculated by first obtaining the Γ. An equivalent approach is to let Tn4 = 1 for the nth surface and T = 0 for all other surfaces. Then, Equations (5.34) and (5.36) yield F kn = -qk ( k ¹ n ) (5.38) F nn = -qn + e n .



For the same conditions, Equation (5.21) yields N



æ dkj

å çè e j =1

N

- Fk - j

j

æ dnj

å çè e



j =1

1- ej ö ÷ q j = - Fk - n ej ø

- Fn - j

j

1- ej ej

( k ¹ n ) . (5.39)

ö ÷ q j = 1 - Fn - n . (5.40) ø

If Equations (5.39) and (5.40) are solved for the q, these are obtained as a function of the F and ε. The F are found from the q by using Equation (5.38); thus, the F are functions of F and ε and do not depend on the T [provided that ε ≠ ε(T)]. Example 5.17 will show how the F values are obtained by using matrix inversion to solve Equations (5.39) and (5.40).

5.3.4 Matrix Inversion for Enclosure Equations When many surfaces are present in an enclosure, a large set of simultaneous equations results such as Equations (5.21), (5.30), or (5.39) and (5.40). The equations can be solved using computation algorithms that accommodate hundreds of simultaneous equations. A set of equations such as Equation (5.21) can be written in a shorter form. Let the right side be Ck and the quantities in parentheses on the left be akj. Then, the k equations can be written as N

åa q



kj

j

= Ck (5.41)

j =1

where

akj =

dkj 1- ej ; Ck = - Fk - j ej ej

N

åF s (T k- j

4 k

)

- T j4 . (5.42)

j =1

For an enclosure of N surfaces, the set of equations then has the form

224

Thermal Radiation Heat Transfer

a11q1 + a12q2 +  + a1 j q j +  + a1N qN = C1 a21q1 + a22q2 +  + a2 j q j +  + a2 N qN = C2











(5.43)

ak1q1 + ak 2q2 +  + akj q j +  + akN qN = Ck 









aN 1q1 + aN 2q2 +  + aNj q j +  + aNN qN = C N . The array of akj coefficients is the matrix of coefficients a:



é a11 ê ê a21 ê  a º éë akj ùû º ê ê ak 1 ê  ê êë aN 1

     

a12 a22  ak 2  aN 2

a1 j a2 j  akj  aNj

     

a1N ù ú a2 N ú  ú ú . (5.44) akN ú  ú ú aNN úû

A method for solving a set of equations such as Equation (5.43) is to obtain a second matrix a−1, called the inverse of matrix a



é A11 ê ê A21 ê  a -1 º éë Akj ùû º ê ê Ak1 ê  ê êë AN 1

     

A12 A22  Ak 2  AN 2

A1 j A2 j  Akj  ANj

     

A1N ù ú A2 N ú  ú ú . (5.45) AkN ú  ú ú ANN úû

The inverse matrix is comprised of Akj elements corresponding to each akj in the original matrix. The Akj elements are found by operating on the a as follows: The kth row and jth column that contain element akj in a square matrix a are deleted, and the determinant of the remaining square array is called the minor of element akj and is denoted by Mkj. The cofactor of akj is defined as (−1)k + jMkj. To obtain the inverse of a square matrix [akj], each element akj is first replaced by its cofactor. The rows and columns of the resulting matrix are then interchanged. The elements of the matrix thus obtained are then each divided by the determinant |akj| of the original matrix [akj]. The elements obtained in this fashion are the Akj. For more detailed information on matrix inversion, the reader should refer to a mathematics text. Most computer mathematics packages such as MATLAB®, Mathcad, COMSOL, and others can also be used to obtain the inverse coefficients Akj from the matrix of akj values. After the inverse matrix is obtained, the qk values in Equation (5.21) are found as the sum of products of A and C: N



qk =

å A C . (5.46) kj

j =1

j

225

Radiation Exchange in Enclosures

(

)

Thus, the solution for each qk is in the form of a weighted sum of Tk4 - T j4 ; this is the same form as Equation (5.34). For a given enclosure, the configuration factors Fk−j remain fixed. If in addition the εk are constant, then the elements akj, and hence the inverse elements Akj, remain fixed for the enclosure. The fact that the Akj remain fixed has utility when it is desired to compute radiation quantities within an enclosure for many different T values. The matrix is inverted only once, and then Equation (5.46) is applied for different values of C. The previous two sections have presented alternative ways of formulating the radiation exchange equations. These are equivalent approaches for obtaining the set of Equations (5.21). For any appreciable number of simultaneous equations, the solution would be found with a computer subroutine or solver. The following example gives the solution to the same problem using three different methods. This illustrates the matrix operations and shows how the various transfer factors are related. Example 5.17 Figure 5.14 is a two-dimensional (2D) rectangular enclosure that is long in the direction normal to the cross-section shown. All the surfaces are diffuse-gray and their temperatures and emissivities are given. For simplicity, the four sides are not subdivided. The configuration factors are obtained by the crossed-string method as F1–3 = F3–1 = 0.2770, F2–4 = F4–2 = 0.5662, F1–2 = F1–4 = F3–2 =  F3–4 = 0.3615, and F4–1 = F4–3 = F2–1 = 0.2169. In what follows, an abbreviated notation is used: E =  (1 − ε)/ε, ϑ = T4, and Fkj = Fk − j.

FIGURE 5.14  Rectangular geometry for Example 5.17.

226

Thermal Radiation Heat Transfer The four equations from Equation (5.21) become



q1 - q2F12E2 - q3F13E3 - q4F14E4 = s ( J1 - F12J2 - F13J3 - F14J4 ) e1 q -q1F21E1 + 2 - q3F23E3 - q4F24E4 = s ( -F21J1 + J2 - F23J3 - F24J4 ) e2 q -q1F31E1 - q2F32E2 + 3 - q4F34E4 = s ( -F31J1 - F32J2 + J3 - F34J4 ) e3 q -q1F41E1 - q2F42E2 - q3F43E3 + 4 = s ( -F41J1 - F42J2 - F44J3 + J4 ) . e4

All quantities except the q are known. Substitution and solution by a computer software package give q1 = −2876.52, q2 = 1612.53, q3 = 1508.92, q4 = −791.97 W/m2. Now, the method in Section 5.3.2 is used. In Equation (5.30), it is noted that the Γjk have the same matrix of coefficients for all k. There are four sets of four equations, one set for each k, and within each set j has values 1, 2, 3, and 4. The matrix of the Γ coefficients is (note that the Fkk = 0 for this example)



-r2F12 1

é 1 ê -r F 1 21 m=ê ê -r1F31 ê êë -r1F41

-r2F32 -r2F42

-r3F13 -r3F23 1

-r4F14 ù -r4F24 úú . -r4F34 ú ú 1 úû

-r3F43

When the values are substituted into this matrix, it becomes



1 é ê ê -0.06507 m=ê ê -0.08310 ê ê -0.06507 ë

- 0.25306

- 0.04155

1

- 0.03254

- 0.25306 - 0.39633

- 0.19883 ù ú - 0.31140 ú ú. - 0.19883ú ú ú 1 û

1 - 0.03254

The matrix of the four columns of values on the right-hand sides of the four sets of equations is, in symbolic and numerical form,





éF11e1 êF e 21 1 f=ê êF31e1 ê êëF41e1 0 é ê 0.15183 f=ê ê0.19389 ê êë 0.15183

F12e 2 F22e 2 F32e 2 F42e 2

F13e3 F23e3 F33e3 F43e3

F14 e 4 ù F24 e 4 úú F34 e 4 ú ú F44 e 4 úû

0.10845

0.23544

0 0.10845

0.18437

0.16986

0 0.18437

0.16268ù 0.25479úú . 0.16268ú ú 0 úû

The matrix of Γkj factors is obtained by matrix inversion as [Γkj ] = m−1f:



é0.13233 ê0.25579 G=ê ê0.32377 ê êë0.27236

0.18271 0.08738

0.39315 0.32299

0.19000 0.222.56

0.18278 0.34391

0.29181 ù 0.33384 úú . 0.30345ú ú 0.16117 úû

227

Radiation Exchange in Enclosures

By summing values in each row, it is evident that

å

4 j=1

G kj = 1. The q values are then found from

Equation (5.33), and they agree with those given previously. The solution is now obtained by the method in Section 5.3.3. The coefficient matrix for the set of Equations (5.39) and (5.40) is



é 1/e1 ê -r 1 ( F21 /e1 ) D=ê ê -r1 (F31 /e1 ) ê êë -r1 (F41 /e1 )

-r2 (F12 /e 2 ) 1/e 2 -r2 (F32 /e 2 ) -r2 (F42 /e 2 )

-r3 (F13 /e3 ) -r3 (F23 /e3 ) 1/e3 -r3 (F43 /e3 )

- 0.84352

- 0.04888

3.33333

- 0.03828

- 0.84352

1.17647

- 1.32111

- 0.03828

-r4 (F14 /e 4 ) ù ú -r4 (F24 /e 4 ) ú -r4 (F34 /e 4 ) ú ú 1/e 4 úû

and in numerical form, this gives



é 1.42857 ê ê -0.09296 D=ê ê -0.11871 ê ê -0.09296 ë

- 0.44184 ù ú - 0.69201 ú ú. - 0.44184 ú ú 2.22222úû

The matrix for the right-side coefficients of Equations (5.39) and (5.40) to determine −q is (note that Fnm = 0 for this example): é -1 ê F21 M=ê êF31 ê êëF41



To relate the −q and F

F12 -1 F32 F42

F14 ù ú F24 ú . F34 ú ú -1úû

F13 F23 -1 F43

from Equation (5.38), the emissivity matrix is ée1 ê 0 e=ê ê0 ê êë 0



0 e2 0 0

0 0 e3 0

0ù ú 0ú . 0ú ú e 4 úû

Then, according to Equation (5.38), the F factors are obtained from the matrix operations F = D-1M + e.

This yields the matrix of F



values:

é 0.09263 ê0.07674 F =ê ê0.27520 ê êë0.12256

0.12790 0.02621

0.27520 0.09690

0.16150 0.10015

0.15537 0.15476

0.20427 ù 0.10015úú . 0.25793ú ú 0.07253úû

From these values, it is evident that there is the relation Fkj = εk Γkj as in Equation (5.35). The q values are computed from Equation (5.34) and are the same as before.

Another procedure proposed for enclosure analysis, which was pointed out in Section 4.3.5, is to use the discrete ordinates method (DOM) where the geometric aspects are included by following rays of radiation. The rays are in a specified number of angular directions from each specified finite area

228

Thermal Radiation Heat Transfer

element. This method is described in Section 13.3 for enclosures that contain a gas such as carbon dioxide that is not transparent and hence interacts with radiation. By omitting the radiative interaction with the gas, discrete ordinates can be used for an enclosure that is evacuated or contains a transparent medium. This has been examined by Sanchez and Smith (1992) and may be utilized as further developments are made on the DOM. Possible applications would be for complex enclosures that have internal obstructions and shadowing of surfaces. Results are obtained by Sanchez and Smith for a 2D square enclosure with either an internal obstruction or two rectangular protrusions on one wall. The boundary temperatures were all specified and radiative heat fluxes were computed. As demonstrated by comparisons with the net-radiation method described in this chapter, good results can be obtained if enough area elements and angular divisions are used for the discrete ordinate integrations.

5.4 RADIATION ANALYSIS USING INFINITESIMAL AREAS 5.4.1 Generalized Net-Radiation Method Using Infinitesimal Areas In the previous section, the enclosure was divided into finite areas. The accuracy of the results is limited by the assumptions that the temperature and the energy incident on and leaving each surface are uniform over that surface. If these quantities are nonuniform over part of the enclosure boundary, that part must be subdivided until the variation over each area in the analysis is not too large. Several calculations can be made in which successively smaller areas (and hence more simultaneous equations) are used until the results no longer change significantly when the area sizes are further diminished. In the limit, the enclosure boundary, or a portion of it, is divided into infinitesimal parts; this allows large variations in T, q, G, and J to be accounted for in calculations. The formulation in terms of infinitesimal areas leads to energy balances in the form of integral equations. By using exact or approximate mathematical techniques for integral equations, it may be possible to obtain an analytical solution. Usually, this is not possible, and the integral equations are solved numerically. Analytical solution methods for integral equations are addressed in the online Appendix I.2 at www.ThermalRadiation.net. Consider an enclosure of N finite areas. The areas would usually be the major geometric divisions of the enclosure or the areas on which a specified boundary condition is held constant. Some or all of these areas are further subdivided into differential area elements as in Figure 5.15. The surfaces are diffuse-gray, and for simplicity, the additional restriction is made that the radiative properties are independent of temperature. An energy balance on element dAk at rk gives

qk ( rk ) = J k ( rk ) - Gk ( rk ) . (5.47)

The outgoing flux is composed of emitted and reflected energy:

J k ( rk ) = e k sTk4 ( rk ) + (1 - e k ) Gk ( rk ) . (5.48)

The incoming flux in Equation (5.48) is composed of portions of the outgoing fluxes from the other area elements of the enclosure. This is a generalization of Equation (5.12) in that an integration is performed over each finite surface to determine the contribution that the local flux leaving that surface makes to Gk:

ò

dAk J k ( rk ) = J1 ( r1 ) dFd1- dk ( r1, rk ) dA1 +  A1



ò ( )

( )

+ Gk rk* dFdk* - dk rk*, rk dA* k + Ak

+

òJ

AN

N

( rN ) dFdN - dk ( rN , rk ) dAN .

(5.49)

229

Radiation Exchange in Enclosures

FIGURE 5.15  Enclosure composed of N discrete surface areas with areas subdivided into infinitesimal elements.

The second integral is the contribution that other differential elements dAk* on surface Ak make to the incident energy at dAk. By using reciprocity, a typical integral can be transformed to obtain

ò J ( r ) dF j

Aj

j

dj - dk

( rj , rk ) dAj = dAk ò J j ( rj ) dFdk - dj ( rj , rk ). Aj

By operating on all the integrals in Equation (5.49) in this manner, the result becomes

Gk ( rk ) =

N

å ò J ( r ) dF j

j

dk - dj

( rj ,rk ). (5.50)

j =1 A j

Equations (5.48) and (5.50) provide two expressions for Gk(rk). These are each substituted into Equation (5.47) to provide two expressions for qk(rk) comparable to Equations (5.16) and (5.17):

qk ( rk ) =

ek ésTk4 ( rk ) - J k ( rk ) ù (5.51) û 1 - ek ë

230



Thermal Radiation Heat Transfer

qk ( rk ) = J k ( rk ) -

N

å ò J ( r ) dF j

j

dk - dj

( rj , rk ). (5.52)

j =1 A j

As shown by Equation (4.9), the differential configuration factor dFdk−dj contains the differential area dAj. To place Equation (5.52) in a form where the variable of integration is explicitly shown, it is convenient to define a quantity K(rj, rk) by K ( rj , rk ) º



dFdk - dj ( rj , rk ) dA j

. (5.53)

Then, Equation (5.52) becomes the integral equation

qk ( rk ) = J k ( rk ) -

N

å ò J ( r ) K ( r , r ) dA . (5.54) j

j

j

k

j

j =1 A j

The K(rj, rk) is the kernel of the integral equation. As in the previous discussion for finite areas, there are two paths that can be followed: (1) When the temperatures and imposed heat fluxes are important, Equations (5.51) and (5.52) can be combined to eliminate the J. This gives a set of simultaneous relations relating the surface temperatures T and the heat fluxes q, and the problem is determinate if half of the total T’=s and qs are specified. Along each surface area, T or q can be specified, and the unknown T and q are found by solving the simultaneous relations. (2) Alternatively, when J is an important quantity, the unknown q can be eliminated by combining Equations (5.51) and (5.52) for each surface that does not have its q specified. For a surface where q is known, Equation (5.52) can be used to directly relate the Js to each other. This yields a set of simultaneous relations for the J in terms of the specified q and T. After solving for the J, Equations (5.51) can be used, if desired, to relate the q and T, where either the q or the T is known at each surface from the boundary conditions. Each of these procedures is now examined. 5.4.1.1 Relations between Surface Temperature T and Surface Heat Flux q To eliminate the J in the first solution method, Equation (5.51) is solved for Jk(rk), giving J k ( rk ) = sTk4 ( rk ) -



1 - ek qk ( rk ) . (5.55) ek

Equation (5.55) in the form shown and with k changed to j is then substituted into Equation (5.52) to eliminate Jk and Jj, which yields qk ( rk ) ek

N

å j =1

1- ej q j ( rj ) dFdk - dj ( rj , rk ) ej

ò

Aj

= sTk4 ( rk ) -

N

å òsT

4 j

( rj ) dFdk - dj ( rj , rk )

j =1 A j

N

=

å òs éëT

4 k

j =1 A j

( rk ) - T j4 ( rj )ùû dFdk - dj ( rj , rk ) .

(5.56)

231

Radiation Exchange in Enclosures

Equation (5.56) relates the surface temperatures to the heat fluxes supplied to the surfaces. It corresponds to Equation (5.21) in the formulation for finite uniform surfaces. Example 5.18 An enclosure of the general type in Figure 5.9 consists of three plane surfaces. Surface 1 is heated uniformly, and surface 2 has a uniform temperature. Surface 3 is black and at T3 ≈ 0. What are the equations needed to determine the temperature distribution along surface 1? With T3 = 0, ε3 = 1, and the self-view factors dFdj−dj = 0, Equation (5.56) is written for the two plane surfaces 1 and 2 having uniform q1 and T2: q1 1 - e 2 q2 ( r2 ) dFd1-d 2 ( r2 , r1 ) = sT14 ( r1 ) - sT24 dFd1-d 2 ( r2 , r1 ) (5.18.1) e1 e2

ò



ò

A2

A2

q2 ( r2 ) 1- e1 - q1 dFd 2- d1 ( r1, r2 ) = sT24 - sT14 ( r1 ) dFd 2- d1 ( r1, r2 ) . (5.18.2) e1 e2

ò



ò

A1

A1

An equation for surface 3 is not needed, since Equations (5.18.1) and (5.18.2) do not involve the unknown q3(r3) as a consequence of ε3 = 1 and T3 = 0. From the definitions of F factors,

ò

A2

dFd1-d 2 = Fd1-2 and

ò

A1

dFd 2-d 1 = Fd 2-1. Equations (5.18.1) and (5.18.2) simplify to the following

relations where the unknowns are on the left:

sT14 ( r1 ) +

q 1- e2 q2 ( r2 ) dFd1-d 2 ( r2 , r1 ) = sT24Fd1-2 ( r1 ) + 1 (5.18.3) e2 e1

ò

A2



òsT

4 1

A1

( r1) dFd 2-d1 ( r1, r2 ) +

q2 ( r2 ) 1- e1 = sT24 + q1 Fd 2-1 ( r2 ) . (5.18.4) e1 e2

Equations (5.18.3) and (5.18.4) can be solved simultaneously for the distributions T1(r1) and q2(r2). Some solution methods are in Section (5.4.2) for these types of simultaneous integral equations.

5.4.1.2 Solution Method in Terms of Outgoing Radiative Flux J Another method results from eliminating the qk(rk) from Equations (5.51) and (5.52) for the surfaces where qk(rk) is unknown. This provides a relation between J and the T specified along a surface:

J k ( rk ) = e k sTk4 ( rk ) + (1 - e k )

N

å òJ ( r ) dF j

j

dk - dj

( rj , rk ) . (5.57)

j =1 A j

When qk(rk), the energy rate supplied to surface k, is known, Equation (5.52) can be used directly to relate qk and J. The combination of Equations (5.52) and (5.57) thus provides a complete set of relations for the unknown J in terms of known T and q. This set of equations for the J is now formulated more explicitly. In general, an enclosure can have surfaces 1, 2, …, m with specified temperature distributions; for these surfaces, Equation (5.57) is used. The remaining N–m surfaces m + 1, m + 2, …, N have an imposed energy flux distribution specified; for these surfaces, Equation (5.52) is applied. This results in N equations for the unknown J distributions:

232



Thermal Radiation Heat Transfer

J k ( rk ) - (1 - e k )

N

å òJ ( r ) dF j

dk - dj

j

( rj , rk ) = ek sTk4 ( rk )

1 £ k £ m. (5.58)

j =1 A j

J k ( rk ) -



N

å ò J ( r ) dF j

j

dk - dj

( rj , rk ) = qk ( rk )

m + 1 £ k £ N . (5.59)

j =1 A j

After the J is found from these simultaneous integral equations, Equation (5.51) is applied to determine the unknown q or T distributions: ek ésTk4 ( rk ) - J k ( rk ) ù 1 £ k £ m (5.60) û 1 - ek ë



qk ( rk ) =



sTk4 ( rk ) =

1 - ek qk ( rk ) + J k ( rk ) m + 1 £ k £ N . (5.61) ek

5.4.1.3 Special Case When Imposed Heating q Is Specified for All Surfaces An interesting special case is when the imposed energy flux q is specified for all surfaces of the enclosure except one and it is desired to determine the surface temperature distributions. Note that all but one q can be specified independently, since å k qk Ak = 0 for steady state; however, at least one enclosure surface temperature must be given, or the temperatures become indeterminate. In many cases, the environmental temperature provides this anchor. For this case, the use of the method of the previous section where the J is first determined has an advantage over the method given by Equation (5.56) where the T is directly determined from the specified q. This advantage arises from Equation (5.59) being independent of the radiative surface properties. For a given set of q, the J need only be determined once from Equation (5.59). Then the temperature distributions are found from Equation (5.61), which introduces the emissivity dependence. This has an advantage when it is desired to examine temperature variations for various emissivity values when there is a fixed set of q. A useful relation is obtained by first considering the case in which the surfaces are all black, εk = 1. Equation (5.61) shows that J k ( rk ) = sTk4 ( rk )black . Because the Jk are independent of the emissivities, the solution in Equation (5.61) can then be written for εk ≠ 1 as sTk4 ( rk ) =



1 - ek qk ( rk ) + sTk4 ( rk )black . (5.62) ek

This relates the temperature distributions in an enclosure for εk ≠ 1 to the temperature distributions in a black enclosure having the same imposed heat fluxes, qk(rk). Thus, once the temperature distributions have been found for the black enclosure, the sTk4 ( rk ) for gray surfaces are found by simply adding [(1 − εk)/εk]qk(rk). Example 5.19 A heated enclosure is the circular tube in Figure 5.16, open at both ends and insulated on the outside surface (Usiskin and Siegel 1960). For a uniform heat addition (such as by electrical heating in the tube wall) to the inside surface of the tube wall and a surrounding environment at 0 K, what is the temperature distribution along the tube? If the surroundings are at Te, how does this influence the temperature distribution? Since the open ends of the tube are nonreflecting, they can be assumed to act as black disks at the surrounding temperature 0 K. Then with ε1 = ε3 = 1, Equation (5.61) gives J1 =  J3 = sT14 = sT34 = 0 .

233

Radiation Exchange in Enclosures

FIGURE 5.16  Uniformly heated tube insulated on the outside and open to the environment at both ends, (a) geometry and coordinate system, and (b) distribution of J on inside of the tube for L/D = 4. Consequently, the summation in Equation (5.59) provides only radiation from surface 2 to itself. Since the tube is axisymmetric, the two differential areas dAk and dAk* can be rings located at x and y. Then, Equation (5.59) yields

J2 ( x ) -

h= l

ò J (h) dF 2

d x -d h

( h - x ) = q . (5.19.1) 2

h=0

where ξ = x/D, η = y/D, l = L/D, and dFdξ − dη(|η − ξ|) is the configuration factor for two rings a distance |η − ξ| apart and is given by factor 15 in Appendix C as



3 3 ì ü ï h-x + 2 h-x ï dFd x - d h ( h - x ) = í1d h. (5.19.2) 32 ý é( h - x )2 + 1ù ï ï ë û þ î

234

Thermal Radiation Heat Transfer

Absolute-value signs are used because the configuration factor depends only on the magnitude of the distance between the rings. When |η − ξ| = 0, dF = dη, and this is the configuration factor from a differential ring to itself. Equation (5.19.1) can be divided by the constant q2 and the dimensionless quantity J2(ξ)/q2 found by numerical or approximate solution methods for linear integral equations, as discussed in Section 5.4.2. The J2(ξ)/q2 distribution is in Figure 5.16b for a tube four diameters in length. From Equation (5.61), the T24 ( x ) along the tube is given by sT24 ( x ) =



1- e 2 q2 + J2 ( x ) . e2

Since q2 is constant, the T24 ( x ) has the same shape as J2(ξ). The wall temperature is high in the central region of the tube and low near the ends where energy is radiated more readily to the low-temperature environment. Now, consider the environment at Te ≠ 0. The open ends of the tube can be regarded as perfectly absorbing disks at Te, and the integral Equation (5.59) yields J2 ( x ) -



l

ò J (h) dF

d x -d h

2

( h - x ) - sT F ( x ) - sT F ( l - x ) = q 4 e d x -1

4 e d x -3

2

h= 0

where Fdξ

(ξ) is the configuration factor from a ring element at ξ to disk 1 at ξ = 0,

− 1

{(

Fdx-1 ( x ) = x + 2

1 2

) (x

2

)

+1

1/ 2

} - x. Since the integral equation is linear in J (ξ), let a trial solution 2

be the sum of two parts where, for each part, either Te = 0 or q2 = 0: J2 ( x ) = J2 ( x )



Te = 0

+ J2 ( x )

q2 = 0

.

Substitute the trial solution into the integral equation to get J2 ( x )

Te = 0

+ J2 ( x )

l

q2 = 0

l

-



ò J ( h) 2

h=0

q2 = 0

-

ò J ( h) 2

h=0

Te = 0

dFd x - d h ( h - x )

dFd x - d h ( h - x ) - sTe4Fd x -1 ( x )

- sTe4Fd x - 3 ( l - x ) = q2. For Te = 0, Equation (5.19.1) applies; subtract this equation to obtain

J2 ( x )

l

q2 = 0

-

ò J ( h) 2

h=0

q2 = 0

dFd x - d h ( h - x ) - sTe4Fd x -1 ( x ) - sTe4Fd x - 3 ( l - x ) = 0.

The solution is J2 q2 =0 = sTe4 , and this would be expected physically for an unheated surface in a uniform temperature environment. The tube temperature distribution is found from Equation (5.61) as 1- e2 q2 + J2 ( x ) + J2 ( x ) Te =0 q2 =0 e2 1- e2 4 = q2 + J2 ( x ) + sTe Te =0 e2

sT24 ( x ) =

where J2 ( x ) was found in the first part of this example. The environment has thus added sTe4 Te =0 to the solution for sT24 ( x ) found previously for Te = 0.

235

Radiation Exchange in Enclosures

To arrive at a general conclusion for the effect of Te in this type of problem, the result of Example 5.19 is written in the form s éëT24 ( x ) - Te4 ùû =



1 - e2 q2 + J 2 ( x ) . Te = 0 e2

Thus, for nonzero Te, the quantity T24 ( x ) - Te4 equals T24 ( x ) for the case when Te = 0. This illustrates a general way of accounting for a finite environment temperature. For energy transfer only by radiation, the governing equations are linear in T4. As a result, in a cavity, the wall temperature Tw T e=0 can be calculated for a zero-temperature environment. Then, by superposition, the wall temperature for any finite Te is Tw4 = Tw4 + Te4 . Hence, the thermal characteristics of a cavity having a wall Te ¹ 0

Te =0

temperature variation Tw and an external environment at Te are the same as a cavity with wall tem-

(

perature variation Tw4 - Te4

)

1/ 4

and a zero environment temperature.

Example 5.20 Consider emission from a long cylindrical hole drilled into a material at uniform temperature Tw (Figure 5.17a). The hole is long, so the portion of its internal surface at its bottom end can be neglected in the radiative energy balances. The outside environment is at Te. As in the previous discussion, the solution is obtained by using the reduced temperature

(

Tr = Tw4 - Te4

)

1/ 4

. If a position is viewed at x on the cylindrical side wall, the energy leaving is J(x).

FIGURE 5.17  Radiant emission from a cylindrical hole at uniform temperature, (a) geometry and coordinate system, and (b) apparent emissivity of the cylindrical wall.

236

Thermal Radiation Heat Transfer

An apparent emissivity is defined as e a ( x ) = J ( x ) /sTr4 . The analysis determines how εa(x) is related to the surface emissivity ε, where ε is constant over the cylindrical side of the hole. The integral equation governing the radiation exchange within a hole was first derived by Buckley (1928) and later by Eckert (1935); both obtained approximate analytical solutions. Results were evaluated numerically by Sparrow and Albers (1960). Using the reduced temperature, the opening of the hole is approximated by a perfectly absorbing disk at zero reduced temperature. Then, from Equation (5.61) (because ε = 1 and Tr = 0 for the opening area), J = 0 from the opening into the cavity. Hence, the governing equation for the enclosure is Equation (5.58), written for the cylindrical side wall and including only radiation from the cylindrical wall to itself. As in Example 5.19, the configuration factor is for a ring of differential length on the cylindrical enclosure exchanging radiation with a ring at a different axial location. Equation (5.58) then yields along the surface of the hole

J ( x ) - (1 - e )

¥

ò J (h) dF

d x-d h

( h - x ) = esT (5.20.1) 4

h= 0

where ξ = x/D, η = y/D, and dFdξ−dη(|η−ξ|) is in Equation (5.19.2). After division by the constant sTr4 , the apparent emissivity εa(ξ) is governed by the integral equation

e a ( x ) - (1- e )

¥

ò e (h) dF a

d x- d h

( h - x ) = e. (5.20.2)

h=0

The solution of Equation (5.20.2) was carried out for various surface emissivities ε, and the results are in Figure 5.17b. The radiation leaving the surface approaches that of a blackbody, as the wall position is at greater depths into the hole. At the mouth of the hole ea » e , as shown in Buckley (1927, 1928). Radiation from a hole of finite depth was analyzed by Lin and Sparrow (1965) and results are in Figure 5.18. Approximate solutions are in Kholopov (1973). The effective hemispherical emissivity εh in Figure 5.18 is a different quantity from that in Figure 5.17b; it gives the total energy leaving

FIGURE 5.18  Apparent emissivity of cavity opening for a cylindrical cavity of finite length with diffuse reflecting walls at constant temperature. (From Lin, S.H. and Sparrow, E.M., J. Heat Trans., 87(2), 299, 1965).

Radiation Exchange in Enclosures

237

the mouth of the cavity ratioed to that emitted from a black-walled cavity. The latter is the same as the energy emitted from a black area across the mouth of the cavity. For each surface emissivity ε, the εh increases to a limiting value as the cavity depth increases; the limiting values are less than one. Unless ε is small, a cavity more than only a few diameters deep radiates the same amount as an infinitely deep cavity. The cavity opening can be partially closed to provide near-blackbody emission for use in calibrating measuring equipment, as in Figure 5.19 (Alfano 1972, Alfano and Sarno 1975). The apparent emissivity εa at the center of the bottom of the cavity is in Table 5.1 as a function of the diaphragm opening to outer radius ratio and the cavity depth to radius ratio. The εa increases toward one as the cavity is made deeper and the diaphragm opening decreases. Similar trends are shown in Heinisch et al. (1973) for partially baffled conical cavities. In Masuda (1973), cavities between radial fins are analyzed. Very detailed results for radiating cavities are in Bedford (1988). Some information on directional emission from a cylindrical cavity is given in Sparrow and Heinisch (1970) where emission normal to the cavity opening as would be received by a small detector along the cavity centerline and facing the opening is calculated. The normal emissivity εn of the cavity is defined as the energy received by the detector divided by the energy that would be received if the cavity walls were black. The εn depends on the detector distance from the cavity opening; the values in Table 5.2 are for large distances where εn no longer changes with distance. The table compares εn with εh from Figure 5.18 (ε is the emissivity of the cavity walls). The εn is larger than εh except for small values of cavity L/R.

FIGURE 5.19  Cylindrical cavity with annular diaphragm partially covering the opening. (From Alfano, G., Int. J. Heat Mass Trans., 15(12), 2671, 1972; Alfano, G. and Sarno, A., J. Heat Trans., 97(3), 387, 1975).

238

Thermal Radiation Heat Transfer

Example 5.21 What are the integral equations for radiation exchange between two parallel opposed plates finite in 1D and infinite in the other, as in Figure 5.20? Each plate has a specified temperature variation that depends only on the x- or y-coordinate, and the environment is at Te. From the discussion in Example 4.2, the configuration factors between the infinitely long parallel strips dA1 and dA 2 are dFd1- d 2 =

a2 1 1 d ( sinb ) = dy 2 2 é y - x 2 + a2 ù 3 2 ( ) ë û

a2 1 dFd 2- d1 = dx. 2 é y - x 2 + a2 ù 3 2 ( ) ë û



239

Radiation Exchange in Enclosures

FIGURE 5.20  Geometry for radiation between two parallel plates infinitely long in one direction and of finite width, (a) parallel plates of width L and infinite length, and (b) coordinates in cross-section of gap between parallel plates. The distribution of heat flux added to each plate is found by applying Equation (5.56) to each plate. As discussed before Example 5.20, reduced temperatures are used to account for Te. With

(

T1 = Tw41 - Te4

)

1/ 4

(

and T2 = Tw42 - Te4

)

1/ 4

q1 ( x ) 1- e 2 e1 e2



, the governing equations are L/2

ò

q2 ( y )

-L / 2

a2 1 dy 2 é y - x 2 + a2 ù 3 / 2 ) êë( úû

(5.21.1)

L/2

a2 1 = sT14 ( x ) - sT24 ( y ) dy. 2 é y - x 2 + a2 ù 3 / 2 ) -L / 2 êë( úû

ò

q2 ( y ) 1- e1 e2 e1



L/2

a2

1

ò q ( x ) 2 é( y - x ) 1

ëê

-L / 2

= sT24 ( y ) -

L/2

ò sT

4 1

-L / 2

(x)

2

+ a2 ùú û

3/ 2

2

dx

a 1 dx. 2 é y - x 2 + a2 ù 3 / 2 ( ) ëê ûú

(5.21.2)

240

Thermal Radiation Heat Transfer

An alternative formulation using Equation (5.57) yields two equations for J1(x) and J2(y):

J1 ( x ) - (1 - e1 )

L/2

2

-L / 2



J2 ( y ) - (1- e 2 )

a2

1

ò J ( y ) 2 é( y - x )

L/2

ò

ë

J1 ( x )

-L / 2

2

+a ù û 2

3/ 2

dy = e1sT14 ( x ) (5.21.3)

1 a2 dx = J2sT24 ( y ) . (5.21.4) 2 é y - x 2 + a2 ù 3 / 2 ( ) ë û

After the J1(x) and J2(y) are found, the desired q1(x) and q1(y) are obtained from Equation (5.60) as

q1 ( x ) =

e1 ésT14 ( x ) - J1 ( x ) ù (5.21.5) û 1 - e1 ë



q2 ( y ) =

e2 ésT24 ( y ) - J2 ( y ) ù . (5.21.6) û 1- e 2 ë

The use of enclosure theory as described in this chapter is applied to building energy transfer in Ficker (2019).

5.4.2 Boundary Conditions Specifying Inverse Problems Throughout this chapter, it has been required that a boundary condition be prescribed on every surface making up the enclosure boundary and that the radiative properties of each surface be known. If this is done, then there is automatically an equal number of equations and unknowns to allow solution of the radiative exchange problem. However, there are classes of problems where two boundary conditions (perhaps heat flux and temperature) are prescribed on one or more surfaces, and the conditions on one or more other surfaces that allow this prescription are to be determined. These problems are inverse boundary condition problems. Such problems and others where properties or other conditions are to be found may require special treatment and are covered in Chapter 18.

5.5 COMPUTER PROGRAMS FOR ENCLOSURE ANALYSIS So far, we have addressed radiative energy exchange within enclosures having black and/or diffuse-gray surfaces. The surface areas can be of finite or infinitesimal size. Each enclosure surface can have a specified net energy flux added to it by some external means, such as conduction or convection, or it can have a specified surface temperature. Various methods were presented for solving the array of simultaneous linear equations or the linear integral equations that result from the formulation of these interchange problems. Inverse problems were also discussed that considered other specifications of conditions that are more general and are more difficult to deal with. It was pointed out that most practical problems become so complex that numerical techniques are required for solution. To solve these problems, several computer programs have been developed. For example, the program Thermal Synthesizer System (TSS) (Panczak et  al. 1991, Chin et  al. 1992), developed under NASA sponsorship, incorporates solvers for enclosures with a very large number of surfaces as used for satellite and space vehicle design. Some other commercially available computer programs for thermal analysis also incorporate solvers for radiative enclosures in addition to including computation of configuration factors (see Section 4.3.6). Computer subroutines such as matrix solvers and integration subroutines can be used and are available in mathematics software packages.

Radiation Exchange in Enclosures

241

HOMEWORK PROBLEMS (Additional problems available in the online Appendix P at www.ThermalRadiation.net) 5.1

What is the net radiative energy transfer from black surface dA1 to black surface A2?

Answer: dQ1–2 = 2.79 W. 5.2 A person holds her open hand (approximated by a circular disk 8 cm in diameter) 6 cm directly above and parallel to a black heater element in the form of a circular disk 20 cm in diameter (such as an electric range element). The heater element is at 750 K. How much radiant energy from the heater element is incident on the hand? Answer: 54.3 W. 5.3 A black electrically heated rod is in a black vacuum jacket. The rod must dissipate 110 W without exceeding 950 K. Calculate the maximum allowable jacket temperature (neglect end effects).

Answer: 666 K. 5.4 A regular tetrahedron of side length 0.5 m has black internal surfaces with the following characteristics:

What are the values of T1, Q2, Q3, and T4? Answers: T1 = 885 K; Q2 = −3562 W; Q3 = 3162 W; T4 = 902 K.

242

5.5

Thermal Radiation Heat Transfer

A rectangular solar collector resting on the ground has a black upper surface and dimensions 1.5 × 3 m. The lower surface is very well-insulated. The collector is tilted at an angle of 40° from the horizontal. Fluid at 95°C is passed through the collector at night. The night sky acts as a blackbody with an effective temperature of 210 K and the ground around the collector is at 5°C. Neglecting conduction and convection to the surroundings, at night, (a) At what rate is energy lost by the collector (W)? (b) If flow through the collector is stopped, what temperature (K) will the collector attain?

Answers: (a) 4070 W; (b) 222 K. 5.6 (a)  A circular cylindrical enclosure has black interior surfaces, each maintained at uniform interior temperature as shown. The outside of the entire cylinder is insulated so that the outside does not radiate to the surroundings. How much Q (W) is supplied to each area as a result of the interior radiative exchange? (Perform the calculation without subdividing any surface areas.)

(b) For the same enclosure and the same surface temperatures, divide A3 into two equal areas A4 and A5. What is the Q to each of these two areas, and how do they and their sum compare with Q3 from part (a)?

Radiation Exchange in Enclosures

243

(c) What are Q6/A6 and Q7/A7 for the same enclosure and surface temperatures? How do they compare with Q3/A3 from part (a) and Q4/A4 and Q5/A5 from part (b)?

Answers: (a) Q1 = 1295 W; Q2 = −1207 W; Q3 = −87.3 W. (b) Q1 = 1295 W; Q2 = −1207 W; Q4 = −853 W; Q5 = 765 W. (c) Q1 = 1295 W; Q2 = −1207 W; Q6 = −261 W; Q7 = 240 W. 5.7

A 3 cm diameter, very long heater rod at the center of a long box furnace of 10 cm × 10 cm cross-section is heated in vacuum to 1000°C, as shown in the following figure. The top and bottom walls of the box furnace are thermally insulated on the back. The two side walls of the box furnace are covered by metallic panels with the temperature maintained

244

Thermal Radiation Heat Transfer

at 700°C. All surfaces are diffuse and gray. The emissivity is 0.8 for the heater rod, 0.5 for the top and bottom walls of the box furnace, and 0.2 for the two metallic panels. Assuming uniform irradiation and temperature on each surface, find the rate of electrical heating applied to the heater rod and the temperature of the top and bottom walls.

Answers: qheater = 29.7 kW/m2; qwall = −14.0 kW/m2; Ttop = 1200 K (925°C). 5.8 Consider a diffuse-gray right circular cylindrical enclosure. The diameter is the same as the height of the cylinder, 3.00 m. The top is removed. If the remaining internal surfaces are maintained at 900 K and have an emissivity of 0.7, determine the radiative energy escaping through the open end. Assume uniform irradiation on each surface so that subdivision of the base and curved wall is not included. The outside environment is at Te = 0 K. How do the results compare with Figure 5.18? Explain any difference. If Te = 500 K, what percentage reduction occurs in radiative energy loss? Answers: −2.04 × 105 W; 90.5% reduction in energy rate. 5.9 A 2D diffuse-gray enclosure (infinitely long normal to the cross-section shown) has each surface at a uniform temperature. Compute the energy added per meter of enclosure length at each surface to account for the radiative exchange within the enclosure, Q1, Q2, Q3. (Assume for simplicity that it is not necessary to subdivide the three areas.) The conditions are T1 = 1100 K, ε1 = 0.6, T2 = 300 K, ε2 = 0.5, T3 = 800 K, ε3 = 0.7.

Answers: 267,400 W/m; −126,900 W/m; −140,470 W/m.

Radiation Exchange in Enclosures

245

5.10 Two infinite parallel black plates are at temperatures T1 and T2. A thin perforated sheet of gray material of emissivity εS is inserted between the plates. Let SF, the shading factor, be defined as the ratio of the solid area of the perforated sheet to the total area of the sheet, Ag/A. Determine the temperature of the perforated plate and find the ratio of the energy transfer Q1 between the black plates when SF = 1 (i.e., with a single solid gray radiation shield) to the energy transfer with a perforated shield, QSF.

Answer: Tg4 =

T14 + T24 e 1 ; Q1,shield Q1,perforated = s . 2 2 éë1 + ( es SF / 2 ) - SF ùû

5.11 A frustum of a cone has its base heated as shown. The top is held at 800 K while the side is perfectly insulated. All surfaces are diffuse-gray. What is the temperature attained by surface 1 as a result of radiative exchange within the enclosure? (For simplicity, do not subdivide the areas.)

Answer: 1087 K. 5.12 An enclosure has four sides that are all equilateral triangles of the same size (i.e., it is an equilateral tetrahedron). The sides are of length L = 4.5 m and have conditions imposed as follows: Side 1 is black and is at uniform temperature T1 = 1100 K. Side 2 is diffuse-gray and is perfectly insulated on the outside. Side 3 is black and has a uniform heat flux of 8 kW/m2 supplied to it. Side 4 is black and is at T4 = 0 K. Find q1, T2, and T3. For simplicity, do not subdivide the surface areas. Answers: 51.4 kW/m2; 941 K; 972 K.

246

Thermal Radiation Heat Transfer

5.13 A rod 2 cm in diameter and 20 cm long is at temperature T1 = 1200 K and has a hemispherical total emissivity of ε1 = 0.30. It is within a thin-walled concentric cylinder of the same length having a diameter of 8 cm. The emissivity on the inside of the cylinder is ε2 = 0.21, and on the outside is εo = 0.15. All surfaces are diffuse-gray. The entire assembly is suspended in a large vacuum chamber at Te = 300 K. What is the temperature T2 of the cylindrical shell? For simplicity, do not subdivide the surface areas. (Hint: F2–1 = 0.225, F2–2 = 0.617.)

Answer: 722 K. 5.14 Three parallel plates of finite width are shown in cross-section. The plates are maintained at T1 = 700 K and T2 = 400 K. The surroundings are at Te = 300 K. The plates are very long in the direction normal to the cross-section shown. What are the values of q1 and q2? All plate surfaces are diffuse-gray. (For simplicity, do not subdivide the plate areas.)

Answers: q1 = 5941.2 W/m2; q2 = 97.5 W/m2.

Radiation Exchange in Enclosures

247

5.15 Two infinitely long, directly opposed, parallel diffuse-gray plates of width W have the same uniform heat flux q supplied to them. The environment is at temperature Te. Set up the governing equation for determining the temperature distribution T(x) on the bottom plate. There is no energy loss from the top side of the upper plate or from the bottom side of the lower plate.

5.16 For the geometry specified in the preceding problem, solve for the temperature distribution T(x) = T(y) on either plate if q = 3000 W/m2, ε = 0.25, H = 0.25 m, and W = 1 m. A numerical solution is required. Present your results graphically. The environment temperature is Te = 300 K. Answer: Peak T(x/W = ½) of 796 K and a temperature at the plate edge of 712 K. 5.17 Two diffusely emitting and reflecting parallel plates are of infinite length into and out of the cross-section shown in the following figure. The lower plate has uniform temperature T1 = 1000 K, and the upper plate has uniform temperature T2 = 500 K. The surroundings have a temperature of Te = 300 K. Surface 1 has gray emissivity ε1 = 0.8 and surface 2 has gray emissivity ε2 = 0.2.

(a) Find the average net radiative heat fluxes q1 and q2 on surfaces 1 and 2 assuming uniform irradiation over each surface. (b) Partition each of the two parallel plates into four equal segments (upper: 1, 2, 3, 4; lower: 5, 6, 7, 8). Find the net radiative heat flux for each segment, assuming uniform irradiation on each segment individually. Compare the results with those of part (a). (c) Do not assume uniform irradiation on either of the parallel plates and find the distribution of net radiative heat flux q1(x1) and q2(x2) on each surface. Plot these results and the results of parts (a) and (b) on the same graph. Answers: (a)  30,390 W/m, −5,970 W/m; (b) q1 = q3 = 32,190 W/m, q2 = 28,120 W/m, q5 = q7 = −5,200 W/m, q6 = −6,770 W/m.

248

Thermal Radiation Heat Transfer

5.18 Two plates are joined at 90° and are very long normal to the cross-section shown. The vertical plate (plate 1) is heated uniformly with a heat flux of 300 W/m2. The horizontal plate has a uniform temperature of 400 K. Both plates have an emissivity of 0.6. The environment is at Te = 300 K. The dimensions are shown on the figure.

Derive the equations necessary for finding the temperature distribution on surface 1 and the net radiative heat flux on surface 2. 5.19 A three-surface enclosure has the properties shown in the following. Assuming each surface can be treated as having uniform radiosity, find the unknown temperatures and heat fluxes.

Answers: T2 = 1116 K; q1 = 37.5 kW/m2; q3 = −25.0 kW/m2. 5.20 A bakery oven will burn biomass for heat to produce organic nonfat sugarless gluten-free whole-grain cookies. The geometry and conditions are shown in the following figure. The oven is quite long in the dimension into the paper.

Radiation Exchange in Enclosures

249

(a) What radiative heat flux q1 must the biomass fuel produce on the heated wall at T1 = 530 K if the cookie tray is to be heated to 425 K? (b) What will the temperature of the sidewalls (T3) be under these conditions? Answers: (a) 1895 W/m; (b) 480 K. 5.21 For the geometry shown with non-conducting walls, (a) Provide the final governing equations necessary for finding T1(x1) and q2(x2) in two forms (1) explicit in the required variables and (2) in nondimensional form using appropriate nondimensional variables. (b) Find the temperature distribution T1(x1) and the heat flux distribution q2(x2) and show them on appropriate graphs. Boundary conditions are q1 ( x1 ) = éë100 x1 - 50 x12 ùû ( kW/m) where x1 is in meters, T 2 = 475 K, and T3 = T4 = 300 K. Properties for gray (or black) diffuse surfaces are shown in the figure.

Show that your solution is grid independent and meets overall energy conservation. Answers: T1,max = 1117 K; q2,max = 10.4 kW/m.

6

Exchange of Thermal Radiation among Nondiffuse Nongray Surfaces Joseph Thomas Gier (1910–1961) completed his undergraduate degree in Mechanical Engineering at UC Berkeley in 1933, followed by a Masters of Engineering in 1940. After serving as a lecturer and researcher, Gier was promoted to Associate Professor of Electrical Engineering at UC Berkeley in 1951, becoming the first tenured African-American professor in the University of California system. He formed a fruitful partnership with Robert V. Dunkle, a professor in the Mechanical Engineering department, in 1943. Together they developed instrumentation to characterize the radiative properties of surfaces and conceived of spectralselectivity to improve the performance of solar collectors.

6.1 INTRODUCTION In Chapter 5, we presented an analysis and solution procedure for radiation exchange within enclosures made up of black or gray diffuse boundary surfaces. An additional restriction was that the radiative properties were independent of temperature. However, as discussed in Chapter 3, most materials deviate from the idealizations of being black, gray, diffuse, or having temperature-independent radiative properties. The assumption of idealized surfaces is made to simplify the computations. This is often reasonable because the radiative properties may not be known to high accuracy, especially their detailed dependence on wavelength and direction. Detailed computations are not meaningful if only very approximate property data are available. Also, the many reflections and rereflections in an enclosure tend to average out radiative nonuniformities, for example, emitted plus reflected radiation leaving a directionally emitting surface may be diffuse if it is chiefly composed of reflected energy from radiation incident from many directions. In some applications more precision is required, and the gray and/or diffuse assumption cannot be trusted to be sufficiently accurate. It is then necessary to carry out exchange computations using as exact a procedure as reasonably possible. To provide techniques for more accurate computations, some methods are examined here for radiative exchange between realistic surfaces. Such analyses are inherently more difficult than for ideal surfaces, and a treatment of real surfaces including all types of variations, while possible in principle, is not usually attempted or justified; however, rapidly increasing computational capacity and speed are making such analyses more feasible. For a detailed solution, the directional spectral properties must be available. Property variations with wavelength for the normal direction are available for some materials, as given by the references in Chapter 3, but data are usually sparse at short (λ  ~15 μm) wavelengths and for other than the normal direction. For many materials there is little detailed information available. Certain solutions obviously must include spectral property variations, such as for spectrally selective coatings used for temperature control of space vehicles exposed to solar radiation and for solar concentrating power systems (see Chapter 19). Directional variations for some materials with optically smooth surfaces can be computed using electromagnetic theory, but real materials can deviate from ideal behavior.

251

252

Thermal Radiation Heat Transfer

As noted in Chapter 2, both directional and spectral surface properties can be interpreted in terms of probabilities that radiation will be emitted, absorbed, or reflected, including directional and spectral variations. This means that analysis based on a probabilistic interpretation can be carried out for the cases examined in this chapter. This is the basis of the Monte Carlo method presented in Chapter 14, and it can be used to treat these problems in detail.

6.2 ENCLOSURE THEORY FOR DIFFUSE NONGRAY SURFACES By considering diffusely emitting and reflecting surfaces, directional surface effects are eliminated. By assuming a surface is diffuse, spectral effects can be separated from directional ones and one can understand how spectral property variations need to be accounted for. Under the diffuse assumption, emissivity, absorptivity, and reflectivity are independent of direction, but properties must be available as functions of λ and T to evaluate the radiative exchange. For diffuse spectral surfaces, configuration factors are valid because they involve only geometry and are for diffuse radiation leaving a surface. The energy-balance equations and methods in Chapter 5 remain valid if they are written for energy in each wavelength interval dλ. Usually, however, the boundary conditions involve total (including all wavelengths) energy. Total energy boundary conditions cannot be applied to the spectral energies. To illustrate this, consider the surface of Figure 6.1 with locally incident radiation heat flux G, and total radiosity J leaving by combined emission and reflection. If the surface is perfectly insulated, the G and J must be equal, J - G = q = 0 (6.1)



However, for an insulated surface with q = 0, in each dλ the incident and outgoing spectral fluxes are not generally equal, so that J l d l - Gl d l = ql d l ¹ 0. (6.2)



An adiabatic surface only has a total radiation gain of zero, or, with Equation (6.1) restated in terms of spectral quantities, ¥



q=

¥

ò q dl = ò ( J l

l =0

l

- Gl ) d l = 0. (6.3)

l =0

The qλdλ is the net energy flux supplied in dλ at λ as a result of the radiative exchange process. For an adiabatic surface, the qλdλ can vary substantially with λ, depending on the spectral property variations and the spectral distribution of incident energy.

FIGURE 6.1  Spectral (or total) energy fluxes at a surface element.

253

Radiation among Nondiffuse Nongray Surfaces

Now consider a nonadiabatic surface where a total energy flux q is supplied by some means such as heat conduction or convection. Then, from Figure 6.1, ¥

ò q dl = ò ( J

q=



¥

l

l =0

l

- Gl ) d l. (6.4)

l =0

The q may be specified or may be a quantity to be determined, if the surface is at a specified temperature. In any dλ the net spectral energy (Jλ − Gλ)dλ is unknown and may be positive or negative. The usual boundary conditions state only that the integral of all such spectral energy values must locally equal the total q.

6.2.1 Parallel-Plate Geometry To build familiarity while avoiding geometric complexity, these concepts are applied for a geometry of two infinite parallel plates. Then relations for a general geometry are given. Example 6.1 Two infinite parallel plates of tungsten at specified temperatures T1 and T2 (T1 > T2) are exchanging radiation. Branstetter (1961) determined the temperature-dependent hemispherical spectral emissivity of tungsten by using electromagnetic theory relations to extrapolate limited experimental data, and some of his results are in Figure 6.2. Using these data, compare the net energy exchange between the tungsten plates with that for gray parallel plates using total emissivities. The solution for gray plates is in Example 5.6, and the present analysis follows that example by writing the equations spectrally and solving for the desired quantities. The spectral result is Equation (5.6.3) written for a wavelength interval dλ,

ql ,1d l = -ql ,2d l =

Elb,1 (T1 ) - Elb,2 (T2 ) d l. (6.1.1) 1 1 -1 + el,1 (T1 ) el ,2 (T2 )

Note that for each dλ the overall spectral energy balance is zero, that is, (qλ,1 + qλ,2)dλ = (qλ,1 − qλ,1)dλ = 0. The total heat flux exchanged (supplied to surface 1 and removed from surface 2) is found by substituting the property data of Figure 6.2 into Equation (6.1.1) and integrating over all wavelengths: ¥



q1 = -q2 =

ò

¥

ql ,1d l =

l =0

ò

l =0

Elb,1 (T1 ) - Elb,2 (T2 ) d l. (6.1.2) 1 1 + -1 el,1 (T1 ) el ,2 (T2 )

The integration is performed numerically for each set of specified plate temperatures T1 and T2. Additional results from Branstetter are shown in Figure 6.3, where the ratio of gray to nongray exchange is given. The gray results were obtained using Equation (5.6.3) with total emissivities computed from the spectral emissivities of Figure 6.2. In the gray computation, the emissivity of the colder surface 2 was inserted at the mean temperature TT 1 2 rather than at T2, which is a modification based on electromagnetic theory that is sometimes recommended for metals (Eckert and Drake 1959). Over the range of surface temperatures shown, the gray results deviate up to 25% below the nongray exchange.

254

Thermal Radiation Heat Transfer

FIGURE 6.2  Hemispherical spectral emissivity of tungsten. (From Branstetter, J. R.: Radiant Heat Transfer between Nongray Parallel Plates of Tungsten, NASA TN D-1088, Washington, DC, 1961.)

FIGURE 6.3  Comparison of effect of gray and nongray surfaces on computed energy exchange between infinite tungsten plates. (From Branstetter, J. R., Radiant heat transfer between nongray parallel plates of tungsten, NASA TN D-1088, Washington, DC, 1961).

The following example illustrates how Equation (6.1.2) can be expressed more simply if the spectral properties can be approximated by constant values in a stepwise fashion over a few spectral regions (spectral bands). The evaluation is then done in a few spectral bands that each extend over a wavelength range; this may be useful for making quick estimates of spectral effects. More generally, the evaluation with variable properties is done by computer integration using numerical integration software.

255

Radiation among Nondiffuse Nongray Surfaces

Example 6.2 Two infinite parallel plates and their approximate spectral emissivities at their respective temperatures are in Figure 6.4. What is the heat flux q transferred across the space between the plates? From Equation (6.1.2), q1 =

¥

Elb,1 (T1 ) - Elb,2 (T2 ) Elb,1 (T1 ) - Elb,2 (T2 ) Elb,1 (T1 ) - Elb,2 (T2 ) dl + dl + d l. 1 1 1 1 1 1 + -1 + -1 + -1 l =3 l =5 l =0 0 .4 0 .7 0 .8 0 .7 0 .8 0 .3 5

3



ò

ò

ò

This can be directly evaluated by numerical integration, or it can be written over bands as ¥ 5 é 0.341 3 ù 0.279 0.596 q1 = sT14 ê l + l + E T d E T d Elb,1 (T1 ) d l ú ( ) ( ) l b , l b , 1 1 1 1 4 4 4 sT1 sT1 ê sT1 ú l =0 l =3 l =5 ë û

ò



ò

ò

¥ 5 é 0.341 3 ù 0.596 0.279 -sT24 ê E T d l + E T d l + Elb,2 (T2 ) d l ú . ( ) ( ) b , l b , l 2 2 2 2 4 4 4 sT2 sT2 ê sT2 ú l =0 l =3 l =5 ë û

ò

(

An integral such as 1 / sT14

ò

)ò

5

l =3



ò

Elb,1 (T1 ) d l is the fraction of blackbody radiation at T1 between

λ = 3 and 5 μm, which is F3T1®5T1 = F5040®8400 , which can be readily evaluated from Equation (1.36) or (1.37). This yields



q = sT14 ( 0.341F0®3T1 + 0.596F3T1®5T1 + 0.279F5T1®¥ ) - sT24 ( 0.341F0®3T2 + 0.596F3T2 ®5T2 + 0.279F5T2 ®¥ ) = 140, 500 W/m2 .



The examples illustrate the additional complication of including spectrally dependent surface properties. The enclosure calculations must be carried out in wavelength intervals, and the total

FIGURE 6.4  Example of radiative transfer between infinite parallel plates having spectrally dependent emissivities approximated in spectral bands.

256

Thermal Radiation Heat Transfer

energy quantities obtained by integration of the spectral energy values. The integration is done rather easily for parallel plates since the formulation can be algebraically reduced to only one exchange equation, Equation (6.1.2). However, when many surfaces are present in an enclosure, the computations increase substantially, since there are multiple exchange equations that need to be evaluated for each wavelength interval. Assuming some of the surfaces are gray provides some simplification. When specifying wavelength intervals, the most important spectral regions for radiative transfer are where both Eλb and ελ are large; thus, the wavelength range should be divided such that most of the wavelength intervals lie within these regions. If the number of wavelength intervals is increased, the exact results for energy transfer are approached. Dunkle and Bevans (1960) presented results that show errors, compared with an exact numerical result, of less than 2% when using a spectral band approximation as compared to about 30% error for the gray-surface approximation. Examples are given for enclosures with specified temperatures or specified net energy fluxes. Additional analyses of energy exchange between spectrally dependent surfaces were given by Goodman (1957) and Rolling and Tien (1967).

6.2.2 Spectral and Finite Spectral Band Relations for an Enclosure For an enclosure of N diffuse surfaces as in Figure 5.5, the relation in Equation (5.21) is written spectrally for dλ for each of the k surfaces: N



å j =1

é dkj 1 - el, j (T j ) ù - Fk - j ê ú ql, j d l = el, j ( T j ) úû êë el, j ( T j )

N

å (d

kj

- Fk - j ) Elb, j ( T j ) d l

( k = 1, 2,¼, N ) . (6.5)

j =1

If the N surface temperatures are all specified, the right sides of these simultaneous equations are known, as are the quantities in the square brackets on the left. The solution yields the qλ,k at λ for each of the surfaces. As a check on the calculations, the overall spectral energy balance within each dλ yields the summation

å

N k =1

Ak ql,k = 0. For parallel plates this is evident from Equation (6.2.1).

The total energy supplied to each surface by other means is then Qk = Ak

ò

¥

l =0

ql,k d l. If some of the

qk are specified, and hence some of the surface temperatures are unknown, an iterative procedure may be required, as discussed following Equation (5.21). This can be especially difficult for a spectrally dependent case because the specified qk is a total energy quantity and represents an integral of the values found from Equation (6.5). The boundary condition does not provide the spectral values of qλ,k. Equation (6.5) is usually applied in finite spectral bands. The bands are selected so that the properties for all surfaces are approximated as constant within each band. Equation (6.5) is then integrated over a band Δλ to yield æç after noting qDl, j º ql, j d l ö÷ Dl è ø

ò

N

å

j =1

é dkj 1 - e Dl, j ( T j ) ù - Fk - j ê ú qDl, j d l e Dl, j ( T j ) úû êë e Dl, j ( T j ) N

=

å (d

kj

- Fk - j ) FlT j ®( l+Dl )T j Elb, j ( T j ) Dl

(6.6)

( k = 1, 2,¼, N ) .

j =1

If the values of T are specified, Equation (6.6) provides N equations for the N unknown qΔλ,k for each Δλ band. The solution is carried out for each Δλ. Then for each of the k surfaces, the total qk is found by summing over all wavelength bands

257

Radiation among Nondiffuse Nongray Surfaces

qk =



åq

Dl ,k

. (6.7)

Dl

If for some surfaces neither q nor T is specified, and for others both are specified, the problem may become indeterminate using the methods of this chapter. Such inverse or indeterminate cases are treated in Chapter 18. Example 6.3 An enclosure consists of three diffuse surfaces of finite width and infinite length normal to the cross-section shown in Figure 6.5. One surface is concave. The radiative properties of each surface depend on wavelength and temperature, and the specified surface temperatures are T1, T2, and T3. Provide a set of band equations for the radiative energy exchange among the surfaces. From Equation (6.6) for k = 1, 2, and 3, and for wavelength band Δλ, the following equations are written. As a more compact notation let EDl , j (Tj ) º éë1 - e Dl , j (Tj ) ùû /e Dl , j (Tj ) and FDlTj º FlTj ®( l + Dl )Tj to yield





é ù 1 - F1-1EDl ,1 (T1 ) ú qDl ,1 - F1-2EDl ,2 (T2 ) qDl ,2 - F1-3EDl ,3 (T3 ) qDl ,3 ê (6.3.1) êë e Dl ,1 (T1 ) úû 4 4 4 = (1 - F1-1 ) FDlT1sT1 - F1-2FDlT2 sT2 - F1-3FDlT3 sT3 -F2-1EDl ,1 (T1 ) qDl ,1 +

1 qDl ,2 - F2-3EDl ,3 (T3 ) qDl ,3 e Dl ,2 (T2 ) (6.3.2)

= -F2-1FDlT1sT14 + FDlT2 sT24 - F2-3FDlT3 sT34



-F3-1EDl ,1 (T1 ) qDl ,1 - F3-2EDl ,2 (T2 ) qDl ,2 +

1 qDl ,3 e Dl ,3 (T3 ) . (6.3.3)

= -F3-1FDlT1sT14 - F3-2FDlT2 sT24 + FDlT3 sT34 These are three simultaneous equations for qΔλ,1, qΔλ,2, and qΔλ,3. The solution is carried out for the qΔλ,k values in each band Δλ. The qΔλ,k is the energy supplied to surface k in wavelength interval Δλ as a result of external heat addition to the surface (e.g., conduction and/or convection) and energy transferred in from other wavelength bands by radiation exchange within the enclosure. Such boundary conditions are discussed in more detail in Chapter 7 for cases including conduction, convection, or imposed heating. Finally, qk at each surface is found by summing qΔλ,k for that surface over all wavelength bands, as in Equation (6.7). This is the heat flux that must be supplied to surface k by some means other than internal radiation, to maintain its specified surface temperature.

FIGURE 6.5  Radiant interchange in enclosure with surfaces having spectrally varying radiation properties.

258

Thermal Radiation Heat Transfer

Example 6.4 Consider the geometry of Figure 6.5. Total energy flux is supplied to the three surfaces at rates q1, q2, and q3. Determine the surface temperatures. The equations are the same as in Example 6.3. Now, however, the prescribed boundary conditions have made the solution much more difficult. Because the surface temperatures are unknown, the FΔλT and εΔλ are unknown because they depend on temperature. The solution is carried out as follows: A temperature is assumed for each surface, and qΔλ,k(T) for each surface is computed. This is done for each of the wavelength bands. The qΔλ,k(T) values are then summed over all the Δλ to find q1, q2, and q3, which are compared to the specified boundary values. New temperatures are chosen and the process repeated until the computed qk values agree with the specified values. The new temperatures for successive iterations are guessed based on the FΔλT and εΔλ variations and the trends of how changes in Tk produce changes in qk throughout the system. As noted in Section 5.4.1.3, the temperature must be specified on at least one surface of the enclosure. Thus, there may be many solutions to this problem.

6.2.3 Semigray Approximations In some practical situations, the radiant energy in an enclosure has two well-defined spectral regions. An important example is an enclosure with an opening that is admitting solar energy. Solar energy is mostly in the short-wavelength spectral region, while emitted energy from the lower-temperature surfaces within the enclosure is at longer wavelengths. One way used to treat this situation is to define, for each surface in the enclosure, a hemispherical total absorptivity for incident solar radiation, and a second hemispherical total absorptivity for incident energy from emission that originates within the enclosure. More generally, N different absorptivities can be defined for each surface Ak, one for incident energy from each of the N enclosure surfaces. An assumption in these approximate analyses is that each absorptivity αk(Tk, Tj) is based on an incident blackbody spectrum at the temperature of the originating surface Tj. The incident spectrum may actually be quite different from the blackbody form, and this is a weakness of the method. Often the dependence of αk on Tk is small, and its principal dependence is on Tj through the spectral distribution of the incident energy. Because the absorptivity αk(Tk, Tj) and emissivity εk(Tk) of surface k are not in general equal, this approach is often called the semigray enclosure theory. Bobco (1964) provides the formulation for a general enclosure. The semigray and exact solutions for the temperature profiles along the surface of a nongray wedge cavity exposed to incident solar radiation are in Figure 6.6 (Plamondon and Landram 1966). The cavity is in vacuum with an environment at ~0 K except for the solar radiation source. The cavity surface properties are independent of temperature and the surfaces are diffuse. Three solution techniques are given, labeled in Figure 6.6b as Exact, Method I, and Method II. The first is an exact solution of the complete integral equations. Method I is the semigray analysis that assigns an absorptivity αsolar for direct and reflected radiation originating from the solar energy, and a second absorptivity αinfrared (equal to the surface emissivity) for radiation originating by emission from the wedge surfaces. Method II is a poorer approximation that retains these same two absorptivities, but applies αsolar only for the incident solar energy and uses αinfrared for all energy after reflection, regardless of its source. The results in Figure 6.6b are for a polished aluminum surface in a 30° wedge. Method I is in excellent agreement with the exact solution, while Method II underestimates the temperatures by about 10%.

6.3 DIRECTIONAL-GRAY SURFACES Radiant exchange between gray surfaces with directional properties is now considered. Many radiation analyses assume diffuse emitting and reflecting surfaces, and some treatments include the effect of specular reflections with diffuse emission (Section 6.6). Diffuse or specular surface conditions

Radiation among Nondiffuse Nongray Surfaces

259

FIGURE 6.6  Effect of semigray approximations on computed temperature distribution in a wedge cavity: (a) Geometry of wedge cavity; (b) temperature distribution along wedge, αsolar = 0.220; αinfrared = 0.099.

FIGURE 6.7  Radiant interchange between parallel directional surfaces of finite width L that are infinitely long in the direction normal to the plane of the drawing.

are the most convenient to treat analytically, and in many instances the detailed consideration of directional emission and reflection effects is unwarranted. However, certain materials and special situations require the examination of directional effects. The difficulty in treating the general case of directionally dependent properties is illustrated by performing an energy balance in a simple geometry: The radiative exchange between two infinitely long parallel nondiffuse gray surfaces of finite width L (Figure 6.7). The radiation intensity leaving element dA1 in direction (θr,1,ϕr,1) is composed of emitted and reflected intensities:

I1 ( qr ,1 , fr ,1 ) = I e,1 ( qr,1 , fr,1 ) + I r ,1 ( qr,1 , fr,1 ) . (6.8)

260

Thermal Radiation Heat Transfer

These two components are given by modifications of Equations (2.4) and (2.40) to yield Equation (6.8) in the form

ò

I1 ( qr,1 , fr,1 ) = e1 ( qr,1 , fr,1 ) I b,1 ( T1 ) + r1 ( qr,1 , fr,1 , q1 , f1 ) I 2 ( q2 , f2 )



A2

cos2 q2 dA2 . (6.9) S2

In the second term on the right of Equation (6.9), the energy incident on dA1 from each element dA2 is multiplied by the bidirectional total reflectivity to give the contribution to the intensity reflected from dA1 into direction (θr,1,ϕr,1). This is then integrated over A2 to include all energy incident on dA1 from A2. A similar equation is written for an arbitrary element dA2 on surface 2. The result is a pair of coupled integral equations to be solved for I(θ, ϕ) at each position and for each direction on the two surfaces. The integral equations are analogous to Equations (5.21.3) and (5.21.4) for diffuse-gray surfaces. Detailed property data for ε(θ, ϕ) and ρ(θr, ϕr, θi, ϕi) are often not available. For the case when T1 and T2 are not known and the temperature dependence of the properties is considerable, the solution for the entire energy-exchange distribution becomes very tedious. Approximations can be made, such as analytically simulating the real properties with simple functions, omitting certain portions of energy deemed negligible, or ignoring directional effects except those expected to provide significant changes from diffuse or specular analyses. Some of these methods are outlined in Bevans and Edwards (1965), Hering (1966a, b), Viskanta et al. (1967), Toor (1967), and Naraghi and Chung (1984, 1986). A method using discrete ordinates was developed by Brockmann (1997) (as in Section 13.3, but without an absorbing gas in the enclosure) to examine directional effects for radiation between the ends of a cylindrical enclosure. An example is now given illustrating the effect of a directional-gray surface on radiative exchange. Example 6.5 Two parallel isothermal plates of infinite length and finite width L are arranged as in Figure 6.8a. The upper plate is black, while the lower is a highly reflective gray material with parallel deep grooves of open angle 1° in its surface extending along the infinite direction. Such a surface might be constructed by stacking polished razor blades. The surroundings are at zero temperature. Compute the net energy gain by the directional surface if T2 > T1 and compare the result to the net energy gain if the directional-gray surface is replaced by a diffuse-gray surface with an emissivity equivalent to the hemispherical emissivity of the directional surface. In Howell and Perlmutter (1963), the directional emissivity is calculated at the opening of an infinitely long groove with specularly reflecting walls of surface emissivity 0.01. This is given by the dot-dashed line in Figure 6.8b. The angle β1 is measured from the normal of the opening plane of the grooved surface and is in the cross-sectional plane perpendicular to the length of the groove as in Figure 6.8a. The ε1(β1) has already been averaged over all circumferential angles for a fixed β1. Thus, it is an effective emissivity for radiation from a strip on the grooved surface to a parallel infinitely long strip element on an imaginary semicylinder over the groove and with its axis parallel to the grooves. The angle β1 is different from the usual cone angle θ1. The actual emissivity ε1(β1) of Figure 6.8b is approximated for convenience by the analytical expression ε1(β1) ≈ 0.830 cos β1. Using cylindrical coordinates to integrate over all β1, the hemispherical emissivity of this surface is



e1

ò =

p2

-p 2

e1 (b1 ) cos b1db1

ò

p2

-p 2

cosb1db1

and this is the dashed line in Figure 6.8b.

p2

= 0.830

ò cos b db = 0.652 2

0

1

1

261

Radiation among Nondiffuse Nongray Surfaces

FIGURE 6.8  Interchange between grooved directional-gray surface and black surface: (a) Geometry of problem (environment at zero temperature); (b) emissivity of directional surface. The energy gained by surface 1 will first be determined when surface 2 is black and surface 1 is diffuse with ε1 = 0.652. The energy emitted by the diffuse surface 1 per unit of the infinite length and per unit time is Qe,1 = 0.652 sT14 L. Because surface 2 is black, none of this energy is reflected to surface 1. The energy per unit length and time emitted by black surface 2 that is absorbed by surface 1 is

Qa,1 = a1sT24

òòdF

dA2 = e1sT24

d 2-d 1

A2 A1

òòdF

dA1.

d 1-d 2

A1 A2

The configuration factor between infinite parallel strips, from Example 4.2, is dFd1 − d2 = d(sinβ1)/2 so that



L æ ö ç dFd1-d 2 ÷ dA1 = 1 ( sin b1,max - sin b1,min ) dx. ç ÷ 2 A1 è A2 x =0 ø

òò

ò

262

Thermal Radiation Heat Transfer 1/ 2

2 From Figure 6.8a, sinb1 = ( x - x ) é( x - x ) + D 2 ù , which gives ë û

é L-x ê ê 2 x - 2xL + L2 + D 2 x =0 ê ë 1/ 2 = 0.652sT24 é L2 + D 2 - Dù êë úû

Qa,1 = e1sT24

L

1 2

ò ( (

+

)

1/ 2

(

ù ú dx 1/ 2 ú 2 2 x +D ûú . x

)

)

The net energy gained by surface 1, Qa,1 − Qe,1, divided by the energy emitted by surface 2, is a measure of the efficiency of the surface as a directional absorber. For surface 1, being diffuse, this ratio is (l = L/D) Effdiffuse =



Qa,1 - Qe ,1 0.652 é 2 = ê 1+ l l ë sT24 L

(

)

12

- 1-

T14 T24

ù l ú . û

When surface 1 is a directional (grooved) surface, the amount of energy emitted from surface 1 is the same as for a diffuse surface (although it has a different directional distribution) since both have the same hemispherical emissivity. The energy absorbed by the grooved surface is, by using α1(β1) = ε1(β1) for a gray surface, Qa,1 = sT

4 2

0.830sT24 a1(b1)dFd 2-d1dA2 = 2

òò

A2 A1



=

0.830sT 4

4 2

L

é

ò êë x

2

x =0

L b1,max

òò

cos2 b1db1dx

x =0 b1,min

D(L - x ) xD xù æL-xö + tan-1 ç + tan-1 ú dx ÷+ 2 2 D D - 2xL + L2 + D 2 + x D è ø û

0.830sT24 L tan-1 . = 2 D The absorption efficiency of the directional surface is then 4



Effdirectional =

Qa,1 - Qe ,2 0.830 æT ö = tan-1 l - 0.652 ç 1 ÷ . 2 sT24 L è T2 ø

The absorption efficiencies of the directional-gray and diffuse-gray surfaces are shown in Figure  6.9 as a function of l with (T1/T2)4 as a parameter. The Eff for the directional surface is higher than that for the diffuse surface for all values of l. As l approaches zero, the configuration approaches that of infinite elemental strips, and emission from surface 1 becomes much larger than absorption from surface 2. Thus, Effdirectional and Effdiffuse are nearly equal, since the surfaces always emit the same amount. As l approaches infinity, the directional effects are lost. At intermediate l a 10% difference in absorption efficiency is attainable.

The effects of directional properties on the local energy loss can be considerable for some geometries. In Figure 6.10 some directional distributions of reflectivity are examined for their influence on local energy loss from the walls of an infinitely long groove. The results are from Viskanta et al. (1967), where for comparison the curves were gathered from original work and various sources (Eckert and Sparrow 1961, Sparrow et al. 1961, Hering 1966a, Toor 1967). The walls of the groove are at 90°, and the surface emissivity distributions are all normalized to give a hemispherical emissivity of 0.1. Curves are presented for diffuse reflectivity ρ, specular reflectivity assumed independent of incident angle ρs, specular reflectivity dependent on incident angle ρs(θi) based on electromagnetic theory, and three distributions of bidirectional reflectivity ρ(θr, θi). The bidirectional distributions

Radiation among Nondiffuse Nongray Surfaces

263

FIGURE 6.9  Effect of directional emissivity on absorption efficiency of surface.

FIGURE 6.10  Local radiative energy loss from surface of isothermal groove cavity. Hemispherical emissivity of surface is assumed with ε = 0.1.

are based on Beckmann and Spizzichino (1963) for rough surfaces having various combinations of the ratio of rms optical-surface roughness amplitude to radiation wavelength, σ0/λ, and the ratio of roughness autocorrelation distance to radiation wavelength, a0/λ. Note that the results in Figure 6.10 for the specular and diffuse models do not provide upper and lower limits to all the solutions, as is sometimes claimed. Additional information on surface roughness as it affects the directional properties of surfaces is in Section 3.4.3. Energy transfer was studied in a groove with two rough

264

Thermal Radiation Heat Transfer

sides, each at uniform temperature. The roughness has a greater influence on the radiation exchange between the two sides than on the net radiation from the groove. Howell and Durkee (1971) compared analysis and experiment for collimated radiation entering a low-temperature very long three-sided cavity. Because of the low temperature, surface emission was not important. The cavity had two surfaces with reflectivities that were diffuse with a specular component, while the third surface was honeycomb material with a strong bidirectional reflectivity. It was found necessary to include all surface characteristics in the analysis to obtain agreement with the experiment. This is characteristic of geometries interacting with collimated incident radiation. Black (1973) analyzed the directional emission characteristics of two types of grooves with some black and some specular sides, as in Figure 6.11a and b. The V-groove tends to emit in the normal direction, while the rectangular groove emits more in the grazing direction β → 90°. Some emission results for the rectangular shape are in Figure 6.11c, and they exhibit a strong directional characteristic for small h/w. Mulford et al. (2018) give the apparent absorptivity of a diffusely irradiated specular-walled V-groove, the apparent emissivity of an isothermal V-groove and the apparent absorptivity of a V-groove subject to collimated irradiation. Grooves and enclosures having specular surfaces are analyzed in Section 6.5.3.

FIGURE 6.11  Directional emission from grooves: (a) V-groove with specular sides; (b) rectangular groove with specular base; and (c) directional emissivity for rectangular groove with specular base (εbase = 0.05) and black sides.

Radiation among Nondiffuse Nongray Surfaces

265

6.4 SURFACES WITH DIRECTIONALLY AND SPECTRALLY DEPENDENT PROPERTIES The general case of radiative transfer in enclosures where surface properties depend on both direction and wavelength, and where properties can be temperature dependent, is complex to treat fully. When those dependencies must be included, numerical techniques are necessary. Toor (1967) used the Monte Carlo method to study radiation interchange for various simply arranged surfaces with directional properties. Zhang et  al. (1997) derived the directional-spectral relation for radiative transfer between parallel plates that is a generalization (with properties independent of angle ϕ) of Equation (6.1.2) (also see Example 6.9): p/2 ¥



q=2

òò

q= 0 l = 0

Elb,1 ( T1 ) - Elb,2 ( T1 ) d l sin q cos qd q. (6.10) 1 1 + -1 el,1 ( q1 , T1 ) el,2 ( q2 , T2 )

A difficulty for such an evaluation is in finding the detailed radiative properties to sufficient accuracy. The technology of interest was for evaluating the insulating performance of a double glass window with a vacuum between the two panes. The glass surfaces are opaque in the infrared region so that, for the temperatures involved, Equation (6.10) could be used for transfer across the vacuum space. Comparisons were made with experiments. In this section, the general integral equations are formulated for radiation in such systems, and a considerably simplified example problem is solved. The procedure is a combination of the previous diffuse-spectral and directional-gray analyses. The equations are formulated at one wavelength, as in Section 6.2, and in terms of intensities for each direction, as in Section 6.3; this accounts for both spectral and directional effects. The interaction between two plane surfaces is developed first; this can be generalized to a multisurface enclosure, as for gray surfaces in Chapter 5. The energy balance is now developed for an area element dA at location r in an x, y, z coordinate system as in Figure 6.12. The Iλο(θr, ϕr, r) is the outgoing spectral intensity from dA in the direction

FIGURE 6.12  Geometry for incoming and outgoing intensities at a differential surface area.

266

Thermal Radiation Heat Transfer

θr, ϕr as the result of both emission and reflection. The spectral intensity emitted by dA in this direction is I le (qr , fr , r ) = el (qr , fr , r )I lb (r ). (6.11)



These quantities also depend on TdA, but this functional designation is omitted to simplify the notation. The intensity reflected from dA into the θr, ϕr direction results from the incident intensity from all directions of a hemisphere above dA. If the spectral intensity incident on dA within dΩi is Iλi(θi, ϕi, r), the intensity reflected from dA into direction θr, ϕr is 2p



I l ,r ( q r , f r , r ) =

ò r (q , f , q , f , r ) I l

r

r

i

i

l ,i

( qi , fi , r ) cos qi dWi . (6.12)

Wi = 0

The net energy flux supplied to dA for steady state is the difference between the outgoing and incoming radiative fluxes:

¥

q (r ) =

òJ

¥

l

l =0

( r ) dl - ò Gl ( r ) dl. (6.13) l =0

The Jλ(r)dλ is the angular integration of the emitted and reflected spectral fluxes over all outgoing directions: 2p p/2

J l ( r ) d l = I lb ( r ) d l

ò ò e ( q , f , r ) sin q cos q dq df l

r

r

r

r

r

fr = 0 qr = 0



2p p/2

+

ò òI

l ,r

r

(6.14)

( qr , fr , r ) dl sin qr cos qr dqr dfr .

fr = 0 qr = 0

The Gλ(r)dλ is the result of spectral fluxes incident from all dΩi directions:

Gl ( r ) d l =

2 p p/2

ò òI

l ,i

( qi , fi , r ) dl sin qi cos qi dqi dfi . (6.15)

fi =0 qi =0

Equations (6.11) through (6.15) provide an exact formulation to obtain the heat flux q(r) that must be supplied by other means to area dA to maintain its temperature at TdA in the presence of incident and emitted radiation. To develop an enclosure theory, various degrees of approximation can be made. If the enclosure is very simple, such as having only two infinite plane surfaces, it may be feasible to include variations of properties and surface temperature across each surface. To develop the required integral equations, consider the two surfaces in Figure 6.13 and let the surrounding environment be at a low temperature so that it does not contribute to incident radiation. The spectral energy leaving dA2 at r2 that reaches dA1 at r1 is Iλo,2(θ2,ϕ2,r2)dλdA2cosθ2(dA1cosθ1/S2). In terms of the incident intensity, the incident spectral intensity in dΩ1 is Iλi,1(θ1,ϕ1,r1)dλdA1cosθ1dΩ1, where dΩ1 = (dA2cosθ2)/S2. Thus, Iλi,1(θ1,ϕ1,r1) = Iλo,2(θ2,ϕ2,r2) and, by using Equations (6.11) and (6.12), I lo,1 ( qr ,1 , fr,1 , r1 ) = el,1 ( qr ,1 , fr ,1 , r1 ) I lb,1 ( r1 )

ò

+ rl,1 ( qr ,1 , fr ,1 , qi,1 , fi,1 , r1 ) I lo,2 ( q2 , f2 , r2 ) A2

cos q1 cos q2 r2 - r1

2

(6.16) dA2 .

267

Radiation among Nondiffuse Nongray Surfaces

FIGURE 6.13  The interchange between surfaces having directional spectral properties (environment at ~0 K).

Similarly, for surface 2, the outgoing intensity is I lo,2 ( qr ,2 , fr ,2 , r2 ) = el,2 ( qr ,2 , fr ,2 , r2 ) I lb,2 ( r2 )

ò



+ rl,1 ( qr ,2 , fr ,2 , qi,2 , fi,2 , r2 ) I lo,1 ( q1 , f1 , r1 )

cos q2 cos q1

A1

r1 - r2

2

dA1.

(6.17)

Equations (6.16) and (6.17) are both in terms of outgoing intensities, and they provide a set of simultaneous integral equations for Iλο,1 and Iλο,2, where the subscript o denotes the outgoing direction. An iterative numerical solution is required that can be quite complex as both unknowns are functions of position and angle. After Iλο,1(θ1, ϕ1, r1) and Iλο,2 (θ2, ϕ2, r2) are obtained, the total energy can be determined that must be supplied to each surface element to maintain the specified local surface temperature. The total energy supplied is the difference between the total emitted and absorbed energies:



dQ1 = dA1

¥

ò òe

l ,1

( qr,1, fr,1, r1 ) I lb,1 ( r1 ) cos q1dW1dl

l =0 Ç

(6.18)

¥

-

ò òa

l = 0 A2

l ,1

( qi,1, fi,1, r1 ) I lo,2 ( q2 , f2 , r2 )

cos q1 cos q2 r2 - r1

2

dA2 dl.

To develop an enclosure theory with more than a few surfaces, the Iλo, temperature, and surface properties are usually assumed uniform over each enclosure surface. In addition, a finite number of

268

Thermal Radiation Heat Transfer

angular intervals must be specified. If we let Ak and Aj be the kth and jth surfaces of an enclosure with N surfaces, then, by integrating Equation (6.17) over Ak and summing the contributions from all of the Aj surfaces, I lo , k ( q r , k , f r , k ) = e l , k ( q r , k , f r , k ) I lb , k

1 + Ak

N

å òòr

l ,k

( q r , k , f r , k , q i , k , f i , k ) I lo , j ( q j , f j )

j =1 Ak A j

cos qk cos q j rj - rk

2

dA j dAk .

(6.19)

When written out for each surface k and for a sufficient number of directions (θr,k, ϕr,k) to obtain acceptable accuracy in the angular integrations that follow, this yields a set of simultaneous equations for Iλo,k(θr,k,ϕr,k) for k = 1,…,N. With this degree of approximation, which is characteristic for enclosure analyses, consider the interaction of the two plane surfaces in Figure 6.13. The surroundings are at low temperature relative to the surface temperature, so radiation from the surroundings is neglected; this provides a two-surface enclosure. Then, writing Equation (6.19) for k = 1 and 2,



I lo,1 ( qr ,1 , fr ,1 ) = el,1 ( qr ,1 , fr ,1 ) I lb,1 cos q1 cos q2 1 (6.20) + dA j dAk . rl,1 ( qr ,1 , fr ,1 , qi,1 , fi,1 ) I lo,2 ( q2 , f2 ) A1 S2

òò

A1 A2



I lo,2 ( qr ,2 , fr ,2 ) = el,2 ( qr ,2 , fr ,2 ) I lb,2 cos q2 cos q1 1 (6.21) + dA1dA2 . rl,2 ( qr ,2 , fr ,2 , qi,2 , fi,2 ) I lo,1 ( q1 , f1 ) S2 A2

òò

A2 A1

Equations (6.20) and (6.21) are in terms of outgoing intensities in each direction θr,k, ϕr,k; they are simultaneous integral equations for Iλo,1 and Iλο,2. A numerical solution can be obtained by writing these equations for a number of discrete angular intervals to develop a set of simultaneous equations. After Iλo,1(θr,1,ϕr,1) and Iλo,2(θr,2,ϕr,2) are obtained for a sufficient number of angular intervals to yield good accuracy, the total energy can be determined that must be supplied to each surface to maintain its specified temperature. This is the difference between energies carried away from and carried to the surface; for A1 this gives Q1 = A1

¥

ò òI

lo,1

( qr,1 , fr,1 ) cos qr,1dWr,1dl

l =0 Ç

1 A1

¥

ò òò

l =0 A1 A2

(6.22) cos q1 cos q2 I lo,2 ( q2 , f2 ) dA2 dA1d l S2

and similarly for A2. For diffuse-gray surfaces, so that Iλο,1 and Iλο,2 are independent of λ, θ, and ϕ, this simplifies to Q1/A1 = πIο,1−πIο,2F1−2 = J1−J2F1−2, as given by Equation (5.17). If Q1 rather than T1 is specified, T1 must be determined and the solution becomes more difficult. A temperature is assumed for A1, and the enclosure equations of the form Equations (6.20) and (6.21) are solved to find the Iλo. The outgoing intensities are substituted into Equation (6.22), and the computed Q1 is compared to the given value. The T1 is then adjusted and the procedure repeated until an agreement between given and computed Q1 is attained. If Q is specified for more than one surface, the solution is even more difficult.

269

Radiation among Nondiffuse Nongray Surfaces

Example 6.6 For an example that can be carried out in analytical form, a small area element dA1 is considered on the axis of, and parallel to, a black circular disk, as in Figure 6.14. The element is at T1, the disk at T2, and the environment at Te ≈ 0 K. The dA1 has a directional spectral emissivity independent of ϕ and approximated by el ,1(q1, T1) = 0.8 cos q1(1 - e -C2 / lT1 ),



where C2 is one of the constants in Planck’s spectral distribution. As will be evident, this dependence on λ and T1 was devised to simplify this illustrative example and obtain an analytical result. More generally, numerical integration can be used. Find the energy dQ1 added to dA1 to maintain T1. The energy balance in Equation (6.18) is emitted energy minus absorbed incident energy. The energy emitted by dA1 is ¥

dQe ,1 = dA1



ò òe

(q1)Ilb,1 cos q1d W1d l.

l ,1

l =0 Ç

Insert the expressions for ελ,1Iλb,1 (Equation (1.15)), and dΩ1 = sinθ1dϕ1 to obtain ¥



dQe ,1 = 0.8dA1

2p p / 2

ò ò ò cos

2

q1(1 - e -C2 / lT1 )

l =0 f1=0 q1=0

2C1 sin q1d q1d f1d l. l 5 (eC2 / lT1 - 1)

Integrating over ϕ1 and θ1 gives

dQe ,1 = 0.8dA1

2p 3

ò

¥

0

2C l e

1 5 C2 / lT1

d l.

FIGURE 6.14  Radiative energy exchange involving directional spectral surface element (cold environment at ~0 K).

270

Thermal Radiation Heat Transfer

Using the transformation ζ = C2/λT1, the relation

ò

¥

0

z 3e - z d z = 3! from a table of definite integrals,

and the Stefan–Boltzmann constant σ from Equation (1.34) yields dQe,1 =



48 sT14 dA1 = 0.493sT14 dA1 p4

Thus, the total hemispherical emission is about half that of a blackbody. The energy absorbed by dA1 is ¥



dQa,1 = dA1

ò òa

l ,1

(q1, f1)Il0 ,2 (q2 , f2 )

l =0 A2

cos q1 cos q2 dA2d l S2

From Kirchhoff’s law, the directional spectral absorptivity and emissivity are equal. For dA 2 taken as a ring element, the solid angle cos θ2dA 2/S2 is written as 2π sin θ1dθ1. This is used to write the absorbed energy as (where Iλo,2 = Iλb,2, since A 2 is a black surface) ¥ q1,max

dQa,1 = 2p(0.8)dA1

ò ò

(cos2 q1 sin q1d q1)Ilb,2 (1 - e -C2 / lT1 )d l

l =0 q1=0 q1,max ¥



= -1.6pdA1

=

cos3 q1 3 q1=0

ò 0

2C1(1 - e -C2 / lT1 ) dl l 5 (eC2 / lT2 - 1)

ù D3 3.2pC1dA1 é ê1 - 2 2 3/ 2 ú 3 ë (D + R ) û

¥

ò 0



1 - e -C2 / lT1 d l. l 5 (eC2 / lT2 - 1)

Using the transformation ζ = C2/λT1, this is placed in the form

dQa,1 =

where r = R/D and G(T2 /T1) = (1/6)

ò

¥

0

48 é 1 æT ö ù 4 sT2 dA1G ç 2 ÷ 1p4 êë (1 + r 2 )3/ 2 úû è T1 ø z 3e - z (1 - e zT2 /T1 ) /(1 - e - z )d z . This integral was evaluated num­

erically, giving G(1.0) = 1.000, G(1.5) = 1.045, and G(2.0) = 1.063; hence, the effect of the temperature ratio is small. Finally, the energy added to dA1 to maintain it at T1 is given by

dQ1 = dQe ,1 - dQa,1 =

48s ìï 4 1 é ù æ T2 ö üï T - T24 ê1 G ç ÷ ý dA1. 4 í 1 2 3/ 2 ú p îï ë (1 + r ) û è T1 ø þï

As shown by Example 6.6, it is difficult to devise an analytical function for ελ(θ, T) that can be integrated in closed form over both angle and wavelength. Numerical methods are required to solve problems of this type for ελ(θ, T) functions that represent experimental data. The development in this section has shown that, although the formulation of radiation-exchange problems involving directional and/or spectral properties is not conceptually difficult, it is usually tedious to solve the resulting integral equations. To simplify the equations, it is usually necessary to make assumptions and approximations that can vary from case to case. Numerical techniques such as iteration are used for directional spectral formulations, since closed-form analytical solutions can rarely be obtained. An alternative numerical technique is the Monte Carlo method presented in Chapter 14. For complicated directional and spectral effects, this is often a better approach than using an integral equation formulation.

Radiation among Nondiffuse Nongray Surfaces

271

Some of the original research is now summarized. A surface with part specular and part diffuse reflectivity and a semigray analysis were used by Shimoji (1977) to find local temperatures in conical and V-groove cavities exposed to incident solar radiation parallel to the cone axis or V-groove bisector plane. Toor and Viskanta (1972) compared with experiment various analytical models using diffuse, specular, semigray, nongray, and combinations of these characteristics. They found, for the particular geometries and materials studied, that spectral effects were less important than directional effects and that the presence of one or more diffuse surfaces in an enclosure made the presence of specularly reflecting surfaces unimportant. Hering and Smith (1970, 1971) and Edwards and Bertak (1971) applied various models of surface roughness in calculating radiant exchange between surfaces. If grooves on a surface have a size that is comparable to or smaller than the wavelength of the incident or emitted radiation, there can be complex interactions of the electromagnetic waves within the grooves. This can produce unusual spectral and directional effects. The radiation behavior of materials with a grooved microstructure was studied by Hesheth et al. (1988), Glass et al. (1982), Wirgin and Maradudin (1985), Sentenac and Greffet (1994), Hajimirza et al. (2011, 2012), and Krishna and Lee (2018). The specular properties of spacecraft thermal control materials were reported by Drolen (1992). Innovative treatments of directional properties are required in modeling the illumination of scenes in computer graphic representations. Representative approaches are given by Buckalew and Fussell (1989), Immel et al. (1986), Kajiya (1985), Wolff and Kurlander (1990), He et al. (1991), and Sillion et al. (1991). Billings et al. (1991a) employed two-parameter Markov chains for treating bidirectional surface properties.

6.5 RADIATION EXCHANGE IN ENCLOSURES WITH SPECULARLY REFLECTING SURFACES In Chapter 5, the enclosure surfaces are all diffuse emitters and reflectors, and in Sections 6.1 through 6.4 of this chapter, surfaces with general directional and reflection characteristics are analyzed. This section considers a particular class of reflections that was not specifically treated. Except for one special example, all of the surfaces are still assumed to emit diffusely. Some of the surfaces in an enclosure are assumed to reflect diffusely; the remaining surfaces are assumed to be specular, that is, to reflect in a mirror-like manner. From the discussion of roughness in Chapter 3, an important surface parameter for radiative properties is the optical roughness (the ratio of the rootmean-square roughness height to the radiation wavelength.) As the radiation wavelength increases, a smooth surface tends toward being optically smooth, and reflections at larger wavelengths tend to become more specular. Thus, although a surface may not appear mirror-like to the eye for the short wavelengths of the visible spectrum, it may be specular for the longer wavelengths in the infrared. When reflection is diffuse, the directional history of the incident radiation is lost upon reflection, and the reflected energy has the same directional distribution as if it had been absorbed and diffusely re-emitted. For specular reflection, the reflection angle relative to the surface normal is equal in magnitude to the angle of incidence. Hence, the directional history of the incident radiation is not lost upon reflection. When dealing with specular surfaces, it is necessary to consider the specific directional paths that the reflected radiation follows between surfaces. The specular reflectivity used in this chapter is assumed independent of incidence angle; the same fraction of the incident energy is reflected for any angle of incidence. In addition, all the surfaces are assumed gray (properties do not depend on wavelength). A spectral or nongray analysis can be done as in Section 6.2; heat flow calculations are carried out in each wavelength band that is significant in the radiative transfer, and the results are summed over all bands to obtain total energy quantities.

6.5.1 Representative Cases with Simple Geometries Radiation exchange between infinite parallel plates, concentric cylinders, and concentric spheres, as shown in Figure 6.15, is of practical importance for predicting the heat transfer performance of

272

Thermal Radiation Heat Transfer

radiation shields, Dewar vessels, and cryogenic insulation applications. Specular exchange for these geometries is well understood, having been discussed and analyzed long ago by Christiansen (1883) and Saunders (1929). Consider radiation between two infinite, gray, parallel specularly reflecting surfaces, as in Figure 6.15a. Although the reflections are specular, the emission is diffuse. All emitted and reflected radiation leaving surface 1 directly reaches surface 2; similarly, all emitted and reflected radiation leaving surface 2 directly reaches surface 1. This is true whether the surfaces are specular or diffuse reflectors. For diffuse emission the surface absorptivity is independent of direction, so there are no directional considerations. Hence, for specular reflections and diffuse emission, Equation (5.6.3) applies and the net radiative heat transfer from surface 1 to surface 2 is

Q1 = -Q2 =

(

A1s T14 - T24

)

éë1/e1 ( T1 ) ùû + éë1/e2 ( T2 ) ùû - 1

. (6.23)

Now consider radiation between the concentric cylinders or spheres in Figure 6.15b and c, assuming that the emission is diffuse. Typical radiation paths for specular reflections are in Figure 6.15d. As shown by path a, all radiation emitted by surface 1 will directly reach surface 2. A portion will be reflected from surface 2 back to surface 1, and a portion of this will be re-reflected from 1. This sequence of reflections continues until insignificant energy remains because radiation is partially absorbed on each contact with a surface. From the symmetry of the concentric geometry and the equal incidence and reflection angles for specular reflections, no part of the radiation following path a can ever be reflected directly from surface 2 to another location on surface 2. Thus, the exchange process for radiation emitted from surface 1 is the same as though the two concentric surfaces were infinite parallel plates. However, the radiation emitted from the outer surface 2 can travel along either of two types of paths, b or c, as shown in Figure 6.15d. Since emission is assumed to

FIGURE 6.15  Radiation exchange for specular surfaces having simple geometries: (a) Infinite parallel plates; (b) gap between infinitely long concentric cylinders; (c) gap between concentric spheres; and (d) paths for specular radiation in gap between concentric cylinders or spheres.

273

Radiation among Nondiffuse Nongray Surfaces

be diffuse, the fraction F2−2 will follow paths of type c (F is a diffuse configuration factor). From the geometry of specular reflections these rays will always be reflected along surface 2, with none reaching surface 1. The fraction F2−1 = A1/A2 is reflected back and forth between the surfaces along paths b in the same fashion as radiation emitted from surface 1. The amount of radiation following this type of path is A2 e2 F2-1sT24 = A2 e2 ( A1 /A2 ) sT24 = A1e2 sT24 . The fraction of the radiation leaving surface 2 that impinges on surface 1 thus depends on A1 and not on A2. Hence, for specular surfaces the exchange behaves as if both surfaces were equal portions of infinite parallel plates with size equal to the area of the inner body. The net radiative heat transfer from surface 1 to surface 2 is thus given by Equation (6.23). Example 6.7 A spherical vacuum bottle consists of two silvered, concentric glass spheres, the inner being 15 cm in diameter and the evacuated gap between the spheres being 0.65 cm. The emissivity of the silver coating is 0.02. If hot coffee at 368 K is in the bottle and the outside temperature is 294 K, what is the initial radiative heat leakage rate from the bottle? Equation (6.23) applies for concentric specular spheres. For the small rate of heat leakage expected, it is assumed that the surfaces will be close to 368 and 294 K. This gives 2



Q1 =

(

p ( 0.15) 5.6704 ´ 10 -8 3684 - 2944

(1 / 0.02) + (1 / 0.02) - 1

) = 0.440 W.

If, instead of using the specular formulation, both surfaces are assumed to be diffuse reflectors, with ε still 0.02, then Equation (5.8.4) applies. The denominator of the above equation for Q1 becomes 2



1 A1 æ 1 1 ö æ 15 ö æ 1 ö + - 1 = 91.50 + ç - 1÷ = e1 A2 è e1 ø 0.02 çè 16.3 ÷ø çè 0.02 ÷ø

1 æ1 1 ö æ 1 ö + ç - 1÷ = - 1 = 99.0. +ç e1 è e1 ø 0.02 è 0.02 ÷ø For diffuse surfaces, the heat loss is increased to 0.476 W. instead of for the specular case where the denominator is

Example 6.8 For the previous example, to illustrate the insulating ability of a vacuum bottle, how long will it take for the coffee to cool from 368 to 322 K if the only heat loss is by radiation? The cooling rate of the coffee is ρMVc(dT1/dt). Assuming coffee is always mixed well enough that it is at a uniform temperature, the cooling rate is equal to the instantaneous radiation loss. The energy loss by radiation at any time t, given by Equation (6.23), is related to the change of internal energy of the coffee by -rMVc



4 4 dT1 A1s éëT1 ( t ) - T2 ùû = dt 1/e1 + 1/e 2 - 1

where it is assumed that surface 1 is at the coffee temperature and surface 2 is at the outside environment temperature. Then T1=TF



-

ò

T1=TI

t

dT1 A1s dt = T14 - T24 rMVc (1/e1 + 1/e 2 - 1)

ò 0

274

Thermal Radiation Heat Transfer

where TI and TF are the initial and final temperatures, respectively, of the coffee ε1 and ε2 are assumed independent of temperature Carrying out the integration and solving for t gives the cooling time from TI to TF as rMVc (1/e1 + 1/e 2 - 1) é 1 (TF + T2 ) / (TF + T2 ) + 1 æ tan-1 TF - tan-1 TI ö ùú . ê 3 ln T2 T2 ÷ø ú A1s (TI + T2 ) / (TI + T2 ) 2T23 çè êë 4T2 û

t=



1 3 p ( 0.15) m3 , c = 4195 J/(kg ⋅ K), ε1 = ε2 = 0.02, A1 = π · (0.15)2 m2, 6 σ = 5.6704 × 10 −8 W/(m2⋅K4), T2 = 294 K, TI = 368 K, and TF = 322 K gives t = 374.7 h. The coffee would require about 16 days to cool if heat losses were only by radiation. Conduction losses through the glass wall of the bottle neck usually cause the cooling to be considerably faster. (This is left as an exercise). Substituting ρM = 975 kg/m3, V =

The following is a summary for exchange between two surfaces that both have diffuse emission. When both surfaces are specular reflectors, Equation (6.23) applies for infinite parallel plates, infinitely long concentric cylinders, and concentric spheres. For infinite parallel plates, Equation (6.23) also applies when both surfaces are diffuse reflectors, or when one surface is diffuse and the other is specular. For cylinders and spheres, Equation (6.23) will apply when the surface of the inner body (surface 1) is a diffuse reflector if the outer body (surface 2) is specular. This is because all radiation leaving surface 1 propagates directly to surface 2 for surface 1 either specular or diffuse. When surface 2 is diffuse, Equation (5.8.4) applies and may be used when surface 1 is either specular or diffuse. The relations are summarized in Table 6.1. The previous development was for surfaces with diffuse emission, which is a common assumption, and it will be used in the more general enclosure theory that will be developed later in this chapter. Before continuing with more general geometries, a special case is considered for two parallel surfaces where it is possible to conveniently examine the effect of directional emission for surfaces that are specularly reflecting. Example 6.9 Two infinite parallel plates as shown in Figure 6.15a are specular reflectors but have emissivities (and hence absorptivities) that depend on angle θ (but not on ϕ), such as for polished metals in Section 3.2.2. What is the equation for radiative exchange between the surfaces? The radiation per unit area emitted in solid angle dΩ from surface 1 in direction θ is

(

)

e1 ( q ) sT14 /p cos qd W = 2e1 ( q ) sT14 cos qsinqd q for ϕ1 independent of ϕ, and similarly for sur-

face 2. For specular reflections, the directionality of the emitted energy is retained throughout the exchange process for the parallel-plate geometry. Hence, for each direction the energy is exchanged as in Equation (6.23). Then, by integrating over the exchanges within all solid angles, the energy transfer for nondiffuse emission is (all reflections are specular)

(

p/ 2

q d q. ) ò 1/e ( qsin) +q1cos /e ( q ) - 1

q1, nd = s T14 - T24 2

1

0

2

For diffuse emission the ε1 and ε2 are each independent of θ, so the ratio of transfer for nondiffuse emitting surfaces to diffuse emitting surfaces is p/ 2



q1, nd =2 q1, d

sin q cos q

ò éë1/e (q)ùû + éë1/e 0

1

2

( q )ùû - 1

dq

1 . (1/e1 ) + (1/e2 ) - 1

Radiation among Nondiffuse Nongray Surfaces

275

As a simple illustration, let ε1(θ) = ε2(θ) = C cos θ, as in Example 2.3. The C ≤ 1 is constant, and from Example 2.3 the hemispherical ε1 = ε2 = 2C/3. The integration yields

é 8 æ Cö 4 ù æ C ö q1,nd = - ê 2 ln ç 1 - ÷ + + 1ú ç ÷ . q1,d 2ø C è ëC û è 3-C ø

For C = 0.5, 0.8, 1.0 this gives, respectively, radiative energy flux ratios of q1,nd/q1,d = 1.029, 1.060, 1.090. The specified ε(θ) can increase the heat exchange up to 9%; the effect is smaller when C, the maximum ε(θ), is small.

6.5.2 Ray Tracing and Image Formation When there are one or more specular surfaces in an enclosure so that mirror-like reflections occur, the procedures of geometric optics can be applied to the radiative exchange process. The basic ideas are outlined here; more advanced ideas are in Stone (1963) and Born et al. (1999). An incident ray striking a specular surface is reflected symmetrically about the surface normal, with the angle of reflection equal in magnitude to the angle of incidence. This is used to formulate the concept of images. An image is an apparent point of origin for an observed ray. In Figure 6.16a, an observer views an object in a mirror. The object appears to be an image behind the mirror in the

276

Thermal Radiation Heat Transfer

FIGURE 6.16  Ray tracing and image formation by specular reflections: (a) Image formed by single reflection; (b) image formed by multiple reflections; and (c) contributions due to emission from specular surfaces.

position shown by the dotted object. This concept is readily extended to cases in which a series of reflections occur, as in Figure 6.16b. An interesting example of this is the “barber-chair” geometry, where there are planar mirrors on opposite walls of a barber shop. If the mirrors are parallel, many reflections occur and a person receiving a haircut can view many images of himself or herself. To this point, mirrors have been discussed only for their ability to change the direction of the rays. For thermal-radiation analysis, the specular surfaces have a finite absorptivity and will thus attenuate the energy of the rays for each reflection. A mirror surface will also emit energy. Emission can be conveniently analyzed using an image system, because in the image system all radiation will act along straight lines without the complexity of directional changes at each reflecting surface. The attenuation at each surface is included by multiplying the ray intensity by the specular reflectivity at each reflection. The emission from three surfaces is illustrated in Figure 6.16c. For example, emitted energy reaching the viewer from surface 3 may be coming directly from the image of 3, with attenuation due to reflections at surfaces 2 and 1.

Radiation among Nondiffuse Nongray Surfaces

277

6.5.3 Radiative Transfer among Simple Specular Surfaces for Diffuse Energy Leaving a Surface Methods such as Monte Carlo (Chapter 14) make amenable the treatment of specular surfaces even in complex enclosure geometries, but the information presented in this section gives useful insight into the behavior of such enclosures. As an introduction to radiative analysis in an enclosure with some specularly reflecting surfaces, a few examples are considered for plane surfaces. The emission from all surfaces is assumed diffuse. This is a reasonable simplifying assumption in many cases, as indicated by the electromagnetic theory predictions of the emissivity of specular surfaces in Figure 3.2. The reflected energy is assumed either diffuse (for the nonmirror-like surfaces) or specular. In the enclosure theory that is developed, the diffuse reflected energy is combined with the diffuse emitted energy, and it is then necessary to know how the transfer of this diffuse energy is influenced by the presence of specularly reflecting surfaces. The exchange factors Fs that will be obtained by considering the presence of the specular surfaces will already account for the specularly reflected energy, which has a specific directional behavior in contrast to the diffuse energy. Hence, it is necessary to consider here only the transfer of diffuse energy leaving a surface in the presence of specular surfaces. Figure 6.17a shows a diffusely emitting and reflecting plane surface A1 facing a specularly reflecting plane surface A2. Surface 1 cannot directly view itself; the ordinary configuration factor from any part of A1 to any other part of A1 is thus zero. However, with A2 specular, A1 can view its image, and a path exists by means of specular reflection for diffuse radiation to travel from the differential area dA to dA1* . By using a ray tracing approach, the radiation arriving at dA1* from dA is 1

1

FIGURE 6.17  Radiation exchange between a diffuse surface and itself by means of a specular surface: (a) Radiation exchange between two differential areas with one intermediate specular reflection; (b) radiation from differential area to finite area by means of one intermediate specular reflection; (c) radiation from finite area that is reflected back to that area by means of one specular reflection; and (d) radiation from dA1 can reach only a portion of A1 by means of specular reflection from A2.

278

Thermal Radiation Heat Transfer

determined and appears to come from the image dA1(2). Thus, the configuration factor between dA1 and dA1* resulting from one reflection can be obtained as dF for diffuse radiation leaving dA . d1(2)–d1*

1

The subscript notation refers to dF from the image of dA1 (as seen in A2) to dA1*. There are points of similarity that should be noted when comparing specular and diffuse reflecting cases. When A1 and A2 in Figure 6.17a are both diffuse reflectors, the diffuse radiation leaving dA1 is received at dA1* by means of diffuse reflection from A2 and is governed by dFd1–d2 and then dFd2–d1 for each element dA2 on A2. The portion of the diffuse energy from dA1 that reaches dA1* after one reflection from A2 is dA1 J1r2

ò

A2

dFd1-d 2 Fd 2-d1* for a diffuse A2, and is dA1J1ρs,2dFd1(2)−d1* for

a specular A2. This reveals that, for ρ2 = ρs,2, the difference in the two exchanges is incorporated in the configuration factors resulting from the nature of the reflection; this is a purely geometric effect. Figure 6.17b describes the diffuse radiation from dA1 that reaches the entire area A1 by means of one specular reflection. The reflected radiation appears to originate from the image dA1(2). Thus, the geometric configuration factor involved from dA1 to A1 is Fd1(2)−1. From the symmetry revealed by the dot-dash lines in Figure 6.17b, Fd1(2)−1 = Fd1−1(2). Thus, the radiative transfer from dA1 to A1 can also be expressed using a configuration factor from the first surface to the image of the second surface. Figure 6.17c shows typical rays leaving A1 that are reflected to A1; the rays appear to originate from the image A1(2). The configuration factor from A1 back to itself by means of one specular reflection is then F1(2)−1. For the geometry of Figure 6.17c, all of the image A1(2) is visible in A2 from any position on A1. In some instances, however, this is not true. For the displaced A2 in Figure 6.17d the radiation from dA1 must be within the limited solid angle shown shaded for the radiation to be reflected to A1. The configuration factor between dA1 and A1 is still Fd1(2)−1, but this factor is evaluated only over the portion of A1 that receives reflected rays. The Fd1(2)−1 is the factor by which dA1(2) views A1 and the view may be a partial one. The factor from dA1 to A1 has a different value, as the location of dA1 changes along A1. This means that the energy from A1 that is reflected to A1 will have a nonuniform distribution along A1. The reflected part of this energy from A1 will provide a nonuniform J from A1; this violates the assumption in the enclosure theory of uniform J from each surface. When partial images are present, the area is subdivided into sufficiently small portions that the solution accuracy is adequate. Now consider the geometry when there are additional specular surfaces involved in the radiation exchange; this provides multiple specular reflections. At each reflection, the radiation is modified by the ρs of the reflecting surface. Two specular surfaces are shown in Figure 6.18. Energy is emitted diffusely from A2 and travels to A1. The fraction arriving at dA1 is given by the diffuse factor dF2−d1. This direct path is illustrated in Figure 6.18a. A portion of the energy intercepted by A1 is reflected to A2 and then reflected to A1. Hence, A2 views dA1 not only directly but also by means of an image formed by two reflections, as constructed in Figure 6.18b. First the image A1(2) of A1 reflected in A2 is drawn. Then, A2 is reflected into this image to form A2(1−2). The notation A2(1−2) is read as the image of area 2 formed by reflections in area 1 and area 2 (in that order). The radiation paths and the shaded area in Figure 6.18b reveal that the solid angle within which radiation leaving A2 reaches dA1 by means of two reflections is the same as the solid angle by which dA1 views the image A2(1−2). Thus, the configuration factor involved for two reflections is dF2(1−2)−d1. This is interpreted as the factor from the image of surface 2 formed by reflections in surfaces 1 and 2 (in that order) to area element d1. Consider the possibility of additional images. The geometric factor involved is always found by viewing dA1 from the appropriate reflected image of A2 as seen through A2 and all intermediate images. In the case of Figure 6.18c, the images of A2 after four reflections A2(1−2−1−2) cannot view dA1 by looking through A2. Hence, there is no radiation leaving A2 that reaches dA1 by means of four reflections, and no additional images need be considered.

Radiation among Nondiffuse Nongray Surfaces

279

FIGURE 6.18  Radiative energy exchange between two specularly reflecting surfaces. (a) Energy emitted from A2 that directly reaches dA1; (b) energy emitted from A2 that reaches dA1 after two reflections; and (c) none of the energy emitted by A2 reaches dA1 by means of four reflections.

Example 6.10 The infinitely long (normal to the cross-section) groove in Figure 6.19 has specularly reflecting sides that emit diffusely. What fraction of the energy emitted from A2 reaches the black receiver surface element dA3 (an infinitely long differential strip)? Express the result in terms of diffuse configuration factors. Consider first the energy reaching dA3 directly from A 2 and by means of an even number of reflections. The fraction of emitted radiation that reaches dA3 directly from A 2 is dF2−d3, as illustrated in Figure 6.19a. A second portion is that emitted from A 2 to A1, reflected back to A 2, and then reflected to dA3. From the image diagram in Figure 6.19b, only part of the reflected image A 2(1−2) can be viewed by dA3 through A 2. The fraction of emitted energy reaching dA3 by this path is the configuration factor evaluated only over the part of A 2(1−2) visible to dA3 multiplied by the two specular reflectivities, ρs,1ρs,2dF2(1−2)–d3. This is not an ordinary configuration factor but takes into account the view through the image system. Similarly, there is a contribution after two reflections from each of A1 and A 2. This is illustrated by the shaded solid angle in Figure 6.19c. The third

280

Thermal Radiation Heat Transfer

FIGURE 6.19  Diffuse emission from one side of a specularly reflecting groove that reaches a differentialstrip receiving area outside the groove opening: (a) Geometry of direct exchange from A2 to dA3; (b) geometry of exchange for radiation from A2. image of A 2, A 2(1−2−1−2−1−2), cannot be viewed by dA3 through A 2 and hence does not make a contribution. Also, the third image of A 2 cannot view A1 through A 2, so there are no additional images of A 2. For the energy emitted by A 2, the sum of the fractions that reach dA3 both directly and by means of the images of A 2 resulting from an even number of reflections is then

dF2-d 3 + r s,1r s,2dF2(1-2)-d 3 + r2s,1r2s,2dF2(1-2-1-2)-d 3 .

Now consider the energy that reaches dA3 from A 2 by means of an odd number of reflections. Using Figure 6.19d and arguments similar to those for an even number of reflections results in

r s,1dF2(1)-d 3 + r2s,1r2s,2dF2(1-2-1)-d 3 + r3s,1r2s,2dF2(1-2-1-2-1)-d 3 .

281

Radiation among Nondiffuse Nongray Surfaces

The first two F factors are evaluated over only the portions of the images of A 2 that can be viewed by dA3. The diffuse energy emitted by surface A 2 that reaches dA3 directly and by all interreflections from both A1 and A 2 is then governed by the summation



dF2-d 3 + r s,1dF2(1)-d 3 + r s,1r s,2dF2(1-2)-d 3 + r2s,1rs,2dF2(1-2-1)-d 3 + r2s,1r2s,2dF2(1-2-1-2)-d 3 + r3s,1r2s,2dF2(1-2-1-2-1)-d 3 .



The notation adopted for the specular configuration factors allows a check on the form of the equations for radiant interchange among specular surfaces. The numbers within the subscripted parentheses designate the sequence of reflections from the specular surfaces. The configuration factor is multiplied by a reflectivity for each of these specular surfaces to account for energy absorption at the surfaces. For example, the factor FA(B−C−D)−E is multiplied by ρs,Bρs,Cρs,D. Details about the absorption and emission of radiation by specular grooves were given by Howell and Perlmutter (1963) and Mulford et al. (2018). Such grooves may be useful in the collection and concentration of solar energy. For this purpose an absorbing surface is placed at the bottom of each groove, which can have sides in the form of parabolic segments (Winston 1974), as in Figure 6.20a, or a series of straight sections (Mannan and Cheema 1977), as in Figure 6.20b. These configurations provide a concentration of the incident energy onto the absorber plane, which provides the elevated temperatures needed for efficient energy conversion. The groove shapes give good concentration even when incident radiation is not well aligned normal to the absorber plate. This type of solar collector is designed to perform well as the angle of the sun changes throughout the day, even though the collector remains in a fixed position.

6.5.4 Configuration-Factor Reciprocity for Specular Surfaces; Specular Exchange Factors Reciprocity relations analogous to those for configuration factors between diffuse surfaces apply for the factors involving specular surfaces under certain conditions. Consider a three-sided isothermal enclosure at temperature T, made up of two black surfaces 1 and 2 and a specular surface 3 of reflectivity ρs,3 (Figure 6.21a). The energy emitted by black surface 1 that reaches black surface 2 directly and by reflection from specular surface 3 is

Q1-2 = sT 4 ( A1 F1-2 + A1rs,3 F1(3)-2 ) . (6.24)

The energy leaving surface 2 that reaches surface 1 directly and by specular reflection from surface 3 is

Q2-1 = sT 4 ( A2 F2-1 + A2 rs,3 F1(3)-1 ) . (6.25)

Since A1F1−2 = A2F2−1 and, for the isothermal enclosure, Q2−1 = Q1−2, the reciprocity relation is obtained for one specular surface in the enclosure:

A1 F1(3)-2 = A2 F2(3)-1. (6.26)

A second type of reciprocity relation exists for configuration factors involving specular surfaces. Consider the energy exchange between two surfaces A1 and A2 of an isothermal enclosure. If both surfaces are specular, the images in Figure 6.21b can be constructed for radiation from surface 2 to 1 by means of a reflection at surface 1 and at surface 2. An analogous system can be constructed in which a plate with an aperture is substituted for the restraints on the ray paths that are present, as in

282

Thermal Radiation Heat Transfer

FIGURE 6.20  Concentrating solar collectors: (a) Parabolic trough concentrator; (b) concentrator made with flat segments.

Figure 6.21c. The aperture allows passage of only those rays that pass through the image system by which A2(1−2) can view at least a portion of A1 through A2 and A1(2). The emitted energy leaving specular surface A2 in the analog system and absorbed by A1 is

Q2(1-2 )-1 = Qe,2 rs,1rs,2 F2(1-2 )-1a1 = A2(1-2 ) e2 sT 4 rs,1rs,2 F2(1-2 )-1e1 (6.27)

for the assumed gray surfaces where α1 = ε1, and there are two intermediate specular reflections. The F2(1−2)−1 is the diffuse-surface configuration factor for the paths through the aperture (see Example 6.11). These paths are exactly those through the image system, so this is also the specular configuration factor. Similarly, the energy absorbed for the reverse path is

Radiation among Nondiffuse Nongray Surfaces

283

FIGURE 6.21  Reciprocity of configuration factors involving specular surfaces: (a) Three-sided enclosure with one specular reflecting surface; (b) system of images of surfaces 1 and 2; (c) energy-exchange analog of image system in (b); and (d) enclosure with two specular and two black surfaces.

Q1-2(1-2 ) = A1e1sT 4 rs,2 rs,1 F1-2(1-2 ) e2 . (6.28)



Equating the energy exchanges in either direction between A1 and A2(1−2) for the isothermal enclosure gives the reciprocity relation A1 F1-2(1-2 ) = A2(1-2 ) F2(1-2 )-1 = A2 F2(1-2 )-1. (6.29)



For nongray surfaces this still applies, as shown by considering the energy in each spectral region. For many intermediate reflections from surfaces A, B, C, D, …, Equation (6.29) can be generalized to A1 F1-2( A- B-C - D...) = A2 F2( A- B-C - D...)-1. (6.30)



For two-dimensional (2D) areas, the crossed-string method (Section 4.3.3.1) can be used to obtain the configuration factors. In Figure 6.21b, the A2 and A1(2) are regarded as apertures in the view between A1 and A2(1−2), and the F1−2(1−2) is found by having the crossed and uncrossed strings pass through these apertures. Example 6.11 A black surface A1 faces a smaller parallel mirror A 2 as in Figure 6.22. Determine the configuration factor F1−1(2) between A1 and the image of A1 formed by one specular reflection in A 2. The surfaces are infinitely long in the direction normal to the plane of the drawing.

284

Thermal Radiation Heat Transfer

FIGURE 6.22  Configuration-factor analysis involving partial views of surface and image (Example 6.11). (a) Portion of A1(2) in view from dA1 through entire A2; (b) limiting x for portion of A1(2) to be in view through entire A2; and (c) portion of A1(2) in view through part of A2. The factor is determined from the integral F1-1( 2) = (1/A1 )

òF A1

d 1-1( 2)

dA1 . Consider the element dA1

at location x on A1. The factor for radiation from dA1 to the portion of A1(2) in view through A 2 is (see Example 4.2)



Fd1-1( 2) =

é 1 1 ( sin b¢ - sin b¢¢) = êê 2 2 êë

x+a

( x + a)

2

+ b2

-

x-a

( x - a)

2

ù ú . 2 ú +b ú û

285

Radiation among Nondiffuse Nongray Surfaces

This is valid until position x = l − 2a is reached (Figure 6.22b). For larger x, the geometry is as shown in Figure 6.22c. Then



Fd1-1( 2) =

é 1 1 ( sin b¢ - sin b¢¢) = êê 2 2 ëê

x+l

(x + l)

2

-

+ 4b 2

x-a

( x - a)

2

ù ú . ú + b2 ú û

The desired configuration factor is obtained by integrating to give F1-1( 2) =



ì l - 2a é l 1ï 1 1 ê 2 Fd1-1( 2)dx = í ê 2l l ï2 0 î 0 ëê

ò

+

1 2

ò

é ê ê l - 2a ê ë l

ò

x+l

(x + l)

2

+ 4b 2

2

2

-

x+a

( x + a)

2

x-a

( x - a)

2

+ b2

-

x-a

( x - a)

2

ù ú dx ú + b2 ú û

ù ü ú dx ï ý 2 ú +b ú ï û þ



2

aö æbö æbö æ = 1+ ç ÷ - ç1- ÷ + ç ÷ . lø èl ø èl ø è This result can be found more easily by the crossed-string method. The surface views its image through the aperture of A 2. Using the crossed-string method (Section 4.3.3.1), the crossed and uncrossed strings are passed through the aperture (Figure 6.23). Then 2lF1-1( 2) =

1 2

( å crossed - å uncrossed)

1 F1-1( 2) = 4l

é 4 l 2 + b2 - 4 êë

(l - a)

2

+ b ùú . û



2

Next, reciprocity is considered when there is more than one specular surface in an isothermal enclosure at temperature T. For simplicity, an enclosure such as Figure 6.21d is used, where there are two specular and two black surfaces. The rate of radiative energy exchange between the two black surfaces by direct exchange and by all specular reflection paths is



Q1-2 = A1 ( F1-2 + rs,3 F1(3)-2 + rs,4 F1( 4 )-2 sT 4 + rs,3rs,4 F1(3-4 )-2 +  + r r F

m m s,3 s,4 1( 3m - 4 n )-2

FIGURE 6.23  Geometry for the crossed-string method.

)

+ .

(6.31)

286

Thermal Radiation Heat Transfer

Q2-1 = A2 ( F2-1 + rs,3 F2(3)-1 + rs,4 F2( 4 )-1 sT 4



+ rs,3rs,4 F2( 4-3)-1 +  + r r F

m m s,3 s,4 2 ( 4 n -3m )-1

)

(6.32)

+ .

The shorthand notation (3m − 4n) means m reflections in 3 and n in 4; hence F1(3m–4n)–2 is the configuration factor to area 2 from the image of 1 formed by these m and n reflections. For an isothermal enclosure, Q1−2 = Q2−1, so Equations (6.31) and (6.32) can be written as Q1-2 Q2-1 = = A1 F1s-2 = A2 F2s-1 (6.33) sT 4 sT 4



where the Fs are specular exchange factors equal to the quantities in parentheses in Equations (6.31) and (6.32).

F1s-2 = F1-2 + rs,3 F1(3)-2 + rs,4 F1( 4 )-2 + rs,3rs,4 F1(3-4 )-2 +  + rsm,3rs,n 4 F1(3m -4n )-2 + . (6.34)

The exchange factor expresses how diffuse energy leaving a surface arrives at a second surface directly and by all possible intermediate specular reflections. In contrast to a configuration factor that cannot exceed 1, a specular exchange factor can be larger than 1. Equation (6.34) gives s F1-2 in terms of energy going from images of 1 to surface 2. An alternative form is in terms of radiation from surface 1 directly to surface 2 and by means of reflections to the images of 2; from Equation (6.33),

F1s-2 =

(

)

A2 s A F2-1 = 2 F2-1 + rs,3 F2(3)-1 + rs,4 F2( 4 )-1 + rs,3rs,4 F2( 4-3)-1 +  . A1 A1

Equation (6.30) is applied to each F factor in the series. The area ratio cancels and the desired result is

F1s-2 = F1-2 + rs,3 F1-2(3) + rs,4 F1-2( 4 ) + rs,3rs,4 F1-2( 4-3) + . (6.35)

Now, looking at Equations (6.31) and (6.32) in more detail, since A1F1−2 = A2F2−1, and from Equation (6.26) for one reflection A1F1(3)−2 = A2F2(3)−1 and A1F1(4)−2 = A2F2(4)−1, the equality in Equation (6.33) reduces to (after dividing by ρs,3ρs,4)



æ ö A1 ç F1(3-4 )-2 +  + rsm,3-1rsn,-41 F1 3m -4n -2 +  ÷ ( ) è ø ö æ = A2 ç F2( 4-3)-1 +  + rsm,3-1rsn,-41 F2 4n -3m -1 +  ÷ . ( ) è ø

(6.36)

This equality must hold in the limit as ρs,3 and ρs,4 approach zero so that

A1 F1(3-4 )-2 = A2 F2( 4-3)-1 (6.37)

which is a geometric property of the system. A continuation of this reasoning leads to the general reciprocity relation:

A1 F1( A- B-C - D)-2 = A2 F2(D-C - B- A)-1. (6.38)

287

Radiation among Nondiffuse Nongray Surfaces

An additional relation is found by combining Equations (6.30) and (6.38) to give A1 F1( A- B-C - D)-2 = A2 F2(D-C - B- A)-1 = A1 F1-2(D-C - B- A) (6.39)

which shows that

F1( A- B-C - D)-2 = F1-2(D-C - B- A) . (6.40)



This can also be deduced directly from the fact that an image system can be constructed either starting with the real surface 1 and working toward image 2(… D − C − B − A), or starting with image 1(A − B − C − D …) and working toward real surface 2; the geometry of the construction is the same in both systems, so the configuration factors between the initial and final surfaces are the same. A specular exchange factor Fs can be larger than 1 as the infinite number of reflections between two parallel mirrors of infinite extent takes place. This can be shown by considering two mirrors, designated as 1 and 2. For infinite parallel areas, the configuration factor between the mirrors and any of its images is 1, so that the exchange factor is given by

F1s-2 = 1 + rs,2 rs,1 + (rs,2 rs,1 )2 + (rs,2 rs,1 )3 + (rs,2 rs,1 )4 +  =

1 1 - rs,1rs,2

which is larger than 1. This does not violate the law of conservation of energy.

6.6 NET-RADIATION METHOD IN ENCLOSURES HAVING BOTH SPECULAR AND DIFFUSE REFLECTING SURFACES 6.6.1 Enclosures with Planar Surfaces In this section, radiation exchange theory is developed for an N-surfaced enclosure, where d surfaces are diffuse reflectors and N − d are specular. All the surfaces emit diffusely and are gray. Let the diffusely reflecting surfaces be 1 through d and the specularly reflecting surfaces be d + 1 through N. If there are no external sources of unidirectional energy entering through a window and striking a specular surface, all the energy within the enclosure originates by surface emission that is diffuse. At each diffuse reflecting surface, the reflected diffuse energy is combined with the emitted energy to form J, which is all diffuse. At each specular surface, the only diffuse energy leaving is εσΤ4, since reflected energy is directional and cannot be combined with the diffuse emission to give J. The transport between surfaces of the diffuse energies J and εσΤ4 is determined by the specular exchange factors Fs. The FAs-B expresses how the diffuse energy leaving surface A arrives at surface B by the direct path and by all possible paths involving specular reflections. The reflected energy from the specular surfaces is already included in the Fs and does not have to be considered after the Fs have been obtained. Hence, by accounting for all paths for J from the diffusely reflecting surfaces and for εσΤ4 from the specularly reflecting surfaces to reach a particular surface, all of the incident energy has been accounted for, including the effects of both diffuse and specular reflections. Then, at any surface, the incident energy is N

d



Gk Ak =

å

J j A j Fjs-k + s

j =1

åe T A F j

4 j

j

s j -k

1 £ k £ N .

j = d +1

After applying reciprocity (Equation (6.33)), the areas cancel and Gk becomes d



Gk =

å

J j Fks- j + s

j =1

N

åe T F j

j = d +1

4 s j k- j

1 £ k £ N . (6.41)

288

Thermal Radiation Heat Transfer

A set of enclosure relations is now derived in terms of G and J. For the diffuse reflecting surfaces, it is convenient to start with the energy-balance Equations (5.10) and (5.11):

Qk = qk Ak = ( J k - Gk ) Ak 1 £ k £ d (6.42)



J k = e k sTk4 + (1 - e k ) Gk 1 £ k £ d. (6.43)

For the specular reflecting surfaces, although these same energy balances apply, the Jk is eliminated as it consists of diffuse emission e k sTk4 , and specular reflection (1 − εk)Gk hence has an inconvenient directional character. Eliminating Jk from Equations (6.42) and (6.43) gives

(

Qk = Ak e k sTk4 - Gk



)

d + 1 £ k £ N . (6.44)

Equations (6.41) through (6.44) can be combined in various ways to obtain equations for the desired unknowns, depending on which quantities are specified. Consider when the temperatures are specified for all the surfaces and it is desired to obtain the net external energy Qk added to each surface. Equation (6.41) is substituted into Equation (6.43) to eliminate Gk and obtain the following equation for each diffuse surface:

J k - (1 - e k )

d

å

J j Fks- j = e k sTk4 + (1 - e k ) s

j =1

N

åe T F j

4 s j k- j

1 £ k £ d. (6.45)

j = d +1

This set of equations is solved for the J for the diffuse reflecting surfaces. This is somewhat simpler than for an enclosure having all diffuse surfaces, as there are only d simultaneous equations rather than N. For each specular surface, the J for the diffuse surfaces are used to obtain Gk from Equation (6.41): N

d



Gk =

å

J j Fks- j + s

j =1

åe T F j

d + 1 £ k £ N . (6.46)

4 s j k- j

j = d +1

The net external energy added to each diffuse reflecting surface is obtained by eliminating Gk from Equations (6.42) and (6.43),

Qk = Ak

ek sTk4 - J k 1 - ek

(

)

1 £ k £ d (6.47)

and the Qk to each specular surface is found from Equation (6.44). Equations (6.44) through (6.47) are general energy-interchange relations for enclosures of diffuse and specular reflecting surfaces. If the kth diffuse surface is black, then J k = sTk4 and 1 − εk = 0, so Equation (6.47) is indeterminate. In this case, from Equation (6.44),

(

)

Qk = Ak sTk4 - Gk (6.48)

where Gk is found from Equation (6.46) with 1 ≤ k ≤ d. If the energy input Qk rather than Tk is specified for a diffuse reflecting surface 1 ≤ k ≤ d, then Tk is unknown in Equation (6.45). Equation (6.47) can be used to eliminate this unknown in terms of Jk and the known Qk. If the heat input Qk is specified for a specular surface, d + 1 ≤ k ≤ N, then one of the T j4 in the last term of Equation (6.45) is unknown. Equation (6.44) is combined with Equation (6.46) to eliminate Gk, to give

289

Radiation among Nondiffuse Nongray Surfaces

sk Tk4 -



Qk = Ak e k

N

d

å

J j Fks- j + s

j =1

åe T F j

4 s j k- j

d + 1 £ k £ N . (6.49)

j = d +1

Since Qk is known, Equation (6.49) can be combined with Equation (6.45) to yield a simultaneous set of equations to determine the J of the diffusely reflecting surfaces and the T for the specularly reflecting surfaces having specified Q. If all the surfaces are specular, a simultaneous solution is not required. The Gk are given by Equation (6.46) as N

åe T F

Gk = s



1 £ k £ N (6.50)

4 s j k- j

j

j =1

and the Qk are then found from Equation (6.44). By substituting Equation (6.50) into Equation (6.44), the Qk are found directly from the specified surface temperatures: æ Qk = sAk e k ç Tk4 ç è



ö e j T j4 Fks- j ÷ . (6.51) ÷ j =1 ø N

å

A useful form for the enclosure equations is found by using Equations (6.47) and (6.44) to eliminate G and J from Equations (6.47) and (6.48). This gives a set of equations of the same form that directly relate the Q and T:

1 Qk e k Ak

d

å j =1

Qj 1 - e j s Fk - j = sTk4 - s Aj e j

N

d

å

T j4 Fks- j - s

j =1

åe T F j

4 s j k- j

1 £ k £ N . (6.52)

j = d +1

Equation (6.51) is the special case of Equation (6.52) for d = 0. Equation (6.52) can be used to obtain relations between the Fs exchange factors analogous to the relation for diffuse surfaces

å

N j =1

Fk - j = 1. Let the entire enclosure be at uniform temperature. Then

there is no net energy exchange, and all the Q are zero, so Equation (6.52) reduces to N

d

å

Fks- j +



j =1

åe F

s j k- j

= 1. (6.53)

j = d +1

If all the surfaces in the enclosure are specular (d = 0), this further reduces to N



å

e j Fks- j =

j =1

N

å(1 - r

s, j

)Fks- j = 1. (6.54)

j =1

The set of enclosure Equations (6.52) is not difficult to solve after the exchange factors have been found; determining these factors, however, may not be easy, depending on the complexity of the enclosure geometry. To illustrate the calculations, some specific enclosures are considered. Figure 6.24a shows an enclosure of three plane surfaces at different uniform specified temperatures; two sides are diffuse reflectors, and the third is specular. In Figure 6.24b, the energy arriving at Α1 comes directly from the diffuse A2 and A3, without any intermediate specular reflections. Hence, F2s-1 = F2-1 and F3s-1 = F3-1. By applying reciprocity, F1s-2 = F1-2 and F1s-3 = F1-3. For A2, the incoming

290

Thermal Radiation Heat Transfer

FIGURE 6.24  Enclosure having one specular reflecting surface and two surfaces that are diffuse reflectors: (a) General geometry; (b) energy fluxes to A1, and energy fluxes that contribute to flux incident upon A2; and (c) enclosure for Example 6.12.

radiation is composed of four parts that originate as shown in Figure 6.24b. The first is the diffuse energy originating from A3 and going directly to A2, which is J3A3F3−2. The remaining three parts arrive by means of A1 and consist of an emitted portion e1sT14 A1 F1-2 and two specularly reflected portions. The latter arise from the energy leaving A2 and A3 that is specularly reflected to A2 and appears to come from the images A2(1) and A3(1) in Figure 6.24a. The specularly reflected portions are J2ρs,1A2F2(1)−2 + J3ρs,1A3F3(1)−2. Multiple specular reflections cannot occur when only one planar specular surface is present. The specular exchange factors are then F1s-2 = F1-2 , F2s-2 = rs,1 F2(1)-2 , and F2s-3 = F2-3 + rs,1F3(1)-2 . By using reciprocity, F2s-1 = F2-1 , F2s-2 = rs,1F2-2(1) , and F2s-3 = F2-3 + rs,1 F2-3(1) . Similarly, for surface 3, F3s-1 = F3-1 , F3s-2 = F3-2 + rs,1 F3-2(1) and F3s-1 = rs,1 F3-3(1) . A numerical example using these factors is given in Example 6.12.

291

Radiation among Nondiffuse Nongray Surfaces

Example 6.12

2 = F3-2

An enclosure with three sides is in Figure 6.24b. The length L is long enough that the triangular ends can be neglected in the radiative energy balances. Two surfaces are black, and the third is a gray diffuse emitter with ε1 = 0.05. Determine the energy added per meter of enclosure length to each surface for the two cases: (1) A1 is a diffuse reflector and (2) A1 is a specular reflector. From symmetry, F1−2 = F1−3. Also, F1−2 + F1−3 = 1, so F1−2 = F1−3 = 1/2. From reciprocity,F2-1 = A1F1-2 /A2 = 2 / 2 = F3-1. In addition, F2-1 + F2-3 = 1, so that F2-3 = 1 - 2 / 2 = F3-2 = F2-2(1) = F3-3(1) . = F2-2(1) = F3-3(1) . Finally, F3-2(1) = F2-3(1) = 1 - F3-2 = 2 - 1. For case 1, apply Equations (5.11.1) through (5.11.3) to obtain 1 1 1 Q1 æ ö = s ç 5254 - 5254 - 7504 ÷ . 2 2 0 05 0 .3 2 è ø



-

é ù æ 2 1 - 0.05 Q2 2ö Q1 2 4 = s ê+ 5254 + 5254 - çç 1 ÷÷ 750 ú 2 ø 0.3 0.3 2 2 0.05 êë 2 úû è

-

é ù æ Q1 2 1 - 0.05 Q3 2ö 2 4 4 5254 - çç 1 + = s ê÷÷ 525 + 750 ú . 2 ø 0 .3 0.3 2 2 0.05 êë 2 úû è

The solution of these three equations yields (Qs are per meter of enclosure length): Q1 = −144.6 W/m, Q2 = −2571.7 W/m, and Q3 = 2716.4 W/m. The energy supplied to A3 is removed from A1 and A 2. The amount removed from A1 is small, because A1 is a poor absorber. For case 2, apply Equation (6.45) to compute J2 and J3. Since ε2 = ε3 = 1, this yields J2 = sT24 and J3 = sT34 for the black surfaces. Then Equation (6.41) yields the G for each surface as 1 é1 ù G1 = s ê 5254 + 7504 ú = 11,125 W/m2 2 ë2 û é ù ïü æ 2 2ö 2 ïì 4 + (1 - 0.05)( 2 - 1)ú ý + 5254 (1 - 0.05) çç 1 G2 = s í0.05(525)4 ÷÷ + 750 ê1 2 2 2 êë úû ïþ ïî ø è



= 13, 666 W/m2



ìï é ù æ 2 ö üï 2 2 G3 = s í0.05(525)4 + 5254 ê1 + (1 - 0.05)( 2 - 1)ú + 7504 (1 - 0.05) çç 1 ÷ý 2 ÷ø ïþ 2 2 êë úû ïî è = 8,101 W/m2 . Equations (6.44) and (6.48) are applied to find the Qs (per meter of enclosure length) as: Q1 = −144.6 W/m, Q2 = −2807.4 W/m, and Q3 = 2952.1 W/m. By making A1 specular, the heat transferred from A3 to A 2 is increased by 9% from 2572 to 2807 W/m. An alternative for case 2 is to use Equation (6.52) to give



Q1 = e1A1s(T14 - T24F1- 2 - T34F1-3 ) Q2 = A2s[e1T14F2-1 + T24 (1 - r s,1F2(1)- 2 ) - T34 (F2-3 + r s,1F2-3(1) )] Q3 = A3s[-e1T14F3-1 - T24 (F3- 2 + r s,1F3- 2(1) ) + T34 (1 - r s,1F3-3(1) )]. Substituting values yields the same results as before.

To further demonstrate enclosure analysis with specularly reflecting surfaces, the rectangular geometry in Figure 6.25 is used. All surfaces are diffuse emitters; two surfaces are diffuse reflectors,

292

Thermal Radiation Heat Transfer

and two are specular. The reflected images are shown dashed. Reflections continue until all the outer perimeter enclosing the composite of the original enclosure plus the reflected images consist of either diffuse reflecting (or nonreflecting, such as an opening) surfaces, or images of diffuse surfaces. For the enclosure in Figure 6.25, the first step is to use Equation (6.45) to obtain J1 and J2 for the two diffuse areas. For the exchange factors, consider first F1s-1. Part of the energy leaving A1 returns to A1 by each of three paths: direct reflection from A3, reflection from A3 to A4 and then to A1, and reflection from A4 to A3 and then to A1. Thus, the energy leaving A1 that returns to A1 is expressed by the specular exchange factor F1s-1 = rs,3 F1(3)-1 + rs,3rs,4 F1(3-4 )-1 + rs,4 rs,3 F1( 4-3)-1 . The F1(3−4)−1 is the configuration factor by which A1(3−4) is viewed from A1 through A4 and then A3(4), which are the reflection areas by means of which the A1(3−4) image was formed. Similarly, F1(4−3)−1 is the factor by which the same area A1(3−4) is viewed from A1 through A3 and then A4(3). Radiation leaving A2 reaches A1 along four paths: direct exchange, reflection from A3, reflection from A4, and reflection from A3 to A4. No energy from A2 reaches A1 by means of reflections from A4 and then A3. This is because A1 cannot view the image A2(4−3) through area A3. This gives F2s-1 = F2-1 + rs,3 F2(3)-1 + rs,4 F2( 4 )-1 + rs,3rs,4 F2(3-4 )-1. The diffuse energy leaving the specular surface A3 (and similarly for A4) consists only of emitted energy e3 A3sT34 . There are two paths by which some of this reaches A1: direct exchange, and by means of specular reflection from A4. The result is F3s-1 = F3-1 + rs,4 F3( 4 )-1 and F4s-1 = F4-1 + rs,3 F4(3)-1 .

FIGURE 6.25  Rectangular enclosure and reflected images when two adjacent surfaces are specular reflectors and the other two are diffuse reflectors.

Radiation among Nondiffuse Nongray Surfaces

293

After applying F factor reciprocity, the factors are substituted into Equation (6.45) to give:



J1 - (1 - e1 ){J1[rs,3 F1-1(3) + rs,3rs,4 ( F1-1(3-4 ) + F1-1( 4-3) )] + J 2 ( F1-2 + rs,3 F1-2(3) + rs,4 F1-2( 4 ) + rs,3rs,4 F1-2(3-4 ) )} (6.55) = e1sT14 + (1 - e1 )s[e3T34 ( F1-3 + rs,4 F1-3( 4 ) ) + e 4T44 ( F1-4 + rs,3 F1-4(3) )].

Similarly, considering J2 for surface 2 gives:

J 2 - (1 - e2 ){J1 ( F2-1 + rs,3 F2-1(3) + rs,4 F2-1( 4 ) + rs,3rs,4 F2-1( 4-3) ) + J 2 [rs,4 F2-2( 4 ) + rs,3rs,4 ( F2-2( 4-3) + F2-2(3-4 ) )]} (6.56) = e2 sT24 + (1 - e2 )s[e3T34 ( F2-3 + rs,4 F2-3( 4 ) ) + e 4T44 ( F2-4 + rs,3 F2-4(3) )].

Equations (6.55) and (6.56) are solved simultaneously for J1 and J2. For the two specular surfaces, the G3 and G4 can be found as soon as the J for the diffuse surfaces are known. From Equation (6.46),

G3 = J1 ( F3-1 + rs,4 F3-1( 4 ) ) + J 2 ( F3-2 + rs,4 F3-2( 4 ) ) + e 4 sT44 F3-4 (6.57)



G4 = J1 ( F4-1 + rs,3 F4-1(3) ) + J 2 ( F4-2 + rs,3 F4-2(3) ) + e3sT34 F4-3 . (6.58)

With the J for the diffuse surfaces and the G for the specular surfaces known, the Q to maintain the specified surface temperatures are found from Equations (6.47) and (6.44). The solution can also be found from Equation (6.52), directly relating the Q and T. Clearly, determining the specular exchange factors can become tedious in enclosures with many surfaces and particularly in 3D enclosures. A more direct approach is the Monte Carlo method, wherein the direction of reflected energy can be specified for each incident small “bundle” of energy, since it approaches a specular surface from a definite direction (Chapter 14). A similar statistical method involving Markov chain theory has been developed by Naraghi and Chung (1984) and extended by Billings et al. (1991b).

6.6.2 Curved Specular Reflecting Surfaces When specular reflecting surfaces are curved, the reflected image geometry can become quite complex. To demonstrate some ideas, consider the exchange of radiation within a specular tube (Perlmutter and Siegel 1963), as in Figure 6.26. It is assumed that the imposed temperature or heating conditions depend only on axial position and are independent of location around the tube circumference. To compute the radiative exchange within the tube for axisymmetric heating, the configuration factor between two ring elements on the tube wall is needed. The direct exchange (Figure 6.26a) is governed by the factor (see Appendix C, Factor 15):

ì ( X /D)3 + 3 X / 2 D ü dFdX1 -dX = í1 ý dX . [( X /D)2 + 1]3/ 2 þ î

Figure 6.26b illustrates the configuration factor for one specular reflection. All the radiation from dX1 that reaches dX by one reflection has been reflected from a ring element halfway between dX1

294

Thermal Radiation Heat Transfer

FIGURE 6.26  Radiation exchange within a specularly reflecting cylindrical tube: (a) Direct exchange between two ring elements; (b) exchange by one reflection; and (c) exchange by two reflections.

and dX. The ring at X/2 is only dX/2 wide, so the solid angle subtending it will spread to a width dX at the location X. The configuration factor for one reflection is then the factor between dX1 and the dashed element dX/2:

ì ( X / 2 D)3 + 3 X / 4 D ü dX dFdX1 -dX / 2 = í1 . ý [( X / 2 D)2 + 1]3/ 2 þ 2 î

Similarly, the factor for exchange by two reflections (Figure 6.26c) is

ì ( X / 3D)3 + 3 X / 6 D ü dX dFdX1 -dX /3 = í1 ý [( X / 3D)2 + 1]3/ 2 þ 3 î

and for n reflections

æ [ X / (n + 1) D]3 + 3 X / 2(n + 1) D ö dX dFdX1 -dX /( n+1) = ç 1 . ÷ {[( X / (n + 1) D]2 + 1}3/ 2 è ø n +1

295

Radiation among Nondiffuse Nongray Surfaces

In general, the geometric factor for any number of reflections is found by considering the exchange between the originating element (dX1 in this case) and the element (call it dX2) from which the first reflection is made (the dashed element in Figure 6.26b and c). This is because the fraction of energy leaving dX1 in the solid angle subtended by dX remains the same through the succeeding reflections along the path to dX. At each reflection the energy is multiplied by the specular reflectivity ρs. If all the contributions are summed, the energy leaving dX1 that reaches dX by direct exchange and all reflection paths is expressed by the specular exchange factor for the inside of a tube that is open at both ends (no reflections from end walls): ¥



s dFdX = 1 - dX

æ

å r èç1 n s

n =0

[ X / (n + 1) D]3 + 3 X / 2(n + 1) D ö dX . (6.59) ÷ {[( X / (n + 1) D]2 + 1}3/ 2 ø n +1

For a complete energy transfer formulation, the energy entering through the tube ends must be included, which requires the exchange factor from the end openings. This factor is obtained by a derivation similar to Equation (6.59). The energy is considered that leaves an element on the tube wall and travels to the circular disk opening at the end of the tube by a direct path or by one or more reflections along the tube wall between the element and the end opening. The exchange factor was obtained by Perlmutter and Siegel (1963), where it was shown that the energy exchange in the tube can be divided into two separate parts with the complete solution given by superposition. One part is for the tube being heated along its length, such as by electric heaters in the tube wall, but with the end environments at zero temperature (Figure 6.27). The outer surface of the tube is insulated, so the only energy loss is by radiation leaving through the tube ends. The energy balance on an element at x states that the heat addition q(x) equals the emitted energy minus that absorbed as a result of incident energy arriving by specular reflections. Since there is no diffuse reflection in this analysis, all of the arriving specular energy is obtained by using the specular exchange factor, Equation (6.59). The energy equation governing the wall temperature at x is then given by (a gray tube wall is assumed so that α = ε)

L éx ù s 4 ê q( x ) = esT ( x ) - e sTw ( y)dF ( x - y) + sTw4 ( y)dF s ( y - x ) ú (6.60) ê ú x ë0 û 4 w

2

ò

ò

where dFs(x − y) is given by Equation (6.59) with X = (x − y).

FIGURE 6.27  Heated tube with internal specular reflections; the outside surface of the tube is insulated.

296

Thermal Radiation Heat Transfer

Equation (6.60) can be solved numerically as an integral equation for Tw(x) if q(x) is specified, or it can be directly integrated to give q(x) if Tw(x) is specified. Effects of the differing reflectivities of the polarized components on transmission through channels are investigated by Edwards and Tobin (1967) and Miranda (1996). The radiative energy transmission through a polished stainless steel tube is investigated by Miranda, and comparisons are made of analysis with experiment. The tube surface was cooled to 77 K so radiation from the tube wall could be neglected. One end of the tube was cooled to 77 K, while a blackbody source at 300 K was placed at the other end. For a tube diameter of 0.067 m, measurements were made for tube lengths up to 1.2 m, and agreement with analysis was always within 8%. Additional comparison with experiment is in Qu et al. (2007a). When the configuration is more involved than the cylindrical geometry, the reflection patterns can become quite complex. Some examples for radiation within a specular conical cavity and a specular cylindrical cavity with a specular end plane are in Lin and Sparrow (1965); a more generalized treatment of nonplanar reflections is in Plamondon and Horton (1967). An approximate analysis of the transfer through specular passages is in Rabl (1977). It is based on estimating the average number of reflections that occur during transmission. The analyses by Tsai and Strieder (1985, 1986) for radiation within a cylindrical and a spherical enclosure consider a more general reflection behavior obtained by dividing the reflectivity into specular and diffuse components. These components can be treated using the methods in the enclosure theory given in this chapter. This reflection model was for application to radiation within high-temperature porous materials. The results of Tsai and Strieder (1985) show that, for a spherical cavity, the many specular reflections tend to eliminate directional effects and cause the radiation to behave diffusely. Example 6.13 A cylindrical cavity open at one end has a specularly reflecting cylindrical wall and base (Figure 6.28a). Determine the fraction of radiation from ring element dX1 that reaches dX by means of one reflection from the base with reflectivity ρs,1 and one reflection from the cylindrical wall with reflectivity ρs,2.

FIGURE 6.28  Reflection in cylindrical cavity with specular curved wall and base: (a) Cavity geometry; (b) image of dX1 formed by reflection in cavity base.

Radiation among Nondiffuse Nongray Surfaces

297

As shown in Figure 6.28b, for this geometry the reflected radiation from the base can be regarded as originating from an image of dX1. The second reflection occurs from an element of width dX/2 midway between the image dX1 and dX. The desired radiation fraction is given by the configuration factor from the image dX1 to the dashed ring area dX/2 multiplied by the two reflectivities:

æ [( X + X1) / 2D]3 + 3( X + X1) / 4D ö dX r s,1r s,2dFdX1-dX = r s,1r s,2 ç 1 . ÷ {[( X + X1) / 2D]2 + 1}3/ 2 è ø 2

Another type of curved specular surface of practical importance is a parabolic mirror such as in a solar furnace. The mirror axis is aligned in the direction of the sun and a receiver is placed at the mirror focal plane. It is desired to estimate the receiver temperature. Information on concentrators for solar furnaces is in Cobble (1961) and Kamada (1965). An inverse analysis for designing a mirror to achieve a specified intensity distribution at the receiver is in Zakhidov (1989). In Maruyama (1991, 1993), the angular distribution of intensity emitted from systems of cylindrical black emitters with various reflector shapes (circular arc, parabolic, and involute) are found by ray tracing. The reflector surfaces are assumed to be either perfect or metallic reflecting surfaces with reflectivity predicted by electromagnetic theory. In Maruyama (1993), there is experimental verification of predictions for circular arc and involute reflectors. The involute is effective for providing a uniformly distributed radiation source.

6.7 MULTIPLE RADIATION SHIELDS The results in Table 6.1 can be extended to obtain the performance of multiple radiation shields, as shown in Figures 6.29 and 6.30. The shields are thin, parallel, highly reflecting sheets, placed between radiating surfaces to reduce energy transfer between them. Highly effective insulation can be obtained by using many sheets separated by vacuum to provide a series of alternate radiation and conduction barriers. One construction approach is depositing highly reflecting metallic films such as aluminum or silver on both sides of thin sheets of plastic spaced apart by placing between them a cloth net having a large open area between its fibers. A stacking of 20 radiation shields per centimeter of thickness can be obtained. An important use of this multilayer insulation is in providing

FIGURE 6.29  Parallel walls separated by N radiation shields.

298

Thermal Radiation Heat Transfer

FIGURE 6.30  Radiation shields between concentric cylinders or spheres.

emergency protective blankets for forest firefighters and in low-temperature applications such as insulation of cryogenic storage tanks. It is used in satellites and other vehicles operating in outer space (Lin et al. 1996). For the results here, the spaces between the shields are evacuated so that heat transfer is only by radiation, and all emission is assumed diffuse. To analyze shield performance, consider N radiation shields between two surfaces at temperatures T1 and T2 with emissivities ε1 and ε2. As a general case, let a typical shield n have emissivity εn1 on one side and εn2 on the other, as in Figure 6.29. As a result of the heat flow, the nth shield will be at temperature Tsn. Because for steady state, the same q passes through the entire series of shields, Equation (6.23) can be written for each pair of adjacent surfaces as æ1 ö 1 qç + - 1 ÷ = s T14 - Ts14 è e1 e11 ø

(

æ 1 ö 1 - 1 ÷ = s Ts41 - Ts42 qç + e e 21 è 12 ø 

(



) )

æ 1 ö 1 qç + - 1 ÷ = s Ts4( N -1) - Ts4N ç e( N -1)2 e N 1 ÷ è ø ö æ 1 1 qç + - 1 ÷ = s Ts4N - T24 . e e N 2 2 ø è

(

(

)

)

299

Radiation among Nondiffuse Nongray Surfaces

Adding these equations and dividing by the resulting factor multiplying q on the left-hand side gives, after some rearrangement,

(

s T14 - T24

q=



1/e1 + 1/e2 - 1 +

)

å n=1 (1/en1 + 1/en2 - 1) N

. (6.61)

In most instances ε is the same on both sides of each shield, and all the shields have the same ε. Let all the shield emissivities be εs; then, q becomes q=



(

s T14 - T24

)

1/e1 + 1/e2 - 1 + N ( 2 /es - 1)

. (6.62)

If the wall emissivities are the same as the shield emissivities, ε1 = ε2 = εs, Equation (6.62) further reduces to q=



(

s T14 - T24

)

( N + 1)( 2 /es - 1)

. (6.63)

In this instance q decreases as 1/(N + 1) as the number of shields N increases. When there are no shields, N = 0, and Equation (6.62) reduces to q = s T14 - T24 éë(1/e1 ) + (1/e2 ) - 1ùû as in Table 6.1. To illustrate shield performance, if in Equation (6.62) ε1 = ε2 = 0.8 and εs = 0.05, then

(

(

)

)

q = s T14 - T24 (1.5 + 39 N ) and the ratio q(N shields)/q(no shields) = 1.5/(1.5 + 39 N). Table 6.2 shows the effect of adding shields. The factor 1/(N + 1) is the q ratio (independent οf ε) when the walls have the same ε value as for the shields, as in Equation (6.63). This illustrates that the fractional reduction in q is larger when the wall ε are large compared to the ε for the shields, since for small wall ε the unshielded energy transfer is already low. As for flat plates, the expressions in Table 6.1 can be used to derive the energy transfer through a series of concentric cylindrical or spherical radiation shields, as in Figure 6.30. If the walls A1 and A2 and all the shields Asn are diffuse reflectors, the energy transfer is (emission is diffuse)

Q=

(

A1s T14 - T24 1/e1 + ( A1 /A2 )(1/e2 - 1) +

)

å n=1 ( A1 /Asn ) (1/en1 + 1/en2 - 1) N

. (6.64)

If the walls are diffuse reflectors and all the shields are specular, then Q=

(

A1s T14 - T24 ì í1/e1 + 1/e11 - 1 + î

å

N -1 n =1

( A1 /Asn ) (

)

)

ü 1/e n 2 + 1/e( n +1)1 - 1 + ( A1 /AsN ) éë1/e N 2 + ( AsN /A2 )(1/e2 - 1) ùû ý þ

. (6.65)

300

Thermal Radiation Heat Transfer

If all the walls and shields are specular, Equation (6.65) applies if AsN/A2 is replaced by 1 in the last term in the denominator. In this instance, if all shield emissivities are the same and equal to εs, the result is

(

A1s T14 - T24

Q= 1/e1 + 1/es - 1 +

å

N -1 n =1

)

éë( A1 /Asn )( 2 /es - 1) ùû + ( A1 /AsN ) (1/es + 1/e n 2 - 1)

. (6.66)

The sum in Equations (6.65) and (6.66) is zero if N = 1. When using radiation shields to insulate a cryogenic tank, one or more of the shields can be cooled by using vapor boil-off from the tank. The resulting decrease of temperature within the insulation helps further reduce the energy loss from the tank. The optimum utilization of vapor cooling can be examined thermodynamically by minimizing the entropy production resulting from the heat losses and the controlled cryogen boil-off used for vapor cooling of one or more shields. This optimization has been analyzed by Chato and Khodadadi (1984) using the thermodynamic ideas of Schultz and Bejan (1983), and the idea is used in the design of the sun shield for the Webb infrared telescope (Chapter 19).

6.8 CONCLUDING REMARKS This chapter outlines the treatment of radiative exchange between directional and specularly reflecting surfaces, and in enclosures containing both specularly and diffusely reflecting surfaces. In many instances, as in Example 6.10, the radiative exchange of energy in enclosures is modified only slightly by considering specular in place of diffuse reflecting surfaces. In other applications, however, such as solar concentrators and solar furnaces, and radiative transmission through highly reflecting tubes to guide radiative energy to a detector, specular reflection is a dominant requirement. The design for these types of devices is often done by ray tracing; this has not been treated here, where the emphasis was on the analysis of enclosures using configuration factors. A ray-tracing technique by the Monte Carlo method is presented in Chapter 14. For enclosures, Bobco (1964), Sparrow and Lin (1965), Sarofim and Hottel (1966), Mahan et al. (1979), Tsai and Strieder (1986a,b), and Qu et al. (2007a, b) have examined radiative exchange involving surfaces with reflectivities that have both diffuse and specular components. Schornhorst and Viskanta (1968) compared experimental and analytical results for radiant exchange among various types of surfaces and found that, regardless of the presence of specular surfaces, the diffuse-surface analysis agreed best with experimental results. Bobco and Drolen (1989) provide a model to represent the reflectivity of a surface by combined diffuse and specular components. Jamaluddin and Fiveland (1990) analyzed the effect in a furnace of having walls with diffuse and specular reflectivity components. The enclosures were 2D and 3D in nature and were filled with radiating combustion products. It was found that for some applications the radiative transfer performance in furnaces could be improved by using highly reflecting specular walls. The multiple specular reflections during radiative transmission within a highly reflecting circular tube or similar device can lead to significant polarization effects. As in Figure 6.28, radiation in a given direction can undergo multiple reflections at the same incidence angle. As discussed in connection with the Fresnel reflection equations, the reflectivity differs for the two components of polarization, as in Equations (3.22) and (3.23). This provides different absorption for each of the components, and hence after reflection an initially unpolarized intensity becomes polarized. After multiple reflections at the same incidence angle during transit through a specular tube, for example, the radiation can become strongly polarized (Edwards and Tobin 1967, Miranda 1996, Qu et al. 2007a). In contrast, for diffuse reflections, a multiple reflection

Radiation among Nondiffuse Nongray Surfaces

301

process involves many different incidence angles, and polarization effects are not a concern. Polarization is considered in Chapter 17 for the specular reflections at the surfaces of glass windows as in Example 17.1. It is sometimes implied that the energy transfer between two real surfaces can be bracketed by calculating two limiting magnitudes: (1) Exchange between diffuse surfaces of the same total hemispherical emissivities as the real surfaces, and (2) interchange between specularly reflecting surfaces of the same total hemispherical emissivities as the real surfaces. This is not generally correct. Consider, for example, a surface that has a reflectivity as given by Figure 6.31a where there is strong reflection into the near-specular direction. Now consider the radiative exchange between the real surface 2 and black surface 1 as in Figure 6.31b. If surface 2 is specular, it will not return any reflected energy to the black surface (Figure 6.31c). If 2 is diffuse, it will return a portion of the incident energy by reflection (Figure 6.31d). For a rough surface, however, it may reflect more energy to the black surface than for either of the ideal surfaces (Figure 6.31e). The ideal directional surfaces (specular or diffuse) therefore do not constitute limiting cases for energy transfer in general. Figure 6.10 demonstrates another case in which diffuse and specular properties do not provide limiting solutions. At best, enclosure calculations based on idealized specular and diffuse assumptions for the surface characteristics give some indication of the possible importance of directional effects. Within enclosures, these directional effects may be small because the multiple reflections between the surfaces tend to make the overall behavior diffuse.

FIGURE 6.31  Radiant energy reflection from various idealizations of surface properties: (a) Bidirectional reflectivity of a real surface; (b) geometry of surfaces; (c) surface 2 reflects specularly; (d) surface 2 reflects diffusely; and (e) surface 2 reflects as typical of corrugated or very rough surface.

302

Thermal Radiation Heat Transfer

HOMEWORK PROBLEMS (Additional problems available in the online Appendix P at www.ThermalRadiation.net) 6.1

An anti-satellite missile has a total radiation sensor built into its nose. The sensor detects total radiation emitted by a hot satellite, and normally tracks to the satellite, where it impacts. A target satellite that is to be protected from the missile is gray with ε = 0.10, has a surface temperature of Ts = 450 K, and is spherical with diameter Ds = 3.2 m. You are to design a dummy countermeasure spherical satellite that is to be ejected from the real satellite as a decoy. The decoy material has the hemispherical spectral emissivity shown. The diameter of the decoy will be Dd = 0.48 m.

(a) What surface temperature Td should be used for the decoy? (b) How large a heat source (kW) is necessary in the decoy? Answers: (a) 671.0 K; (b) 7.49 kW. 6.2 Radiative energy is being transferred across the space between two large parallel plates with hemispherical spectral emissivities approximated as shown. What is the net heat flux q1 transferred from 1 to 2?

Answer: 107,530 W/m2.

Radiation among Nondiffuse Nongray Surfaces

6.3

303

The two plates in Homework Problem 6.2 have a flat plate radiation shield placed between them so the geometry is now three large parallel plates. The shield is gray, and has an emissivity of 0.10 on both sides. What is the shield temperature, and what is the heat flux being transferred from plate 1 to plate 2? The spectral emissivities of 1 and 2 are in Homework Problem 6.2.

Answers: Ts = 1194 K; q1 = 9,525 W/m2. 6.4 A polished aluminum tank in vacuum in orbit is surrounded by one thin aluminum radiation shield. The shield is polished on the side facing the tank, and is painted with white paint on the outside facing the sun with paint reflectivity of ρλ,n = 0.85 for λ1.9 µm. Treating the geometry as infinite parallel plates, what are the temperature of the shield and the heat flux into the tank for normally incident solar radiation? Assume the tank is maintained at 100 K.

Answers: 267 K; 14.0 W/m2. 6.5 Estimate the heat flux leaking into a liquid hydrogen container from an adjacent liquid nitrogen container. The glass walls are coated with polished aluminum. Use electromagnetic theory to estimate the radiative properties.

Answer: q = 0.0030 W/m2.

304

6.6

Thermal Radiation Heat Transfer

Two circular disks are parallel and directly facing each other. The disks are diffuse, but their emissivities vary with wavelength. The properties are approximated with step functions as shown. The disks are maintained at temperatures T1 = 1200 K and T2 = 800 K. The surroundings are at Te = 400 K. Compute the rate of energy that must be supplied to or removed from the disks to maintain their specified temperatures. The outer surfaces of the disks are insulated so there is radiation interchange only from the inner surfaces that are facing each other.

Answers: Q1 = 180,750 W; Q2 = −58,570 W. 6.7 An area element dA at temperature T is radiating out through a circular opening of radius r in a plate above and parallel to dA. The directional total emissivity of dA is ∊(θ) = 0.85 cos θ as in Figure 2.4. Obtain a relation for the energy emission Qnondiffuse from dA through the opening. From Example 2.3, the hemispherical total emissivity of dA is 0.57. Using this ε and the diffuse configuration factor, compute Qdiffuse through the opening. Plot Qnondiffuse/Qdiffuse as a function of h/r.

Radiation among Nondiffuse Nongray Surfaces

(

)

305

3/ 2

H2 + 1 - H3 Q h Answer: nondiffuse = ; H= 1/ 2 2 Qdiffuse r H +1

(

)

6.8

Two diffusely emitting and reflecting parallel plates are of infinite length normal to the cross-section shown in the figure. The lower plate has uniform temperature T1 = 1000 K, and the upper plate has uniform temperature T2 = 500 K. The surroundings have a temperature of Te = 450 K. The surfaces are nongray with properties shown in the figure. Do NOT assume uniform irradiation on either surface, and find the distribution of net radiative heat flux on each surface.

6.9

Obtain an analytical expression for the configuration factor F1−1(2) between the cylinder A1 and its image A1(2) for each of the two situations shown (the geometries are 2D in nature and are shown in cross-section).

306

Thermal Radiation Heat Transfer

Answers:

(a) F1-1(2 )

ì ü é ù ï ï ê ú 2 2 ïï ê ú 1 ïï æ L ö 1 R R L 3 æ ö æ ö æ ö -1 -1 = í ç ÷ - 1 + cos-1 ê ú + tan ç ÷ - cos ç L ÷ - ç R ÷ - 4 ý 2 2 L pï èRø è ø è ø è ø ï ê æLö 1ú ï ï ê çR÷ + 4 ú ïî ïþ ë è ø û

(b) F1-1( 2 ) =

2 é ù p 1 ê æLö æRö L - 1 + - cos-1 ç ÷ - ú . ç ÷ 2p ê è R ø 2 è L ø Rú ë û

6.10 A long black circular cylinder is partially surrounded by two parallel long plane specular surfaces as shown in cross-section (geometry is 2D in nature). What is the rate of heat loss from the cylinder in terms of the quantities shown?

Answer:

ìï 2 é 1/ 2 ù ïü Q æ1ö = 2pRsT14 í1 - ê X 2 - 1 + sin -1 ç ÷ - X ú ý ; X = L /R. length èXø ïî p ë û ïþ

(

)

Radiation among Nondiffuse Nongray Surfaces

307

6.11 A long enclosure is made up of two specular and two diffuse surfaces as shown in crosssection. Draw a diagram of the images that are needed to determine the energy exchange process. Then write the equations for F1s-2 and F1s-3 in terms of the required specular cons figuration factors and reflectivities (i.e. F1-2 = F1- 2 + rs,3 F1- 2(3) + ). Now write the set of energy exchange equations for finding Q1, Q2, Q3, Q4.

6.12 An equilateral triangular enclosure has sides that extend in the normal direction infinitely far into and out of the plane of the cross-section shown in the figure. Side 3 is specular and perfectly insulated on the outside. Find: (a) All necessary exchange factors Fs needed to solve for the unknown temperatures. (b) The temperatures of surfaces A2 and A3.

Answers: T2 = 1140 K; T3 = 1121 K. 6.13 An enclosure of equilateral triangular cross-section and of infinite length normal to the cross-section shown has two diffuse reflecting interior surfaces and one specularly reflecting interior surface that is perfectly insulated on the outside (i.e. q3 = 0). All surfaces are gray and are diffuse emitters. Compute T3 and the Q added to each of A1 and A2 as a result of radiative exchange within the enclosure for the conditions shown. (For simplicity, do not subdivide the surface areas).

Answers: T3 = 1090 K; −q2 = q1 = 16.9 kW/m2.

308

Thermal Radiation Heat Transfer

6.14 Compute the specular exchange factor F2s-1 for the 2D rectangular enclosure shown. All surfaces are gray and are diffuse emitters. Surfaces A1 and A2 are diffuse reflectors, while A3 and A4 are specular reflectors with ρs,3 = 0.5 and ρs,4 = 0.8.

Answer: 0.311. 6.15 An infinitely long square bar of size 2 m on a side is enclosed by an infinitely long concentric circular cylinder of 2 m radius. The temperatures and emissivities of the bar and cylinder are, respectively, Tb, εb, and Tc, εc. Find the rate at which radiant energy is exchanged between A1 and A2 per unit of enclosure length if (a) Both are diffuse. (b) Both are specular. (c) The bar is diffuse, and the cylinder is specular.

Answers: (a) Q =

(

8s Tb4 - Tc4

)

; (b), (c) Q =

(

8s Tb4 - Tc4

).

1 1 ö 1 2æ 1 + -1 - 1÷ + e b ec e b p çè ec ø 6.16 Derive Equations (6.64) and (6.65) for concentric diffuse cylindrical or spherical surface containing N radiation shields when the shields are either (a) diffuse, or (b) specular. Derive an equation for the temperature of the nth shield in each case. Answers: Q is given by Equations (6.64) and (6.65). For all diffuse shields, the temperature of shield n is

309

Radiation among Nondiffuse Nongray Surfaces

(T

4 1





T =T 4 n

4 1

ì1 ö A æ 1 ï - T24 í + 1 ç - 1÷ + ø ïî e1 As1 è es11 ö 1 A1 æ 1 + - 1÷ + e1 A2 çè e2 ø

)

å

and for all specular shields with diffuse boundary surfaces, the temperature of any shield n is N -1 A æ 1 ö ïü ö 1 1 ïìæ 1 1 + - 1÷ý - T24 íç + - 1÷ + ç 1 = n Asn è esn 2 es( n+1)1 ø ïîè e1 es11 ø ïþ Tn4 = T14 . N 1 ö A1 é 1 æ1 ö öù 1 A1 æ 1 1 AsN æ 1 + + - 1÷ú - 1÷ + - 1÷ + ç ê ç + n =1 Asn A2 çè e2 øû è e1 es11 ø è esn 2 es( n+1)1 ø AsN ë esN 2

(T

4 1



å

ö ù üï A1 é 1 A æ 1 ê + sn ç - 1÷ú ý ÷ú n =1 Asn ê e n 2 As( n+1) çè e( n+1)1 ø û ïþ ë N ù 1 A1 é 1 1 + ê ú n =1 Asn ë e n1 en2 û N -1

)

å

å

6.17 Two infinite gray parallel plates are separated by a thin gray radiation shield. What is Ts, the temperature of the shield? What radiative energy flux is transferred from plate 2 to plate 1 with the shield in place? What is the ratio of the heat transferred from 2 to 1 with the shield to that transferred without the shield?

Answers: 852 K; 2,578 W/m2; 0.113. 6.18 A radiative energy flux q0 is transferred across the gap between two gray parallel plates having the same emissivity ε that are at temperatures T1 and T2 and ε independent of temperature). A single thin radiation shield also having emissivity ε on both sides is placed between the plates. Show that the resulting heat flux is q0/2. Show that adding a second identical shield reduces the heat flux to q0/3. Show that for n shields the heat flux is q0/(n + 1) when all the surface emissivities are the same.

310

Thermal Radiation Heat Transfer

6.19 (a)  What is the effect of a single thin radiation shield on the transfer of energy between two concentric cylinders? Assume the cylinder and shield surfaces are diffuse-gray with emissivities independent of temperature. Both sides of the shield have emissivity εs, and the inner and outer cylinders have respective emissivities ε1 and ε2. (b) What is the effect of a single thin radiation shield on the transfer of energy between two concentric spheres? Assume the sphere and shield surfaces are diffuse-gray with emissivities independent of temperature. Both sides of the shield have emissivity εs, and the inner and outer spheres have respective emissivities ε1 and ε2.

Answers: (a)

Q1,shield = Q1,no shield

Q G12 G12 ; (b) 1,shield = A1Gs2 AG Q1,no shield G1s + G1s + 1 s2 As As

where ö ö 1 A æ1 1 A æ 1 G1s º + 1 ç - 1 ÷ ; Gs2 º + s ç - 1 ÷ . A e1 As è es e e 2 è 2 s ø ø 6.20 Derive a relation for the energy transfer rate q through an N-layer set of concentric circular cylindrical shields placed between an inner surface at T1 and an outer layer at T2. The outer surface has diameter D1, and the inner surface has diameter D2. All surfaces are diffuse and have an emissivity of ε. Answers: q =

(

s T14 - T24 1 é 1 D1 æ 1 ö ù -1 ú + ê + pD1 ë e DS1 çè e ÷ø û

where Rn-( n+1) =

1 pDn

å

N -1 n =1

)

Rn-( n+1) +

ìï 1 é æ Dn ö ù æ Dn ö üï í ê1 + ç ÷ú - ç ÷ý . îï e ë è Dn+1 ø û è Dn+1 ø þï

1 é 1 DSN æ 1 ö ù -1 ú ê + pDSN ë e D2 çè e ÷ø û

311

Radiation among Nondiffuse Nongray Surfaces

6.21 Two large gray parallel plates have 20 gray radiation shields between them. The plate temperatures and emissivities are shown in the figure, and all of the shield surfaces have εs = 0.035. Determine the temperatures of the 20 shields.

1/ 4

ìï æ2 ö ù üï qé1 1 Answers: Tn = íT14 - ê + - 1 + ( n - 1) ç - 1 ÷ ú ý s ë e1 es è es ø û þï îï T20 = 569.9 K.

resulting in T1 = 1043 K and

7

Radiation Combined with Conduction and Convection at Boundaries Ernst Rudolph George (ERG) Eckert (1904–2004) researched radiation from solids and gases, and published measurements of directional emissivity from various materials as well as directional reflectivity of blackbody radiation. He also developed optical methods for obtaining configuration factors. In 1937, he turned to measurement of the emissivity of CO2-N2 mixtures as well as water vapor at various temperatures and partial pressures.

7.1 INTRODUCTION In the preceding chapters, enclosure theory was formulated for radiative exchange between surfaces. The local net radiation loss at a surface was balanced by energy supplied by “some other means” that were not explicitly described. This chapter is concerned with this energy at the surface either by conduction from within the volume interior to the surface (such as from within a wall of an enclosure) or by convection or conduction at the surface from a surrounding medium. At each location along the surface, the radiation, convection, and conduction combine to form a thermal boundary condition. The solution to the energy equations subject to this condition provides the surface temperature and heat flux distributions. The analysis has the same restrictions as in the previous theory: The surfaces are opaque, and the medium between the radiating surfaces is perfectly transparent. The medium between the radiating surfaces may be conducting or convecting energy, but it does not interact with radiation passing through it. One example of combined-mode energy transfer is a vapor-cycle power plant operating in outer space. Waste heat must be rejected by radiation. In the space radiator in Figure 7.1a, the vapor of the working fluid in a thermodynamic cycle is condensed, releasing its latent heat. This energy is conducted through the condenser wall and into fins that radiate the energy into outer space. The temperature distribution in the fins and their radiating efficiency depend on combined radiation and conduction. A fin-tube geometry is also commonly used for the absorber in a flat-plate solar collector. Solar energy is incident on the absorber plate through one or more transparent cover glasses that reduce convective losses to the atmosphere. A fluid is heated as it flows through tubes attached to the absorber plate. The collector design requires analysis of combined radiation, conduction, and convection. In one type of steel-strip cooler in a steel mill (Figure 7.1b), a sheet of hot metal moves past a bank of cold tubes and loses energy to them by radiation. At the same time cooling gas is blown across the sheet. A combined radiation and convection analysis is required to determine the temperature distribution along the steel strip. Controlled radiative and convective cooling is also used in the tempering of sheets of high-strength glass for automobile windows. In a possible design for a nuclear rocket engine, illustrated by Figure 7.1c, hydrogen gas is heated by flowing through a high-temperature nuclear reactor. The hot gas then passes out through the rocket nozzle. The interior surface of the rocket nozzle receives energy by radiation from the exit face of the reactor core and by convection from the flowing gas. Cooling the nozzle depends on conducting this energy through the nozzle wall and removing it to a flowing coolant. 313

314

Thermal Radiation Heat Transfer

FIGURE 7.1  Heat transfer devices involving combined radiation, conduction, and convection: (a) Space radiator or absorber plate of flat-plate solar collector, (b) steel-strip cooler, and (c) nuclear rocket.

These examples involve energy transfer by two or more modes. The modes can be in series, such as conduction through a wall followed by radiation from its surface. Energy transfer can also be by parallel modes, such as simultaneous conduction and radiation through a transparent material such as glass, or simultaneous radiation and convection from a hot surface. Series and parallel modes may both be present. The interaction of the modes can be simple in some cases. For example, if the amounts of energy transferred from a surface by radiation and convection are independent, they can be computed separately and added. In other instances, the interaction can be complex, such as when coupled surface radiation and free convection interact. The various heat transfer modes depend on temperature and/or temperature differences to different powers. When radiation exchange between black surfaces is considered, the energy fluxes depend on surface temperatures to the fourth power. For nonblack surfaces, the temperature dependence may differ somewhat because of emissivity variations with temperature. Heat conduction depends on the local temperature gradient. Convection depends approximately on the first power of the temperature difference. The exact power depends on the type of flow; for example, free convection depends on temperature difference to a power from 1.25 to 1.4. Physical properties that vary with temperature introduce additional temperature dependencies. The various powers and dependencies of temperature produce nonlinear energy transfer relations, and it is usually necessary to use numerical solution techniques. This chapter provides methods for setting up the energy-balance relations, and physical behavior is illustrated, and some common solution methods are presented.

7.2 ENERGY RELATIONS AND BOUNDARY CONDITIONS 7.2.1 General Relations In the analyses developed for enclosures, the net radiative energy flux at any position on the boundary was balanced by the energy flux q supplied by “some other means.” The means considered

315

Combined Mode Boundary Conditions

here are conduction, convection, or wall internal energy sources such as electric heaters or nuclear reactions. Since the enclosure walls are assumed opaque, the absorption of radiation is at the surface, and the energy balance provides a boundary condition. Although there can be conduction or convection in a medium between radiating surfaces, the medium is assumed here to be perfectly transparent, so radiation passes through with undiminished intensity. The radiation exchange relations developed previously for an enclosure are unchanged. If convection is expressed in terms of a heat transfer coefficient, Equation (7.1) for q is

q = h(Tg - Tw ) - k

¶T ¶n

= J - G (7.1) wall

where all quantities are at r on the surface of the enclosure wall in Figure 7.2. The previous enclosure relations are valid as they are written in terms of q. For example, Equation (5.56) relates T and q along the enclosure boundaries. If T is given, Equation (5.56) is solved for the q; then Equation (7.1) yields ∂T/∂n|wall. T and ∂T/∂n at the wall surface are the boundary conditions for the heat conduction equation within the wall interior:

rc

¶T = Ñ × (kÑT ) + q. (7.2) ¶t

The form of the energy equation inside the enclosure in the space between the radiating surfaces depends on the type of convection, such as forced convection in a channel, a boundary layer flow, or free convection. If the convection depends significantly on the boundary temperatures or heat flux distributions, the solution may require simultaneous solution of the radiation exchange, heat conduction in the wall, and convection relations. In some problems, the net energy added to the surface by external means is specified more directly than by the normal derivative in Equation (7.1). If electric heating generates an energy flux qe in the wall with insulation on one boundary and negligible energy conduction along the wall (Figure 7.3), all the qe appears at the radiating boundary, and Equation (7.1) becomes, at each location along the boundary,

q = h(Tg - Tw ) + qe = J - G. (7.3)

The qe may be uniform over the surface area, or it can have a specified variation with location. The heating might be by passage of electric current within the wall, such as for an electrically heated wire.

FIGURE 7.2  Boundary condition at location r on the surface of an opaque-walled enclosure.

316

Thermal Radiation Heat Transfer

FIGURE 7.3  Boundary condition at the surface of an opaque wall with specified heat flux.

7.2.2 Uncoupled and Coupled Energy Transfer Modes In the simplest situations, the radiation, conduction, and convection contributions to an unknown quantity, such as heat flux, are independent. The separate contributions are computed and the results combined. The energy transfer modes are uncoupled with regard to the desired quantity. Example 7.1 Consider the region between two large gray parallel walls with a transparent gas between them (Figure 7.4). The internal surface temperatures T1 and T2 are specified. There is free convection in the gas, and the free-convection heat transfer coefficient hfc depends on T1 and T2. What is the steady-state energy transfer from wall 1 to wall 2? The energy transfer is the net radiative exchange and the transfer by free convection. It is equal to the flux q1 that must be added to wall 1 to maintain it at its specified temperature. Since T1

FIGURE 7.4  Parallel wall geometry for Examples 7.1 and 7.2 (T1 > T2).

317

Combined Mode Boundary Conditions

and T2 are given, the hfc can be computed from free-convection correlations and the net energy transfer is, by use of Equation (5.6.3) plus the convection term, q1 =



(

s T14 - T24

)

1 / e1 (T1 ) + 1 / e 2 (T2 ) - 1

+ hfc (T1,T2 )(T1 - T2 ) .

The radiative and convective components are uncoupled. The q for each mode is computed independently, and the contributions added. The methods of radiative computation developed earlier can be applied without modification.

Coupled problems are more common than uncoupled problems. In coupled problems, the desired unknown quantity cannot be found by adding separate solutions; the energy relations must be solved with the transfer modes simultaneously included, usually resulting in highly nonlinear equations. In some situations, it may be possible to assume that the modes are uncoupled because only weak coupling occurs. Example 7.2 Let T0 and T2 in Figure 7.4 be specified. Since energy must be conserved in crossing surface 1 of the lower wall, the conduction through the lower wall must equal the transfer from surface 1 to surface 2 by combined radiation and free convection. Then, for constant thermal conductivity kw,

q1 =

(

)

s T14 - T24 kw + hfc (T1,T2 ) (T1 - T2 ) (To - T1) = a 1 / e1 (T1 ) + 1/e 2 (T2 ) - 1

The problem is coupled since the unknown T1 must be found from an equation that simultaneously incorporates all energy transfer processes. The equation for T1 is highly nonlinear and T1 can be obtained by iteration or by a computer math package root solver.

These examples demonstrate that the type of boundary condition governs the possibility of uncoupling the calculations. When all temperatures are specified, the energy fluxes can often be uncoupled. If energy fluxes are specified, the entire problem must be treated simultaneously because of the nonlinear coupling of the unknown temperatures.

7.2.3 Control Volume Approach for One- or TwoDimensional Conduction along Thin Walls In some situations, the radiating wall is thin and temperature variations are principally along the length and width of the wall rather than across its thickness. An important example is energy dissipation by radiating fins in devices that operate in outer space. Energy is conducted along the fin and radiated from the fin surface. The determination of the fin temperature distribution and performance requires a coupled conduction–radiation solution. The analysis is usually simplified by assuming a uniform temperature across the thin fin thickness at each location. A control volume across the thickness can be used to derive the heat balance equation. A volume element of area dx dy and thickness a is shown in Figure 7.5. The thickness is small, so T(x, y) is considered uniform within the z-axis of the element. Transparent fluids at Tm,1 and Tm,2 are flowing across the upper and lower surfaces providing convective heat transfer coefficients h1 and h2. The temperature can change with time, and there can be internal heat generation within the element such as by electric heating. An energy balance shows that the change with time of the internal

318

Thermal Radiation Heat Transfer

FIGURE 7.5  Element of the thin plate for control volume derivation.

energy of the element equals the energy gains by radiation exchange, conduction, convection, and internal energy sources: rca

¶T ¶ æ ¶T = G1 - J1 + G2 - J 2 + ç ka ¶t ¶x è ¶x

ö ¶ æ ¶T ö ÷ + ¶y ç ka ¶y ÷ ø è ø (7.4)

 . + h1 ( Tm,1 - T ) + h2 ( Tm,2 - T ) + qa This is used in the analyses of thin fins that follow.

7.3 RADIATION TRANSFER WITH CONDUCTION BOUNDARY CONDITIONS Combined conduction and radiation is fairly common, such as energy losses from radiating fins, energy transfer through the walls of a vacuum Dewar, energy transfer through insulation made of many separated layers of highly reflective material, energy losses from radiators in outer space, and temperature distributions in satellite and spacecraft structures. The sophistication of the radiative portion of the analysis can vary considerably, depending on the accuracy required and the importance of radiation relative to heat conduction. If conduction dominates, stronger approximations can be made in the radiative portion of the analysis, and vice versa.

7.3.1 Thin Fins with 1D or 2D Conduction 7.3.1.1 1D Energy Flow The heat transfer performance of a thin circular fin is now considered. From circular symmetry, the heat flow is one-dimensional (1D) in the radial direction.

319

Combined Mode Boundary Conditions

Example 7.3 A thin annular fin in vacuum is embedded in insulation, so it is insulated on one face and around its outside edge (Figure 7.6a, b). The disk has thickness a, inner radius ri, outer radius ro, and thermal conductivity k. Energy is supplied to the inner edge from a solid rod of radius ri that fits the central hole and maintains the inner edge at Ti. The exposed annular surface, which is diffusegray with emissivity ε, radiates to the environment at Te ~ 0 K to investigate performance in the cold environment of outer space. Find the temperature distribution as a function of radial position across the disk. The results also apply for the more general annular fin in Figure 7.6c if the heat loss through the end edge of the fin is neglected. There is no heat flow across the symmetry plane of the fin, and hence this plane acts as the insulated boundary in Figure 7.6a. Assume the disk is thin enough that the local temperature can be considered constant across the thickness a, which is the usual thin fin assumption. For surroundings at zero temperature, there is no incoming radiation. If a and k are constant, the control volume Equation (7.4) for a ring element of width dr (Figure 7.6d) gives

ka

1 d æ dT ö r - esT 4 = 0. (7.3.1) r dr çè dr ÷ø

This is to be solved for T(r) subject to two boundary conditions: At the inner edge T = Ti at r = ri, and at the insulated outer edge where there is no heat flow dT/dr = 0 at r = ro. Using dimensionless variables ϑ = T/Ti and R = (r − ri)/(ro − ri) and two parameters δ = ro/ri and g = ( ro - ri ) esTi 3 /ka results in 2



d 2J 1 dJ + - gJ4 = 0 (7.3.2) dR 2 R + 1 ( d - 1) dR

FIGURE 7.6  Geometry for finding temperature distribution in a thin radiating annular plate insulated on one side and around the outside edge: (a) Heat flow path through fin, (b) disk geometry, (c) application to the annular fin, and (d) portion of ring element on the annular disk.

320

Thermal Radiation Heat Transfer

with the boundary conditions ϑ = 1 at R = 0 and dϑ/dR = 0 at R = 1. Equation (7.3.2) is a secondorder nonlinear differential equation where ϑ(R) depends on the two parameters δ and γ. Solutions can be obtained by numerical methods, and solvers are available in computer mathematics software packages. A design parameter for cooling fins is the fin efficiency, η. This is the energy radiated by the fin divided by the energy that would be radiated if the entire fin were at the maximum temperature Ti, which would occur for a fin with infinite thermal conductivity. The fin efficiency for the circular radiating fin is then 2pes h=



(

ò

ro

ri

rT 4 ( r ) dr

)

p ro2 - ri 2 esTi 4

=

1

ò éëR (d - 1) + 1ùû J

2

0

d +1

4

(R ) dR



and is evaluated after ϑ(R) has been determined from the differential Equation (7.3.2). The η has been obtained by Chambers and Somers (1959) and is in Figure 7.7. Keller and Holdredge (1970) extended the results to fins of radially varying thickness. The annular fin is a model for a circular foil heat flux sensor, and solutions for that application that also include convection are in Kuo and Kulkarni (1991).

In a more general situation, if the environment is at Te and the fin is nongray with a total absorptivity α for the incoming radiation spectrum (such as for incident solar radiation in a space application), the energy balance in Equation (7.3.1) becomes (ε is the total emissivity for the spectrum emitted by the fin)

ka

a ö 1 d æ dT ö 1 d æ dT ö æ - s eT 4 - aTe4 = ka r r - es ç T 4 - Te4 ÷ = 0 (7.5) r dr çè dr ÷ø r dr çè dr ÷ø e è ø

(

)

where (a e)Te4 is an additional parameter. For a gray fin, α = ε; hence, a nongray fin acts like a gray fin in an effective radiating environment of ( a /e ) Te4 . By using this effective environment, results for gray fins can be utilized for nongray fins. Design results for rectangular fins, including incident radiation from the environment, are in Mackay (1963).

FIGURE 7.7  Radiation fin efficiency for fin of Example 7.3. (From Chambers, R.L. and Somers, E.V., JHT, 81(4), 327, 1959.)

321

Combined Mode Boundary Conditions

For transients where the fin temperature distribution changes with time, the energy storage term in Equation (7.4) must be included. The partial differential equation for T(r, τ) is then rca



a ö ¶T 1 ¶ æ ¶T ö æ r - es ç T 4 - Te4 ÷ . (7.6) = ka ç ÷ r ¶r è ¶r ø e ¶t è ø

Results for transient radiating fin behavior are in Eslinger and Chung (1979). Fins of various shapes are treated in Kraus et al. (2001). Example 7.4 A thin plate of thickness a and length 2L is between two tubes in a radiator used to dissipate energy in orbit as in Figure 7.8. The dimension is long in the direction normal to the cross-section shown. Both sides of the plate have the same emissivity and are losing energy by radiation to space. Radiation from the surroundings, such as from the sun, earth, or a planet, is incident on the plate surfaces, and the fluxes absorbed on the top and bottom sides are qabs,t and qabs,b. The plate is diffuse-gray with emissivity ε on both sides and has constant thermal conductivity. Find an expression for the plate temperature distribution in the x direction. Neglect radiative interaction with the tube surfaces. From the control volume relation Equation (7.4), the energy equation for a plate element of width dx is -ka



d 2T + 2esT 4 = qabs,t + qabs,b . (7.4.1) dx 2

The boundary conditions for the thin plate are T = Ttube specified at x = 0 and, from symmetry, dT/dx = 0 at x = L. To find T(x), multiply Equation (7.4.1) by dT/dx and integrate to give 2



-

ka æ dT ö 2 + es éT 5 - T 5 ( L ) ùû = ( qabs, t + qabs,b ) éëT - T ( L ) ùû 2 çè dx ÷ø 5 ë

where T(L) (which is unknown) is inserted to satisfy the boundary condition at x = L. Solve for dT/dx to yield 1/ 2



dT æ 4es ö = -ç ÷ dx è 5ka ø

1/ 2

5 ì 5 5 ( qabs, t + qabs,b ) éëT - T ( L )ûù üý íT - T ( L ) 2es î þ

FIGURE 7.8  Flat-plate fin geometry for Example 7.4.

.

322

Thermal Radiation Heat Transfer

The minus sign was chosen for the square root because T(x) must be decreasing with x. Separate variables and integrate again to obtain 1/ 2 Ttube



æ 5ka ö x=ç ÷ è 4es ø

ò {T T

dT

5

-T

5

( L ) - ( 5/ 2es ) ( qabs, t + qabs,b ) éëT - T ( L )ùû}

1/ 2

(7.4.2)

which satisfies T = Ttube at x = 0. To obtain T(L), the relation is used that at the known length L, 1/ 2 Ttube



æ 5ka ö L=ç ÷ è 4es ø

ò( ) {T

T L

dT

5

-T

5

( L ) - ( 5/2es ) ( qabs, t + qabs,b ) éëT - T ( L )ùû}

1/ 2

. (7.4.3)

A numerical root finder in mathematics software packages can be used to obtain T(L) from Equation (7.4.3). The temperature distribution is then found by evaluating the integral in Equation (7.4.2) numerically to find x for various T values (in the lower limit of the integral) between Ttube and T(L).

Examples 7.3 and 7.4 considered a single radiating fin. When there are multiple fins that have radiative exchange among them, integral terms are introduced into the energy equations as shown in the next example. Example 7.5 An infinite array of identical thin fins of thickness a, width W in the x direction, and infinite length in the z direction is attached to a black base maintained at a constant temperature Tb, as in Figure 7.9. The fin surfaces are diffuse-gray and are in vacuum. Set up the equation describing the local fin temperature, assuming the environment is at Te ≈ 0 K. Because the fins are thin, their local temperature is assumed constant across the thickness a, and the control volume Equation (7.4) is used for the circled differential element in Figure 7.9. Since there is an infinite array of fins, the surroundings are identical for each fin and are the same on both sides of each fin. From symmetry, only half the fin thickness need be considered. All the

FIGURE 7.9  Geometry for determination of local temperatures on parallel fins.

323

Combined Mode Boundary Conditions

fins are the same, so the energy balance need be considered for only one fin, and the fin temperature distributions are all the same. The net conduction into the element dx per unit time and per unit length of fin in the z direction is, for constant thermal conductivity, (ka/2)(d2Tf/dx2)dx. The radiation relations are formulated from the net-radiation method as given by the enclosure Equation (5.56). Writing this for an element dx along the fin gives q ( x ) 1- e e e



W

ò q ( x) dF

= sTf

= Fdx -b ,

q( x ) =

dx - d x

4

( x ) - sT

4 b

x =0

W

b

òdF

dx - db

-

ò sT

f

4

( x ) dFdx -d x . (7.5.1)

x =0

0

Using the relations b

ò dF



dx -db

0

ka d 2Tf ( x ) , 2 dx 2

q(x) =

ka d 2Tf (x) 2 d x2

gives, after rearrangement,

-

2 ka d Tf ( x ) + sTf 4 ( x ) = sTb4Fdx -b + 2e dx 2

é 1 - e ka d 2Tf ( x ) ù + sTf 4 ( x ) ú dFdx -d x . (7.5.2) ê2 e 2 x d ê úû x =0 ë W

ò

In dimensionless form this becomes

where J( X ) =

-m

d 2J ( X ) dX 2

Tf ( X ) , Tb

B=

+ J4 ( X ) = FdX -B +

b , W

m=

é ù d 2J ( Z ) + J4 ( Z ) ú dFdX -dZ (7.5.3) ê -m (1 - e ) 2 dZ úû ê Z =0 ë 1

ò

ka , 2esTb3W 2

X=

x , W

Z=

x . W

Equation (7.5.3) is a nonlinear integrodifferential equation and can be solved numerically. Two boundary conditions are needed. At the base of the fin, Tf(x = 0) = Tb, so J = 1 at X = 0. (7.5.4)



At the tip of the fin, x = W, the conduction to the tip boundary must equal the energy radiated: -k ¶Tf / ¶x x =W = esTf 4 (W ) . In terms of ϑ,

-

d J esTb3W 4 1 a 4 = J = J dX k 2m W

at X = 1 (7.5.5)

and the fin thickness-to-width ratio a/W enters as another parameter. If (a/W)/2μ is very small, dϑ/dX can be approximated as zero (since the maximum ϑ is 1). The configuration factors in Equation (7.5.3) are found by the methods of Examples 4.2 and 4.5.

A black base surface without fins provides the maximum radiative emission. Adding an infinite array of fins to a nonblack plane surface produces a series of radiating cavities that can approach the performance of a black surface. The fins provide additional weight and complexity, and hence it is better to simply use a plane surface with a high emissivity than to use a plane surface with a large array of fins. This conclusion is also reached in the analysis and discussion in Krishnaprakas (1996, 1997). However, a finned surface can provide directional emission or absorption characteristics that make it attractive for some applications, as discussed in Chapter 3. Because of the interest

324

Thermal Radiation Heat Transfer

in radiator design for application in power systems operating in the vacuum of Earth orbit or outer space, many conduction–radiation systems have been analyzed. Typical are the previous references in this chapter and Heaslet and Lomax (1961), Sparrow et al. (1962), Stockman and Kramer (1963), Masuda (1973), Schnurr et al. (1976), Frankel and Silvestri (1991), Chung and Zhang (1991), and Ma et al. (2016). Many other references are in the literature. In contrast to a plane surface, if fins are placed on a cylindrical pipe, the radiating area is increased compared with the pipe surface area alone, and radiative dissipation can be substantially improved. The fins can be circular and normal to the pipe axis or axial as in Figure 7.10a. Fins of uniform thickness were analyzed by Frankel and Silvestri (1991), but to save weight, the fins can have a nonuniform cross-section that decreases with x (Figure 7.10b) because the energy that must be conducted within the fin also decreases. The optimization of the fin cross-sectional shape has been analyzed by Chung and Zhang (1991) and Krishnaprakas (1997). These include radiative interactions between the fins and between the fins and the base surface. 7.3.1.2 2D Energy Flow The configuration and boundary conditions for many fin applications is such that the fins can be analyzed as having 1D heat flow. Some applications use fins with 2D heat flow, such as for removing excess heat from electronic equipment in satellites and other space devices. The 2D fin is typically a thin metal plate, such as aluminum, with heat-generating equipment in good thermal contact with a portion of the plate area. The heat transferred to the plate is conducted away in 2D and is dissipated by radiation to cooler surroundings. In Badari Narayana and Kumari (1988), the cooling modes are radiation and conduction; in Bobco and Starkovs (1985), convection is also included. The fin analyzed in Badari Narayana and Kumari is shown in Figure 7.11. It has a uniform thickness and constant thermal properties, and its radiative properties are diffuse-gray. Energy to be dissipated to cool equipment is transferred to the shaded area on one side and is radiated away from both sides. The radiating plate is exposed to the sun on the outside and to surroundings at Ti on the inside. The energy equation within the zone (shown shaded) over which energy from the equipment is being transferred to the plate is given by Equation (7.4) as



æ ¶ 2T ¶ 2T ka ç 2 + 2 ¶y è ¶x

ö 4 ÷ + as qsolar cos q + qe - eo sT ( x, y ) = 0 ø

(7.7)

net conduction + absorbed solar energy + internal generaation - emission loss = 0. In this region, an element of the plate receives energy by 2D conduction within the plate, by absorption of solar radiation, and by heat addition from electronic equipment and/or other heat sources.

FIGURE 7.10  Axial fin array on a cylindrical pipe: (a) Array with four tapered fins, and (b) energy balance for tapered fin.

325

Combined Mode Boundary Conditions

FIGURE 7.11  Two-dimensional radiating fin with heat flux addition qe to the area on one side.

The plate loses energy by radiation from its outside surface. For the other portions of the plate, where qe = 0, the energy equation becomes

æ ¶ 2T ¶ 2T ö ka ç 2 + 2 ÷ + as qsolar cos q + se i éëTi4 - T 4 ( x, y ) ùû - eo sT 4 ( x, y ) = 0 (7.8) ¶y ø è ¶x

where there is a term for the net radiation loss from the surface that is inside the enclosure. Although this was the term used in Badari Narayana and Kumari, it is written more generally in Equation (7.4) as (G − J) to account for a more complex heat exchange with a surrounding enclosure. Heat losses from the end edges of the fin were assumed small, so insulated edge boundary conditions are used:

¶T = 0 at x = ( 0, L ) ; ¶x

¶T = 0 at y = ( 0, W ) . ¶y

Along the boundary between the area of heat addition (shaded area) and the remaining area of the plate, there is continuity of fin temperature and heat flow in the x and y directions. Solutions to other fin problems involving mutual interactions are in Heaslet and Lomax (1961), Nichols (1961), Sparrow et al. (1962), Sparrow and Eckert (1962), Stockman and Kramer (1963), Frankel and Wang (1988), and Frankel and Silvestri (1991). The optimization of a fin array with respect to minimum weight is in Wilkins (1962) and Chung and Zhang (1991). The radiant interchange is analyzed by Masuda (1972) between diffuse-gray external circular fins on cylinders, extending normal to the cylinder axis, and the local heat flux distribution is obtained on the fins and cylinders as well as the fin effectiveness.

7.3.2 Multidimensional and Transient Heat Conduction with Radiation For a thin radiating fin, the local fin temperature is assumed uniform across the fin thickness, and temperature variations are only in directions along the radiating surfaces. If the conducting solid

326

Thermal Radiation Heat Transfer

is thick, however, the temperature will also vary normal to the radiating surface. The conduction can be steady or transient. The surfaces are assumed opaque, so without external convection, the net radiation at the surface is the boundary condition for conduction within the solid. If n is the outward normal from the surface, the conduction heat flux at the surface, flowing outward from within the solid, is −k(∂T/∂n). This is the local flux supplied to the surface to balance the net radiative loss. Hence, in the absence of convection, there is the boundary condition, Equation (7.1), −k(∂T/∂n)|wall = (J − G) = qr. The governing partial differential equation in the solid is Equation (7.2). The distribution of (J − G) along the conducting surfaces is found from the radiative enclosure methods described previously. For a nongray surface, the radiative qr is the integral of the spectral fluxes as developed in Chapter 6. Subject to these boundary conditions, the multidimensional heat conduction equations can be solved by finite-difference or finite-element methods (FEM), as in Section 7.7. When the temperature distributions within the solid are transient as well as spatially dependent, relatively few analytical solutions can be obtained in view of the complexity of the radiative boundary conditions. Some analytical investigations were made by Abarbanel (1960). Transient solutions are more feasible if the geometry is 1D. An example is electric heating of a thin wire where the transient temperature distribution varies only along the wire length (Carslaw and Jaeger 1959). Transient radiative cooling of a wire was used in Masuda and Higano (1988) to obtain the hemispherical total emittance of a metal by measuring the cooling rate. From the control volume approach, the 1D energy equation for a wire of radius r is

r 2 rc

¶T ¶ 2T = kr 2 2 - 2r éë J ( x, t ) - G( x, t ) ùû . (7.9) ¶t ¶x

The properties have been assumed constant. The J and G depend on the radiative exchange with the surroundings and are functions of axial position x and time t. The J value varies considerably as the wire temperature changes with time and position. The solution for T(x,t) requires an initial temperature distribution and two boundary conditions in x. For example, the boundary conditions could be fixed electrode temperatures at the ends of the wire. A 2D transient solution was carried out numerically in Sunden (1989). A hollow cylinder, insulated on its internal surface, is heated on its exterior by a time-varying radiation flux from one direction. Absorbed energy is then conducted within the cylinder in radial and circumferential directions. During the transient heating, the outer surface loses energy by radiation and convection. When the cylinder has low thermal conductivity and low thermal diffusivity, the temperature distributions are quite nonuniform and the surface temperatures are high, so radiative cooling is important. For highconductivity materials, the temperature levels are lower and the temperature distributions are more uniform. Torabi et al. (2019) provide an approximate method for treating transient energy transfer through multi-layer insulation and compare with experimental data.

7.4 RADIATION WITH CONVECTION AND CONDUCTION Interactions of radiation, convection, and conduction are found in a wide variety of situations, such as convective and radiative cooling of high-temperature components in air, cooling of hypersonic and re-entry vehicles, interactions of incident solar radiation with the earth’s surface to produce complex free-convection patterns, convection cells and their effect on radiation within and from stars, and marine environment studies for predicting free-convection patterns in oceans with absorption of solar energy. The energy equations and boundary conditions contain temperature differences from convection and temperature derivatives from conduction. Results must usually be obtained by using numerical solution methods. The basic ideas are developed here for transparent/ nonparticipating media between surfaces by using some illustrative examples of practical interest. Additional information and results are in Kuo and Kulkarni (1991), Perlmutter and Siegel (1962),

Combined Mode Boundary Conditions

327

Siegel and Perlmutter (1962), Cess (1961), Keshock and Siegel (1964), Aziz and Benzies (1976), Siegel and Keshock (1964), Okamoto (1964, 1966a, b), Sohal and Howell (1973, 1974), Campo (1976), Shouman (1965, 1968), and Razzaque et al. (1982). Hadley et al. (1999) studied the transient response of a thermocouple with radiation and convection using a lumped-parameter model. Ma et al. (2016) study a radial fin with temperature-dependent conductivity, and Mallick et al. (2019) use an inverse approach to estimate parameters in a functionally graded annular fin. Singhal et al. (2020) use a semi-analytic method to study a composite wall with temperature-dependent properties and internal generation.

7.4.1 Thin Radiating Fins with Convection Thin fins with multiple heat transfer modes are used extensively for providing effective energy dissipation. Example 7.6 Examine the performance of the fin in Figure 7.12; it depends on combined conduction, convection, and radiation. A gas at Te is flowing over the fin and removing energy by convection. The environment to which the fin radiates is also assumed to be at Te. The fin cross-section has area A and perimeter P. The fin is nongray with total absorptivity α for radiation incident from the environment. Using a control volume, an energy balance on a fin element of length dx gives

kA

d 2T dx = s éëeT 4 ( x ) - aTe4 ùû Pdx + hPdx éëT ( x ) - Te ùû . (7.6.1) dx 2

The term on the left is the net conduction into the element, and on the right are the radiative and convective losses. The radiative exchange between the fin and its base is neglected here, but could be included in more detailed analyses. This equation is to be solved for T(x), which can then be used to obtain the energy dissipation. Multiply by [1/(kA dx)] dT/dx, and integrate once to yield 2



ö esP æ T 5 a 4 ö hP æ T 2 1 æ dT ö - TTe ÷ + C (7.6.2) - TTe ÷ + = ç ç ç ÷ e 2 è dx ø kA è 5 kA 2 è ø ø

where C is a constant of integration. For convection and radiation from the end surface of the fin, or for the end of the fin assumed insulated, the dT/dx in Equation (7.6.2) is expressed in terms of the appropriate end conditions.

FIGURE 7.12  Fin of constant cross-sectional area transferring energy by radiation and convection. (Flowing gas and environment are both at Te).

328

Thermal Radiation Heat Transfer

For a case that leads to an analytical solution, let Te ≈ 0 and let the fin be long. For large x, T(x) → 0 and dT/dx → 0, and from Equation (7.6.2), C = 0. Solving for dT/dx gives 1/ 2

dT æ 2 P es 5 hP 2 ö T T + = -ç (7.6.3) kA ÷ø dx è 5 kA



The minus sign is used for the square root, since T decreases as x increases. The variables in Equation (7.6.3) are separated and the equation integrated with the condition that T(0) = Th, x

T

ò

dx = -

ò

Tb

0



( (

dT é2 ù T ê (P es /kA)T 3 + hP /kAú ë5 û

é GT 3 + M b 1 x = M -1/ 2 êln ê 3 GTb3 + M ëê where G =

) )

1/ 2

1/ 2

-M

1/ 2

+M

1/ 2

- ln

(GT (GT

3 3

1/ 2

(7.6.4)

) + M) +M

1/ 2

1/ 2

-M ù ú ú + M1/ 2 ú û 1/ 2

2 Pes/kA, M = hP/kA. 3

For this simplified limiting case, an analytical relation for T(x) is obtained. The solution can be carried somewhat further, as considered in Homework Problem 7.2. A detailed treatment of this type of fin is found in Shouman (1965, 1968).

An array of pin fins extending from a surface is useful for heat transfer augmentation, and the optimization of a triangular array is analyzed by Gerencser and Razani (1995). The fins are circular in cross-section, and the cross-section is variable with length, as in Figure 7.13. The convective heat transfer coefficient hc is assumed constant throughout the array, and for a constant fin thermal conductivity, the energy balance on an element at x of a typical fin is

k

d é dT ( x ) ù = 2pr ( x ) hc éëT ( x ) - Te ( x ) ùû + J ( x ) - G( x ) (7.10) pr ( x )2 ê dx ë dx úû

{

}

where the differential surface area has been approximated as 2πrdx. The incident radiative flux, G(x), is obtained by the interaction of a fin with the base surface, which is assumed to be at uniform temperature, and with the surrounding fins. It is assumed in Gerencser and Razani that all of the fins surrounding any one fin may be approximated as a surrounding circular cylinder of constant radius that has a temperature distribution in the x direction that is the same as along the fins. The optimization for minimum fin volume to provide the required energy dissipation yields fins with a curved profile along their length. It is found that good performance can also be achieved by using tapered pin fins with a triangular profile, which are more economical to manufacture.

FIGURE 7.13  An array of tapered pin fins with external radiation and convection.

329

Combined Mode Boundary Conditions

7.4.2 Channel Flows Flow of a transparent gas through a heated radiating tube is now considered. Solutions of this type are in Cess (1961), Perlmutter and Siegel (1962), Siegel and Perlmutter (1962), Keshock and Siegel (1964), Siegel and Keshock (1964), Aziz and Benzies (1976), and Razzaque et al. (1982). This chapter treats convective media that are completely transparent to radiation. The enclosure energybalance equations, such as Equations (5.16) and (5.17), can be used as before, and the qk at the wall surface will contain convective heat addition to the wall. Example 7.7 A transparent gas flows through a black circular tube (Figure 7.14). The tube wall is thin, and its outer surface is perfectly insulated. The wall is heated electrically to provide uniform energy input qe per unit area and time. The wall temperature along the tube length is to be determined. The convective heat transfer coefficient h between the gas and the inside of the tube is assumed constant. The gas has a mean velocity um, specific heat cp, and density ρf. Axial conduction in the thin tube wall is neglected. If radiation were not considered, the local heat addition to the gas would equal the local electric heating (since the outside of the tube is insulated) and hence would be invariant with x along the tube. The gas temperature and wall temperature would both rise linearly with x. If convection were not considered, the only means for heat removal would be by radiation out of the tube ends, as in Example 5.19. In this instance, for equal environment temperatures at the tube ends, the wall temperature is a maximum at the center of the tube and decreases toward each end. The solution for combined radiation and convection is expected to exhibit trends of both limiting solutions. Consider a ring element dAx of length dx on the interior of the tube wall at x, as in Figure 7.14. The energy supplied per unit time is composed of electric heating, energy radiated to dAx by other wall elements of the tube interior (see Example 5.19), and energy radiated to dAx through the tube inlet and exit: qe pDdx +

ò

l

z =0

sTw4 ( z)dFdz - dx ( z - x ) pDdz

pD 2 pD 2 + sT dF2- dx (l - x ). dF1- dx ( x ) + sTr4,2 4 4



4 r ,1

FIGURE 7.14  Flow through tube with uniform internal energy input to the wall and the outer surface insulated.

330

Thermal Radiation Heat Transfer

The tube ends are assumed to act as black disks at the inlet and outlet reservoir temperatures, which are assumed equal to the inlet and outlet gas temperatures. The energy leaving the ring ele-

{

}

ment at x by convection and radiation is h éëTw ( x ) - Tg ( x )ùû + sTw4 ( x ) pDdx . Neglecting axial heat conduction in the wall, the energy quantities are equated to yield (reciprocity was used on the F factors so that dx could be divided out) l



h éëTw ( x ) - Tg ( x )ùû + sTw4 ( x ) = qe +

ò sT (z)dF 4 w

dx - dz

(x -z)

(7.7.1)

z =0

+sTr4,1Fdx -1( x ) + sTr4,2Fdx - 2(l - x ). This has the form of Equation (5.8) with Qk /Ak = qe + h[Tg(x) − Tw(x)]. Equation (7.7.1) has two unknowns, Tw(x) and Tg(x); a second equation is needed before a solution can be found. This is obtained from an energy balance on a volume element of length dx in the tube. The energy carried into this volume by the gas is umρfcpTg(x)(πD2/4) and that added by convection from the wall is

{

}

πDh[Tw(x) − Tg(x)]dx. The energy carried out by the gas is umrf cp ( pD 2 / 4) Tg ( x ) + éëdTg ( x ) /dx ùû dx . An energy balance gives umrf cp



D dTg ( x ) = h éëTw ( x ) - Tg ( x )ùû . (7.7.2) 4 dx

By defining the dimensionless quantities, 1/ 4

1/ 4



St =

h æ qe ö 4h 4Nu = ; H= qe çè s ÷ø umrf cp RePr

;

æ sö J=Tç ÷ è qe ø



and X = x/D, Z = z/D, and L = l/D; the energy balances on the wall and fluid are X



L

ò

ò

J4w ( x ) + H éëJw ( x ) - Jg ( x )ùû = 1+ J4w ( Z )dFdX - dZ ( X - Z ) + J4w ( Z )dFdX - dZ ( Z - X ) X

0

(7.7.3)

+Jr4,1Fdx -1( X ) + Jr4,2Fdx - 2(L - X ) d Jg ( X ) = St éëJw ( X ) - Jg ( X )ùû . (7.7.4) dX



The two equations have the unknowns ϑw(X) and ϑg(X) and five parameters: St, H, L, ϑr,1, and ϑr,2. Equation (7.7.4) can be solved by using an integrating factor. The boundary condition is that ϑg(X) has a specified value ϑg,1 at X = 0. The solution is X

ò

Jg ( X ) = Ste -StX eStZ Jw ( Z )dZ + Jg ,1e -StX . (7.7.5)



0

This is substituted into Equation (7.7.3) to yield an integral equation for ϑw(X) X

4 w

J ( X ) + HJw ( X ) - HSte

- StX

òe

StZ

Jw ( Z )dZ - HJg ,1e -StX

Z =0 X



= 1+

ò

Z =0

L

J4w ( Z )dFdX - dZ ( Z - X ) +

ò J (Z)dF 4 w

Z =X

+ Jr4,1FdX -1( X ) + Jr4,2FdX - 2(L - X ).

dX - dZ

( Z - X ) (7.7.6)

331

Combined Mode Boundary Conditions

Solutions to Equation (7.7.6) for flow inside a tube were obtained by Perlmutter and Siegel (1962) and representative results calculated by numerical integration are in Figure 7.15. Note that the predicted temperatures for combined radiation and convection fall below the temperatures predicted for either convection or radiation acting independently. For a short tube, radiation effects are significant over the entire tube length, and for the parameters shown, the combined-mode temperature distribution is similar to that for radiation alone. For a long tube, the combined-mode distribution is close to that for convection alone over the central portion of the tube. The heat transfer resulting from combined convection–radiation is more efficient than by either mode alone. Hence, the wall temperature distribution in the combined problem is below distributions predicted by using either mode alone. Example 7.8 What are the governing energy equations if the tube interior in Example 7.7 is diffuse-gray with emissivity ε rather than being black? A convenient derivation is by use of the enclosure Equation (5.56). The energy flux added to the interior surface of the wall by means other than internal radiative exchange is qw(x) = h[Tg(x) − Tw(x)] + qe. The enclosure equation yields



qw ( x ) 1- e e e

l

ò

l

qw ( z ) dFdx - dz ( z, x ) = sTw4 ( x ) -

z =0

ò sT

4 w

( z ) dFdx -dz ( z, x )

z =0 4 r ,1 dx -1

- sT F

(7.8.1)

4 r , 2 dx - 2

- sT F

where qw(x) can be substituted to yield an equation with Tw(x) and Tg(x). Equation (7.7.5) is unchanged by having the wall gray. Thus, Equations (7.8.1) and (7.7.5) are two equations for the unknowns Tw(x) and Tg(x). Numerical solutions are in Siegel and Perlmutter (1962).

FIGURE 7.15  Tube wall temperatures resulting from combined radiation and convection for transparent gas flowing in a uniformly heated black tube for St = 0.02, H = 0.8, ϑr,1 = ϑg,1 = 1.5, and ϑr,2 = ϑg,2. (a) Tube length, l/D = 5; (b) tube length, l/D = 50.

332

Thermal Radiation Heat Transfer

Example 7.9 Consider again the tube in Example 7.7 that is uniformly heated and perfectly insulated on the outside and has a black interior surface. Gas flows through the tube, and the convective heat transfer coefficient is assumed constant. Axial heat conduction in the tube wall is now included. The tube wall has thermal conductivity kw, thickness b, and inside and outside diameters Di and Do = Di + 2b. The desired result is Tw(x) along the tube length. The wall is assumed sufficiently thin that the local Tw(x) is constant across the wall thickness. The energy balance in Equation (7.7.1) is modified to include the net gain of energy by an element of the tube wall from axial wall heat conduction. This provides the energy balance



h éëTw ( x ) - Tg ( x )ùû + sTw4 ( x ) = qe + kw

Do2 - Di2 d 2Tw ( x ) + dx 2 4Di

l

ò sT (z)dF 4 w

dx - dz

(x -z)

z =0

(7.9.1)

+ sTr4,1Fdx -1( x ) + sTr4,2Fdx - 2(l - x ). As in connection with Equation (7.7.1), all lengths are nondimensionalized by dividing by the internal tube diameter, and dimensionless parameters are introduced. The conduction term yields a new parameter

NCR =

kw 4qeDi

éæ Do ö 2 ù æ qe ö1/ 4 êç ÷ - 1ú ç s ÷ . êëè Di ø úû è ø

(

)

For thin walls where (Do − Di)/2 = b ≪ Di, this reduces to NCR = kw b/qeDi2 ( qe /s ) references. The dimensionless form of the energy equation is J4w ( X ) + H éëJw ( X ) - Jg ( X )ùû = 1+ NCR

d 2Jw ( X ) + dX 2

1/ 4

, used in some

X

ò J (Z)dF 4 w

dX - dZ

(X - Z)

Z =0

L

+

ò J (Z)dF 4 w

dX - dZ

(7.9.2)

( Z - X ) + Jr4,1FdX -1( X ) + Jr4,2FdX - 2(L - X ).

Z =X

The energy equation for the fluid is still Equation (7.7.5); these equations can be combined as in Equation (7.7.6). Hottel discussed this problem in terms of slightly different parameters. He obtained an early numerical solution for five ring-area intervals on the tube wall, before the common use of computers. Results are in Figure 7.16 in terms of the parameters derived here.

If the formulation includes axial conduction, there are two additional conduction boundary conditions. The solution of Equation (7.9.2) requires two boundary conditions because of the constants introduced by integrating the d2ϑw/dX2 term. The boundary conditions depend on the physical construction at each end of the tube that determines the amount of conduction. In Siegel and Keshock (1964), some detailed results were obtained where it was assumed for simplicity that the tube end edges were insulated, (d Jw /dx ) x =0 = (d Jw /dx ) x =l = 0. The extension was also made in Siegel and Keshock to have the convective heat transfer coefficient inside the tube vary with position along the tube as in a thermal entrance region. The FEM (Section 7.7.2) was used by Razzaque et al. (1982) to extend the results to large tube lengths and to include a sinusoidal heat flux addition along the tube length. Ganesan et al. (2015) give experimental results for airflow in a horizontal duct with axial temperature variation and show that surface–surface radiation has a significant effect on total heat transfer.

Combined Mode Boundary Conditions

333

FIGURE 7.16  Wall temperature distribution for flow of transparent fluid through black tube with combined radiation, convection, and conduction for L = 5, St = 0.005, NCR = 0.316, H = 1.58, ϑr,1 = ϑg,1 = 0.316, and ϑr,2 = ϑg,2.

7.4.3 Natural Convection with Radiation At moderate temperatures radiative fluxes are small, but in conjunction with natural convection in air that generally produces small convective heat transfer coefficients, the radiative transfer may be comparable to convection. If a single vertical plate in air is internally heated, there can be an interaction of radiation with free convection, depending on the heating condition of the plate. For a specified amount of heating along the plate, such as by electric heating, the local temperatures along the plate for steady state must be such that the local heating is dissipated by radiation, convection, and conduction within the plate to adjacent elements. The behavior of a very thin vertical electrically heated stainless-steel foil with negligible lengthwise heat conduction was investigated analytically and experimentally by Webb (1990). With radiative dissipation included with natural convection, the foil tends to have a more uniform temperature distribution than for natural convection alone. For a fin on a surface, the net heat conduction along the fin is balanced by local radiative dissipation and natural convection. An analysis for a single vertical fin on an isothermal base is in Balaji and Venkateshan (1996) for combined radiation and natural convection. From symmetry about the vertical center plane of the fin, half of the geometry can be considered for analysis with the center plane of the fin perfectly insulated to provide the symmetry boundary condition. With the base surface included, this results in a study of combined natural convection and radiation inside an L-shaped corner, with the temperature distribution in the vertical wall depending on heat conduction in the fin, free convection, and radiation exchange with the base surface of finite width and with the surrounding environment. Radiative exchange with adjacent fins was not considered. The results were verified by comparison with experiments from Rodigheiro and de Socio (1983). A proposed method is discussed in Guglielmini et al. (1987) for cooling electronic components attached to a base plate. A series of staggered fins can be attached to a base plate to provide cooling by radiation and free convection. Enclosure theory was used to evaluate the radiative interaction between the fins and the base surface. Predicted results agreed well with an experimental study.

334

Thermal Radiation Heat Transfer

Natural convection inside a closed 2D horizontal rectangular enclosure is analyzed in Dehghan and Behnia (1996) (rectangular cross-section in the vertical x–y plane and a large length in the horizontal z direction). The two horizontal boundaries have no external energy supplied to them and are insulated on the outside (adiabatic top and bottom boundaries). The two vertical boundaries are each maintained at a different uniform temperature. The radiative transfer within the enclosure filled with transparent gas was formulated using the net radiation method in Chapter 5. The enclosure was divided into 20 zones on each boundary, and the configuration factors were evaluated with the crossed-string method (Section 4.3.3.1). The resulting simultaneous equations were solved directly by the Gaussian elimination method. For the gas, the steady 2D laminar free-convection flow equations were used with the Boussinesq approximation. The flow and energy equations in the gas were solved by using a finite-volume numerical method given in Gosman et al. (1969). In a closed rectangular space, as in Balaji and Venkateshan (1994a, b), there is heat flow across the space if the two vertical walls are at different temperatures. Interferometric measurements are given in Ramesh and Venkateshan (1999). An enclosed air space can be used to provide insulation; for this application the heat transfer across the space must be reduced. This can be done by placing vertical partitions in the space, thus providing radiation barriers and suppressing natural convection, as has been shown by Nishimura et al. (1988). Radiation was not included, and to examine radiative effects, an analysis was made by Sri Jayaram et al. (1997) of a closed rectangular space with a single vertical partition dividing the interior into two equal rectangular enclosures. The finite-volume numerical method and enclosure theory were used, as in Balaji and Venkateshan (1994a). Including radiation between the surfaces was found to suppress free convection and to augment the total heat transfer compared with the results without radiation. The vertical partition had a strong influence on the radiative transfer, and results in the partitioned enclosure could not be predicted from results in an enclosure without a partition. This is a result of the nonlinear behavior of the radiative transfer. Natural convection induced by heated walls forming a vertical 2D channel was studied in Carpenter et al. (1976) and Moutsoglou et al. (1992). Two parallel walls of length L are spaced b apart as in Figure 7.17, and the bottom and top of the channel are open. The internal surfaces A1 and A2 can have specified temperatures or specified heat fluxes. Because the walls are unequally heated, an asymmetric free-convection velocity distribution develops. The radiative exchange tends to equalize the convective heat transfer from the walls, which leads to an improvement in the overall convective cooling. In addition, there is radiative energy dissipation through the end openings of the channel. The radiative exchange can considerably alter the free-convection behavior. Heat transfer

FIGURE 7.17  Natural convection between parallel walls exchanging radiation.

Combined Mode Boundary Conditions

335

away from the heated walls can be augmented by placing an unheated vertical plate between the walls. Energy is radiated from A1 and A2 to this additional surface and is then transferred away by natural convection and radiation. A two-wall geometry as in Figure 7.17 was analyzed by Moutsoglou et al. (1992) to determine the effect of placing one or more vents in one wall with that wall unheated and insulated on the outside. The other wall was uniformly heated on the outside. Radiation is exchanged between the walls, thereby heating the vented insulated wall. The increased temperatures of both walls induce free convection, and air enters through the bottom space between the walls and through the vents in the insulated wall. The radiative boundary conditions are formulated by standard enclosure theory, as discussed in Chapter 5. The natural convection flow equations using the Boussinesq approximation were solved numerically. It was found that the presence of vents degrades the natural convection cooling process; an unvented channel provides the best performance. In electronic equipment, there can be isolated heat sources that are cooled by combined radiation and free convection. In Dehghan and Behnia (1996), the heat transfer was analyzed and compared with experiments for a heat source on the boundary of a cavity formed by two parallel walls, as in Figure 7.17, but with the opening at the bottom closed by an adiabatic boundary. The two side walls have a finite thickness and are each adiabatic on the outside. There is heat conduction within these walls along their vertical length. A heat source is located at the mid-height of one vertical wall. The radiation exchange was formulated by subdividing the vertical walls and using enclosure theory. The natural convection flow is 2D and laminar, and the Boussinesq approximation was used. A finite-difference solution was obtained using a pseudo- time-dependent iteration with the alternating direction implicit method. Radiation was found to have a significant effect on the flow as it caused a recirculation zone to form, and including radiation in the analysis was necessary to obtain good comparisons with experiment. Lage et al. (1992) consider a cavity formed by two parallel vertical walls with a horizontal adiabatic boundary at the bottom. The vertical walls are at differing uniform temperatures. Based on some previous work that was found to provide sufficiently accurate results, a simplifying approximation is made by obtaining the solution in two steps. First, an analysis for natural convection in the cavity is carried out without radiation. A finite-difference solution is obtained that provides the free-convection heat transfer coefficients at the walls. Then net radiation enclosure analysis is used to add the radiation exchange. A more complex geometry is considered by Zhao et al. (1992) to determine the cooling behavior of three electrically heated power cables inside a horizontal rectangular conduit. The conduit is a closed rectangular channel that is long, so the geometry can be considered 2D, and the three cables are internally heated horizontal cylinders along the bottom of the rectangular enclosure. The stream function, vorticity, and energy equations were used for the gas in the conduit, and a finite-difference solution was obtained using a pseudo transient convergence method (DeVahl Davis 1986). The radiative portion was formulated by using the net radiation enclosure theory (Chapter 5). The effect of radiative transfer was found to be important in the results, and it must be included in the theoretical modeling. The heat transfer behavior for each of the three cylindrical power cables in the conduit was significantly different from that for a single cable in a horizontal conduit. Natural convection instabilities can be produced or modified by radiation exchange. Instabilities were analyzed by Lienhard (1990) for a plane layer of transparent fluid between two horizontal walls with the lower wall heated; the walls radiate and conduct energy. The radiation exchange between the walls tends to partially equalize temperature nonuniformities and thereby stabilize the fluid against the development of free-convection circulation cells. The stability of confined plane horizontal fluid layers has application to the design of flat-plate solar collectors. The radiation exchange between surfaces can have significant effects on crystals being grown from the vapor phase within an enclosure. An analysis by Kassemi and Duval (1990) showed that the radiative exchange can induce natural convection for a normally stable heating configuration, thus altering the vapor transport to the crystal growth interface. Another study (de Groh and Kassemi 1993) considered a short circular cylindrical enclosure closed at both the top and bottom with flat

336

Thermal Radiation Heat Transfer

plates. The top plate is heated, and both analytical and experimental information is obtained for combined radiation and free convection in the enclosure. If an enclosure is heated on the top and cooled at the bottom, the buoyancy associated with the temperature distribution in the gas tends to make the gas stable. Radiation exchange in the enclosure changes this; energy is radiated to the side walls and free-convection patterns are thereby initiated. The radiative exchange in this analysis was formulated for diffuse-gray surfaces using the net radiation enclosure theory (Chapter 5). The analysis in the gas was carried out with the finite-element computer code FIDAP. It was found by comparison with experiments that radiation effects are important even at temperature levels as low as 300°C. If radiation is not included, the numerical predictions can be far from reality. Radiative effects resulted in a double annular circulation pattern within the enclosure. Yamala and Rao (2017) simulated mixed convection and surface radiation from a vertical plate simulating an electronic board with flush-mounted heat sources. Ganesan et al. (2018) did experimental measurements of laminar developing free convection in a vertical channel with two isothermal and two adiabatic walls. The effect of inclination angle on surface radiation-natural convection transfer in an open cavity is studied in Msaddak et al. (2018) using a lattice-Boltzmann method. These studies have applications in flat plate solar collectors and vertical wall cavities. Because of the bifurcation/chaos characteristics of natural convection analysis, care should be taken in assuming that 2D solutions are valid, as 3D cell structures can form in apparently 2D geometries (e.g., long channels with asymmetrically heated walls.) These structures can be augmented or retarded by the presence of radiation. Chaurasia et al. (2017) examine 3D laminar mixed convection of air in a rectangular channel with protruding heat sources on the lower channel wall with a geometry typical of a printed circuit board and conclude that radiation is significant at lower Reynolds numbers and lower circuit board thermal conductivity.

7.5 NUMERICAL SOLUTION METHODS Numerical solution methods are now presented and illustrated for radiation combined with conduction and/or convection. The methods also apply to pure radiation problems, which are usually easier to solve; for example, for gray surfaces, the equations are linear in temperature to the fourth power when there is only radiative transfer. In the previous chapters, energy equations were derived from an energy balance on each element of the system used to model the real configuration. For some pure radiation solutions, detailed temperature distributions may not be needed and an enclosure may be divided into relatively few elements. With conduction and convection included, there is usually a need for finer detail because these transfer modes depend on local temperature derivatives; accurate and detailed temperature distributions must be obtained so that derivatives can be evaluated accurately. Sometimes the difference between large incoming and outgoing radiation is needed to determine a relatively small amount of conduction and/or convection, and this leads to difficulties with convergence and/or accuracy. The local energy balance on each element of the system involves the net radiation that is absorbed at the surface, and this depends on the summation of all the contributions from the surroundings. In a detailed formulation, the radiative portion may be set up as integrals involving the temperature distributions and configuration factors from the surrounding surfaces. Some of the solution methods using numerical integration are presented here and are illustrated with a few examples. Integration subroutines are available in computer mathematics software packages. The benefits of nondimensionalization are discussed for multimode analyses. The relative sizes of the dimensionless parameters may provide insight into the best numerical approach. Some illustrative examples are set up using the finite-difference method and FEM. The result of the numerical formulation is a set of nonlinear algebraic equations, and solution methods are discussed such as by successive substitutions or iteration. Because of the nonlinearity of the combined-mode equations, it is often necessary to use damping or under-relaxation factors to obtain convergence. Problems involving only radiation will sometimes permit over-relaxation to speed convergence. Further, the

337

Combined Mode Boundary Conditions

Newton–Raphson method is presented as another useful technique for solving a set of nonlinear algebraic equations. The Monte Carlo method of Chapter 14 may also be used to obtain radiative or multimode heat transfer solutions. This is a statistical method in which many small quantities of radiant energy are individually followed along their paths during radiative transfer. This method is relatively easy to set up for complex problems that involve spectral effects and/or directional surfaces. The solutions may require relatively long computer running times (although this is rapidly becoming immaterial with increasing computer speeds and the solving capability of GPUs), and the Monte Carlo method may be the only reasonable way to attack some complex problems.

7.5.1 Numerical Integration Methods for Use with Enclosure Equations Integration is needed for numerical solution of pure radiation or combined-mode problems. For radiative exchange, the integrals are often functions of two position variables, and integration is over one or both of them. For example, the configuration factor dFdi–dj from position ri on surface i to position rj on surface j appears in the integral over surface j to obtain Fdi–j in the form (see Equations (4.11) and (5.53))

Fdi - j (ri ) =

ò dF

di - dj

Aj

ò

(ri , rj ) = K (ri , rj )dA j . (7.11) Aj

Many ways can be used to numerically approximate an integral. Because the integrands in radiative enclosure formulations are usually well-behaved at the end points, closed numerical integration forms are often used that include the end points. Open methods do not include the end points and can be used when end-point values are indeterminate, such as for improper integrals that yield finite values when integrated. In analyses including convection and/or conduction, the numerical integration will usually use the grid spacing imposed by the differential terms. In some situations it is sufficient to use numerical integration methods that have regular grid spacing. However, uneven spacings are often advantageous for placing more points in regions where functions have large variations, or to adequately follow irregular boundaries. Gaussian quadrature can be used for variable grid spacing. Simpler schemes such as the trapezoidal rule or Simpson’s rule may be adequate for some problems. These often employ uniform grid spacing and are closed, whereas Gaussian quadrature is open. The trapezoidal rule can readily be used with a nonuniform grid size. The standard numerical integration methods are discussed in detail in Appendix I of the online appendices at www.ThermalRadiation.net. Most are included in the standard mathematical packages such as MATLAB® and Mathcad.

7.5.2 Numerical Formulations for Combined-Mode Energy Transfer Figures 7.2 and 7.3 illustrated the boundary condition for an opaque surface where the net radiation (G − J) provides the local addition of radiative heat flux. The (G − J) is found by analyzing the radiative enclosure surrounding the surface area. This boundary condition can be applied to obtain the 3D conduction solution within the wall. As a simplified case, Figure 7.5 gave a control volume derivation for a thin wall in which the temperature distribution is 2D, as it is assumed not to vary significantly across the wall thickness. The control volume approach is further developed here to illustrate combined-mode solutions by using thin-walled enclosures as an example. Consider an enclosure, as in Figure 5.15, and let one or more of the walls be thin and heatconducting. There is uniform heat generation within each wall and convection at the inside surfaces, as in Figure 7.18. The outside of the enclosure is assumed to be well-insulated for simplicity, but external heat flows can be added in a similar way to those included here.

338

Thermal Radiation Heat Transfer

FIGURE 7.18  Radiative enclosure with thin walls in which there is 2D heat conduction.

A local rectangular coordinate system is positioned along a typical wall Ak (Figure 7.18). For a wall element at rk, an energy balance is written where the net radiative loss is balanced by the decrease in internal energy and by the energy added by conduction, convection, and internal heat generation: é æ ¶ 2T ¶ 2T ¶T J k (rk ) - Gk (rk ) = qk (rk ) = ê -rca k + ka ç 2k + 2k ¶t ¶y êë è ¶x



ù ö  ú . (7.12) ÷ + h ( Tm - Tk ) + qa úû rk ø

The local radiative heat loss qk(rk) is found from the enclosure Equation (5.56):

qk ( rk ) ek

N

-

å j =1

1- ej q j ( rj ) dFdk -dj ( rj , rk ) = sTk4 ( rk ) ej

ò

Aj

N

åòsT

4 j

( rj ) dFdk -dj ( rj , rk ) . (7.13)

j =1 Aj

The local temperature of the convecting medium, Tm(rk), is obtained from additional convective heat transfer relations, as we illustrate. The equations are placed in dimensionless form to yield é ¶J æ ¶ 2 Jk ¶ 2 Jk q k (R k ) = ê - k + N CR ç + 2 ¶Y 2 êë ¶t è ¶X





q k ( R k ) ek

N

-

å j =1

1- ej ej

ò

Aj

ù ö ÷ + H (Jm - Jk ) + S ú (7.14) ø ûú Rk

q j ( R j ) dFdk -dj ( R j , R k ) = Jk4 ( R k ) -

N

å òJ ( R ) dF 4 j

j

dk - dj

( R j , R k ) (7.15)

j =1 A j

where

 /sTref4 q = q /sTref4 , N CR = k /asTref3 , H = h /sTref3 , S = qa



t = sTref3 /rca, R = r /a, X = x /a, Y = y /a, J = T /Tref .

In Equations (7.14) and (7.15), the dimensionless temperature ϑ and the dimensionless net radiative heat flux q are the dependent variables, and X, Y, and t are independent variables. The dimensionless

339

Combined Mode Boundary Conditions

parameters N CR , S , and H (or slight modifications of them) appear in combined-mode problems involving radiative transfer. They provide a measure of the importance, relative to radiation, of conduction, internal energy generation, and convection, and their relative magnitudes can help indicate the best solution method. If H is large, the problem could be solved as convective heat transfer with a small effect of radiation. The solution will probably converge best if ϑ is chosen as the dependent variable. For small H, the problem might best be solved as a radiative transfer problem using ϑ4 as the dependent variable. In a transient problem, the variation in temperature level may shift the relative importance of the modes. Equation (7.15) is the enclosure equation for gray surfaces. If the enclosure has surface properties that vary with wavelength, Equations (6.6) and (6.7) can be used. This is solved for qΔλ,k in each wavelength band for each surface. The q k ( R k ) in Equation (7.14) is then found as the summation q Dl,k . Equation (7.14) is otherwise unchanged. Once the best arrangement of the equations

å

Dl

is determined, they must be placed in a form for numerical solution. The application of the finitedifference method and FEM are described in some examples. 7.5.2.1 Finite-Difference Formulation The solution technique by finite differences is described by two examples. Example 7.10 The fin temperature distribution in the array of fins in Figure 7.9 is considered in Example 7.5 and is governed by the dimensionless energy Equation (7.5.3)

d 2J( X ) -m + J4 ( X ) = FdX -B ( X ) + dX 2

where FdX -B ( X ) =

1

é ù d 2J(Z ) + J4 (Z )ú × dFdX - dZ ( X , Z ) (7.10.1) ê -m(1- e) 2 dZ ë û z =0

ò

B2 1 1é X ù 1- 2 , dFdX -dZ ( X , Z ) = dZ , m = ka/2esTb3W 2 , and B = b/W . 2 1/ 2 ú ê 2 ë (B + X ) û 2 é B 2 + ( Z - X ) 2 ù 3/ 2 ë û

m = ka/2esTb3W 2 , and B = b/W . The boundary conditions are ϑ = 1 at the fin base X = 0, and dϑ/dX = 0 at X = 1 as it is assumed for simplicity that the end edge of each fin has negligible energy loss. To evaluate the integral on the right-hand side of Equation (7.10.1), the entire distributions of temperature and its second derivative must be known. Hence, if this equation is written at each of a set of X values, each equation will involve the unknown ϑ(X) at all of the X values, and all of the equations must be solved simultaneously. To proceed with the solution of Equation (7.10.1), the fin is divided into N small elements, so that Xi = iΔX and Zj = jΔZ, where 0 ≤ i ≤ N and 0 ≤ j ≤ N. The second derivative is approximated by d 2J Ji +1 - 2Ji + Ji -1 = ( DX )2 dX 2



The integral on the right-hand side of Equation (7.10.1) can be approximated using the trapezoidal rule (see Appendix G of the online appendices at www.ThermalRadiation.net): é1 f ( X , Z )dZ » DZ ê f0 ( X ) + ê2 0 ë 1



where f ( X , Z ) =

ò

N -1

ù

å f (X ) + 2 f (X )úúû j

1

j =1

1 ù B2 é d 2J + J4 ( Z ) ú 3/ 2 . ê -m (1 - e ) 2 2 2 2 ë dZ û éB + ( Z - X ) ù ë û

N

340

Thermal Radiation Heat Transfer Using ΔZ = ΔX and substituting the finite-difference form of the second derivative results in f j ( Xi ) =



ù J - 2J j + J j -1 B2 é 1 ê -m (1- e ) j +1 + J4j ú 2 2 2 ê ú 2 X D ( ) ë û B + éë( j - i ) DX ùû

{

}

3/ 2

.

The two limits where j = 0 and j = N are evaluated by applying the boundary conditions. For j = i = 0, ϑ0 = 1 and d2ϑ/dZ 2 = 0. The latter condition is needed because the ϑi−1 term would otherwise be undefined; the condition arises because the energy entering the fin at the base is all by conduction, so the temperature gradient is linear at that location. Then, for j = 0, f0 ( X i ) =



B2 2 éëB2 + (i DX )2 ùû

3/ 2

.

For j = N, ϑN + 1 = ϑN−1 from (dϑ/dx)X = 1 = (ϑN−1−ϑN−1)/(2ΔX) = 0, where N + 1 is a symmetric image point of N − 1. Then fN ( X i ) =



ù B2 é 2(JN -1 - JN ) 1 ê -m (1- e ) + JN4 ú 2 2 2 ê ú 2 D X ( ) ë û B + éë( N - i ) DX ûù

{

}

3/ 2

.

The energy Equation (7.10.1) for element i at Xi = iΔX can now be written in finite-difference form as



ì ü é1 ï Ji +1 - 2Ji -1 1ï 1 4 ê f0 ( Xi ) + + = -m J + D X i í ý ( DX )2 2 ï B /(i DX ) 2 + 1 1/ 2 ï ê2 [ ] ë î þ

{

}

N -1

å j =1

f j ( Xi ) +

ù 1 fN ( Xi )ú . (7.10.2) 2 ú û

This equation is written for each element i for the range 1 ≤ i ≤ N, giving N equations for the N unknown temperatures ϑi. Each equation contains every unknown ϑi, which appears in the fj terms. If the resulting set of equations is written in matrix form, the coefficient matrix is full. This is in contrast to 1D pure conduction problems where the coefficient matrix is usually tridiagonal. This illustrates that in radiative transfer problems, the temperature of every element can be influenced by the temperature of all of the surrounding elements. Equation (7.10.2) written for element i = 1 gives

-m

ö æ J2 - 2J1 + 1 1 1 1ì ü DXB2 ç 1 ÷ + + J14 = í13 / 2 2 2 1/ 2 ý ( DX ) 2 çç 2 éB2 + ( DX )2 ù ÷÷ 2 î [(B / DX ) + 1] þ û ø è ë N -1



-

å éêëm(1- e) j =1

+

J j +1 - 2J j + J j -1 1 ù - J4j ú ( DX )2 2 û B + [( j - 1)DX ]2

{

ù 2 ( JN -1 - JN ) 1é 1 + JN4 ú ê -m(1- e) 2 2ë ( DX ) û B2 + [(N - 1)DX ]2

{

}

}

3/ 2

3/ 2

(7.10.3)

.

For the infinite array of fins, ϑi = fj, so that, gathering terms that have ϑj and J4j , and using ϑ0 = 1, Equation (7.10.3) is written as (a repeated subscript denotes a summation over the values of that subscript, e.g., A1j J j =

å

N j =1

A1j J j )

341

Combined Mode Boundary Conditions

A1j J j + B1j J4j = C1 (7.10.4)

where éëlet B º B2m (1 - e )/2DX )ùû

4 2 1 ü ì ï ï + 3/ 2 A11 = B í (1 - e ) DXB2 B3 é 2 2 ý ù D + B X ( ) ë û þï îï



-2 1 2 1 ì ü A12 = B í + 3 - 2 + 2 2 2 3/ 2 2 3/ 2 ý XB B B X B X e ( ) D [ + ( D ) ] [ + ( D ) ] 1 2 î þ 

æ ö 1 2 1 A1j = B ç 2 - 2 + 2 2 3/ 2 ÷ 2 3/ 2 2 3/ 2 {B + [( j - 1)DX ] } [B + ( j DX ) ] ø è {B + [( j - 2)DX ] } 

2 < j < N - 1

1 1 2 æ ö - 2 + 2 A1(N -1) = B ç 2 2 3/ 2 2 3/ 2 2 3/ 2 ÷ + D + 2 ) D X ] } { B + [( N 1 ) D X ] } B N X B N { [( 3 ) ] } { [( è ø æ 1 A1N = B ç çç 2 B + [(N - 2)DX ]2 è

{



B1j = d1j C1 =

}

3/ 2

-

B2DX ; b j {B + [( j - 1)DX ]2}3 / 2 2

ö ÷ 3/ 2 ÷ 2 2 ÷ B + [(N - 1)DX ] ø 1

{

}

b j = 2, 1£ j £ N b j = 4, j = N



DX m m(1- e) 1ì B2( DX ) ü + + . í1- 2 2 1/ 2 ý 2 2 2 î [B + ( DX ) ] þ ( DX ) 4[B + ( DX )2 ]3 / 2 2B( DX)

This is done for each element i, 1 ≤ i ≤ N, and a matrix equation is generated of the form éë Aij ùû éëJ j ùû + éëBij ùû éëJ4j ùû = [Ci ] . (7.10.5)



This is a set of nonlinear algebraic equations for the unknown temperatures ϑi = ϑj. Solution methods are in Section 7.8.

Example 7.11 A transparent gas with mean velocity um and constant physical properties flows through a circular tube of inner diameter Di and length l (Figure 7.19). A specified heat flux qe(x) = qmax sin(πx/l) is applied along the tube length. The heat transfer coefficient h between the gas and the tube interior surface is assumed independent of x. The tube wall is thin and has thermal conductivity kn. The gas enters the tube from a large plenum at Tg,1. The gas leaves the tube at Tg,2 and enters a mixing plenum that is also at Tg,2. The tube interior surface is diffuse-gray with emissivity ε. Set up the energy equations and boundary conditions to determine the wall temperature T(x). Put the equations into a finite-difference form for numerical solution. Following Examples 7.8 and 7.9, the governing energy equations are X ìï é 1- e ù 4 e í J4 ( X ) + ê e f ( Z ) - J ( Z ) ú dFdX - dZ ( X - Z ) ë û ïî Z =0

ò

L



+

é 1- e ù f ( Z ) - J4 ( Z ) ú dFdX - dZ ( Z - X ) e û

ò êë

Z =X

}

- Jr4,1FdX -1 ( X ) - Jr4,2FdX - 2 ( L - X ) = f ( X )

(7.11.1)

342

Thermal Radiation Heat Transfer

FIGURE 7.19  Cylindrical tube geometry with gas flow and internal radiation exchange.



d 2J æ pX ö + H[Jg ( X ) - J( X )] + NCR f( X ) = sin ç (7.11.2) ÷ dX 2 è L ø



Jg ( X ) = St e -StX eStZ J( Z )dZ + Jg ,1e -StX . (7.11.3)

X

ò 0

In this form, ϑ = T(σ/qmax)1/4 and the other parameters are as defined in Examples 7.7 and 7.9 (the reference value of qe is qmax and the reference length is Di). Substituting the last two equations to eliminate Φ(X) and ϑg(X) in Equation (7.10.4) results in L é Z é ù ì æ pZ ö - StZ ê H e eStxJ(x)d x + Jg ,1e -StZ - J( Z )ú eJ4 ( X ) + ê(1- e) ísin ç St + ÷ ê ê ú î è L ø 0 ë 0 ë û

ò



ò

+ NCR

= N CR

ù d 2J ü 4 ý - eJ ( Z )ú dFdX - dZ ( X - Z ) dZ 2 þ úû

(7.11.4)

X é ù d 2J æ pX ö StX ê H e eStZ J( Z )dZ + Jg ,1e -StX - J( X )ú + + St sin ç ÷ 2 dX ê ú è L ø 0 ë û

ò

+ Jr4,1FdX -1( X ) + Jr4,2FdX - 2(L - X ). For the boundary conditions required by the d2ϑ/dX2 term, both end edges of the tube are assumed to have negligible heat losses, so (dϑ/dX)x = 0 = (dϑ/dX)x = L = 0. The condition ϑg(X = 0) = ϑg,1 was used in deriving Equation (7.11.3). To proceed with the numerical solution, define Z æ ì ù é 1- e ï æ pZ ö - StZ ê H e g ( x ) d x + Jg ,1e -StZ - J ( Z ) ú f (X, Z ) = ç sin ç St + í ÷ çç e ê ú ïî è L ø 0 û ë è (7.11.5)

ò



+ NCR

ö dFdX - dZ ( X - Z ) d 2J ü - J4 ( Z ) ÷ 2ý ÷ dZ þ dZ ø

343

Combined Mode Boundary Conditions where g(ξ) = eStξϑ(ξ). Equation (7.11.4) becomes



L X é ù é d 2J æ pX ö - StX ê H e e êJ4 ( X ) + f ( X , Z ) dZ ú = NCR sin St g ( Z ) dZ + + ç L ÷ dX 2 ê ú ê è ø (7.11.6) 0 0 ë û ë - StX 4 4 - J ( X ) ùû + Jr ,1FdX -1 ( X ) + Jr ,2FdX - 2 ( L - X ) . + Jg ,1e

ò

ò

Numerical integration is applied to each integral, and a set of nonlinear algebraic equations is obtained as in Example 7.5.1. At Xi = iΔX, Equation (7.11.6) becomes, by use of the trapezoidal rule and having ΔX = ΔZ where ΔX = L/I (I is the number of ΔX increments), ì é1 ï e íJi4 + DX ê f0 ( Xi ) + ê2 ïî ë

I -1

å j =1

fi ( Xi ) +

ù üï 1 J - 2Ji + Ji -1 æX ö fI ( Xi )ú ý = NCR i +1 + sin ç i ÷ 2 2 ( D ) X úï è L ø ûþ ì é1 ï + H íSte -StXi DX ê g0 + ê2 ïî ë



4 r ,1 dXi -1

+J F

( Xi ) + J F

i -1

ù

å g + 2 g úú + J

4 r , 2 dXi - 2

j

j =1

1

i

ü ï e -StXi - Ji ý (7.11.7) ïþ

g ,1

û

(L - Xi ).

Note that for the reservoir temperatures for the specified conditions of this example, ϑr,1 = ϑg,1 and ϑr,2 = ϑg,2. By gathering terms after expansion, as in Example 7.5.1, the full set of equations can be written as ( A00J0 + B00J04 ) ( A10J0 + B10J04 )  ( Ai0J0 + Bi0J04 ) ( AI 0J0 + BI 0J04 )

+( A01J1 + B01J14 ) +( A11J1 + B11J14 ) + + +

+ + + +( AijJj + BijJ4j ) +( AIjJj + BIjJ4j )

+ + + + +

+( A0I JI + B0I JI4 ) +( A1I JI + B1I JI4 ) + +( AiI JI + BiI JI4 ) +( AII JI + BII JI4 )

= C0 = C1 =  . (7.11.8) = Ci = CI

or in matrix form

éë Aij ùû éëJj ùû + éëBij ùû éëJ4j ùû = [Ci ] . (7.11.9)

Once ϑj is determined from the solution of Equations (7.11.8) and (7.11.9), the heat flux ϕj can be found from Equation (7.11.7) if needed.

7.5.2.2 Finite Element Method Formulation The finite element method (FEM) has the advantage that the temperature in a volume or surface element can vary across the element. Finite-difference formulations assign a single uniform temperature to each element. The temperature variation in the FEM can be specified to increased degrees of approximation (constant, linear, parabolic, etc.) at the cost of increasing the computation time. Temperatures at the boundaries of adjacent elements can be matched, temperature gradients can be forced to match, and, with increased complexity and computer time, the second or higher derivatives can be forced to match. Various approaches to the FEM formulation are used, but the most prevalent is the Galerkin method. It is only feasible to provide a brief description here. After this brief description oriented toward solving the energy equation, the use of the FEM is illustrated by formulating Example 7.12. Further developments using FEM for other types of radiative energy transfer problems are in Section 13.6.

344

Thermal Radiation Heat Transfer

Consider the energy equation for transient heat conduction and for constant properties, with the local radiative energy source −∇ ⋅ qr combined with the local source by internal heat generation q into a term qs (see Section 7.2):

rcp

¶T - kÑ 2T - qs (r, T , t ) = 0. (7.16) ¶t

The volume in which the energy equation is to be solved is divided into finite subregions; these are the finite elements. For 1D geometries, the elements are plane, cylindrical, or spherical layers that can have different thicknesses. In two dimensions, triangles are usually used, as in Figure 7.20, or irregular quadrilaterals. In three dimensions, tetrahedrons or rectangular prisms are often used. These types of finite elements are used to divide geometries of irregular shape; the treatment of irregular volumes is one advantage of using the FEM. Nodes are assigned to locations in the elements at which the unknown function, such as temperature, is to be determined. Nodes are often placed only at the corners of the elements, but additional nodes can be placed internally or along the element boundaries. 7.5.2.2.1 Shape Function The next step in the FEM is to choose shape or interpolation functions to provide an approximate variation of the dependent variable within each element between the values at the nodes. The simplest shape function is linear, but quadratic and higher-order variations can be used. Each shape function is a local interpolation function that is finite only within elements containing a particular node. To describe the shape function in more detail, consider a planar 1D problem with coordinate X. The shape function can be derived by using a series expansion of the unknown function, T(X) =  a0 + a1X + a2X2 + a3X3 + ⋯. The number of terms used in the series is determined by two considerations: Retaining more terms allows a more accurate representation of the temperature distribution within each element, but using fewer terms reduces computation time. The choice of the form of the shape function is a trade-off between accuracy within an element, and thus the number of elements required, and the average computation time per element. Most solutions have used linear or quadratic forms.

FIGURE 7.20  Two-dimensional region represented by triangular finite elements having nodes at vertexes.

345

Combined Mode Boundary Conditions

FIGURE 7.21  One-dimensional finite elements and linear shape functions: (a) Plane layer with elements of unequal size, and (b) linear shape functions for typical elements.

For a linear shape function in one dimension, there is a node at each end of the element; these nodes are designated here by subscripts 1 and 2 on T and X (note: The a0 and a1 are coefficients, and their subscripts do not refer to the nodes). The values of the dependent variable at the nodes are T1 = T(X1) = a0 + a1X1 and T2 = T(X2) = a0 + a1X2. Solving for a0 and a1 gives a0 = (T1X2−T2X1)/ (X2−X1) and a1 = (T2−T1)/(X2−X1). Then, in the element, T(X) = Φ1(X)T1(X1) + Φ2(X)T2(X2), where Φ1(X) = (X2−X)/(X2−X1) and Φ2(X) = (X−X1)/(X2−X1). The Φ are the shape or interpolation functions. These functions are each equal to 1 at the node designated by their subscript, and they decrease linearly to zero at the neighboring node. Each shape function is zero outside of the element that contains its node (Figure 7.21). For a quadratic shape function in a 1D geometry, three nodes are used per element. For this and higher-order shape functions, to evaluate the am coefficients, the first- and higher-order derivatives of T(X) are matched at the nodes, in addition to the T values. For 2D problems, the form of the linear shape function for triangular elements is found by using the expansion T(X,Y) = a0 + a1X + a2Y. For square elements, the biquadratic form can be used: T(X,Y) = a0 + a1,1X + a1,2X2 + a2,1Y + a2,2Y2. The shape functions are then derived by substituting the T values at the nodes and solving simultaneously for the unknown am or am,n. For the five coefficients in the biquadratic form, there can be a node at each corner of the square and one in the center. 7.5.2.2.2 Galerkin Form for the Energy Equation By using the shape functions, an approximate solution Tˆ ( r, t ) for T(r,t) is assumed in the form

T ( r,t ) » Tˆ ( r,t ) =

N

åT ( t ) F ( r) (7.17) j

j =1

j

346

Thermal Radiation Heat Transfer

where the Tj are the values at the nodes desired from the solution Φj(r) are the shape functions that are equal to 1 at each node. When this approximation is substituted into the energy equation, Equation (7.16), there is a residual that depends on r and t: rcp



¶Tˆ - kÑ 2Tˆ - qs r, Tˆ , t = Res ( r, t ) . (7.18) ¶t

(

)

It is sought to obtain a solution that, in an average sense over the entire volume, is as close as possible to the exact solution at each time. Variational principles are applied to minimize the residual. A set Wj(r) of independent weighting functions is applied, and the residual is made orthogonal with respect to each of the weighting functions. This provides the following integral, which is evaluated for each of the set of independent weighting functions,

òRes ( r, t ) W ( r ) dV = 0. (7.19)



i

V

The integration is over the whole volume in which the solution is being obtained. Carrying out the integration for each of the set of weighting functions yields a set of simultaneous equations that can be solved for the Tˆi values at the nodes. In the Galerkin method, the weighting functions are chosen to be the same function set as the shape functions. Since each shape function Φj is zero except within an element containing Tˆj , the resulting matrix for solving the simultaneous equations for Tˆj is banded and sparse. By using the residual from Equation (7.18), Equation (7.17) for Tˆ , and the Φ (r) j

as the set of weighting functions, Equation (7.19) provides the Galerkin form of the energy equation for each of the weighting functions: é êrcp êë

N

ò å



V

j =1

F j (r )

¶T j (t ) -k ¶t

ù Ù T j (t )Ñ 2F j (r ) - qs (r,T,t ) ú F i (r )dV = 0. (7.20) j =1 ûú N

å

Evaluating Equation (7.20) for each i provides N simultaneous equations for the Tˆj . This method is now illustrated through an example. Example 7.12 Set up Example 7.11 for numerical solution using the FEM. For simplicity, let the tube interior surface be black. Following the analysis in Razzaque et al. (1982), it is advantageous to use the independent variable ω(X) ≡ ϑ4(X), so Equation (7.16) becomes, with ε = 1, L

ò

w( X ) - w( Z )dFdX - dZ (|X - Z|) = 0



NCR d æ 1 d w ö æ pX ö + sin ç ÷ 4 dX çè w3 / 4 dX ÷ø è L ø

X é ù + H êSte -StX eStZ w1/ 4 ( Z )dZ + Jg ,1e -StX - w1/ 4 ( X )ú (7.12.1) ê ú 0 ë û

ò

+ Jr4,1FdX -1( X ) + Jr4,2FdX - 2(L - X )

347

Combined Mode Boundary Conditions with boundary conditions dω/dX = 0 at X = 0, L. If we define A(ω) ≡ NCR /4ω3/4 and X é ù æ pX ö - StX ê H e e -StZ w1/ 4 ( Z )dZ + Jg ,1e -StX - w1/ 4 ( X )ú y( X , w) º sin ç + St ÷ ê ú è L ø 0 ë û

ò



L

(7.12.2)

ò

+ Jr4,1FdX -1( X ) + Jr4,2FdX - 2(L - X ) + w( Z )dFdX - dZ (|X - Z|) 0

then Equation (7.12.1) has the form -



d dX

dw ù é ê A(w) dX ú + w = y( X , w). (7.12.3) ë û

Note that ψ(X, ω) contains nonlinear terms in the variable ω. To solve Equation (7.12.3) by Galerkin FEM, the ω(X) is required to satisfy a variational form of Equation (7.12.3) and its boundary conditions, which has the form (using Equation (7.19) with the residual of Equation (7.12.3)) L

ò



0

ì d íî dX

L

ü dw ù é ê a(w) dX ú + wý W ( X )dX - y( X , w)W ( X )dX = 0. (7.12.4) ë û þ 0

ò

The W(X) is a weighting function defined by (see Equation (7.17)) N

W (X ) =



å W F (X ). (7.12.5) i

i

i =1

The Φi(X) is the shape function, and the Wi are coefficients at the nodes. The first term in the first integral in Equation (7.12.4) is integrated by parts, and the boundary conditions of the insulated end edges of the tube wall are used: dϑ/dX = dω/dX = 0 at X = 0 and L. This gives L

ò



0

L

d w dW é ù ê A(w) dX dX + wW ú dX - y( X , w)W ( X )dX = 0. (7.12.6) ë û 0

ò

Now, as in Equation (7.17), an approximate solution is sought of the form N

w( X ) » W( X ) =



å W F (X ). (7.12.7) j

j

j =1

In the Galerkin method, the shape functions are the same as in the weighting function, Equation (7.12.5), and Ωj are the values of Ω at the nodes. Substituting Equations (7.12.5) and (7.12.7) into Equation (7.12.1) results in N



æ Wi ç ç è

N

ìï L í ïî 0

å åò i =1

j =1

L ö é ù üï dFi dF j ÷ = 0. (7.12.8) dX X dX A ( W ) + F F W y ( , W ) F i j j i ý ê ú ÷ dX dX ë û ïþ 0 ø

ò

If we now define L



Kij =

ò 0

é ù dFi dF j ê A(W) dX dX + F i F j ú dX ë û

L

and

ò

y i y( X , W)F i dX 0

348

Thermal Radiation Heat Transfer

Equation (7.12.8) becomes N



é Wi ê ê ë

N

å åK W i =1

ij

j =1

j

ù - y i ú = 0. (7.12.9) ú û

To satisfy this equation, the quantity inside the square brackets must equal zero, since the Wi are constant coefficients. Then the required Ωj values can be obtained by solving the equivalent set of equations K11W1 + K12W 2 +  +  + K1N WN = y1 K 21W1 + K 22W 2 +  +  + K 2N WN = y 2

 Ki1W1 +  + Kij W j +  + KiN WN = y i

(7.12.10)

 KN1W1 +  + KNj W j +  + KNN WN = y N . In matrix form, this is

[Kij ][W j ] = [y i ]. (7.12.11)

Because each shape function (and hence, each weighting function in the Galerkin method) is usually zero, except within elements containing a particular node, the [Kij ] is a sparse matrix that is banded along the diagonal. As noted in the introduction to this section, the shape function Φ can have many forms. For example, a linear form could be used for Φi(X) within each element Xi ≤ X ≤ Xi + 1 (where Xi + 1 = Xi + ΔX) to yield Φi(X) = (Xi + 1 − X)/(Xi + 1 − Xi). The FEM solution can now be carried out by solving Equations (7.12.10) and (7.12.11) using a numerical technique, as in Section 7.8. The equation for numerical solution in the FEM, Equations (7.12.10) and (7.12.11), has the same form as Equations (7.11.8) and (7.11.9) for finite differences. Numerical methods that work well with finite-difference formulations will usually apply for solving FEM formulations. After the Ωj is obtained, Equation (7.12.7) is used to find the required temperature values from Ω(X) ≈ ω(X) = ϑ4(X). Note, however, that both Kij and ψi are functions of Ω and thus of Ωj, so the solution is iterative.

Razzaque et al. (1982) applied the FEM to the situation in Examples 7.12 and 7.13. They used a quadratic shape function and were able to extend previous results to a dimensionless tube length of L = 20 (limited in other methods to less than 10, and in some cases to less than 5, by numerical instabilities). Some results are in Figure 7.22. The FEM was applied by Altes et al. (1986) to determine the 3D temperature distribution inside a conducting solid exposed to combined conduction and radiation boundary conditions. Complex geometries representing circuit board chips and a magnetoplasmadynamic propulsion system were modeled using this approach. Radiation interaction among surface elements was included in the solutions, which were obtained using a modified version of a commercially available computer code.

7.5.3 Numerical Solution Techniques Examples 7.11 through 7.13 result in a matrix of nonlinear algebraic equations of the form of Equation (7.11.9). It is important to examine the relative values of the elements Aij and Bij. If the Aij are comparatively large, the problem can be treated as linear in ϑj; conversely, for large Bij, the problem can be treated as linear in J4j . When the coefficients A and B are approximately equal, other treatments are in order.

Combined Mode Boundary Conditions

349

FIGURE 7.22  Comparison of solutions with sinusoidal wall heat flux to illustrate the effects of radiation and axial wall conduction: L = 20, St = 0.01, ϑr,1 = ϑg,1 = 1.5, ϑr,2 = ϑg,2, H = 0.8, and NCR is defined in Example 7.9.

If we define Aij* = Aij + Bij J3j , Equation (7.11.9) becomes

éë Aij ùû éëJ j ùû + éë Bij ùû éëJ4j ùû = éë Aij + Bij J3j ùû éëJ j ùû = é Aij* ù éëJ j ùû = éëCi ùû . (7.21) êë úû

This is a set of linear algebraic equations with coefficients A*ij that are variable and nonlinear. The equations cannot be solved by elimination or direct matrix inversion, because theA*ij are temperature dependent and thus are not known. Many techniques are available for numerical solution of the nonlinear equations typical of combined mode problems with radiation. Some of these are gathered and explored in the online Appendix I.3 at www.ThermalRadiation.net.

7.5.4 Verification, Validation, and Uncertainty Quantification Computer solution of radiative transfer problems, particularly multimode problems, can be quite challenging. To determine the quality of a numerical code, we must first define what it is you really want to know (the quantity of interest, or QoI). Is it the heat flux at a boundary? The temperature at a location on the boundary? Secondary factors might be allowed to have imprecise prediction if the prediction of the QoI is accurate. In all computational problems, it is necessary to establish a priori conditions for the convergence and insensitivity to grid resolution of the QoI. But these are not enough! It should be recognized that any computer model or code ideally should be subjected to three categories of tests: verification, validation, and uncertainty quantification (UQ). This applies to any code, not just those for radiative transfer. 7.5.4.1 Verification A software engineer needs to ask the following questions to verify his or her code: Is the code bug-free, and is it providing results that reflect the governing equations within the assumptions of the analysis of the physical problem being addressed? To ensure that a code is bug-free and giving

350

Thermal Radiation Heat Transfer

correct results within the physical assumptions, one possibility is comparison of code predictions with known benchmark solutions. Benchmarks can be well-accepted solutions from the literature or can be generated. For example, the radiation code can be tested against known simple limiting cases: 1. If all surfaces are isothermal at the same T, is the predicted energy transfer among all elements = 0? 2. If ε = 0 on a surface, is the predicted radiative energy flux = 0 at that surface? 3. For a combined-mode problem, does the code give correct results for each mode independently when the other modes are set to zero? 7.5.4.2 Validation Even if verified, the numerical code may not reflect reality, since various assumptions are usually implicit in the code (gray and/or diffuse surfaces, temperature-independent properties, etc.) The code should therefore be compared with “reality.” Usually, that is a comparison with available experimental data. For comparison of predictions with experiment, we must know both the error limits on the experimental data and the limits on the accuracy in the code predictions to determine whether the predictions lie within the range of the experimental measurements. Because perfect agreement between measurement and prediction is seldom obtained, a priori limits on what constitutes acceptable validation in the prediction of the QoI should be available. Must the prediction be within 1%, 5%, or 20% before the code predictions are acceptable? The answer will depend on the particular problem and the QoI being predicted. 7.5.4.3 Uncertainty Quantification What is the uncertainty in the results of the code (the QoI)? A code prediction usually has at least three types of error source. First, input data to the code such as physical properties, dimensions, and energy source values will have some degree of uncertainty. (Also, see Hively et al. 2013.) Second, the model itself has uncertainties due to the assumptions within the model (e.g., angular and spectral discretization errors, gray and diffuse assumption, assumptions as to whether a 1D, 2D, or 3D analysis is acceptable). Finally, error due to increment size in space and/or time and machine accuracy (number of allowed significant figures) can introduce numerical uncertainty. Having defined possible sources of uncertainty in both input data, the model itself, and numerical uncertainties, how do we quantify how these uncertainties propagate through the code so that the uncertainty in the QoI prediction can be quantified? This is a quite difficult question to answer and is an ongoing research area. It is, however, a very important question, because unless some bounds are available on the uncertainty in prediction of the QoI, the code results are not useful. For example, if the code is predicting global warming effects, but the uncertainty allows predictions that range from a new ice age to immediate loss of the polar ice caps, then the code is not of practical benefit. If we find that the uncertainty in the QoI is so large that the code results are not useful, how do we proceed to reduce prediction uncertainty? It is possible to find the sensitivity of the code to the input data, the modeling uncertainties, and machine errors and to then improve the data or code to reduce the uncertainty that is inherent in the most sensitive factors. This is a daunting task when there are many parameters and much data to be considered. The field of verification, validation, and UQ for complex codes is an active research field (Helton 2009; Oden et al. 2010a, b).

7.6 CONCLUDING REMARKS This chapter has provided extensive information on modeling and numerical solution methods for radiative transfer either alone or in combined modes with other heat transfer mechanisms. There is

Combined Mode Boundary Conditions

351

an extensive and rapidly growing body of literature on this class of problems. The highly nonlinear nature of the governing equations makes them challenging. The need for verification, validation, and uncertainty quantification is especially necessary to assure solutions are correct and useful.

HOMEWORK PROBLEMS (Additional problems available in the on-line Appendix P at www.ThermalRadiation.net) 7.1

7.2

A thin 2D fin in vacuum is radiating to outer space, which is assumed at Te ≈ 0 K. The base of the fin is at Tb, and the heat loss from the end edge of the fin is negligible. The fin surface is gray with emissivity ε. Write the differential equation and boundary conditions in dimensionless form for determining the temperature distribution T(x) of the fin. (Neglect any radiant interaction with the fin base.) Can you separate variables and indicate the integration necessary to obtain the temperature distribution? (Hint: ∫(d2θ/dx2) (dθ/dx) = (1/2) (dθ/dx)2 + constant.)

Consider the fin in Figure 7.12 as analyzed in Example 7.6. The heat transfer coefficient at the tip of the fin is hL, and the emissivity of the end area is ε as for the rest of the fin surface. Formulate the boundary condition for the end face of the fin and apply this condition to the general solution of the fin energy equation. Formulate the analytical relations and describe how you would obtain the fin efficiency. 7.3 A very small-diameter pipe is at Tpipe = 650 K. The pipe is thin-walled polished copper, has diameter D = 0.2 cm, and is in a large room at Te = 300 K. The radiative emissivity of the copper is εc = 0.04. A cylindrical opaque insulation layer with thickness t and thermal conductivity k = 0.07 W/m·K is added to the surface of the pipe. The emissivity of the outer insulation surface is εi = 0.85. The free convective heat transfer coefficient on the surface of the insulation is h = 15 W/m2·K. For simplicity, h is assumed to be independent of insulation diameter and surface temperature, but a more precise analysis should include these effects. It is found that adding the insulation increases the rate of energy loss from the pipe. Find the thickness of insulation t = tmax that maximizes the heat loss from the pipe. Answer: tmax = 3.4 mm 7.4 Consider Homework Problem 6.17. The radiation shield now consists of a layer of opaque plastic 0.10 cm thick coated on each side with a thin layer of metal having the same emissivity, εs = 0.1, as in Homework Problem 6.17. The thermal conductivity of the plastic is 0.200 W/(m·K). What is the heat transfer from plate 2 to plate 1, and how does it compare with that for the very thin shield analyzed in Homework Problem 6.17?

352

Thermal Radiation Heat Transfer

Answer: 2497 W/m2. 7.5 A space radiator is composed of a series of plane fins of thickness 2t between tubes with radius R. The tubes are at uniform temperature Tb. The tubes are black and the fins are gray with emissivity ε. The radiator operates in a vacuum with an environment temperature of Te ≈ 0 K. Formulate the differential equation (including the analytical expressions for the configuration factors) and boundary conditions to obtain the temperature distribution T(x) along the fin. Include the interaction between the fin and the tubes.

7.6

Steam is condensing inside a thin-walled tube of radius ri. The tube has a coating of emissivity εt = 1 on the outer surface. The saturation temperature of the steam is Tb. Identical annular fins of outer radius ro and emissivity εf are evenly spaced a distance L (between fin faces) along the tube. The fins are of thickness δ and thermal conductivity k. The environment surrounding the fin-tube assembly is at Te ≈ 0 K. Convection can be ignored. The configuration factor from a ring element on the tube to a ring element on fin 1 is dFdt – d1 (x, ρ1), and from a ring element on fin 2 to a ring element on fin 1 is dFd2 – d1(L, ρ1, ρ2).

Set up the governing equation for the temperature distribution of the fin, T(ρ).

Combined Mode Boundary Conditions

353

7.7

Two directly opposed parallel diffuse–gray plates of finite width W have a uniform heat flux qe supplied to each of them. The plates are infinitely long in the direction normal to the cross-section shown. They are separated by a distance H. The plates are each of thickness t (t ≪ W) and thermal conductivity k, and both have emissivity ε. The plates are in vacuum, and the surroundings are at Te. Set up the governing integral equation for finding T(x), the temperature distribution across both plates. The outer surface of each plate is insulated, so all radiative heat exchange is from the inner surfaces.

7.8

A copper–constantan thermocouple (ε = 0.15) is in a transparent gas stream at 300 K adjacent to a large blackbody surface at 900 K. The heat transfer coefficient from the gas to the thermocouple is 35 W/(m2·K). Estimate the thermocouple temperature if it is (a) bare, or (b) surrounded by a single polished aluminum radiation shield with εshield = 0.075 in the form of a cylinder open at both ends. The heat transfer coefficient from the gas to both sides of the shield is 15 W/(m2·K).

Answers: (a) 376 K; (b) 358 K. 7.9 A 1 cm diameter thin cryogenic electronic device is glued by epoxy to the bottom surface of a 0.5 cm deep, 1 cm diameter cavity of a high-thermal conductivity ceramic package. The ceramic package is kept at liquid nitrogen temperature of 77 K inside a relatively large vacuum enclosure at the ambient temperature of 300 K. The epoxy provides a thermal interface conductance of 1 × 104 W/m2·K between the electronic device and the ceramic package. All surfaces are gray with diffuse emission. The emissivity is 0.6 on the device surface and 0.3 on the ceramic surface. Calculate the temperature of the electronic device when the device is turned off.

354

Thermal Radiation Heat Transfer

Answer: Td = 77 K. 7.10 For the cryogenic device in Homework Problem 7.9, a thin metallic specularly reflecting foil with reflectivity of 0.1 is inserted to cover the cavity as shown in the figure. Calculate the temperatures of the electronic device and the thin metallic foil when the device is turned off. Compare with the results of Homework Problem 7.9.

Answers: Td = 77 K and Tf = 253 K. 7.11 A rod of circular cross-section extends out from a slender space vehicle in Earth orbit into surroundings at Te. The rod axis is normal to the direction from the sun. The rod is coated so that its infrared emissivity is εIR and its solar absorptivity is αs. The base of the rod is at Tb > Te. Derive a differential equation to predict the rod temperature distribution, T(x). State the boundary conditions, including radiation at the circular end face. Neglect temperature variations within the rod cross-section at each x. Neglect radiation to the rod from the slender vehicle surface and neglect any emitted or reflected radiation from the Earth.

7.12

A1 and A2 are diffuse concentric spheres. The inner surface of the inner sphere is heated with nonradiating combustion products at Tcomb = 1100 K with a convective heat transfer coefficient to the surface of 50 W/(m2·K). Calculate the temperature T1 of the inner sphere for the surface spectral emissivities shown, where ελ,1 is assumed independent of temperature.

Combined Mode Boundary Conditions

355

Answer: 869 K. 7.13 A wire between two electrodes is heated electrically with a total of Qe Watts. The wire resistivity re Ohm-cm is constant, and the current is I. One end of the wire is at T1 and the other is at T2. The immediate surroundings are a vacuum, and the surroundings have a radiating temperature of T0. The wire is gray with emissivity ε, diameter D, thermal conductivity k, and length L. (a) Set up the differential equation for the steady-state temperature distribution along the wire, neglecting radial temperature variations within the wire. Integrate the equation, and find an expression (in the form of an integral) for the wire temperature as a function of x. The final results should contain only the quantities given and should not contain dT/dx. Explain how you would evaluate the expression to find T(x). (b) Let both the emissivity and the electrical resistivity be proportional to T (see Equations (3.33) and (3.34)). Derive the solution for this case and describe how it can be evaluated for a fixed value of Qe.

7.14 A spherical temperature sensor, 0.15 cm in diameter, is on the axis of a short pipe, halfway between the open ends. Air at 400 K is flowing through the pipe, and the convective heat transfer coefficient on the sensor is 23 W/(m2·K). Calculate the sensor temperature. All surfaces are gray. Neglect blockage (shadowing) by the sensor when computing configuration factors between boundaries of the pipe. Do not subdivide surfaces.

Answer: 909 K.

356

Thermal Radiation Heat Transfer

7.15 Two plates are joined at 90° and are very long normal to the cross-section shown. The vertical plate (plate 1) is heated uniformly with a heat flux of 300 W/m2. The horizontal plate has a uniform temperature of 400 K. Both plates have an emissivity of 0.6. The environment is at Te = 300 K. Both plates are made of tungsten, with thermal conductivity k = 174 W/m·K. The plates are 0.056 cm in thickness. At the edge where the plates join, a thin layer of ideal insulation provides an infinite contact resistance. The dimensions are shown in the figure. Derive the equations necessary for finding the temperature distribution on surface 1 and the net radiative heat flux on surface 2. Determine reasonable boundary conditions for the plate ends; justify your choice. (This is a continuation of Homework Problem 5.18).

7.16 An electrically heated nickel wire is suspended in vacuum between two water-cooled electrodes maintained at 810 K. The wire diameter is 0.15 cm and the wire length is 4.15 cm. The surroundings are at a uniform temperature of Te = 900 K and act as a black environment. Air at a temperature of Te flows over the wire, causing a heat transfer coefficient between the wire and the air of h = 35 W/m2·K. Set up the combined radiation and conduction relations to determine the wire temperature T(x) assuming that the radial temperature distribution within the wire is uniform at each x. Determine how many Watts must be generated within the wire for its center temperature to be 1050 K. (The wire thermal conductivity is constant with a value of 85 W/(m·K) and the wire emissivity is constant with value εw = 0.13.) What is the required wattage if the nickel becomes oxidized so that εw = 0.47?

Combined Mode Boundary Conditions

357

Answers: εw

Q (W)

0.13

14.5

0.47

16.0

7.17 A long solid rectangular region in vacuum has the cross-section shown and thermal conductivity kw. One half of one of the long sides is heated by contact with an opaque source of uniform flux qe. The surroundings, which act as a black environment, are at a uniform temperature Te. The exposed surfaces of the region are gray and have emissivity εw. Using the grid shown (for simplicity), set up the finite-difference relations to be solved for the steady temperature distribution in the rectangular solid.

7.18 A long stainless-steel tube with a thick wall is filled with a highly insulating material and is near an infinite black hot wall. Divide the circumference of the tube symmetrically about θ = 0 into eight increments and obtain an expression for the radiant energy from the hot wall to each tube increment. Then, using a finite-difference approximation, obtain an approximate temperature distribution around the tube wall. Neglect radial temperature distributions in the tube wall. The tube wall material has thermal conductivity kw = 34 W/ (m·K) and its outer surface is gray with emissivity εw = 0.187. The surroundings are at low temperature that can be neglected in the analysis, Te ≈ 0 K.

358

Thermal Radiation Heat Transfer

Answers: T1 1021.9 K

T2

T3

T4

1014.3 K

1003.5 K

995.9 K

7.19 A tube has length L = 0.50 m and has an inside diameter of Di = 0.5 m. The tube has a wall thickness of b = 1 cm, and the tube wall material has a thermal conductivity of k = 300 W/m·K. An electric heating tape is wrapped around the outside of the tube and is uniformly and carefully insulated on its outer surface. The tape heater imposes a uniform heat flux of qe = 6000 W/m2 at the inner tube surface. A transparent fluid at Tf, in = 300 K enters the tube from a large plenum at the same temperature. The fluid flows through the tube at a mass flow rate of 0.2 kg/s. The fluid has specific heat cp = 4100 J/(kg·K) and density ρ = 1000 kg/m3. At the given flow rate, the heat transfer coefficient between the fluid and the tube surface is given by hx = 1000/(x + 0.01)1/2 W/(m2·K), where x is the distance from the tube entrance in meters. Four diffuse-gray coatings are available to cover the inside of the tube. These have emissivities of ε = 0.05, 0.35, 0.65, and 0.95. (a) Derive the energy equations that govern the heat transfer behavior to determine the tube wall and fluid temperatures along the tube length, and place them in dimensionless form. (b) Find the maximum tube surface temperature, the position of that temperature, x(Tmax), and the mean fluid temperature at the tube exit, Tf(x = L) for each of the three possible emissivities. Show a plot of the tube inner surface and fluid temperature distributions versus x for each emissivity. These may be in dimensionless form. (c) Discuss the temperature results, giving some physical discussion of the relative effects of the various heat transfer mechanisms on the shapes of both the tube wall and fluid temperature profiles. Discuss the numerical accuracy of the results. Are they converged, and are they within acceptable accuracy? Estimate the possible error in your solutions.

7.20 A high-temperature nuclear reactor is cooled by a transparent gas. The gas flows through tubular cylindrical fuel elements. The fuel elements are of length L m and inside diameter D m, and the gas has specific heat cp kJ/(kg·K), density ρ kg/m3 (both temperature independent), and a mass flow rate of m kg/s through each fuel element. The tube has wall thickness b. Energy is generated in a sinusoidal distribution along the length x of æ px ö the tube at a rate q( x ) = qmax sin ç ÷ W/m 2 based on the inside tube area. The heat è L ø transfer coefficient h between the gas and the tube surface is assumed constant with x and has units of W/(m2·K). The gas enters the tube at temperature Tg,i from a large chamber at

Combined Mode Boundary Conditions

359

temperature Tr,i. The gas leaves the tube at Tg,e and enters a mixing plenum that is at temperature Tr,e. The tube interior surface is diffuse-gray with emissivity ε. The tube exterior surface is perfectly insulated. The tube wall material has thermal conductivity k W/(m·K). (a) Set up the governing equations for determining the wall and gas temperature distributions along the tube length (see Examples 7.7 and 7.8 for some help). (b) Using the dimensionless groups given in Example 7.7 or modifications appropriate for this problem, put the equations in dimensionless form. (c) For values of the dimensionless parameters St = 0.02, N = 5, H based on qmax = 0.08, ϑr,i = ϑg,i = 1.5, and ϑr,e = ϑg,e = 5, solve for the wall temperature distribution along the tube for the cases ε = 1 and ε = 0.5. You may wish to obtain the pure radiation and pure convection results as limiting cases.

7.21 A square enclosure of length 1 m on a side has the conditions shown below and is filled with a transparent stagnant gas. Water is flowing upward along the right-hand vertical surface and enters the channel at the bottom of that wall at Tw = 90°C. The flow rate of the water is large, so that its temperature changes by a negligible amount in passing along surface A1. There is a heat transfer coefficient between the water and the thin enclosure wall of 50 W/m2·K. Surface 4 is specular and perfectly reflecting, surface 3 is black, and the remaining surfaces are diffuse-gray. Surface 2 has a specified temperature and surface 3 has a specified heat flux.

Find the temperature distribution T1(r1) along surface 1. 7.22 A 2D problem is to be solved by the FEM in a rectangle that has a width significantly larger than its height. A rectangular finite element is to be used with unequal width and height so

360

Thermal Radiation Heat Transfer

the elements will conveniently scale into the problem geometry. Linear shape functions are used, so the temperature distribution is represented as T(x,y) = a + bx + cy + dxy. Obtain the shape functions at each of the four nodes for the element and geometry shown.

7.23 To provide a more accurate representation of the functional variation, the previous problem for a rectangular element can be extended to use quadratic shape functions. An intermediate node is inserted at the center of each side as shown in the figure. The interpolating polynomial is chosen to have the quadratic form, T(x, y) = a + bx + cy + dxy + ex2 + fy2 + gx2y +  hxy2. Obtain the shape functions at the eight nodes. Make a 3D plot of the shape function for a corner node.

7.24 A microelectronic system is to be cooled convectively. The system is shown below.

The flow channel is composed of very wide parallel plates separated by a distance D = 0.25 mm, and the channel length is L = 1 mm. Argon enters the channel from a large plenum at Tg,in = 27°C. The mass flow rate of the argon is very high. The heat transfer coefficient between the argon and either channel wall is given by h(x) = 20/[0.1+ (1000x)0.2] (W/m2·K)

Combined Mode Boundary Conditions

361

where x is in meters. The top thin channel wall is heated uniformly by electric heaters at a rate of 50 kW/m2. The lower channel wall is carefully insulated. The hemispherical-spectral emissivity of both channel walls are shown in the graph below.

Provide plots of the temperature of each surface versus x and show the grid independence of your results. Answers: Ttop(x = 0) = 550 K; Ttop, max = 995 K; Ttop (x = 1) = 991 K; Tbottom(x = 0) = 340 K; Tbottom, max = 575 K; Tbottom (x = 1) = 530 K.

8

Electromagnetic Wave Theory James Clerk Maxwell (1831–1879) described the propagation of an electromagnetic wave through a system of 20 equations, and showed that the speed of wave propagation equals the speed of light, implying that light was itself an electromagnetic wave. Among his many accomplishments, Maxwell, in collaboration with Clausius, used a statistical approach to find the velocity distribution in an assembly of gas molecules at a given temperature (later derived using the maximum entropy principle by Boltzmann, and now called the Maxwell–Boltzmann distribution). Oliver Heaviside (1850–1925) was a self-taught engineer/mathematician/physicist who invented methods for solving differential equations and made many contributions to vector calculus. In 1888/1899, he reformulated Maxwell’s 20 field equations into a more manageable set of four equations in terms of four variables. He also coined many terms used to describe the electromagnetic properties of a medium, including permittivity and permeability. Heaviside was often at odds with his employer and the scientific community, but made important contributions to physics, astronomy, and mathematics.

8.1 INTRODUCTION Light is an electromagnetic (EM) wave, and so is thermal radiation. As we discussed in Chapter 1, thermal radiative transfer takes place between 0.1 and 100 μm, although for most practical engineering applications, we consider the range between 0.3 and 20 μm, which includes the visible spectrum from 0.4 to 0.7 μm. These wavelengths are quite small compared to many physical objects we study. For example, the thickness of a human hair is about 100 μm, and most engineered objects are larger than that. Given this, the wave nature of thermal radiation does not need to be explicitly addressed when solving most engineering problems; instead, the radiative transfer equation (RTE) as discussed in the previous chapters can be suitably employed. It should be understood, however, that the wave nature of thermal radiation is crucial for determining the spectral properties of different dielectric and metallic media, as described in Chapter 3. Particles and gases also absorb and scatter radiation in a way that depends strongly on wavelength. Wave mechanics must also be considered when modeling thermal radiation between microand nanoscale objects, which have length scales on the order of the wavelength. Only through EM theory can we consider polarization, coherence, and radiation tunneling, along with the propagation of radiative energy between such structures. In this chapter, we provide general expressions for EM wave theory and show how they can be used to determine surface properties. Finally, we briefly describe how the EM theory can be used to derive the RTE. Sir James Clerk Maxwell established the coherent concept of EM waves with his seminal paper published in 1864. This paper, a crowning achievement of classical physics, shows the relation between electric and magnetic fields and concludes that EM waves propagate with the speed of light, indicating that light itself is an EM wave (Maxwell 1890). Even though later studies have shown that quantum mechanics is a more general theory for energy transfer at all scales, EM wave theory is sufficient for understanding light and radiative energy propagation and its interaction with matter at most size ranges. The Maxwell equations can explain almost all fundamental physics, from nanoscales to stellar systems (Jackson 1998, Mishchenko et al. 2006). In the context of radiative 363

364

Thermal Radiation Heat Transfer

transfer, the concepts of reflection, refraction, transmission, and scattering can be explained using these equations. In addition, values for these parameters, as well as absorptivity of materials can, in certain cases, be calculated from their optical and electrical properties, as shown for the applications in Chapter 3. However, the Maxwell equations do not account for blackbody emission, which needs to be considered separately and is discussed in Chapter 16. Here, relations between radiative, optical, and electrical properties are developed by considering wave propagation in a medium and the interaction between the EM wave and matter. An ideal interaction is considered for optically smooth, clean surfaces that reflect, refract, and transmit the incoming wave in a specular, i.e. mirror-like, behavior. Most real surfaces are not mirrorlike since almost all have surface roughness, contamination, impurities, and crystal-structure imperfections, requiring the development of different approximations to account for diffuse or diffuse-specular reflection, refraction, and transmission characteristics. The departures of real materials from the ideal conditions assumed in the theory can produce large variations of measured property values from theoretical predictions. Although the Maxwell equations cannot completely predict the radiative properties of real surfaces, they serve a number of useful purposes. For example, they explain why there are basic differences in the radiative properties of insulators and electrical conductors and reveal general trends that help unify the presentation of experimental data. These trends become crucial for engineering calculations when limited experimental data must be extrapolated into other spectral and thermal ranges. The classical theory of EM waves also explains the angular behavior of directional reflectivity, absorptivity, and emissivity. Since the theory applies to pure substances with ideally smooth surfaces, it can be used to compute the limits of attainable properties, such as the maximum reflectivity or minimum emissivity of a metallic surface. In this chapter, we start by presenting Maxwell equations and then introduce a number of simplifications. We then use these equations to explore how EM waves propagate through dielectric and conducting media in Section 8.3. Section 8.4 focuses on using Maxwell’s equations to predict the radiative properties of interfaces between perfect dielectrics and conductors, while Section 8.5 presents classical theories for the optical constants needed for these calculations. Finally, in Section 8.6, we discuss the relationship between the RTE and the EM wave theory, as outlined by Mishchenko et al. (2006) and Mishchenko (2014a).

8.2 THE ELECTROMAGNETIC WAVE EQUATIONS The Maxwell equations describe the propagation of EM waves within any medium. In their most general form as reformulated by Heaviside (1889, 1892) ¶B ¶t



Ñ´E = -



Ñ´H = J +



(Faraday’s law) (8.1)

¶D (Ampère’s law) ¶t

Ñ × D = re

(Gauss’s law)

Ñ × B = 0 (Gauss’s law)

(8.2) (8.3) (8.4)

where E, D, H, and B are, respectively, the electric field, electric displacement, magnetic field, and magnetic induction, ρe is the free charge density, J is the current density, and t represents time. For simplicity we will assume our medium has a neutral charge, so ρe = 0. Electric displacement and magnetic induction are related to the electric and magnetic fields. Faraday’s law of induction tells us that the induced electromotive force in any closed circuit is equal

365

Electromagnetic Wave Theory

to the rate of change of the magnetic flux through the circuit. This law clearly shows that the timevarying electric field will produce a time-varying magnetic field and vice versa. Equations (8.1)–(8.4) are augmented by constitutive relationships. The electric field is related to the current density via J = se E



(8.5)

where σe is the electrical conductivity of the material, which is the inverse of the resistivity. In a dielectric material the presence of an electric field causes the atomic nuclei and their electrons, that is, the bound charges in the material, to slightly separate. This induces a local electric dipole moment. In linear materials, the displacement field is linearly related to the electric field and the dipole moment,

D = g 0 E + P = g 0 E + g 0 ce E = g 0 (1 + ce ) E = gE. (8.6)

where γ is the electrical permittivity, γ0 is its value in a vacuum, P is the induced dipole, and χe is the susceptibility, which indicates the ease with which a dipole moment is induced by an electric field in the medium. (Loosely speaking, how “tightly” the electrons are bonded to their atoms). Likewise, the magnetic field is related to the magnetic induction by B = mH



(8.7)

where μ is the magnetic permeability. From these assumptions, and further assuming that there is no charge accumulation (i.e. ρe = 0), the Maxwell equations simplify to

Ñ ´ E = -m Ñ´H = g

¶H ¶t

(Faraday’s law)

¶E + se E (AmpŁre’s law) ¶t

(8.8) (8.9)



Ñ × E = 0 (Gauss’s law)



Ñ × H = 0 (Gauss’s law). (8.11)

(8.10)

The corresponding quantities are listed in Table 8.1. The solutions to these equations reveal how EM waves travel within a medium and how electric and magnetic fields interact with matter. Expressions that describe reflection and transmission at interfaces are obtained by knowing how waves move in each of two adjacent media and applying coupling relations at the interface. Using these expressions, we can obtain the relations for absorption, which can be correlated with emission via Kirchhoff’s law.

8.3 WAVE PROPAGATION IN A MEDIUM Here, we discuss EM wave propagation within an infinite, homogeneous, isotropic medium. A perfect dielectric (defined as a nonconductor or perfect insulator) is considered in Section 8.3.1, where the energy of the propagating EM waves is not attenuated. A medium having finite electrical conductivity is then analyzed in Section 8.3.2; examples of such media are the imperfect dielectrics (poor conductors), metals (good conductors), or semiconductors (materials with intermediate conductivity). In these media, EM waves are attenuated as their energy is absorbed by the medium itself. For simplicity we initially assume that the permittivity, permeability, and susceptibility are real-valued and independent of wavelength.

366

Thermal Radiation Heat Transfer

8.3.1 EM Wave Propagation in Perfect Dielectric Media Generally, EM waves propagating in a perfect dielectric medium do not lose any of their energy, as perfect dielectrics are, by definition, nonabsorbing (nonattenuating). Wave propagation through such media is similar to that in vacuum except that the speed of propagation is reduced in proportion to the index of refraction. Here, we first consider a vacuum or a dielectric material (or insulator) with large electrical resistivity such that the σeE term in Equation (8.9) can be neglected. Suppose the coordinate system is aligned with the x-coordinate pointing in the direction of wave propagation. Under this condition the electric and magnetic field strengths can only be functions of x and t, so derivatives ∂(∙)/∂y = ∂(·)/∂z = 0. Accordingly, from Gauss’s law,

¶E x = 0, ¶x

¶H x = 0 ¶x

(8.12)

so only Ey and Ez vary with respect to x. From Faraday’s law and Ampère’s law, we can further show 0=





-

¶H y ¶Ez = -m ¶x ¶t

¶E y ¶H = -m z ¶x ¶t 0=



¶H x ¶t

-

¶E x ¶t

¶E ¶H z =g y ¶x ¶t

(8.13)

(8.14) (8.15) (8.16) (8.17)

367

Electromagnetic Wave Theory



¶H y ¶E = g z . (8.18) ¶x ¶t

Hence, the electric and magnetic field components in the direction of propagation are steady and independent of the propagation direction x. This result shows that monochromatic EM waves can be envisioned as transverse waves that oscillate in mutually orthogonal directions and perpendicular to the direction of propagation, as opposed to longitudinal waves (like sound waves) which oscillate in the direction of propagation. By taking the derivative of Equations (8.14) and (8.15) with respect to x, and the derivative of Equations (8.17) and (8.18) with respect to t, we obtain the wave equations

¶2 Ey ¶2 E = mg 2 y 2 ¶x ¶t

(8.19)



¶ 2 Ez ¶2 E = mg 2 z 2 ¶x ¶t

(8.20)



mg

¶2 Hz ¶2 Hz = ¶x 2 ¶t 2

(8.21)

mg

¶2 H y ¶2 H y . (8.22) = ¶x 2 ¶t 2

and

At this point, we further simplify the derivation by assuming that the electric and magnetic fields are plane-polarized so they only fluctuate in the x–y and x–z planes, respectively, as shown in Figure 8.1. The general solution of Equation (8.19) is

æ æ t ö t ö Ey = f ç x ÷+ gç x + ÷ ç ÷ ç mg ø mg ÷ø è è

(8.23)

FIGURE 8.1  Wave propagation through a dielectric material. The electric field wave is polarized in the x-y plane, traveling in x direction, with the companion magnetic field wave polarized in the x-z plane.

368

Thermal Radiation Heat Transfer

where f and g are any differentiable functions. The f provides propagation in the positive x direction, while g accounts for propagation in the negative x direction. The present discussion is for a wave moving in the positive direction, so only the f function is present. We can obtain the speed of the propagating wave from the wave equation. Consider an observer moving with the wave, so the observer is always at a location of constant Ey on Figure 8.1. The x location of the observer must then vary with time, which means that the argument of f, as given in Equation (8.23), must be constant. The speed of the propagating wave is expressed as dx dt = 1/ mg . Then, the relation æ t ö Ey = f ç x ÷ ç mg ÷ø è



(8.24)

represents a wave with y component Ey, propagating in the positive x direction with speed c = 1/ mg. In vacuum, light propagates with the speed c0; therefore, c0 = 1/ m0 g 0 . Independent measurements of μ0, γ0, and c0 validate this result. Indeed, this basic deduction led Maxwell to conclude that light must be an EM wave. For a medium other than a vacuum, the refractive index is n = c0 c = mg / m0 g 0 . For most conventional materials, n > 1. The f function in Equation (8.24) can have any complex and arbitrary-shaped waveform. Such a function can be represented using a Fourier series as a superposition of harmonic waves, each having a different fixed wavelength. We consider only one such monochromatic wave, as more complex waveforms can be constructed from others using superposition due to the linearity of Maxwell’s equations. With this idea in mind, we write the expression for the Ey component of the electric field at the origin x = 0 as E y = E y 0 cos(w t )



(8.25)

An observer riding with this wave and leaving the origin at time t1 will arrive at location x after a time increment of x/c, where c is the wave speed. Hence, the time of arrival is t = t1 + (x/c), so that the time that the observer departed the origin becomes t1 = t – (x/c). A wave traveling in the positive x direction is then given by

é æ nx ö ù é æ x öù E y = E y 0 cos êw ç t - ÷ ú = E y 0 cos éêw t - mg x ùú = E y 0 cos êw ç t - ÷ ú . (8.26) û ë ë è c øû ë è c0 ø û

(

)

This is a solution to the governing wave Equation (8.19), as is shown by comparison with Equation (8.24). Other forms of the solution can be obtained by using relations between the angular frequency ω, the linear frequency ν, or the wavelength, λ: ω = 2πν = 2πc/λ = 2πc0/λ0, where λ and λ0 are the wavelengths in the medium and in vacuum. This result can be substituted into Equation (8.15) to find the corresponding magnetic field strength,

-m

é æ nx ö ù w n ¶H z ¶E y = = E y0 sin êw ç t - ÷ ú . (8.27) ¶t ¶x ë è c0 ø û c0

Then, noting the t dependence and integrating yields

Hz =

é æ nx ö ù n g E y 0 cos êw ç t - ÷ ú = Ey mc0 m ë è c0 ø û

(8.28)

369

Electromagnetic Wave Theory

where the integration constant is set equal to zero, assuming that there is no background magnetic field beyond the one induced by Ey. This result shows that, if the electric field is plane-polarized so that it oscillates in the x–y plane, the magnetic field oscillates in the x–z plane at the same phase as shown in Figure 8.1 and at an amplitude of g m E0 y . The coefficient g m = n mc0 is the impedance of the material. A more general solution can be written for an arbitrarily oriented coordinate system using vector notation

 = 0 Ñ 2 E - mgE

(8.29)

which is a generalization of Equation (8.19). If the field at the origin fluctuates according to

E = E 0 cos ( wt ) , (8.30)

the solution to Equation (8.29) is

E ( r , t ) = E 0 cos ( w t - q × r )

(8.31)

where r is the position vector and the wave vector, q, has a magnitude w mg = 2pn c = 2p l = 2pn l 0 and points in the direction of wave propagation. In the special case of a plane-polarized wave propagating in the x-direction, Equation (8.31) reduces to Equation (8.26). Equation (8.31) can also be written using Euler’s formula,

ei×q = cos q + i sin q, (8.32)

in which case

E = Re éë E c ùû = Re é E 0 e ( ë

i wt - q×r )

ù û

(8.33)

where Re denotes the real component. Likewise

H = Re éëH c ùû = Re éH 0 e ( ë

i wt - q×r )

ù . (8.34) û

The complex vectors Ec and Hc are general mathematical solutions to the wave equations, but the “physical” solution corresponds to the real component of these vectors. However, complex notation is useful, especially when analyzing conducting media, as discussed below. As shown by Equations (8.26) and Figure 8.1, the wave amplitude does not decrease in vacuum or in a perfect dielectric medium, which has zero electrical conductivity. In many real materials the electrical conductivity is significant; therefore, the last term on the right of Equation (8.9) cannot be neglected. This causes the propagating wave to be attenuated.

8.3.2 Wave Propagation in Isotropic Media with Finite Electrical Conductivity In this section we discuss EM wave propagation in conducting media, including imperfect insulators (imperfect dielectrics) that have a low electrical conductivity, in semiconductors, and in good conductors (metals). As a special case, let us again consider a wave propagating in the x-direction, and polarized so that E z = Hy = 0. Following similar steps as above, we obtain

¶2Ey ¶E ¶2Ey = mg + mse y . (8.35) 2 2 ¶x ¶t ¶t

370

Thermal Radiation Heat Transfer

In this case it is convenient to assume a complex sinusoidal wave having the general form E y = E y 0 Re éë Ec ùû = E y 0 Re ée ( ë

i wt - qx )



ù û

(8.36)

Taking the derivatives with respect to x and t and substituting them into Equation (8.35) results in q 2 + mseiw - mgw2 = 0



(8.37)

which shows that, for a medium having finite conductivity, q must be complex. If we substitute q = a + ib into Equation (8.37) we get a 2 - b2 - mgw2 = 0



(8.38)

and b=



msew . (8.39) 2a

Solving Equations (8.38) and (8.39) simultaneously gives a = ωn/c0 and b = ωk/c0 where n2 =



2ù é æs ö mgc02 ê 1 ± 1 +ç e ÷ ú 2 ê è gw ø ú û ë

(8.40)

and k2 =



2ù é æs ö mgc02 ê -1 ± 1 +ç e ÷ ú . (8.41) 2 ê è gw ø ú û ë

The positive sign is chosen for most cases in Equations (8.40) and (8.41), although the negative sign is required for materials having negative electrical permittivity. The terms n and k are the real and imaginary components of the complex refractive index, n = n - ik. (8.42)



The imaginary part k is called the absorption index and is related to the absorption coefficient κ of the RTE. Substituting the result for q into Equation (8.36) gives

E y = E y 0 Re ée ( ë

i wt - ax + ibx )

ù = E y 0 Re éei( wt -ax ) ù e - bx = E y 0 cos éêw æ t - nx ö ùú e ç ÷ û ë û ë è c0 ø û

-wkx c0

. (8.43)

This result shows that the electric field intensity of an EM wave propagating through a conducting medium will diminish exponentially with the distance traveled, as shown in Figure 8.2. The magnetic field can then be found through Faraday’s law, Equation (8.15). After some manipulation, one arrives at

Hz =

é æ nx öù n2 + k 2 E y 0 cos êw ç t + f ÷ú e mc0 øû ë è c0

- wkx c0



(8.44)

371

Electromagnetic Wave Theory

FIGURE 8.2  Instantaneous electric and magnetic field in a weakly absorbing medium. In this case the phase shift between the electric and magnetic waves is small.

where ϕ =tan−1(k/n). This result shows that the magnetic field also diminishes exponentially with distance, but there is a phase shift between the electric and magnetic fields caused by the finite conductivity of the medium. In the case of perfect dielectrics, σe = 0. In this scenario, Equations (8.40) and (8.41) simplify to n = mg c0 = c0 c and k = 0. Substituting these results into Equations (8.43) and (8.44) gives Equations (8.26) and (8.28), respectively. The above results apply to a plane-polarized wave propagating in the x-direction. More generally the electric and magnetic field intensities are given by Equations (8.33) and (8.34) where the wave vector is complex q = q I - iq II . (8.45)



Vector qI is perpendicular to surfaces of constant phase and points in the direction of wave propagation, while qII is perpendicular to surfaces of constant amplitude and points in the direction of wave attenuation. In order for the wave defined by the wave vector to satisfy Maxwell’s equations, it can be shown that (e.g., Bohren and Huffman, 1983)

(

q × q = w2 gm = w2 n 2 c02 = w2 n2 - k 2 - 2ink

)

c02 . (8.46)

In the special case of a homogeneous wave, qI and qII are collinear, and the wave vector can be written

q = q × eˆ = wn c0 × eˆ = ( qI - iqII ) × eˆ

(8.47)

where qI = ωn/c0, qII = ωk/c0 and eˆ is a unit vector pointing in the direction of wave propagation. If qI and qII are not parallel, the wave is heterogeneous, and the respective magnitudes of these vectors depend on n and k in a more complicated way.

8.3.3 Energy of an EM Wave The energy carried by an EM wave per unit time and per unit area is expressed by the cross product of the electric and magnetic field vectors, which are orthogonal to each other:

S = E ´ H

(8.48)

372

Thermal Radiation Heat Transfer

where S is the Poynting vector, which propagates in the direction of the wave vector and perpendicular to the oscillating E and H fields. In the case of complex notation

S = Re ( E c ) ´ Re ( H c )

(8.49)

The direction of energy propagation can be determined starting from the E and H vectors and using the right-hand rule. As an example, for the plane wave shown in Figure 8.1, the electric field fluctuates in the y direction; therefore, the corresponding magnetic field fluctuates in the z direction. Consequently, the EM wave propagates in the positive x direction, and its magnitude is

S = S = E y H z . (8.50)

When Hz is substituted into Equation (8.50), the magnitude of Poynting vector is

S=

n 2 E y . (8.51) mc0

The energy transmitted by the wave, per unit time and area, is proportional to the square of the amplitude of the electric field. The |S| is a spectral quantity that is proportional to spectral radiation intensity, Iλ, introduced in Chapter 1. In the case of conducting media, both the electric and magnetic field intensities, Equations (8.43) and Equation (8.44), decay exponentially in the direction of wave propagation, according to exp(−2ωkx/c) or exp(−4πkx/λ), as shown in Figure 8.2. As a result of absorption, the spectral intensity Iλ decays as exp(−κλx), so the relation between the absorption coefficient κλ and the absorption index k is

kl =

4pk . (8.52) l

This equation provides a procedure to determine the spectral absorption coefficient κλ from optical data for the extinction index k or the complex index of refraction as a function of wavelength.

8.3.4 Polarization In Sections 8.3.1 and 8.3.2 the electric and magnetic field fluctuations were assumed to be contained in the x–y and x–z planes, respectively, to simplify the treatment of Maxwell’s equations. In many cases the electric and magnetic fields fluctuate in a more complex manner as described by the polarization state. (By convention the polarization state describes the electric field fluctuations, which are related to magnetic field fluctuations by Maxwell’s equations). While the polarization state of a wave does not affect the propagation of energy through a homogeneous medium, it plays a strong role in determining how an incident wave is reflected or transmitted at an interface, or how a wave is scattered by a particle (cf. Chapter 10). A brief overview of polarization is provided here; for a more detailed description the reader is referred to the excellent discussion by Bohren and Huffman (1983). For simplicity, consider a wave propagating through a perfect dielectric in the x direction, but with electric field fluctuations no longer confined to the x–y plane. The fluctuation of the electric field with time can be described by

E = Re ( E c ) = A cos ( kx - wt ) - B sin ( kx - wt )

(8.53)

where A and B are real-valued vectors. Furthermore, consider a plane perpendicular to the direction of propagation, e.g., the y–z plane at x = 0 as shown in Figure 8.3. Over time the tip of E will trace out an ellipse, which is called the vibration ellipse. If either A = 0 or B = 0, the electric field

373

Electromagnetic Wave Theory

FIGURE 8.3  Trajectory of the electric field vector and vibration ellipse of an: (a) Elliptically polarized wave; and (b) a plane-polarized wave. In all cases the wave is propagating through a dielectric medium.

fluctuations are contained within a single plane; in this case the wave is plane-polarized (also called linearly polarized) and the vibration ellipse is a straight line oriented at some angle with respect to the y and z axes. The wave described by Equation (8.26) and plotted in Figure 8.1 is a special case where B = 0 and A = [0, Ey,0]T. More generally, the vibration ellipse is elliptical, and the electric field vector will follow a helical trajectory as the wave propagates. The vibrational ellipse is fully characterized by the direction of rotation (handedness), the ratio of major to minor axes (ellipticity) and the angle-of-tilt (azimuth). While these three attributes provide a clear geometric interpretation of the polarization state of the wave, they are difficult to measure experimentally and to manipulate as the wave undergoes changes (e.g., if two waves are superimposed, or if a wave is reflected by a surface). The Stokes parameters provide an alternative means to represent polarization state. In this approach, the electric field is parameterized as the superposition of two orthogonal components

ikx -iwt ) ù E = Re é E 0 e( , E 0 = Eeˆ  + E ^ eˆ ^ ë û

(8.54)

where eˆ  and eˆ ^ are orthogonal unit basis vectors pointing in the “parallel” and “perpendicular” directions, e.g., relative to the plane-of-incidence for a wave incident upon a surface, and both are orthogonal to the wave propagation direction. The coefficients E∥ and E⊥ are complex vectors,

E = ae -i d

E ^ = a^ e - i d ^

that oscillate with a phase difference δ = δ∥ − δ⊥. The Stokes parameters are then given by

(8.55)

374

Thermal Radiation Heat Transfer



I = a2 + a^2

(8.56)



Q = a2 - a^2

(8.57)



U = 2a a^ cos d

(8.58)



V = 2a a^ sin d. (8.59)

These parameters can be combined into a single Stokes vector, s = [I, Q, U, V]T. Since the four Stokes parameters depend on three variables, a⊥,, a∥, and δ, the Stokes parameters are not independent; it can be shown that I2 = Q2 + U2+V2. Also, after some manipulation, the Stokes parameters can be related to the handedness, ellipticity, and azimuth attributes described above. The main advantages offered by the Stokes parameters are that: (1) They can be derived from experimentally measurable attributes, e.g., by passing the wave through a sequence of polarization filters and measuring the intensity of the transmitted EM wave; (2) if several independent waves propagating in the same direction are superimposed, the Stokes parameters of the resulting wave are the sums of the Stokes parameters of the constituent waves; and (3) the change in polarization as a wave undergoes various transformations (e.g., reflection, scattering) can be expressed by a linear transformation. For example, if a wave incident on a particle or surface is given by Equation (8.54), the scattered wave can be related to the incident wave by the scattering amplitude matrix

é E , s ù é S2 ê ú=ê ë E ^,s û ë S1

S3 ù é E,i ù ú úê S4 û ë E ^,i û

(8.60)

where subscripts s and i denote the scattered and incident wave. The complex elements of the matrix depend on the scattering angle, as well as the nature of the scatterer. By expressing the polarization states using the Stokes parameters, the wave scattered in a prescribed direction can be computed from the Mueller matrix



é I s ù é S11 ê ú ê Qs S21 ss = ê ú = ê êU s ú ê S31 ê ú ê ë Vs û ë S41

S12 S22 S32 S42

S13 S23 S33 S43

S14 ù é I i ù úê ú S24 ú êQi ú = si S34 ú êUi ú úê ú S44 û ë Vi û

(8.61)

where S11, S12, etc. can be found from the elements of the amplitude scattering matrix. The Mueller matrix for an ensemble of scatterers (e.g., an aerosol or colloid suspension) is found by summing the Mueller matrices of the scatterers due to the superposition property described above. The above discussion considers an ideal monochromatic wave where E∥ and E⊥ are constant with respect to time. In many scenarios these coefficients will vary as a function of time, albeit slowly relative to 2π/ω. In this case Equation (8.54) becomes

E = E 0 (t ) e(

ikx -iwt )

, E 0 (t ) = E (t ) eˆ  + E ^ (t ) eˆ ^ . (8.62)

If E∥(t) and E⊥(t) are uncorrelated in time, which is the case for thermal emission, the EM wave is unpolarized. In this scenario the shape of the vibration ellipse evolves over time in a random way, and there is no preferential shape.

375

Electromagnetic Wave Theory

8.4 LAWS OF REFLECTION AND REFRACTION Next, we focus on how EM waves behave at the interface of dissimilar media. This allows us to derive the laws of reflection and refraction in terms of the real and imaginary parts of the complex index of refraction, which are related to the electric and magnetic properties of the two media. Here, our focus is on a single ideal planar interface, which is important for many engineering applications. These ideas can be extended to absorption and scattering of EM waves by finite-size particles and agglomerates, which are the subject of Chapter 10. We must first consider boundary conditions at the interface between medium 1 and medium 2. Suppose the incident ray originates in medium 1 at an angle θi relative to the normal vector of ˆ pointing into medium 1. A portion of the wave energy will be reflected back into the interface, n, medium 1 at an angle θr, while a portion will be transmitted into medium 2 at an angle χ. The electric fields of these components can be written using Equation (8.31) as

i× w t - q ×r E i ( r, t ) = Re é E 0,i e ( i i ) ù ë û

(8.63)



i× w t - q ×r E r ( r, t ) = Re é E 0,r e ( r r ) ù û ë

(8.64)

i× w t - q ×r E t ( r, t ) = Re é E 0,t e ( t t ) ù ë û

(8.65)

and

The electric field in medium 1 is the superposition of incident and reflected components, E1 = Ei(r,t) + Er(r,t), while the electric field in medium 2 is the transmitted component, E2 = Et(r,t). In order to conserve energy across the interface

ò S × nˆ dA =ò S × nˆ dA 1

A

2

(8.66)

A

where S1 and S2 are the Poynting vectors of the EM waves in these media. Equation (8.66) is satisfied by

éë E i ( r, t ) + E r ( r, t ) ùû ´ nˆ = E t ( r, t ) ´ nˆ

(8.67)

éëH i ( r,t ) + H r ( r,t ) ùû ´ nˆ = H t ( r, t ) ´ nˆ

(8.68)

and

which means that the tangential wave components are conserved across the interface. Because this boundary condition must be satisfied at all times, the waves must have the same time dependence, so ωi = ωr = ωt. Consider a coordinate system at the interface oriented so that z is aligned with nˆ and x points in the incidence plane defined by projecting qi onto the boundary, as shown in Figure 8.4; thus y is perpendicular to the incident wave direction and qi = [qisin θi, 0, qicos θi]T. Set the position vector to point at any point on the interface, p = [x, y, 0]T. In order to satisfy the above boundary conditions, and since ωi = ωr = ωt, the phase shift of the waves must also be equal so

q i × p = q r × p = q t × p. (8.69)

376

Thermal Radiation Heat Transfer

FIGURE 8.4  Incident and reflected waves at the boundary between two materials. In this scenario the wave is polarized so that the electric field fluctuates parallel to the plane of incidence.

Since qi,y is zero, this condition is only satisfied if the y-components of the reflected and transmitted wave vectors are also zero, so the transmitted and reflected wave vectors must also lie in the incidence plane. Examining the x-components of the wave vectors along the boundary with y = 0 gives

qi, x = qi sin qi = qr , x = qr sin qr = qt , x = qt sin c

(8.70)

Since the incident and reflected waves propagate in the same medium, it follows that qi = qr so θi = θr. Thus, for a smooth surface, the incident and reflected angles are the same and EM waves are always reflected in a specular (mirror-like) manner. Furthermore, Equation (8.70) shows that

sin c qi n1 = = sin qi qt n2

(8.71)

which is Snell’s law. If the refractive indices are complex, the angles θi and χ are also complex, and do not have a simple geometric interpretation. This scenario is discussed in Section 8.4.2. While these relationships indicate the direction of the transmitted and reflected waves in terms of the incident wave, it is also possible to calculate their amplitudes and thus the transmitted and reflected energy. The cases for waves crossing the interface between two perfect dielectrics, and the interface between a dielectric and a conductor, are considered below.

8.4.1 Reflection and Refraction at the Interface between Perfect Dielectrics First, consider an EM wave incident upon the interface between two dielectrics. In general, the polarization state of the incident wave is described by a Stokes vector, but any wave can be decomposed into two orthogonal plane-polarized waves, according to Equation (8.54), where the parallel and perpendicular components are defined relative to the incidence plane. We start with the wave component having an electric field oscillating parallel to the incidence (x–z) plane (“parallel-polarized”), as shown in Figure 8.5(a). As described above, the tangential components of the electric and magnetic fields must be conserved. By taking the appropriate projections with respect to the incident and transmitted angles

E0, i  cos qi - E0 ,r  cos qi = E0, t  cos c

(8.72)

377

Electromagnetic Wave Theory

FIGURE 8.5  Incident, reflected, and transmitted waves between two perfect dielectrics for the cases where the waves are polarized so that the electric fields oscillate (a) parallel to and (b) perpendicular to the plane of incidence.

and

H 0, i  + H 0 ,r  = H 0, t  . (8.73)

The amplitudes of the electric and magnetic fields are related by the impedance, so Equation (8.73) can be rewritten as

E0 , i  + E0 , r  E0 , t  = . (8.74) m1c0 n1 m2 c0 n2

The magnetic permeability of most materials is close to μ0 so

n1 ( E0 ,i  + E0 ,r  ) = n2 E0 ,t  . (8.75)

Solving Equations (8.72) and (8.75) simultaneously results in

E0 , r  cos c cos qi - n1 n2 = r = E0 , i  cos c cos qi + n1 n2

(8.76)

where r∥ is the reflection coefficient for the electric field amplitude. If we repeat the analysis for an incident EM wave polarized perpendicular to the incident plane, as shown in Figure 8.5 (b), (see Homework Problem 8.5)

E0 , r ^ n n - cos c cos qi = r^ = 1 2 . (8.77) E0 , i ^ n1 n2 + cos c cos qi

Substituting Snell’s law, Equation (8.71), into Equations (8.76) and (8.77) to eliminate the n1/n2 terms gives

tan ( qi - c ) E0 , r  = r = E0 , i  tan ( qi + c )

(8.78)

378

Thermal Radiation Heat Transfer

and sin ( qi - c ) E0 , r ^ . (8.79) = r^ = sin ( qi + c ) E0 , i ^



The energy carried by a wave is proportional to the square of the wave amplitude, per Equation (8.51). Then, taking the square of the reflection coefficients results in the ratio of reflected energy to incident energy as a function of incident direction. Thus, the values of ρλ,s(θi, ϕi) for incident parallel and perpendicular polarized components are obtained by æ E0 , r  rl,s, ( qi , fi ) = r2 = ç ç E0 , i  è



2

ö ÷÷ ø

(8.80)

2

æE ö rl,s,^ ( qi , fi ) = r = ç 0 ,r ^ ÷ . (8.81) è E0 , i ^ ø 2 ^



Because all reflectivities predicted by EM theory are specular, the subscript s is not carried from this point on to simplify the notation. There is also no dependence on the azimuthal angle, ϕ, which is also omitted to simplify the notation. The wavelength notation is also dropped, but it is important to remember that the reflectance is a function of wavelength through the refractive index, as discussed in Section 8.5. As explained above, for unpolarized incident radiation, the time-averaged perpendicular and parallel components are equal and the reflectivity is simply the average of the two components r ( qi ) =

r^ ( qi ) + r ( qi ) 2



=

2 2 1 é tan ( qi - c ) sin ( qi - c ) ù + ê ú 2 êë tan 2 ( qi + c ) sin 2 ( qi + c ) úû

(8.82)

2 1 sin ( qi - c ) é cos ( qi - c ) ù = ú. ê1 + 2 2 2 sin ( qi + c ) êë cos ( qi + c ) úû 2

Equation (8.82) is the Fresnel equation and gives the reflectivity for an unpolarized ray incident upon a surface for two perfect (nonattenuating) dielectric media. The relationship between χ and θi is given by Equation (8.71), which is later used to eliminate χ to obtain an expression in terms of θi only. When the incident radiation is normal to the interface, cos θi = cos χ = 1 and Equations (8.80) and (8.81) yield

E0 , r  E 1 - n1 n2 n2 - n1 = r = 0 ,r ^ = r^ = = . (8.83) E0 , i  E0 , i ^ 1 + n1 n2 n2 + n1

The normal reflectivity is then 2



2

æ 1 - n1 n2 ö æ n2 - n1 ö r n = r ( qi = 0 ) = ç ÷ =ç ÷ . (8.84) è 1 + n1 n2 ø è n2 + n1 ø

379

Electromagnetic Wave Theory

8.4.2 Reflection and Refraction at the Interface of a Perfect Dielectric and a Conducting Medium Next consider the important case where a wave propagating in a perfect dielectric (k1 = 0) is incident on a conductor (k2 > 0). The incident and reflected wave vectors are real-valued and given by Equations (8.63) and (8.64), while the transmitted wave vector is complex and given by

E t ( r, t ) = Re é E 0,t e ëê

i éëwt t -( q t,I -iq t,II )×r ùû

ù ûú

(8.85)

with a similar equation for Ht(r, t). Using the same coordinate system as before, it can be shown that energy across the interface is conserved if the tangential components of the incident, reflected, and transmitted wave vectors are equal, but because qt is complex the real and imaginary components must be addressed separately. Considering the real components of qi, qr, and qt, and remembering that qi = qr = ωn1/c0

qi sin qi = qt,I sin qt

(8.86)

which means that θi = θr, while only the transmitted wave vector is complex,

0 = q t,II × p. (8.87)

Since the terms in Equation (8.86) are not zero, it follows that qt,I and qt,II are not collinear, so the transmitted wave is heterogeneous as shown in Figure 8.6. Recalling the definitions of the real and imaginary wave vector components, qt,I is perpendicular to planes of constant phase, and the wave

FIGURE 8.6  The transmitted component of an EM wave incident on the interface between a perfect dielectric and a conductor is a heterogeneous wave. Snell’s law, Equation (8.71), still applies, but the angle χ is complex and does not have a convenient geometric interpretation.

380

Thermal Radiation Heat Transfer

propagates at a transmitted angle θt relative to the normal vector. The imaginary component, qt,II, is perpendicular to planes of constant amplitude and must be perpendicular to the surface in order to satisfy Equation (8.87). The real and imaginary components of the transmitted wave vector and the direction of wave propagation can be calculated from Equation (8.86) and

cos qt =

q t,I × q t,II qt,I × qt,II

(8.88)

which follows from Equation (8.87) and Figure 8.6, along with Equation (8.46)

(

q t × q t = qt,I qt,I - qt,II qt,II - 2iq t,I × q t,II = w2 n22 - k22 - 2in2 k2

)

c02 . (8.89)

Combining these equations and considering the real and imaginary components separately results in a system of three equations that can be solved simultaneously for the three unknowns: qt,I, qt,II, and θt (See Homework Problem 8.4). In the case of normal incidence, θi = θt = 0 and the transmitted wave is homogenous, so qt,I = ωn2/c0 and qt,II = ωk2/c0. Otherwise, the transmitted wave is heterogeneous, and the propagation speed and wave attenuation of the transmitted wave are nonlinear functions of θt, n2, and k2. The fractions of incident wave energy that are transmitted and reflected can be obtained using the same equations presented in Section 8.4.2, but replacing the real-valued refractive index n2 with the complex refractive index n2 = n2 – ik2 . This results in

E0 , r  cos qi cos c - n1 n2 = r = E0 , i  cos qi cos c + n1 n2

(8.90)



E0 , r ^ cos c cos qi - n1 n2 = r^ = E0 , i ^ cos c cos qi + n1 n2

(8.91)

where the transmitted angle is found from Snell’s law

sin c =

n1 sin qi . (8.92) n2

Because n2 is complex, χ, r∥, and r⊥ are also complex. For the special case of normal incidence,

E0 , r  E n -n n - ik2 - n1 = r = 0 ,r ^ = r^ = 2 1 = 2 . (8.93) E0 , i  E0 , i ^ n2 + n1 n2 - ik2 + n1

Recall, however, that the wave energy depends on |E|2, and for a complex quantity z, |z2| = zz* where z* is the complex conjugate. Then, because the relationships for r∥ and r⊥ are the same at normal incidence 2 æ n - ik2 - n1 ö æ n2 + ik2 - n1 ö ( n2 - n1 ) + k2 rn = ç 2 . (8.94) = ÷ ç ÷ 2 è n2 - ik2 + n1 ø è n2 + ik2 + n1 ø ( n2 + n1 ) + k22 2



The same procedure can be followed for oblique incidence, but this requires considerable manipulation because cos χ is a complex number. In the case of an EM wave incident from a vacuum or air (n1 = 1, k1 = 0, n2 = n = n – ik ). Hering and Smith (1968) show that

381

Electromagnetic Wave Theory

( ng - a cos qi ) + ( n2 + k 2 ) a - n2 g 2 r ( qi ) = 2 ( ng + a cos qi ) + ( n2 + k 2 ) a - n2 g 2

(8.95)

( nb - cos qi ) + ( n2 + k 2 ) a - n2b2 r ^ ( qi ) = 2 ( nb + cos qi ) + ( n2 + k 2 ) a - n2b2

(8.96)

2

and

2

where

2





æ sin 2 qi ö 4n2 a2 = ç1 + 2 ÷ 2 2 2 è n +k ø n +k b2 =

n2 + k 2 2n2

æ sin 2 qi ö ç 2 2 ÷ èn +k ø

æ n2 - k 2 sin 2 qi ö - 2 + a ÷ ç 2 2 2 n +k èn +k ø

(8.97)

(8.98)

12



g=

ö 2nk æ n2 + k 2 n2 - k 2 b + a - b2 ÷ . (8.99) 2 2 2 2 ç 2 n +k n +k è n ø

Harpole (1980) considers the more complex scenario of oblique incidence from an adjacent absorbing material rather than vacuum or a material with n1 = 1, such as air. An important case concerns an EM wave incident on a metal surface. For metals, σe >> μω ≈ μ0ω for the wavelengths important to thermal radiation, and it follows from Equations (8.40) and (8.41) that n >> 1 and k >> 1 and therefore n2 + k2 >> sin2θi. In this case α = β = γ = 1 and Equations (8.95) and (8.96) simplify to

r ( qi ) =

( n cos qi - 1) + ( k cos qi ) 2 2 ( n cos qi + 1) + ( k cos qi ) 2

2

(8.100)

and

( n - cos qi ) + k 2 . (8.101) r ^ ( qi ) = 2 ( n + cos qi ) + k 2 2

At normal incidence

( n - 1) + k 2 2 ( n + 1) + k 2 2



rn =

(8.102)

which is the same result as Equation (8.94) with n1 = 1.

8.5 CLASSICAL MODELS FOR OPTICAL CONSTANTS EM wave propagation through a medium, and how the wave behaves at the interface between dissimilar media (e.g., reflection, transmission, emission, scattering) depends on the electromagnetic properties of the media. These properties are assembled into the complex refractive index,

382

Thermal Radiation Heat Transfer

n = n - ik , through Equations (8.40) and (8.41). It is sometimes more convenient to work with the complex dielectric function, ε = εI - iεII , also called the reduced permittivity, ε = εI - iεII =



s ε - i e = n 2 . (8.103) ε0 2wε0

The refractive index and complex dielectric function are related to each other according to εI = n2 - k 2 , εII = 2nk



(8.104)

and

n2 =

(

)

(

)

1 1 εI + εI2 + εII2 , k 2 = -εI + εI2 + εII2 . (8.105) 2 2

When describing wave propagation, it is more convenient to use n , which relates directly to wave speed and attenuation, while ε is more closely linked with how waves interact with matter. While the wavelength dependence of these parameters is often omitted in the nomenclature, it is important to remember that they do, in fact, vary with wavelength. The distinct, wavelength-dependent behavior of metallic and nonmetallic surfaces can be explained, in large part, by the fundamental difference in how these materials interact with EM waves as summarized by n or ε. Electromagnetic waves are perturbations of the electronic and magnetic field, so it follows that the manner through which an EM wave interacts with a material depends on the polarizability of the material. Attenuating matter can be broadly categorized according to whether the matter contains bound charges or free charges (free electrons). Figure 8.7 shows a schematic of the electron band structure for non-conductors and conductors, which indicates the range of energy states that electrons may have within the material. These states are grouped into “bands” due to the periodic nature of a crystal lattice. Each band is composed of closely spaced orbitals, and empty regions denote “forbidden” energy states. The innermost orbitals are occupied by valence electrons, which are bound to particular atoms, while electrons that occupy the outermost, incompletely filled orbits are conduction electrons, which are non-localized. In a conductor the conduction and valence bands overlap, so it is possible for an EM wave (or photon) to energize a valence electron into

FIGURE 8.7  Electron energy diagram for non-conductors and conductors. Adapted from Bohren and Huffman (1983).

383

Electromagnetic Wave Theory

the conduction band, which allows the electron to travel through the medium. In the case of nonconductors, the valence and conduction bands are separated by a bandgap. For semiconductors, the bandgap is small, so photons/EM waves having an energy greater than this threshold can promote a valence band electron into the conduction band, while in insulators the bandgap is large and the electrons remain bound to atoms. A number of classical models have been developed to connect Maxwell’s equations to the polarizability of matter. These models predate quantum mechanics, and do not really represent what is going on at the atomic scale. Nevertheless, they are quite robust and provide a useful conceptual foundation for how EM waves interact with matter.

8.5.1 Lorentz Model (Non-conductors) In the case of a non-conductor, the EM wave interacts with bound charged particles (electrons or ions) within the matter. In the Lorentz model, developed at the end of the 19th and the beginning of the 20th century by Hendrik Lorentz, the bound charges have a prescribed mass and are attached to a lattice by a spring, as shown in Figure 8.8. In the presence of an oscillating EM field, the charge will oscillate due to the Coulomb forces imposed by the field. There is also a damper (dashpot) that accounts for the transfer of energy between the moving charge and the lattice, through which the wave energy is converted into internal energy of matter. The equation of motion for the charge is

m x + bx + Kx = eE local

(8.106)

where m is the mass of the charge (e.g., the rest mass of an electron), b is the damping coefficient, K is the spring constant, and e is the charge. The product eElocal, which fluctuates with time, is the Columbic force that causes the simple harmonic motion of the charge. (The distinction between Elocal, the local EM field experienced by an individual oscillator, and E, the average macroscopic field experienced by an ensemble of oscillators, is not considered here.) For simplicity, consider a plane-polarized monochromatic wave so that the electric field strength fluctuates in the y direction. The solution to Equation (8.106) is the superposition of a transient harmonic term due to damping, which diminishes with increasing time, and a steady-state harmonic term. After sufficient time, only the steady-state term needs to be considered, which is given by

y (t ) =

e m Ey ( t ) w - w2 - izw 2 0



(8.107)

FIGURE 8.8  Schematic of the Lorentz oscillator model. The dashpot attenuates the amplitude of the oscillator and causes a phase shift with respect to the driving EM field.

384

Thermal Radiation Heat Transfer

where ω0 = K/m and ζ = b/m. The damping term causes a phase shift between the driving force field and charge motion, and the factor between y(t) and Ey(t) is complex. The dipole moment of an oscillator, cf. Equation (8.6), is ey, where y is the displacement, and if there are N oscillators per unit volume

P ( t ) = N ey ( t ) =

n e2 m E y ( t )

=

w20 - w2 - izw

w2p g 0 Ey ( t ) w20 - w2 - izw

(8.108)

The plasma frequency, ωp = [Ne2/(mγ0)]1/2, is proportional to the square root of the number density of oscillators, and is the frequency of motion that would occur were the charges initially displaced and then allowed to reach equilibrium without a driving force field. Comparing Equation (8.108) with Equation (8.6) shows that the coefficient of γ0Ey(t) is the susceptibility, χe, which, in turn, gives the complex dielectric function

ε = 1 + ce = 1 +

w2p . (8.109) w20 - w2 - izw

This can be separated into real and imaginary components

εI = 1 + Re ( ce ) = 1 +

(

w2p w20 - w2

(w

2 0

- w2

)

2

)

+ z 2 w2



(8.110)

and

εII = Im ( ce ) = 1 +

(w

2 0

w2p zw - w2

)

2

+ z 2 w2

. (8.111)

Representative plots of the dielectric function and refractive index in Figure 8.9 show that n is constant with respect to ω, and k is approximately zero at very low and very high frequencies, but not in the vicinity of the resonant frequency, ω0. The spectral variation of n causes the waves to “disperse” or spatially separate according to their wavelengths (e.g., as occurs when light is shone through a prism); hence, theories that describe this behavior are dispersion theories. At low frequencies, Lorentz theory predicts n = 1 + Ne2/(2mγ0ω02) and k = 0. In this range the speed of light depends on the number density of oscillators, their mass, and the spring constant (i.e. how “tightly” they are localized.) This behavior is called “normal dispersion.” At frequencies slightly higher than ω0, n  0. This is called “anomalous dispersion,” and coincides with the absorption of light by the material. Most materials exhibit anomalous dispersion in the infrared and ultraviolet spectra, and those that absorb in the visible spectrum appear tinted. It is important to recognize that n  5 atm and T > 2500 K.) Figures 9.17 and 9.18 show the emittance of water vapor and CO2 from the Alberti et al. correlations. Observe that the emittance for water vapor increases with the pressure-path length product as expected. The trend with temperature is

428

Thermal Radiation Heat Transfer

FIGURE 9.17  Computed emittance of water vapor from line-by-line data. Courtesy Dr. Michael Alberti (2019).

that emittance generally decreases with increasing temperature for water vapor. In contrast, CO2 tends to go through a peak in emittance at about 1200 K (Figure 9.18). The correlations are for properties of the absorbing gases at essentially zero pressure mixed with air at a total pressure of one bar. If the total pressure differs considerably from one bar, then a pressure correction must be applied to the predicted one bar emittance of the individual gases because of increased pressure broadening of the individual lines that make up the bands that are summed to obtain the total emittance. Hottel (1954) and Alberti et al. (2018) present graphs for this correction, and Leckner (1972) has provided algebraic expressions. The individual emittances for H2O and CO2 in air must be modified when both gases are present in a mixture, which is commonly the case. This is because the individual spectral lines and absorption bands for the two gases overlap in some spectral regions, and simple addition of the individual emittances will overpredict the emittance of the mixture. In some cases, a simple addition predicts a gas absorptance and emittance that are greater than unity at certain wavelengths. This can be seen as follows: If two absorbing gases have spectral absorption coefficients κλ,1 and κλ,2, then from Equation (9.13), e=

1 sTg4

=

1 sTg4

¥

ò éë1 - e

-( kl ,1 + kl ,2 )

l =0

(9.63)

¥

ò éë1 - e

l =0

ù Elb (Tg )d l û

-kl ,1

(

)(

)

+ 1 - e -kl ,2 - 1 - e -kl ,1 1 - e -kl ,2 ù Elb (Tg )d l. û

429

Properties of Participating Media

FIGURE 9.18  Computed emittance of CO2 from line-by-line data. Courtesy Dr. Michael Alberti (2019).

The first four terms integrate to give the total emittances of the two individual gases, resulting in

1 e = e1 + e2 sTg4

¥

ò (1 - e )(1 - e ) E -kl ,1

-kl ,2

lb

(Tg )d l = e1 + e2 - De. (9.64)

l =0

The final term on the right is the band overlap correction. Hottel (1954) and Alberti et al. (2018) present graphs of the band overlap correction (Appendix F at www.ThermalRadiation.net). The final emittance equation including the pressure corrections and overlap correction Δε is

e( pLe ) = CH2O eH2O ( pH2O Le ) + CCO2 eCO2 ( pCO2 Le ) - De. (9.65)

The complete graphs and spread sheet provided by Alberti et al. (2018) use the pressure correction and overlap correction relations proposed by Hottel along with correlations to the emittance from line-by-line integration. Correlations of the emissivities shown for H2O (Figure 9.17), CO2 (Figure 9.18), CO, and their mixtures modified by the pressure and overlap corrections are within +/− 1% of the line-by-line calculations. Calculation of H2O-CO2-CO-N2 mixture emissivities is available in an Excel spreadsheet at​ https​://do​i.org​/10.1​016/j​.jqsr​t.201​8.08.​008. Alberti et al. (2020) have examined earlier methods for predicting the absorptance of uniform CO, CO2, and H2O mixtures and present a more accurate method based on correlations of line-byline data from the HITRAN 2010 database.

430

Thermal Radiation Heat Transfer

Example 9.5 A container with effective radiation thickness of Le = 2.4 m contains a mixture of 15 volume % of CO2, 20% H2O vapor, and the remainder air. The total pressure of the gas mixture is 1 atm, and the gas temperature is 1200 K. What is the emittance of the gas? Using either the Alberti graphs at www.ThermalRadiation.net or the spreadsheet at https​://do​i. org​/10.1​016/j​.jqsr​t.201​8.08.​008 gives ε = 0.401.

9.6 ABSORPTION COEFFICIENT NEGLECTING INDUCED EMISSION Equation (1.46) gives the attenuation of intensity passing through an absorbing, nonemitting, nonscattering medium as would be observed by detectors of the incident and emerging radiation. Such information could be used in determining κλ. Actually, as radiative energy passes through a translucent medium, not only is it absorbed, but there is an additional phenomenon where the radiation field stimulates some of the atoms or molecules to emit energy. This is not ordinary or spontaneous emission caused by the temperature of the medium, as discussed in Section 1.9.4. The spontaneous emission is the result of an excited energy state of the medium being unstable and decaying spontaneously to a lower energy state. Emission resulting from the presence of the radiation field is termed stimulated or induced emission and acts like negative absorption. For induced emission, the radiation field encounters a particle, such as an atom or molecule, that is in an excited energy state. There is a probability that the incident electromagnetic wave will trigger a return of the particle to a lower energy state. If this occurs, the particle emits radiation at the same frequency and in the same direction as for the incident EM wave. Thus, the incident radiation is not absorbed but is joined by a radiation of identical energy and wavelength. Induced emission constitutes a portion of the intensity that is emerging from a translucent volume. Consequently, the energy actually absorbed by the medium is greater than the difference between the entering and leaving intensities. This is because the observed emerging intensity is the result of the actual absorption modified by induced emission added along the path. The actual absorbed energy depends on the absorption coefficient neglecting induced emission kl+ (T , P ) , which is larger than the absorption coefficient κλ(Τ, P) calculated from observed attenuation data. Statistical-mechanical considerations give the relation between κλ(Τ, P) and kl+ (T , P ) for a gas with refractive index n = 1 as

é é æ hc ö ù æ C öù kl (T , P ) = ê1 - exp ç - 0 ÷ ú kl+ (T , P ) = ê1 - exp ç - 2 ÷ ú kl+ (T , P ). (9.66) è k lT ø û è lT ø û ë ë

Because of the negative exponential term, kl+ is always larger than κλ (hence the use of the superscript +). Because induced emission depends on the incident radiation field, it is usually combined, as is done here, with the true absorption, thereby yielding the absorption coefficient κλ, and it is κλ that is used in the radiative energy calculations. The emission term in the equation of radiative transfer is then only spontaneous emission and consequently depends only on the local conditions of the medium; it is then not necessary to include directional effects in the emission term. The exponential term in Equation (9.66) is small unless λT is large. Thus, κλ and kl+ are nearly equal except at large λT (long wavelengths and/or high temperatures). The values are within 1% for λΤ less than 3120 μm ⋅ Κ, and within 5% for λT less than 4800 μm⋅Κ. For calculations of radiative transfer in absorbing–emitting media, the properties in the literature give κλ with few exceptions; hence, the absorption coefficient neglecting induced emission kl+ does not need to be considered further in this development.

431

Properties of Participating Media

9.7 DEFINITIONS AND USE OF MEAN ABSORPTION COEFFICIENTS 9.7.1 Use of Mean Absorption Coefficients Chapter 13 will consider methods for analyzing radiative transfer. Only passing reference is made to the treatment of the spectral dependence of the radiating medium treated in this chapter. As shown, the properties of many translucent materials can vary considerably with the spectral variable. To try to overcome the need to include detailed property variations in an analysis, there have been attempts to use mean property values over a spectral range. A mean absorption coefficient that is spectrally averaged over all wavelengths provides a simplification that sidesteps the tedious but precise procedure of carrying out a spectral analysis and then integrating the spectral energy over all wavelengths or using a k-distribution or SLWSGG methods to obtain the total energy. A difficulty is whether it is possible to decide in advance for a particular situation if using a mean absorption coefficient will yield a sufficiently accurate solution.

9.7.2 Definitions of Mean Absorption Coefficients Local total emission by a volume element dV in a material is given by the integral

4dV

ò

¥

0

kl (T , P )pI lb (T )d l = 4dV

ò

¥

0

kl (T , P )Elb (T )d l. (9.67)

For the emission integral, it is convenient to define the Planck mean absorption coefficient κP(T, P) as



k P (T , P ) º

ò

¥

0

kl (T , P )Elb (T )d l

ò

¥

0

=

ò

¥

0

kl (T , P )Elb (T )d l sT 4

Elb (T )d l

. (9.68)

The κP is the mean of the spectral coefficient weighted by the blackbody emission spectrum (Planck distribution). It is useful for considering emission from a volume and for certain special cases of radiative transfer. Further, the Planck mean κP is convenient since it depends only on the local properties at dV. It can be tabulated and is especially useful where the pressure is constant over the volume of a gaseous medium. Fraga et al. (2019b) examine the use of the Planck mean evaluated from line-by-line data in various WSGG formulations for CO2/H2O mixtures. The local absorption in a material depends on the integral

ò

¥

0

kl (T , P )I l d l . For this absorption

integral, an incident mean absorption coefficient κi(T, P) can be defined



k i (T , P ) =

ò

¥

0

kl (T , P )I l d l

ò

¥

0

. (9.69)

I l dl

A tabulation of κi would require many combinations of incident spectral distributions and spectral variations of local absorption coefficients. Except in very limited special cases, this would not be warranted. The incident mean can be used when the incident intensity has a spectral form that remains fixed so that κi can be evaluated and tabulated. For example, incident energy having a solar spectrum occurs sufficiently often that κi might be tabulated for the solar distribution. The κi is useful in the transparent approximation when the incident spectral intensity is known, as this spectrum

432

Thermal Radiation Heat Transfer

remains unchanged while radiation travels through the medium. If the mean intensity I l,i is proportional to a blackbody spectrum at the temperature of the position for which κλ(Τ, P) is evaluated, that is, I l,i µ I lb ( T ) , then the incident mean is equal to κP:

k i (T , P ) =



ò

¥

0

kl (T , P )I lb (T )d l

ò

¥

0

= k P (T , P ). (9.70)

I lb (T )d l

The Rosseland mean attenuation coefficient arises from treating radiation transfer in optically thick media, where it acts as a diffusion process. For only absorption (no scattering), -1

é¥ 1 ¶Elb (T ) ùú k R (T , P ) = ê d l . (9.71) ê kl (T , P ) ¶Eb (T ) ú ë0 û

ò



At first glance, the Rosseland mean appears to be entirely different from κP and κi, which are weighted by spectral distributions of energy or intensity. However, for 1D diffusion, radiative spectral flux is found to depend only on the local blackbody emissive power gradient and absorption coefficient (see Section 13.2.1): ql,z d l = -

¥

Then

ò 0

4 ¶Eb kl ql,z d l = and 3 ¶z

¥

ò 0

4 dElb (T ) 4 ¶Eb ¶Elb dl = d l (9.72) 3k l dz 3kl ¶z ¶Eb

4 ¶Eb qz d l = 3 ¶z

¥

1 dElb

ò k dE l

0

dl

b

Substituting into Equation (9.71) gives for the diffusion case ¥



k R (T , P ) =

ò k q dl . (9.73) ò q dl 0

l l ,z

¥

0

l ,z

The Rosseland absorption coefficient is thus a mean value of κλ weighted by the local spectral energy flux qλ,zdλ through the assumption that the local flux depends only on the local gradient of emissive power and the local κλ. For a gray gas, the absorption coefficient is independent of wavelength, κλ(T,P) = κ(T,P), and the mean values reduce to κP(T,P) = κi(T,P) = κR(T,P) = κ(T,P). Determining any of the mean coefficients from spectral absorption coefficients usually requires detailed line-by-line numerical integrations or use of the k-distribution or SLWSGG methods.

9.7.3 Approximate Solutions of the Radiative Transfer Equation Using Mean Absorption Coefficients Some references incorporate mean absorption coefficients in radiative transfer calculations. Solving the transfer and energy equations is then considerably simplified because integrations over wavelength are not needed. The most common approximation is that gray–gas relations are applied to a real gas by substituting an appropriate mean absorption coefficient in place of the κ in the gray solution. By examining 40 cases, Patch (1967a, b) showed that simple substitution of the Planck mean in gray–gas solutions leads to errors in total intensities that varied from −43% to 881% from the

433

Properties of Participating Media

solutions obtained by using spectral properties in the transfer equations and integrating the spectral results. Reductions in error result from dividing the intensity into multiple spectral bands and using an individual Planck mean for each band. Other mean absorption coefficients have been introduced. Sampson (1965) synthesized a coefficient that varies from the Planck mean to the Rosseland mean as the optical depth increases along a path. Agreement was obtained within a factor of two with exact solutions for various problems. Abu-Romia and Tien (1967) applied a weighted Rosseland mean over optically thick portions of the spectrum and a Planck mean over optically thin regions and obtained relations for energy transfer between bounding surfaces. Planck and Rosseland mean absorption coefficients for carbon dioxide, carbon monoxide, and water vapor are given to aid such computations. Patch (1967a, b) defined an effective mean absorption coefficient as



ke (S, T , P ) =

ò

¥

0

kl (T , P )I lb (T ) exp[ - kl (T , P )S ]d l

ò

¥

0

(9.74)

I lb (T ) exp[ - kl (T , P )S ] d l

The values of κe(S, T, P) can be tabulated as a function of temperature and pressure; in addition, κe depends on the path length and must be tabulated as a function of S. For small S, κe approaches κP. For large S, the exponential term in the integrals causes κe to approach the minimum value of κλ in the spectrum considered. In radiative transfer calculations, the approximation is made that the real gas, with T and P known variables along S, is replaced along any path by an effective uniform gas with absorption coefficient κe. Computations are then performed using κe in the gray–gas equation of transfer. The κe value used is found by equating κeS at the T and P of the point to which S is measured, to the optical depth of that point in the real gas. For 40 cases, Patch shows the agreement of total intensities within −25%–28% of the integrated spectral solutions. Other methods of using mean coefficients are in Lick (1963), Stewart (1964), Thomas and Rigdon (1964), Grant (1965), Howe and Scheaffer (1967), and Mengüc (1985). Solution methods based on k-distributions have almost completely replaced the use of mean absorption coefficients for most analyses, as they are little more time-consuming and much more accurate.

9.8 RADIATIVE PROPERTIES OF TRANSLUCENT LIQUIDS AND SOLIDS Many solids and liquids are opaque to thermal radiation, but the exceptions are important. Figure 9.19 shows κλ for diamond (Garbuny 1965). Strong absorption peaks exist due to crystal lattice vibrational energy states at certain wavelengths. However, the spectral variations are more regular than for gases. This is true for most solids and liquids that are translucent in various spectral regions. Consider the selective transmission and absorption behavior of glass and water. Figure 9.20 shows the overall spectral transmittance of glass plates for normally incident radiation. As shown in Section 17.2.1, the overall transmittance Tλ includes the effect of absorption (related to the absorption coefficient κλ) within the glass of thickness d, and multiple surface reflections (related to the surface reflectivity, ρλ); it is given by Equation (17.2) as Tλ = τλ (1− ρλ)2/(1− ρλ2 τλ2) where τλ = exp(−κλd) and ρλ= [(n − 1)/(n + 1)]2. For small κλd this reduces to Tλ = (1 − ρλ)/(1 + ρλ). Typically for glass n ≈ 1.5, so ρλ = (0.5/2.5)2 = 0.04. Then including only reflection losses gives Tλ = (1 − 0.04)/(l + 0.04) = 0.92. In Figure 9.20 the fused silica has very low absorption in the range λ = 0.2–2 μm, and Tλ ≈ 0.9 in this region as a result of surface reflections. Ordinary glasses typically have two strong cutoff wavelengths beyond which the glass becomes highly absorbing and Tλ decreases rapidly to near-zero except for very thin plates. The measured curve for fused silica in Figure 9.20 shows this. There are strong cutoffs in the ultraviolet at λ ≈ 0.17 μm and in the near-infrared at λ ≈ 2.5 μm. The glass is therefore a strong absorber or emitter for λ  2.5 μm. Figure 9.21 shows the overall

434

Thermal Radiation Heat Transfer

FIGURE 9.19  Spectral absorption coefficient of diamond. (From Garbuny, M., Optical Physics, Academic Press, New York, 1965).

FIGURE 9.20  Normal overall spectral transmittance of a borosilicate or fused silica glass plate (includes surface inter-reflections) at 298 K. (Replotted from Touloukian, Y.S. and Ho, C.Y. (eds.), Thermophysical properties of matter, TRPC data services, Thermal Radiative Properties: Nonmetallic Solids, vol. 8, Touloukian, Y.S. and DeWitt, D.P., Plenum Press, New York, 1972a).

transmittance for various thicknesses of soda-lime glass, which is more absorbing than fused silica. The effect of absorption is illustrated quite well as the thickness increases. Typical optical constants for glass are in Hsieh and Su (1979). Nicolau and Maluf (2001) provide measured values of tinted commercial glass. For windows in high-temperature devices, such as furnaces or solar-cavity receivers, emission from within the windows can be significant. From Kirchhoff’s law, the overall spectral emittance

Properties of Participating Media

435

FIGURE 9.21  Effect of plate thickness on normal overall spectral transmittance of soda-lime glass (includes surface inter-reflections) at 298 K. (From Hsieh, C.K. and Su, K.C., Sol. Energy, 22(1), 37, 1979).

Eλ that includes the effect of surface reflections is equal to the spectral absorptance. Hence, from Equation (17.3) for an isothermal window, Eλ = (1−ρλ)(1−τλ)/(1−ρλτλ). For a thick window, beyond the cutoff wavelength, τλ → 0 and Eλ ≈ 1 − ρλ. In this instance, reflection from only one surface is significant because all the radiation passing through the first surface is absorbed before it can be transmitted to the second surface of the window. If the n for glass is 1.5, then ρλ ~ 0.04 for incidence from the normal direction; hence, Eλ = 0.96 in the normal direction for the highly absorbing spectral regions of the glass. In a fashion like Example 3.3, the hemispherical value is found as Eλ = 0.90. This is the upper value in Figure 9.22, which shows the hemispherical emittance of window-glass sheets of various thicknesses (Gardon 1956). The transmission behavior in Figures 9.21 and 9.22 shows why glass windows have the important ability to trap solar energy. The sun radiates a spectral energy distribution very much like a blackbody at 5780 K (10,400°R). Considering the range 0.3 p

(1 - e p )1/ 2 C l m 0.95 - (1 - e p )1/ 2

(e p = porosity). (10.85)

482

Thermal Radiation Heat Transfer

The dependent scattering region (results deviate more than 5% from independent scattering) is demarcated when C/λm = 0.5 is inserted into Equation (10.85); this is based on far-field effects alone. Figure 10.21 shows the regions of dependent and independent scattering by this criterion (Tien and Drolen 1987). This affects the contribution of soot agglomerates to radiative balance in combustion chambers. To understand this effect, Ivezic and Mengüç (1996) and Ivezic et al. (1997) used DDA and showed that an agglomerate of N particles must be modeled as dependent scatterers if the effective size parameter of N particles (ξe = Nπd/λ) is between 0.2 and 2. They explored the effects of agglomeration using DDA. They conclude that independent scattering depends on the single particle size parameter and the distance between two particles, such that c > 2/xs = 2.4/xe. Later, a similar problem was explored by Wang and Zhao (2020). Another effect of increasing particle density is dependent absorption (Kumar and Tien 1990). For highly absorbing particles, the particle absorption efficiency was found to increase as the spacing between the particles decreased, while the scattering efficiency declined. The total extinction by combined absorption and scattering increased. About a 5% increase in absorption efficiency was found for a particle volume fraction of 0.06. Dependent effects tend to increase the absorption over that predicted by using independent absorption, while dependent scattering tends to decrease scattering from that predicted using independent scattering. Enhanced absorption resulting from dependent scattering is analyzed by Ma et al. (1990). Closely spaced parallel cylinders are also of interest in applications such as fibrous insulation, filament wound structures, and the determination of nanomaterial properties. In the production of structures by winding closely spaced fibers in a resin, it is possible to initiate resin polymerization by infrared radiation from an external source. The fibers are very closely spaced, and the effects on the scattering phase function of both dependent scattering and the dependence of scattering on the incidence angle must be considered. Near-field effects are used to determine the radiative properties of arrays of parallel cylinders subject to normal irradiation and dependent scattering (Lee 1992, Chern et al. 1995, Lee and Cunnington 1998). They show the ranges of cylinder volume fraction and size parameters where dependent scattering becomes important for various values of the fiber refractive index. The analysis was further extended into coated and uncoated fibers subject to obliquely incident radiation (Lee and Grzesik 1995). For fiber layers, the interaction of wavelength, refractive index, fiber diameter, and fiber spacing is so complex that no simple criterion is available for deciding whether dependent scattering will be important, and maps must be referred to, similar to Figure 10.22. However, for ξ = πD/λm > 2 (based on cylinder diameter D and the wavelength in the medium surrounding the fibers λm), dependent scattering is relatively independent of the fiber refractive index and the size parameter, and dependent scattering is present for porosity  1. Additional analytical relations are provided in Chapter 17 for materials such as glass, water, translucent plastics, or translucent ceramics that have n > 1.

11.2 ENERGY EQUATION AND BOUNDARY CONDITIONS FOR A PARTICIPATING MEDIUM An energy balance on a volume element within a material includes contributions by conduction, convection, internal heat sources such as by electrical dissipation and combustion, compression work, viscous dissipation, energy storage during transients, and the contribution by radiative transfer in a translucent material. Compared with internal energy in the form of heat capacity, the storage of radiant energy by the increase of photons within a volume element is usually negligible, except for some special transients such as are considered in Kumar and Mitra (1995) and Longtin and Tien (1997) where this additional transient term is provided; hence, no modification of the usual transient heat capacity term in the energy equation is considered in this chapter as a result of the thermal radiation. In most problems, radiation pressure is negligible relative to fluid pressure and is not included in the compression work. For heat conduction, the net contribution to a volume element can be written as the negative divergence of a conduction flux vector −∇ · qc = ∇ · (k∇T). Similarly, the net contribution by radiant energy per unit volume within a translucent medium can be written as the negative of the divergence of the radiant flux vector qr, and expressions for this term will be obtained in the subsequent development of the radiative transfer relations. Thus, the conventional energy equation for a single-component translucent fluid can be modified for the effect of radiative transfer by adding –qr to the k∇T to give rcp



DT DP = bT + Ñ × (kÑT - q r ) + q + F d (11.1) Dt Dt

where

¶ (Y) ¶ (Y) ¶ (Y) DY ¶ ( Y ) +u +v +w = Dt ¶t ¶x ¶y ¶z β is the thermal coefficient of volume expansion of the fluid q is the local energy source (electrical, chemical, nuclear) per unit volume and time Φd is the energy production by viscous dissipation.

D(Y)/Dt is the substantial derivative,

491

Fundamental Radiative Transfer Relations

Although traditionally referred to as the energy equation, the terms are each energy rates (or power) per unit volume. An alternative form in terms of enthalpy is r



Dh DP = + Ñ × (kÑT - q r ) + q + F d . (11.2) Dt Dt

For an incompressible fluid with constant properties in rectangular coordinates,



¶T ¶T ¶T ¶T k æ ¶ 2T ¶ 2T ¶ 2T +u +v +w = + + ç ¶t ¶x ¶y ¶z rc è ¶x 2 ¶y 2 ¶z 2 +

ö 1 æ ¶qr , x ¶qr , y ¶qr ,z ö + + ÷- ç ÷ ¶y ¶z ø ø rc è ¶x

(11.3)

1 1 q ( x, y, z, t ) + F d . rc rc

To obtain the temperature distribution in the medium by solving Equation (11.1), an expression for ∇·qr is needed in terms of the temperature distribution. The ∇·qr is also affected by scattered radiation, which will need to be determined within the medium. One approach for determining the required relations is to obtain the qr and then differentiate to obtain ∇·qr; this is demonstrated in Chapter 13 for the specific geometries of a plane layer and a long circular cylinder. Another approach is to obtain ∇·qr directly by considering the local radiative interaction within a differential volume in the medium. The ∇·qr is found from the difference between the radiation emitted from a volume element and the absorption in that same element. This depends on the absorption coefficient and the intensity incident from all directions (note that for a medium with refractive index n > 1, an n2 factor is in the emission term, as given in Chapter 17). The radiative energy required for the energy equation is the total radiation (includes all wavelengths, wave numbers, or frequencies). Hence, as will be developed in detail, relations are required that give the difference between total radiation incident from all solid angles Ωi that is locally absorbed and the locally emitted radiation. These two quantities (absorbed and emitted radiation) are expressed by the two integral terms over all wavelengths on the right side of the following relation (see Section 11.4.2): ¥ ù é 4p kl ê I l (W i )dW i ú dl - 4p kl I lb dl. (11.4) ê ú l =0 l =0 ë Wi = 0 û Equation (11.4) is valid whether scattering is present or not. The RTE is developed in the next section as needed to obtain the local intensity throughout the ¥



material volume in the

-Ñ × q r =

ò

4p

Wi = 0

ò

ò

ò

I l (W i ) dW i term.

A special case of the energy equation is when radiation dominates over both conduction and convection. Then, for steady state, the equilibrium temperature distribution is achieved only by radia present. This condition is called radiative equilibrium. tive effects with or without heat sources, q, This limit provides useful results for some high-temperature applications, and solutions for radiative equilibrium provide limiting results for comparison with solutions including conduction and/or  such convection combined with radiation. If a medium is considered with internal energy sources q, as chemical, nuclear, or electrical energy release, the energy equation for radiative equilibrium is

Ñ × q r (r ) = q (r ) (11.5)

where r is a position vector in the participating medium. The solution of the energy equation requires boundary conditions. For the general energy relations given by Equations (11.1) through (11.3), the dependent variable is temperature or a temperature-dependent property such as enthalpy. Because the equation is second order in the space

492

Thermal Radiation Heat Transfer

coordinates and first order in time, the equation requires two boundary conditions for each independent space variable, such as x, y, and z in Cartesian coordinates and an initial condition. When radiation is present, there are additional boundary conditions for the radiative intensities. The radiative boundary conditions enter the results through the solution of the RTE that is used to evaluate the ∇·qr term in the energy equation. Thus, the radiative boundary conditions are not explicitly stated for the energy equation, but are incorporated in the solution for the radiative flux divergence, ∇·qr. In this chapter and in Chapters 12 through 14, discussion centers on cases with either radiative  r ). These cases are generally solved as linear equaequilibrium or a prescribed internal source q( tions. Cases with conduction and/or convection become highly nonlinear in temperature and are discussed in Chapter 15.

11.3 RADIATIVE TRANSFER AND SOURCE-FUNCTION EQUATIONS To obtain the radiative intensity throughout the translucent material, the radiative transfer equation (RTE) is used. This is the differential equation that describes the radiation intensity along a path in a fixed direction through an absorbing, emitting, and scattering medium. Bouguer’s law, Equation (1.46), accounts for attenuation of intensity by absorption and scattering. The RTE also includes augmentation of the radiation intensity by emission and scattering into the path direction. We will now discuss the details.

11.3.1 Radiative Transfer Equation As derived in Chapter 1 (Equation (1.74)), the RTE is I l (r + dr , W, t + dt ) - I l (r , W, t )

= kl I lb (r , t )dr - kl I l (r , W, t ) dr - ss,l I l (r , W, t ) dr (11.6) +

s s ,l 4p

ò I (r, W, t )F (W¢, W)dW¢dr. l

l

4p

We can combine the attenuation terms and define the extinction coefficient βλ = κλ + σs,λ. Equation (11.6) is written for a particular propagation direction S and for steady conditions (e.g., over a time interval in which the radiation intensity varies insignificantly due to photon time-of-flight effects, ∂I/c∂t ≈ 0):

s dI l = -bl (S )I l (S ) + kl (S )I lb (S ) + s,l dS 4p

4p

ò I (S, W )F (W, W )dW . (11.7) l

i

l

i

i

Wi =0

In Equation (11.7) and the development that follows, σs,λ is assumed to be independent of the direction of incidence, Ωi. This assumption is generally valid except for a few special cases such as propagation through filament wound structures and other layered scattering materials. Define the single scattering albedo, ωλ, as the ratio of scattering to extinction coefficients,

wl =

s s s ,l kl k = s,l ; 1 - wl = = l . (11.8) k l + s s ,l bl k l + s s ,l b l

If the medium is not absorbing, but scattering alone, ωλ→1. For a purely absorbing medium, ωλ→0. The optical thickness or opacity was defined in Equation (1.55) as

493

Fundamental Radiative Transfer Relations S



tl ( S ) =

S

ò b (S*)dS* = ò éë k (S*) + s l

l

S *= 0

s ,l

(S*) ùû dS *. (11.9)

S *= 0

The RTE (Equation (11.7)) in terms of albedo and optical thickness becomes

dI l w = - I l (tl ) + (1 - wl )I lb (tl ) + l d tl 4p

4p

ò I (t , W ) F (W, W )dW (11.10) l

l

i

l

i

i

Wi =0

where dτλ = βλ(S)dS is the optical differential thickness τλ = τλ(S) ωλ = ωλ(S). The final two terms in Equation (11.10) act to increase the intensity along the direction S by emission and in-scattering. They are often combined into the source function, Iˆl (tl , W),

w Iˆl ( tl , W ) = (1 - wl ) I lb ( tl ) + l 4p

4p

òI

l

( tl , Wi ) F l ( W, Wi ) dWi . (11.11)

Wi =0

For anisotropic scattering, Iˆl (tl , W) is a function of Ω (i.e. of the direction S). The RTE (Equation (11.10)) then becomes dI l + I l ( tl ) = Iˆl (tl , W) (11.12) d tl



where τλ = τλ(S). This appears to be a differential equation; however, because Iλ is within the source function in Equation (11.11), Equation (11.12) is actually an integrodifferential equation. It can be formally integrated after multiplying it by the integrating factor e tl :

dI l tl d é I l ( tl ) e tl ù = Iˆl (tl , W)e tl . (11.13) e + I l ( t l ) e tl = û d tl d tl ë

Integrating over the optical path from τλ = 0 to τλ = τλ (S) and rearranging gives tl



I l (tl , W) = I l (0, W)e

- tl

+

ò I ( t* ) e l

t*l = 0

l

S

- ( tl - t*l )

d t*l ; tl (S ) =

ò b (S*)dS * (11.14) l

*

S =0

where t*l is a dummy optical variable of integration along S (see Equation (11.9)) Iλ(0, Ω) is the intensity in the direction of S at the boundary or location where S = 0. Equation (11.14) is the integrated form of the RTE. Equation (11.14) is interpreted physically as the intensity at optical depth τλ, being composed of two terms. The first is the attenuated initial intensity that arrives at τλ. The second is the intensity at τλ resulting from emission and incoming scattering in the S direction by all thickness elements along the path from 0 to S, reduced by exponential attenuation between each location of emission and incoming scattering t*l and the location τλ.

494

Thermal Radiation Heat Transfer

Without scattering (i.e. for absorption only), the source function defined in Equation (11.11), Iˆl ( tl ) = I lb ( tl ) , reduces to the local blackbody intensity that is isotropic, and the RTE (Equation (11.14)) becomes tl



I l (tl , W) = I l (0, W)e - tl +

ò ( )

S

*

- ( tl - t ) l I lb t*l e d t*l ; tl (S ) =

ò k (S*)dS * . (11.15) l

S *= 0

t*l= 0

For a purely scattering medium, the single scattering albedo ωλ = 1, and the source function Equation (11.11) becomes

1 Iˆl ( tl , W ) = 4p

4p

ò

I l ( tl , W i ) F l ( W, W i ) dW i ; tl (S ) =

Wi =0

S

òs

s ,l

(S*) dS *. (11.16)

S *= 0

The RTE becomes Equation (11.14) with this substitution for the source function. The local intensity for pure scattering is then no longer a function of medium temperature except through the possible temperature dependence of the properties. For isotropic scattering, Φλ = 1, so the source function Equation (11.11), including emission, becomes

w Iˆl ( tl ) = (1 - wl ) I lb ( tl ) + l 4p

4p

òI

l

( tl , Wi ) dWi . (11.17)

Wi =0

The source function is now independent of direction (isotropic). In the limit of no absorption (ωλ = 1) and isotropic scattering, Equation (11.17) shows that the source function reduces to the local mean incident intensity,

1 Iˆl ( tl ) = 4p

4p

òI

l

( tl , Wi ) dWi º I l (tl ). (11.18)

Wi =0

The boundary condition Iλ(0,Ω) needed for the solution of the radiative transfer equation in Equation (11.14) is obtained as in the usual enclosure theory. At an opaque solid boundary, the intensity consists of emitted and reflected intensities. At a translucent boundary, the Iλ(0,Ω) depends on radiation entering from the exterior surroundings of the boundary. Obtaining and applying the boundary condition is illustrated by examples and solutions in succeeding chapters where the equation of transfer is applied; translucent boundaries are considered in Chapter 17 for materials with n > 1.

11.3.2 Source-Function Equation The source function was defined in Equation (11.11). Eliminate Iλ by using Equation (11.14) in Equation (11.11) to obtain an equation for Iˆl ; the result is the source-function equation. As shown in Figure 11.1, the location τλ in Equation (11.11) is along path S, and the source function is being considered along this path. At each location along S, radiation is augmented by scattering coming in from all incident paths such as Si. The optical length along S is τλ,i from its origin τλ,i = 0 to the location τλ along S. Then, from Equation (11.14), the incident intensity in dΩi at τλ in Figure 11.1 provided by radiation along Si is tl , i



I l (tl,i ,W i ) = I l (tl,i = 0,W i )e - tl ,i +

ò I ( t* , W ) e l

tl , i * = 0

l ,i

i

*

- (tl ,i - tl ,i )

d t*l,i . (11.19)

495

Fundamental Radiative Transfer Relations

FIGURE 11.1  Path direction S for intensity and typical incident path Si

Substituting this into Equation (11.11), the source-function equation becomes Iˆl ( tl , W ) = (1 - wl ) I lb ( tl )

w + l 4p

tl ,i é ù - ( tl ,i - t*l ,i ) ê ú - tl ,i ˆ * * + d tl,i ú F l ( W, W i ) dW i. I l tl , i , W i e ê I l (tl,i = 0, W i )e Wi = 0 ê úû t*l ,i = 0 ë 4p

ò (

ò

)

(11.20) This integral equation shows how the radiative source function along a path depends on the source function along all of the other intersecting paths arriving from throughout the medium volume. The solution requires obtaining results for the entire radiation field. Because of the complexity of the source-function equation, simplifying approximations are often made that are reasonable for various physical situations. Some of these are now examined. A common assumption is that the scattering is isotropic (in addition to σs,λ being independent of incidence angle as assumed earlier). This is a limiting approximation that is only valid for random distributions of scattering particles and for conditions without strong directionally incident radiation so that the radiation interacting with the particles tends to be isotropic. Under this assumption, the phase function Φ = 1 and Iˆl is isotropic (see Equation (11.17)) so that Equation (11.20) reduces to Iˆl ( tl ) = (1 - wl ) I lb ( tl )

tl , i é ù (11.21) * 1 - tl , i ê + wl I l (tl,i = 0, W i )e + Iˆl t*l,i e -( tl ,i -tl ,i ) d t*l,i ú dW i . ê 4p ú Wi =0 ê t*l , i = 0 úû ë 4p

ò

ò ( )

496

Thermal Radiation Heat Transfer

For a gray medium with constant properties and isotropic scattering, the equation for Iˆ in terms of the physical position S (rather than the optical length), the incident path length Si, and the extinction coefficient β is Iˆ ( S ) = (1 - w) I b ( S ) Si 4p é ù (11.22) 1 * I (Si = 0, W i )e -bS dW i + b Iˆ S*i e -b( Si -Si ) dSi* ú. + wê ê 4p ú S *= 0 ë Wi =0 û



ò

ò ( )

Without scattering, ωλ = 0 or ω = 0, the source function is simply the blackbody intensity, Iˆl (tl ) = I lb (tl ) or Iˆ(S ) = I b (S ). For a medium with only scattering and no absorption, ωλ = 1 or ω = 1 in Equations (11.21) and (11.22), or the source function is given by Equations (11.16) and (11.18) in terms of local mean incident intensity. Equations (11.19) and (11.20) often cannot be solved simultaneously for the source function and intensity distributions throughout the translucent medium because they contain the unknown temperature distribution within the medium and that affects the blackbody intensity Iλb(T). For temperature-dependent properties, the temperature distribution is also needed to determine the local absorption and scattering coefficients; the local optical depth τλ(S) along a path can then be computed from Equation (11.9) and the physical path length S related to the optical length τλ. The temperature distribution depends on energy conservation within the medium, Equation (11.1), which in turn depends on the total radiative source (radiative flux divergence) in each volume element, −∇ · qr, that is obtained from the intensities and the source function. As will be shown as the development continues, the energy equation is solved, along with the intensity and source-function equations, to yield the temperature distribution and the angular and spatial distributions of intensity. The equations are sufficiently complex that numerical solutions are almost always required. In a few instances analytical techniques have been used to obtain closed-form solutions for a limited range of geometries and conditions. The energy, intensity, and source-function equations provide the set of basic equations required for analysis. Example 11.1 A black surface element dA is 10 cm from an element of gas dV (Figure 11.2). The gas element is a part of a gas volume V that is isothermal and at the same temperature T as dA. If the gas has an absorption coefficient κλ = 0.1 cm−1 at λ = 1 μm and there is no scattering, what is the spectral intensity at λ = 1 μm that arrives at dV along the path S from dA to dV?

FIGURE 11.2  Geometry for Example 11.1

497

Fundamental Radiative Transfer Relations

Because dA is black and at temperature T, the intensity at S = 0 is Iλ(0) = Iλb(T). Since the gas is isothermal, the emitted blackbody intensity in the gas is Iλb(T). Substituting into the integrated RTE, Equation (11.15), that is, for σs,λ = 0, gives tl



Il ( tl ) = Ilb (T )e - tl + Ilb (T )e - tl

ò

e t*l d t*l .

*

tl = 0

After integration, this reduces to Iλb(τλ) = Iλb(T). The intensity arriving at dV along an isothermal path through the gas from a black surface element at the same temperature as the gas is thus equal to the blackbody spectral intensity emitted by the wall and does not depend on κλ or S. The attenuation by gas absorption of the spectral intensity emitted by the wall is exactly compensated by emission from the gas along the path from dA to dV. This is true at each wavelength, so, from Equation (11.4), the radiative source is −∇·qr = 0. If T is specified, the spectral intensity arriving at dV can be calculated using Equation (1.13), since ngas ≈ 1.

Example 11.2 An absorbing–emitting layer of gray gas with κ = 1.6 m−1 is adjacent to a black wall. As a result of internal energy generation, the medium has a parabolic temperature distribution decreasing from 650 K at the wall to 425 K at the boundary x = D (Figure 11.3). What is the total intensity I(D) in the direction normal to the wall? The parabolic temperature distribution is given by T(x) = Tw − (Tw − 425)(x/D)2. From Equation (11.15), the intensity normal to the wall at x = D is given for a gray gas by

sT 4 I(D) = w e - kD + p

kD

ò

kx = 0

sT 4 ( x ) - k(D - x ) kdx. e p

If we insert the T(x) relation and other numerical values, numerical integration yields I(D) = 2510 W/(m2·sr).

FIGURE 11.3  Conditions for Example 11.2

498

Thermal Radiation Heat Transfer

11.4 RADIATIVE FLUX AND ITS DIVERGENCE WITHIN A MEDIUM The required radiative flux vector qr and its divergence −∇·qr are now considered for solving the energy Equation (11.1).

11.4.1 Radiative Flux Vector For an energy balance on a volume element dV, the net radiative energy supplied to dV is needed. For a volume element dx dy dz, the radiative energies exchanged at the dx dz faces are shown in Figure 11.4. The energies are similarly written for the other faces, the outgoing energies are subtracted from the incoming, and the result divided by dx dy dz. The result is the net rate of radiative energy supplied to dV per unit volume: é ¶q ¶q ¶q ù - ê r , x + r , y + r ,z ú = -Ñ × q r . (11.23) ¶y ¶z û ë ¶x



This is the negative of the divergence of the radiative flux vector qr, q r = iqr , x + jqr , y + kqr ,z . (11.24)



The vector form −∇·qr can be used for any coordinate system. The flux vector is now related to the intensity from the equation of transfer. Consider the area dA in Figure 11.5. This could be one of the faces of the volume element in Figure 11.4, and n is then along a coordinate direction. In general, n is any direction, and the flux qr,n is through an area normal to the n direction. Let s be a unit vector in the S direction, which is the direction for the intensity I. The direction cosines for S in the rectangular coordinate system are α, δ, and γ. Then, cos θ = s·n, and in Figure 11.5, the intensity is the energy rate per unit solid angle crossing dA per unit area normal to the direction of I = I(α, δ, γ). Hence, the energy rate crossing dA as a result of I is I = I(α, δ, γ) dA cosθ dΩ. The radiative flux crossing dA as a result of intensities incident from all directions is then 4p



qr,n =

ò

4p

I (a, d, g ) cos qdW =

W =0

FIGURE 11.4  Radiative fluxes for volume element

ò I (a, d, g)s × ndW (11.25)

W =0

499

Fundamental Radiative Transfer Relations

FIGURE 11.5  Quantities in derivation of radiative flux vector

where θ is the angle from the normal of dA to the direction of I(α, δ, γ). The qr,n depends on the direction of n relative to s. The cosθ becomes negative for θ > π/2, so the sign of the portion of the energy flux traveling in the direction opposite to the positive n direction is automatically included. The qr,n is the component in the n direction of the radiative flux vector given by 4p

qr =



ò I (a, d, g)sdW. (11.26)

W= 0

That is, qr,n = n·qr. If the direction cosines of n in the rectangular coordinate system are α′, δ′, and γ′, and those for s (the direction of I) are α, δ, and γ, then n = iα′ + jδ′ + kγ′ and s = iα + jδ + kγ so that s ⋅ n = cosθ = αα′ + δδ′ + γγ′ and, from Equation (11.25), 4p



qr,n =

ò I (a, d, g)(aa¢ + dd¢ + gg¢)dW.

W =0

Thus,

qr,n = a¢qr , x + b¢qr , y + g¢qr ,z . (11.27)

The qr,x, qr,y, and qr,z are fluxes across areas normal to the x, y, and z directions and are the components of the radiative flux vector. Each component is obtained from the integral in Equation (11.25) with n oriented in that coordinate direction. For example, 4p



qr, x =

ò I (a, d, g) adW,

W= 0

where α is the cosine of the angle between the x axis and the direction of I that is in the direction of s. The radiative flux vector can also be written by using Equation (11.26) with a spherical coordinate system as in Figure 11.6. The unit vector s is then

s = i cos f sin q + j sin f sin q + k cos q. (11.28)

500

Thermal Radiation Heat Transfer

FIGURE 11.6  Spherical coordinate system for radiative flux vector

Substituting s and dΩ = sinθdθdϕ into Equation (11.26) gives the vector qr in terms of its three components: 2p

q r = iqr , x + jqr , y + kqr , z = i

p

ò ò I (q, f) cos f sin qdqdf 2

f=0 q=0 2p



+j

p

ò ò I (q, f)sin f sin qdqdf (11.29) 2

f=0 q=0 2p

+k

p

ò ò I (q, f) cos q sin qdqdf.

f=0 q=0

The negative divergence of the radiant flux vector, −∇·qr, is considered in Sections 11.4.2 and 11.4.3 and is used in the energy equation, as discussed in Section 11.2. In some situations, such as extremely rapid transients, it is necessary to account for the radiative energy contained within a volume element (Kumar and Mitra 1995, Longtin and Tien 1997). This is related to the local intensity averaged over all solid angles, as will now be shown. Consider radiation as a collection of energy bundles, and the conditions at any location in a medium are given by an energy distribution function f. Since radiation energy is related more directly to frequency than to wavelength, frequency is used here as the spectral variable. In volume dV at position r, let f(ν, r, S) dν dV dΩ be the number of energy bundles traveling in the direction of S in frequency interval dν centered about ν and within solid angle dΩ about the direction S (see Figure 11.7). Each bundle has energy proportional to hν. The energy per unit volume per unit frequency interval is then hνf(ν, r, S) dΩ integrated over all solid angles. This is the spectral radiant energy density (Equation (1.65)): 4p



U n (n, r ) = hn

ò

W =0

f (n, r, S )dW. (11.30)

501

Fundamental Radiative Transfer Relations

FIGURE 11.7  Schematic for derivation of radiative energy density

To obtain the intensity and show its relation to Uv, the energy flux in the S direction is needed across area dA normal to the S direction, Figure 11.7. The photons have velocity c, and the number density traveling in the normal direction across dA is f dn dΩ. The number of photons crossing dA in the S direction per unit time is then cf dn dΩ dΑ. The spectral radiative energy carried by these photons is hn cf(n,r,S) dn dΩ dΑ. The spectral intensity is the energy in a single direction per unit time, unit frequency interval, and unit solid angle crossing a unit area normal to that direction. This gives the spectral intensity at location r and in direction S as Iν = hncf(r,S). The relation between energy density and the volume integrated intensity is then obtained by using In to eliminate f in Equation (11.30),

U n (n, r ) =

1 c

4p

ò

W =0

I n ( r, S ) d W =

4p ù 4p 4p éê 1 I n ( r, S ) d W ú = I n (r ) (11.31) c ê 4p ú c ë W =0 û

ò

where I n (r ) is the mean incident spectral intensity, Equation (11.18). This is the relation given in Equation (1.65), followed by expressions for radiative pressure in Equation (1.66). As noted before, the c in the denominator makes the value of U n (n, r ) small for most engineering applications.

11.4.2 Divergence of Radiative Flux without Scattering (Absorption Alone) In some important applications, such as for water vapor or CO2 gas without suspended particles, scattering can be neglected, so ωλ → 0 and the spectral source function Equation (11.11) reduces to Iˆl = I lb . Equation (11.15) for the spectral intensity along a path within a medium (as in Figure 11.8) then yields the intensity incident at τλ in solid angle dΩi, tl



I l (tl , W i ) = I l (0, W i )e - tl +

òI

*

lb

(t*l )e -( tl -tl ) d t*l . (11.32)

t*= 0

To obtain the net rate of radiative energy supplied to a volume element dV, consider the energy absorbed and emitted by dV. The rate of energy absorbed from the incident spectral intensity Iλ(τλ, Ωi) arriving within solid angle dΩi in Figure 11.8 is, as in the development of Equation (1.61), κλ(dV)

502

Thermal Radiation Heat Transfer

FIGURE 11.8  Geometry and incident intensity for derivation of energy-conservation relation

Iλ(τλ, Ωi) dV dλ dΩi. The incident intensity Iλ(τλ, Ωi) is given by Equation (11.32) where Iλ(0, Ωi) is the spectral intensity directed toward dV from the boundary. The rate of energy absorbed by dV from all incident directions is the integral over all W i , kl (dV )dl

ò

4p

Wi =0

I l (tl , W i )dW i . The mean incident inten-

sity I l (tl ) at dV as in Equation (11.31) is used for convenience, I l (tl ) º (1 / 4p)

ò

4p

Wi =0

I l ( tl , W i )d W i ,

and the spectral energy absorbed in dV is then 4pkl (dV )I l (tl )dV dl . By integrating over all wavelengths, the rate of total energy absorbed by dV from the radiation field is 4pdV

ò

¥

l =0

kl (dV )I l (tl ) dl .

The rate of total energy emission by dV is obtained from Equation (1.62) by integrating over all wavelengths to give 4pdV

ò

¥

l =0

kl (dV )I lb (T , dV ) dl . Using the Planck mean absorption coefficient,

Equation (9.68), this is also 4κPσT4(S)dV. Since dV is very small, the energy emitted by dV escapes without reabsorption within dV. The net outflow of radiant energy per unit volume, which is the desired divergence of the radiant flux vector, is then the emitted energy rate minus the absorbed energy rate so that, at any location S, 4p é ù 1 k l ( S ) ê I lb ( S ) I l ( S , W i )d W i ú d l ê ú 4p l =0 Wi =0 ë û ¥

Ñ × q r (S ) = 4p

ò

ò

(11.33)

¥ é ù = 4p kl (S ) éë I lb (S ) - I l ( S ) ùû dl = 4 ê k P sT 4 (S ) - p kl (S )I l (S )dl ú . ê ú l =0 l =0 ë û ¥

ò

ò

This is the relation given earlier by Equation (11.4) when the concept of radiative flux divergence was being introduced.

503

Fundamental Radiative Transfer Relations

11.4.3 Divergence of Radiative Flux Including Scattering The relations in the previous section to obtain ∇·qr(S) are now extended to include scattering for media such as gases with suspended particles. Begin with the equation of transfer, Equation (11.7), which applies for an anisotropic phase function. The first term of Equation (11.7) is expanded as dI l ¶I l dx ¶I l dy ¶I l dz ¶I ¶I ¶I + + = a l + d l + g l (11.34) = ¶x dS ¶y dS ¶z dS ¶x ¶y ¶z dS



where α, δ, γ are the direction cosines for Iλ in the S direction in an x, y, z coordinate system. In a small number of situations, such as in Chern et al. (1995a, b, c), a layered embedded structure that is producing the scattering in the translucent material is modeled as an anisotropic scattering medium. For this type of application, the amount of energy that is scattered from an incident direction can also depend on that incident direction. After the energy is scattered, its directional distribution can be anisotropic. To include these more realistic effects, Equation (11.10), modified to have σs,λ depend on incidence direction, is integrated at location S over all Ω, 4p

ò

W =0

dI l dW = dS

4p

æ ¶I l

¶I l ö

W =0

= - kl ( S )



¶I l

ò çè a ¶x + d ¶y + g ¶z ÷ø dW 4p

òI

l

( S, W ) dW -

W =0

+

1 4p

4p

òs

s ,l

( S, W ) I l ( S, W ) dW + kl ( S )

W =0

4p

ò òs

4p

s ,l

4p

òI

lb

( S )dW (11.35)

W =0

( S, Wi ) I l ( S, Wi )F l ( W, Wi ) dWi dW .

W =0 Wi =0

The modeling of the macroscopic anisotropic structure might also lead to directional internal absorption and emission (i.e. κλ(S) = κλ(S, Ω)), but this effect is not included here. From Section 11.4.1,

ò

4p

W =0

aI l (S, W)dW is the radiative flux qrλ,x(S), and similarly in the y and z directions. The Iλb

is isotropic, so it is independent of angular direction Ω. Then, Equation (11.35) becomes 4p

ò

W= 0



¶q ¶q ¶q dI l dW = rl, x + rl, y + rl,z = Ñ × q rl dS ¶x ¶y ¶z = - kl ( S )

4p

ò

W =0

I l ( S, W )dW -

4p

òs

s ,l

( S, W ) I l ( S, W )dW + 4pkl ( S ) I lb ( S ) (11.36)

W =0

æ 1 4p ö s s ,l ( S , W i ) I l ( S , W i ) ç F l ( W, W i )dW ÷ dW i . + ç 4p ÷ Wi =0 W =0 è ø 4p

ò

ò

The phase function Φλ can be normalized as given by Equation (1.72), (1/ 4p)

ò

4p

W= 0

F l (W, W i ) dW = 1.

The two integrals of the local σs,λ(Ω)Iλ(Ω) and σs,λ(Ωi)Iλ(Ωi) are both overall solid angles, so they combine to zero. For use in the energy Equation (11.1), the radiative source must include the energy

504

Thermal Radiation Heat Transfer

for all wavelengths, so that spectrally integrating the remaining terms of Equation (11.36) yields the following result that is equivalent to the relation in Equation (11.33): 4p é ù 1 k l ( S ) ê I lb ( S ) I l ( S, W ) dW ú dl 4p ê ú W =0 l =0 ë û ¥

Ñ × q r (S ) = 4p

ò

ò

¥ ù é = 4p kl ( S ) éë I lb ( S ) - I l ( S ) ùû dl = 4 ê k P sT 4 (S ) - p kl ( S ) I (S ) dl ú . ê ú l =0 l =0 ë û ¥

ò

(11.37)

ò

This relation for ∇·qr is valid for both anisotropic and isotropic scattering. Although the scattering coefficient is not explicitly evident in Equation (11.37), Iλ is a function of the scattering as it is obtained by integrating Equation (11.14) over Ω and λ, where Iˆl is from Equation (11.11) and depends on ωλ. The ∇·qr in Equation (11.37) is in terms of the mean incident intensity, but it can be written in terms of the source function, which makes the scattering albedo evident. For isotropic scattering (and no dependence of σs,λ on incidence angle), Equation (11.11) relates Iˆl (tl )and I l (tl ) by

Iˆl (tl ) = (1 - wl )I lb (tl ) + wl I l (tl ); wl =

ss,l = wl (tl ). (11.38) kl + ss,l

The I l ( tl ) is then eliminated by combining Equations (11.37) and (11.38) to obtain ∇·qr in terms of Iˆl ( tl ) , ¥



Ñ × q r (S ) = 4p

k l ( tl )

ò w (t )

l =0

l

l

é I lb ( tl ) - Iˆl ( tl ) ù dl; tl (S ) = ë û

S

ò b (S*)dS* (11.39) l

S *= 0

where the source function Iˆl ( tl ) is obtained by solving the integral Equation (11.20). Consider the special case of radiative propagation in a purely scattering medium that does not absorb or emit. Then, κλ = 0, and Equations (11.37) and (11.39) show that for any type of scattering, in the absence of absorption, Ñ × q r = 0. (11.40)



Without absorption there is no energy being locally absorbed or emitted, so there is no radiative heat source. Energy is being redirected only by scattering. In this limit, Equation (11.5) cannot apply; there must be conduction and/or convection to yield a steady temperature distribution. Also, in the limit of no scattering (ωλ = 0), Equation (11.39) is indeterminate, and Equation (11.37) applies.

11.5 SUMMARY OF RELATIONS FOR RADIATIVE TRANSFER IN ABSORBING, EMITTING, AND SCATTERING MEDIA 11.5.1 Energy Equation The energy equation for a single-component translucent medium with a radiative internal energy  is source –∇ · qr, and other internal heat sources q,

rcp

DT DP = bT + Ñ × (kÑT - q r ) + q + F d . (11.41) Dt Dt

505

Fundamental Radiative Transfer Relations

11.5.2 Radiative Energy Source For a medium with or without scattering, the radiative source term for the energy equation is absorbed minus emitted energy, 4p é ù 1 k l ( tl ) ê I l (tl , W i ) dW i - I lb ( tl ) ú dl; ú ê 4p l =0 ë Wi =0 û ¥

-Ñ × q r (S ) = 4p

ò

ò

(11.42)

S

ò b (S*) dS *.

tl ( S ) =

l

*

S =0

For pure scattering of any type (no absorption), the radiative energy source is zero, -Ñ × q r (S ) = 0. (11.43)



For isotropic scattering with absorption, the radiative energy source can be obtained from the temperature and source-function distributions by evaluating ¥



-Ñ × q r (S ) = 4p

ò

l =0

k l ( tl ) ˆ é I l ( tl ) - I lb ( tl ) ù dl; tl (S ) = û wl ( tl ) ë

S

ò b (S*)dS*. (11.44) l

S *= 0

For absorption only without scattering, ωλ = 0, and Equation (11.44) becomes singular, so Equation (11.42) should be used.

11.5.3 Source Function The source function for isotropic scattering with σs,λ independent of Ωi is obtained by solving an integral equation that requires the temperature distribution and the radiative boundary condition Iλ(τλ,i = 0, Ωi) for all paths τλ,i in the radiation field (see Figure 11.1), tl ,i 4p é ù -( tl ,i - t*l ,i ) 1 - tl ,i ˆ ˆ ê * * I l ( tl,i = 0,W i )e dW i + I l tl,i e d tl,i ú I l ( tl ) = (1 - wl ) I lb ( tl ) + wl ê 4p ú tl ,i *= 0 ë Wi =0 û

ò ( )

ò

(11.45) where ωλ = ωλ(τλ) and τλ and τλ,i are optical paths to the same location, as shown in Figure 11.1. The τλ,i account for all paths to τλ through the medium. For absorption only (no scattering), ωλ = 0, and the source function is the local blackbody intensity Iˆl ( tl ) = I lb ( tl ) . (11.46)



For scattering only (no absorption), ωλ = 1, and Equation (11.45) becomes

1 Iˆl ( tl ) = 4p

4p

ò

Wi = 0

I l ( tl,i = 0,W i ) e

tl ,i - tl ,i

dWi +

( ò Iˆ ( t* ) e l

tl ,i * = 0

l,i

- tl ,i - t*l ,i

) d t* . (11.47) l ,i

506

Thermal Radiation Heat Transfer

11.5.4 Radiative Transfer Equation If scattering is independent of incidence direction (as for most atmospheric, combustion, and other engineering systems; however, see Section 10.8 for situations where this is not the case), the change of intensity along a path is given by the RTE in the form s s ,l ( S ) dI l = -bl ( S ) I l ( S ) + kl ( S ) I lb ( S ) + dS 4p



4p

òI

l

( S, Wi ) F l ( S, W, Wi ) dWi . (11.48)

Wi =0

An integrated form of the RTE is then

ò Iˆ ( t* ,W ) e

tl

I l (tl ,W) = I l (0,W)e



- tl

+

l

l

- (tl - t*l )

d t*l . (11.49)

t*l = 0

11.5.5 Relations for a Gray Medium In addition to isotropic scattering, if the simplification is made that the medium is gray, the radiative energy source Equation (11.44) and source function Equation (11.45) reduce to -Ñ × q r (S ) = 4p



k ( t) ˆ é I ( t ) - I b ( t ) ù = 4 k(S ) é pIˆ(S ) - sT 4 (S ) ù (11.50) û û w ( t) ë w(S ) ë

where S

t(S ) =



ò b(S*) dS *

S *= 0



ti 4p é ù -( ti - t*i ) - ti ˆI ( t ) = (1 - w) I b ( t ) + w ê 1 I (ti = 0, W i )e dW i + Iˆ t*i e d t*i ú (11.51) ê 4p ú t*= 0 ë Wi =0 û

ò

ò ( )

where ω = ω(τ). For ω → 0 (no scattering), Equation (11.50) is singular, so Equation (11.42) (which is valid for ω ≥ 0) can be used:

4p é ù 1 ê -Ñ × q r (S ) = 4pk ( t ) I (t, W i )dW i - I b ( t ) ú = 4 k ( t ) éë pI ( S ) - sT 4 ( S ) ùû (11.52) ê 4p ú ë Wi =0 û

ò

where t(S ) =

ò

S

S *= 0

k ( S * ) dS *. For a uniform extinction coefficient β, Equation (11.51) can be writ-

ten in terms of the physical position S along a path as



Si 4p é ù -b( Si - Si* ) - b Si ˆI ( S ) = (1 - w) sT 4 ( S ) + w ê 1 I (Si = 0, W i )e dW i + b Iˆ S*i e dSi* ú (11.53) ê 4p ú p S *= 0 ë Wi =0 û

ò

ò ( )

where the Si are all incident paths to location S. In Chapter 12, the basic equations are further developed for specific geometries such as a plane layer, a two-dimensional (2D) rectangular volume, and a cylinder. Some solutions are obtained.

507

Fundamental Radiative Transfer Relations

This clarifies the application of the relations given here and their simultaneous solution subject to both radiative (for determining I and Iˆ ) and thermal (for determining T) boundary conditions for physical situations.

11.6 TREATMENT OF RADIATION TRANSFER IN NON-LTE MEDIA Section 9.3.2 has a discussion of the conditions when local thermodynamic equilibrium (LTE) conditions might not occur. In this book, we generally assume that LTE exists and that the emission from dV is governed by Equation (1.62). However, the remainder of this section briefly examines some engineering approximations that allow basic non-LTE radiation problems to be approached. Non-LTE cases have been treated by using gray gas and emission-dominated approximations (Vincenti and Kruger 1986). A compromise between complete non-LTE calculations (which require detailed balancing among all energy states based on transition rates between them) and the assumption that LTE exists is that a different local equilibrium temperature is applied to each energy mode (translational, vibrational, rotational, electronic, and electron plasma). Then, local properties can be approximately calculated for each mode of the various chemical species present in the gas. This is still a formidable computational task when multiple chemical species and their forms (atoms, molecules, ions, and electrons) are present. Tofurther reduce the complexity, a two-temperature model of local states is often used. Usually, Te (r ) , the electron temperature, is assumed to apply to the vibrational and electronic energy modes of heavy particles (atoms, molecules, and ions) as well  as to the translational energy distribution of the local electron plasma, and Tr (r ), the rotational (and translational) temperature, applies to all rotational line transitions of heavy particles. This assumption allows k-distribution calculation of spectral properties based on only two local temperature states and greatly reduces the computational load. This model reflects the physical observation that energy distributions controlling emission from a lower temperature medium are governed by the translational temperature and the associated closely spaced rotational–vibrational states; these are chiefly influenced by local collisions among molecules. This temperature thus affects emission from the medium. In a non-LTE case, the electronic and bound–free (ionization) states are assumed to be governed by the electron temperature Te, which is influenced by high-energy radiation from more remote locations, causing a higher effective temperature for these distributions, chiefly influenced by radiation absorption. Such a model is clearly not exact, but does capture first-order non-LTE effects. For the particular case of space-to-atmosphere re-entry radiation, this or a similar approach is used by Lamet et al. (2008, 2010), Bansal et al. (2009a, b, 2010a, b, 2011a, b), Sohn et al. (2010), and Maurente et al. (2012). Wilbers et al. (1991) and Trelles et al. (2007) apply a two-temperature model to analyze radiation in an argon arc plasma torch. Bansal (2011b) and Maurente present alternative formulations of the non-LTE RTE applicable to spacecraft re-entry conditions. Both assume that a multiscale full spectrum k-distribution (see Section 9.4.2.3) can be used for the medium properties (Howell 2014). In the Bansal approach, a separate k-distribution-based RTE is written for each species m:

dI gne

b,m

ds

(

= NC I bne,m - I gne

b,m

)

m = 1,2,3,…, M (11.54)

where N is the number density of the absorbing species C is the absorption cross-section (and thus NC = κ) I gne is the intensity for species m at the cumulative distribution function (CDF) value g of the b,m k-distribution ne I b,m is the non-LTE black intensity for species m, based on the temperature of that species.

508

Thermal Radiation Heat Transfer

In this approach, the nonequilibrium effects are contained in the source term intensity, I bne,m . If there is no spectral overlap between the species and the absorption characteristics of a given species are unaffected by the presence of the others, then the total intensity is obtained after solution of the RTE for each species by summation of I gne over m and g. b,m Maurente et al. proposed an alternative form of the RTE for non-LTE atmospheric re-entry conditions, which after some manipulation becomes

å(

)

é M ù NC = g gne I bne,m - I g j ú m = 1,2,3,…, M . (11.55) ê ds êë m =1 b,m úû

dI g j



Here, the g gne is a factor that relates the CDF of the k-distribution for a given species to a reference b,m

CDF, and the j subscript indicates evaluation over the joint cumulative k-distribution for all species (rather than for a single species). This equation need be solved only once rather than for each species, as for the Bansal approach, and I is then found by integrating overall g. Both Bansal and Maurente compare their method with line-by-line calculations for test cases. Further details can be found in the references, as they are outside the scope of this book.

11.7 NET-RADIATION METHOD FOR ENCLOSURES FILLED WITH AN ISOTHERMAL MEDIUM OF UNIFORM COMPOSITION In this and many of the following sections, treatment is limited to non-scattering media. Most of the simplified methods presented do not lend themselves to including scattering, although some attempts have been made to include isotropic scattering. The radiation-exchange equations were developed in Section 6.2 for an enclosure that does not contain an absorbing–emitting medium and has diffuse surfaces with spectrally dependent properties. This section expands that treatment to include an isothermal absorbing medium between the surfaces. Since the absorption properties of gases and other absorbing media are almost always strongly wavelength dependent, the present development is carried out for a differential wavelength interval. Integrations over all wavelengths or over a k-distribution then yield the total radiative behavior. It is assumed that surface directional property effects are sufficiently unimportant in the desired results that surfaces can be treated as diffuse emitters and reflectors. In gas-filled enclosures, such as in industrial furnaces or engine combustion chambers, there is often sufficient mixing that the entire gas is essentially isothermal and of uniform composition. In this instance, the analysis is simplified, as it is not necessary to compute or specify the gas temperature distribution. Sometimes, the uniform gas temperature can be found from the governing energy balances. Even with this uniform gas simplification, a detailed spectral radiation exchange computation between the gas and bounding surfaces can become quite involved. Consider an enclosure of N surfaces, each at a uniform temperature, as in Figure 11.9. Typical surfaces are designated by subscripts j and k. Some of the surfaces can be open boundaries such as a window or hole; these are usually modeled as a perfectly absorbing (black) surface at the temperature of the surroundings outside the opening. The enclosure is filled with an absorbing–emitting medium, such as a radiating gas at uniform temperature Tg. The quantity Qg is the energy rate that it is necessary to supply by means other than radiation to the entire absorbing–emitting medium to maintain its uniform temperature. A common source for Qg is combustion. If in a problem solution, the Qg is found to be negative, the medium is gaining a net amount of radiative energy from the enclosure boundaries, and energy must be removed from the medium to maintain it at its steady temperature Tg. The Qg is analogous to the Qk at a surface, which is the energy rate supplied to area Ak by means other than radiation inside the enclosure. Enclosure theory yields equations relating Qk and Tk for each surface to Qg and Tg for the gas or other absorbing uniform isothermal medium filling the enclosure. Considering all the surfaces and

Fundamental Radiative Transfer Relations

509

FIGURE 11.9  Enclosure composed of N discrete surface areas and filled with uniform medium at Tg (enclosure shown in cross-section for simplicity)

FIGURE 11.10  Spectral energy quantities incident on and leaving a typical surface area of enclosure

the medium, if half of the Q and T are specified, there are sufficient radiative heat balance equations to solve for the remaining unknown Q and T values. Either Q or T must be specified for each boundary surface, or the final matrix of equations may become singular. Further, at least one boundary temperature or the medium temperature must be specified to anchor the solution. Problems where an unspecified boundary condition is to be found are addressed in Chapter 18. For some applications, the energy input to the gas from external sources Qg is given, and the analysis yields the steady gas temperature Tg. Conversely, if a Tg is specified, the analysis yields the energy that must be supplied to the gas to maintain its temperature. The net radiation method in Chapters 5 and 6 is now extended to include radiation exchange with the medium. At the kth surface of an enclosure (Figure 11.10), an energy balance, as in Equation (7.1), gives

Ql,k dl = ql,k Ak dl = ( J l,k - Gl,k ) Ak dl (11.56)

510

Thermal Radiation Heat Transfer

The Jλ,k dλ and Gλ,k dλ are, respectively, the outgoing and incoming radiative energy fluxes (radiosity and irradiation) in wavelength interval dλ. The Qλ,k dλ is the energy rate supplied to the surface Ak in the wavelength region dλ. The external energy supplied to Ak by some means such as conduction and/or convection is equal to

ò

¥

l =0

Ql,k dl.

The radiosity Jλ,k is composed of emitted and reflected energy, as in Equation (5.11), written spectrally

J l,k dl = el,k (Tk )Elb,k (Tk )dl + rl,k (Tk )Gl,k dl. (11.57)

The functional notation is often omitted in what follows, to shorten the equations. The Eλb,k(Tk) dλ is the blackbody spectral emission at Tk in wavelength region dλ about wavelength λ. The Gλ,k dλ in Equation (11.56) is the incoming spectral flux to Ak. It is the sum of the contributions, from all the surfaces, that reach the kth surface after partial absorption while passing through the intervening medium, plus the contribution by emission from the medium. The equation of transfer allows for both attenuation and emission as radiation passes along a path through a medium. A typical path from Aj to Ak within incident solid angle dΩk is in Figure 11.9. If all such paths and solid angles by which radiation can pass from all the surfaces (including Ak itself if it is concave) to Ak are accounted for, the solid angles dΩk will encompass all of the medium that can radiate to Ak. Thus, by using the equation of transfer to compute the energy transported along all paths between surfaces, the emission by the medium is included. Consider radiation passing from one surface to another through a non-scattering medium, including emission and absorption by the intervening medium. In enclosure theory, Jλ is assumed uniform over each surface. Since the surfaces are assumed to be diffuse, the spectral intensity leaving dAj is Iλo,j = Jλ,j/π. From the RTE given in Equation (11.15), the intensity arriving at dAk after traversing path S is, for a non-scattering medium with uniform temperature and composition (constant κλ throughout the volume),

(

)

I l,i, j - k = I l,o, j e - kl S + I lb,g 1 - e - kl S (11.58)

where the subscript j–k means j to k (not a minus sign). Using the definitions tλ(S) ≡ e -kl S (spectral transmittance of the medium along the path length S) and αλ(S) ≡ 1−e -kl S (spectral absorptance along the path),

I l,i, j -k = tl (S )I l,o, j + a l (S )I lb,g (Tg ). (11.59)

This intensity arriving at dAk in solid angle dΩk provides the energy Iλ,i,j−kdAkcosθkdΩkdλ. Using dΩk = dAjcosθj/S2, the arriving spectral energy is

d 2Qli, j -k dl = éëtl (S )I l,o, j + a l (S )I lb,g (Tg ) ùû

dAk dA j cos qk cos q j dl. (11.60) S2

For a diffuse surface, Jλ,j = πIλ,o,j and Eλb,g = πIλb,g, so

d 2Qli,j -k dl = éëtl (S )J l, j + a l (S )Elb,g (Tg ) ùû dl

cos qk cos q j dAk dA j . (11.61) pS 2

Equation (11.61) is integrated over both Aj and Ak to give the spectral energy transmitted and emitted along all paths from Aj that are incident on Ak:

511

Fundamental Radiative Transfer Relations



Qli, j -k dl =

ò ò éët (S)J l

+ a l (S )Elb,g (Tg ) ùû dl

l, j

Ak A j

cos qk cos q j dAk dA j . (11.62) pS 2

11.7.1 Definitions of Spectral Geometric-Mean Transmission and Absorption Factors The double integral in Equation (11.62) is similar to the double integral in Equation (4.16) for the configuration factor between two surfaces. By analogy, define tl, j -k such that tl, j -k º



1 A j Fj -k

òò

Ak Al

tl (S ) cos qk cos q j dA j dAk (11.63) pS 2

where Fj–k is the configuration factor. For no absorbing medium, tλ(S) = 1, and the double integral on the right side of Equation (11.63) becomes Fj–k so, for no absorption, tl, j -k = 1. For complete absorption in the medium between Aj and Ak, tl, j -k = 0. tl, j -k is the geometric-mean transmittance from Aj to Ak. From the second quantity in brackets in Equation (11.62), a geometric-mean absorptance a l, j -k is defined, a l, j -k º



1 A j Fj -k

òò

Ak A j

a l (S ) cos qk cos q j dA j dAk (11.64) pS 2

For a nonabsorbing medium, a l, j -k = 0, while for complete absorption, a l, j -k = 1. From the definitions of tλ and αλ, tl and a l are related by a l, j -k = 1 - tl, j -k . (11.65)

Equation (11.62) becomes

Qli, j -k dl = éë tl (S )J l, j + a l (S )Elb,g (Tg ) ùû A j Fj -k dl. (11.66)

An alternative terminology is also used in which A j Fj -k tl, j -k is the geometric transmission factor and A j Fj -k a l, j -k the geometric absorption factor. To compute the heat exchange in an enclosure, it is necessary to determine each tl and a l ; only one double integration is needed because of relation Equation (11.65).

11.7.2 Matrix of Enclosure Theory Equations For an enclosure with N surfaces bounding a uniform isothermal medium at Tg, the incident spectral energy on any surface Ak is that arriving from all the surrounding surfaces and the enclosed isothermal medium: N



Gl,k Ak =

å éë t (S)J l

l, j

+ a l (S )Elb,g (Tg ) ùû A j Fj -k . (11.67)

j =1

From reciprocity, AjFj−k = Ak Fk−j; so, Ak is eliminated to give N



Gl,k =

å éë t (S)J l

j =1

l, j

+ a l (S )Elb,g (Tg ) ùû Fk - j . (11.68)

512

Thermal Radiation Heat Transfer

Equations (11.56), (11.57), and (11.68) relate Jλ, Gλ, and qλ for each of the surfaces in the enclosure to Eλb for that surface and to Eλb,g. The Gλ is eliminated by combining Equations (11.56) and (11.57) and by substituting Equations (11.56) and (11.68). This yields two equations for Jλ,k and qλ,k for each surface in terms of Eλb,k and Eλb,g, ql,k =



e l ,k ( Elb,k - J l,k ) (11.69) 1 - e l ,k

N

ql,k = J l,k -



å éë t

l, j -k

(S )J l, j + a l, j -k (S )Elb,g (Tg ) ùû Fk - j . (11.70)

j =1

Equation (11.69) is the same as for an enclosure without an absorbing medium; if ελ,k = 1, Equation (11.69) becomes (after multiplying through by 1- el,k ) Jλ,k = Eλb,k. Equations (11.69) and (11.70) are analogous to Equations (5.16) and (5.17) for a gray enclosure with no absorbing medium. From the symmetry of the integrals in Equations (11.63) and (11.64), and from AjFj−k = Ak Fk−j, it is found that tl, j -k = tl,k - j (11.71)

and

a l, j -k = a l,k - j . (11.72)



Then, Equation (11.70) can also be written as N

ql,k = J l,k -



å éë t

l ,k - j

(S )J l, j + a l,k - j (S )Elb,g (Tg ) ùû Fk - j . (11.73)

j =1

As in Section 5.3.1, Equations (11.69) and (11.70) can be further reduced by solving Equation (11.69) for Jλ and inserting it into Equation (11.70). This gives N



å j =1

æ dkj ö 1 - e l ,k - Fk - j tl,k - j ÷ ql, j = ç e e l ,k è l ,k ø

N

å éë(d

kj

- Fk - j tl,k - j )Elb, j - Fk - j a l,k - j Elb,g ùû . (11.74)

j =1

The Kronecker delta is δkj = 1 when k = j, and δkj = 0 when k ≠ j. This equation is analogous to Equation (5.21). If Equation (11.74) is written for each k from 1 to N, a set of N equations is obtained relating the 2N quantities qλ and Eλb for all the surfaces. If the medium temperature (and hence Eλb,g) is known, then one-half of the qλ and Eλb values need to be specified as boundary conditions, and the equations can be solved. (If both qλ and Eλb are specified on the same surface, the solution may become indeterminate. Such problems are addressed in Chapter 18.) To find the total energy quantities, the equations must be solved in many wavelength intervals, and integration of each energy quantity then performed over all wavelengths. This is the same as for the band equations in Section 6.2.2. If Tg is unknown, an additional equation is needed, as given in the next section.

11.7.3 Energy Balance on a Medium An energy balance on the medium relates the medium temperature, Tg, and the energy that is supplied to the medium, Qg, by means other than radiative exchange within the enclosure. From an energy balance on the entire enclosure, the energy that must be supplied to the medium, for example, by combustion, is equal to the net energy escaping from all the N boundary surfaces,

513

Fundamental Radiative Transfer Relations ¥

N

Qg = -



åA ò q k

k =1

l ,k

dl. (11.75)

l =0

This can be evaluated after the qλ,k are found for each surface from Equation (11.74) in a sufficient number of wavelength intervals. These equations can be solved if Tg, and hence Eλb,g is known. If Tg is unknown, and instead Qg is specified, a less direct solution is required. The Tg is guessed and Qg is found. The solution is iterated on Tg and the resulting relation between Qg and Tg yields the Tg corresponding to the specified Qg. Example 11.3 Consider a well-mixed gas at Tg completely surrounded by a single wall at uniform temperature T1 such as in a simple cooled combustion chamber. The heat transfer from the gas to the wall is to be found in terms of T1 and Tg. For an enclosure with a single wall, Equation (11.74) yields

æ 1 ö 1- el ,1 - F1-1 tl ,1-1 ÷ ql ,1 = (1- F1-1tl ,1-1)Elb,1 - F1-1a l ,1-1Elb, g . ç e e l ,1 è l ,1 ø

Using F1−1 = 1 and 1 - tl ,1-1 = a l ,1-1, this equation is simplified and then integrated over all λ to obtain the energy rate added to the gas that is transferred to the wall, ¥

Qg = -Q1 = - A1



ò

¥

ql ,1d l = A1

l =0

ò 1/ e 0

Elb, g - Elb,1 d l. (11.3.1) + 1 / a l ,1-1 - 1

l ,1

Example 11.4 Obtain relations for the energy transfer in an enclosure of two infinite parallel plates at temperatures T1 and T2 bounding a well-mixed gas at uniform temperature Tg. Equation (11.74), applied to a two-surface enclosure, gives for k = 1 and 2 (note that F1–1 = F2–2 = 0):



1 1- el,2 ql ,1 - F1- 2 tl ,1- 2ql ,2 = Elb,1 - F1- 2tl ,1- 2Elb,2 - F1- 2a l ,1- 2Elb, g (11.4.1) e l ,2 el ,1 -F2-1

1- el ,1 1 tl ,2-1ql ,1 + ql ,2 = -F2-1tl ,2-1Elb,1 - F2-1a l ,2-1Elb,2 + Elb, g . (11.4.2) el ,1 e l ,2

For infinite parallel plates, F1–2 = F2–1 = 1, and from Equations (11.71) and (11.72), tl ,2-1 = tl ,1- 2 and a l ,2-1 = a l ,1- 2 . For simplicity, the numerical subscripts on t and a are omitted. Then, Equations (11.4.1) and (11.4.2) become 1 1- e l , 2 ql ,1 tlql,2 = Elb,1 - tlElb,2 - a lElb, g (11.4.3) e l ,2 el ,1





-

1- el ,1 1 tlql ,1 + ql ,2 = tlElb,1 + Elb,2 - a lElb, g . (11.4.4) el ,1 e l ,2

Equations (11.4.3) and (11.4.4) are solved for qλ,1 and qλ,2. After using the relation a l = 1 - tl , this results in

514

Thermal Radiation Heat Transfer

{

}

{

}

ql ,1 =

1 e l ,1e l ,2tl (Elb,1 - Elb,2 ) + e l ,1(1- tl ) éë1+ (1- el ,2 )tl ùû (Elb,1 - Elb, g ) (11.4.5) 1- (1- el ,1)(1- el,2 )tl2

ql ,2 =

1 e l ,1e l ,2tl (Elb,2 - Elb,1) + e l ,2(1- tl ) éë1+ (1- el ,1)tl ùû (Elb,2 - Elb, g ) . (11.4.6) 1- (1- el ,1)(1- el,2 )tl2

The total energy fluxes added to surfaces 1 and 2 are ¥

q1 =



ò

¥

ql ,1d l

and

q2 =

l =0

òq

d l. (11.4.7)

l ,2

l =0

The total energy added to the gas to maintain its temperature Tg is equal to the net energy leaving the parallel plates. Hence, per unit area of the plates, qg = -(q1 + q2 ). (11.4.8)



When the medium between the plates does not absorb or emit radiation, then tl = 1, and Equations (11.4.5) and (11.4.6) reduce to Equation (5.5). With an absorbing radiating gas, the numerical integration of Equations (11.4.5) and (11.4.6) over all λ to obtain q1 and q2 is difficult because of the very irregular variations of the gas absorption coefficient with λ; some methods for avoiding the detailed integrations were discussed in Chapter 9.

11.7.4 Spectral Band Equations for an Enclosure One approach for integrating over λ is to divide the spectrum into bands in which the gas is either absorbing or essentially nonabsorbing. k-distribution methods or the spectral line-based weighted sum of gray gases approaches for computing band properties (Chapter 9) can be used. For situations where both the geometry and thermal boundary conditions are simple, the enclosure equations can be solved in closed form as in Examples (11.3) and (11.4). In Equation (11.3.1), for example, if a l » 0 within a wavelength region, this nonabsorbing band does not contribute to Qg. Then, if l designates an absorbing band, Qg = A1



æ

Elb,g - Elb,1 ö ÷ Dl l . (11.76) l ,1 + 1/a1-1 - 1 øl

å çè 1/e l

The a1-1 for each band is found from one of the methods in Chapter 9. Usually, the bandwidth is small, so that Eλb,g, Eλb,1, and Eλ,1 can be considered constant over each band. If the Eλb variation is significant over a band, the integrated blackbody functions can be used, as in Example 6.2. For more complex enclosure geometries, Equation (11.74) is used. For a band of width Δλ, the integration of Equation (11.74) over the band gives N



òå

Dl j =1

æ dkj ö 1 - el, j - Fk - j tl,k - j ÷ ql, j dl = ç el, j è el, j ø Dl

N

ò å éë( d j =1

kj

- Fk - j tl,k - j ) Elb, j - Fk - j a l,k - j Elb,g ùû dl. (11.77)

It is assumed that the bands are sufficiently narrow that ql, j , el, j , tl,k - j , a l,k - j , Elb, j , and Elb,g can be regarded as constants over the bandwidth, being characteristic of some mean wavelength within the band or, in the case of t and a, being averaged over the band. Then, Equation (11.77) is written for band l as

515

Fundamental Radiative Transfer Relations N



å j =1

æ dkj ö 1 - el , j - Fk - j tl ,k - j ÷ ql , j = ç el , j è el , j ø

N

å éë( d

kj

j =1

- Fk - j tl ,k - j ) Elb, j - Fk - j al ,k - j Elb,g ùû . (11.78)

In a spectral region where the gas can be approximated as nonabsorbing, tl = 1 and a l,k–j = 0, and Equation (11.78) reduces to N

å



j =1

æ dkj 1 - el , j ö - Fk - j ç ÷ ql , j = el , j ø è el , j

N

å(d

kj

- Fk - j )Elb, j . (11.79)

j =1

Equations (11.78) and (11.79) provide simultaneous equations for ql in each band at each boundary. The al ,k - j in Equation (11.78) is found from Equation (11.64) by taking an integrated average over the band:



a l ,k - j

1 = Ak Fk - j

òò

é ê(1/Dl l ) ë

ò

Dll

Al Ak

ù a l (S )dl ú cos q j cos qk û dAk dA j (11.80) pS 2

For each band, tl ,k - j = 1 - al ,k - j , and only a single evaluation is needed: a l ,k - j =



1 Ak Fk - j

òò

al ( S ) cos q j cos qk

Al Ak

pS 2

dAk dA j (11.81)

where αl(S) is the integrated band absorption, al (S ) =



1 Dl l

1 ò a (S)dl = Dl ò (1 - e ) dl. (11.82) - kl S

l

l

Dll

Dll

If desired, the αl can be expressed from Equation (9.26) in terms of the effective bandwidth as al (S ) =



Al (S ) . (11.83) Dl l

A detailed development is given by Nelson (1984) of the band relations for a nongray isothermal gas in an enclosure with diffuse walls. The net radiation enclosure equations are examined in detail in Stasiek (1998) for an enclosure filled with a uniform gas. An extension is made for some of the walls being windows so that the enclosure interior can interact directly with its surrounding environment.

11.7.5 Gray Medium in a Gray Enclosure If a gas contains many suspended particles or droplets, it may be reasonable to neglect spectral variations of the properties of the suspension. In addition, the walls in a furnace or combustion chamber may be partially soot covered, so an examination of the radiation transfer with gray boundaries may yield accurate results. From Example 11.3, if a gray gas at Tg is bounded by a chamber consisting of one gray wall with area A1 at T1, the energy supplied to the gas by combustion or other nonradiative means and transferred to the wall is

Qg =

(

A1s Tg4 - T14

)

1/e1 + 1/a1-1 - 1

=

e1a1-1 A1s Tg4 - T14 . (11.84) 1 - (1 - e1 )(1 - a1-1 )

(

)

516

Thermal Radiation Heat Transfer

From Example 11.4, for a gray medium between infinite gray parallel plates, the q1 from Equation (11.4.5) becomes

q1 =

(

)

(

e1e2 t s T14 - T24 + e1 (1 - t ) éë1 + (1 - e2 ) t ùû s T14 - Tg4 1 - (1 - e1 )(1 - e2 ) t

2

) , (11.85)

and similarly for q2 from Equation (11.4.6). Now, consider a chamber made up of two finite gray surfaces that enclose a gray medium. From Equation (11.78),





æ1 ö 1 - e1 1 - e2 ç e - F1-1 e t1-1 ÷ q1 - F1-2 e t1-2 q2 (11.86) 1 2 è 1 ø 4 4 4 = (1 - F1-1 t1-1 ) sT1 - F1-2 t1-2 sT2 - ( F1-1a1-1 + F1-2 a1-2 ) sTg - F2 -1

ö æ 1 1 - e1 1 - e2 t2 -1q1 + ç - F2 -2 t2 -2 ÷ q2 e1 e e 2 è 2 ø

(11.87) = - F2 -1 t2 -1sT + (1 - F2 -2 t2 -2 ) sT - ( F2 -1a 2 -1 + F2 -2 a 2 -2 ) sT . 4 1

4 2

4 g

If T1, T2, and Tg are specified, these equations can be solved for q1 and q2. Then, the energy supplied to the medium by some means other than radiation within the enclosure is Qg = −(q1A1 + q2 A2). If one of the walls (wall 2) is adiabatic (e.g., an uncooled refractory wall), then q2 = 0. The T2 can be eliminated from Equations (11.86) and (11.87) and the result solved for q1 in terms of T1 and Tg to yield -1

æ 1 - e1 ö q1 A1 1 =ç + ÷ (11.88) 4 4 A e 1 / f 1 / ( f f ) + + s T1 - Tg 12 1g 2g ø è 1 1



(

)

where f1g = éë A1 ( F1-1a1-1 + F1-2 a1-2 ) ùû f12 = ( A1F1-2 t1-2 )



-1

-1

-1

f2g = éë A2 ( F2 -1a 2 -1 + F2 -2 a 2 -2 ) ùû . Returning to Equations (11.86) and (11.87), if the medium is in radiative equilibrium (Qg = 0), then the two walls are exchanging energy through a well-mixed medium that will come to an equilibrium temperature. Then Q2 = −Q1, so that q2 = −q1A1/A2. q2 is then eliminated from Equations (11.86) and (11.87) and the two equations are combined to eliminate Tg. The result is

æ 1 - e1 q1 A1 1 1 - e2 =ç + + 4 4 ç s T1 - T2 è A1 e1 1/f12 + 1/ ( f1g + f2 g ) A2 e2

(

)

-1

ö ÷ (11.89) ÷ ø

where the f quantities are defined in Equation (11.88). With Q1 = q1A1 known and q2 = −q1A1/A2, Equations (11.86) and (11.87) can be solved for the temperature of the medium if desired. For the general case of a well-mixed gray medium in a gray enclosure where there are more than a few surfaces, a closed-form algebraic equation often cannot be obtained. Rather, the enclosure

517

Fundamental Radiative Transfer Relations

simultaneous equations would be solved numerically. For the general situation, Equation (11.78) is written for the gray case (one spectral band) as N



å j =1

æ dkj ö 1- e j - Fk - j tk - j ÷ q j = ç e e j è j ø

N

å(d

kj

- Fk - j tk - j ) sT j4 - Fk - j a k - j sTg4 . (11.90)

j =1

11.8 EVALUATION OF SPECTRAL GEOMETRIC-MEAN TRANSMITTANCE AND ABSORPTANCE FACTORS To compute thermal variable values from the enclosure equations, the property and geometric quantities t and a or AF t and AFa must be evaluated. These quantities depend on both geometry and wavelength. The integrations over the surface areas of certain geometries can be carried out analytically using the defining Equations (11.63), (11.64), and the relation of Equation (11.65). The derivation of some of the resulting algebraic relations is provided in Appendix B, Derivation of Geometric Mean Beam Length Relations, contained on the book’s website www.ThermalRadiation. net. Factors are evaluated for the hemisphere to the differential area at the center of its base, the top of the right circular cylinder to the center of its base, the side of the cylinder to the center of its base, the entire sphere to any area on its surface or to its entire surface, the infinite plate to any area on the parallel plate, and the rectangle to the directly opposed parallel rectangle. The results are summarized in Table 11.1. Also in the web Appendix B, Section B.7 Geometric-Mean Beam Length for Spectral Band Enclosure Equations, the derivation is given for the integrated average mean beam length between pairs of surfaces assuming that the integrated band absorption αl(S) varies linearly between the surfaces. The geometric-mean beam length can be used as the length in the effective bandwidth correlations; the methodology for this approach is in web Appendix B. Algebraic equations and tables of values for geometric-mean beam lengths from Dunkle (1964) are given for directly opposed parallel equal rectangles and for rectangles at right angles sharing a common edge, allowing computation for radiative transfer in rectangular parallelepipeds containing an absorbing/emitting isothermal medium.

518

Thermal Radiation Heat Transfer

11.9 MEAN BEAM LENGTH APPROXIMATION FOR SPECTRAL RADIATION FROM AN ENTIRE VOLUME OF A MEDIUM TO ALL OR PART OF ITS BOUNDARY Some practical situations require evaluating the energy radiated from a volume of isothermal medium with uniform composition to all or part of its boundaries, without considering emission and reflection from the boundaries. Such an approach allows assessing the relative importance of radiation in a complex problem before embarking on a more comprehensive computation. An example is radiation from hot furnace gases to walls that are cool enough for their emission to be small and that are rough and soot covered so they are essentially nonreflecting. In this section, the energy is considered in dλ; in the next section, an integration will include all λ to provide the total radiant energy. For the specified conditions, the Jλ,j dλ in Equation (11.67), which is the spectral outgoing energy flux from Aj, is zero. The incoming spectral energy at Ak is then N

Ak Gl,k dl =



åE

lb,g

dlA j Fj -k a l, j -k . (11.91)

j =1

If the geometry is a hemisphere of medium with radius R radiating to an area element dA k at the center of its base, Equation (11.91) has an especially simple form. Since the hemispherical boundary is the only surface in view of dAk, Equation (11.91) reduces to dAk Gl,k = Elb,g A j dFj -dk a l, j -dk = Elb,g dAk Fdk - j a l, j -dk (11.92)



where a l, j -dk = 1 - tl, j -dk = 1 - e - kl R . For radiation between dAk and the surface of a hemisphere, Fdk–j = 1, so Equation (11.92) reduces to the following simple expression giving the incident energy flux from a hemisphere of medium emitting to the center of the hemisphere base:

(

)

Gl,k dl = 1 - e - kl R Elb,g dl. (11.93)



(

)

The term 1- e - kl R is the spectral emittance of the gas εk(T, P, R) for path length R. Then, Equation (11.93) becomes

Gl,k dl = el ( kl R )Elb,g dl

el ( kl R ) = 1 - e -kl R . (11.94)

The gas emittance and thus the incident spectral energy depend on the optical radius of the hemisphere, κλR. It would be very convenient if a relation having the simple form of Equation (11.94) could be used to determine the value of Gλ,k dλ on Ak for any geometry of radiating medium volume and for Ak being all or part of its boundary. Because the geometry of the medium enters Equation (11.93) only through ελ(κλR), we can define a fictitious equivalent value of R, say Le, that would yield a value of ελ(κλLe) such that Equation (11.94) would give the correct Gλ dλ for a given geometry. This fictitious length Le is called the mean beam length. Then, for an arbitrary geometry of gas, let

(

)

Gl,k dl = el ( kl Le )Elb,g dl = 1 - e -kl Le Elb,g dl. (11.95)

The mean beam length is the required radius of an equivalent hemisphere of a medium that radiates a flux to the center of its base equal to the average flux radiated to the area of interest by the actual volume of the medium.

519

Fundamental Radiative Transfer Relations

11.9.1 Mean Beam Length for a Medium between Parallel Plates Radiating to Area on Plate Consider two black infinite parallel plates at T1 = T2 = 0 separated by a distance D. The plates enclose a uniform medium at Tg with absorption coefficient κλ. The rate at which spectral energy is incident upon Ak on one plate is, from Equation (11.91),

Gl,k Ak dl = Elb,g dlA j Fj -k a l, j -k = Elb,g dlA j Fj -k éë1 - 2 E3 ( kl D ) ùû (11.96)

where the expression a l, j -k = [1 - 2 E3 ( kl D)] for infinite parallel plates is from Table 11.1 (This is derived in web Appendix B, Equation B.13, available on the book’s website at www.ThermalRadiation. net). Here, E3 is the exponential integral function, tabulated in Appendix D of this book. For infinite parallel plates, Fj−k = 1, and by reciprocity, Fj−k = Ak /Aj, so Equation (11.96) reduces to

Gl,k dl = éë1 - 2 E3 ( kl D ) ùû Elb,g dl. (11.97)

Comparing Equations (11.94) and (11.97) provides the mean beam length for infinite parallel plates as

Le = -

1 ln[2 E3 ( kl D)] (11.98) kl

or, in terms of optical thickness τλ = κλD,

Le 1 = - ln[2 E3 (tl )]. (11.99) D tl

11.9.2 Mean Beam Length for the Sphere of a Medium Radiating to Any Area on Its Boundary Consider a medium within a nonreflecting sphere of radius R with the sphere boundary Aj at Tj = 0. From Equation (11.91), the spectral radiation flux incident on an element dAk is Gl,k = Elb,g ( A j /dAk )Fj -k a l, j -k . For a sphere, Equation (5.65) gives dFj−dk = dAk /Aj. Substituting into Equation (11.92) for a l, j -k from Table 11.1 for this geometry gives

ì ü 2 é1 - (2 kl R + 1)e-2 kl R ù ý . (11.100) Gl,k = Elb,g í1 2 ë û î (2 k l R ) þ

Equating Equations (11.95) and (11.100) gives the mean beam length for this geometry as

ì 2 ü Le 1 é1 - (2 kl R + 1)e -2 kl R ù ý . (11.101) =ln í 2 ë û 2R 2 k l R î (2 k l R ) þ

Because the expression used for a l, j -k is quite general, Equation (11.101) gives the correct mean beam length for the entire sphere of medium radiating to any portion of its boundary. Additional results for spheres are in Koh (1965).

11.9.3 Radiation from the Entire Medium Volume to Its Entire Boundary for Optically Thin Media Because of the integrations involved, the mean beam length for an entire medium volume radiating to all or part of its boundary is usually difficult to evaluate except for the simplest shapes. It is

520

Thermal Radiation Heat Transfer

fortunate that some practical approximations can be found by looking first at the optically thin limit. For a small optical path length κλS, the transmittance becomes

é ù ( k S )2 lim tl = lim e - kl S = lim ê1 - kl S + l - ú ® 1. kl S ® 0 kl S ® 0 kl S ® 0 2! ë û

Each differential volume of the uniform temperature medium emits spectral energy 4κλEλb,gdλdV. Since tλ = 1, there is no attenuation of emitted radiation, and it all reaches the enclosure boundary. For the entire radiating volume, the energy reaching the boundary is 4κλEλb,gdλV, so the average spectral flux received at the entire boundary of area A is, in the optically thin limit,

Gl = 4 kl Elb,g

V . (11.102) A

By use of the mean beam length, the average spectral flux reaching the boundary is given by Equation (11.93). For the optically thin case, let Le be designated by Le,0. Then, from Equation (11.93), for small κλLe,0,

ìï é ù üï ( k L )2 Gl = í1 - ê1 - kl Le,o + l e,o - ú ý Elb,g » kl Le,0 Elb,g . (11.103) 2! ïî ë û ïþ

Equating this to Gλ in Equation (11.102) gives the desired result for the mean beam length of an optically thin medium radiating to its entire boundary: Le,0 =



4V . (11.104) A

To give a few examples, for a sphere of diameter D,

Le,0 =

4Vs 4pD3 / 6 2 = = D. (11.105) As 3 pD 2

For an infinitely long circular cylinder of diameter D,

Le,0 =

4pD 2 / 4 = D. (11.106) pD

For a medium between infinite parallel plates spaced D apart,

Le,0 =

4Vg 4 DA = = 2 D. (11.107) As 2A

For an infinitely long rectangular parallelepiped with cross-sectional dimensions h and w,

Le,0 =

4hw 2hw = . (11.108) 2(h + w) h + w

11.9.4 Correction to Mean Beam Length When a Medium Is Not Optically Thin For a medium that is not optically thin, it would be convenient if Le could be obtained by applying a simple correction factor to the Le,0 from Equation (11.104). A useful technique is to introduce a correction coefficient C so that Le is given by

521

Fundamental Radiative Transfer Relations

Le = CLe,0 . (11.109)



Then, the incoming spectral flux in Equation (11.95) is

(

)

Gl dl = 1 - e - klCLe,0 Elb,g dl. (11.110)

Consider a radiating medium between infinite parallel plates spaced D apart. Using Equation (11.107) in Equation (11.110) gives Gλdλ = [1−exp(−2CκλD)]Eλb,gdλ. From Equation (11.97), the actual flux received is Gλdλ = [1−2E3(κλD)]Eλb,gdλ. To compare these fluxes, the ratio [1−2E3(κλD)]/ [1−exp(−2CκλD)] is shown in Figure 11.11 for a range of κλD using C = 0.9. This value of C was found to yield a ratio close to 1 for all κλD, and hence is a valid correction coefficient for this geometry. Table 11.2 gives the mean beam length Le,0 for various geometries, along with Le values that provide reasonably accurate radiative fluxes for nonzero optical thicknesses. The values of C are found for each case to be in a range near 0.9 (Hottel 1954, Eckert and Drake 1959, Hottel and Sarofim 1967). Hence, it is historically recommended that, for a geometry for which exact Le values have not been calculated, the approximation

Le » 0.9Le,0 » 0.9

4V (11.111) A

be used for an entire uniform isothermal medium volume radiating to its entire boundary. It is shown in Cartigny (1986) that the concept of mean beam length can also be used for scattering media, and Equation (11.111) applies in the limit of pure scattering and small optical thickness.

11.10 EXCHANGE OF TOTAL RADIATION IN AN ENCLOSURE BY USE OF MEAN BEAM LENGTH The mean beam length was obtained in the previous section at one wavelength. This concept is now applied to obtain the exchange of total energy within an enclosure. The use of the mean beam length simplifies the geometric considerations, but it remains to integrate the spectral relations to obtain total energy transfer.

11.10.1 Total Radiation from the Entire Medium Volume to All or Part of Its Boundary The mean beam length was found to be approximately independent of κλ, as evidenced by Equation (11.111). This means that Le can be used as a characteristic dimension of the gas volume and

FIGURE 11.11  Ratio of emission by layer of medium to that calculated using a mean beam length, L e = 1.8D. Deviation from 1.00 corresponds to an error in the approximation

522

Thermal Radiation Heat Transfer

(Continued)

523

Fundamental Radiative Transfer Relations

approximated as constant during integration over wavelength. The total heat flux from the medium that is incident on a surface is found by integrating Equation (11.95) over all λ: ¥

G=



ò (1 - e

- kl Le

)Elb,g dl (11.112)

l =0

where Le is independent of λ. Now, define a total emittance εg for the medium such that G = eg sTg4 . (11.113)

Equating the last two relations gives ¥

eg



ò (1 - e ) E = -kl Le

l =0

sTg4

lb,g

dl . (11.114)

The εg in Equation (11.114) is a convenient quantity that can be provided for each medium in terms of Le and Tg. Values of εg are available for the important radiating gases, and also analytical forms are available that are convenient for computer use; these results for εg are in Section 9.4. Then, for a particular geometry and medium temperature and pressure, the εg is applied by use of Equation (11.113). An example illustrates how εg is obtained and used. Example 11.5 A cooled right cylindrical tank 4 m in diameter and 4 m long has a black interior surface and is filled with hot gas at a total pressure of 1 atm. The gas is composed of CO2 mixed with a transparent gas that has a partial pressure of 0.75 atm. The gas is uniformly mixed at Tg = 1100 K. Compute

524

Thermal Radiation Heat Transfer

how much energy must be removed from the tank surface to keep it cool if the tank walls are all at low temperature so that only radiation from the gas is significant. The geometry is a finite circular cylinder of gas, and the radiation to its walls will be computed. Using Table 11.2, the corrected mean beam length is Le = 0.60D = 2.4 m. The partial pressure of the CO2 is 0.25 atm. Using the Alberti et al. (2018) spread sheet from https​://do​i.org​/10.1​016/j​.jqsr​t.201​ 8.08.​008, eCO2 ( pCO2Le ,Tg ) = 0.174 . From Equation (11.113), the energy to be removed is

Qi = GA = eCO2 sTg4 A = 0.174 ´ 5.6704 ´ 10 -8 (1100)4 24p = 1089 kW.

11.10.2 Exchange between the Entire Medium Volume and the Emitting Boundary In the previous section, the temperature of the black enclosure wall was small enough that emission from the wall could be neglected. If wall emission is significant, the average heat flux removed at the wall is the emission of the medium to the wall, which is all absorbed because the wall is black, minus the average flux emitted from the wall that is absorbed by the medium. For steady state, the total energy removed from the entire boundary equals the energy supplied to the medium by some other means, such as by combustion in a gas. An energy balance gives, for an enclosure with its entire boundary black and at Tw,

-

Qw Qg = = s éëeg (Tg )Tg4 - a g (Tw )Tw4 ùû . (11.115) A A

This also follows from integrating Equation (11.3.1) when the wall is black (ελ,1 = 1). The αg(Tw) is the total absorptance by the medium for radiation emitted from the black wall at temperature Tw. The αg(Tw) depends on the spectral properties of the medium and on Tw, as this determines the spectral distribution of the radiation received by the medium. For radiative exchange calculations in furnaces, an approximate procedure for determining αg is in Hottel and Sarofim (1967). The αg is obtained from the gas total emittance values by using

a g = a CO2 + a H2O - Da (11.116)

where 0.5



æ Tg ö + a CO2 = CCO2 eCO 2 ç ÷ (11.117) è Tw ø



æT ö a H2O = CH2O eH+ 2O ç g ÷ (11.118) è Tw ø



Da = (De)at Tw . (11.119)

0.5

+ The eCO and eH+ 2O are, respectively, eCO2 and eH2O obtained from Equation (9.64) evaluated at Tw and 2 at the respective parameters pCO2 Le¢ and pH2O Le¢ where Le¢ = LeTw /Tg . It is pointed out in Edwards and Matavosian (1984) that the exponent 0.5 is now becoming more accepted to replace the values 0.65 and 0.45 originally used in Equations (11.117) and (11.118). At high temperatures and pressures

when there is overlapping of absorption lines in the infrared spectrum, Le¢ = Le ( Tw /Tg ) , which is discussed by Edwards and Matavosian. The curve fits and spread sheet provided by Alberti et al. (2018) allow the inclusion of CO in the analysis (see Appendix F in the online book appendix at www.ThermalRadiation.net). 3/ 2

525

Fundamental Radiative Transfer Relations

Section 11.11.1 considered a uniform isothermal gas in a black enclosure. If the bounding walls are not black and hence are reflecting, radiation can pass through the gas by means of multiple reflections from the boundary. For an enclosure with a single wall, this can be included by integrating Equation (11.114) over all wavelengths. In Edwards and Matavosian, a procedure is proposed to use total emittance values for situations with multiple reflections. The solution of enclosure heat transfer problems with reflecting walls was treated in Section (11.7.4) by integrating spectral relations over the wavelength absorption bands. Yuen (2015) extends the mean beam length to three-dimensional enclosures with a nongray N2/ CO2/H2O gas and includes the presence of soot. An example that considers a black enclosure with walls at differing temperatures is in Example B.2 in Appendix B on the book website www.ThermalRadiation.net.

11.11 OPTICALLY THIN AND COLD MEDIA The spectral intensity along a path depends on attenuation by absorption and scattering and on augmentation by emission and incoming scattering. From the radiative transfer Equation (11.14), we can write the intensity as

æ I l (S ) = I l (0)exp ç è

S

ö æ bl S dS ÷ + bl (S*)I l (S*)exp ç S *=0 ø 0 è

ò

S

( ) *

*

ò

ö bl S ** dS ** ÷ dS* (11.120) S **=0 ø

ò

S

( )

where the spectral properties βλ = κλ + σs,λ may vary along the path S. The source function Iˆl ( S ) depends on local blackbody emission and scattering and is found from an integral equation such as Equation (11.11). Since Iˆl ( S ) depends on Iλb(S), the temperature distribution must be found to evaluate Iλb(S), so a simultaneous solution is required with the energy equation. Various limiting cases can may be important in some situations, such as an optically thin or thick medium, weak or strong scattering relative to absorption, and weak or strong internal emission relative to radiation incident at the boundaries.

11.11.1 Nearly Transparent Medium When the optical depth along a path is small,

ò

S

S *= 0

bl (S *)dS * ® 0, Equation (11.120) is simplified

as the exponential attenuation terms each approach one, so that S



I l (S ) = I l (0) +

ò b (S*) Iˆ (S*)dS*. (11.121) l

l

S *=0

The intensity that enters at S = 0 is not appreciably attenuated, and there is no attenuation along the path of the intensity that is locally emitted or scattered. In the special case when the optical depth is small and the Iλ(0) is not small, so that the integral in Equation (11.121) is small relative to Iλ(0), Equation (11.121) further reduces to

I l (S ) = I l (0) (11.122)

as in Table 11.3. The incident intensity is dominant and is essentially unchanged along its path through the medium. This approximation is now used to examine a limiting case where conduction is very small compared with radiation and convection is absent.

526

Thermal Radiation Heat Transfer

Example 11.6 Two infinite parallel black walls at T1 and T2 are a distance D apart, and the space between them is filled with an absorbing, emitting, and scattering gas with absorption coefficient κλ(x). Assuming the nearly transparent approximation is valid, derive an expression for the gas temperature distribution for the limit where radiation is dominant so that heat conduction can be neglected. For steady state and no heat conduction, the medium is in radiative equilibrium without internal energy sources, so qr is constant and Equation (11.37) gives (this includes scattering) ¥

ò



¥

ò k ( x)I

kl ( x )Ilb éëT ( x ) ùû d l =

l =0

l

l ,i

( x ) d l (11.6.1)

l =0

where the x coordinate is normal to the walls. The Il,i is obtained from the intensities reaching a volume element along paths in the positive and negative coordinate directions,

ò

ò

4p Il ,i ( x ) = Il+ ( x , Wi ) d Wi + Il- ( x , Wi ) d Wi. (11.6.2)



Ç

Ç

Since the walls are black, the nearly transparent approximation gives, at any location between the walls, Il+ (Wi ) = Ilb (T1) and Il- (Wi ) = Ilb (T2 ). Then, since the blackbody intensity is independent of angle, Equation (11.6.2) reduces to 4p Il ,i = 2p éëIlb (T1 ) + Ilb (T2 ) ùû . (11.6.3)



Substituting Equation (11.6.3) into Equation (11.6.1) gives, at any x position between the walls, ¥



ò

l =0

kl ( x )Ilb éëT ( x ) ùû d l =

1 2

¥

ò k (x) éëI l

lb

(T1) + Ilb (T2 )ùû d l. (11.6.4)

l =0

If κλ(x) depends on local temperature, Equation (11.6.4) is solved iteratively for T(x). If κλ can be assumed independent of gas temperature,

527

Fundamental Radiative Transfer Relations ¥

ò



kl ( x )Ilb éëT ( x ) ùû d l =

l =0

1 2

¥

òk

l

éëIlb (T1 ) + Ilb (T2 ) ùû d l. (11.6.5)

l =0

The right side is evaluated for the specified T1 and T2. Then, the gas temperature T, which is independent of x for the specified conditions, can be found to satisfy Equation (11.6.5) by using a root solver. For a gray gas with temperature-independent properties, κ is constant and Equation (11.6.5) gives T 4 = (T14 + T24 )/ 2. In this optically thin limit without heat conduction, the entire medium approaches a fourth-power temperature that is the average of the fourth powers of the black boundary temperatures.

Le Dez and Sadat (2015) present the solution for an optically thin gray medium between concentric cylinders.

11.11.2 Optically Thin Media with Cold Boundaries or Small Incident Radiation: The Emission Approximation In the nearly transparent approximation, the gas is optically thin, and the local intensity within the medium is dominated by intensities from the boundaries. In the emission approximation, the gas is again optically thin, but there is negligible energy from the boundaries. There is only energy emission within the medium and no attenuation by either absorption or scattering. The radiative transfer Equation (11.120), simplified for these conditions and integrated over all wavelengths, becomes ù é¥ ê kl (S*)I lb (S*)dl ú dS *. (11.123) I (S ) = ê ú S *= 0 ë l = 0 û S

ò ò



The I(S) is the integrated contribution by all the emission along the path as the emitted energy travels through the medium without attenuation. Example 11.7 Use the emission approximation to find the energy flux emerging from an isothermal gas layer at Tg with an integrated mean absorption coefficient weighted by the blackbody spectrum at Tg (Planck mean, Equation (9.68)) of κP = 0.010 cm−1 and thickness D = 1.5 cm, if the layer is bounded by transparent nonradiating walls and cold surroundings (Figure 11.12a). If I(θ) is the total intensity emerging from the layer in direction θ, the emerging flux is q = 2p

ò

p/2

q =0

I(q) cos θ sin θ dθ. The layer is isothermal with constant temperature Tg, so for the inner

integral of Equation (11.123), the quantities are independent of S*, and the Planck mean of κλ(Tg) gives p

ò

¥

l =0

kl (Tg )Ilb (Tg )d l = kP (Tg )sTg4 . Then, Equation (11.123) can be integrated over any path

S = D/cos θ through the layer to yield I(q) = (1/p)kp (Tg )sTg4D / cos q. This gives the radiative flux leaving the gas layer at each boundary as p/2

q=2



ò k (T )sT P

g

4 g

D sin qd q = 2kP (Tg )sTg4D (11.7.1)

q =0 4 g

which gives q = 0.03 sT for the specified numerical values. This is not a precise result, even though the layer thickness is optically thin: κPD = 0.015 ≪ 1. This is because the radiation reaching the layer boundary along each path has passed along a thickness D/cos θ. For θ approaching π/2, the optical path length becomes very large, so the emission approximation cannot hold. A more accurate solution including effects of the proper path lengths gives q = 1.8 kP (Tg )sTg4D (see Section 11.9.4), which is a 10% decrease compared with Equation (11.7.1).

528

Thermal Radiation Heat Transfer

FIGURE 11.12  Examples for emission approximation, (a) layer geometry for Example 11.7, (b) emission from spherical gas-filled balloon with transparent skin

Example 11.8 A plane layer of gray medium with thickness D and constant properties is initially at uniform temperature T0. It has attenuation coefficient β, density ρ, and specific heat cv. At time t = 0, the layer is subjected to a very cold environment. Consider only radiation transfer and obtain the transient temperature of the layer by using the emission approximation. From the results of Example 11.7, the instantaneous transient heat flux emerging from both boundaries of the layer is q(t) = 4βσT4(t)D. The energy equation for the layer then becomes ρcvdT/dt = −4βσT4(t) or, in dimensionless form, d J /dt = -4J4 ( t ) where ϑ = T/T0 and t = (bsT03 /rcv )t . Integrating with the condition that ϑ = 1 at t = 0 gives the transient uniform temperature throughout the layer for the emission approximation as J( t ) =



1 . (1 + 2t )1/ 3

Example 11.9 A spherical balloon of radius R is in orbit around the earth and enters the earth’s shadow. The balloon has a perfectly transparent wall and is filled with a gray gas with constant absorption coefficient κ, such that κR = 1. Neglecting radiant exchange with the earth, derive a relation for the initial rate of radiant energy loss from the balloon if the initial temperature of the gas is T0. Using the emission approximation, Equation (11.123), Figure 11.12b, shows that the intensity at the surface for a typical path S is I(q) = ( ksT04 /p)S = ( ksT04 /p)2R cos q. The radiative flux leaving the surface is p/2



q = 2p

ò

p/2

I(q) cos q sin qd q = 4ksT04R

q =0

4

ò cos q sin qdq = 3 ksT R. 2

4 0

q =0

The initial rate of energy loss from the entire sphere is then Q = ( 4 / 3)ksT04R( 4pR 2 ) = 4ksT04Vs , where Vs = (4/3)πR3 is the sphere volume. This is what is expected, as it was found in Section 1.10.4 that any isothermal gas volume with negligible internal absorption radiates according to this relation.

11.11.3 Cold Medium with Weak Scattering This approximation applies when both the local blackbody emission and scattering within a medium are small. Scattering is small enough that scattering into the S direction can be neglected. The radiative transfer Equation (11.120) reduces to

529

Fundamental Radiative Transfer Relations

æ I l (S ) = I l (0) exp ç è



ö bl (S*)dS * ÷ , (11.124) S *= 0 ø

ò

S

so along each path the local intensity consists only of the attenuated incident intensity such as Iλ(0) being provided by a heated boundary. Example 11.10 Radiant energy of 100 W leaves a frosted glass spherical light bulb 10 cm in diameter enclosed in a fixture having a flat glass plate as in Figure 11.13. If the glass is 2 cm thick and has a gray extinction coefficient of 0.05 cm−1, find the intensity directly from a location on the bulb surface, leaving the fixture at an angle of θ = 60° as shown in the figure. Neglect interface–interaction effects resulting from the difference in refractive index between the glass and surrounding air; these effects are included in Chapter 17. Integrating Equation (11.124) over λ and S results in the total intensity I(S,θ) = I(0,θ)e−(κ + σs)S. To obtain I(0, θ), consider the bulb to be a diffuse sphere. The emissive power at the sphere surface is 100 W divided by the sphere area. The intensity is this diffuse emissive power divided by π, I(0,θ) =  

100 W/(π102 cm2 × πsr) = 0.101 W/(cm2 · sr) Then, I( S, q) = 0.101e -0.05( 2/cos 60 ) = 0.0827 W/(cm2 × sr) .

HOMEWORK PROBLEMS (Additional problems available in the online Appendix P at www.ThermalRadiation.net) 11.1 A layer of isothermal gray gas with Tg = 1000 K and with uniform properties is 1 m thick. The layer boundaries are perfectly transparent. Intensity I(x = 0) is normally incident on the left boundary; there is no incident intensity from the environment on the boundary at x = 1. For each set of conditions shown in the following figures, what is the value of the intensity in the x-direction normal to the layer at x = 1 m?

Answer: 0, 0, 13,249 W/(m2 · sr).

530

Thermal Radiation Heat Transfer

11.2 A slab of non-scattering solid material has a gray absorption coefficient of κ = 0.20 cm−1 and refractive index n ≈ 1. It is 5 cm thick and has an approximately linear temperature distribution within it, as established by thermal conduction. 1. For the condition shown, what is the emitted intensity normal to the slab at x = D? What average slab temperature would give the same emitted normal intensity? 2. If the temperature profile is reversed (i.e. T(x = 0) = 800 K, T(x = D) = 300 K), what is the normal intensity emitted at x = D?

Answers: (a) 1849 W/(m2 · sr); 634 K (b) 1149 W/(m2 · sr). 11.3 The radiation property of a gas is measured with the use of a blackbody radiation source at temperature 2000 K and an optical detector, as shown in the figure. The gas at temperature 300 K fills the 2 m spacing between the 1 cm diameter aperture of the radiation source and the 1.5 cm diameter detector surface. Scattering by the gas can be ignored. Derive an expression to relate the spectral radiation (Qλ,abs) absorbed by the opaque detector with the spectral absorption coefficient (κλ) of the gas, the directional-hemispherical spectral reflectivity [ρλ(θi)] of the detector surface, the angle θ1 measured from the normal to the aperture, and the angle θ2 measured from the normal to the detector surface.

Answer: dQl = éë1 - rl ( q2 ) ùû I l (S ) cos(q1 )dA1

dA2 cos(q2 ) . S2

11.4 An isothermal enclosure is filled with a non-scattering absorbing and emitting medium. The medium and enclosure are at the same temperature. Show that ∇·qr must be zero for this condition. 11.5 Pure carbon dioxide at 1 atm and 2000 K is contained between parallel plates 0.3 m apart. What is the radiative flux received at the plates as a result of radiation by the gas? (Use the Alberti et al. (2018) spreadsheet from https​://do​i.org​/10.1​016/j​.jqsr​t.201​8.08.​008 for CO2 total emittance.) Answer: 99.8 kW/m2.

Fundamental Radiative Transfer Relations

531

FIGURE 11.13  Directional intensity of radiation from light fixture (Example 11.10)

11.6 A rectangular furnace of dimensions 0.6 × 0.5 × 2.2 m has soot-covered interior walls that can be considered black and cold. The furnace is filled with well-mixed combustion products at a temperature of 2200 K composed of 40% by volume CO2, 30% by volume water vapor, and the remainder N2. The total pressure is 1 atm. Compute the radiative flux and the total energy rate to the walls using the Alberti et al. (2018) spreadsheet from https​:// do​i.org​/10.1​016/j​.jqsr​t.201​8.08.​008 for the total emittance of the CO2 and H2O mixture. Answer: 914 kW. 11.7 A pipe 15 cm in diameter is carrying superheated steam at 1.5 atm pressure and at a uniform temperature of 1300 K. What is the radiative flux from the steam received at the pipe wall? Answer: 64.5 kW/m2. 11.8 A furnace at atmospheric pressure with an interior in the shape of a cube having an edge dimension of 1.0 m is filled with a 50:50 mixture by volume of CO2 and N2. The gas temperature is uniform at 1800 K and the walls are cooled to 1100 K. The interior surfaces are black. At what rate is energy being supplied to the gas (and removed from the walls) to maintain these conditions? Use the method in Section 11.11.2. Answer: 355 kW. 11.9 Consider the same conditions and furnace volume as in Homework Problem 11.8. The furnace is now a cylinder with height equal to 2 times its diameter. What is the energy rate supplied to the gas? Answer: 317 kW. 11.10 A cubical black-walled enclosure with 2 m edges contains a mixture of gases at T = 1600 K and a pressure of 2 atm. The gas has volume fractions of 0.4/0.4/0.2 for CO2/H2O/N2. Cooling water is passed over one face of the cube. If the water is initially at 25°C and has a maximum allowable temperature rise of 10°C, what mass flow rate of water (kg/s) is required? (Neglect re-radiation from the wall, and assume all other walls are cool.) Answer: 13.1 kg/s. 11.11 A large furnace has within it a large number of parallel tubes arranged in an equilateral triangular array. The tubes are of 2.5 cm outside diameter, and the tube centers are spaced 7.5 cm apart. The furnace gas is composed of 75% CO2 and 25% H2O by volume, and the combustion process in the gas maintains the gas temperature at 1350 K and a total pressure of 0.9 atm. What is the radiative flux incident on a tube in the interior of the bundle (i.e. completely surrounded by other tubes) per meter of tube length? Answer: 2.50 kW/m.

532

Thermal Radiation Heat Transfer

11.12 A sphere of gray gas at uniform temperature is situated above a surface with the region between the sphere and surface being nonabsorbing. Derive a relation for the radiative energy incident on the circular area as shown. (Hint: Consider the circular area as being a cut through a concentric sphere surrounding the gas sphere, and make use of the symmetry of the geometry.)

æ h Answer: Qi = 2pR 2 sTg4 ç 1 ç 2 h + r2 è

öì ü 2 é1 - (2 kR + 1)e -2 kR ù ý . ÷÷ í1 2 ë û k R ( ) 2 þ øî

11.13 For the radiating sphere of gas in Homework Problem 11.12, derive an expression for the local energy flux incident along the plane surface as a function of distance r. ì ü dQi hr 1 2 1[1 - (2kR + 1)e -2kR ]ý . = 4pR 2 sTg4 2 2 3/2 í 2 drˆ 2 (h + rˆ ) î (2kR) þ 11.14 Two infinite diffuse-gray parallel plates at temperatures T1 and T2, with respective emissivities ε1 and ε2, are separated by a distance D. The space between them is filled with a gray-absorbing and gray-scattering medium having a constant extinction coefficient β. Obtain expressions for the net radiative heat flux being transferred between the plates and the temperature distribution in the medium, by using the nearly transparent approximation. Heat conduction can be neglected compared with radiative transfer. Answer: qi (rˆ ) =

(

Answer: q = s T14 - T24

)

ö æ1 1 ö T 4 - T24 æ 1 1 ö æ 1 1 ç e + e - 1÷ ; T 4 - T 4 = ç e - 2 ÷ ç e + e - 1÷ . 2 2 2 ø è 1 ø 1 è 2 ø è 1

11.15 A sphere of high-temperature optically thin gray gas of fixed volume is being cooled by radiative loss to cool black surroundings (neglect emission from the surroundings). At any instant the entire gas may be considered isothermal and the emission approximation used to compute the radiative loss. Heat conduction is neglected. Write the transient energy equation and solve to obtain the gas temperature as a function of time, starting from an initial temperature Ti. æ 12sk 1 ö t+ 3÷ Answer: T = ç rc T i ø v è

-1/ 3

.

11.16 The sphere in Homework Problem 11.15 now consists of a gas that radiates spectrally in three wavelength bands. The sphere is 0.6 m in diameter and has an initial uniform

Fundamental Radiative Transfer Relations

533

temperature of Ti = 1150 K. The three absorption bands have absorption coefficients κλ1 = 1.1 m−1, κλ2 = 1.6 m−1, and κ λ3 = 0.9 m−1 in the wavelength bands from λ = 0.9–2.0 μm, 2.9–4.0 μm, and 6.0–9.0 μm. In the other portions of the spectrum, the gas is transparent. Write the transient energy equation and calculate the transient gas temperature for cooling of the sphere in cool black surroundings. Assume that the emission approximation can be used.

12

Participating Media in Simple Geometries

John William Strutt, Lord Rayleigh (1842–1919) entered Cambridge in 1861 where he read mathematics and devoted his full time to science. In 1859, John Tyndall had discovered that bright light scattering off nanoscopic particulates was faintly blue-tinted. He conjectured that a similar scattering of sunlight gave the sky its blue hue, but he could not explain the preference for blue light, nor could atmospheric dust explain the intensity of the sky’s color. In 1871, Rayleigh published two papers on the color and polarization of skylight to quantify Tyndall’s effect in water droplets. In 1881, with the benefit of James Clerk Maxwell’s 1865 proof of the electromagnetic nature of light, he showed that his equations followed from electromagnetism.

12.1 INTRODUCTION The conservation of energy equation was introduced in Chapter 11 (Equations (11.1) through (11.3)) along with relations for the associated radiative flux divergence (Equation (11.4)) and the radiative transfer equation (Equations (11.6) through (11.10)). In this chapter, these relations are applied to the cases of radiative transfer in plane layers and simple multidimensional geometries, with and without internal generation. Solution techniques are covered in Chapter 13 and 14, and the incorporation of additional energy transfer modes (conduction, convection) is considered in Chapter 15.

12.2 RADIATIVE INTENSITY, FLUX, FLUX DIVERGENCE, AND SOURCE FUNCTION IN A PLANE LAYER To evaluate the influence of some of the many variables for heat transfer in a radiating gas or other translucent medium, it is helpful to use a simple geometry such as a plane layer of large extent relative to its thickness and with uniform conditions along each boundary. There is considerable literature for this one-dimensional (1D) geometry for engineering applications, atmospheric physics, and astrophysics, since the earth’s atmosphere and the outer radiating layers of the sun can be approximated as plane layers (Kourganoff 1963, Chandrasekhar 1960, Goody and Yung 1989, Bohren and Clothiaux 2006, Petty 2006).

12.2.1 Radiative Transfer Equation and Radiative Intensity for a Plane Layer A plane layer between two boundaries is shown in Figure 12.1. It is a 1D system, so the temperature and properties of the medium vary only in the x direction. The boundaries of the plane layer form a two-surface enclosure, and, referring to Figures 11.1 and 11.2, all paths S and Si for the radiation intensity originate at the lower or upper surface. As shown in Figure 12.1, the path direction is given by the angle θ measured from the positive x direction. The superscript notation + or − indicates, respectively, the directions with positive or negative cosθ, so that I l+ corresponds to 0 ≤ θ ≤ π/2 and I l- to π/2 ≤ θ ≤ π.

535

536

Thermal Radiation Heat Transfer

FIGURE 12.1  Plane layer between infinite parallel boundaries.

The optical depth τλ(x) within the layer is defined along the x coordinate across the thickness with the extinction coefficient β as

x

ò

tl ( x ) =

éë kl ( x* ) + ss,l ( x* ) ùû dx* =

x* = 0

x

ò b ( x*) dx* (12.1) l

x *= 0

which accounts for variable properties, βλ(x). For constant properties, τλ(x) = βλx. For the positive x direction, the relation between optical positions along the S and x directions is S



tl ( S ) =

ò

S *=0

1 bl ( S * ) dS * = cos q

x

tl ( x )

ò b ( x*) dx* = cos q . (12.2) l

x *=0

For the negative direction, Equation (12.2) still applies because dS = −dx/cos(π − θ) = dx/cosθ. With dτλ(S) = dτλ(x)/cosθ, the radiative transfer Equation (11.12) becomes, for I l+ and I l- ,

cosq

¶I l+ I l- [ tl (x ),q] = Iˆl [ tl (x ),q] p /2 £ q £ p (12.3) ¶tl (x )



cosq

¶I l+ + I l+ [ tl (x ),q] = Iˆl [ tl (x ),q] 0 £ q £ p /2. (12.4) ¶tl (x )

The source function Iˆl , which consists of emission and in-scattering terms, depends on θ because of the angular dependence of the phase function Φ for anisotropic scattering, as in Equation (11.12). The partial derivatives in Equations (12.3) and (12.4) emphasize that I l+ and I l- depend on both τλ(x) and θ. It is convenient to let μ ≡ cosθ; then Equations (12.3) and (12.4) become

m

¶I l+ + I l+ éë tl (x ),m ùû = Iˆl éë tl (x ),m ùû 1 ³ m ³ 0 (12.5) ¶tl (x )



m

¶I l+ I l- [ tl (x ),m ] = Iˆl [ tl (x ),m ] 0 ³ m ³ -1. (12.6) ¶tl (x )

537

Participating Media in Simple Geometries

Using an integrating factor as in Equation (11.13), Equations (12.5) and (12.6) are integrated subject to the following boundary conditions that need to have their values specified:

I l+ éë tl , m ùû = I l+ éë0, m ùû at tl = 0 (12.7)



I l- éë tl , m ùû = I l- ëé tl,D , m ùû at tl = tl,D (12.8)

ò

x

x *=0





I

I

l

+ l

bl ( x ) dx =

ò

D

éë kl ( x ) + ss,l ( x ) ùû dx is the layer optical thickness. Integration gives the intensities as a function of location and angle in the plane layer as

where tl,D =

x =0

( tl ,m ) = I ( 0,m ) e + l

( tl ,m ) = I ( tl,D ,m ) e l

- tl /m

- tl / - m

tl

ò Iˆ ( t*,m ) e

1 + m

l

(

)

(

)

- tl - t*l /m

l

d t*l ; 1 ³ m ³ 0 (12.9)

t*l = 0 t l ,D

ò Iˆ ( t*,m ) e

1 m

l

- t*l - tl / -m

l

d t*l ; 0 ³ m ³ -1. (12.10)

t*l = 0

In Equation (12.10) the θ is between π/2 and π, so that μ = cos θ is negative. The use of the source function Iˆl in Equations (12.9) and (12.10) applies for both scattering and non-scattering media; however, without scattering, Equation (11.45) gives Iˆl ( tl , m ) = I lb ( tl ) , which is isotropic, so Equations (12.9) and (12.10) simplify for a medium with absorption only to give



I l+ ( tl , m ) = I l+ ( 0, m ) e - tl / m +

I l- ( tl , m ) = I l- ( tl,D , m ) e - tl / -m -

1 m 1 m

tl

ò ( )

-( tl - tl ) / m I lb t*l , m e d t*l ; 1 ³ m ³ 0 (12.11) *

t*l = 0 tl , D

- ( tl - t l ) / - m d t*l ; 0 ³ m ³ -1. (12.12) I lb t*l , m e

ò ( )

*

t*l =0

12.2.2 Local Radiative Flux in a Plane Layer At each x location the total radiative energy flux in the positive x direction is the integral of the spectral flux over all λ. With the notation that qrλ dλ is the spectral flux in dλ,

qr ( x ) =

¥

òq

l =0

¥

rl

( x )d l =

òq

rl

(tl )d l (12.13)

l =0

where τλ(x) is the optical depth at x. For a fixed x, the optical coordinate τλ depends on λ since βλ is a function of λ. The net spectral flux in the positive x direction crossing dA in the plane at x in Figure 12.1 is obtained in two parts, one from I l+ and one from I l- . Since intensity is energy per unit solid angle crossing an area normal to the direction of I, the projection of dA must be considered normal to either I l+ or I l- . The spectral energy flux in the positive x-direction from I l+ is then (using dΩ = 2π sin θ dθ and μ = cos θ)

538

Thermal Radiation Heat Transfer p/ 2



q

+ rl

1

( tl ) dl = 2pdl ò I ( tl , q ) cos q sin qdq = 2pdl ò I l+ ( tl , m ) mdm, (12.14) + l

q= 0

m =0

which agrees with the z component in Equation (11.29). The q ( tl ) is equivalent to the local spectral radiosity in the positive x-direction, J l+ ( tl ) . The total flux from I l+ in the positive x-direction is + rl



q

+ r

¥

òq

( x) = J ( x) = +

+ rl

( tl ) dl. (12.15)

l =0

Similarly, in the negative x-direction, p/2

ò

qrl- ( tl ) d l = J l- ( tl ) d l = 2pd l

I l- ( tl , q ) cos ( p - q ) sin ( p - q ) d ( p - q )

p - q= 0



p

= -2pd l

ò

I l- ( tl , q ) cos q sin qd q = 2pd l

q= p / 2

(12.16)

1

òI

l

( tl , -m ) mdm.

m =0

The net spectral flux in the positive x-direction is then qrl ( tl ) d l = éëqr+l ( tl ) - qr-l ( tl ) ùû d l = éë J l+ - J l- ùû d l

1

= 2pd l

ò

é I l+ ( tl , m ) - I l- ( tl , m ) ù mdm. ë û

(12.17)

m =0

The intensities can be substituted from Equations (12.9) and (12.10) or from Equations (12.11) and (12.12) when there is no scattering. Using Equations (12.9) and (12.10) that include scattering gives 1 é 1 - t -t m + - tl / m ê qrl ( tl ) dl = 2p I l ( 0, m ) e mdm - I l- ( tl, D , -m ) e ( l , D l ) mdm ê m =0 ë m =0

ò

1



+

tl

ò

òò ( )

-( tl - tl ) m d t*l dm Iˆl t*l , m e *

(12.18)

m = 0 t*l = 0

ù -( t*l - tl ) m Iˆl t*l , -m e d t*l dm ú dl. ú m = 0 t*l = tl ûú 1

+

tl , D

ò ò (

)

The total net radiative flux at x is found by integrating Equation (12.18) from λ = 0 to λ = ∞ using the τλ at each λ corresponding to the fixed x.

12.2.3 Divergence of the Radiative Flux: Radiative Energy Source The radiative flux divergence ∇·qr(x) is needed for the conservation of energy Equation (11.1). For a plane layer with uniform conditions over each boundary, the qr(x) depends only on x so that

539

Participating Media in Simple Geometries



Ñ × qr ( x ) =

dqr ( x ) dx

¥

d dx

=

¥

ò

qrl ( x )dl =

l =0

ò

l =0

dqrl ( x ) dl = dx

¥

ò b (t ) l

l

l =0

dqrl (tl ) dl (12.19) d tl

where dτλ = βλ(x) dx and, throughout the integration over λ, the τλ corresponds to the specified x. Differentiating Equation (12.18) with respect to τλ yields the quantity in the integral of Equation (12.19) as dqr l ( x ) dx



dqr l ( tl )

d l = bl ( tl )

d tl

1 é 1 - t -t /m = -2pbl ( tl ) ê I l+ ( 0, m ) e - tl / m dm + I l- ( tl, D , -m ) e ( l ,D l ) dm ê m =0 ëm = 0 (12.20)

ò

m =0

1 m

I l t*l , m e

m =0

(

)

- tl - t*l / m

1

d t*l dm -

tl , D

1 m

ò I ( t , m ) dm l

)

- ( tl - tl ) / m I l t*l , -m e d t*l dm -

t*l = tl

l

m =0

t*l = 0

ò ò ( 1

+

tl

ò ò ( ) 1

+

ò

*

ù I l ( tl , -m ) dm ú . ú m =0 ûú 1

ò

Another form is to use Equation (11.42) to obtain

dqr ( x ) = 4p dx

¥

ò k ( x ) éëI l

lb

( x ) - I l ( x )ùû dl (12.21)

l =0

which applies with or without scattering. The forms in Equations (12.19) and (12.20) or in Equation (12.21) are equivalent, because the I l ( l, x ) in Equation (12.21) must be found from Equations 1 1 + é I l ( x, m ) + I l- ( x, -m ) ù dm. (12.9) and (12.10) by using I l ( x ) = û 2 0ë ˆ In the limit of absorption only (no scattering), I l ( tl , m ) ® I lb ( tl ) and Equations (12.11) and (12.12) apply for I l+ and I l- . In the limit of scattering only (no absorption), Equation (11.43) gives dqr(x)/dx = 0. This is valid for any type of scattering. In this case radiation is decoupled from other modes, that is, conduction and convection heat transfer.

ò

12.2.4 Equation for the Source Function in a Plane Layer The equation for the source function is obtained from Equation (11.20). Substitutions are made that τλ(S) = βλ(x)/cosθ and dΩi = 2πsinθidθi. Then, following the procedure in the derivation of the previous plane layer relations, the source-function equation becomes (where ωλ = ωλ(τλ)) é 1 ˆI l ( tl , W ) = (1 - wl ) I lb ( tl ) + wl ê I l+ ( 0, m i ) e - tl / mi F l ( m, m i ) dm i 2 ê ëmi = 0

ò

1



+

òI

+ l

( tl , D , m i ) e - ( t

l ,D - tl

) / mi

F l ( m, -m i ) dm i

mi = 0

mi = 0 1

+

tl

ò ò ( ) 1

+

ò

mi = 0

1 mi

1 mi

- ( tl - tl ) / m i Iˆl t*l , mi e d t*l F ( m, mi ) dmi

*

t*l = 0

ù - ( t*l - tl ) / mi Iˆl t*l , -mi e d t*l F ( m, -mi ) dmi ú . ú úû t*l = tl tl , D

ò (

)

(12.22)

540

Thermal Radiation Heat Transfer

12.2.5 Relations for Isotropic Scattering Isotropic scattering is an assumption that provides a substantial reduction in the complexity of the radiative transfer relations with a possible loss of considerable accuracy. For isotropic scattering, the source function is given by Equation (11.21) as w Iˆl ( tl ) = (1 - wl ) I lb ( tl ) + l 4p



4p

òI

l

( tl , Wi ) dWi (12.23)

W i =0

where ωλ = ωλ(τλ). The source function Iˆl ( tl ) then consists of isotropic scattering and isotropic emission, so it is independent of direction. For isotropic scattering the spectral radiative flux Equation (12.18) reduces to 1 é 1 - t -t /m + - tl / m ê qrl ( tl ) d l = 2p I l ( 0, m ) e mdm - I l- ( tl, D , -m ) e ( l ,D l ) mdm ê m =0 ëm =0

ò

ò

tl



1

)

( ò ( )ò e

- tl - t*l / m

I l t*l

+

m =0

t*l = 0 tl , D

1

ò ( )ò I l t*l

+

(12.24)

dmd t*l

e

)

(

- t*l - tl / m

m =0

t*l = tl

ù dmd t*l ú d l. ú úû

This relation and others that follow are written much more conveniently by using the exponential integral function 1

ò

En ( x ) º m n-2 e - x m dm. (12.25)



0

The Εn(ξ) functions are discussed in detail by Kourganoff (1963) and Chandrasekhar (1960), and the important relations for radiative transfer are in Appendix D. Numerical values for En ( x ) are available in handbooks and in computational software packages. The spectral radiative flux for isotropic scattering then becomes 1 é 1 - t -t /m qrl ( tl ) dl = 2p ê I l+ ( 0, m ) e - tl / m mdm - I l- ( tl, D , -m ) e ( l , D l ) mdm ê m =0 ë m =0

ò



ò

tl

+

ò I ( t* )E ( t l

l

2

t*l = 0

l

)

- t*l d t*l +

- tl , D

ù t*l E2 t*l - tl d t*l ú dl. ú úû

òI( ) ( l

t*l = tl

(12.26)

)

Similarly, by use of En(ξ), Equation 12.20 for the flux divergence for isotropic scattering (the negative of the radiative heat source) reduces to dqrl ( tl ) dx



1 é 1 - t -t /m = -2pbl ( tl ) ê I l+ ( 0, m ) e - tl / m dm + I l- ( tl, D , -m ) e ( l ,D l ) dm ê m =0 ëm = 0

ò

ò

ù + I l t*l E1 çæ t*l - tl ö÷ d t*l ú + 4pbl ( tl ) I l ( tl ) . ú è ø úû t*l = 0 tl , D

ò ( )

(12.27)

541

Participating Media in Simple Geometries

For isotropic scattering, the source-function Equation (12.21) reduces to 1 é1 w - t -t /m Iˆl ( tl ) = (1 - wl ) I lb ( tl ) + l ê I l+ ( 0, m ) e - tl /m dm + I l- ( tl,D , -m ) e ( l ,D l ) dm 2 ê m =0 ë m =0

ò



ò Iˆ ( t* ) E ( t* - t )

tl

+

l

l

l

1

l

t*l =0

ò

ù d t*l ú ú úû

(12.28)

The expressions in square brackets are the same in Equations (12.27) and (12.28). The two equations are combined to obtain, as in Equation (11.37), -



k l ( tl ) ˆ dqrl é I ( tl ) - I lb ( tl ) ù . (12.29) = 4p û dx wl ( tl ) ë

Equation (12.21), which is in terms of I l , can also be used. This applies with or without scattering. For absorption only, ωλ → 0 and Iˆl ® I lb , so Equation (12.29) becomes singular. Equation (12.21) can then be used



-

dqr ( x ) = 4p dx

ü ì 1 ï1 é + ù dm - I lb ( x ) ïý dl. (12.30) kl ( x ) í m + m I x , I x , ( ) ( ) l l ë û ïî 2 m=0 ïþ l =0 ¥

ò

ò

12.2.6 Diffuse Boundary Fluxes for a Plane Layer with Isotropic Scattering An absorbing and emitting gas would usually be confined by an enclosure of solid boundaries. If a plane layer has opaque boundaries that can be assumed diffuse, the intensities leaving the boundaries I l+ ( 0, m ) and I l- ( tl,D , -m ) do not depend on angle (are independent of μ) and can be expressed in terms of diffuse outgoing fluxes

I l+ ( 0, m ) = I l+ ( 0 ) =

J l,1 J ; I l- ( tl,D , -m ) = I l- ( tl,D ) = l,2 (12.31) p p

where 1 and 2 correspond to boundaries at τλ = 0 and τλ,D. Then, in Equations (12.26) through (12.28), I l+ ( 0 ) and I l- ( tl,D ) are taken out of the integrals over μ. The integrals are expressed in terms of exponential integral functions to give 1

òI



+ l

( 0, m ) e- t /m mdm = l

m =0 1



ò éëI

l

( tl,D , -m ) e-( t

l , D - tl

m =0

1



òI

m =0

+ l

)/m ù

û

mdm =

( 0, m ) e- t /m dm = l

J l,1 E3 ( tl ) (12.32) p J l,2 E3 ( tl,D - tl ) (12.33) p

J l,1 E2 ( tl ) (12.34) p

542

Thermal Radiation Heat Transfer 1

òI



l

( tl,D , -m ) e-( t

l , D - tl

m =0

)/m

dm =

J l,2 E2 ( tl,D - tl ) . (12.35) p

To obtain equations for Jλ,1 and Jλ,2 in terms of boundary emissivities, Equation (12.26) is evaluated at surface 1, τλ = 0, to give (note that E3(0) = 1/2) tl ,D é ù ê qrl,1 = J l,1 - 2J l,2 E3 ( tl,D ) + 2p Iˆl t*l E2 t*l d t*l ú ê ú êë úû t*l = 0

ò ( ) ( )



By comparison with Equation (6.2), which gives qrλ,1 = Jλ,1−Gλ,1, the quantity in the square brackets is Gλ,1. Using the emitted plus reflected incident energy for the outgoing energy (radiosity) gives the equation for Jλ,1 as tl , D é ù ê * * * J l,1 = el,1pI lb,1 + 2(1 - el,1 ) J l,2 E3 ( tl,D ) + p I l tl E2 tl d tl ú ê ú (12.36) úû êë t*l = 0 = el,1Elb,1 + (1 - el,1 )Gl,1.

ò ( ) ( )



Similarly, at surface 2 (by use of Equation (12.26) evaluated at τλ,D)



tl , D é ù * ê * * J l,2 = el,2 pI lb,2 + 2(1 - el,2 ) J l,1E3 ( tl,D ) + p I l tl E2 tl,D - tl d tl ú ê ú (12.37) úû êë t*l = 0 = el,2 Elb,2 + (1 - el,2 )Gl,2 .

ò ( ) (

)

12.3 GRAY PLANE LAYER OF ABSORBING AND EMITTING MEDIUM WITH ISOTROPIC SCATTERING A gray medium has absorption and scattering coefficients that are independent of wavelength. From the discussion of gas-property spectral variations, Section 9.2.5, it is evident that gases are usually far from being gray. However, in some cases, gases may be considered gray over all, or a portion of, the spectrum. If the temperatures are such that this spectral region contains an appreciable portion of the energy being exchanged, the approximation is reasonable. When particles of soot or other materials are present in the medium, the gas–particle mixture may act nearly gray. Examination of the radiative behavior of a gray medium provides an understanding of some features of a real medium without the complications that real media introduce. The equations for local flux and the source function are now written for a gray medium with isotropic scattering (properties can vary with x). The equations are expressed in terms of the total ¥

quantities I =

ò



sT Iˆ ( t ) = (1 - w)

l =0

I l d l, and wl and tl become w and t. Integrating Equation (11.17) over all λ gives 4

p

( t) +

w 4p

4p

ò

W i =0

I (t, W i )d W i = (1 - w)

sT 4 ( t ) p

+ wI ( t ) . (12.38)

In the flux Equation (12.26), the boundary values I l+ ( 0, m ) and I l- ( tD , -m ) can still be spectrally dependent, since the boundaries have not been assumed gray. Integrating over all λ gives

543

Participating Media in Simple Geometries 1 é 1 - t -t /m qr ( t ) = 2p ê I + ( 0, m ) e - t/m mdm - I - ( tD , -m ) e ( D ) mdm ê m =0 ë m =0

ò



ò

ù + Iˆ t* E2 t - t* d t* Iˆ t* E2 t* - t d t* ú . ú t* =0 t* = t û t

ò ( ) (

tD

)

ò ( ) (

(12.39)

)

If the boundaries are diffuse and gray, I + (0,μ) = J1/π and I−(τD,−μ) = J2/π, and the first two integrals in Equation (12.39) become exponential integral functions qr ( t ) = 2 éë J1 E3 ( t ) - J 2 E3 ( tD - t ) ùû tD ù (12.40) é t Iˆ t* E2 t* - t d t* ú . + 2p ê Iˆ t* E2 t - t* d t* ú ê úû êë t* =0 t* = t



ò ( ) (

)

ò ( ) (

)

For a gray medium, Equation (12.27) for the divergence of the radiative flux becomes dqr ( t ) dx

1 é 1 - t -t /m = -2pb ( t ) ê I + ( 0, m ) e - t/m dm + I - ( tD , -m ) e ( D ) dm ê m =0 ë m =0

ò

ò

ò Iˆ ( t* ) E (

tD

+

1

*

t =0

ù t* - t d t* ú + 4pb ( t ) Iˆ ( t ) . ú û

(12.41)

)

If the boundaries are diffuse and gray, the flux divergence is

dqr ( t ) dx

tD é ù ê b t J E t J E t t p Iˆ t* E1 t* - t d t* ú + 4pb ( t ) Iˆ ( t ) . (12.42) = -2 ( ) 1 2 ( ) + 2 2 ( D - ) + ê ú t* =0 ë û

ò ( ) (

)

For a gray medium, the integral Equation (12.28) for the source function is (where ω = ω(τ)) sT Iˆ ( t ) = (1 - w)

4

p

( t ) + w éê

ò Iˆ ( t* ) E (

tD

1

t* =0

ò

2ê ë m =0

+

1

I + ( 0, m ) e - t/m dm +

1

òI

-

( tD , -m ) e-( t

D -t

)

m =0

m

dm (12.43)

ù t* - t d t* ú . ú û

)

If the boundaries are diffuse and gray, the integral equation becomes tD ù w é ê J E t J E t t p Iˆ t* E1 t* - t d t* ú . (12.44) 1 2( )+ 2 2( D - )+ p 2p ê ú t* =0 ë û By eliminating the integrals from Equations (12.42) and (12.44), or by using Equation (12.29), the flux divergence has the convenient form (as shown by Equation (11.50))

sT Iˆ ( t ) = (1 - w)

4

( t) +

ò ( ) (

)

544

Thermal Radiation Heat Transfer

dqr k(t) k( x ) ésT 4 (t) - pI (t)ù = 4 ésT 4 ( x ) - pI ( x )ù (12.45) =4 û û dx w(t) ë w( x ) ë

where t( x ) =

x

ò b( x* )dx*. By use of Equation (12.38), Iˆ ( t) can be eliminated to provide the form 0

corresponding to Equation (12.21),

dqr = 4 k( x ) éësT 4 ( x ) - pI ( x ) ùû . (12.46) dx



In the limit of zero absorption, so there is only scattering of any type, dqr/dx = 0, as is evident from Equation (12.46) (see also Equation (11.43)). To illustrate the use of the radiative transfer equation for a plane layer, the dimensionless emission Q is calculated for a plane layer of gray medium at uniform temperature Tg with emission, absorption, and isotropic scattering, in surroundings at Te, as in Figure 12.2. This has application to the dissipation of waste energy in outer space by radiation from sheets of hot liquid droplets traveling through space (Taussig and Mattick 1986, Siegel 1987b, c). The absorption and scattering coefficients are constant throughout the layer, and Q = qr és Tg4 - Te4 ù . This definition is used ë û as it gives Q independent of T . The q can be found by evaluating Equation (12.40) at τ = τ and

(

e

)

r

d

assuming that the surroundings act as a black background at Te so that J1 (0) = J 2 (tD ) = sTe4 . The result is arranged into the form Q = 2



tD

pIˆ(t) - sTe4

ò s (T

t=0

4 g

- Te4

)

E2 (tD - t)d t. (12.47)

The Iˆ ( t ) is obtained from the integral Equation (12.44), which becomes tD sTg4 w ìï sTe4 éë E2 ( t ) + E2 ( tD - t ) ùû + Iˆ t* E1 t* - t d t* Iˆ ( t ) = (1 - w) + í p 2ï p t* =0 î This is arranged into

ò ( ) (





pIˆ ( t ) - sTe4

(

s Tg4 - Te4

)

w =1- w+ 2

tD

pIˆ ( t* ) - sTe4

ò s (T

t* =0

4 g

- Te4

)

(

)

ü ï. ý ïþ

)

E1 t* - t d t* . (12.48)

FIGURE 12.2  Plane layer of emitting, absorbing, and scattering medium at uniform temperature.

545

Participating Media in Simple Geometries

This can be solved numerically by iteration for the dimensionless variable éë pIˆ(t) - sTe4 ùû / éës(Tg4 - Te4 ùû . Te4 ùû / éës(Tg4 - Te4 ùû . The results are inserted into Equation (12.47) to obtain Q . The Q is a function of τd and ω, where τd = (κ + σs)D. When ω = 0, Equation (12.48) shows that Iˆ(t) = sTg4 /p, so Q = 2

ò

kD

0

E2 (tD - t)d t = 1 - 2 E3 (tD ). Some results from Siegel (1987b) are in Table 12.1. For each

ω the Q values increase with κd asymptotically to a maximum value that decreases as ω increases. Another situation of interest is transient radiative cooling of a layer such as in Figure 12.2. This shows how the layer emissive ability decreases as the outer portions of the layer become cool and then do not radiate as well as the inner regions. If the layer consists of a dispersion of particles or drops, heat conduction in the medium is small, and if convection is also small, radiative transfer dominates. From the energy Equation (11.1), rc



¶T ¶q ¶q = - r = -b r . (12.49) ¶t ¶x ¶t

Using Equation (12.45) for ∂qr/∂τ at each t yields rc ¶T ( t, t ) 1- w é =4 pI (t, t ) - sT 4 (t, t ) ùû . (12.50) ¶t b w ë



The Iˆ(t) is related to T 4(τ) by the integral Equation (12.44)

sT Iˆ ( t, t ) = (1 - w)

4

( t, t ) + w ìï sTe4 é E ( t ) + E ( t - t )ù + 2 2 D í û p ë

p

2ï î

ü ï Iˆ t* E1 æç t* - t ö÷ d t* ý. (12.51) è ø ïþ t* = 0 tD

ò ( )

Equations (12.50) and (12.51) can be solved starting from a specified initial temperature distribution Τ(τ,0). This is inserted into Equation (12.51), which is solved numerically by iteration for Iˆ(t, 0). By using this along with T4(τ,0), Equation (12.50) is used to extrapolate to T(τ, t + Δt). This is inserted into Equation (12.51) to find Iˆ(t, t + Dt ) and thereby continue the transient solution.

546

Thermal Radiation Heat Transfer

Some numerical techniques are discussed in Appendix I in the online book appendix at www. ThermalRadiation.net. A similarity solution is found when the surroundings are at low temperature, (Te/T)4 ≪ 1. This solution yields constant layer emittance values that are ultimately achieved during transient radiative cooling. A solution is tried of the form

ésT 4 ( t, t ) ù ë û



ù é ˆ ë pI (t, t ) û

1/ 4

1/ 4

= Q(t )F ( t ) (12.52)

= Q(t )G(t). (12.53)

Then, Equations (12.50) and (12.51) become 4 4 rc 1 dQ 1 - w é G ( t) - F ( t) ù = 4 ê ú (12.54) F ( t) w êë s1/ 4b Q 4 ( t ) ¶t úû





w G ( t ) = (1 - w) F ( t ) + 2 4

ò G ( t*) E ( t* - t ) dt*. (12.55)

tD

4

4

1

t* =0

In Equation (12.54) the functions of τ and t have been separated, so the functions on each side of the equation must be a constant. Then, from the right side of Equation (12.54),

G4 ( t) - F 4 ( t) F ( t)

=

G4 ( 0 ) - F 4 ( 0 ) F (0)

. (12.56)

This is solved simultaneously with Equation (12.55) to obtain F(τ) and Γ(τ); the numerical solution details are in Siegel (1987b). The emittance reached in this “fully developed” transient region is defined as efd = qr (t = tD , t ) /sTm4 (t ), where Tm (t) is the integrated mean temperature across the layer. εfd is independent of time and is a function only of τD and ω. εfd is lower than ε for uniform temperature, which are equal to Q in Table 12.1, because of the relatively larger cooling of the outer portions of the layer during transient cooling. This uneven cooling results in poor radiative dissipation compared with that expected from the mean temperature of the layer.

12.4 GRAY PLANE LAYER IN RADIATIVE EQUILIBRIUM 12.4.1 Energy Equation In some situations, radiation dominates over other means of energy transfer. This condition also provides a limiting case to compare with energy transfer by combined modes, where heat conduction and/or convection is present with radiation. In this section, the special case of steady state without significant heat conduction, convection, viscous dissipation, or internal heat sources is considered. For simplicity, the development is for a gray medium. When all energy sources and transfer mechanisms are negligible compared with radiation, the total energy emitted from each volume element must equal its total absorbed energy. This is termed radiative equilibrium and is steadystate energy conservation in the absence of any transfer but radiation. With only radiation present, Equation (11.1) becomes, for steady state without energy sources (q = 0),

Ñ × qr =

dqr ( x ) = 0 or dx

dqr (t) = 0. (12.57) dt

qr is the total heat flux being transferred, since radiation is the only means of transfer.

547

Participating Media in Simple Geometries

12.4.2 Absorbing Gray Medium in Radiative Equilibrium with Isotropic Scattering With dqr/dτ = 0, Equation (12.45) shows that sT 4 (t) Iˆ(t) = , (12.58) p



so for radiative equilibrium with a nonzero absorption coefficient, the source function equals the local blackbody intensity. With this relation used to eliminate Iˆ ( t ) and with dqr/dτ = 0, the same integral equation for T 4(τ) results from either Equation (12.41) or (12.43): 1 é 1 - t -t /m + - t/m ê 4sT ( t ) = 2 p I ( 0, m ) e dm + p I - ( tD , -m ) e ( D ) dm ê m =0 ë m =0

ò

4



ò

ò sT ( t* ) E (

tD

+

4

1

t* =0

ù t* - t d t* ú . ú û

(12.59)

)

The radiative flux Equation (12.39) becomes, with Iˆ(t) = sT 4 (t) /p , 1 é 1 - t -t /m qr = 2 ê p I + ( 0, m ) e - t/m mdm - p I - ( tD , -m ) e ( D ) mdm ê m =0 ë m =0

ò



ò

tD

t

+

ù t* E2 t* - t d t* ú . ú û

ò sT ( t* ) E ( t - t* ) dt* - ò sT ( ) ( 4

4

2

t* = t

t* =0

(12.60)

)

Thus, for a gray medium with q = 0 in radiative equilibrium with isotropic scattering or without scattering, and in which the absorption coefficient is nonzero, a single integral Equation (12.59) governs the temperature distribution. After the temperature distribution has been obtained, it is inserted into Equation (12.60) to obtain qr. Since dqr/dτ = 0, qr is constant (does not depend on τ). Hence Equation (12.60) can be evaluated at any convenient τ, such as τ = 0. Now the special case is considered in which there is scattering, but absorption is zero.

12.4.3 Isotropically Scattering Medium with Zero Absorption Consider an isotropically scattering medium with zero absorption coefficient. Although this case is included here because dqr/dτ = 0, as given by Equation (12.60), it does not require radiative equilibrium. If ω → l, then from Equation (12.38), Iˆ(t) = I (t) , where I (t) is the mean scattered intensity at τ, and t =



x

ò s dx. Equation (12.43) reduces to an integral equation for I (t) : 0

s

tD 1 é 1 ù 1ê -( t D - t ) / m + - t/m I ( t) = I ( 0, m ) e dm + I ( tD , -m ) e dm + I t* E1 t* - t d t* ú . (12.61) ú 2ê m =0 t* =0 ë m =0 û

ò

ò

The radiative flux Equation (12.39) becomes

ò ( ) (

)

548

Thermal Radiation Heat Transfer 1 é 1 - t -t /m qr = 2p ê I + ( 0, m ) e - t/m mdm - I - ( tD , -m ) e ( D ) mdm ê m =0 ë m =0

ò



ò

(12.62)

ù I t* E2 t* - t d t* ú . + I t* E2 t - t* d t* ú t* =0 t* = t û t

ò ( ) (

)

tD

ò ( ) (

)

Equation (11.40) shows that the divergence of the radiative flux is zero, dqr/dτ = 0, so qr in Equation (12.62) is constant in space. This result for pure scattering does not require radiative equilibrium; it does not rely on the absence of other heat transfer modes. Within a nonabsorbing medium, the radiative transfer by scattering does not interact with the energy equation.

12.4.4 Gray Medium with dqR/dx = 0 between Opaque Diffuse-Gray Boundaries For diffuse-gray boundaries, Equation (12.40) applies. For an absorbing medium with or without scattering, Iˆ ( t ) ® sT 4 ( t ) /p for dqr/dx = 0 as in Equation (12.58). For a scattering medium without absorption, Iˆ ( t ) ® I ( t ) as for Equation (12.62). The net radiative flux in the x direction is then tD t é ù * * * qr = 2 ê J1 E3 ( t ) - J 2 E3 ( tD - t ) + G t E2 t - t d t G t* E2 t* - t d t* ú (12.63) ê ú t* =0 t* = t ë û

ò ( ) (

)

ò ( ) (

)

where



ìïsT 4 ( t ) G ( t) = í îï pI ( t )

ü ì x ï b( x*)dx * when k > 0, ss ³ 0 ï ïï ïï x *= 0 t=í x ý . (12.64) ï ï ï ss ( x*)dx * when k = 0, ss > 0 ï ïî x*=0 ïþ

ò

ò

qr is constant across the layer, so it can be evaluated at any convenient location, such as τ = 0. Γ(τ) is found from the integral equation

tD ù 1é ê G ( t) = J1 E2 ( t ) + J 2 E2 ( tD - t ) + G t* E1 t* - t d t* ú . (12.65) 2ê ú t* =0 ë û

ò ( ) (

)

J1 and J2 are found starting with Equation (12.36) and (12.37).

tD é ù ê J1 = e1sT + 2(1 - e1 ) J 2 E3 ( tD ) + p G t* E2 t* d t* ú (12.66) ê ú t* =0 ë û



tD ù é J 2 = e2 sT24 + 2(1 - e2 ) ê J1 E3 ( tD ) + G t* E2 tD - t* d t* ú . (12.67) ê ú êë úû t* =0

4 1

ò ( ) ( )

ò ( ) (

)

549

Participating Media in Simple Geometries

By using Equation (12.63) evaluated at τ = 0 for Equation (12.66) and at τ = τD for Equations (12.36) and (12.37), Equations (12.66) and (12.67) become

J1 = e1sT14 - (1 - e1 ) éëqr - J1 ùû ; J 2 = e2 sT24 + (1 - e2 ) ëéqr + J 2 ùû .

Solving for J1 and J2 gives J1 = sT14 -



(1 - e1 ) (1 - e2 ) qr ; J 2 = sT24 + qr . (12.68) e1 e2

12.4.5 Solution for Gray Medium with dqr/dx = 0 between Black or Diffuse-Gray Boundaries at Specified Temperatures A layer of gray medium with absorption and scattering coefficients κ(T) and σs(T) is between opaque infinite parallel boundaries at specified temperatures T1 and T2, as in Figure 12.3. It is desired to obtain the temperature distribution in the medium and the energy transfer qr between boundaries for energy transfer by radiation only. 12.4.5.1 Gray Medium between Black Boundaries For black boundaries ε1 = ε2 = 1, J1 = sT14 and J 2 = sT24 , from Equation (12.68). Evaluating Equation (12.63) at τ = 0 gives the net heat flux from boundary 1 to boundary 2 as qr = sT - 2sT E3 ( tD ) - 2 4 1



4 2

tD

ò G ( t* ) E (t*)dt* (12.69) 2

t* =0

where, from Equation (12.65), Γ(τ) is found by solving the integral equation

G ( t) =

tD ù 1 éê 4 sT1 E2 ( t ) + sT24 E2 ( tD - t ) + G t* E1 t* - t d t* ú (12.70) 2ê ú t* =0 ë û

ò ( ) (

)

Equations (12.69) and (12.70) are placed in dimensionless form by defining

yb º

(

qr

s T14 - T24

)

, f b ( k) º

G (t) s - T24 (12.71) T14 - T24

FIGURE 12.3  Plane layer of medium in radiative equilibrium between infinite parallel diffuse surfaces.

550

Thermal Radiation Heat Transfer

where the b subscript emphasizes that this is for black boundaries. This gives



fb ( t ) =

tD é ù 1ê E2 ( t ) + fb t* E1 t - t* dt*ú (12.72) ú 2ê t* =0 ë û

ò ( ) (

)

tD



yb = 1 - 2

ò f ( t* ) E ( t* ) dt*. (12.73) b

2

t* =0

The ϕb(τ) is obtained by solving Equation (12.72) and is then used in Equation (12.73) to obtain ψb. For the limiting case of both absorption and scattering in the medium becoming very small, τD → 0 and E3(τD) → 1/2, so Equation (12.69) reduces to

qr

tD ®0

(

)

= s T14 - T24 (12.74)

(or ψb = 1), which is the solution for black infinite parallel plates with a transparent medium between them. For this limit, since E2(0) = 1, Equation (12.70) yields

G(t) T 4 + T24 = 1 (12.75) 2 s tD ®0

(or ϕb = 1/2). For a nearly transparent gray medium with κ > 0, Γ(κ) = σT4(κ), and the temperature of the medium to the fourth power approaches the average of the fourth powers of the boundary temperatures. In an isotropically scattering medium without absorption, the [G(t) /s]tD ®0 in Equation (12.75) equals pI (t)/s within the medium, Equation (12.64). In the following discussion, let κ > 0 so that the results can be interpreted in terms of the medium temperature. Numerical results from Equations (12.72) and (12.73) for the temperature distribution and heat flux in a gray gas with constant properties between infinite black parallel boundaries have been obtained by many investigators. Solution methods are discussed in succeeding chapters. Heaslet and Warming (1965) presented results accurate to four significant figures. These are in Figure 12.4 and values for ψb are also in Table 12.2. Figure 12.4a shows there is a discontinuity between each specified boundary temperature and the temperature of the medium at the boundary. This is called a temperature “jump” or “slip.” If the jump were not present, the curves would all go to 1 at τ/τd = 0 and to 0 at τ/τd = 1. There is no jump when heat conduction is present; the jump is a limit as transfer only by radiation is approached. To determine the magnitude of the jump, the temperature of the medium is evaluated at τ = 0. Using Equation (12.70),

T14 - T 4 (t = 0) 1 é T14 T4 = ê 4 - 4 2 4 E2 ( t D ) 4 4 4 2 ê T1 - T2 T1 - T2 T1 - T2 ë

tD

ò 0

ù T 4 (t* ) E1 t* d t* ú . (12.76) 4 4 T1 - T2 ú û

( )

As τD → 0 the integral vanishes and E2(τD) → 1, so this limit gives

T14 - T 4 (t = 0) T14 - T24 t

= D ®0

1 . (12.77) 2

The magnitude of the jump for a gray medium with constant absorption coefficient is in Figure 12.5 as a function of the layer optical thickness. From symmetry, T14 - T 4 (t = 0) = T 4 (t = tD ) - T24 .

551

Participating Media in Simple Geometries

FIGURE 12.4  Temperature distribution and energy flux in gray medium contained between infinite black parallel boundaries. (a) Temperature distribution; (b) energy flux. (From Heaslet, M.A. and Warming, R.F., Int. J. Heat Mass Trans., 8(7), 979, 1965.)

12.4.5.2 Gray Medium between Diffuse-Gray Boundaries An extension can now be made for gray rather than black boundaries. From Equations (12.63) and (12.65), the equations have the same form as Equations (12.69) and (12.70), except that the outgoing fluxes J1 and J2 replace sT14 and sT24 . Hence, as in Equation (12.71), for gray boundaries, let

y=

G ( t) - J2 qr , f ( t) = J1 - J 2 J1 - J 2



where

4 ïìsT ( t ) G ( t) = í ïî pI ( t )

when k > 0, ss ³ 0 . when k = 0, ss > 0

552

Thermal Radiation Heat Transfer

FIGURE 12.5  Discontinuity at the boundary between gray gas and black boundary temperatures for radiative equilibrium. (From Heaslet, M.A. and Warming, R.F., Int. J. Heat Mass Trans., 8(7), 979, 1965.)

The equations for ϕ(τ) and ψ are then the same as in Equations (12.72) and (12.73), so ϕ = ϕb, and ψ = ψb, and for gray boundaries the Γ(τ) and qr are given by

G ( t ) = fb ( t ) ( J1 - J 2 ) + J 2 (12.78)



qr = y b ( J1 - J 2 ) . (12.79)

Hence, assuming that ϕb and ψb have been obtained for black boundaries, only J1 and J2 are needed to solve the gray boundary case. From Equations (12.68),

J1 = sT14 -

1 - e1 qr (12.80) e1

553

Participating Media in Simple Geometries

J 2 = sT24 +



1 - e2 qr . (12.81) e2

Substitute these relations into Equation (12.79) and solve for qr to obtain

(

qr

s T -T 4 1

4 2

)

=

yb . (12.82) 1 + y b (1/e1 + 1/e2 - 2 )

Then substitute J1 and J2 from Equations (12.80) and (12.81) into Equation (12.78) and eliminate qr by using Equation (12.82) to give G(t) /s - T24 fb (t) + éë(1 - e2 ) /e2 ùû y b . (12.83) = 1 + y b (1/e1 + 1/e2 - 2 ) T14 - T24



Thus, the solutions for plane layers with any combination of gray boundaries can be found conveniently from the black boundary solutions. 12.4.5.3 Extended Solution for Optically Thin Medium between Gray Boundaries Section 11.11 provides some solutions for optically thin media contained within black or transparent boundaries. Here, we provide more comprehensive relations for a nearly transparent medium between infinite parallel boundaries. Consider the heat flux Equation (12.26) used in conjunction with Equations (12.36) and (12.37) for diffuse boundaries that are not black. From the series expansions in Appendix D, the exponen1 tial integrals are approximated for small arguments by E2(x) = 1 + O(x) and E3 ( x ) = - x + O( x 2 ). 2 Then Equation (12.26) becomes, after substituting Equations (12.32) and (12.33) for diffuse boundaries in the first two integrals on the right, qrl ( tl ) = (1 - 2tl ) J l,1 - (1 - 2tl,D + 2tl ) J l,2 + 2p

tl

ò I ( t* ) éëê1 + O ( t l

l

t*l = 0



)

- t*l ù d t*l ûú (12.84)

tl , D

- 2p

l

ò I ( t* ) éëê1 + O ( t* - t )ùûú dt*. l

l

l

l

l

t*l = tl

If τλ,D ≪ 1, the terms of order τλ are neglected and this reduces to qrλ(τλ) = Jλ,1−Jλ,2. The local spectral flux in the medium is the difference between the fluxes leaving the boundaries; the fluxes are not attenuated by the medium in the nearly transparent approximation. In a similar fashion, the source-function Equation (12.28) reduces to

w Iˆl ( tl ) = (1 - wl )I lb ( tl ) + l ( J l,1 + J l,2 ) . (12.85) 2p

The flux derivative Equation (12.27) simplifies to dqrl = -2 ( J l,1 + J l,2 ) + 4pI l ( tl ) d tl





dqrl ( x ) dx

{

}

= 2 kl ( x ) 4pI lb ( x ) - ( J l,1 + J l,2 ) . (12.86)

554

Thermal Radiation Heat Transfer

The diffuse spectral fluxes at the boundaries, Equations (12.36) and (12.37), reduce to J l,1 = el,1 Elb,1 + (1 - el,1 ) J l,2 J l,2 = el,2 Elb,2 + (1 - el,2 ) J l,1.



Solving simultaneously for the spectral fluxes leaving the boundaries yields, for the nearly transparent approximation,

J l,1 =



J l,2 =

el,1 Elb,1 + el,2 Elb,2 (1 - el,1 ) 1 - (1 - el,1 )(1 - el,2 )

el,2 Elb,2 + el,1 Elb,1 (1 - el,2 ) 1 - (1 - el,1 )(1 - el,2 )

(12.87)

. (12.88)

These relations are now applied in examples. Example 12.1 A nearly transparent medium with extinction coefficient βλ is between two diffuse parallel boundaries separated by a distance D and temperatures T1 and T2. What total heat flux is being transferred between the boundaries in the absence of heat conduction and convection? For the nearly transparent approximation, qrλ = Jλ,1 − Jλ,2. Substituting Jλ,1 and Jλ,2 from Equation (12.36) yields qrld l =



el ,1Elb,1 + el ,2Elb,2 (1- el ,1 ) - el ,2Elb,2 + el ,1Elb,1 (1- el ,2 ) d l. 1- (1- e l ,1 )(1- el ,2 )

Simplifying and integrating with respect to λ gives the required result, ¥

qr =



Elb,1 - Elb,2 d l. (12.1.1) l ,1 + 1/e l , 2 - 1

ò 1/e

l =0

This is the same as Equation (6.1.2); the radiant energy flux transferred is uninfluenced by the nearly transparent medium between the boundaries. The medium temperature is obtained in the next example.

Example 12.2 What is the temperature distribution in the medium for the conditions in Example 12.1? For radiative equilibrium without internal heat sources (no conduction or convection), Ñ × qr =

ò

¥

0

(dqrl /dx )d l = 0. Then, from Equation (12.86), ¥



òk

l

( x ) éë 2pIlb ( x ) - ( Jl ,1 + Jl ,2 )ùû d l = 0.

l =0

Substituting Equations (12.87) and (12.88) yields

555

Participating Media in Simple Geometries ¥



2p

ò

kl ( x ) Ilb éëT ( x ) ùû d l =

l =0

¥

òk

l

(x)

l =0

2 ( el ,1Elb,1 + el ,2Elb,2 ) - el ,1el ,2 ( Elb,1 + Elb,2 ) d l. (12.2.1) el ,1 + el ,2 - el ,1el ,2

This can be solved for T(x) by iteration, noting that κλ(x) can be a function of T and x; Equation (12.2.1) reduces to Equation (10.6.4) when ελ,1 = ελ,2 = 1. If all properties are independent of both wavelength and temperature, Equation (12.2.1) reduces to the uniform temperature T4 =



(

)

(

)

4 4 4 4 1 2 e1T1 + e 2T2 - e1e 2 T1 + T2 . (12.2.2) 2 e1 + e 2 - e1e 2

12.5 MULTIDIMENSIONAL RADIATION IN A PARTICIPATING GRAY MEDIUM WITH ISOTROPIC SCATTERING Sections 12.2 through 12.4 developed radiative relations for plane layers. Other geometries are considered next, including rectangular regions. Since the relations become more complicated, the development is for a gray medium, with constant properties, and isotropic scattering. With these simplifications, Equations (11.17), (11.26), (11.38), (11.45), and (11.52) become 4p

qr =



ò IsdW (12.89)

W =0



æ sT 4 ö k Ñ × q r = 4pk ç - I ÷ = 4pb ( I - I ) = 4pb p s s è ø 1 æ sT 4 Iˆ = ç k + ss I bè p





I =

1 4p

4p

ò

W =0

Id W =

1 4p

æ sT 4 ˆ ö - I ÷ (12.90) ç è p ø

ö sT 4 + wI (12.91) ÷ = (1 - w) p ø

S ù é - b S - S* ê I ( 0 ) e -bS + b Iˆ(S*)e ( ) dS *ú dW (12.92) ê ú S *= 0 W =0 ë û 4p

ò

ò

where κ and σs are constants β = κ + σs. The temperature is a function of position within the radiating medium and is found from the energy conservation equation that requires ∇ ⋅ qr.

12.5.1 Radiation Transfer Relations in Three Dimensions A 3D geometry is shown in Figure 12.6. Location vectors r0, r, and r* in Figure 12.6b give the positions of dA, dV, and dV*. Then the path length in Figure 12.6a is S = |r − r0|, and the unit vector along S is s = (r − r0)/| r − r0| or (r − r*)/| r − r*|. The cosθ is the dot product of the unit vector n and the unit vector s along S, so cosθ = n·(r − r0)/|r − r0|. The solid angle that dA subtends when viewed from dV is dΩ = dA cos θ/S2 = dA n·(r − r0)/|r − r0|3. In the integral of Iˆ ( S * ) in Equation (12.92),

556

Thermal Radiation Heat Transfer

FIGURE 12.6  Geometry for 3D translucent medium.

the integration over all dΩ includes all volume elements of the medium dV * = dS*d W(S - S* )2 . By using these vector relations, and dS* dΩ = dV*/|r − r*|2, Equation (12.92) becomes

I (r ) =

n × ( r - r0 ) -b r -r0 b ˆ e 1 I ( r0 ) e dA + I (r*) dV *. (12.93) 3 2 4p 4p r - r0 r - r* A V

ò

ò

-b r -r*

The expression for qr is obtained in the same manner. Equation (12.89) contains the unit vector s and is otherwise like the definition of I . Using Equation (12.93) and inserting the additional s, qr is

ò

q r ( r ) = I ( r0 ) éën × ( r - r0 ) ùû A

r - r0 r - r0

Relations of this type are given in Lin (1988).

4

e

-b r -r0

r - r * -b r -r* dA + b Iˆ(r*) e dV *. (12.94) 3 r - r* V

ò

557

Participating Media in Simple Geometries

12.5.2 Two-Dimensional Transfer in an Infinitely Long Right Rectangular Prism To illustrate the geometrical manipulations for applying Equations (12.93) and (12.94), radiative transfer is considered in a rectangular region with uniform conditions along the axial, z, direction (Figure 12.7). The length in the z direction is large compared with dimensions b and d, and the axial location of dV is arbitrary; hence it is specified to be at z = 0. Then, the volume integral in Equation (12.94) becomes

e -b|r -r*| Iˆ ( r * ) dV * = 2 | r - r* |2

ò V

d

b

òò

Iˆ ( x*, y* )

x *=0 y*=0

¥

2

e -b[( x - x*)

ò ( x - x* )

z *=0

2

+( y - y*)2 + z*2 ] 1/ 2

+ ( y - y* ) + z*2 2

dz* dx* dy* . (12.95)

12 Let x* = z* [( x - x*)2 + ( y - y*)2 ] = z* r* to obtain the z* integral as ¥



ò

x =0

*2 1 2

1 e -br*(1+x ) r* d x* = 2 2 * * r* r (1 + x )

¥

ò

t =1

1 p e -br*t dt = S1 ( br* ). (12.96) 2 12 r* 2 t (t - 1)

The transformation t = (1 + x*2 )1 2 has been used, and the function S1 is one of the class of functions

2 Sn ( x ) º p

¥

ò 1

e - xt 2 dt = n 2 12 p t (t - 1)

p2

òe

- x cos q

cosn-1 q d q n = 0,1, 2,…. (12.97)

0

The Sn are akin to exponential integral functions En that were used for plane layers. Their mathematical properties are examined by Yuen and Wong (1983) and Altaç (1996), and some of their characteristics are in Appendix D along with a table of values. Djeumegni et  al. (2019) further explored these relations to provide analytical expressions for the intensity, flux, and temperature fields in a non-scattering gray medium enclosed between a square internal body and a rectangular 2D enclosure, all with black boundaries. The first (area) integral on the right side of Equation (12.93) consists of four parts corresponding to the four sides of the rectangle. Consider the top side. From the triangle in Figure 12.7,

FIGURE 12.7  Geometry for radiation in 2D-translucent rectangular medium.

558

Thermal Radiation Heat Transfer

(

cos q = n × ( r - r0 ) | r - r0 | = ( b - y ) r20 + z02

)

12

, where r0 ( x0 , y0 ) = éë( x - x0 )2 + ( y - y0 )2 ùû

12

.

Then for the upper surface, the area integral becomes, after integration over the z variable, ¥

d



2

ò I ( x, b ) ò e

x0 =0

b-y dz0 dx0 = p(b - y) 2 (r0 + z02 )3 2

-b( r20 + z02 )1 2

z0 =0

d

ò

x0 =0

I ( x0 , b ) S2 (br0 ) dx0 . (12.98) r20

The I ( r ) in Equation (12.93) is then given by d

4 I ( x, y ) = ( b - y )

ò

I ( x0 , b )

S2 éëbr0 ( x0 , b) ùû r ( x0 , b ) 2 0

x0 =0 d

+y



ò

I ( x0 , 0 )

x0 =0 d

+b

b

ò ò

x *=0 y*=0

S2 éëbr0 ( x0 , 0) ùû r ( x0 , 0 ) 2 0

b

dx0 + (d - x )

ò

I ( d, y0 )

S2 éëbr0 (d, y0 ) ùû

y0 =0 b

dx0 + x

ò

I ( 0, y0 )

r20 (d, y0 )

S2 éëbr0 ( 0, y0 ) ùû

y0 =0

r20 ( 0, y0 )

dy0

dy0

(12.99)

S1 ( br* ) Iˆ ( x*, y* ) dx* dy*. r*

When calculating results from these equations, it is sometimes advantageous to transform them into cylindrical coordinates. Then a dxdy increment becomes ρdρdθ, and this cancels the ρ that is in the denominator. This eliminates any computational difficulty with the denominator becoming zero or close to zero; this is illustrated by Yuen and Ho (1985) for a rectangular gray radiating medium with internal heat generation. Some examples are now considered for radiative equilibrium. Radiation combined with other modes is considered in Chapter 15. Example 12.3 Consider radiative cooling from within a translucent material in the limit when heat conduction is negligible compared with radiation. A hot rectangular bar of emitting, absorbing, and scattering material with nonreflecting boundaries, such as in Figure 12.7, is initially at uniform temperature Ti. It is suddenly placed in a very low temperature environment. The medium is gray with constant radiative properties, scatters isotropically, and has density ρ and specific heat c. Derive the energy relation to obtain the transient temperature distribution. Using Equation (12.90) for ∇ ⋅ qr, the energy equation for T(x, y, t) is

rc

¶T ( x , y , t ) k ésT 4 ( x , y , t ) - pIˆ ( x , y , t )ù = -4b û ¶t ss ë

T ( x , y , 0) = Ti . (12.3.1)

This is solved simultaneously with the integral equation found by substituting I from Equation (12.99) into Equations (12.91) and (12.92). The incoming intensities at the boundaries are zero, as the surroundings are at low temperature and the boundaries are nonreflecting. Then

k sT 4 ( x , y , t ) s s Iˆ( x , y , t ) = + b p 4

2 2 where r* = é( x - x * ) + ( y - y * ) ù ë û

12

.

d

b

ò ò Iˆ ( x*, y*, t )

x * =0 y * =0

S1 (br* ) r*

dx * dy * (12.3.2)

559

Participating Media in Simple Geometries

Equations (12.3.1) and (12.3.2) are placed in dimensionless form and solved numerically. Starting at t = 0, the T(x, y, 0) in Equation (12.3.2) is set equal to Ti. Then the integral equation is solved numerically for Iˆ ( x , y , 0 ) . T(x, y, 0) and Iˆ ( x , y , 0 ) are substituted into the right side of Equation (12.3.1), and the resulting temperature derivative is used to extrapolate forward in time to obtain T(x, y, Δt). With this temperature distribution, Iˆ ( x , y , Dt ) distribution is found by iteration of Equation (12.3.2), and the process is continued to move forward in time. The numerical method is discussed in Appendix I in the online appendix at www.ThermalRadiation.net.

Example 12.4 For the transient cooling problem in Example 12.3, derive an expression for the instantaneous emittance of the rectangular region based on its instantaneous heat loss and mean temperature. The emittance is obtained from the local heat fluxes leaving the region boundaries. For this example, the externally incident intensities are zero, so only the volume integral in Equation (12.94) is needed. Along the upper boundary, d



qr ( x , b) = 2b

b

¥

ò ò ò

x * =0 y * =0 z * =0

2

2

*2 1 2

(b - y*)e -b[( x - x*) +( b-y *) + z ] Iˆ ( x*, y * ) dz * dx * dy * . (12.4.1) [( x - x*)2 + (b - y*)2 + z *2 ]3 2

Using the same transformation as in Equation (12.96), the integration over z* is expressed as an S2 function, d



qr ( x , b) = pb

b

ò ò Iˆ ( x*, y *)

x * =0 y * =0 2 2 where r*1 = é( x - x * ) + ( b - y * ) ù ë û

12

qr (d , y ) = pb

r*1 2

( )

S2 br*1 dx * dy * (12.4.2)

. Similarly, along boundary x = d, d



b - y*

b

ò ò Iˆ(x*, y*)

x * =0 y * =0

d - x* r*2 2

S2 (br*2 ) dx * dy * (12.4.3)

12

2 2 where r*2 = é( d - x * ) + ( y - y * ) ù . The overall emittance of the rectangle will be based on its ë û instantaneous mean temperature

é 1 Tm (t ) = ê ê bd ë



ù T 4 ( x , y , t )dxdy ú ú x =0 y =0 û d

b

òò

1/ 4

. (12.4.4)

The total heat loss Q(t) is obtained by integrating the local qr over their respective boundaries around the rectangle and using symmetry of the upper and lower boundaries and the two vertical sides. Then, the overall emittance is



e(t ) =

d d ù Q(t ) 2 éê = q ( x , b ) dx + qr (d , y )dy ú . (12.4.5) r 4 4 sTm (t ) sTm (t ) ê ú 0 ë0 û

ò

ò

A useful case for testing the accuracy of numerical solutions is to evaluate the radiation to cold black boundaries by a 2D translucent rectangular region at a uniform temperature Tm. The region contains an absorbing–emitting medium without scattering, and with uniform properties. The fluxes at the boundary can be found from Equations (12.4.2) and (12.4.3) by using the source function Iˆ = sTm4 /p, which is constant within the volume for this case, and, without scattering, β = κ. Then, Equations (12.4.2) and (12.4.3) become

560

Thermal Radiation Heat Transfer

qr ( x, b ) =k sTm4



qr ( d , y ) =k sTm4



d

b

ò ò

x *= 0 y *= 0 d

b

ò ò

x *=0 y*=0

( )

b - y* S2 kr*1 dx* dy* (12.100) r*1 2

( )

d - x* S2 kr*2 dx* dy*. (12.101) r*2 2

In Siegel (1991a), a series of transformations was used, and the integrations in Equations (12.100) and (12.101) were carried out to yield the following readily evaluated forms: qr ( x, b ) = 1 - S1 ( BX ) + S3 ( BX ) - S1 éë B( R - X ) ùû + S3 éë B( R - X ) ùû sTm4

R

12 12 ü ì æ é 2 2 ö æ ö -B íS0 ç B êë( X - X * ) + 1ùúû ÷ - S2 ç B éêë( X - X * ) + 1ùúû ÷ ý dX * è ø è øþ X * =0 î

(12.102)

ò

qr ( d , y ) = 1 - S1 ( BY ) + S3 ( BY ) - S1 éë B(1 - Y ) ùû + S3 éë B(1 - Y ) ùû sTm4

1

(

)

12 12 ü ì æ é 2 2 ö 2 2 -B íS0 ç B ëê R + (Y - Y * ) ùûú ÷ - S2 B éë R + (Y - Y *) ùû ý dY * è ø þ Y *= =0 î

ò

(12.103)

where B = ab (optical dimension) R = d/b (aspect ratio) X = x/b, Y = y/b. Tables of local heat flux along the boundary are in Siegel (1991a) for a wide range of aspect ratios and optical thicknesses. Figure 12.8 shows results for a square region. For large optical thicknesses, the dimensionless heat flux is equal to 1 along the boundary away from the corners and equals 1/2 at the corners. An extension for a nongray medium is in Siegel (1992a). References that provide solutions for radiative transfer in rectangular geometries are Thynell and Özişik (1987), Fiveland (1984), Yuen and Wong (1984), Razzaque et al. (1984), Siegel (1989b, 1990, 1991a, 1992a, 1993), Yuen and Takara (1990b), Güven and Bayazitoglu (2003), and Tencer (2014).

12.5.3 One-Dimensional Transfer in a Cylindrical Region A nonplanar geometry that has been extensively studied is the cylindrical shape, which is of interest for radiation in tubular furnaces, cylindrical combustion chambers, and radiating hot gas flows in pipes. Studies in the literature include infinitely long cylinders (Men 1973, Thynell 1990b), concentric cylindrical regions (Greif and Clapper 1966, Pandey 1989, Wu and Wu 1993), and 2D analyses in cylinders of finite length (Reguigui and Dougherty 1992, Hsu and Ku 1994, Pessoa-Filho and Thynell 1996, Zhang and Sutton 1996). The analysis in Fernandes and Francis (1982) is for an infinite cylinder with isotropic scattering (some transient cases are also included), and Thynell (1992) and Azad and Modest (1981a, b) include models for anisotropic scattering. An inverse problem based on the forward solution for a cylindrical system is given by Mengüç and Manickavasagam (1993). For simplicity, the development here is for the 1D behavior in an infinitely long axisymmetric cylinder of a gray medium, in which isotropic scattering is included. The energy equation requires a

Participating Media in Simple Geometries

561

FIGURE 12.8  Local radiative flux along the boundary of square absorbing–emitting region as a function of optical length of side. (From Siegel, R., J. Heat Trans., 113(1), 258, 1991a.)

value for ∇ ⋅ qr. The radiative heat flow at or across a boundary is obtained from qr. These quantities can be found by integrating, at any location, the contributions supplied by the radiation intensity as a result of radiation leaving boundary and volume elements. For a cylinder, the integrations become geometrically complex. This derivation is limited to axisymmetric conditions with no variations along the cylinder axis and is for constant radiative properties. The results are consequently 1D, and qr depends only on the radius. However, the required area and volume integrations are 3D. The geometric aspects of the cylindrical shape are given by Heaslet and Warming (1966) and Kesten (1968) and the results have been used in various forms to obtain solutions in Thynell (1992), Fernandes and Francis (1982), Azad and Modest (1981a, b), and Siegel (1988, 1989a, c). The approach in Kesten is used to illustrate the derivation of the radial radiative heat flux. A typical path AOB is considered, Figure 12.9, and the intensities along this path are integrated at location O in the required manner to obtain the radiative flux, qr(r). dτ along the path is related to the distance dx projected on a cross-section normal to the cylinder axis by dτ = dx/cos α = dx/μ, where μ = cos α and dτ and dx, and also r and R, are optical coordinates (physical coordinates multiplied by the attenuation coefficient β). From this point on in the derivation, the β in the equations is the angle shown in Figure 12.9, and not the attenuation coefficient. Then, from Equation (11.12), the radiative transfer equation is

m

Ù dI + I (x ) = I(x ). (12.104) dx

The law of cosines is used to relate x to the radial coordinate r,

r 2 = x 2 + R 2 - 2 xR cos b. (12.105)

562

Thermal Radiation Heat Transfer

FIGURE 12.9  Geometry for radiative energy transfer in an infinitely long cylinder.

Angle β remains constant along the path x, so, by differentiating Equation (12.105), dx and dr are related along the path by rdr = ( x - R cos b)dx. (12.106)



Equations (12.105) and (12.106) are combined to eliminate x, with the result that dx = ±



(

(r

rdr 2

- R 2 sin 2 b

Letting F (r, b) = r 2 - R 2 sin 2 b

)

12

)

12

ìï-for 0 £ x < R cos b üï í ý . (12.107) ïî+for R cos b < x £ 2 R cos b ïþ

r , Equation (12.104) becomes, in terms of r, dI 1 I ± I =± (12.108) dr mF (r, b) mF (r, b)



with the signs used as previously defined. Equation (12.108) can be integrated as for Equation (11.14). It is assumed that a diffuse intensity I(R), in the direction of τ, is leaving the inside of the boundary at location A. To shorten the relations, define G(a, b) º I along the path x is given by r



I (r ) = I (R)e(1 m)G (R,r ) -

b

ò dx F(x, b). Then, from Equation (11.14), a

1 (1 m)G (R,r ) e -(1 m)G (R,r *) Iˆ(r*) e dr* (0 £ x < Rcosb) (12.109) m F (r*,b)

ò R

563

Participating Media in Simple Geometries



I (r ) = I (R sinb)e

- (1 m ) G ( Rsin b,r )

1 + e -(1 m)G(Rsin b,r ) m

r

ò

Rsinb

e(1 m)G(Rsinb,r *) Iˆ(r*) dr* F (r*,b) (12.110)

(Rcosb < x £ 2Rcosb) where I(R sinβ) is Equation (12.109) evaluated at r = R sinβ. The heat flux in the radially outward direction is obtained by integrating the intensities passing through a cylindrical surface with constant r, with each intensity weighted by the projected area of the surface normal to it, qr (r ) =

ò

4p

W =0

I (r ) cos qd W. The angle θ is between I(r) and the outward

normal to the cylindrical surface. Using cos θ = −cosα cosγ and the solid angle as obtained by projecting an element of surface area onto a unit sphere, dΩ = sin(π/2−α)dγd(π/2−α) = −cosαdγdα, qr(r) becomes p2 p

qr (r ) = -4



ò ò I (r ) cos a cos gd gda. (12.111) 2

a =0 g =0

qr(r) can be expressed in terms of β rather than γ by using the relation from Figure 12.9, p ì ü ï0 £ g < 2 , 0 £ x < R cos b ï ï ï r sin g = R sin b í ý . (12.112) ïp ï ïî 2 < g £ p, R cos b < x £ 2 R cos b ïþ



Let I– and I+ be the intensities corresponding to Equations (12.109) and (12.110). Then R qr (r ) = -4 r

0 æ sin-1 (r R ) ö + ç I cos bdb + I cos bdb ÷ cos2 ad a çç ÷÷ a =0 è 0 sin -1 ( r R ) ø p2

ò ò

ò

é sin-1 (r R ) ù R ê ( I + - I - ) cos bdb ú cos2 ad a. =4 ê ú r a =0 ê úû ë 0

(12.113)

p2

ò ò

Equations (12.109) and (12.110) are now substituted for I+ and I–. The integration over α can be carried out as previously done for the rectangular cross-section. This accounts for all axial locations along the cylinder and results in Sn functions as defined in Equations (12.97). The expression for qr(r) becomes R qr (r ) = 2p r

sin -1 ( r R )

r



+

ò

r *= R sin b

3

3

b=0

I ( r * )

F ( r*, b )

S2 éëG ( r*, r ) ùû dr * -

R

I ( r * )

ò F(r*, b) S ( éëG(r, r*)ùû ) dr* (12.114) 2

r *= r

ü ï S2 éëG( R sin b, r ) + G( R sin b, r*) ùû dr *ý cos bdb. F ( r*, b ) ïþ r *= R sin b R

+

ò {I (R) ( S éë2G(R sin b, r ) + G(r, R)ùû - S éëG(r, R)ùû )

ò

I ( r * )

564

Thermal Radiation Heat Transfer

The physical interpretation of the four terms in Equation (12.114) can be visualized by considering the effect on the heat flux at r of radiation traveling along a diameter of the cylinder. The first term is the heat flux from the cylinder boundary attenuated as it passes through the medium. The positive portion is the radiation passing from the boundary through the center region of the cylinder to r, while the negative portion is from the boundary on the opposite side. The second term is the positive contribution by the emitting and scattering medium between the center of the cylinder and r, while the third term is the negative heat flux contribution by the medium between the boundary and r. The fourth term is the radiation emitted and scattered by the medium between the boundary and the center of the cylinder and then attenuated as it passes from the source location through the center of the cylinder to r. ∇⋅qr for substitution in the energy equation can then be obtained in cylindrical coordinates for the 1D axisymmetric case by carrying out the differentiation (1/r)d(rqr)/dr. ∇⋅qr can also be derived by carrying out the integral form for I in Equation (12.93) and then using Equation (12.90). This approach was used by Heaslet and Warming (1966). Various forms for qr and ∇⋅qr are also given by Fernandes and Francis (1982), Azad and Modest (1981a, b), and Siegel (1988).

12.5.4 Additional Information on Non-planar and Multidimensional Geometries The quantities for the 1D axisymmetric cylindrical case in the previous section depend only on radius. An extension to another radially dependent case is the axisymmetric cylindrical annulus (Habib 1973, Viskanta and Anderson 1975, Wu and Wu 1993). In Wu and Wu, there are two concentric regions around an opaque cylinder. Anisotropic scattering in an annular region is treated in Harris (1989) by using the P1 and P3 approximations. A cylindrical region with variable radius is analyzed by Song et al. (1998). The more complex 2D analysis of a cylinder of finite length is considered by Yucel and Williams (1987), Reguigui and Dougherty (1992), Hsu and Ku (1994), Zhang and Sutton (1996), and Pessoa-Filho and Thynell (1996). Mengüç and Manickavasagam (1993) presented a formulation for an axisymmetric cylindrical system and then used it for an inverse solution. Another radially dependent geometry is a sphere (Thynell and Özişik 1985, Dombrovsky 2000, Liu et al. 2002). The region between two uniform concentric spheres is analyzed by Viskanta and Merriam (1968), Tong and Swathi (1987), and El-Wakil and Abulwafa (2000). In Viskanta and Merriam, the bounding spheres are diffuse-gray, and the medium can emit, absorb, and scatter isotropically. The same geometry is considered in Tong and Swathi and in El-Wakil and Abulwafa, anisotropic scattering is included; additional references are also given for spherical geometries. Radiative transfer within a 2D corner region was analyzed by Wu and Fu (1990). The geometry is a layer of fixed thickness that extends around the corner along the exterior surface of a rectangle. An application is the radiative performance of a coating placed on the outside of a corner. For 3D situations, a cylinder can be considered with both circumferential and axial variations. In Mengüç et al. (1985) an axisymmetric cylinder of finite length was analyzed, including anisotropic scattering. The long rectangular region discussed previously is further generalized to a 3D rectangular region in Mengüç and Viskanta (1985) and Fiveland (1988), and some exact radiative transfer relations for 3D rectangular geometries are in Crosbie and Schrenker (1982) and Lin (1987). Modeling of more general multidimensional geometries is developed by Carvalho et al. (1993) and Martynenko et al. (1998). An extensive review of transient solutions in translucent materials including radiation is in Siegel (1998), and further discussion is in Chapter 17. A solution for the penetration of radiation is in Manohar et al. (1995). Transient heat transfer in a layer with Arrhenius heat generation is studied by Crosbie and Pattabongse (1987). Numerical solutions for spherical geometries are found in Viskanta and Lall (1965, 1966) and Tsai and Özişik (1987), and results for the region between coaxial cylinders are in Chang and Smith (1970). Wendlandt (1973) derives expressions for the transient temperature distribution in a semi-infinite absorbing medium exposed to a laser pulse. An analysis for a very short laser pulse is in Guo et al. (2000), including the effect of the speed of radiant propagation. The transient heating of some semitransparent solid geometries is reviewed by Viskanta and Anderson

Participating Media in Simple Geometries

565

(1975). Guo and Kumar (2002) use the discrete ordinates method to treat transient radiation effects in 3D systems and apply it to a rectangular enclosure. Tan et  al. (2006) apply a meshless leastsquares collocation technique to transient radiative transfer. The body of the literature is growing rapidly in both journal articles and conference proceedings. Recent papers are summarized in online Appendix M on the website for this book, www. thermalradiation.net.

HOMEWORK PROBLEMS 12.1 A semi-infinite, isotropically scattering, absorbing–emitting medium maintained at uniform temperature Tg is in contact with a black boundary at Tw. The medium is gray with constant scattering and absorption coefficients σs and κ. The medium is not moving, and heat conduction is neglected. Show that the heat flux transferred to the boundary can be expressed in the form q = H s(Tg4 - Tw4 ). Provide the relations needed to obtain values for H.

12.2 In Homework Problem 12.1, the plane boundary at x = 0 is modified to be diffuse-gray instead of black. Derive the integral equation relations that can be numerically evaluated to determine the heat flux transferred to the boundary. Place the equations in a convenient dimensionless form. 12.3 A non-scattering stagnant gray medium with absorption coefficient κ = 0.15 cm−1 is contained between large black parallel plates 20 cm apart as shown. Assume the medium has constant density, negligible heat conduction, and n ≈ 1. What is the net energy flux being transferred by radiation from the lower to the upper plate? If the black plates are replaced with gray surfaces with ε1 = 0.6 and ε2 = 0.1, what is the energy flux being transferred? Plot the temperature distribution T(x) for black boundaries and for gray boundaries.

Answers: 6567 W/m2; 1677 W/m2. 12.4 A plane layer of absorbing and non-scattering gray gas is between black parallel plates at temperatures T1 and T2 as in Homework Problem 12.3. Heat conduction in the gas is neglected. A chemical reaction is producing a uniform energy generation per unit volume in the gas. Derive the equations for the temperature distribution in the gas and the local energy flux in the x direction. What is the equation to obtain the net energy flux q1 and q2 supplied to each plate? What does the temperature distribution become for the limiting case when τD = κD → 0 (optically thin layer)?

566

Thermal Radiation Heat Transfer

12.5 A scattering, nonabsorbing, non-conducting medium is contained between large parallel plates 8 cm apart. The scattering is assumed to be isotropic and independent of wavelength, and the scattering coefficient is σs = 0.25 cm−1. The plate temperatures are T1 = 950 K and T2 = 630 K. Compute the net radiative heat flux transferred from plate 1 to plate 2 if the plates are black or if the plates are gray with ε1 = 0.85 and ε2 = 0.32. How do these transfers compare with the values for the plates in a vacuum? Answers: 37,250 W/m2 (black, vacuum); 11,280 W/m2 (gray, vacuum); 11,280 W/m2 (black, medium); 7657 W/m2 (gray, medium). 12.6 A gray absorbing and scattering medium is contained between gray parallel plates 8 cm apart with T1 = 950 K, ε1 = 0.85, T2 = 630 K, ε2 = 0.32. The scattering is isotropic and independent of wavelength, and heat conduction is neglected. Compute the net energy transfer from plate 1 to plate 2 if the scattering and absorption coefficients are, respectively, σs = 0.25 cm−1 and κ = 0.1 cm−1. Compare the result with that of Homework Problem 12.5 to show the effect of adding absorption in the medium. Answer: 6828 W/m2. 12.7 A rectangular enclosure that is very long normal to the cross-section shown has diffusegray boundaries at conditions shown in the following and encloses a uniform gray gas at Tg = 1600 K. The gas has an absorption coefficient of 0.25 m−1. Find the average net radiative flux at each surface and the energy necessary to maintain the gas at 1600 K. (Some necessary relations are given in Appendix B on the book website http://www. ThermalRadiation.net.)

Answers: q1 = 670 kW/m2; q2 = −99.9 kW/m2; q3 = −43.5 kW/m2; energy added = 2306 kW/m. 12.8 Two infinite diffuse-gray parallel plates at temperatures T1 and T2, with respective emissivities ε1 and ε2, are separated by a distance D. The space between them is filled with a gray absorbing and scattering medium having a constant extinction coefficient β. Obtain expressions for the net radiative heat flux being transferred between the plates and the temperature distribution in the medium, by using the nearly transparent approximation. Heat conduction can be neglected compared with radiative transfer.

(

Answers: q = s T - T 4 1

4 2

)

1 1 æ1 1 ö T 4 - T24 e2 2 + 1 ; = . çe ÷ 4 4 1 1 è 1 e2 ø T1 - T2 + -1 e1 e2

Participating Media in Simple Geometries

567

12.9 Two gases have constant extinction coefficients β1 and β2 that are both much smaller than 1. The gases are separated by a thin metal barrier with low emissivity εs. The gases are in the form of two layers bounded by large parallel diffuse boundaries as shown. Radiation exchange is large compared with heat conduction that can be neglected. Using the nearly transparent approximation, obtain relations for the temperature distributions within the gases and the net radiative energy transfer from boundary 1 to boundary 2. Evaluate the results for ε1 = 0.78, εs = 0.04, ε2 = 0.88, T1 = 980 K, and T2 = 550 K. Determine the temperature jump across the thin metal barrier and determine the size of the jump as a function of εs.

Answers: q = 934.4 W/m2, Tg1 = 977 K, Tg2 = 567 K, ΔTjump = 410 K. 12.10 A gray medium is contained between parallel black boundaries. The optical thickness based on plate spacing is τD = 1.0. Find the dimensionless radiative heat flux between the plates, Ψb, and compare the result with the value computed by Heaslet and Warming (1965) in Table 12.2. Use the exact relations in Section 12.4.5.2. Answer: 0.5532.

13

Numerical Solution Methods for Radiative Transfer in Participating Media Subrahmanyan Chandrasekhar (1910–1995) won the Nobel Prize in 1938 for his work on the structure and evolution of stars, later showing star progression toward becoming a black hole. His book Radiative Transfer (1960) outlines the discrete ordinates method (DOM) now used extensively in thermal radiation transfer and introduces methods for treating scattering using Stokes parameters. Photo courtesy of University of Chicago Photographic Archive, [apf1-09456], Special Collections Research Center, University of Chicago Library.

Svein Rosseland (1894–1985) was a theoretical astrophysicist who followed early education in Norway with a fellowship at the Institute of Physics in Copenhagen in 1920, where he met many pioneers in atomic physics, including Niels Bohr. In 1924, he published the paper describing the opacity coefficient of stellar matter, now known as the Rosseland coefficient. Photo courtesy of Creative Commons-Oslo Museum

13.1 INTRODUCTION In the previous chapters, governing relations were presented for analyzing radiative transfer and energy conservation. These relations were applied to one- and two-dimensional (1D and 2D) systems, along with some approximate methods for optically thin media. In this chapter, methods are given for treating more general cases of radiative transfer, along with numerical methods for carrying out solutions. The presence of massively parallel computer architectures now makes the solution of very complex problems possible. Some methods outlined in earlier editions of this text have been superseded by more accurate techniques. In particular, the Curtis–Godson method, which seeks to handle spectral properties by substituting a fictitious medium that obeys the radiative transfer equation (RTE) at optically thin and thick limits, the exponential kernel approximation, which replaces the exponential integral functions En with simpler functions to allow the analytical solution in some cases, and the YIX method for conveniently treating homogeneous media, have been removed from the book. However, the details remain available in Appendices C, D, and E at www.ThermalRadiation.net. As noted in Chapter 1, the radiation intensity is dependent on seven independent variables: Time, a spectral variable, three spatial variables, and two angular variables. In this text, we assume that the dependence of intensity on time can be neglected, as the time of flight of photons at the speed of light is usually negligibly small in comparison with other time-varying changes encountered through coupling with the energy equation. For applications where the time-dependent term in the RTE cannot be neglected, see Olson et al. (2000), Frank (2007), and Kim et al. (2010). The solution of the RTE for the local intensity thus requires treatment of six variables: I(λ, x, y, z, θ, ϕ). The spectral variable and methods for finding accurate spectral properties are discussed in Chapters 9 and 10. In this chapter, we discuss general numerical methods for solving the RTE to determine the

569

570

Thermal Radiation Heat Transfer

local spectral intensity, the local radiative flux, and the flux gradient. Chapter 14 provides an indepth discussion of the Monte Carlo method. Chapter 15 extends the discussion to combined-mode problems that include radiation. Each of the methods in this chapter is applied to the solution of the general RTE (Equation (11.12)) repeated here: ¶I l (S, W) ¶I l (S, W)  = = I l (S ) - I l (S, W). (13.1) bl ¶S ¶tl (S )



13.2 SERIES EXPANSION AND MOMENT METHODS Many methods for the solution of the differential form of the RTE are based on moment methods, and these are often combined with using a series expansion of the intensity along a particular direction vector. The series is then truncated to some small number of terms, resulting in final formulations that are mathematically and numerically tractable. Five such methods are discussed in this section: The diffusion solution, the Milne–Eddington approximation, the PN (spherical harmonics expansion) method, the SPN (simplified PN ) method, and the MN method. Further discussion is in Frank (2007) and Tencer (2013). First, define the radiative moment equations. These equations are developed by multiplying the local intensity at location S by powers of the direction cosines li(i = 1, 2, 3), either individually or in combination, and then integrating over all solid angles. This results in the zeroth, first, second, and general moments as I (0) ( S ) =



4p

ò I ( S, W )dW = 4pI (S) = cU (S) (13.2) l

W =0



(i )

I

4p

ò l I ( S, W ) d W = q (S )

( i = 1, 2, 3) (13.3)

ò l l I ( S, W )dW = cP (S)

( i, j = 1, 2, 3) (13.4)

(S ) =

i

i

W =0



I

( ij )

(S ) =

4p

i j

ij

W =0



I (ijkl…) ( S ) =

4p

ò l l l l … I ( S, W )dW i j k l

( i, j, k, l,… , = 1, 2, 3) . (13.5)

W =0

The zeroth moment is 4π times the average local intensity or, if divided by c, it is the radiative energy density (Equation (11.31)). The first moment is the radiative energy flux in the i coordinate direction (Equation (11.26)), and the second moment divided by c is the local radiation stress and pressure tensor. Higher-order moments have no physical meaning. Taking moments of the RTE by integrating with respect to the direction cosines gives the RTE moment equations: Zeroth moment RTE 4p



ò

W =0

¶I l (S, W) dW = ¶tl

4p

ò

W =0

4p

Il (S ) d W -

ò I (S, W)dW (13.6) l

W =0

571

Numerical Solution Methods

or, using the moment definitions, 4p

¶I l(1) (S ) = ¶tl



ò I (S )dW - I l

(0) l

(S ) (13.7)

W =0

or ¶qi,l (S ) = ¶tl



4p

ò I (S, W)dW - 4pI (S). (13.8) l

l

W =0

For the special case of isotropic scattering, the integral of the source function over all directions is zero, and Equation (13.8) reduces to 1 ¶qi,l (S ) = - I l (S ). (13.9) 4p ¶tl

The first moment RTE is 4p



ò

W =0

li l j ¶I l (S, W) dW = bl ¶S

4p

4p

ò

ò l I (S, W)dW (13.10)

l j Il (S ) d W -

j l

W =0

W =0

or 4p



I (S ) = (2) l

ò l I (S)dW - I j

l

(1) l

(S ) (13.11)

W =0 4p



cPl,ij (S ) =

ò l I (S)dW - q j

l

l ,i

(S ). (13.12)

W =0

Further moment equations are generated in a similar way, but the zero and first moment equations are sufficient for most solution methods. Equations (13.1), (13.8), and (13.11) form a set of three partial differential equations involving the local intensity Iλ(S) and the first three moments Il(0), Il(1), and Il(2) (or alternatively, Iλ(S), qλ,i(S), I l (S ), and Pl,ij (S )). Thus, some approximation must be made to provide a fourth equation to close the set, called a closure condition. Usually, this is in the form of some approximation for the radiative pressure tensor, Pl,ij (S ).

13.2.1 Optically Thick Media: Radiative Diffusion In an optically thick medium, radiation travels only a short distance before being scattered or absorbed. The local intensity results from radiation originating from the nearby surroundings where emission and scattering terms are close to those of the location under consideration. Radiation from locations where conditions are appreciably different is greatly attenuated in an optically thick medium before reaching the location being considered. For this situation, it is possible to transform the expressions for radiative energy into a diffusion relation similar to that for heat conduction. Energy transfer then depends only on the conditions in the immediate vicinity of the position being considered and is expressed in terms of the gradient of the conditions at that position. The diffusion approximation provides a substantial simplification. Standard techniques such as finite-difference schemes can be used for solving the radiative diffusion differential equation.

572

Thermal Radiation Heat Transfer

Real gases are usually transparent in some wavelength regions. The diffusion approach can be applied with accuracy only in wavelength regions where the optical thickness of the medium is greater than about 10; this value depends on the geometry and conditions of the problem and may need to be larger in some instances. The fact that some mean optical thickness meets this criterion is not sufficient. Sometimes good results for some quantities are predicted for much smaller optical thicknesses. Wavelength-band applications of the diffusion method can be made to use the method in the optically thick spectral regions (Siegel and Spuckler 1994a). In a high-temperature gas core nuclear reactor (Slutz et al. 1994), the gaseous uranium fuel has a high opacity, so the diffusion approximation is adequate for the desired thermal radiation heat transfer calculations. The diffusion approximation requires the intensity to be nearly isotropic within the medium. This can occur in the interior of an optically thick medium with small temperature gradients but is not valid near boundaries for certain types of conditions. For example, at a boundary of a hot medium adjacent to much lower or much higher temperature surroundings, radiation leaving and entering the medium will be much different. As a result of this large anisotropy, the diffusion approximation loses accuracy near the boundary. A temperature jump boundary condition may be used to improve accuracy. If heat conduction is present and dominant near a boundary, the inaccuracy of the radiative diffusion can be less important. 13.2.1.1 Simplified Derivation of the Radiative Diffusion Approximation A simplified derivation for a 1D layer is used to show the basic ideas in the diffusion approximation for an absorbing–emitting medium with isotropic scattering. A more general derivation is then discussed. Let H be a path length along a path direction S over which the radiant energy density changes appreciably, and let the extinction mean free path (Equation (1.54)) be lm,λ = 1/(κλ + σsλ) = 1/βλ. For the diffusion approximation to apply, lm,λ/H = 1/βλH ≪ 1. In Figure 13.1a, using dS = dx/cosθ in Equation (13.1), the RTE for the change of Iλ with x for a fixed θ direction is

-

cos q ¶I l ( x, q) = I l ( x, q) - Il ( x ) (13.13) bl ¶x

FIGURE 13.1  Geometry for the derivation of diffusion equations: (a) 1D plane layer of medium, and (b) general 3D region.

573

Numerical Solution Methods

or, defining μ = cos θ, using H to nondimensionalize so X = x/H, and defining the optical thickness as τλ = βλH -



m ¶I l ( X , m) = I l ( X , m) - Il ( X ) (13.14) tl ¶X

where the source function Iˆl does not depend on the angle for isotropic scattering. Now, solve Equation (13.14) by expanding the intensity in a series of unknown functions 1 (n) I l , n = 0,1, 2,… , multiplied by powers of  1: tl 2

Il = I



(0) l

æ 1 ö 1 + I l(1) + ç ÷ I l( 2 ) +  (13.15) tl è tl ø

where the (n) superscript denotes the terms of the expanded intensity function, not to be confused with superscripts for moments. Insert Equation (13.15) into (13.14) to obtain ( Iˆl is given by Equation (11.17)) -m

ù 1 é ¶I l( 0 ) 1 ¶I l(1) 1 + + ú = I l( 0 ) + I l(1) +  - (1 - wl ) I lb ê tl ë ¶X tl ¶X t l û



ù é æ ( 0 ) 1 (1) ö 1 ú - wl ê I + I +  d W i l ç l ÷ ê 4p ú tl è ø ë Wi = 4 p û

(13.16)

ò

where ωλ = σsλ/βλ. In addition to the expansion parameter 1/τλ, there is the quantity ωλ, the scattering albedo, which characterizes extinction by scattering relative to total extinction. For small ωλ, there is diffusion by absorption alone, as in a dense gas containing few scattering particles. For ωλ→1, there is scattering alone, as in a poorly absorbing gas with many suspended scattering particles. In Equation (13.16), collect the terms of zero order in 1/τλ to obtain

I

(0) l

= (1 - wl ) I lb

The terms Iλb and the first moment of I l( 0 ),

ò

4p

W i =0

w + l 4p

4p

òI

(0) l

d Wi . (13.17)

W i =0

I l( 0 ) d W i , on the right do not depend on the incidence

angle dΩi. Hence, I l( 0 ) on the left cannot depend on Ωi. Using this fact, the first moment (integral) term in Equation (13.17) reduces to wl I l( 0 ) so that Equation (13.17) becomes I l( 0 ) = I lb . (13.18)



Now collect the terms in Equation (13.16) of first order in 1/τλ, and substitute Equation (13.18) for I l( 0 ) to obtain

-m

w dI lb = I l(1) - l 4p dX

4p

òI

(1) l

d W i . (13.19)

W i =0

To find I l(1), multiply by dΩi = 2πsinθdθ = −2πμdμ and integrate over all dΩi:

574

Thermal Radiation Heat Transfer

dI lb dX



æ 4p ö 4p wl ç (1) ÷ 2pmdm = d Wi . I d Wi I l d Wi ÷ 4p ç m =-1 W i =0 W = 0 W = 0 ø i è i 4p

1

ò

ò

ò

(1) l

ò

The integral on the left is zero, so 4p

òI

0=



4p (1) l

d W i - wl

W i =0

For any ωλ ≠ 1, the

ò

4p

W i =0

òI

(1) l

d W i .

W i =0

I l(1) d W i = 0 , and Equation (13.19) reduces to I l(1) = -m



dI lb . (13.20) dX

Substitute Equations (13.18) and (13.20) into Equation (13.15) to obtain I l = I lb -



m dI lb . (13.21) tl dX

This reveals the important feature that in the diffusion limit, the local intensity depends only on the magnitude and the gradient of the local blackbody intensity. Since temperature gradients are small and τλ is large, the last term on the right is small and Iλ is nearly isotropic, as for Iλb. The local spectral energy flux at X flowing in the X direction is found by multiplying Iλ by μ dλ and integrating over all solid angles:

ql ( X ) d l = 2pd l

ò

1

m =-1

I l ( m ) mdm. (13.22)

Because the blackbody intensity Iλb does not depend on μ, using Equation (13.21) in Equation (13.22) gives

ql ( X ) d l = 2pI lb ( X ) d l

1

ò

m =-1

2pd l dI lb mdm tl ( X ) dX

1

4p

dI lb

ò m dm = - 3t ( X ) dX dl. (13.23) 2

m =-1

l

Equation (13.23) is the Rosseland diffusion equation. The local radiative energy flux depends only on local conditions. In a gray medium, since πIλb dλ integrated over all λ is σT4, Equation (13.23) gives the total radiative flux as

q(X) = -

4s dT 4 16sT 3 dT =. (13.24) 3t ( X ) dX 3t ( X ) dX

For κ = 0 (ω = 1, pure scattering), energy must be supplied by an external source since the medium does not emit radiation; then σT 4 in Equation (13.24) is replaced by pI as in Section 11.4.3. 13.2.1.2 General Radiation–Diffusion Relations in a Medium In the previous section, to obtain the diffusion equation, only first-order terms were retained in the series Equation (13.15), and boundaries were not considered. More general equations, including second-order terms, were considered in Deissler (1964) and boundary conditions were introduced

575

Numerical Solution Methods

so that the diffusion equations could be applied to finite regions. The intermediate equations in the derivation become somewhat complex because of their general form in 3D Taylor series expansions. The final results are provided here, and Deissler can be referred to for details. 13.2.1.2.1 Rosseland Diffusion Equation for Local Radiative Flux A control volume is used in a 3D region, as in Figure 13.1b. The energy from dV to dV0 is derived from the equation of transfer. In the general diffusion approximation, the radiation at dV0 originates only from locations close to dV0. The intensities are expanded about dV0 in a 3D Taylor series and then truncated after second-order terms; this is sufficient in an optically dense medium where the diffusion approximation is assumed to apply. The intensities are used to integrate over all directions for energy approaching dV0 from the positive and negative coordinate directions. The second-order terms cancel when the diffusion approximation is imposed. Hence, the Rosseland diffusion equation obtained in the general second-order derivation is found to be the same as in Equation (13.24). In a 3D geometry (Figure 13.1b) the general relation for the local radiative energy flux (at any position r) in each coordinate direction, including isotropic scattering (the z direction is used here), is given by

ql,z r d l = -

4p æ ¶I lb ö 4 æ ¶Elb ö dl = d l. (13.25) ç ÷ 3bl ( r ) è ¶z ør 3bl ( r ) çè ¶z ÷ør

This is the general relation for local spectral radiative energy flux in terms of the local emissive power gradient. Equation (13.25) has the same form as the Fourier heat conduction law, permitting solution of some radiation problems by the same solution methods used for heat conduction. A mean of the attenuation coefficient βλ may be defined for the medium. To obtain the energy flux in a wavelength range, integrate Equation (13.25) over Δλ (the parentheses and r subscript are omitted for simplicity): qDl,z = -

Dl



4 æ ¶Elb ç l è ¶z

ò 3b

4 ¶ =3bR,Dl ¶z

ò

Dl

4 ö ÷ d l º - 3b ø R ,Dl

ò

Dl

¶Elb dl ¶z

4 ¶EDlb El b d l = . 3bR,Dl ¶z

(13.26)

This defines the mean attenuation coefficient βR,Δλ as



1 bR,Dl

º

ò

Dl

(1/bl ) ( ¶Elb / ¶z ) dl

ò

Dl

( ¶Elb /¶z ) dl

=

ò

Dl

(1/bl ) ( ¶Elb /¶Eb ) dl

ò

Dl

( ¶Elb /¶Eb ) dl

. (13.27)

For the entire λ range, there is the important result that the local total radiative flux is

qz = -

(

)

4 4 ¶Eb 4 ¶ sT 16 ¶T ==sT 3 (13.28) 3bR ¶z 3bR ¶z 3bR ¶z

where

1 º bR

¥

æ 1 ç è bl l =0

ò

ö æ ¶Elb ÷ç ø è ¶Eb

ö ÷ d l. (13.29) ø

576

Thermal Radiation Heat Transfer

The βR is the local Rosseland mean attenuation coefficient, named after Svein Rosseland, who was the first to use diffusion theory in studying radiation effects in astrophysics (Rosseland 1936). The ∂Eλb/∂Eb is found by differentiating Planck’s law (Equation (1.20)) after letting T = (Eb/σ)1/4: 2pC1 æ ¶Elb ¶ ç 1/ 4 = 5 ¶Eb ¶Eb ç l exp éêë( C2 / l )( s/Eb ) ùûú - 1 è

{

p C1C2 s1/ 4 = 2 l 6 Eb5 / 4



=

}

ö ÷ ÷ ø

1/ 4 exp éê( C2 /l )( s/Eb ) ùú ë û

{

}

1/ 4 exp éê( C2 /l )( s/Eb ) ùú - 1 ë û

2



exp ( C2 /lT ) p C1C2 1 . 6 5 2 l sT éexp ( C2 /lT ) - 1ù 2 ë û

Then,

1 º bR

¥

æ 1 ö æ ¶Elb ç ÷ç è bl ø è ¶Eb l =0

ò

ö p C1C2 ÷ dl = 2 sT 5 ø

¥

æ 1 ç è bl l =0

ò

æ lT ) exp ( C2 /l öç ÷ç 6 2 ø ç l éëexp ( C2 /lT ) - 1ùû è

ö ÷ d l. (13.30) ÷÷ ø

If the spectral absorption coefficient βλ has spectral regions of transparency, this definition breaks down and special steps must be taken. 13.2.1.2.2 Emissive Power Jump Boundary Condition To this point, the location in the medium was considered to be far enough (in optical thickness) from any boundary that the boundary conditions did not enter the diffusion relations. Now the interaction with a diffuse wall is considered for the limit when radiation is the only means of energy transfer (no conduction or convection). Unlike the cases where conduction or convection is present, the boundary temperature and the medium temperature adjacent to the boundary are not necessarily equal. Let the wall bounding the medium from earlier, Figure 13.2, have a hemispherical spectral emissivity ελw2. All quantities evaluated on the wall have the subscript w to distinguish them from quantities in the medium at the wall; these do not have a w subscript. Consider area dA2 in the medium parallel and immediately adjacent to the wall. The spectral energy quantities passing through dA2 are shown in Figure 13.2, so the net spectral flux across dA2 in the positive z direction is

( ql,z )2 dl = el,w2 éëGl,2 - El,bw2 ùû dl. (13.31)

Rearranging gives

- El,bw 2 =

( ql,z )2 e l ,w 2

- Gl,2 . (13.32)

The incident radiative flux (irradiation) Gλ,2 is expanded in a Taylor series and terms through second order are retained (Deissler 1964). The diffusion equation (Equation (13.24)) is substituted for the first-order terms, and the result is the following relation for the discontinuity (“jump”) in emissive power at dA2 in the limit of only radiative transfer:

æ 1 1ö 1 æ ¶ 2 El,b 1 ¶ 2 El,b 1 ¶ 2 El,b ö + + El,b 2 - El,bw 2 = ç - ÷ ( ql, z )2 - 2 ç ÷ . (13.33) 2bl è ¶z 2 2 ¶y 2 2 ¶x 2 ø2 è el,w 2 2 ø

Numerical Solution Methods

577

FIGURE 13.2  Geometry for the derivation of the energy-jump condition at the opaque boundary.

All quantities without a w subscript are evaluated at dA2 in the medium adjacent to the wall. The quantities with a w subscript are on wall 2, and qλ,zdλ is the net flux in dλ in the positive z direction. Similarly, the discontinuity in emissive power at dA1 in Figure 13.2 is

æ 1 1ö 1 æ ¶ 2 El,b 1 ¶ 2 El,b 1 ¶ 2 El,b ö + El,bw1 - El,b1 = ç - ÷ ( ql,z )1 + 2 ç + ÷ (13.34) 2 ¶y 2 2 ¶x 2 ø1 2bl è ¶z 2 è el,w1 2 ø

where quantities with a w1 subscript are on wall 1 and those without w are in the medium adjacent to wall 1. Equations (13.33) and (13.34) are boundary conditions relating the emissive power Eλ,b in the medium adjacent to the wall to the wall emissive power Eλ,bw. In the limit when radiation is the only means of energy transport in the medium, there is a discontinuity in emissive power in passing from the medium to each wall. The use of the diffusion approximation in the derivation of these boundary relations implies that the proportionality between local radiative flux and emissive power gradient in the medium is valid in the medium very near a bounding surface. Although this is not strictly true, the use of jump boundary conditions corrects, to a good approximation, for wall effects if radiation dominates over conduction and/or convection. The general radiation–diffusion equation has been provided at λ as Equation (13.25) and, for a wavelength band, as Equations (13.26) and (13.27). The boundary conditions at solid boundaries

578

Thermal Radiation Heat Transfer

with normals into the medium in the negative and positive coordinate directions are given at λ by Equations (13.33) and (13.34). When the diffusion equation is used, it is assumed to apply throughout the entire medium, including the region adjacent to a boundary. The boundary effect is accounted for by using a jump boundary condition. 13.2.1.2.3 Gray Stagnant Medium between Parallel Gray Walls Many molecular gases have strong property variations with wavelength, and it is necessary to solve the diffusion equation in many wavelength regions. For some situations, such as soot-filled flames and high-temperature uranium gas, a total gray-medium approximation can be made. The equations reduce considerably in this case. For illustration, consider a gray medium between infinite parallel gray walls at unequal temperatures (Figure 13.3) for the limit where radiation is strongly dominant over conduction and/or convection. For a gray medium, the absorption and scattering coefficients are independent of wavelength and Equation (13.27) gives qz = −4/[3β(z)](dEb/dz). This can be integrated directly because with no conduction, convection, or heat sources in the medium, the radiative flux qz is constant in a 1D planar system. With the additional assumption that β does not depend on temperature and is therefore independent of z, integrating from 0 to z gives

Eb ( z ) - Eb1 = -

3b qz z. (13.35) 4

Evaluating Equation (13.35) at z = D yields

Eb 2 - Eb1 3bD =. (13.36) qz 4

The Eb1 and Eb2 are in the medium adjacent to the walls. To connect the unknown Eb1 and Eb2 with the specified wall conditions, the jump boundary conditions are applied. Differentiating Equation (13.35) twice with respect to z shows that the second derivative terms are zero in the boundary condition Equations (13.33) and (13.34), so they become, for a gray medium with gray walls,

Eb 2 - Ebw 2 1 1 = ew2 2 qz

(13.37)

FIGURE 13.3  Schematic for radiative interchange between infinite gray boundaries enclosing a gray medium with absorption and isotropic scattering.

Numerical Solution Methods



579

Ebw1 - Eb1 1 1 = - . (13.38) qz e w1 2

To eliminate the unknown medium emissive powers at the bounding surfaces, Eb1 and Eb2, add Equations (13.37) and (13.38), and then substitute Eb2 − Eb1 from Equation (13.36) to give

qz 1 = . (13.39) Ebw1 - Ebw 2 3bD / 4 + 1 / e w1 + 1 / e w 2 - 1

Equation (13.39) gives the radiative energy transfer (for radiative equilibrium without internal heat sources) through a layer of gray medium as a function of the optical thickness βD and the wall emissivities. It is relative to the difference in the black emissive powers of the walls, which is the maximum possible energy transfer. A comparison of this diffusion solution with the solution of the exact integral equation solution (Heaslet and Warming 1965) is in Figure 13.4 for equal wall emissivities. Agreement is excellent for all optical thicknesses. The distribution of emissive power Eb(z) across the layer is found from Equation (13.35) by eliminating the unknown Eb1 by use of Equation (13.38) or in another form by eliminating Eb1 and Eb2 from Equations (13.35) through (13.38). This result, which is for radiative equilibrium without internal heat sources, is shown in Table 13.1. 13.2.1.2.4 Other Radiative Diffusion Solutions for Gray Media Table 13.1 provides diffusion solutions for the temperature distribution and energy transfer in simple geometries for gray media with constant properties between gray walls in the limit of radiative equilibrium (see Example 13.1 for further analytical details). Media such as real gases are usually not gray and are not optically thick in all wavelength regions, so caution is advised. For cylindrical or spherical geometries, agreement with exact solutions is often not as good as for infinite parallel plate boundaries. Good agreement has been found in cylindrical and spherical geometries if the optical thickness is greater than about 10, with improved agreement as wall emissivities become lower and diameter ratios Dinner/Douter approach unity. A comparison for the cylindrical geometry is discussed later in connection with Figure 13.9.

FIGURE 13.4  Validity of a diffusion solution for energy transfer through a gray medium with absorption and isotropic scattering between parallel gray walls for the limit of radiative equilibrium without internal heat sources.

580

Thermal Radiation Heat Transfer

Example 13.1 The space between two diffuse-gray spheres (Figure 13.5) is filled with an optically dense stagnant medium having constant attenuation coefficient β. For the limiting condition of radiative equilibrium, compute the radiative energy flow Q1 across the gap from sphere 1 to sphere 2 and the temperature distribution T(r) in the medium, using the diffusion method with jump boundary conditions. For a gray medium with constant β, Equation (13.28) gives the net heat flux in the positive r direction as qr = −(4/3β) dEb/dr. From energy conservation, with no internal heat sources, qr varies with r as qr = Q1/4πr2. Combine these relations and integrate from R1 to R 2 to obtain



Q1 4p

R2

ò

r =R1

dr 4 =r2 3b

Eb 2

ò

dEb (13.1.1)

Eb = Eb1

Q1 æ 1 1 ö 4 = (Eb2 - Eb1) (13.1.2) 4p çè R2 R1 ÷ø 3b

or

Q1 16p 1 . (13.1.3) = E E 3 b 1 1ö æ ( b1 b2 ) çR - R ÷ 2 ø è 1

581

Numerical Solution Methods

The Eb1 and Eb2 are in the gas adjacent to the boundaries and jump boundary conditions express these quantities in terms of wall values. The jump boundary conditions are in Equations (13.33) and (13.34) and involve second derivatives that are now found. By integrating Equation (13.1.1) from R1 to r,

Eb ( r ) - Eb1 =

3bQ1 æ 1 1 ö . (13.1.4) 16p çè r R1 ÷ø

Substitute r = (x2 + y2 + z2)1/2 and differentiate twice with respect to x to obtain

(

)

3/ 2

(

2 2 2 - 3x 2 x 2 + y 2 + z 2 ¶ 2Eb ( r ) 3bQ1 x + y + z = 3 16p ¶x 2 x2 + y 2 + z2

(

)

)

1/ 2

(13.1.5)

and similarly for the y and z directions. In the boundary condition Equation (13.33), point 2 can be conveniently taken in Figure 13.5 at x = y = 0 and z = R 2. This gives

é ¶ 2Eb ( r ) ù é ¶ 2Eb ( r ) ù 3bQ1 1 ; ê ú =ê ú =2 2 16p R23 ¶ ¶ x y êë úû 2 êë úû 2

é ¶ 2Eb ( r ) ù 3bQ1 1 . ê ú = 2 8p R23 ¶ z êë úû 2

Also, qz,2 = Q1/4pR22 . Substituting this into Equation (13.33) gives

Q1 1 ö Q1 3Q1 1 æ 1 - ÷ = Eb 2 - Ebw 2 = ç 2 3 2 è ew 2 2 ø 4pR2 32bp R2 4pR2

éæ 1 3 ù 1ö - ÷êç ú . (13.1.6) ëè ew 2 2 ø 8bR2 û

Similarly, at the inner sphere boundary, from Equation (13.34),

Ebw1 - Eb1 =

Q1 4pR12

éæ 1 1 ö 3 ù - ÷+ êç ú . (13.1.7) e 2 8 b R1 û w 1 è ø ë

FIGURE 13.5  Radiation across the gap between concentric spheres with an intervening medium of constant attenuation coefficient.

582

Thermal Radiation Heat Transfer

Adding Equations (13.1.6) and (13.1.7) gives

Eb 2 - Eb1 = Ebw 2 - Ebw1 +

Q1 éæ 1 3 æ 1 1 öù 1ö 1 æ 1 1ö 1 + + ê ú. 4p ëçè ew 2 2 ÷ø R22 çè ew1 2 ÷ø R12 8b çè R13 R23 ÷ø û

After substituting this into the right side of Equation (13.1.2), the result is solved for Q1 to give the ψ in the last entry in Table 13.1. To obtain the temperature distribution, integrate Equation (13.1.1) from R 2 to r to obtain Eb(r)− Eb2 = (3βQ1/16π)(1/r−1/R 2). Add Equation (13.1.6) to eliminate Eb2:

Eb ( r ) - Ebw 2 =

3bQ1 æ 1 1 ö æ 1 1 ö Q1 3Q1 1 . (13.1.8) + 16p çè r R2 ÷ø çè ew 2 2 ÷ø 4pR22 32bp R23

This gives the last expression for ϕ in Table 13.1. The ϕ is valid only if κ > 0, since temperatures are indeterminate in the limit of pure scattering when there is only radiative transfer.

Example 13.2 A plane layer of gray medium with constant properties is originally at a uniform temperature T0. The attenuation coefficient is β, and the layer thickness is D. The heat capacity of the medium is cV and its density is ρ. At time t = 0, the layer is placed in surroundings at zero temperature. Neglecting conduction and convection, discuss the solution for the transient temperature profiles for cooling only by radiation when β is very large. Contrast this with the solution for β being very small, as in Section 11.11. At the layer center, x = D/2 (boundaries are at x = 0 and D), the symmetry condition provides that at any time, (∂T/∂x)t = 0 for x = D/2. At t = 0 for any x, T = T0. For only radiation being included, there is a temperature jump at the boundaries x = 0, D, so the boundary temperatures are finite rather than being equal to the zero outside temperature. For large β, the diffusion approximation can be employed, and from Equation (13.24), the heat flux in the x direction is

q ( x, t ) = -

4 4 ¶Eb ( x , t ) 4s ¶T ( x , t ) =. 3b ¶x 3b ¶x

Energy conservation gives −∂q(x,t)/∂x = ρcV∂T/∂t. Combining these two equations to eliminate q gives the transient energy diffusion equation for the temperature distribution in the layer with constant absorption coefficient: rcV



¶T 4s ¶ 2T 4 ( x , t ) = . 3b ¶t ¶x 2

Defining dimensionless variables as t = bsT03t /rcv , t = bx , and J = T /T0 gives

( )

2 4 ¶J 4 ¶ J t, t = . ¶t 3 ¶t2



The initial condition and the symmetry condition at x = D/2 are ϑ(τ,0) = 1 and

¶J = 0. At the ¶t t= kD ,t

boundary τ = βD, a slip condition must be used. For surroundings that are empty space at zero temperature, Ebw = 0 and εw = 1, so that at the exposed boundary of the medium for any time, Equation (13.33) gives

sT 4

x =D

=

1 æ 4s ¶T 4 ö s ¶ 2T 4 - 2 ç÷ 2 è 3b ¶x ø x =D 2b ¶x 2

x =D

583

Numerical Solution Methods or

æ 4 ¶J4 4 ¶ 2J4 ö + 0 = ç 2J4 + . ÷ 3 ¶t 3 ¶t2 ø t=bD è

Similar relations apply at x = 0. For these conditions, a numerical solution is necessary. For comparison, for the limit of a small β, an analytical solution is obtained in Example 11.6.

Viskanta and Bathla (1967) obtained numerical solutions to the transient energy equation, along with limiting solutions. Results for an optical thickness of βD = 2 are in Figure 13.6 for transient profiles in one-half of the symmetric layer. The results in this section are for the important limit where radiation is the dominant mode of energy transfer; conduction and convection have been neglected. Combining heat conduction and convection with radiation using the diffusion solution is considered in Chapter 15.

13.2.2 Moment-Based Methods Moment-based methods are developed by multiplying the differential equation of radiative transfer by various powers of the direction cosines of the intensity to form a simultaneous set of moment equations (Equations (13.1), (13.8), and (13.11)) that are then solved. The general procedure provides one less equation than the number of unknowns generated. To overcome this difficulty, the local intensity is approximated by a series expansion. If the series is in terms of spherical harmonics, denoted by P, and the series is truncated after a selected number of terms, N, this procedure provides a closure relation so that a solution can proceed. When the series is truncated after one or three terms, the method is called P1 or P3; in general,

FIGURE 13.6  Dimensionless temperature profiles as a function of time for radiative cooling of a gray slab; optical thickness τ(x = D) = 2. (From Viskanta, R. and Bathla, P.S., Z. Angew. Math. Phys., 18(3), 353, 1967).

584

Thermal Radiation Heat Transfer

it is the PN method. It is also referred to as the moment method and the differential approximation. This will serve as an introduction to moment methods. First, a plane layer is considered. The first two RTE moment equations (zeroth and first) are solved simultaneously. Then, a general development is provided. 13.2.2.1 Milne–Eddington (Differential) Approximation This approximate method is developed by starting with the RTE Equation (11.3) in a 1D plane layer with scattering that is assumed isotropic. The optical coordinate τ is derived in Equation (12.1). To simplify the notation, the symbols for wavelength are omitted by carrying out the derivation for a gray medium. The Milne–Eddington equations that are obtained also apply to spectral calculations if the same forms are written using spectral quantities. The RTE Equation (11.7), with the source function substituted from Equation (12.23), gives m

where I ( t ) = (1/4p )

ò

4p

W i =0

dI = - I ( t ) + (1 - w) I b ( t ) + wI ( t ) (13.40) dt

I ( t, W i ) d W i and t ( x ) =

ò

x

x *=0

b ( x * ) dx * . To obtain the zeroth moment

equation (equivalent to Equation (13.6)), Equation (13.40) is multiplied by dΩ and integrated over all solid angles to give 4p

ò



W =0

Because

ò

4p

W=0

dI d m dW = dt dt

4p

4p

W =0

W =0

4p

4p

W =0

W=0

ò mIdW = - ò IdW + (1 - w) I ò dW + wI ò dW. (13.41) b

mIdW is the radiative flux, and Ib and I do not depend on direction, this integrates to

become the Milne–Eddington equation:

dqr ( t ) dt

= -4pI + 4p (1 - w) I b + 4pwI = 4p (1 - w) éë I b ( t ) - I ( t ) ùû . (13.42)

To obtain the first moment equation (equivalent to Equation (13.10)), the RTE equation (Equation (13.40)) is multiplied by μdΩ and integrated over all solid angles to give 4p



ò

W =0

dI m dW = dt 2

4p

4p

4p

W =0

W =0

ò I mdW + (1 - w) I ò mdW + wI ò mdW. (13.43) b

W =0

Because dΩ = −2πdμ = 2πsinθdθ, the integrals involve products of the orthogonal functions sinθ and cosθ, which simplifies their evaluation. Orthogonal functions will also be used in the more general formulation that follows. To evaluate Equation (13.43), the approximation made independently by Eddington (1926) and Milne (1930) is that, for radiation crossing a unit area oriented normal to the x direction, all intensities with positive directional components in x have a value independent of angle, and all intensities with a negative x directional component have a different constant value. Hence, the local radiation in each coordinate direction is assumed to be isotropic, as in Figure 13.7. After making this assumption, Equation (13.43) is integrated to yield the second Milne– Eddington equation:

qr ( t ) = -

2p + 4p dI ( t ) I + I- = . (13.44) 3 3 dt

(

)

585

Numerical Solution Methods

FIGURE 13.7  Approximation of intensities being isotropic in positive and in negative directions as used in the Milne–Eddington and two-flux methods.

The I+ and I− are isotropic in each of the +x and −x directions, so there are the following relations for the radiative flux qr and the mean intensity I (see below Equation (13.40)) at each τ location:



(

)

qr = p I + - I - = qr+ - qr- (13.45) I =

I + + I - qr+ + qr= . (13.46) 2 2

Solving Equations (13.45) and (13.46) for qr+ and qr- yields the auxiliary relations:

qr+ = pI +

qr (13.47) 2



qr- = pI -

qr . (13.48) 2

The two Milne–Eddington flux equations (Equations (13.42) and (13.44)) can be combined to eliminate either I (t) or qr(τ). This yields two differential equations for qr(τ) and I (t) as

dEb ( t ) d 2 qr - 3 (1 - w) qr ( t ) = 4 (1 - w) (13.49) 2 dt dt



d2I - 3 (1 - w) I ( t ) = 4 (1 - w) Eb ( t ) . (13.50) d t2

586

Thermal Radiation Heat Transfer

These relations also apply in spectral form. For example, for radiative properties that depend on λ but not on x, the optical coordinate τ becomes τλ = (κλ + σsλ)x = βλx, and Equation (13.49) can be written as (because the relation between τλ and x varies with λ, it is sometimes easier to use the physical coordinate) dElb ( x ) d 2 qrl - 3kl ( kl + ssl ) qrl ( x ) = 4 kl . (13.51) 2 dx dx



To obtain a solution, the two equations, Equations (13.42) and (13.44), or Equations (13.49) and (13.51), are used with the energy equation to provide three equations with three unknowns, qr, I , and T, that are in the blackbody function Eb = πIb. For the special case of steady-state energy transfer only by radiation, the energy equation (Equation (11.5)) without convection or conduction (radiative equilibrium) gives, for a 1D plane layer and a gray medium, dqr ( x )



dx

= q ( x ) or

dqr ( t ) dt

=

1 1 q ( t ) = q ( t ) . (13.52) k + ss b

If, in addition, there are no internal heat sources, so q = 0, the radiative flux qr is a constant, and Equation (13.49) simplifies to qr = -



4 dEb ( t ) 3 dt

or qr = -

4 dEb ( x ) (13.53) 3b( x ) dx

where for the total flux, Eb = σΤ4. For this special case, Equation (13.53) for the radiative heat flux is the same as for the diffusion approximation (Equation (13.28)). In spectral form, qr ,l = - éë 4/3bl ( x ) ùû éë dElb ( x )/dx ùû for radiative equilibrium without internal heat sources. Two sets of boundary conditions are needed: One set for solving the RTE and the second for solving the energy equation. At an opaque solid boundary for the translucent medium, the outgoing radiation (radiosity) consists of emission by the boundary and reflected incoming radiation. Hence, for an opaque gray surface at each boundary of a gray translucent layer, 0 ≤ x ≤ D as in Figure 13.3,

J ( 0 ) = e w1 Ebw1 + (1 - e w1 ) G ( 0 ) (13.54)



J ( D ) = e w 2 Ebw2 + (1 - e w 2 ) G ( D ) . (13.55)

For use with the Milne–Eddington equations, these boundary conditions are placed in terms of I and qr by the use of Equations (13.47) and (13.48), noting that J ( 0 ) = qr+ (0) and J ( D ) = qr- ( D), with the result that, at x = 0 and x = D,

I (0) +

1æ 1 1ö 1 - ÷ qr ( 0 ) = Ebw1 (13.56) ç p è e w1 2 ø p



I ( D) -

1æ 1 1ö 1 - ÷ qr ( D ) = Ebw 2 . (13.57) ç p è ew2 2 ø p

Example 13.3 A plane layer of gray gas with constant absorption and scattering coefficients and isotropic scattering is between two opaque parallel walls spaced D apart, which are at specified temperatures T1 and T2 and have gray emissivities ε1 and ε2. Uniform energy generation is occurring throughout the

587

Numerical Solution Methods

 gas at a rate qper unit volume. The gas is not moving and heat conduction in the gas is neglected to obtain the limiting result for only radiative transfer. Use Milne–Eddington relations to find the distributions of radiative transfer and temperature within the layer. Integrating the energy Equation (13.52), the distribution of radiative flux in the layer is qr ( t) = q t/b + C1, where C1 is an integration constant. With the functional form of qr(τ) known, the second Milne–Eddington equation (Equation (13.44)) can be integrated to obtain I ( t ) = - 3q t2 / 8pb - ( 3C1t / 4p ) + C2 . The two integration constants are obtained by

(

)

substituting the qr(τ) and I ( t) evaluated at the boundaries into the two boundary conditions in Equation (13.54) and solving the results simultaneously for C1 and C2. The C1 needed in the distribution qr ( t) = q t/b + C1 is found as

C1 =

- ( q tD /2b ) éë( 3tD /4 ) + ( 2 /e 2 ) - 1ùû + Eb1 - Eb 2

( 3tD /4 ) + (1/e1) + (1/ee2 ) - 1

, (13.3.1)

where τD = βD. Then from Equation (13.56), I (0) = C2 and qr(0) = C1, C2 = −(1/π) (1/ε1 − 1/2)C1 + Eb1/π. With I ( t) and qr(τ) known, the temperature distribution is then obtained from the first Milne– Eddington equation (Equation (13.42)) as

sT 4 ( t ) = pIb( t ) =Eb1 +

q æ 1 3t2 ö æ 3t 1 1 ö + C1. (13.3.2) ç ÷2 è 2k 4b ø çè 4 e1 2 ÷ø

13.2.2.2 General Spherical Harmonics (PN ) Method In the general PN method, the integral equations of radiative transfer are reduced to a set of differential equations by approximating the transfer relations by a finite set of moment equations. As discussed in Section 13.2, the moments are generated by multiplying the equation of transfer by powers of the cosine of the angle between the coordinate direction and the direction of the intensity and then integrating over all solid angles. This is a generalization of Milne–Eddington method in which Equations (13.42) and (13.44) are from the equation of transfer multiplied by (cos θ)0 and (cos θ)1 and then integrated over all dΩ. These are equivalent to the more general moment Equations (13.8) and (13.11). The PN development is in 3D so that general geometries can be treated. The treatments in Higenyi (1979), Bayazitoglu and Higenyi (1979), Ratzel and Howell (1982), Liu et al. (1992a, b), and Mengüç and Viskanta (1985, 1986) are followed; other pertinent references are Mark (1945), Marshak (1947), Stone and Gaustad (1961), Adrianov and Polyak (1963), Traugott and Wang (1964), Rhyming (1966), Chou and Tien (1968), Dennar and Sibulkin (1969), Cheng (1972), Selçuk and Siddall (1976), Shvartsburg (1976), Modest (1990), Tencer (2013), and Bruno (2017). The formulation of the PN approximations is addressed in Modest et al. (2014) and Ge et al. (2015). A rectangular coordinate system x1, x2, x3 is shown in Figure 13.8a. The variation of intensity at position r along the S direction in the direction of the unit vector s is given by the RTE equation (Equation (11.10)). If the medium is assumed gray with uniform scattering and absorption coefficients, and assumed to scatter isotropically, Equation (11.10) is integrated over all λ to give

s dI = kI b - ( k + ss ) I + s 4p dS

4p

ò I ( S, W ) dW (13.58) i

i

W i =0

where I = I(S, Ω) and Ib = Ib(S). At each location in a medium, radiation is traveling in all directions. It is useful to express the direction S of the intensity I in terms of the angles θ and ϕ in a spherical coordinate system or in terms of direction cosines li (i = 1, 2, 3) of the coordinate system (Figure 13.8a), so that

dI ¶I ¶I ¶I ¶I ¶I ¶I + sin q cos f + sin q sin f = l1 + l2 + l3 (13.59) = cos q ¶x1 ¶x2 ¶x3 ¶x1 ¶x2 ¶x3 dS

588

Thermal Radiation Heat Transfer

FIGURE 13.8  PN approximation: (a) Coordinate system showing intensity as a function of position and angle for PN approximation, and (b) heat fluxes in boundary condition.

where l1 = cos θ, l2 = cos δ, and l3 = cos γ. Using the optical coordinate dτi = (κ+σs)dxi, the albedo ω, and incident solid angle Ωi, the RTE, Equation (13.59), becomes 3



å i =1

li

w ¶I + I = (1 - w) I b + 4p ¶ti

4p

ò I ( S, W ) dW (13.60) i

i

W i =0

where I is a function of location S and direction Ω. To develop the PN method, the intensity at each S is expressed as an expansion in a series of orthogonal harmonic functions:

589

Numerical Solution Methods ¥

I ( S , W) =



l

å å A (S)Y m l

l

m

(W). (13.61)

l =0 m =- l

In the limit of an infinite number of terms in the series, l → ∞, the spherical harmonics approximation is exact. TheAlm ( S ) are position-dependent coefficients to be determined by the solution, and Yl m ( W ) are the angularly dependent normalized spherical harmonics: é 2l + 1 (l - m )! ù Yl (W) = ê ú êë 4p (l + m )! úû

1/ 2 m

e jmf Pl (cos q) (13.62)

m



m

where j = -1 so that ejmϕ provides the harmonics cosmϕ and sinmϕ.* The Pl (cos q) are associated Legendre polynomials of the first kind, of degree l and order m, m

Pl (m) =



(1 - m 2 ) 2l l !

m /2

d

l- m

dm

l- m

(m2 - 1)l (13.63)

where μ = cosθ*. The Pl (m) º 0 for |m| > l and Pl 0 (m) º Pl (m) . Values Pl m ( cos q ) for 0 ≤ l ≤ 3 are in Table 13.2. To apply the PN method, Equation (13.61) is truncated after a finite number of N terms. Generally, in engineering radiative transfer problems, terms are retained for l = 0 and 1 (P1 approximation) or for l = 0, 1, 2, and 3 (P3 approximation). It is possible to retain higher-order terms, but for thermal radiation problems, the P3 approximation usually has been found adequate. A higher-order approximation adds considerable complexity and may not be practical. Even-order approximations (P2, P4, etc.) give little increase in accuracy over the next lower-order odd-numbered expansion, and they are difficult to apply for boundary conditions of specified temperature or energy flux. Thus, odd-order expansions are most often used for radiative transfer. To determine I(S, Ω), the coefficients Alm ( S ) in the series expansion Equation (13.61) must be evaluated. To do this, the local intensity in Equation (13.61) is substituted into the integrals in the moment equations (Equations (13.2) through (13.4)), the series is truncated at the desired approximation, and the integrations are carried out. For the P3 approximation, this gives 20 coupled algebraic equations in 20 moments of intensity. Ratzel and Howell (1982) solved this set and present forms for the Alm ( S ) coefficients in terms of the moments. Substituting these expressions for Alm ( S ) m

* Two definitions of normalized spherical harmonics are commonly used. One definition (not used here) includes a factor of (−1)n. This factor introduces an alternating sign in harmonics with positive m. If this definition is used, Equations (13.62) and (13.63) must correctly incorporate the factor.

590

Thermal Radiation Heat Transfer

into Equation (13.61) gives the relation for the P3 approximation for the local intensity I(S, θ, ϕ) in terms of its moments as 0 1 2 3 4pI ( S, q, f ) = I ( ) + 3I ( ) cos q + 3I ( ) sin q cos f + 3I ( ) sin q sin f

+

)(

(

)

)

) 7 + ( 5I ( ) - 3I ( ) ) ( 5 cos q - 3 cos q ) 4 21 + é( 5I ( ) - I ( ) ) cos f + ( 5I ( ) - I ( ) ) sin f ù ( 5 cos q - 1) sin q ûú 8 ëê +



(

5 11 0 12 13 3I ( ) - I ( ) 3 cos2 q - 1 + 15 I ( ) cos f + I ( ) sin f cos q sin q 4

(

15 é ( 22 ) (33) 23 cos 2f + 2 I ( ) sin 2f ù sin 2 q I -I ê ë ûú 4 111

1

211

3

2

311

(

) ) ) cos 3f - ( I (

3

2

+

105 é (122 ) (133) 123 cos 2f + 2 I ( ) sin 2f ù cos q sin 2 q I -I úû 4 êë

+

35 é ( 222 ) 233 - 3I ( I 8 ëê

(

333)

(13.64)

)

322 - 3I ( ) sin 3f ù sin 3 q ûú

along with the identities

0 11 22 33 1 111 222 333 2 211 222 233 I ( ) = I ( ) + I ( ) + I ( ); I ( ) = I ( ) + I ( ) + I ( ); I ( ) = I ( ) + I ( ) + I ( )

( 3)

( 311)

( 322 )

( 333)



. I =I +I +I The P1 approximation is much simpler; for the local intensity, only the first four terms in Equation (13.64) appear:

I ( S, q, f ) =

(

)

1 (0) 1 2 3 I + 3I ( ) cos q + 3I ( ) sin q cos f + 3I ( ) sin q sin f . (13.65) 4p

This illustrates why the P1 approximation has been used much more than the P3. To continue the solution, expressions for the moments of intensity must be developed so that explicit relations for intensity for the P3 and P1 approximations can be obtained from Equation (13.64) or (13.65). This is done by generating moment differential equations from the differential RTE, Equation (13.60). The integral in Equation (13.60) is the zeroth moment I(0), so the RTE, Equation (13.60), can be written as 3



¶I

å l ¶t + I = (1 - w) I i

i

i =1

b

+

w (0) I . (13.66) 4p

Equation (13.66) is multiplied by powers of the direction cosines individually and in combination, and the results are then integrated over all solid angles, resulting in the first- and higher-order RTE moment equations (such as Equation (13.10) for the first-order moment equation). Because the derivative terms on the left of Equation (13.66) are with respect to coordinate position while the integrals are over solid angle, the derivative and integral of each term can be interchanged. The results of carrying out these operations for the P3 approximation are the differential equations: 3



å i =1

¶I ( ) 0 = (1 - w) 4pI b - I ( ) (13.67) ¶ti i

(

)

591

Numerical Solution Methods 3

å



i =1

3

å



i =1

ij

(3 equations: j = 1, 2, 3) (13.68)

w (0) ù ¶I ( ) 4p é jk d jk ê(1 - w) I b + ons: j, k = 1, 2, 3) (13.69) = -I ( ) + I ú (9 equatio p ¶ti 3 4 ë û ijk

3

å



¶I ( ) j = -I ( ) ¶ti

i =1

ijkl ¶I ( ) jkl = -I ( ) ¶ti

(27 equations: j, k, l = 1, 2, 3). (13.70)

The δjk is the Kronecker delta, δjk = 1 for j = k, δ = 0 for j ≠ k. For the P1 approximation, only Equations (13.67) and (13.68) are used. Equation (13.70) for the P3 approximation contains a new set of unknowns, the fourth-order moments I(ijkl). For the P1 approximation, the second-order moments I(ij) are present. To close the set of equations, values for these moments must be found. This is done by substituting Equation (13.64) (or Equation (13.65) for the P1 approximation) into the general moment equation (Equation (13.63)) to generate a relation for the fourth moment I(ijkl) (or I(ij) for P1). This relation is not exact because Equations (13.64) and (13.65) were truncated to N = 3 and N = 1, respectively. However, an approximate closure condition is generated. For P3, the result is

(

)

1 ij 1 ijkl ik jk il kl jl I ( ) = I ( ) dkl + I ( ) d jl + I ( ) dil + I ( ) d jk + I ( ) dij + I ( ) dik - ( dij dkl + dil d jk + dik d jl ) . (13.71) 7 35 For P1,

1 ij 0 I ( ) = dij I ( ) (13.72) 3

so that I(11) = I(22) = I(33) = I(0)/3 (which is consistent with the first identity following Equation (13.64)) and all the other I(ij) = 0. The formulation is now complete, in that the number of equations equal to the number of unknowns is available for the moments of intensity. Once these are determined, Equation (13.64) or (13.65) provides the local angular values of the intensity and the problem is, in principle, solved. The boundary conditions needed to solve the moment differential equations are now formulated, and applications of the PN method are then presented to illustrate the use of these relations. 13.2.2.2.1 Boundary Conditions for the PN Method Useful boundary conditions for engineering applications are due to Marshak (1947); they work well for odd-order expansions. Other discussions of boundary conditions are in Mark (1945), Shokair and Pomraning (1981), and Liu et al. (1992b). For an opaque directional-gray surface with emissivity ε(Ω) in the direction of solid angle dΩ, the outgoing intensity Io(Ω) from a boundary is produced by emission and by reflection of incident intensities from Ωi as

I o ( W ) = e ( W ) I b,w +

1 p

2p

ò r ( W, W ) I ( W ) l dW (13.73) i

i

i

i

W i =0

where li is the direction cosine of I(Ωi) relative to the surface normal, the integration for the reflected intensity is over the hemisphere of all incident solid angles Ωi, and ρ(Ω, Ωi) is the bidirectional reflectivity. For a nondiffuse surface, the general Marshak boundary condition is

592

Thermal Radiation Heat Transfer 2p é ù 1 ê e ( W ) I b,w + r ( W, W i ) I ( W i ) li d W i ú d W. (13.74) I o ( W ) Yl ( W ) d W = ê ú p W =0 W =0 ë W i =0 û 2p

2p

ò



ò

m

ò

For 1D cases (no dependence on ϕ in Equation (13.62)), Yl m may be replaced by Yl m = li2 n-1 (n = 1, 2, 3,…) . In this notation 2n − 1 = N. For multidimensional rectangular geometries, Yl m = li (i = 1, 2, 3) and Y3m = li l j lk (i, j, k = 1, 2, 3). These equations provide more relations than are needed, and it is suggested in Ratzel and Howell (1982) that combinations be chosen with moments of the intensity normal to the boundary. Some suggestions for removing ambiguity in the boundary conditions and improving accuracy are in Liu et al. (1992). For the 1D case with n = 1, corresponding to the P1 approximation, the left side of Equation (13.74) reduces to 2p

ò I ( W )Y



o

l

m

( W ) dW =

W =0

2p

ò I ( W )l

2 n -1 i

o

2p

dW =

W =0

ò I ( W )l dW = J (13.75) o

i

W =0

so this condition gives the radiosity J (Figure 13.8b). The other boundary terms do not have a physical interpretation. The use of the PN relations is now demonstrated in an example for conditions where radiation dominates so that conduction and/or convection can be neglected. The P1 relations are used, but the P3 relations can be applied. Example 13.4 Using the P1 approximation, derive relations for the temperature distribution and energy transfer between parallel plane walls at Tw1 and Tw2 for the limit of radiative equilibrium without internal heat sources. Each wall has the same diffuse-gray emissivity εw, and they are separated by an emitting, absorbing, and isotropically scattering medium with albedo ω and optical thickness τD based on the spacing between the walls. Because the geometry requires the radiative energy flux to be only in the l1 direction, it follows that the first moments (equal to the fluxes) I(i) = 0 for j = 2, 3. Then, Equation (13.65) for the P1 approximation becomes

I ( S, q, f ) =

(

)

1 (0) 1 I + 3I ( ) cos q (13.4.1) 4p

and the moment equations (Equations (13.67) and (13.68)) used for the P1 approximation become dI ( ) 0 = (1- w) 4pIb - I ( ) (13.4.2) d t1

(

1



dI ( ) 1 = -I ( ) , d t1 11



dI ( ) = 0, d t1 12

)

dI ( ) = 0. (13.4.3) d t1 13

The closure condition for P1, Equation (13.72), gives I(11) = (1/3)I(0) and I(12) = I(13) = 0. Substituting into the second moment differential equation (Equation (13.4.3)) gives 1 dI ( ) 1 = -I ( ) . (13.4.4) 3 d t1 0



593

Numerical Solution Methods

Now, because I(1) = qr and qr is constant in this geometry for radiative equilibrium with q = 0, it follows from Equation (13.4.2) that I(0) = 4πIb = 4σT4(τ1). Substituting this into Equation (13.69) gives qr = -



4s dT 4 . (13.4.5) 3 d t1

This is the same relation as for the diffusion solution Equation (13.24). However, for other geometries, the P1 solution does not generally provide the diffusion result. Integrating Equation (13.4.5) results in a linear T4 distribution in the medium: sT 4 ( t1 ) = -



3qr t1 + C. (13.4.6) 4

The boundary conditions are applied to relate the temperature distribution to the known temperatures and obtain the integration constant C. Measuring τ1 from the wall at Tw1, Equation (13.74) at this boundary becomes (using Equation (13.75) and Y10 = (3 /4p)1/ 2 cos q) J ( t1 = 0 ) = 2p

p/2

p/2

q =0

q =0

ò

4 Io ( t1 = 0 ) cos q sin qd q = 2ew sTw,1

ò cos q sin qd q

é p/2 ù + 2 (1- e w ) ê 2p I ( t1 = 0, qi ) cos qi sin qi d qi ú cos q sin qd q. ê ú q =0 ë qi = 0 û p/2

ò

(13.4.7)

ò

The incident intensity I(τ1 = 0, θ) in Equation (13.4.7) is expressed by Equation (13.4.1), so that, from the moments that have been found,

I ( t1 = 0, qi ) =

(

)

1 (0) 1 1 I + 3I ( ) cos qi = 4sT 4 ( t1 = 0 ) + 3qr cos qi . 4p 4p

(

)

Substituting into the boundary condition, Equation (13.4.7) gives for the diffuse boundary

J ( t1 = 0 ) = e w sTw41 +

(1- ew ) 2

p/2

ò éë4sT

4

( t1 = 0 ) + 3qr cos qi ùû cos qi sin qi d qi . (13.4.8)

qi = p

From the net-radiation equation (Equation (5.18)) at boundary 1 (note: qr = J − G = q1),

J ( t1 = 0 ) = sTw41 -

(1- e w ) q . (13.4.9) ew

r

Integrating Equation (13.4.8) and using Equation (13.4.9) to eliminate J results in

æ 1 1ö 4 sT 4 ( t1 = 0 ) = sTw1 -ç - ÷ qr . (13.4.10) è ew 2 ø

A similar analysis applied at wall 2 gives

æ 1 1ö 4 sT 4 ( t1 = tD ) = sTw2 +ç - ÷ qr . (13.4.11) è ew 2 ø

594

Thermal Radiation Heat Transfer

Equations (13.4.10) and (13.4.11) are the same as for the jump boundary conditions for diffusion, Equations (13.37) and (13.38). Equation (13.4.10) is used to evaluate the constant in Equation (13.4.6) where σT4(τ1 = 0) = C, resulting in

3q æ 1 1ö 4 sT 4 ( t1 ) = sTw1 -ç - ÷ qr - r t1. (13.4.12) 2 4 e è w ø

Evaluating Equation (13.4.12) at τ1 = τD, substituting Equation (13.4.11) to eliminate T4(τ1 = τD), and solving the result for qr yields

qr 1 = . (13.4.13) 4 3 4 t 2 / ew - 1 / + s T - Tw2 D

(

4 w1

)

This expression for qr can be substituted into Equation (13.4.12) to obtain T4(τ1) in terms of the known boundary conditions (see Table 13.3). This completes the solution.

Although the P1 approximation provides the same result as the diffusion solution for a translucent medium between infinite parallel diffuse-gray walls, for other geometries, it generally provides a different and more accurate solution than the diffusion result. Table 13.3 has P1 predictions for a medium between parallel plane walls, concentric cylinders, and concentric spheres. Figures 13.9 and 13.10 compare the exact solution for energy transfer with the diffusion, P1, and P3 approximations for concentric cylinders and concentric spheres (Bayazitoglu and Higenyi 1979).

Numerical Solution Methods

595

FIGURE 13.9  Comparison of solutions of energy transfer between infinitely long concentric black cylinders enclosing a gray medium in radiative equilibrium with q = 0; Dinner /Douter = 0.5.

FIGURE 13.10  Comparison of solutions of energy transfer between black concentric spheres enclosing a gray medium.

Physically, ψ cannot be larger than 1 and should approach 1 for small optical thicknesses when the bounding surfaces are black. This is the limit for the Monte Carlo solution (Perlmutter and Howell 1964) in Figure 13.9 and the exact analytical solution (Rhyming 1966) in Figure 13.10. The diffusion approximation requires an optically thick medium and is not expected to give good results for small optical thicknesses. It gives good results for optical thicknesses greater than 1 for Dinner/Douter = 0.5. For smaller Dinner/Douter, the diffusion results are not as good, especially for spheres; larger optical thicknesses are required for good agreement with the exact solution. The P3 approximation is better than the P1 and provides good results for Dinner/Douter = 0.5. However, the PN results are poor for

596

Thermal Radiation Heat Transfer

smaller diameter ratios, as shown in Figure 13.10. The PN approximations are based on series expansions about each location, and hence would be expected to give less accurate results as the optical thickness decreases, and each location is influenced to an increasing extent by distant surroundings and boundaries. Additional information is in Heaslet and Warming (1966), Kesten (1968), Loyalka (1969), Schmid-Burgk (1974), Dua and Cheng (1975), Modest (1979, 1989), Mengüç and Viskanta (1985, 1986), Liu et al. (1993a), Tencer (2013), Modest et al. (2014), and Ge et al. (2015). For radiative equilibrium with q = 0, an engineering solution for ψ can sometimes be obtained by using the optically thin solution, which is exact in the limit of the small optical dimension and either the diffusion or the differential approximation at the large optical dimension. A curve faired between these solutions may provide acceptable accuracy for ψ over the entire range of optical thicknesses. However, this approach will not provide the temperature distribution. P1 and P3 solutions are in Ratzel and Howell (1982) for radiative transfer in a rectangular enclosure with diffuse-gray or specular walls at differing isothermal temperatures containing a gray isotropically scattering gas. Profiles of wall heat flux distributions and gas temperature distributions are given. Figure 13.11 shows the error in wall heat flux on the wall opposite the hot wall in the P1 solution relative to the exact analytical solution (Crosbie and Schrenker 1984) for the case of a square black-bounded enclosure with one hot boundary enclosing a cold medium as a function of the medium optical thickness and albedo. The results are from Tencer (2013) and Tencer and Howell (2013b). Good agreement is present for all values of albedo if the optical thickness is less than 1 and for all values of optical thickness if the albedo exceeds 0.9. Errors will be smaller if gray diffuse rather than black boundaries are present, as this will tend to reduce the gradients in intensity.

FIGURE 13.11  Error in P1 solution relative to the exact solution for a square enclosure with one hot and three cold black boundaries and a cold medium. (From Tencer, J. and Howell, J.R., A parametric study of the accuracy of several radiative transport solution methods for a set of 2-D benchmark problems, Proceedings of the ASME 2013 Summer Heat Transfer Conference, Minneapolis, MN, July 14–19, 2013.)

597

Numerical Solution Methods

Sun et al. (2019) use the P1 method with three different spectral models to treat non-gray gases in a spherical geometry and compare with discrete ordinates solutions. 13.2.2.3 Simplified PN (SPN) Method A modified PN method known as the simplified PN (SPN ) method shows superior accuracy and computational efficiency over the conventional PN method for multidimensional problems at low orders. It was proposed on a heuristic basis by Gelbard (1961) and was originally applied to neutron diffusion. Larsen et al. (1993) provided a theoretical justification for the simplification and showed that the SPN approach can be derived from an asymptotic expansion of the intensity in the transport equation. The expansion results in the diffusion solution as the leading order term, with the SPN terms providing higher-order approximations. To obtain the SPN solution, following the derivation in Larsen et al. (2002), the local intensity is expanded in a Neumann series around the local blackbody intensity for the non-scattering case as 2 3 4 é ù 4 2 3 æ d ö æ d ö æ d ö d I l = ê1 m×Ñ + ç ÷ (m × Ñ ) + ç ÷ ( m × Ñ ) - ú I lb . (13.76) ÷ (m × Ñ ) - ç êë kl úû è kl ø è kl ø è kl ø



Here δ is a small expansion parameter defined by δ = 1/κrefLref = 1/τref, so the derivation assumes an optically thick medium with τref ≫ 1 at all wavelengths. Using this scaling, the radiative flux divergence Equation (11.4) in dimensionless terms becomes -Ñ × q r =



¥ é¥ ù 1 ê ú . (13.77) k l p k l I ( W ) d W d I d 4 l l i i l lb ú d2 ê l =0 ë l =0 f û

ò ò

ò

Equation (13.76) is now integrated over the sphere of solid angles, noting that é1 + ( -1)n ù 2p Ñ n . This results in ëê ûú n + 1

ò (m × Ñ ) dW = n

f

é ù d2 d4 d6 4pI l = I l dW = 4p ê1 + 2 Ñ 2 + 4 Ñ 4 + 6 Ñ6 + ú I lb + O(d8 ) (13.78) 3 k 5 k 7 k l l g ë û f

ò



where ∇2 = ∇ ⋅ ∇, etc. The expansion for Iλb then becomes -1

I lb



2 3 4 é ù 4 2 3 æ d ö æ d ö æ d ö d = ê1 m × Ñ + ç ÷ ( m × Ñ ) - ç ÷ ( m × Ñ ) + ç ÷ ( m × Ñ ) - ú éë I l - O(d8 ) ùû êë kl úû è kl ø è kl ø è kl ø

ù é d2 ù ïì é d2 d4 d6 d4 d6 = í1 - ê 2 Ñ 2 + 4 Ñ 4 + 6 Ñ6 + ú + ê 2 Ñ 2 + 4 Ñ 4 + 6 Ñ6 + ú 5k l 7kl 5k l 7kl û ë 3k l û îï ë 3kl

2

(13.79)

3 é d2 ù ïü d4 d6 - ê 2 Ñ 2 + 4 Ñ 4 + 6 Ñ6 + ú ý éë I l - O(d8 ) ùû . 5k l 7kl ë 3k l û ïþ

Neglecting terms of order greater than O(δ6) and gathering the remaining terms gives the SPN expansion for intensity:

æ 4d4 4 44d6 6 ö d2 I lb = ç 1 - 2 Ñ 2 Ñ Ñ ÷ I l . (13.80) 45kl4 945k6l è 3k l ø

598

Thermal Radiation Heat Transfer

13.2.2.3.1 SP1 Solution Restricting Equation (13.80) to terms of order lower than δ4 gives

k l ( I lb - I l ) = - d 2 Ñ ×

1 ÑI l + O(d4 ). (13.81) 3k l

Integrating Equation (13.81) over wavelength and inserting it into Equation (13.78) gives

Ñ × qr =

4p d2

¥

ò

k l ( I lb - I l ) d l = -

l =0

¥

4p

ò Ñ × 3k

l =0

ÑI l dl. (13.82)

l

Equation (13.82) is the SP1 approximation for the local radiative flux divergence, accurate to order δ2. To order δ2, Equation (13.79) indicates that I l = I lb , so Equation (13.82) predicts the local radiative flux divergence as ¥



Ñ × qr = -

ò

Ñ×

l =0

4p ÑI lb dl = -Ñ × 3k l

¥

4p

ò 3k

l =0

ÑI lb dl = 0 (13.83)

l

indicating that the local radiative flux is ¥



qr = -

4

ò 3k

l =0

n ×ÑElb dl = l

4s n ×ÑT 4 , (13.84) 3k R

which is the same as the Rosseland diffusion result of Equation (13.24) as well as the P1 solution of Equation (13.4.5). 13.2.2.3.2 SP1 Boundary Conditions Larsen et al. (2002) note that the boundary conditions for the SP1 solution in general geometries reduce to the same Marshak conditions found for the P1 solution (Equations (13.4.9) through (13.4.12)), which are in turn the same as those found for the diffusion solution (Equations (13.33) and (13.34) or (13.37) and (13.38)). The SP1 solution and boundary conditions for radiative transfer in general geometries are seen to be equivalent to the P1 and diffusion solutions. This is not the case for higher-order SPN solutions. 13.2.2.3.3 Higher-Order Solutions Retaining one more term in Equation (13.80) than for the SP1 solution gives the SP2 relation accurate to order δ4:



æ 4d 4 4 ö d2 I lb = ç 1 - 2 Ñ 2 Ñ ÷ Il 45kl4 è 3k l ø 1 = Il - d Ñ × 2 3k l 2

(13.85)

é ù 4 2 ê I l + d Ñ × 15k2 ÑI l ú l ë û

or

I l = I lb + d 2 Ñ ×

1 ÑI l + O(d4 ). (13.86) 3k2l

599

Numerical Solution Methods

Substituting Equation (13.86) into Equation (13.85) results in I lb = I l - d 2 Ñ ×

1 4 1 4 é ù ÑI l - d 2 Ñ × Ñ ( I l - I lb ) = I l - d2 Ñ × 2 Ñ ê I l + ( I l - I lb ) ú . (13.87) 2 2 5 3k l 15kl 3k l ë û

Integrating as in Equation (13.82) gives

Ñ × qr =

4p d2

¥

ò

k l ( I lb - I l ) d l = -

l =0

¥

4p

ò Ñ × 3k

l =0

l

4 é ù Ñ ê I l + ( I l - I lb ) ú dl + O(d4 ). (13.88) 5 ë û

The kernel of the integral in the SP2 result is the same as for the SP1 solution (Equation (13.82)) modified by the addition of the term 4/5 ( I l - I lb ) , and the local radiative flux is

4p n ´ qr = 3

¥

ò

l =0

1 4 4p é ù n × Ñ ê I l + ( I l - I lb ) ú d l = kl 5 3 ë û

¥

1

òk

l =0

l

4 ù é9 n × Ñ ê I l - I lb ú d l. (13.89) 5 û ë5

Noting that Equation (13.89) gives the local spectral radiative flux as qr ,l = -



4p Ñ é9I l - 4 I lb ùû , (13.90) 15kl ë

then Equation (13.87) gives the local value for I l as I l = I lb + d 2 Ñ ×



1 d2 é ù 9 4 Ñ I I = I n ×Ñq r ,l . (13.91) l l b l b û 15kl2 ë 4pkl

Retaining one additional term in Equation (13.80) than for the SP1 result of Equation (13.81) gives the SP3 relation accurate to order δ6: æ 4d4 4 44d6 6 ö d2 I lb = ç 1 - 2 Ñ 2 Ñ Ñ ÷ Il 45kl4 945k6l è 3k l ø

= Il -

d2 2 é 4d 2 2 44d4 4 ù Ñ I I + Ñ + Ñ Il ú l l ê 3k2l 15k2l 315kl4 ë û

= Il -

æ d2 2 é 11d2 2 ö æ 4d2 2 ö ù Ñ ÷ç Ñ Il ÷ú . I Ñ + 1 + ê l ç 2 21k2l 3k2l êë è ø è 15kl ø úû

(13.92)

æ 11d2 2 ö æ 4d2 2 ö Defining I l( 2 ) = ç 1 + Ñ ÷ç Ñ I l ÷ , Equation (13.92) becomes 2 21kl2 è ø è 15kl ø I lb = I l - d 2 Ñ ×



1 Ñ é I l + I l( 2 ) ùû . (13.93) 3kl2 ë

Integrating Equation (13.93) over wavelength and the sphere of solid angles gives ¥



òò

f l =0

k l ( I lb - I l ) d l d W = - d 2

¥

1

ò ò Ñ × 3k f l =0

l

( )

Ñ éë I l + I l( 2 ) ùû d ld W + O d6 . (13.94)

600

Thermal Radiation Heat Transfer

Substituting into Equation (13.82) to determine the radiative flux divergence gives 4p Ñ × qr = 2 d



¥

ò k (I l

l =0

lb

- I l ) dl = -

¥

4p

ò Ñ × 3k

l =0

l

Ñ éë I l + I l( 2 ) ùû dl. (13.95)

Comparison with Equation (13.83) for the SP1 (diffusion) prediction of radiative flux divergence shows that the additional term I l( 2 ) has now entered the radiative flux divergence. Equations (13.93) and (13.95) and associated boundary conditions comprise the SP3 solution. 13.2.2.3.4 Boundary Conditions for Higher-Order SPN Solutions The emissive power jump boundary conditions of Marshak (Equations (13.35) through (13.38)) are used with the SPN equations to generate appropriate boundary conditions for the higher-order solutions. Tomašević and Larsen (1975) and Brantley and Larsen (2000) derive the boundary conditions for black boundaries, and Larsen et al. (2003) extend these to nongray boundaries with directional reflectivity. For a gray diffuse surface at x = 0, the SP2 boundary relations reduce to 5I l ( 0 ) +



2-e 4 n ×ÑI l ( 0 ) = éë6 I lb,w - I lb ( 0 ) ùû . (13.96) e kl

Substituting Equation (13.91) for I l ( 0 ) and rearranging the equation gives

El b , w - El b ( 0 ) æ 1 1ö çe -2÷ è l ø

=

æ 1 ö 1 -5 4 Ñ ×ç Ñq r ,l ÷ . (13.97) n ×Ñq r ,l + n ×ÑElb ( 0 ) kl 3k l æ 1 1ö è 3k l ø 24 kl ç - ÷ e 2 l è ø

For the case of no sources in the medium, ∇qr,λ = 0 and Equation (13.97) reduces to Equation (13.37). The boundary conditions for the SP3 relations result in a pair of weakly coupled differential equations that are derived in Larsen et al. (2002). These authors observe that the SP1 equation and boundary conditions are identical with P1 solutions for all geometries, and that for planar geometries the SPN and PN equations and boundary conditions are also identical. However, for multidimensional geometries, the SP2 and higher-order SPN solutions differ from the PN solutions, and the SP2 and SP3 relations are found to be more accurate and easier to program and implement than the corresponding PN equations. Morel et al. (1979) have implemented the SPN method into a general unstructured grid code. Cai et al. (2012) employ various SPN orders with k-distributions to study spectral effects in hydrogen–air diffusion flames. Zhang et al. (2013b) compare the convergence rate among various formulations of the SPN method. 13.2.2.4 MN Method The MN method for solving the RTE uses entropy maximization as the closure principle for the first two (zeroth and first) RTE moment equations (Equations (13.6) and (13.10)) (Minerbo, 1978). The derived closure relation is exact in the diffusion and transport limits; however, the method itself is not exact, in that it uses the first two moment equations plus closure, rather than the RTE itself (Ripoll and Wray 2003, Berthon et al. 2007, González et al. 2008, 2009). The maximum entropy closure condition gives a nonlinear coefficient on the radiative flux equation. The MN method is quite useful in neutron transport solutions, where the time of flight of neutrons in transient problems requires that the transient terms in the neutron transport equation be retained. Because this term is generally negligible in photon transport, the MN solutions have been found to be less useful. In the first moment equation, Equation (13.12), for isotropic or no scattering, or in fact for any source function that is symmetric around the propagation direction, the integral term is zero. With this assumption, the closure condition is given by assuming the radiative pressure term to

601

Numerical Solution Methods

be proportional to the gradient in energy density with a variable coefficient ζ(R). Substituting into Equation (13.12), with the assumption noted and treating the medium as gray,

4 4p qr (S ) = cPij (S ) = - z( R)cÑU (S ) = z( R)ÑI (S ). (13.98) b b

For no scattering, this becomes

qr ( S ) = -

4 z( R)ÑEb (S ). (13.99) k

Various forms have been suggested for the variable coefficient ζ(R), called the flux limiter (Pomraning 1982, Levermore 1984, Brunner and Holloway 2001, Frank 2007, Fan 2012). Flux limiters are required to avoid nonphysical solutions from arising in diffusion-based solutions for certain combinations of extinction coefficient and any constant coefficient on the diffusion relation, such as on the Rosseland equation, Equation (13.25). The basic constraint can be seen by examining the first and second moment equations, Equations (13.2) and (13.3), which require that qr (S ) < 4pI (S ). Using entropy maximization and this constraint for 1D M1 solutions, the value of the variable flux limiter is found by solving the transcendental equations (Minerbo 1978, Tencer 2013a): z( R ) =

1 - csch 2 ( Z ) Z2

coth( Z ) - (1/Z ) R= z( R )

(13.100)

 where R is a function of the effective albedo wand R is given by Pomraning (1982) as

éæ ¶I (i ) (S ) ö ù  -Ñ × qr 1- w  I ( 0 ) (S ) ú = = R = êç . (13.101) ÷ bw  I (S )  w êëè ¶S ø úû 4pbw

The parameter R is thus in the range 0  n2, as discussed

Radiative Effects in Translucent Solids

787

FIGURE 17.9  Reflection and transmission of electromagnetic wave in gap between two dielectrics; T3  χ and r⊥ is negative; that is, there is a phase change of π upon reflection. The tan(θ − χ) is positive, but tan(θ + χ) becomes negative for θ + χ > π/2, and r|| then yields a phase change of π. In a similar fashion, for transmitted radiation



Eçç,t 2 sin c cos q = tçç = Eçç,i sin(q + c) cos(q - c)

(17.48)

E ^ ,t 2 sin c cos q = t^ = . (17.49) E ^ ,i sin(q + c)

In going from medium 2 to medium 1, the χ and θ values are interchanged in these relations (the χ and θ remain the angles in 2 and 1, respectively), and the reflection coefficients are equal to −r|| and −r⊥. In this instance, the transmission coefficients are called tʹ and are equal to



t||¢ =

2 sin q cos c (17.50) sin(c + q) cos(c - q) t ^¢ =

2 sin q cos c . (17.51) sin(c + q)

For the simplified case of normal incidence, and for radiation going from medium 1 to medium 2, these reduce to

Er n -n = r|| = r^ = r = 1 2 (17.52) Ei n1 + n2



2n1 Et = t|| = t ^ = t = . (17.53) n1 + n2 Ei

Formally, r|| has a negative sign as E||,r and E||,i point in opposite directions. This sign is not significant in the present discussion. For normally incident radiation going from medium 2 into medium 1

r=

n2 - n1 (17.54) n2 + n1



t¢ =

2n2 . (17.55) n2 + n1

For simplicity, the following discussion is limited to normal incidence on the thin film. Figure 17.14 shows the radiation reflected from the first and second interfaces (note: for clarity in showing each path, the paths are drawn at an angle). The beams a and b can interfere with each other. For normal incidence, beam b reflected from the second interface travels 2D farther than beam a, which is reflected from the first interface. Hence reflected beam b originated at time −2D/c1 earlier than reflected beam a, where c1 is the propagation speed in the film. If beam a originated at time 0, then beam b originated at time −2D/c1. If the two waves originated from the same vibrating source, the phase of b relative to a is eiωτ = e−iω2D/c1. The circular frequency can be written as ω = 2πc0/λ0, where λ0 is the wavelength in vacuum. Then, eiωτ = e−i4πn1D/λ0 where n1 = c0/c1 is the film refractive index.

794

Thermal Radiation Heat Transfer

FIGURE 17.14  Reflection from the first and second interfaces of a thin film. This is for normal incidence; paths are drawn at an angle for clarity.

FIGURE 17.15  Multiple reflections within a thin nonattenuating film for normal incidence; paths are drawn at an angle for clarity.

Consider a thin nonabsorbing film of refractive index n1 on a substrate with index n2. For a normally incident wave of unit amplitude, the reflected radiation is shown in Figure 17.15. Accounting for the phase relationships and defining γ1 ≡ 4πn1D/λ0, the reflected amplitude is

RM = r1 + t1t1¢r2 e -ig1 - t1t1¢r1r22 e -2ig1 + t1t1¢r12r23e -3ig1 -  = r1 +

t1t1¢r2 e -ig1 . (17.56) 1 + r1r2 e -ig1

Note that t1t1¢ = 1 - r12 , so this can be reduced to

RM =

r1 + r2 e -ig1 . (17.57) 1 + r1r2 e -ig1

An important application for a thin coating is to obtain low reflection from a surface to reduce reflection losses during transmission through a series of lenses in optical equipment. To have zero reflected amplitude, R M = 0, requires r1 = −r2e−iγ1. This can be obtained if r1 = r2 and e−iγ1 = −1. Since e−iπ = −1, this yields D = λ0/4n1. The quantity λ0/n1 is the wavelength of the radiation within the film. Hence the film thickness for zero reflection at normal incidence is one-quarter of the wavelength within the film. The required condition is (ns−n1)/(ns+n1) = (n1−n2)/(n1+n2), which reduces to n1 = ns n2 . Thus, for normal incidence onto a quarter-wave film from a dielectric medium with

795

Radiative Effects in Translucent Solids

index of refraction ns, the index of refraction of the film for zero reflection should be n1 = ns n2 , the geometric mean of the n values on either side of the film. The thin film provides better performance than the thick film, as it is possible to achieve zero reflectivity. However, this result is only for normal incidence at a single wavelength. To obtain more than one condition of zero reflectivity, it is necessary to use multi-layer films. The optimization of the design of low-reflectivity multi-layer coatings is discussed by Thornton and Tran (1978). For a system of two nonabsorbing quarter-wave films, zero reflection is obtained for n12 n3 = n22 ns, where n1 and n2 are for the coatings (the coating with n2 is next to the substrate) and n3 is for the substrate. The previous expressions have been for the reflected amplitude from a thin film; now the reflected energy for normal incidence is considered. From Section 8.4, the reflected energy depends on |E|2. 2 Since R M = Er/Ei, the reflectivity for energy is R = RM = RM RM* where RM* is the complex conjugate of R M. From Equation (17.57), the reflectivity of the film is R=



r1 + r2 e -ig1 r1 + r2 eig1 . 1 + r1r2 e -ig1 1 + r1r2 eig1

After multiplication and simplification, this becomes R=



r12 + r22 + 2r1r2 cos g1 . (17.58) 1 + r12r22 + 2r1r2 cos g1

Inserting r1 = (ns−n1)/(ns+n1) and r2 = (n1−n2)/(n1+n2), using the identity cosγ1 = 1−2sin2(γ1/2), and simplifying gives

R=

n12 (ns - n2 )2 - (ns2 - n12 )(n12 - n22 ) sin 2 (2pn1 D /l 0 ) . (17.59) n12 (ns + n2 )2 - (ns2 - n12 )(n12 - n22 ) sin 2 (2pn1 D /l 0 )

For a quarter-wave film, D = λ0/4n1 and this reduces to 2



æ n n - n2 ö R = ç s 2 12 ÷ . (17.60) è ns n2 + n1 ø

The reflectivity becomes zero when n1 = ns n2 . If n1 is high, R is increased, and this behavior can be used to obtain dielectric mirrors. For various film materials of refractive index n1 on glass (n2 = 1.5), the reflectivity becomes, for incidence in air (ns ≈ 1),

Multilayer films can be used to obtain reflectivities very close to 1. An application is for reflection of high-intensity laser beams where the fractional absorption of energy must be kept very small to avoid damage from heating the mirror. Thin films of superconducting material are predicted to have absorptivity approaching zero out to very large wavelengths (Zeller 1990). This high reflectivity gives some promise of producing nearly ideal (zero heat loss) radiation shields, particularly for preventing energy loss from cryogenic

796

Thermal Radiation Heat Transfer

storage systems in outer space. This technology will be of increasing usefulness as the critical temperature Tc of superconductors reaches higher values with new materials. A thin film on a rough surface was investigated by Tang et al. (1999a, b). Paretta et al. (1999) provide measurements of bidirectional spectral reflectivity for various surface treatments to the surface of photovoltaic cells. 17.4.2.2 Absorbing Thin Film on a Metal Substrate In this instance, the complex refractive indices n1−ik1 and n2−ik2 replace the n1 and n2 of the previous section. For normal incidence, the R = |R M|2 becomes, by use of Equation (17.57) 2



r + r e -ig1 R = 1 2 -ig1 (17.61) 1 + r1r2 e

where r1 and r2 now contain n1 and n2 and γ1 contains n1. The behavior of Equation (17.61) is studied by Snail (1985) to obtain a selective surface for absorption of solar energy. The R values of some coatings on an aluminum substrate are in Figure 17.16. The two complex indices of refraction of the coating in Figure 17.16 illustrate a trade-off between a low reflectivity for solar energy (high solar absorption) and the width of the transition region between the visible and infrared regions. Coating 1, which is thicker and has a lower n, has a significantly higher solar absorptance than coating 2, but it also has a wider transition region and would re-emit more in the infrared region, thereby increasing energy loss by a solar collector. The lower n for coating 1 provides lower reflection losses at the front surface of the coating, but its higher value of κD produces a wider transition region. A method for improving the performance of the film–substrate combination for use as a solar collection surface is to have the refractive index n vary within the film, from the substrate value at the interface with the substrate to the value for the incident medium (air) at the front surface. This is explored by Snail (1985), and references are given for analytical solutions for the use of a graded index of refraction. A similar strategy can be used to produce fiber-optic cables with low losses by using radially varying refractive index. Zhu et al. (2011) use curved ray tracing to handle a spatially varying refractive index in an isotropically scattering medium. Degheidy et al. (2016) apply a Galerkin technique to the same situation and predict the

FIGURE 17.16  Spectral reflectivity for two homogeneous films on an aluminum (n = 1.50 - 10i ) substrate for normal incidence (from Snail, K.A., Solar Energ. Mater., 12, 411, 1985).

Radiative Effects in Translucent Solids

797

reflection and transmission coefficients of slabs. Hosseini-Sarvari (2016, 2018) investigates radiative transfer in semitransparent media with a variable refractive index. This development has shown how the net-radiation method and ray tracing can be used to analyze the radiative behavior of partially transparent coatings. Extensive information is available on coatings and thin films. In addition to the references already discussed, further information is in Hsieh and Coldewey (1974), Musset and Thelen (1970), Forsberg and Domoto (1972), Taylor and Viskanta (1975), Heavens (1965, 1978), Roux et al. (1974), Palik et al. (1978), and Palik (1998). Palik has reported extensive spectral data for n and k for metals, semiconductors, and dielectrics. By use of the relation κ = 4πk/λ, the spectral absorption coefficient can be found from the k values in the complex refractive index.

17.4.3 Films with Partial Coherence When an incident wave enters a thin film, it is refracted and propagates into the film. The incident wave is joined at the first interface by the wave that has been reflected from the substrate and re-reflected from the front surface interface. This re-reflected wave has a time lag relative to the refracted incident wave. The two portions of the forward-propagating wave are thus partially coherent. In the limits of a thick film, the two portions become completely incoherent, and the theory presented in Section 17.4.1 (geometric optics) applies. If the film is thin, with thickness ≈ λ or less, the theory in Section 17.4.2 (and the more complete treatment in Chapter 16) for a coherent wave applies. However, there is a significant portion of the wavelength range for a given film thickness where neither approach provides accurate results. This is the region of partial coherence. This region is analyzed by Chen and Tien (1992), and relations are presented for determining the film transmittance and absorptance. The region of partial coherence for nearly monochromatic incident radiation with frequency range Δn is bounded by 1.13 ≤ f ≤ 2.59, where f = 4πnDΔn/c0 and c0 is the speed of light in a vacuum. Above this range for f, geometric optics can be used; below this value of f, coherent optics needs to be considered, meaning that Maxwell and Fresnel equations should be used. Information is in Phelan et al. (1992) on the effect of thickness on radiative performance of a superconducting thin film on a substrate, and Fu et al. (2006) provide an analytical formulation. For thin films or coatings on substrates exposed to radiation with very narrow spectral spans, such as a laser beam, the partially coherent theory becomes important in predicting the reflectance and absorptance of the films. To examine other wave and tunneling effects in thin films, particularly when the gap between the plates is smaller than the wavelength of the incident radiation, it is necessary to include near-field effects (Chapter 16).

17.5 REFRACTIVE INDEX EFFECTS ON RADIATION IN A PARTICIPATING MEDIUM Each of the partially transmitting layers in Sections 17.2 through 17.4 is assumed to have a uniform temperature throughout their thicknesses. It is also assumed that their temperatures are low enough that internal radiation emission from the layers is not significant. The behavior of layers at elevated temperatures and with internal temperature variations can be analyzed by using the radiative transfer equations given in Chapters 11 through 14, but modifying them for n ≠ 1, along with appropriate conditions at the boundaries that include refractive index effects. These effects are now discussed for use in detailed heat transfer analyses.

17.5.1 Effect of Refractive Index on Intensity Crossing an Interface Consider radiation with intensity Iλ,1 in an ideal dielectric medium of refractive index n1. Let the radiation in solid angle dΩ1 pass into an ideal dielectric medium of refractive index n2 as in Figure 17.17. As a result of the differing refractive indices, the rays change direction as they pass into medium 2. The radiation in the solid angle dΩ1 at incidence angle θ1 passes into solid angle dΩ2

798

Thermal Radiation Heat Transfer

FIGURE 17.17  Radiation with intensity Iλ,1 crossing interface between two ideal dielectric media having unequal refractive indices.

at angle of refraction θ2. After allowing for reflection, the radiative energy is conserved in crossing the interface. From the definition of intensity, this energy conservation is given by

I l,1[1 - rl (q1 )]cos q1dAd W1d l1 = I l,2 cos q2 dAd W2 d l 2 (17.62)

where ρλ(θ1) is the directional–hemispherical reflectivity of the interface dA is an area element in the plane of the interface. The λ1 and λ2 are related by λ2 = (n1/n2)λ1. Note that Iλ,2 does not include internal radiation reflected from the inside of the interface. Using the relation for solid angle, dΩ = sinθdθdϕ, Equation (17.61) can be modified as (noting that the increment of circumferential angle dϕ is not changed in crossing the interface)

I l,1[1 - rl (q1 )]sin q1 cos q1d q1d l1 = I l,2 sin q2 cos q2 d q2 d l 2 . (17.63)

From Equation (3.3), Snell’s law relates the indices of refraction to the angles of incidence and refraction by n1/n2 = sinθ2/sinθ1 and by differentiation n1cosθ1dθ1 = n2cosθ2dθ2. Substituting into Equation (17.63) gives

I l,1[1 - rl (q1 )]d l1 I l,2 d l 2 = . (17.64) n12 n22

For spectral calculations with variable n, it is better to work with frequency than with wavelength; this avoids introducing relations between λ and n. Frequency does not change with n, so Equation (17.64) becomes

I n,1[1 - rn (q1 )] I n,2 = 2 . (17.65) n12 n2

799

Radiative Effects in Translucent Solids

17.5.2 Effect of Angle for Total Reflection Consider a volume element dV inside a medium with refractive index n2 as in Figure 17.18. Suppose that diffuse radiation of intensity I1 is incident upon the boundary of this region from a medium having refractive index n1, where n1  n /n . From Snell’s 2

2

1

2

law (Equation (3.3)), this means that such a ray would enter medium 1 at an angle given by sin q1 = (n2 /n1 ) sin q*2 > (n2 /n1 )(n1 /n2 ) = 1. Since sin θ1 cannot be greater than 1, any ray incident on the interface from medium 2 at an angle greater than θ2,max = sin−1(n1/n2) cannot enter medium 1 and has total internal reflection at the interface. The θ2,max defined by Equation (17.66) is the angle for total internal reflection.

FIGURE 17.18  Effect of refraction on radiation transport in media with non-unity refractive index.

800

Thermal Radiation Heat Transfer

From Section 1.9.8, the blackbody spectral intensity emitted locally inside a medium with n ≠ 1 and with n a function of frequency is

I nb,m d n =

2nn2C1n3 d n (17.67) co4 (eC2n / coT - 1)

where C1 and C2 are the coefficients defined in Chapter 1. If spectral variations of no are important, integration over frequency should be performed during the solution to obtain total energy quantities, provided that data for nv as a function of n are available. If the refractive index is constant with frequency, integrating Equation (17.67) over all n yields the local total emitted blackbody intensity inside a medium:

I b,m = n2 I b (17.68)

where Ib in this relation is for n = 1. Consequently, for an absorbing–emitting gray medium (n not a function of frequency) with absorption coefficient κ, the total energy dQe emitted by a volume element dV at temperature T has an n2 factor:

dQe = 4n2 ksT 4 dV . (17.69)

From Equation (17.68), it might appear that because n > 1, the intensity radiated from a dielectric medium into air could be larger than blackbody radiation Ib = σΤ4/π. This is not the case, as some of the energy emitted within the medium is reflected back into the emitting body at the inside surface of the medium–air interface. Consider a thick ideal dielectric medium (k = 0) at uniform temperature and with refractive index n. The maximum intensity received at an element dA on the interface from all directions within the medium is n2Ib. Only the energy received within a cone having a vertex angle θmax relative to the normal of dA can penetrate through the interface; for incidence angles larger than θmax, the energy is totally reflected into the medium. Hence, the maximum amount of energy received at dA that can leave the medium is

ò

qmax

q= 0

(

)

2pn2 I b dA cos q sin qd q = 2pn2 I b dA sin 2 qmax / 2 .

From Equation (17.66), with n1 = 1 and n2 = n in this case, sinθmax = 1/n, so the total hemispherical emissive power leaving the interface is 2πn2Ib/2n2 = πIb. Dividing by π gives Ib, as the maximum diffuse intensity that can leave the interface, which is the expected blackbody radiation intensity. For a real interface, there is partial internal reflection for the angles where 0 ≤ θ ≤ θmax, and the intensity leaving the outside of the interface is less than Ib.

17.5.3 Effects of Boundary Conditions for Radiation Analysis in a Plane Layer In Section 12.2, the plane layer of semitransparent medium with n = 1 was analyzed. Several results presented in Chapter 12 were for a layer bounded by diffuse or black walls, so the intensities at the boundaries of the translucent layer with n = 1 were 1/π times the diffuse fluxes leaving the walls. Here, we consider a plane layer within a surrounding medium with a different refractive index, such as a glass plate in air or in water, or a layer of ceramic in a high-temperature gaseous environment. Radiation is incident from the surrounding medium, and some of it crosses the boundaries into the plane layer. Expressions are obtained for the intensities inside the plane layer at the boundaries. The analysis of Chapter 12 can then be applied using these internal boundary conditions. 17.5.3.1 Layer with Nondiffuse or Specular Surfaces The layer in Figure 17.19 has smooth surfaces that are not diffuse reflectors. As discussed in connection with Equation (17.65), it is preferable to use frequency as the spectral variable when radiation

801

Radiative Effects in Translucent Solids

FIGURE 17.19  Intensities in a plane layer surrounded by a medium with a different refractive index.

crosses an interface between media having different refractive index n. The reason for this was discussed in Chapter 1, where we noted that the frequency is independent of the medium properties; therefore, the same value of frequency is valid in all media. On the other hand, the wavelength within a medium changes inversely with n, as λ = λ0/n. Both the plane layer and the surrounding medium are assumed to be ideal dielectrics with regard to interface behavior. A subscript s (for “surroundings”) is used to designate conditions outside the layer. The angles θ and ϕ give the direction within the medium. The θi and ϕi fix the incident directions at the boundaries. The qv,s dv are the spectral fluxes incident on the layer from within the surrounding medium; these fluxes are assumed uniform over the layer boundary. The intensity I n+ (0, q, f) leaving boundary 1 inside the medium in Figure 17.19 is composed of a transmitted portion from In,s(0,θi,ϕi) and the reflected portion of I n- (0, qi , fi ) . The bidirectional spectral reflectivity Equation (2.40) relates the reflected and incident intensities by

rn (q, f, qi , fi ) =

I n,r (q, f, qi , fi ) . (17.70) I n,i (qi , fi ) cos qi dW

Similarly, a bidirectional transmissivity of the interface is defined (Equation (2.72) as

tn (q, f, qi , fi ) =

I n,t (q, f, qi , fi ) (17.71) I n,i (qi , fi ) cos qi dW i

where Iv,τ is the intensity in the direction θ, ϕ obtained by transmission. The τn contains a factor of (nv/nv,s)2 to account for the effect in Section 17.5.1. By integrating over all incident solid angles at boundary 1, the intensity inside the layer leaving the boundary is

802

Thermal Radiation Heat Transfer 2 p p/2

ò òt

+ n

I (0, q, f) =

n ,1

(q, f, qi , fi )I n,s (0, qi , fi ) cos qi sin qi d qi d fi

fi = 0 qi = 0



2p

ò ò

+

(17.72)

p

rn,1 (q, f, qi , fi )I n- (0, qi , fi ) cos qi sin qi d qi d fi .

fi = 0 qi = p / 2

Similarly, inside the medium at boundary 2 2p



p

ò ò

I n- ( D, q, f) =

tn,2 (q, f, qi , fi )I n,s ( D, qi , fi ) cos qi sin qi d qi d fi

fi = 0 qi = p / 2

(17.73)

2 p p/2

+

ò òr

n ,2

(q, f, qi , fi )I n+ ( D, qi , fi ) cos qi sin qi d qi d fi .

fi = 0 qi = 0

As a special case, let the media be ideal dielectrics and the layer have optically smooth surfaces with incident energy fluxes qn,s(0)dν at x = 0 and qv,s(D)dv at x = D that are diffuse. As discussed in Chapter 3, reflection at an interface depends on the component of polarization. The analysis should consider the portion of radiation in each polarization component and then add the two energies to obtain the total quantity. For simplicity, this effect is neglected at present, and average values of the surface properties are used. The two components of polarization can be included by using Equations (3.6) and (3.7), as illustrated in Example 17.2. The internal reflections from the optically smooth interfaces are specular; hence θ = π−θi and ϕ = ϕi + π, and, from Equation (3.9), at x = D

I n- ( D, q, f) 1 sin 2 (qi + c) é cos2 (qi + c) ù = ê1 + ú (17.74) + I n ( D, qi , fi ) 2 sin 2 (qi + c) ë cos2 (qi - c) û

and similarly, for I n+ (0, q, f) / I n- (0, qi , fi ) where π−θi is used on the right in place of θi. The χ is determined from Snell’s law, so that, for the internal reflections, sin χ/sin(π−θi) = sin χ/sin θi = nn/nn,s at x = 0 (boundary 1) and sin χ/sin θi = nn/nn,s at x = D (boundary 2), where the subscript s designates the surroundings outside the layer. For some conditions and directions, there is total internal reflection, so the intensity reflected from the interface has the same magnitude as the incident intensity. This occurs when the index of refraction inside the medium is greater than that outside, nv > nv,s. There is total reflection when θi > θmax, where θmax = sin−1(nn,s/nn). For the transmitted intensity, the factor (nv/nv,s)2 is included to account for the effect discussed in Section 17.5.1, and incidence is from the surrounding medium onto the layer. Then, using Equation (3.10) for either boundary 1 or 2 gives 2



ìï sin 2 (qi - q) é cos2 (qi + q) ù üï æ nn ö I n+ (0, q, f) I - ( D, q, f) = n = í1 ÷ (17.75) ê1 + úýç I n,s (0, q, fi ) I n,s ( D, qi , fi ) ïî 2 sin 2 (qi + q) ë cos2 (qi - q) û þï è nn,s ø

where θ and θi are related by sin θ/sin θi = nn,s/nn. For diffuse incident fluxes in Figure 17.19, the externally incident intensities are given by

I n,s (0, qi , fi ) =

qn,s (0) (17.76) p

803

Radiative Effects in Translucent Solids

I n,s ( D, qi , fi ) =



qn,s ( D) . (17.77) p

Example 17.2 A volume element dV is located inside glass at x = 3 cm from an optically smooth planar interface in air, as in Figure 17.20. A diffuse–gray radiation flux qi = 40 W/cm2 in the air is incident on the glass surface. The absorption coefficient of the glass is assumed constant at κ = 0.08 cm –1 and the refractive index of the glass is n = 1.52. Scattering is neglected. Determine the energy absorption rate per unit volume in dV. Since the incident energy is diffuse and unpolarized, the incident intensity in each component of polarization is G/2π. The fraction of the intensity transmitted through the interface depends on the angle of incidence and is 1 − ρ(θ), where ρ(θ) is given for each component of polarization by Equations (3.6) and (3.7). From Equation (17.64), the intensity inside the medium for each component of polarization requires multiplying the incident intensity by n2 in addition to a reflectivity factor itself. The intensity in the medium in direction χ then becomes (G/2π)[1−ρ(θ(χ))]n2, where χ is the angle of refraction. The path length traveled from the interface to the volume element is x/cosχ. Using Bouguer’s law, the fraction reaching dV is e−κx /cosχ. The fraction κdx/cos χ of this incident energy at dV, IdV(χ)dA cos χ, is then absorbed in the volume element dA dx. The energy absorption for all directions of the arriving energy is found by integrating over 0 ≤ χ ≤ χmax, where χmax is given by Snell’s law as sin–1(l/n). The integration over all incident solid angles introduces the factor for solid angle, 2π sin χ dχ. The energy absorbed per unit volume in dV is obtained by doing the integration for each component of polarization and summing the results. This yields

dQ = kGn2 dV

cmax

ò [ 2 - r ( c) - r ( c)] e 

^

- kx /cos c

sin cd c.

0

From Equations (3.7) and (3.8), the reflectivities are given by 2



é n2 cos qi - (n2 - sin2 qi )1/ 2 ù r (c) = ê 2 2 2 1/ 2 ú ë n cos qi + (n - sin qi ) û



é (n2 - sin2 qi )1/ 2 - cos qi ù r ^ ( c) = ê 2 ú 2 1/ 2 ë (n - sin qi ) + cos qi û

2

where the θi = θi(χ) is given by Snell’s law as θi(χ) = sin−1(n sin χ). Using the specified values, the integration is carried out numerically to give dQ/dV = 3.38 W/cm3.

FIGURE 17.20  Schematic for Example 17.2.

804

Thermal Radiation Heat Transfer

17.5.3.2 Diffuse Surfaces For high-temperature applications such as in combustion chambers for advanced aircraft engines, it may be necessary to use ceramic parts or ceramic coatings to protect metal components. Some ceramics are partially transparent for thermal radiation, and they may be somewhat crystalline. They scatter strongly and their surface may not be smooth enough to use the specular (mirror-like) reflection assumption. Instead, it is assumed that they reflect randomly in all directions, and for such a surface, the transmitted and reflected external or internal radiation would be diffuse, as in the case of frosted glass. We assume radiation from the interior of the medium that reaches a boundary is diffuse. Note that there is a difference between specular (mirror-like) boundaries and diffuse boundaries. When the boundary is specular, part of the radiation reaching to the surface from a high index of refraction side would be trapped inside the medium due to total internal reflection. However, for the diffuse surface, part of the radiation leaving the rough interface going into a material of a lower refractive index can be from within the angular range where there would be total internal reflection for an optically smooth interface. This affects the path lengths being followed by reflected radiation within the layer. For ceramics, n can be large enough, such as 1.5–2.5, for there to be significant effects of internal reflections. At a diffuse boundary, the outgoing flux inside a layer of medium equals the transmission of externally incident flux and the reflection of internal incoming flux as illustrated in Figure 17.21 (superscripts o and i designate outside and inside an interface):

J n (0) = (1 - rno ,dif )qnr1 + rin,dif Gn (0) (17.78)



J n ( D) = (1 - ron,dif )qnr 2 + rin,dif Gn ( D). (17.79)

The radiative flux at the interior of each surface is related to the outgoing and incoming fluxes within the layer by

qvr ( 0 ) = J v ( 0 ) - Gv ( 0 ) (17.80)



qvr ( D ) = - J v ( D ) + Gv ( D ) . (17.81)

The diffuse reflectivity at the outside of each surface, rodif (n), is estimated from a hemispherically averaged Fresnel relation as rodif (n) = 1 - e(n) , where ε(n) is given by Equation (3.13). This assumes

FIGURE 17.21  Plane layer of thickness D with n > 1 and with specular or diffuse surfaces.

805

Radiative Effects in Translucent Solids

that the medium behaves like a perfect dielectric where the effect on reflectivity of the absorption index k can be neglected in the complex refractive index. This is a good assumption unless the absorption index is large, as shown by Cox (1965) and Hering and Smith (1968). For reflections inside the layer, each roughness facet is assumed to reflect in a specular manner, but the random orientation of the roughness elements results in a diffuse reflection. For each reflection from a roughness element, some of the energy has total internal reflection, and the rest is reflected as predicted by Fresnel relations for the smooth surface of each element. This results in the reflectivity at the inside of the interface being given in terms of the reflectivity at the outside of the interface by o ridif = 1 - (1/n2 ) ´ (1 - rdif ), as derived by Richmond (1963).

17.5.4 Emission from a Translucent Layer (n > 1) at Uniform Temperature with Specular or Diffuse Boundaries To illustrate the application and effect of specular or diffuse boundary conditions, the emission is analyzed from a translucent non-scattering plane layer at uniform temperature with n > 1, into surroundings with n = 1. For a layer at uniform temperature, a layer emittance can be defined as the emission from the layer relative to that from a blackbody at the same temperature. For blackbody radiation, emission is diffuse, but from an emitting translucent layer with smooth specular boundaries, the radiation intensity is not uniform over all directions. From Kirchhoff’s laws, the emittance into a given direction equals the absorptance for energy from that direction. For diffuse incident radiation, the emittance including all directions equals the absorptance including all directions, which provides the proper comparison with blackbody emission, which is diffuse. First, consider the absorptance of a non-scattering layer with refractive index n > 1 and with specular boundaries. The layer has diffuse incident energy that is partially absorbed, as shown in Figure 17.21. For specularly reflecting surfaces, radiation in each incident direction θ is refracted and internally reflected as given by Fresnel relations. The fraction of energy absorbed is given by Equation (17.3) as α(θ) = [1−ρo(θ)][1−τ(θ)]/,[1−ρo(θ)τ(θ)], where the internal transmittance is τ(θ) = e−κD/cosχ(θ) and the angle of refraction, from Equation (3.3), is χ(θ) = sin−1[(sinθ)/n]. The reflectivity ρo(θ) is a function of the polarization component, so α(θ) has different values for the perpendicular and parallel components. For incident diffuse unpolarized radiation, the absorptance of a layer with specular surfaces is obtained by integrating the absorbed energy for all incident solid angles. This gives the following relation for absorptance, which, by virtue of Kirchhoff’s law, equals the emittance of the uniform temperature layer: p/2



as (n, kD) = es (n, kD) =

ò [a (q) + a (q)]cos q sin qdq. (17.82) ^



0

The interface reflectivities ρo(θ) needed to obtain α⊥(θ) and α||(θ) are given by Equations (3.6) and (3.7). With all quantities in Equation (17.82) in terms of θ, the integration is performed numerically for various n and optical thicknesses κD (this expression for αs is also given by Gardon (1956)). The boundaries are now assumed diffuse. The diffuse reflectivity at a boundary is obtained from the hemispherically integrated Fresnel relation in Equation (3.13) by using rodif (n) = 1 - e(n) . The layer emittance εdif, which is a comparison with blackbody emission that is diffuse, is found by εdif being equal to the layer absorptance that is derived for diffuse incident radiation. Consider the interaction of the layer with a unit incident diffuse flux, qrl = 1, on the boundary at x = 0 in Figure 17.21. The outgoing flux from the interior side of each diffuse boundary is, from Equations (17.78) and (17.79)

J (0) = 1 - rodif + ridif G(0); J ( D) = ridif G( D) (17.83)

806

Thermal Radiation Heat Transfer

where ridif is at the interior side of a diffuse boundary. At the interior side, the diffuse reflectivity is obtained by locally accounting for ordinary reflection from each interface element for incidence angles less than those for total internal reflection and totally reflected energy for larger incidence angles (see Siegel and Spuckler 1994b). As a result of transmission within the layer, the incoming and outgoing fluxes at the interior sides of the two boundaries are related by

G(0) = 2 J ( D)E3 ( kD); G( D) = 2 J (0)E3 ( kD) (17.84)

as obtained by using the terms without emission or scattering in Equations (12.36) and (12.37). Equations (17.83) and (17.84) are solved for J(0), J(D), G(0), and G(D). The fraction of incident energy that is absorbed is a dif = 1 - (1 - ridif )[G(0) + G( D)], which yields for the absorptance or emittance of the layer (Siegel and Spuckler 1994b):

o ) a dif (n, kD) = edif (n, kD) = (1 - rdif

1 - 2 E3 ( kD) . (17.85) o 1 - [1 - (1/n2 )(1 - rdif )]2 E3 ( kD)

The εs and εdif for the isothermal layer from Equations (17.82) and (17.85) are spectral quantities since the absorption coefficient, κ, depends on frequency (the subscript ν has been omitted to simplify the notation). For specular or diffuse boundaries, the layer emittance was evaluated from Equations (17.82) and (17.85), and comparisons (from Siegel and Spuckler 1994b) are in Figure 17.22 for refractive indices n = 1–4 and optical thicknesses κD = 0.2–4. For κD > 4, almost all incident energy that is not reflected from the first surface is absorbed in the layer; the layer spectral absorptance and emittance then approach the asymptotic value, 1 - rov . The type of surface reflection is very significant when κD is less than about 3. For the limit n = 1, there are no surface reflections, so the results are the same

FIGURE 17.22  The effect of diffuse or specular surfaces on spectral emittance of a translucent plane layer at uniform temperature as a function of its optical thickness and refractive index (from Siegel, R. and Spuckler, C.M., JHT, 116, 787, 1994b).

Radiative Effects in Translucent Solids

807

for the solid and dashed curves. As n increases, the layer ευ are increasingly different, so for n = 4 with κD  1 (Spuckler and Siegel 1996): 3 d 2 In ( x) - 3bn2 (1 - wn )I n ( x ) = - bn2 (1 - wn )n2 Enb ( x ). (17.96) 2 p dx



The local blackbody spectral emission, n2Evb[T(x)], depends on the local coating temperature where Evb(T)dν is given by Equation (1.22). Two boundary conditions are required for solving Equation (17.95), since a differential representation of the radiative transfer is being used rather than an integral form that would already contain the boundary conditions. For a clean external surface, the boundary condition for I v at x = 0 is derived by starting with Equation (17.78). The incident radiative spectral flux is qνr1dν = Eνb(Ts1)dν, and Milne–Eddington Equations (Section 13.2.2.1) are used that qno (0) = pI n (0) + qnr (0) / 2 , qni (0) = pI n (0) - qnr (0) / 2 , and qnr (0) = -(4p / 3bn )(dI n /dx |0 ) . After substituting into Equation (17.78), the result is rearranged into the boundary condition: -



2 dI n 3bn dx

+ x =0

1 - ri 1 1 - ro I n (0 ) = Enb (Ts1 ). (17.97) p 1 + ri 1 + ri

A somewhat similar derivation gives the boundary condition for I v at x = Dc, but here the opaque surface of the metal that has an emissivity εm enters the relation. The radiative flux in the negative direction is the emission from the metal and the reflected incoming radiation; this gives, for a gray wall, qn-r ( Dc ) = em n2 Enb ( Dc ) + (1 - em )qn+r ( Dc ) . The two-flux relations in terms of I are used to replace the q – and q+, and the result is the boundary condition for I at the interface of the translucent layer and the metal wall (Siegel 1997b): 2 dI n 3bn dx



+ x = Dc

em 1 em I n ( Dc ) = n2 Enb ( Dc ). (17.98) p 2 - em 2 - em

At this interface, continuity of temperature gives Tc(Dc) = Tm(Dc) ≡ Tmh (the metal temperature on the hot side). Within the metal wall, energy is transferred only by conduction, so qtot = k m(Tmh−Tmc)/Dm. At the cooled side of the metal wall, there is convection to the cooling gas and radiation to the sur4 roundings. If large surroundings at Ts2 are assumed, qtot = h2 (Tmc - Tg 2 ) + em s(Tmc - Ts42 ). For zirconia, the translucent spectral region extends from small λ to λ ≈ 5 μm; for larger λ, the extinction coefficient becomes quite large. This behavior is appropriate to demonstrate a two-band calculation; the band for large wavelengths (small frequencies) is considered opaque. To obtain the temperature distribution in the thermal barrier coating, Equation (17.95) is integrated over the translucent band (large frequencies). The I ( x ) was found by solving Equation (17.96) subject to the boundary conditions of Equations (17.97) and (17.98). Siegel (1997a) derived a Green’s function solution for this equation and boundary conditions, and it is applied here using the properties and blackbody energy in the translucent spectral band; this yields I L ( x ), where the subscript L designates the band with large frequencies. An iterative solution can be developed, where the opaque heat conduction solution is used as a first guess for Tc(x) (Siegel 1997a). Then the I L ( x ) can be evaluated since it depends on the temperature distribution. Relations between the temperatures and the total heat flow are developed. For example, at a clean exposed zirconia surface, qtot consists of radiation in the translucent band, radiation at the surface in the opaque band, and convection at the surface. Using the two-flux relation following Equation (17.94) for the radiation in the translucent band gives

dI L dx

= x =0

3 bcL [h1[1 - Tc (0) + (1 - ro )(Ts41 [1 - F (Ts1 )] - Tc (0)4 {1 - F[Tc (0)]}) - qtot ]. (17.99) 4p

812

Thermal Radiation Heat Transfer

These relations can be used to develop an iterative procedure where successive iterative values are adjusted to conform to energy conservation; this provides rapid convergence. For the details of the method, see Siegel (1997a). Illustrative temperatures are in Figure 17.25 for a zirconia coating on a high-alloy steel combustion liner. The solid lines are for an uncoated metal wall; the lower line is for oxidized metal (εm = 0.6) on both sides. In both instances, the temperatures are above the metal melting point, so a thermal barrier protective coating must be used. A limiting calculation is for a zirconia coating assumed to be opaque (long dashed lines). For a 1 mm thick coating, the metal temperature is substantially reduced. If the coating can be kept clean, there may be a further benefit if the coating is semitransparent, although this effect depends on properties such as the index of refraction; an increase in n will shift the positions of the results. The high scattering of zirconia reflects much of the incident radiation so that the temperatures in the zirconia are decreased (lower short dashed line) and the hot side of the metal is reduced about 20 K.

17.6.3 Composite of Two Translucent Layers The radiative analysis is now described for a composite layer of two gray translucent absorbing and scattering materials with thicknesses D1 and D2, as in Figure 17.26, which have unequal refractive indices larger than 1 (Spuckler and Siegel 1994). To provide general boundary conditions, each side of the composite is heated by radiation and convection. The diffuse radiation incident from the

FIGURE 17.25  Combustor liner wall temperature distributions for oxidized metal without a coating, metal with an opaque thermal barrier coating, and metal with a semitransparent thermal barrier coating; results with and without soot on the exposed surface; on the cooled side of the metal, radiation is to large surroundings. Parameters: h1 = 250; h2 = 110 W/(m2 · K); kc = 0.8; km = 33 W/(m · K); Dc = 10 –3; Dm = 0.794 × 10 –3 m; n = 1.58; κ = 30 m−1 and σs = 104 m−1 for λ  qro2 and Tg1 > Tg2 in the present analysis. Energy is transferred internally by conduction, emission, absorption, and isotropic scattering. The layers have absorption and scattering coefficients κ1, σs1 and κ2, σs2. The external surfaces of the composite and its internal interface are diffuse, which is intended to model composite ceramics that have not been polished and are bonded together. Internal reflections are included at the boundaries and at the internal interface. Inside each layer there are outgoing and incoming diffuse fluxes, J and G, at each interface.

17.6.3.1 Temperature Distribution Relations from the Energy Equation Within each layer, energy is transferred by conduction and radiation according to the energy Equation (15.3)

kj

d 2T j dqrj = 0 ( j = 1, 2). (17.100) dx j dx 2j

Since the energy flux by combined radiation and conduction through the composite is a constant, Equation (17.100) can be integrated with respect to χ and then equated to its values at x1 = 0 and x2 = D2 to evaluate the constant of integration; this gives k1

dT1 dx1

- qr1 ( x1 ) = k2 x1

dT2 dx2

- qr 2 ( x2 ) = k1 x2

dT dT1 - qr1 ( x1 = 0) = k2 2 dx2 dx1 0

- qr 2 ( x2 = D2 ). (17.101) D2

After partial reflection at each outer boundary, the externally incident radiation qrjo passes into the composite and interacts internally. There is no absorption at the exact plane of an outer boundary since an interface does not have any volume. Hence, the conduction derivative terms at the

814

Thermal Radiation Heat Transfer

boundaries in Equation (17.101) are equal to only the external convection, and Equation (17.101) can be written as h1[Tg1 - T1 (0)] + qr1 ( x1 = 0) = -k1 = -k2



dT1 dx1 dT2 dx2

+ qr1 ( x1 ) x1

+ qr 2 ( x2 )

(17.102)

x2

= h2 [T2 ( D2 ) - Tg 2 ] + qr 2 ( x2 = D2 ). Equation (17.102) is integrated to give relations for the layer temperature distributions:



T1 ( x1 ) = T1 ( x1 = 0) -

T2 ( x2 ) = T2 ( x2 = 0) -

h1 x 1 [Tg1 - T1 ( x1 = 0)]x1 - 1 qr1 ( x1 = 0) + k1 k1 k1

h2 x 1 [T2 ( x2 = D2 ) - Tg 2 ]x2 - 2 qr 2 ( x2 = D2 ) + k2 k2 k2

x1

ò q ( x* ) dx* (17.103) r1

1

0

x2

ò q ( x* ) dx*. (17.104) r22

2

2

0

Equations (17.103) and (17.104) are evaluated, respectively, at the boundaries x1 = D1 and x2 = D2, and the overall temperature difference for the composite T1(x1 = 0)−T2(x2 = D2) is found by addition noting that, at the internal interface, T1(x1 = D1) = T2(x2 = 0). Then, by using the equality of the first and last sets of terms in Equation (17.102), the T2(x2 = D2) or T1(x1 = 0) is eliminated to yield the two surface temperatures as



é h æD D T1 ( x1 = 0) = Tg1 + ê1 + 1 + h1 ç 1 + 2 k2 è k1 ë h2 -



1 1 qr 2 ( x2 = D2 ) h2 k1

é h æD D T2 ( x2 = D2 ) = Tg 2 - ê1 + 2 + h2 ç 1 + 2 k2 è k1 ë h1 -

1 1 qr1 ( x1 = 0) h1 k1

-1

öù ì é 1 D1 D2 ù ÷ ú íTg 2 - Tg1 + ê h + k + k ú qr1 ( x1 = 0) 1 2 û øû î ë 2 D1

1

ò q ( x )dx - k ò r1

1

1

2

0

-1

1

D2

ò q ( x ) dx - k ò 0

1

0

üï qr 2 ( x2 )dx2 ý ïþ

öù é æ 1 D1 D2 ÷ ú êTg 2 - Tg1 + ç h + k + k 1 2 øû ë è 1

D1

r1

D2

1

2

0

ù qr 2 ( x2 )dx2 ú . ú û

(17.105)

ö ÷ qr 2 ( x2 = D2 ) ø (17.106)

17.6.3.2 Relations for Radiative Energy Flux The temperature relations of Equations (17.103) through (17.106) contain the radiative fluxes. In scattering layers, they depend on the unknown radiative source function Iˆ j ( x j ) and are given in each layer by Equation (12.63): qrj (x j ) = 2 J (x j = 0)E3 (b j x j ) - 2 J (x j = D j )E3[b j (D j - x j )]

Dj ü (17.107) ìxj ï ˆ ï +2pb j í I j (x j *)E2 [b j (x j - x j *)]dx j * - I j (x j *)E2 [b j (x j * - x j )] dx j *ý . ïî 0 ïþ xj

ò

ò

815

Radiative Effects in Translucent Solids

Equation (17.107) contains the diffuse fluxes J(xj = 0) ≡ Ja and J(xj − Dj) = Jd leaving the internal surface of each boundary (Figure 17.26). These fluxes are expressed in terms of the fluxes incident from outside of each layer to provide coupling with the external radiation and with the energy crossing the internal interface. By using the relations given previously between diffuse reflectivities on the two sides of an interface (see information following Equation (17.81)), the outgoing flux is written at each diffuse interface in terms of transmitted and reflected energy fluxes (see Figure 17.26):

J a = qro1n12 (1 - ra ) + Ga ra (17.108)



æn ö J b = Gc ç 1 ÷ (1 - rb ) + Gb rb (17.109) è n2 ø



J d = qro2 n22 (1 - rd ) + Gd rd . (17.110)

2

Because there are four internal sides of interfaces (a, b, c, d), another independent relation is needed. This is obtained from continuity of radiative flux across the internal interface, which gives Gb - J b = J c - Gc . (17.111)



For j = 1, 2, by using qrj(xj = 0) = J − G at interfaces a and c and qrj(xj = Dj) = G − J at interfaces b and d, Equation (17.107) is used to obtain the internal incoming fluxes at the four internal boundaries as D1

Ga = 2 J b E3 (b1D1 ) + 2pb1



ò Iˆ (x )E (b x )dx (17.112) 1

1

2

1 1

1

x1 = 0 D1

Gb = 2 J a E3 (b1D1 ) + 2pb1



ò Iˆ (x )E [b (D - x )]dx (17.113) 1

1

2

1

1

1

1

x1 = 0

D2

Gc = 2 J d E3 (b2 D2 ) + 2pb2



ò Iˆ (x )E (b x )dx (17.114) 2

2

2

2 2

2

x2 = 0 D2

Gd = 2 J c E3 (b2 D2 ) + 2pb2



ò Iˆ (x )E [b (D - x )]dx . (17.115) 2

2

2

2

2

2

2

x2 = 0

The G are eliminated between Equations (17.108) through (17.115), and the resulting equations are solved simultaneously for J at each internal boundary to yield

Jb =

J a = C1 + A1 J b (17.117)



( A2C3 + A3C1 + C2 )[1 - ( A42 / rd )] + A2 A4 [( A1C1 /ra ) + ( A4C3 /rd ) + C4 ] (17.116) (1 - A1 A3 )[1 - ( A42 /rd )] + A2 A4 [1 - ( A12 /ra )]

Jc =

-( A2C3 + A3C1 + C2 )[1 - ( A12 /ra )] + (1 - A1 A3 )[( A1C1 /ra ) + ( A4C3 /rd ) + C4 ] (17.118) (1 - A1 A3 )[1 - ( A42 /rd )] + A2 A4 [1 - ( A12 /ra )]

816

Thermal Radiation Heat Transfer

J d = C3 + A4 J c (17.119)

where

A1 = 2ra E3 (b1 D1 ) (17.120)



æn ö A2 = 2(1 - rb ) ç 1 ÷ E3 (b2 D2 ) (17.121) è n2 ø



A3 = 2rb E3 (b1 D1 ) (17.122)



A4 = 2rd E3 (b2 D2 ) (17.123)

2

D1



C1 = (1 - ra )n12qro1 + 2ra pb1

ò Iˆ (x )E (b x )dx (17.124) 1

1

2

1 1

1

x1 = 0



D2 D1 ì ü ï ï 2 ˆ C2 = 2p í(1 - rb )(n1 /n2 ) b2 I 2 (x2 )E2 (b2 x2 )dx2 + rbb1 Iˆ1 (x1 )E2 [b1 (D1 - x1 )]dx1 ý (17.125) x2 = 0 x1 = 0 îï þï

ò

ò

D2



C3 = (1 - rb )n q + 2rd pb2 2 o 2 r2

ò Iˆ (x )E [b (D - x )]dx (17.126) 2

2

2

2

2

2

2

x2 = 0



D2 ì D1 ü ï ï ˆ C4 = 2p íb1 I1 ( x1 )E2 [b1 ( D1 - x1 )]dx1 + b2 Iˆ2 ( x2 )E2 (b2 x2 )dx2 ý . (17.127) ïî x1 =0 ïþ x2 =0

ò

ò

17.6.3.3 Equation for the Source Function The source function Iˆ j ( x j ) required for the radiative flux is obtained in each layer from Equation

(11.44), with the addition of an n2j factor:

sT j4 ( x j ) w j ì J ( x j = 0) Iˆ j ( x j ) = (1 - w)n2j E2 (b j x j ) + í 2 î p p

J(x j = Dj ) + E2 [b j ( D j - x j )] + b j p

ü ˆI j ( x*j )E1 (b j | x j - x*j |)dx*j ïý ï x* j =0 þ

(17.128)

Dj

ò

( j = 1, 2) .

17.6.3.4 Solution Procedure and Typical Results For diffuse interfaces, the relations in Section 17.5.3 are used for the interface reflectivities. An iterative solution method is used by first assuming temperature and source function distributions in both layers. The A1 … A4 and C1 … C4 are evaluated from Equations (17.120) through (17.127) and are used to calculate the J from Equations (17.116) through (17.119). New Iˆ1 ( x1 ) and Iˆ2 ( x2 ) are then evaluated by iterating Equation (17.128) using the assumed T1(x1) and T2(x2), and the flux distributions qrl(xl) and qr2(x2) are evaluated from Equation (17.107). New boundary temperatures Tl(x1 = 0) and T2(x2 = D2) are calculated from Equations (17.105) and (17.106) and new temperature distributions

Radiative Effects in Translucent Solids

817

Tl(x1) and T2(x2) from Equations (17.103) and (17.104). The new T1(x1) and T2(x2) and Tl(x1) and T2(x2) are used to start a new iteration; the process is continued until they converge. Under-relaxation factors are usually required, as discussed in online Appendix I at www.ThermalRadiation.net. Representative results are displayed in Figure 17.27, where the refractive index for the first layer is n1 = 1.5 and for the second layer is either n2 = 1.5 or n2 = 3. The J = T /Tg1 , N j = k j /sTg31 D j , and H j = h j /sTg31 . For n2 = n1 = 1.5, there are no reflections at the internal interface. Because there is no scattering, the optical thicknesses are the physical thicknesses multiplied by the layer absorption coefficients. The second layer has τD2 = 1.0, while the first layer has τD1 = 0.01, 1, and 100. A large τD1 produces a large temperature decrease in the first layer, giving reduced temperatures in the second layer. For n2 = 3, the temperature profiles become more uniform in the central portion of the second layer as a result of increased internal reflections. Additional steady and transient analysis and results for a two-layer composite with absorption and scattering are available in Tan et al. (2000b). For radiation in translucent solids, a review of the literature and a summary of the theory are in Viskanta and Anderson (1975). The review by Siegel (1998) discusses transient effects that have been investigated in translucent media since the 1950s, and it also summarizes some of the analytical relations used for radiative transfer during transients. Some noteworthy contributions to the analysis of translucent media are those by Gardon (1956, 1958, 1961, 1995b), who was a pioneer in the analysis of radiative transfer in hot glass where the refractive index and surface reflections are

FIGURE 17.27  Effect on temperature distributions of different optical thicknesses and refractive indices in the two translucent layers of a composite: ϑst = ϑg1 = 1; ϑs2 = ϑg2 = 0.25; H1 = H2 = 1; τD1 = 0.01, 1, and 100; τD2 = 1 (from Spuckler, C.M. and Siegel, R., JTHT, 8(2), 193, 1994).

818

Thermal Radiation Heat Transfer

important along with internal emission. Perpendicular and parallel polarization contributions of the radiation, as discussed for the reflectivity relations in Chapter 3, are in the original analysis in Gardon (1956). In Gardon (1958, 1995b), a comprehensive analysis of the heat treatment of glass is given. Heat treatment is important for manufacturing safety glass in automobile windows, and its analysis includes the effects of conduction within the glass and convection at the surface coupled with internal emission and radiative transport. Gardon (1961) reviews radiant heat transfer as studied by researchers in the glass industry with a digest of much of the literature on the subject up to 1961. A later review of radiation effects occurring in the glass industry in early days was given by Condon (1968), and an overview is by Frank and Klar (2008). With regard to the thermal conductivity of a translucent solid, it has been pointed out (e.g., Gardon 1961, Araki 1990) that for heat transfer by combined radiation and conduction in translucent media, care must be taken in an analysis to use values for the thermal conductivity k that include only the molecular conductivity. At elevated temperatures, measurements of heat flux to determine k in translucent materials must be corrected to eliminate the radiative transfer that produces an apparent increase in k that increases strongly as the temperature is raised. A discussion of glass thermal conductivities and their determination from dynamic temperature measurements is in Mann et al. (1992). The emittance is given from layers of translucent materials with n > 1 that are semi-infinite in thickness (Armaly et al. 1973, Isard 1980), of finite thickness (Siegel and Spuckler 1994b), bounded on one side by a substrate (Caren and Liu 1971, Baba and Kanayama 1975, Anderson 1975), and have internal opaque radiation barriers Siegel (1999b). The radiation and conduction heat transfer through a high-temperature semitransparent layer of slag is analyzed by Viskanta and Kim (1980). Refractive index effects were analyzed by Spuckler and Siegel (1992) for plane layers exposed to external radiation and convection. Internal reflections tended to make the temperature distributions more uniform in the central portions of the layer as compared with a layer with n = l, as demonstrated in Figure 17.27. This is also shown by Siegel and Spuckler (1992) for an absorbing and scattering radiating layer; in the limit where radiation is dominant so that conduction and convection are very small, the energy flows and temperatures for n > 1 are shown to be directly related to those for n = 1. This was extended by Siegel and Spuckler (1993) to a multi-layer lamination of translucent materials, which provides a means for examining the effect of a variable refractive index within a plane layer. Inhomogeneous films may provide useful absorption properties for solar energy collection (Fan 1978, Heavens 1978). In Heping et  al. (1991), Andre and Degiovanni (1995), Tan et  al. (2000a), and Sakami et  al. (2002), the response to an external radiation pulse was analyzed. Anisotropic scattering effects are examined by Wu and Wu (2000) and Liu and Dougherty (1999). An interesting example of ray-tracing in a nonplanar geometry with n > 1 is that of radiation absorption by spheres (water droplets) Harpole (1980). The emittance is analyzed by Wu and Wang (1990) of a sphere of scattering medium with Fresnel conditions at its boundary. Transient radiation effects in two-dimensional (2D) rectangular regions are reviewed by Siegel (1998), and an analysis of cooling of a 2D axisymmetric glass disk is in Lee and Viskanta (1998). Transient heat transfer in a highly back-scattering heat shield is analyzed by Cornelison and Howe (1992) for use during re-entry into a planetary atmosphere. Ceramic coatings may be used for thermal protection on the walls in a channel in which there is flow of a high-temperature gas. Some ceramics are translucent, and if the walls are at different temperatures, there is radiative exchange between the coatings; analyses of temperature distributions in the coatings that include these conditions are in Siegel (1999a, b) and Wang et al. (2000). Some situations such as antireflection coatings with graded refractive index require treatment of the variation in refractive index within a medium (Siegel and Spuckler 1993a, b, Ben Abdallah and LeDez 2000a). Lemmonier and LeDez (2005) discuss a varying refractive index in a slab using the discrete ordinates method. Huang et al. (2006) study a layer with sinusoidally varying n. In BenAbdallah et al. (2001) analyze a cylindrical region that has a spatially variable index of refraction.

Radiative Effects in Translucent Solids

819

Liu (2006) presents a solution for a 2D solid with graded refractive index, and Ferwerda (1999) presented the effect of a varying refractive index on scattering within a medium. Zhao et al. (2015) use Monte Carlo to study radiation in planar gradient-index media and include the effects of polarization and of Mie or Rayleigh scattering in the medium. Coupled radiation and conduction in media with varying refractive index is treated in Ben-Abdallah and DeVez (2000a, b), Xia et al. (2002), and Huang et al. (2004, 2006). Hosseini Sarvari (2016, 2018) uses both DOM and Monte Carlo and Li and Wei (2018) uses DOM and a Chebyshev polynomial and collocation spectral method to treat radiative transfer in planar and multidimensional media with varying refractive index.

17.7 LIGHT PIPES AND FIBER OPTICS Perfect internal reflection at an interface within medium 2 with n2 > n1 is predicted to occur (Equation æn ö (17.66)) whenever the incident angle q > sin -1 ç 2 ÷ . The radiation within the higher refractive index è n1 ø material will undergo 100% reflection at the interface. This means that once radiation enters a perfectly transmitting material (a perfect dielectric) such as an optical fiber in which no radiation absorption occurs, the radiation will propagate without loss along the fiber. The perfect wall reflectivity allows no losses by refraction through the fiber surface. This phenomenon is observed in a planar geometry by swimmers (and fish), who can only see through the water surface above them within a cone of angles described by

æ n q < sin -1 ç air è nwater

ö -1 æ 1 ö -1 ÷ = sin ç ÷ = sin ( 0.75 ) = 48.6°. è 1.33 ø ø

At greater angles, the water surface appears to be a mirror; at lesser angles, although all directions in the hemisphere are viewed from within medium 2, the refraction through the surface produces considerable distortion of the view. For a light pipe or fiber optic, radiation enters the flat end of a long circular cylinder. If the refractive index ratio n2/n1 exceeds 2 , all radiation entering the light-pipe end will encounter the internal cylindrical surface at greater than the critical angle (Qu et al. 2007b). An example of the pattern of radiation transmitted through a light pipe is shown in Figure 17.28. Because of the increasing

FIGURE 17.28  (a) Experimental and (b) predicted patterns of laser energy incident at exit of a fused quartz light pipe (n = 1.45843) for laser energy incident on the inlet end at 20° to the light-pipe axis. Light-pipe diameter 4 mm and length 16 cm (from Qu, Y. et al., IEEE Trans. Semiconduc. Manuf., 20(1), 26, 2007b).

820

Thermal Radiation Heat Transfer

reflectivity (decreasing transmissivity) of dielectrics with incident angle (Equations (3.6) and (3.7)), the radiation actually entering the light-pipe end is chiefly from near-normal angles. Barker and Jones (2003a, b) and Frankman et al. (2006) examined the temperature distribution that may arise internally in the light pipe because of absorption by inclusions or the inherent absorption by the light-pipe material. The first two papers looked at the particular case when the light-pipe tip exposed to radiation was coated with a highly absorbing material so that it acts as a blackbody absorber, emitting into the light pipe in proportion to the absorbed signal. Meyer (2002), Kreider et al. (2003), Qu et al. (2007a, b), and Ertürk et al. (2008a, b) have analyzed the errors that arise because of various signal loss mechanisms that can occur in a light pipe, such as the presence of very minor surface imperfections, internal inclusions in the light-pipe material that can act as scattering centers, the effects of blocking and shadowing of reflected energy from the surface being measured by the light pipe itself, changing the temperature of the measured object by radiative losses to the colder probe, and any environmental radiation that may enter through the light-pipe walls (rather than the tip) and enter the internal signal path by encountering surface imperfections or scattering centers within the light pipe. Arnaoutakis et al. (2013) examine the effect of adding claddings or coatings and of tapered entries to light pipes for solar energy applications. The signal entering the detector at the end of a light-pipe radiation thermometer (LPRT) or optical fiber consists of emitted plus reflected energy from the object being viewed. If the environment is cold relative to the viewed object so that reflected energy can be neglected and the detector is only sensitive in a small wavelength range around a particular wavelength λ, then the detected emission is proportional to, from Equation (2.7)

El (Tact ) = el El (Tact ) = el

2pC1 é æ C l 5 êexp ç 2 è lTact ë

ö ù ÷ - 1ú ø û

. (17.129)

However, rather than the actual temperature Tact, the detector reads an apparent temperature Tapp that would appear to originate from a blackbody: El (Tapp ) =



2pC1 é æ C l êexp çç 2 êë è lTapp 5

ö ù ÷÷ - 1ú ø úû

. (17.130)

Equating the two emitted energy rates relates the actual and apparent temperatures as

el

2 pC1 é æ C l 5 êexp ç 2 è lTact ë

ö ù ÷ - 1ú ø û

=

2pC1 é æ C l 5 êexp çç 2 êë è lTapp

ö ù ÷÷ - 1ú ø úû

. (17.131)

For most engineering conditions, the factor of (−1) in the denominator can be neglected relative to the exponential term (resulting in Wien’s spectral distribution (Equation (1.26)), and the actual temperature in terms of the apparent absolute temperature then becomes

Tact =

Tapp . (17.132) é lTapp ù ê1 + C ln el ú 2 ë û

As ελ → 1, the apparent and actual temperatures approach one another. Equation (17.132) can be used for many spectrally based temperature measurements, remembering the proviso that the

Radiative Effects in Translucent Solids

821

environmental temperature must be low in order to get an accurate temperature measurement from an optical pyrometer or light-pipe radiation thermometer.

17.8 FINAL REMARKS In this chapter, we have discussed several models and formulations for radiative transfer through translucent materials, windows, and coatings. Because of space constraints, many other radiative transfer applications and the corresponding details are omitted. A natural extension would be to paints, for example, which are important for energy efficiency applications in buildings. Similarly, there are many other dispersed media, from foams to food preparation coatings (cheese-making or baking), where radiation transfer plays a crucial role. The concepts discussed in this book constitute a good starting point for analysis of such systems, but the details need careful study. A discussion of radiative transfer in porous, dispersed, and foamy materials is in the online appendix (Appendix E) to this book at www.ThermalRadiation.net.

HOMEWORK PROBLEMS (Additional problems available in the online Appendix P at www.ThermalRadiation.net) 17.1 A horizontal glass plate 0.225 cm thick is covered by a plane layer of water 0.625 cm thick. Radiation is incident from within air onto the upper surface of the water at an incidence angle of 60°. What is the path length of this radiation through the glass? (Let nH2O = 1.33, and nglass = 1.57.) Answer: 0.270 cm. 17.2 Do Example 11.10 modified to include surface reflection and refraction effects. (Let nglass = 1.53.) Answer: 750 W/(m2·sr). 17.3 A radiation flux qs is incident from the normal direction on a series of two glass sheets, each 0.35 cm thick, in air. What is the fraction T that is transmitted? (nglass = 1.53, and κglass = 9.5 m−1.) What is the fraction transmitted for a single plate 0.7 cm thick in air?

Answers: 2 plates, T = 0.790; 1 plate, T = 0.857. 17.4 Prove that for an absorbing–emitting layer with L > λ, T + A + R = 1. 17.5 Derive an analytical expression that shows whether the total absorption in a system of n plates and m plates, A(m + n), is independent of whether radiation is first incident on the n or the m plates. 17.6 A double-panel glass window consists of two 5 mm parallel glass plates separated by a 10 mm air gap. The outside glass plate is made of crown glass with a refractive index of 1.52. The inside glass plate is made of flint glass with a refractive index of 1.62. Absorption in the glass plates can be ignored. Calculate the transmittance for normal incidence on the outer plate. Answer: 0.828.

822

Thermal Radiation Heat Transfer

17.7 For the two-plate system in Homework Problem 17.6, determine the maximum transmittance for parallel-polarized radiation and the incidence angle that gives rise to that maximum transmittance. Also, find the transmittance through the plate assembly for unpolarized incident radiation at this angle. Compare with the transmittance found for normal incidence in Homework Problem 17.6. Answer: Ttop + bottom = 0.725. 17.8 The two glass plates in Homework Problem 17.3 are followed by a collector surface with absorptivity αc = 0.95. What is the fraction Ac absorbed by the collector surface for normally incident radiation?

Answer: Ac = 0.756. 17.9 Derive Equation (17.35) for a thick dielectric film on a dielectric substrate. Answer: R =

r1 + r2 (1 - r1 ) t2 1 - r1r2 t2

.

17.10 Extend Homework Problem 17.9 to a system of two differing dielectric layers coated onto a dielectric substrate. Answer: R = r1 +

(1 - r1 )

2

t12 éër2 + (1 - 2r2 ) r3 t22 ùû

(1 - r r t ) (1 - r r t ) - (1 - r ) 2 1 2 1

2 2 3 2

2

2

r1r3 t12 t22

.

17.11 Do Homework Problem P.13.2 of the online Appendix P at www.ThermalRadiation.net with the index of refraction for glass equal to 1.53, and for the liquid equal to 1.35. Answer: 503 W/m2. 17.12 The sun shines on a flat-plate solar collector that has a single 0.5 cm thick glass cover (use the data in Figure 9.22 and assume spectral properties are constant below λ = 1 μm) over a silicon oxide-coated aluminum absorber plate (use data from Figure 3.50 and assume properties are constant below λ = 0.4 μm). Determine the effective solar absorptivity of the collector for normal incidence (i.e. the fraction of incident solar energy that is absorbed by the collector plate). Answer: 0.54. 17.13 A semitransparent layer of fused silica has a complex index of refraction of n − ik = 1.42 − i × 1.45 × 10 −4 at the wavelength λ = 5.20 μm. Unpolarized diffuse radiation in air at this wavelength is incident on the layer. What is the value of the overall transmittance of the layer for this diffuse incident radiation? Answer: T = 0.148.

823

Radiative Effects in Translucent Solids

17.14 A plane layer of semitransparent emitting and absorbing hot material is at uniform temperature. The surfaces of the layer are optically smooth, and the layer material is nonscattering. The simple index of refraction of the layer is n and the absorption coefficient is κ (the layer is assumed gray). Obtain an expression for the emittance of the layer into vacuum.

17.15 A layer of transmitting glass is over a parallel opaque cold substrate with gray absorptivity α. The glass has a surface reflectivity of ρ, and the glass layer has transmittance of τ. Derive an expression for the reflectance R of the glass/substrate assembly. Answer: R = r +

(1 - r)2 t éë1 - r - a + 2ra ùû 1 - r + ra - rt2 éë1 - r - a + 2ra ùû

.

18

Inverse Problems in Radiative Transfer Andrey Nikolayevich Tikhonov (1906–1993) was a Russian mathematician. He received his PhD from Moscow State University in 1927. Upon graduation, he immediately made important contributions to topology and mathematical physics, including a uniqueness proof for solutions to the heat equation. Among many other contributions to mathematical analysis, he developed one of the most widely-used regularization techniques for ill-posed problems, which bears his name.

18.1 INTRODUCTION The majority of problems considered elsewhere in the book can be categorized as forward problems, in which a well-defined system is analyzed to obtain an unambiguous solution. There is a growing appreciation, however, that many radiative transfer problems are inverse problems, which, broadly speaking, involve inferring some parameter or quantity from information that is only indirectly related to the unknown. In the context of radiative transfer, most inverse problems are either parameter estimation or inverse design problems. The goal of parameter estimation problems is to infer some property or property distribution from indirect measurements, e.g., to tomographically reconstructing the absorption coefficient from spectral intensity measurements made along the domain periphery, while inverse design focuses on inferring a design configuration that corresponds to some ideal outcome, such as a furnace geometry that must uniformly irradiate a surface being thermally processed. Inverse problems are ill-posed, to an extent that depends on the “indirectness” of the relationship between the known information and the unknown variables; this makes inverse problems much more difficult to solve compared to forward problems. Ill-posedness arises from an information deficit; the available information is either insufficient to specify a unique solution, or there may be multiple solutions that “almost” explain the available solution. In parameter estimation problems this property sensitizes the inference process to small amounts of measurement noise and model errors, while inverse design problems are ill-posed if multiple candidate designs exist that provide similar outcomes. All inverse solution schemes, without exception, address this information deficit by incorporating additional information into the inference process. Sometimes this is done explicitly, e.g., incorporating the results of previous analyses through a Bayesian prior, or it may be more subtle, like the filtering action of linear regularization schemes. Inverse problems involving parameter estimation have been around for a long time, although they are sometimes not recognized as inverse problems. Examples include absorption and emission tomography (e.g., Cai and Kaminski 2017) and light scattering measurements used to infer particle morphologies (e.g., Huber et al. 2016). Recent decades have seen an explosion of interest in these and other topics, driven by the development of high-fidelity physical models and new experimental equipment (e.g., tunable light sources and hyperspectral imaging, ultrafast laser pulses and sensors). In general, “pushing diagnostics to their limits,” i.e. extracting as much information as possible from the data, leads to ill-posed problems. Inverse design of radiative systems dates from the 1990s (e.g., Harutunian et  al. 1995, Jones 1999) and initially focused on optimizing the heat flux or temperature distribution over surfaces within diffuse-walled enclosures. Subsequent work tackled more complicated problems, including 825

826

Thermal Radiation Heat Transfer

geometric optimization of enclosures containing specularly-reflecting surfaces (e.g., Daun et  al. 2003e, Roger et al. 2005), enclosures containing participating media (e.g., França et al. 1999), and mixed-mode problems (França et  al. 2001), and continues to expand into new frontiers such as design optimization of nanostructures (Hajimirza and Howell 2014a). Inverse problems can be computationally expensive to solve, since this often requires repeated evaluation of the model equations; advancements in high performance computing and algorithms, e.g., artificial neural networks (Ertürk et al. 2002a, Mirsepahi et al. 2013) can be used to incorporate increasingly sophisticated, high-fidelity simulations into inverse analyses. Analytical techniques for calculating sensitivity (Rukolaine 2015, Roger et al. 2005) have also greatly expanded the capabilities of inverse analysis for radiative systems. This chapter starts with an overview of inverse analysis and the mathematical properties of illposed inverse problems, followed by a discussion of solution schemes. The chapter concludes with a summary of parameter estimation and inverse design problems involving radiative transfer.

18.2 INVERSE ANALYSIS AND ILL-POSED PROBLEMS Inverse problems are challenging to solve because they are ill-posed. This term originates from Hadamard (1902), who identified well-posed problems as those that have solutions that: (i) exist; (ii) are unique; and (iii) are stable to small perturbations in the problem definition. Problems that do not satisfy these criteria are ill-posed. Hadamard focused on the properties of differential equations, and viewed those derived from “real world” problems, e.g., the heat equation, as being inherently well-posed. Ill-posed problems, in contrast, were interpreted as “artificial” problems having only theoretical interest. There has been a growing appreciation that this is far from true: Inverse problems are ubiquitous in science, engineering, and daily life. While parameter estimation problems satisfy the existence criterion, Bohren and Huffman (1998) provide a helpful example of why they may violate the uniqueness or stability criterion. Suppose a knight who is hunting dragons comes across a set of footprints, which could have been left by one of several species of dragons (Figure 18.1). The knight could easily predict which sort of footprints would be left by a particular species, e.g., from a Field Guide to Dragons; this is the forward problem. On the other hand, inferring the dragon species from an observed footprint is potentially more difficult if each species leaves a similar footprint; this is the inverse problem. This process is even more difficult if the fine features of the footprint that distinguish the species become smeared in the mud, leading the knight to misidentify the species. In this analogy, the “smearing” represents measurement noise and inconsistencies between

FIGURE 18.1  A knight hunting dragons; an example of an inverse parameter estimation problem. (Courtesy of T. Sipkens.)

827

Inverse Problems in Radiative Transfer

the physical system and the measurement model. To resolve this problem, the knight must specify additional information, e.g., a particular dragon species is known to inhabit the forest. Examples involving inverse design can also be derived from daily life. Suppose the “design problem” consists of purchasing a can of frozen orange juice from the supermarket. If the store stocks two brands of orange juice at the same price, this problem has two optimal solutions, thereby violating Hadamard’s uniqueness criterion. To resolve this ambiguity the shopper must introduce additional information, e.g., a preference for a particular brand. On the other hand, the store may be out of orange juice, which violates the existence criterion.

18.3 MATHEMATICAL PROPERTIES OF INVERSE PROBLEMS When solving an inverse problem, the first step is to classify the type of inverse problem; this step is critical, since the ill-posed characteristics as well as the techniques that should be used to solve the problem depend on the type of problem being solved. Most broadly, inverse problems can be categorized as either linear or nonlinear.

18.3.1 Linear Inverse Problems In many problems the unknown variables are related to the known parameters in a linear way through an integral equation of the first kind (IFK) (Wing 1991, Hansen 1998) b

ò

f ( u ) = g ( v ) k ( u, v ) dv (18.1)



a

where f(u) is the known function, g(v) is the unknown function, and k(u,v) is the kernel. This scenario arises frequently in radiative transfer. In a radiant enclosure design problem, for example, f(u) could represent a desired irradiation of a surface being heated (a “design surface”), g(v) would be the radiosity distribution over a heater surface to be designed (e.g., the “heater surface”), and k(u,v)dv would be the infinitesimal view factor, dFu–v between these locations. Alternatively, in parameter estimation problems g(v) is some unknown parameter to be inferred, f(u) represents measured data, and the kernel function is derived from the measurement equations that relate these two quantities. In the forward problem, g(v) is known and the objective is to calculate the system response, f(u), by carrying out the integration in Equation (18.1); this process is called convolution. Because of the smoothing action of the kernel, the features of g(v) are blended by the integration into smaller variations in f(u). When solving the inverse problem, in principle one could reconstruct g(v) if f(u) were known perfectly through the reverse procedure, which is called deconvolution. However, if f(u) is contaminated by small amounts of noise, deconvolution amplifies this noise into large variations in the inferred g(v). In the case of inverse design, the blending action of the kernel means that the irradiation on each point of the design surface is influenced by the radiosity over the entire heater surface, and multiple radiosity distributions g(v) could be substituted into the integral to provide very similar distributions for f(u). The knowns and unknowns are rarely specified as continuous functions; instead they correspond to discrete values of u and v, respectively, {ui, i = 1,2,…m}, {vj, j = 1...n}, where m and n are the number of knowns and unknowns, respectively. The integral in Equation (18.1) is thus approximated by a summation



bi = f ( ui ) =

n

v j + Dv 2

å ò

j =1 v j - Dv 2

g ( v ) k ( ui , v ) dv »

n

å j =1

g (vj )

v j + Dv 2

ò

v j - Dv 2

k ( ui , v ) dv =

n

å a x . (18.2) ij

j =1

j

828

Thermal Radiation Heat Transfer

Writing this equation for each value of ui results in an m × n matrix equation, Ax = b, where the elements of A are derived from k(u,v). In many problems, u and v can be discretized so that m = n; these types of inverse problems are discrete ill-posed (Hansen 1998). Problems having more unknowns than knowns are rank-deficient. The extent of the ill-posedness depends on the smoothing action of k(u,v) in Equation (18.1), which can be connected to the properties of A via a singular value decomposition (SVD), A = USVT, where U and V are (m × m) and (n × n) orthonormal matrices that satisfy UTU = VTV = I, and S is an (m × n) matrix that contains the singular values along its diagonal. If m = n, S = diag(s1, s2, s3…sn), where s1 ≥ s2 ≥ … ≥ sn, while if m ≠ n, the maximum number of singular values is the minimum of m or n, and the remaining rows or columns of S are zero. Consider first the case where m = n. Well-posed problems can be found by inverting A, x = A−1b. Due to the orthonormality of U and V, it can be shown that x = VS−1UTb, or n

x=



å j =1

u Tj b v j (18.3) sj

where uj and vj are the jth column vectors of U and V, respectively. The summation terms resemble a Fourier series; the first terms carve out the low frequency components of x, while finer details come from the later terms. When A is derived from an IFK, the singular values decay over several orders of magnitude. If x and b are sufficiently smooth, the |ujTb| terms, called “Fourier coefficients”, decay faster than the singular values, and the series is convergent. However, in parameter estimation problems, the theoretical “exact” data bexact is contaminated with δb due to measurement noise or model error, e.g., simplifications used to derive the measurement model, and the error caused by discretizing the integral in Equation (18.2). Equation (18.3) becomes n

x exact + dx =



n

u Tj bexact u Tj db vj + v j . (18.4) sj sj j =1 j =1    dx x exact

å

å

Because δb is uncorrelated with the column vectors of U, they remain approximately constant in magnitude, and beyond a critical value of j, sj  2) corresponding to a fixed ||δb|| indicates the domain of δx, which quantifies the uncertainty in the recovered solution. For ill-conditioned problems, the principle axes corresponding to small singular values are long, so the volume is large. For inverse design problems, the elliptical contours enclose candidate solutions that produce design outcomes within a margin of ||δb|| of the desired outcome. Equation (18.4) can be interpreted as taking “steps” towards bottom of the valley, x = A−1b, as illustrated in Figure 18.2 (d). For n = 2, there are only two steps: The first is aligned with the direction of steepest curvature and identifies the location in the valley having the smallest Euclidean distance to the origin. The second step is aligned with the valley floor and can be large due to the small singular value; this corresponds to the main direction of error propagation. These results highlight

FIGURE 18.2  Contour plots of ||Ax−b||2 for (a) a matrix having a condition number of 2.2 and (b–d) a matrix having a condition number of 27. Figures (a) and (b) show the relationship between the principal axes and the SVD of A. Figure (c) shows that measurement error propagates in the directions of the V column vectors that correspond to the smallest singular values. (d) Equation (18.4) corresponds to taking a series of “steps” from the origin, in progressively shallower directions.

830

Thermal Radiation Heat Transfer

an important feature of linear inverse problems; if n = m ||Ax−b||2 is a convex function, meaning that it only has one global minimum. For rank-deficient problems the valley is completely flat in directions corresponding to the last n−m column vectors of V. The complete set of solutions that solves Ax = b is generated from m

x=



å j =1

n

u Tj b C j v j (18.7) vj + sj j = m +1

å

for any {Cj, j = m + 1…n}. This result highlights that rank-deficient problems violate Hadamard’s uniqueness criterion, a scenario discussed in Example 18.2. Example 18.1 The size distribution of particles in an aerosol can be found by shining a laser beam of width w through the aerosol and measuring the scattered light over a set of angles, as shown in Figure 18.3. Derive the governing measurement equation assuming that the aerosol is homogenous, optically thin, and purely scattering. The detector spectral irradiation at a given measurement angle, qλ,d(θ), is the product of the incident intensity on the detector surface, Iλ,d(θ), the detector solid angle, ΔΩd, and the detector area, Ad. The incident intensity on the detector is found by applying the RTE for nearly transparent media, Equation (11.121), along the detector line-of-sight L

s w Il ,d ( q ) = ss,l Iˆl ( s ) ds » s,l Iˆl ( q ) . sin q

ò



0

The radiative source term is equal to zero except when s lies within the probe volume, where it is 1 Iˆl ( q ) = 4p



4p

ò I ( s, W ) F l

Wi =0

i

l

( W, Wi ) d Wi =

Il ,laser DWlaser F l ( q ) 4p

and ΔΩlaser is the solid angle subtended by the light source as viewed from the probe volume. Combining these terms results in

ql ,d ( q ) =

Ad DWlaser DWdetectorw ss,l Wl ( q ) . sin q 4p

FIGURE 18.3  Schematic of a multiangle light-scattering experiment to infer the size distribution of particles within an aerosol.

831

Inverse Problems in Radiative Transfer

For monodisperse particle sizes σs,λ·Φλ is the product of the particle number density, Np, multiplied by the differential scattering cross-section, dCs/dΩ, which defines the probability that incident radiation will be scattered by a single particle in a given direction, cf. Equation (10.7). The differential scattering cross-section is found using appropriate EM theory. In the case of polydisperse aerosols, ¥



ss,l F l = Np

dCs

ò dW 0

p ( dp ) d(dp ). dp ,q

Finally, grouping all terms that are not functions of dp and θ into a coefficient C results in ¥



ò

ql ,d ( q ) = C p ( dp ) k ( q, dp ) d(dp ) 0

which is a Fredholm IFK, and k(θ, dp) is the differential scattering cross-section. In principle this equation could be discretized into an n × n matrix equation, Ax = b, by choosing a maximum value of dp and then approximating p(dp) using uniformly spaced bins equal in number to the number of measurement angles. In practice, this equation is highly ill-posed because of the “blurring” action of the kernel; the differential scattering cross-section usually increases geometrically with particle size, so a large number of small particles may not have as much influence on the detector signal compared to comparatively few large particles. The presence of small particles corresponds to subtle features in qλ,d(θ) that are overwhelmed by noise and model error. Consequently, it is necessary to introduce more information about p(dp) to stabilize the deconvolution process.

Example 18.2 In line-of-sight absorption (LOSA) tomography the distribution of a species (e.g., fuel in a combustion chamber) is tomographically reconstructed by shining lasers having a wavelength aligned with an absorption feature of the target species across the problem domain and measuring the transmittance. Consider the arrangement shown in Figure 18.4, which consists of four mutually orthogonal sets of six parallel lasers, shining across a domain that is discretized into 10 × 10 pixels. Derive the governing IFK for each beam by invoking the optically thin approximation, assuming that the beams are infinitesimally thin. Show how the problem would be discretized into a matrix equation, and comment on the structure of these equations.

FIGURE 18.4  Schematic of a LOSA tomography experiment. (a) A beam is defined by the perpendicular distance from the origin, s, and angle, θ. If b(s, θ) were known as a continuous function of θ and s, κλ could be found through a Radon transform; (b) the integral equation is discretized by discretizing the domain into pixels, within each of which κλ is assumed uniform; (c) combining information from all the rays results in a rank-deficient matrix equation.

832

Thermal Radiation Heat Transfer

Consider a single beam oriented at an angle θ and an axial location s from the origin. The derivation starts with the RTE for a cold, optically thin medium with no scattering, Equation (11.124),

é L ù Il ,L ( s, q) = Il 0 exp ê - kl [r( s, q, u)] du ú ê ú ë u =0 û

ò

where u is a distance along the beam, r(s, θ, u) points to the corresponding position on the beam, and the spectral absorption coefficient, κλ, is proportional to the species concentration. This is rearranged into L



é I ù ln ê l ,0 ú = b( s, q) = kl [r( s, q, u)] du I s ( , q ) ë l ,L û u =0

ò

which is a Fredholm IFK having a kernel of unity. Self-absorption along the ray is neglected, which is only possible for optically thin media. Including this effect, e.g., Equation (11.120), would make this problem nonlinear. Considering only one beam, an infinitely large set of candidate κλ exists, each of which could be substituted into the integral to produce the observed b. Were b(s,θ) known as a continuous function of s and θ, however, the IFK could be deconvolved using a Radon transform (Bertero and Boccacci 1998), although this process would still be ill-posed since it would violate Hadamard’s stability criterion. In practice b can only be measured using a limited number of laser beams. Assuming κλ is uniform within each pixel transforms each IFK into a ray-sum L



bi =

ò 0

kl éëri (u ) ùû du »

n

åa x ij

j

j =1

where xj is the absorption coefficient within the jth pixel and aij is the chord length of the ith beam subtended by the jth pixel. Writing this equation for each ray results in Ax = b, which has 24 equations and 100 unknowns. The solution space could be visualized by performing an SVD, A = USV T, where the dimensions of U, S, and V are 24 × 24, 24 × 100, and 100 × 100. Equation (18.7) shows that an infinite set of solutions exists. (This is obvious since some pixels are not transected by the beams and do not influence the measurements.) Accordingly, additional information about the distribution of x must be specified in order to obtain a unique solution.

Example 18.3 A rapid thermal processing (RTP) chamber heats a silicon wafer, as shown in Figure 18.5a. All other surfaces in the chamber are black, and all surfaces are cold except for the heater. Calculate the emissive power distribution over the heater surface that produces a uniform irradiation of 10 kW/m2 over the wafer. This is an inverse design problem in which the wafer is the design surface. Because the surrounding surfaces are cold and black, the irradiation of the design surface is related to the heater emissive power by a Fredholm IFK

Rs

Rs

0

0

ò

ò

GDS ( r1 ) = Gtarget = Eb,HS ( r2 ) dFd1-d 2 ( r1, r2 ) = Eb,HS ( r2 ) k ( r1, r2 ) dr2

where dFd1–d2 is the infinitesimal view factor from a ring element dA1 on the wafer to ring element dA 2 on the heater surface, and k(r1,r2) = dFd1–d2/dr2 is the kernel. This problem is ill-posed because each point on the wafer can “see” the entire heater. Consequently, multiple emissive power distributions over the heater surface may exist that can produce the desired irradiation. In practice, the IFK is solved by discretizing the heater surface into ring elements, and following Equation (18.2),

833

Inverse Problems in Radiative Transfer

FIGURE 18.5  (a) Inverse design problem for the emissive power distribution over a heater used to irradiate a silicon wafer; (b) singular values from the A matrix show that it is ill-conditioned; (c) the emissive power distribution over the heater surface recovered from x = A−1b is nonphysical, even though it produces the desired irradiation over the design surface (d).



GDS ( ri ) = Gtarget = bi =

n

å j =1

n

Eb,HS, j Fdi - j =

åx a j ij

j =1

where Fdi–j is the view factor between an infinitesimal ring element on the design surface at r1 = ri and the jth ring element on the heater. Writing this equation for n discrete locations on the design surface produces an n × n matrix equation. Figure 18.5b shows the singular values for n = 50, which decay continuously over several orders of magnitude. This is typical of matrices generated from IFKs. In principle, the emissive power distribution over the heater can be found from x = A−1b, which is plotted in Figure 18.5c. Although substituting this result into Ax = b produces the desired irradiation, Figure 18.5d, and constitutes a mathematical solution to the problem, the emissive power distribution plotted in Figure 18.5c is nonphysical.

18.3.2 Nonlinear Inverse Problems Both parameter estimation and inverse design problems may involve nonlinear measurement models, i.e. the knowns and unknowns are not related in a linear way. For example, if the objective of the design problem in Example 18.3 were to optimize the RTP chamber geometry, the problem would be governed by a nonlinear IFK having the form

834

Thermal Radiation Heat Transfer b

ò

f ( u ) = g ( v ) k éëu, v, h ( v ) ùû dv (18.8)



a

where u and v indicate locations on the design and heater surfaces, g(v) would be the emissive power over the heater surface (which may or may not be known) and the kernel would depend on some intermediate geometric variable h(v), e.g., the local curvature of the heater surface, which must also be solved for. Nonlinear problems are usually discretized into a(x) = b, where a is a vector of nonlinear model equations that relate the unknowns in x to the knowns in b (a discrete set if measuring data or design outcomes). Since this nonlinear equation cannot be inverted directly, it is usually solved as a minimization problem x* = arg min éë F ( x ) ùû = arg min r ( x ) r ( x ) (18.9) T



x

x

where r(x) = a(x) − b is the residual between the measured and modeled quantity, and “arg min F(x)” means “the value of x that minimizes F(x).” Intuitively, minimizing the Euclidean norm of the residual vector finds the value of x that makes the modeled data match the measured data as closely as possible in parameter estimation, or that produces a simulated design outcome that most closely matches the desired outcome in design optimization. In an experimental context, the least-squares formulation has a useful statistical interpretation. Measured data is not “exact,” but instead can be thought of as a particular realization of some underlying probability density; often the data obeys a multivariate normal distribution, N ( m b , G b ) , defined by a mean, μb, and a covariance matrix, Γb, that defines the “reliability” of the measurements. The likelihood of observing the data b for a hypothetical x is given by

( )

p bx =

T T é 1 ù é 1 ù exp ê - r ( x ) G b-1r ( x ) ú µ exp ê - r ( x ) G b-1r ( x ) ú . (18.10) ë 2 û ë 2 û det ( G b )

1

( 2p )

m2

In other words, all datasets are possible, but the one that is most likely to occur corresponds to the “true” value of x. This PDF is called the “likelihood function.” In the absence of other information, the PDF for x conditional on the observed data in b equals the likelihood, so p(x|b) = p(b|x), and the most probable solution of x is the one that maximizes the likelihood of observing the data, i.e.

T x MLE = arg min F ( x ) = arg min éêr ( x ) G b-1r ( x ) ùú . (18.11) ë û x x

If the measurement noise is independent (i.e. the elements of b are uncorrelated), then



x MLE

é = arg min ê ê x ë

m

å i =1

2 éë ai ( x ) - bi ùû ù ú (18.12) ú sb2 ,i û

which is a weighted least-squares regression; intuitively, measurements subject to large uncertainties should have less influence on the recovered parameters. If the measurements have the same amount of uncertainty, the MLE is found by minimizing Equation (18.9). As long the number of independent measurements, m, exceeds the number of unknowns, n, xMLE can be found through multivariate minimization, as discussed in Section 18.4. However, if

835

Inverse Problems in Radiative Transfer

the inference problem is ill-posed, xMLE will be surrounded by a region containing values of x that “almost” explain the data. The size of this region can be estimated from the Jacobian matrix evaluated at xMLE. The (m × n) Jacobian (or sensitivity) matrix has elements defined by

J pq ( x ) =

¶a p . (18.13) ¶xq x

For a set of measurement equations to be well-posed, the Jacobian should be large in magnitude, so the measurements are sensitive to the unknowns, and the rows of J should be dissimilar, so each ai(x) is sensitive to each element of x in a different way. For ill-conditioned measurement equations, this condition is not satisfied. The extent of the ill-posedness can be quantified by Cond[J(xMLE)TJ(xMLE)]; in the case of a linear measurement model, J = A, and the uncertainty scales with Cond(ATA). An elliptical region can be defined that contains the “true” x with a certain probability, P, based on the corresponding contour xP% ∊ x:||a(x)−b||2