Radiative Heat Transfer

Radiative Heat Transfer Second Edition

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RADIATIVE HEAT TRANSFER

Second Edition

Michael F. Modest The Pennsylvania State University

ACADEMIC PRESS An imprint of Elsevier Science

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Sponsoring Editor: Jeremy Hayhurst Publishing Services Manager: Diane Grossman Editorial Coordinator: Nora Donaghy Cover Design: Shawn Girsberger Cover image: Photograph taken by permission of Brian Vanderkolk Copyeditor: Beverley Clarke Packager: Keyword Publisliing Services Ltd Printer: Maple-Vail This book is printed on acid-free paper. @ Copyright 2003, 1993 Elsevier Science (USA) All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone: ( + 44) 1865 843830, fax: ( + 44) 1865 853333, e-mail: [email protected]. You may also complete your request on-line via the Elsevier Science homepage (http://elsevier.com), by selecting "Customer Support" and then '^Obtaining Permissions." Explicit permission from Academic Press is not required to reproduce a maximum of two figures or tables from an Academic Press chapter in another scientific or research publication provided that the material has not been credited to another source and that full credit to the Academic Press chapter is given. Academic Press An imprint of Elsevier Science 525 B Street^ Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com Academic Press 84 Theobald's Road, London WCIX 8RR, UK http://www.academicpress .com Academic Press 200 Wheeler Road, Burlington, Massachusetts 01803, USA http://www.academicpressbooks.com Library of Congress Catalog Card Number: 2002114345 International Standard Book Number:

0-12-503163-7

PRINTED IN THE UNITED STATES OF AMERICA 02 03 04 05 06 9 8 7 6 5 4

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2

1

ABOUT THE AUTHOR

Michael F. Modest was bom in Berlin and spent the first 25 years of his life in Germany. After receiving his Dipl.-Ing. degree from the Technical University in Munich, he came to the United States, and in 1972 obtained his M.S. and Ph.D. in Mechanical Engineering from the University of Cahfomia at Berkeley, where he was first introduced to theory and experiment in thermal radiation. Since then, he has carried out many research projects in all areas of radiative heat transfer (measurement of surface, liquid, and gas properties; theoretical modeling for surface transport and within participating media). Since many laser beams are a form of thermal radiation, his work also encompasses the heat transfer aspects in the field of laser processing of materials. For several years he has taught at Rensselaer Polytechnic Institute and the University of Southem California, and since 1987 has been Professor of Mechanical Engineering at the Pennsylvania State University at University Park, PA. He is a fellow of the American Society of Mechanical Engineers and a member of the American Institute of Aeronautics and Astronautics and the Laser Institute of America. Dr. Modest resides in Boalsburg, PA, with his wife, two children, a dog, a cat, and an assortment of other animals.

To the m&m 's in my life, Monika, Mara, and Michelle

CONTENTS

Preface to the Second Edition

xiv

List of Symbols

xvii

1 Fundamentals of Thermal Radiation

1

1.1 Introduction 1.2 The Nature of Thermal Radiation 1.3 Basic Laws of Thermal Radiation 1.4 Emissive Power 1.5 SolidAngles 1.6 Radiative Intensity 1.7 Radiative Heat Flux 1.8 Radiation Pressure 1.9 Visible Radiation (Luminance) 1.10 Introduction to Radiation Characteristics 1.11 Introduction to Radiation Characteristics 1.12 Introduction to Radiation Characteristics 1.13 Introduction to Radiation Characteristics 1.14 Outline of Radiative Transport Theory References Problems

1 3 4 6 11 13 16 17 18 20 22 24 24 26 27 28

of Opaque Surfaces of Gases of Solids and Liquids of Particles

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

Vll

30 30 31 32 37 42 57 60 60

VUl

CONTENTS

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

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

5 Radiative Exchange Between Gray, Diffuse Surfaces

61 61 62 73 76 84 90 95 96 103 107 121 126

131 131 132 136 137 141 145 149 154 156 158 158

162

5.1 Introduction 5.2 Radiative Exchange Between Black Surfaces 5.3 Radiative Exchange Between Gray, Diffuse Surfaces 5.4 Electrical Network Analogy 5.5 Solution Methods for the Governing Integral Equations References Problems

162 163 168 175 179 189 190

6 Radiative Exchange Between Partially-Specular Gray Surfaces

198

6.1 Introduction 6c2 Specular View Factors 6.3 Enclosures with Partially-Specular Surfaces 6.4 Electrical Network Analogy 6.5 Radiation Shields 6.6 Semitransparent Sheets (Windows) 6.7 Solutionof the Governing Integral Equation 6.8 Concluding Remarks References Problems

198 200 203 216 217 219 223 225 226 227

CONTENTS

7 Radiative Exchange Between Nonideal Surfaces 7.1 Introduction 7.2 Radiative Exchange Between Nongray Surfaces 7.3 Directionally Nonideal Surfaces 7.4 Analysis for Arbitrary Surface Characteristics References Problems

8 Surface Radiative Exchange in the Presence of Conduction and Convection 8.1 Introduction 8.2 Conduction and Surface Radiation—Fins 8.3 Convection and Surface Radiation References Problems

9 The Equation of Radiative Transfer in Participating Media 9.1 Introduction 9.2 Radiative Intensity in Vacuum 9.3 Attenuation by Absorption and Scattering 9.4 Augmentation by Emission and Scattering 9.5 The Equation of Transfer 9.6 Formal Solution to the Equation of Transfer 9.7 Boundary Conditions for the Equation of Transfer 9.8 Radiation Energy Density 9.9 Radiative Heat Flux 9.10 Divergence of the Radiative Heat Flux 9.11 Integral Formulation ofthe Equation of Transfer 9.12 Overall Energy Conservation 9.13 Solution Methods for the Equation of Transfer References Problems

10 Radiative Properties of Molecular Gases 10.1 Fundamental Principles 10.2 Emission and Absorption Probabilities 10.3 Atomic and Molecular Spectra 10.4 Line Radiation 10.5 Spectral Models for Radiative Transfer Calculations 10.6 Narrow Band Models 10.7 Narrov^ Band ^-Distributions 10.8 Wide Band Models 10.9 Total Emissivity and Mean Absorption Coefficient 10.10 Experimental Methods References Problems

IX

233 233 234 238 246 247 248

250 250 251 254 258 261

263 263 264 265 267 269 271 274 275 276 277 279 281 282 284 285

288 288 290 292 297 304 307 317 324 339 346 352 356

CONTENTS

11 Radiative Properties of Particulate Media 11.1 Introduction 11.2 Absorption and Scattering from a Single Sphere 11.3 Radiative Properties ofa Particle Cloud 11.4 Radiative Properties of Small Spheres (Rayleigh Scattering) 11.5 Rayleigh-Gans Scattering 11.6 Anomalous Diffraction 11.7 Radiative Properties of Large Spheres 11.8 Absorption and Scattering by Long Cylinders 11.9 Approximate Scattering Phase Functions 11.10 Experimental Determination ofRadiative Properties of Particles 11.11 Radiation Properties of Combustion Particles References Problems

12 Radiative Properties of Semitransparent Media 12.1 Introduction 12.2 Absorption by Semitransparent SoUds 12.3 Absorption by Semitransparent Liquids 12.4 Experimental Methods References Problems

13 Exact Solutions for One-Dimensional Gray Media 13.1 Introduction 13.2 General Formulation for a Plane-Parallel Medium 13.3 Radiative Equilibrium ofa Nonscattering Medium 13.4 Radiative Equilibrium ofa Scattering Medium 13.5 Plane Medium with Specified Temperature Field 13.6 Radiative Transfer in Spherical Media 13.7 Radiative Transfer in Cylindrical Media 13.8 Nuiherical Solution of the Governing Integral Equations References Problems

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

15 The Method of Spherical Harmonics (/VApproximation) 15.1 15.2 15.3 15.4

Introduction Development ofthe General/^-Approximation Boundary Conditions for the Pv-Method The Pi-Approximation

361 361 362 368 373 375 376 377 383 385 390 394 405 410

413 413 414 416 418 421 422

423 423 424 428 433 434 436 440 444 445 446

449 450 451 456 458 461 463 463

465 465 466 469 472

CONTENTS

XI

15.5 Z^-and Higher-Order Approximations 15.6 Enhancements to the P]-Approximation References Problems

479 483 492 494

16 The Method of Discrete Ordinates (Syv-Approximation)

498

16.1 Introduction 16.2 General Relations 16.3 The One-Dimensional Slab 16.4 One-Dimensional Concentric Spheres and Cylinders 16.5 Multidimensional Problems 16.6 The Finite Volume Method 16.7 Other Related Methods 16.8 Concluding Remarks References Problems

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

18 The Treatment of CoUimated Irradiation

498 499 502 507 513 523 529 530 530 536

539 539 539 545 551 558 561 561 562

565

18.1 Introduction 18.2 Reduction of the Problem 18.3 The Modified Pi-Approximation with CoUimated Irradiation 18.4 Short-Pulsed CoUimated Irradiation with Transient Effects References Problems

565 568 571 574 577 579

19 The Treatment of Nongray Extinction Coefficients

581

19.1 Introduction 19.2 The Mean Beam Length Method 19.3 Semigray Approximations 19.4 The Stepwise-Gray Model (Box Model) 19.5 General Band Model Formulation 19.6 The Weighted-Sum-of-Gray-Gases (WSGG) Model 19.7 ^-Distribution Models 19.8 The FuU-Spectmm ^-Distribution (FSK) Method References Problems

581 583 589 592 603 611 616 617 637 641

XU

CONTENTS

20 The Monte Carlo Method for Thermal Radiation 20.1 Introduction 20.2 Numerical Quadrature by Monte Carlo 20.3 Heat Transfer Relations for Radiative Exchange Between Surfaces 20.4 Random Number Relations for Surface Exchange 20.5 Surface Description 20.6 Ray Tracing 20.7 Heat Transfer Relations for Participating Media 20.8 Random Number Relations for Participating Media 20.9 Overall Energy Conservation 20.10 Efficiency Considerations 20.11 Backward Monte Carlo 20.12 Example Problems References Problems

21 Radiation Combined with Conduction and Convection 21.1 Introduction 21.2 Combined Radiation and Conduction 21.3 Melting and Solidification with Intemal Radiation 21.4 Combined Radiation and Convection in Boundary Layers 21.5 Combined Radiation and Free Convection 21.6 Combined Radiation and Convection in Internal Flow 21.7 Combined Radiation and Combustion 21.8 Interfacing Between Turbulent Flow Fields and Radiation 21.9 Interaction ofRadiation with Turbulence References Problems

22 Inverse Radiative Heat Transfer

644 644 648 649 651 655 655 658 659 666 667 669 673 676 678

680 680 681 689 695 700 700 705 707 710 715 727

729

22.1 Introduction 22.2 Solution Methods 22.3 The Levenberg-Marquardt Method 22.4 The Conjugate Gradient Method 22.5 Inverse Surface Radiation 22.6 Inverse Radiation in Participating Media References

729 730 732 732 733 736 739

Problems

742

A Constants and Conversion Factors

743

B Tables for Radiative Properties of Opaque Surfaces

745

References

758

C Blaclibody Emissive Pov^er Table

759

D View Factor Catalogue

762

References

773

CONTENTS

E Exponential Integral Functions References

F Computer Codes References

XUl

779 781

782 788

Acknowledgments

789

Author Index

792

Subject Index

808

PREFACE TO THE SECOND EDITION

Ten years have passed since the first edition of "Radiative Heat Transfer" was published. Because of continued interest in the field of radiation and due to the emergence of new research topics, the field has seen many significant advances during these ten years. Thus, the contents of the second edition of this book has changed significantly to reflect this additional knowledge and also to improve its general readabihty and usefulness. The objectives of this book remain the same, and are more extensive than to provide a standard textbook for a one-semester core course on thermal radiation, since it does not appear possible to cover all important topics in the field of radiative heat transfer in a single graduate course. A number of important areas that would not be part of a "standard" one-semester course have been treated in some detail. It is anticipated that the engineer who may have used this book as his or her graduate textbook will be able to master these advanced topics through self-study. By including all important advanced topics, as well as a large number of references for further reading, the book may also be used as a reference book by the practicing engineer. Major changes in the second edition include breaking the chapter on radiative interaction with conduction and convection into two. The first deals with surface radiation and is now placed much earlier in the book, giving instructors the opportunity to include it as part of a surface radiation module. The second, dealing with participating media, has been greatly augmented, incorporating the many new developments in the fields of combustion and turbulent flow in the presence of radiation. The chapters on gas properties and on nongray modeling have essentially been rewritten because of the great advances made in these fields, and the chapter on particle properties has been augmented to include fiber properties and the latest knowledge on soot properties. The last ten years have also seen the discrete ordinates method become the solution method of choice, and significant new space has been devoted to it and its new cousin, the finite volume method. Other new developments included are the treatment of transient laser radiation (a new section in the chapter on collimated radiation), reverse Monte Carlo simulations (also resulting in a new section), and a short new chapter has been devoted to the emerging field of inverse radiation. A new appendix describes a number of computer programs, which may be downloaded from a dedicated web site located at www. academicpressbooks. com. Some of the codes are very basic and are entirely intended to aid the reader with the solution to the problems given at the end of the more basic chapters. Others were bom out of research, some basic enough to aid a graduate student with more complicated assignments or a semester project, and a few so sophisticated in nature that they will be useful only to the practicing engineer conducting his or her own research.

XIV

PREFACE TO THE SECOND EDITION

XV

Many smaller changes have also been made, among them the inclusion of many additional problems as well as a comprehensive literature update. Finally, the names of most surface properties have been changed as recommended by the National Institute of Standards and Technology (NIST), While this will cause some grumbling by a few (after all, none of us likes to change old habits!), the NIST standard has gained substantial ground over the past few years, and this book wants to make its contribution toward achieving this uniform standard. As in the first edition, each chapter shows the development of all analytical methods in substantial detail, and contains a number of examples to show how the developed relations may be applied to practical problems. At the end of each chapter a number of exercises are included to give the student additional opportunity to famiUarize him- or herself with the application of analytical methods developed in the preceding sections. The breadth of the description of analytical developments is such that any scientist with a satisfactory background in calculus and differential equations will be able to grasp the subject through self-study—^for example, the heat transfer engineer involved in furnace calculations, the architectural engineer interested in lighting calculations, the oceanographer concerned with solar penetration into the ocean, or the meteorologist who studies atmospheric radiation problems. An expanded Instructor's Solutions Manual is available for adopting instructors who register at www. academicpressbooks. com. The book is now divided into 22 chapters, covering the four major areas in the field of radiative heat transfer. After the Introduction, there are two chapters dealing with theoretical and practical aspects of radiative properties of opaque surfaces, including a brief discussion of experimental methods. These are followed by four chapters dealing with purely radiative exchange between surfaces in an enclosure without a "radiatively participating" medium, and one more chapter examining the interaction of conduction and convection with surface radiation. The rest of the book deals with radiative transfer through absorbing, emitting, and scattering media (or "participating media"). Aft;er a detailed development of the equation of radiative transfer, radiative properties of gases, particulates, and semitransparent media are discussed, again including brief descriptions of experimental methods. The next nine chapters cover the theory of radiative heat transfer through participating media, separated into a number of basic problem areas and solution methods. And, finally, the book ends with two chapters on combined-modes heat transfer and the emerging field of inverse radiation. I have attempted to write the book in a modular fashion as much as possible. Chapter 2 is a fairly detailed (albeit concise) treatment of electromagnetic wave theory, which can (and will) be skipped by most instructors for a first course in radiative heat transfer. The chapter on opaque surface properties is self-contained and is not required reading for the rest of the book. The four chapters on surface transport (Chapters 4 through 8) are also self-contained and not required for the study of radiation in participating media. Similarly, the treatment of participating medixmi properties is not a prerequisite to studying the solution methods. Along the same line, any of the different solution aspects and methods discussed in Chapters 13 through 20 may be studied in any sequence. Whether either of the last two chapters are covered or skipped will depend entirely on the instructor's preferences or those of his or her students. I have not tried to mark those parts of the book that should be included in a one-semester course on thennal radiation, since I feel that different instructors will, and should, have different opinions on that matter. Indeed, the relative importance of different subjects may not only vary with different instructors, but also depend on student background, location, or the year of instruction. My personal opinion is that a one-semester course should touch on all four major areas (surface properties, surface transport, properties of participating media, and transfer through participating media) in a balanced way. For the average U.S. student who has had very little exposure to thermal radiation during his or her undergraduate heat transfer experience, I suggest that about half the course be devoted to Chapters 1, 3, 4, 5, plus parts of Chapters 6, 7 and/or 8 (leaving out the more advanced features). The second half should be devoted to Chapters 9, 10 and 11

XVI

PREFACE TO THE SECOND EDITION

(again omitting less important featxires); some coverage of Chapter 13; and a thorough discussion of Chapter 14. If time permits (primarily, if surface and/or participating media properties are treated in less detail than suggested above), I suggest to cover the Pj-approximation (which may be studied by itself, as outlined in the beginning of Chapter 15), the basic ideas behind the discrete ordinates method, and/or a portion of Chapter 19 (solution methods for nongray media). A second, special-topics course could include detailed discussions of radiative properties (Chapters 2, 3, 10, 11, and 12) and/or in-depth coverage of certain solution methods (such as /^ and 5,v/FVM approximations, nongray media, or zonal and Monte Carlo methods). Michael E Modest

LIST OF SYMBOLS

The following is a list of symbols used frequently in this book. A number of symbols have been used for several different purposes. Alas, the Roman alphabet has only 26 lowercase and another 26 uppercase letters, and the Greek alphabet provides 34 more different ones, for a total of 86, which is, unfortunately, not nearly enough. Hopefully, the context will always make it clear which meaning of the symbols is to be used. I have used what I hope is a simple and uncluttered set of variable names. This usage, of course, comes at a price. For example, the subscript " i " is often dropped (meaning "at a given wavelength," or "per unit wavelength"), assuming that the reader recognizes the variable as a spectral quantity from the context. Whenever appUcable, units have been attached to the variables in the following table. Variables without indicated units have multiple sets of units. For example, the units for total band absorptance depend on the spectral variable used {A, rj or v), and on the absorption coefficient (linear, density- or pressure-based), for a total of nine different possibilities. a semimajor axis of polarization ellipse, [N/C] a plane-polarized component of electric field, [N/C] a particle radius, [cm] a weight fimction for full-spectrum ^-distribution methods, [-] Uk weight factors for sum-of-gray-gases, [-] Un, bn Mie scattering coefficients, [-] A total band absorptance (or effective band width) A* nondimensional band absorptance = Ala), [-] A, An slab absorptivity (of « parallel sheets), [-] A, Ap area, projected area, [cm^] Am scattering phase function coefficients, [~] Aij, Bij Einstein coefficients b line half-width b self-broadening coefficient, [-] b semiminor axis of polarization ellipse, [N/C] B rotational constant Bo convection-to-radiation parameter (Boltzmann number), [-] c, q, speed of light, (in vacuum), [cm/s] c specific heat, [kJ/kg K] C\,Ci. C3 constants for Planck function and Wien's displacement law Ci, C2, C3 wide band parameters for outdated model d line spacing

xvn

XVUl

LIST OF SYMBOLS

D diameter, [cm] A D* detectivity (normalized), [1/W] ([cm Hz^/^W]) Df mass fractal dimension, [-] e unit vector into local coordinate direction, [-] E, Eh emissive power, blackbody emissive power E electric field vector, [N/C] Efi exponential integral of order n, [-] / ^-distribution, [cm] /il, fsi fi volume, solid, liquid fractions, [ - ] f(nAT) fractional blackbody emissive power, [-•] F wide band ^-distribution, [cm] Fi-j (diffuse) view factor, [-] F^. specular view factor, [~] ^^j radiation exchange factor, [-] cjk degeneracy, [-] g nondimensional incident radiation, [-] g cumulative distribution, [-] ^ » 9i9k direct exchange areas in zonal method, [cm^] p , gg direct exchange area matrix, [cm^] G incident radiation = direction-integrated intensity G/5;, G,Gk total exchange areas in zonal method, [cm^] GS, GG total exchange area matrix, [cm^] h Planck's constant = 6.6261 x 10"^"^ J s h convective heat transfer coefficient, [W/cm^ K] H irradiation onto a surface H Heaviside's unit step function, [-] H nondimensional heat transfer coefficient, [-] ^ nondimensional irradiation onto a surface, [-] H magnetic field vector, [C/cm s] / nondimensional polarized intensity, [-] i unit vector into the jc-direction, [-] / intensity of radiation / first Stokes' parameter for polarization, [N^/C^] / moment of inertia, [kg cm^] lb blackbody intensity (Planck function) //, IJ^' position-dependent intensity functions /(), /i modified Bessel functions, [-] 3 imaginary part of complex number j rotational quantum number, [-] j unit vector into the ^/-direction, [-] J radiosity, [W/cm^] ^ nondimensional radiosity, [-] k thermal conductivity, [W/m K] k Boltzmann's constant = 1.3807 x 10"^^ J/K k absorptive index in complex index of refraction, [-] k absorption coefficient variable, [cm~^] kf fractal prefactor, [-] k unit vector into the z-direction, [~] K kernel function

LIST OF SYMBOLS

K /, m, n L L L Le Lo, L,n m m m Jf n n n n A'' Nc Nf Nu (f {} p P /}, Pl^ Pr q, q qji ^lum Q Q g'^' r r r R Ri, R R R, Rr, % Re s § J[Sj, ligk ss, sg S S S S St

luminous efficacy, [Im/W] direction cosines with x-, y-, z-axis, [-] length, [cm] latent heat of fusion, [J/kg] luminance mean beam length, [cm] geometric, or average mean beam length, [cm] mass, [kg] complex index of refraction, [~] mass flow rate, [kg/s] molecular weight, [g/mol] self-broadening exponent, [~] refractive index, [-] number distribution function for particles, [cm""*] unit surface normal (pointing away from surface into the medium), [-] conduction-to-radiation parameter (Stark number), [-] conduction-to-radiation parameter, [~] nimiber of particles per imit volume, [cm"^] Nusselt number, [~] order of magnitude, [-] pressure, [bar]; radiation pressure, [N/m^] probability function, [-] (associated) Legendre polynomials, [-] Prandtl number, [~] heat flux, heat flux vector, [W/cm^] radiative flux, [W/cm^] luminous flux, [Im/m^ = Ix] heat rate, [W] second Stokes' parameter for polarization, [N^/C^] heat production per unit volume, [W/cm^] radial coordinate, [cm] reflection coefiicient, [-] position vector, [cm] radius, [cm] universal gas constant = 8.3145 J/mol K random number, [-] radiative resistance, [cm~^] slab reflectivity (of « parallel sheets), [-] real part of complex number Reynolds number, [~] geometric path length, [cm] unit vector into a given direction, [-] direct exchange areas in zonal method, [cm^] direct exchange area matrix, [cm^] distance between two zones, or between points on enclosure surface, [cm] line-integrated absorption coefiicient = line strength radiative source fimction Poynting vector, [W/cm^] Stanton number, [-]

XIX

XX

LIST OF SYMBOLS

Ste 5/5;, SiGk SS, SG / t t fin t T r, Tn u u u u Uk U V V V V V w Wi W X, y , z X X X X X yj" z a a a a p P y y y y yp d 6 dij dk 6 e 6

Stefan niimber, [~] total exchange areas in zonal method, [cm^] total exchange area matrix, [cm^] time, [s] transmission coefficient, [-] thickness, [cm] unit vector in tangential direction, [-] temperature, [K] slab transmissivity (of « parallel sheets), [~] internal energy, [kJ/kg] radiation energy density velocity, [cm/s] scaling function for absorption coefficient, [-] nondimensional transition vt^avenumber, [ - ] third Stokes' parameter for polarization, [N^/C^] vibrational quantum number, [-] velocity, [cm/s] velocity vector, [cm/s] volume, [cm^] fourth Stokes' parameter for polarization, [ N ^ C ^ ] wave vector, [cm**" ^ ] quadrature weights, [-] equivalent line width Cartesian coordinates, [cm] particle size parameter, [-] line strength parameter, [-] mole fraction, [-] optical path length interface location, [cm] spherical harmonics, [--] nondimensional spectral variable, [-] absorptance or absorptivity, [-] band-integrated absorption coefficient = band strength parameter opening angle, [rad] thermal diffusivity, [m^/s] extinction coefficient line overlap parameter, [~] line overlap parameter for dilute gas, [-] complex permittivity, [C^/N m^] azimuthal rotation angle for polarization ellipse, [rad] oscillation damping factor, [Hz] Euler's constant = 0.57221... Dirac-delta function, [~] polarization phase angle, [rad] Kronecker's delta, [-] vibrational transition quantum step = Ay, [--] emittance or emissivity, [-] energy level, [J] electrical permittivity, [C^/N m^]

LIST OF SYMBOLS

e Tf 77 77 /Tiun^ 0 9 0 K A // fi /i V V ^ ^ p p Py^ (T (Ts 0^, cTdc 07, a? T T (f> 0 O O m= r

imcoseidQi,

(1.35)

Jcos0i0

If the surface is black ( Q = 1), there is no energy reflected from the surface and li = IbA, leading to (^,i)out = EfjA' If the surface is not black, the outgoing intensity consists of contributions firom emission as well as reflections. The outgoing heat flux is positive since it is going into the medium. The net heat flux from the surface may be calculated by adding both contributions, or (^^)net = fe)in + ^^Oout= I

l\(§,) COS 9 dQ.,

(1.37)

J An

where a single direction vector s was used to describe the total range of solid angles, 4n. It is readily seen from Fig. 1-10 that cos6> = n • s and, since the net heat flux is evaluated as the flux into the positive n-direction, one gets (1.38) J An

1.8 RADIATION PRESSURE

17

In order to obtain the total radiative heat flux at the surface, equation (1.38) needs to be integrated over the spectrum, and q.n=

f

q,ihdA=\

\ /.](s)ii-s^QJi.

Jo

Jo

(1.39)

JAn

Example 1.5. A solar collector mounted on a satellite orbiting Earth is directed at the sun (i.e., normal to the sun's rays). Determine the total solar heatfluxincident on the collector per unit area. Solution The total heat rate leaving the sun is Qs = AnRlEh{Ts\ where Rs ^ 6.96 x 10^ m is the radius of the sun. Placing an imaginary spherical shell around the sun of radius SES - 1.496 x 10^ ^ m, where SES is the distance between the sun and Earth, wefindthe heatfluxgoing through that imaginary sphere (which includes the solar collector) as

?soi =

; ;,

= hiTs)-f = h{Ts)a,,

'^ES

where we have replaced the sun's emissive power by intensity, Eh = TT//,, and ^s = 6.80 x 10 ^ sr is the solid angle with which the sun is seen from Earth, as determined in Example 1.3. Therefore, with UTs) = o-r/M and Ts = 5777K, q.^ = -{(TTsVnX^s) = --5.670 x 10"^ x 5777^ x 6.80 x 10"' W/m' n = -1367W/m% where we have added a minus sign to emphasize that the heatfluxis going into the collector. The total incoming heatfluxmay, of course, also be determinedfromequation (1.35) as !(%) COS OidClj. Jcc cos6i