Radiative Heat Transfer, Third Edition [3ed.] 0123869447, 978-0-12-386944-9, 9780123869906, 0123869900

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Radiative Heat Transfer, Third Edition [3ed.]
 0123869447, 978-0-12-386944-9, 9780123869906, 0123869900

Table of contents :
Content: 6.5 Radiation Shields6.6 Semitransparent Sheets (Windows)
6.7 Solution of the Governing Integral Equation
6.8 Concluding Remarks
References
Problems
Chapter 7: Radiative Exchange Between Nonideal Surfaces
7.1 Introduction
7.2 Radiative Ex. 4.8 The Inside Sphere Method4.9 The Unit Sphere Method
References
Problems
Chapter 5: Radiative Exchange Between Gray, Diffuse Surfaces
5.1 Introduction
5.2 Radiative Exchange Between Black Surfaces
5.3 Radiative Exchange Between Gray, Dif. 3.2 Definitions3.3 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 Proper. 1.13 Introduction to Radiation Characteristics of Solids and Liquids1.14 Introduction to Radiation Characteristics of Particles
1.15 The Radiative Transfer Equation
1.16 Outline of Radiative Transport Theory
References
Problems
Chapter 2: R. Front Cover
Radiative Heat Transfer
Copyright Page
About the Author
Dedication
Contents
Preface to the Third Edition
List of Symbols
Chapter 1: Fundamentals of Thermal Radiation
1.1 Introduction
1.2 The Nature of Thermal Radiation
1.3.

Citation preview

RADIATIVE HEAT TRANSFER

Third Edition

Michael F. Modest The University of California at Merced

New York

San Francisco

London

Academic Press is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands First edition 1993 Second edition 2003 Third edition 2013 Copyright © 2013 Elsevier Inc. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Permissions may be sought directly from Elseviers Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material. Notices No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-386944-9 For information on all Academic Press publications visit our website at store.elsevier.com Printed and bound in USA 13 14 15 16 17

10 9 8 7 6 5 4 3 2 1

ABOUT THE AUTHOR

Michael F. Modest was born in Berlin and spent the first 25 years of his life in Germany. After receiving his Dipl.-Ing. degree from the Technical University in Munich, he came to the United States, and in 1972 obtained his M.S. and Ph.D. in Mechanical Engineering from the University of California at Berkeley, where he was first introduced to theory and experiment in thermal radiation. Since then, he has carried out many research projects in all areas of radiative heat transfer (measurement of surface, liquid, and gas properties; theoretical modeling for surface transport and within participating media). Since many laser beams are a form of thermal radiation, his work also encompasses the heat transfer aspects in the field of laser processing of materials. For several years he has taught at Rensselaer Polytechnic Institute and the University of Southern California, and for 24 years was a Professor of Mechanical Engineering at the Pennsylvania State University. Today Dr. Modest is the Shaffer and George Professor of Engineering at the University of California, Merced, the 10th campus of the University of California system, and the first newly established research university of the 21st century. He is a fellow of the American Society of Mechanical Engineers, and an associate fellow of the American Institute of Aeronautics and Astronautics. Dr. Modest and his wife Monika reside in Merced, CA.

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

CONTENTS

Preface to the Third Edition

xiv

List of Symbols 1

xvii

Fundamentals of Thermal Radiation

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1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Nature of Thermal Radiation . . . . . . . . . . . . . . . . . 1.3 Basic Laws of Thermal Radiation . . . . . . . . . . . . . . . . . 1.4 Emissive Power . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Solid Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Radiative Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Radiative Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Radiation Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Visible Radiation (Luminance) . . . . . . . . . . . . . . . . . . . 1.10 Radiative Intensity in Vacuum . . . . . . . . . . . . . . . . . . . 1.11 Introduction to Radiation Characteristics of Opaque Surfaces . 1.12 Introduction to Radiation Characteristics of Gases . . . . . . . 1.13 Introduction to Radiation Characteristics of Solids and Liquids 1.14 Introduction to Radiation Characteristics of Particles . . . . . . 1.15 The Radiative Transfer Equation . . . . . . . . . . . . . . . . . . 1.16 Outline of Radiative Transport Theory . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Radiative Properties of Real Surfaces

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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 . . . . . . . . . . . . . . . . . . . . . . . . . .

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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 . . . . . . . . . . . . . . . . . . . . . . . .

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Radiative Exchange Between Gray, Diffuse Surfaces

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 Radiation Shields . . . . . . . . . . . . . . . . . . . . . . 5.6 Solution Methods for the Governing Integral Equations References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Radiative Exchange Between Partially Specular Gray Surfaces 6.1 Introduction . . . . . . . . . . . . . . . . . . 6.2 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 Solution of the Governing Integral Equation 6.8 Concluding Remarks . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . .

61 62 73 75 83 89 93 95 101 105 118 123

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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 . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Numerical Quadrature by Monte Carlo . . . . . . . . . . . . . . . . 8.3 Heat Transfer Relations for Radiative Exchange Between Surfaces 8.4 Random Number Relations for Surface Exchange . . . . . . . . . . 8.5 Surface Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Ray Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Efficiency Considerations . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Monte Carlo Method for Surface Exchange

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9 Surface Radiative Exchange in the Presence of Conduction and Convection 9.1 Introduction . . . . . . . . . . . . . . . . 9.2 Conduction and Surface Radiation—Fins 9.3 Convection and Surface Radiation . . . . References . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . .

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10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Attenuation by Absorption and Scattering . . . . . . . . . 10.3 Augmentation by Emission and Scattering . . . . . . . . . 10.4 The Radiative Transfer Equation . . . . . . . . . . . . . . . 10.5 Formal Solution to the Radiative Transfer Equation . . . . 10.6 Boundary Conditions for the Radiative Transfer Equation 10.7 Radiation Energy Density . . . . . . . . . . . . . . . . . . . 10.8 Radiative Heat Flux . . . . . . . . . . . . . . . . . . . . . . 10.9 Divergence of the Radiative Heat Flux . . . . . . . . . . . 10.10 Integral Formulation of the Radiative Transfer Equation . 10.11 Overall Energy Conservation . . . . . . . . . . . . . . . . . 10.12 Solution Methods for the Radiative Transfer Equation . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8

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11.9 Narrow Band k-Distributions . . . . . . . . . . . . . 11.10 Wide Band Models . . . . . . . . . . . . . . . . . . . 11.11 Total Emissivity and Mean Absorption Coefficient 11.12 Experimental Methods . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Absorption and Scattering from a Single Sphere . . . . . . . . . . 12.3 Radiative Properties of a Particle Cloud . . . . . . . . . . . . . . . 12.4 Radiative Properties of Small Spheres (Rayleigh Scattering) . . . 12.5 Rayleigh–Gans Scattering . . . . . . . . . . . . . . . . . . . . . . . 12.6 Anomalous Diffraction . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Radiative Properties of Large Spheres . . . . . . . . . . . . . . . . 12.8 Absorption and Scattering by Long Cylinders . . . . . . . . . . . 12.9 Approximate Scattering Phase Functions . . . . . . . . . . . . . . 12.10 Radiative Properties of Irregular Particles and Aggregates . . . . 12.11 Radiative Properties of Combustion Particles . . . . . . . . . . . 12.12 Experimental Determination of Radiative Properties of Particles References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Radiative Properties of Particulate Media

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Radiative Properties of Semitransparent Media

13.1 Introduction . . . . . . . . . . . . . . . . 13.2 Absorption by Semitransparent Solids . 13.3 Absorption by Semitransparent Liquids 13.4 Radiative Properties of Porous Solids . . 13.5 Experimental Methods . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . .

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15.1 The Optically Thin Approximation . . . . . . . . . . . . . . . . . 15.2 The Optically Thick Approximation (Diffusion Approximation) . 15.3 The Schuster–Schwarzschild Approximation . . . . . . . . . . . . 15.4 The Milne–Eddington Approximation (Moment Method) . . . . 15.5 The Exponential Kernel Approximation . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Exact Solutions for One-Dimensional Gray Media

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 General Formulation for a Plane-Parallel Medium . . . . 14.3 Plane Layer of a Nonscattering Medium . . . . . . . . . 14.4 Plane Layer of a Scattering Medium . . . . . . . . . . . . 14.5 Radiative Transfer in Spherical Media . . . . . . . . . . . 14.6 Radiative Transfer in Cylindrical Media . . . . . . . . . 14.7 Numerical Solution of the Governing Integral Equations References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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15 Approximate Solution Methods for One-Dimensional Media

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16 The Method of Spherical Harmonics (PN -Approximation)

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16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 16.2 General Formulation of the PN -Approximation . . . 16.3 The PN -Approximation for a One-Dimensional Slab 16.4 Boundary Conditions for the PN -Method . . . . . . . 16.5 The P1 -Approximation . . . . . . . . . . . . . . . . . 16.6 P3 - and Higher-Order Approximations . . . . . . . . 16.7 Simplified PN -Approximation . . . . . . . . . . . . . 16.8 The Modified Differential Approximation . . . . . . 16.9 Comparison of Methods . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Surface Exchange — No Participating Medium . . . . . . . . 18.3 Radiative Exchange in Gray Absorbing/Emitting Media . . . 18.4 Radiative Exchange in Gray Media with Isotropic Scattering 18.5 Radiative Exchange through a Nongray Medium . . . . . . . 18.6 Determination of Direct Exchange Areas . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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17 The Method of Discrete Ordinates (SN -Approximation)

541

17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 17.2 General Relations . . . . . . . . . . . . . . . . . . . . 17.3 The One-Dimensional Slab . . . . . . . . . . . . . . . 17.4 One-Dimensional Concentric Spheres and Cylinders 17.5 Multidimensional Problems . . . . . . . . . . . . . . 17.6 The Finite Volume Method . . . . . . . . . . . . . . . 17.7 The Modified Discrete Ordinates Method . . . . . . . 17.8 Even-Parity Formulation . . . . . . . . . . . . . . . . 17.9 Other Related Methods . . . . . . . . . . . . . . . . . 17.10 Concluding Remarks . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

19

The Zonal Method

Collimated Irradiation and Transient Phenomena

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585 585 590 596 603 606 606 607

610

20 Solution Methods for Nongray Extinction Coefficients Introduction . . . . . . . . . . . . . . . The Mean Beam Length Method . . . . Semigray Approximations . . . . . . . The Stepwise-Gray Model (Box Model)

541 542 545 550 556 566 572 573 574 576 576 582

585

19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Reduction of the Problem . . . . . . . . . . . . . . . . . . . . . 19.3 The Modified P1 -Approximation with Collimated Irradiation 19.4 Short-Pulsed Collimated Irradiation with Transient Effects . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20.1 20.2 20.3 20.4

495 496 497 498 502 509 522 527 531 534 537

610 613 616 619 622 624

626 . . . .

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626 628 634 637

xii

CONTENTS

20.5 General Band Model Formulation . . . . . . . . . . . . . . . . . . . . . . . 20.6 The Weighted-Sum-of- Gray-Gases (WSGG) Model . . . . . . . . . . . . . 20.7 k-Distribution Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.8 The Full Spectrum k-Distribution (FSK) Method for Homogeneous Media 20.9 The Spectral-Line-Based Weighted Sum of Gray Gases (SLW) . . . . . . . 20.10 The FSK Method for Nonhomogeneous Media . . . . . . . . . . . . . . . 20.11 Evaluation of k-Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.12 Higher Order k-Distribution Methods . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

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The Monte Carlo Method for Participating Media

21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 21.2 Heat Transfer Relations for Participating Media . . 21.3 Random Number Relations for Participating Media 21.4 Treatment of Spectral Line Structure Effects . . . . 21.5 Overall Energy Conservation . . . . . . . . . . . . . 21.6 Discrete Particle Fields . . . . . . . . . . . . . . . . 21.7 Efficiency Considerations . . . . . . . . . . . . . . . 21.8 Backward Monte Carlo . . . . . . . . . . . . . . . . 21.9 Direct Exchange Monte Carlo . . . . . . . . . . . . 21.10 Example Problems . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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694

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22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Combined Radiation and Conduction . . . . . . . . . . . . 22.3 Melting and Solidification with Internal Radiation . . . . 22.4 Combined Radiation and Convection in Boundary Layers 22.5 Combined Radiation and Free Convection . . . . . . . . . 22.6 Combined Radiation and Convection in Internal Flow . . 22.7 Combined Radiation and Combustion . . . . . . . . . . . 22.8 Interfacing Between Turbulent Flow Fields and Radiation 22.9 Interaction of Radiation with Turbulence . . . . . . . . . . 22.10 Radiation in Concentrating Solar Energy Systems . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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22 Radiation Combined with Conduction and Convection

23

694 694 695 700 705 706 712 713 717 717 720 722

724

Inverse Radiative Heat Transfer

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

643 649 654 656 659 660 668 679 686 691

724 724 733 738 743 744 748 751 753 759 763 777

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779 780 785 788 794 795 797 801

xiii

CONTENTS

24

Nanoscale Radiative Transfer

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

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803 803 804 805 807 809 811 814 816 817

A

Constants and Conversion Factors

818

B

Tables for Radiative Properties of Opaque Surfaces

820

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 820

C

Blackbody Emissive Power Table

833

D

View Factor Catalogue

836

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E

Exponential Integral Functions

852

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853

F

Computer Codes

855

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 861

Acknowledgments

863

Index

867

PREFACE TO THE THIRD EDITION

Another ten years have passed since the second edition of “Radiative Heat Transfer” was published. Thermal radiation remains a relatively young field, with basic relations dating back to the early 20th century, and serious heat transfer models only starting to appear in the 1950s. Consequently, continued interest in the field has led to many significant advances and the emergence of new research topics during these past ten years. Therefore, the contents of the third edition of this book has again changed significantly to reflect this additional knowledge, and further attempts have been made to improve its general readability 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 is also intended as a reference book for the practicing engineer. Major changes in the third edition include breaking the chapter on the statistical Monte Carlo method 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 the surface radiation discussion in a single semester course. The second, dealing with participating media, has been greatly augmented, incorporating the many new developments in the field, such as spectrallyresolved schemes, extensions for stochastic media, and others. The fields of inverse radiation and radiative transfer at the micro- and nanoscales have seen particularly much activity during the last 10 years. Therefore, the chapter on inverse radiation has been entirely rewritten, and a new chapter on nanoscale radiative heat transfer has been added. These two chapters should be understood as introductions to these extensive fields, giving the engineer a basic understanding of these new research areas, and a good foundation to embark on further reading of the pertinent literature. The chapters on gas properties and on nongray modeling have again seen very significant changes and additions because of the continued advances made in these fields and because of the growing interest in nonequilibrium radiation. The last ten years have also seen some further development in modern radiative transfer equation solution methods, reflected in the chapters on the spherical harmonics and discrete ordinates methods, in addition to the Monte Carlo method. The appendix describing a number of computer programs has been retained, and the codes may be downloaded from a dedicated web site located at http://booksite.elsevier.com/9780123869449. Some of the codes are very basic and are entirely intended to aid the reader with the solution to the problems given at the end of the early chapters on surface transport. Others were born out of research, some basic enough to aid a graduate student with more complicated assignments or a semester project, and a few so sophisticated in nature that they will be useful only to the practicing engineer conducting his or her own research. Recognizing that many graduate students no longer learn compiler languages, such as Fortran and C++, the more basic programs are now also available in Matlabr . Many smaller changes have also been made, such as omission of some obsolete material, inclusion of many new small developments, and restructuring of material between chapters xiv

PREFACE TO THE THIRD EDITION

xv

to aid readability. And, of course, a comprehensive literature update has been provided, and many new homework problems have been added at the end of the chapters. As in the first two editions, each chapter shows the development of all analytical methods in substantial detail, and contains a number of examples to show how the developed relations may be applied to practical problems. At the end of each chapter a number of exercises are included to give the student additional opportunity to familiarize him- or herself with the application of analytical methods developed in the preceding sections. The breadth of the description of analytical developments is such that any scientist with a satisfactory background in calculus and differential equations will be able to grasp the subject through self-study—for example, the heat transfer engineer involved in furnace calculations, the architectural engineer interested in lighting calculations, the oceanographer concerned with solar penetration into the ocean, or the meteorologist who studies atmospheric radiation problems. An expanded Instructor’s Solutions Manual is available for adopting instructors who register at http://textbooks.elsevier.com/web/product details.aspx?isbn=9780123869449. The book is now divided into 24 chapters, covering the four major areas in the field of radiative heat transfer. After the Introduction, there are two chapters dealing with theoretical and practical aspects of radiative properties of opaque surfaces, including a brief discussion of experimental methods. These are followed by five chapters dealing with purely radiative exchange between surfaces in an enclosure without a “radiatively participating” medium, and one more chapter examining the interaction of conduction and convection with surface radiation. The rest of the book deals with radiative transfer through absorbing, emitting, and scattering media (or “participating media”). After a detailed development of the equation of radiative transfer, radiative properties of gases, particulates, and semitransparent media are discussed, again including brief descriptions of experimental methods. The next eight 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 three chapters on combined-modes heat transfer and the emerging fields of inverse and nanoscale radiative heat transfer. 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 five chapters on surface transport (Chapters 4 through 9) are also self-contained and not required for the study of radiation in participating media. Similarly, the treatment of participating medium properties is not a prerequisite to studying the solution methods. Along the same line, any of the different solution aspects and methods discussed in Chapters 14 through 21 may be studied in any sequence (although Chapter 21 requires knowledge of Chapter 8). Whether any of the last three 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 thermal 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 7, 8 and/or 9, leaving out the more advanced features. While the Monte Carlo method of Chapter 8 may be considered an “advanced feature,” I have found it to be immensely popular with students, and at the same time gives exposure to an engineering tool of fast-growing importance. The second half of the course should be devoted to Chapters 10, 11 and 12 (again

xvi

PREFACE TO THE THIRD EDITION

omitting less important features); some coverage of Chapter 14; and a thorough discussion of Chapter 15. If time permits (primarily, if surface and/or participating media properties are treated in less detail than indicated above), I suggest to cover the P1 -approximation (which may be studied by itself, as outlined in the beginning of Chapter 16), the basic ideas behind the discrete ordinates method, and/or a portion of Chapter 20 (solution methods for nongray media). With the addition of new material, and in spite of omitting outdated items, the third edition has again grown considerably over its previous version. I would like to thank several of my friends and colleagues from around the world who guided me in the decision making process for many of the changes in this book, viz., A. Charette (Quebec), P. Coelho (Lisbon), K. Daun (Waterloo, Canada), L. Dombrovsky (Moscow), W. Lipinski (Minnesota), S. Mazumder (Ohio), ´ K. Mitra (Florida), L. Pilon (California), S. Thynell (Pennsylvania), and R. Viskanta (Indiana). Of course, if you put ten professors into a room, you are bound to get a minimum of twelve different opinions: I hope they will forgive me if not all their suggestions were followed. Z. Zhang (Georgia) supplied a small Matlabr program, from which Figs. 24-7 and 24-8 were generated, which is gratefully acknowledged. And a special thank you goes to two of my young colleagues: Wojciech Lipinski for writing the two sections on Radiative Properties of Porous Solids ´ (Chapter 13) and Radiation in Concentrating Solar Energy Systems (Chapter 22); and also Kyle Daun for many hours spent helping me to rewrite the chapter on inverse radiation (Chapter 23). Thanks also go to two of my postdoctoral researchers, J. Cai and R. Marquez, who converted the more basic computer codes (Appendix F) to Matlabr . Finally, I would like to posthumously acknowledge Eileen Stevenson, my wonderful secretary from many years ago who typed the first edition, and who recently passed away at much too young an age. Michael F. Modest Merced, California November 2012

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 “λ” is often dropped (meaning “at a given wavelength,” or “per unit wavelength”), assuming that the reader recognizes the variable as a spectral quantity from the context. Whenever applicable, units have been attached to the variables in the following table. Variables without indicated units have multiple sets of units. For example, the units for total band absorptance A depend on the spectral variable used (λ, η, or ν), and on the absorption coefficient (linear, density- or pressure-based), for a total of nine different possibilities. a semimajor axis of polarization ellipse, [N/C] a plane-polarized component of electric field, [N/C] a particle radius, [m] a weight function for full-spectrum k-distribution methods, [−] ak weight factors for sum-of-gray-gases, [−] an , bn Mie scattering coefficients, [−] A total band absorptance (or effective band width) A∗ nondimensional band absorptance = A/ω, [−] A, An slab absorptivity (of n parallel sheets), [−] A, Ap area, projected area, [m2 ] Am scattering phase function coefficients, [−] Ai j , Bi j Einstein coefficients b self-broadening coefficient, [−] b semiminor axis of polarization ellipse, [N/C] B rotational constant Bo convection-to-radiation parameter (Boltzmann number), [−] c, c0 speed of light, (in vacuum), [m/s] c specific heat, [J/kg K] C1 , C2 , C3 constants for Planck function and Wien’s displacement law C1 , C2 , C3 wide band parameters for outdated model d line spacing dnij , Dnij Wigner-D functions, [−] D diameter, [m] D, D∗ detectivity (normalized), [1/W] ([cm Hz1/2 /W]) Df mass fractal dimension, [−] eˆ unit vector into local coordinate direction, [−] E, Eb emissive power, blackbody emissive power E molecular energy level, [J] E electric field vector, [N/C] E(m) refractive index function, [−] En exponential integral of order n, [−] f k-distribution, [cm] f probability density function xvii

xviii

LIST OF SYMBOLS

fv , fs , fl f (nλT) F F Fi− j Fi−s j Fi→ j 1k 1 1 1i s j , 1i 1k gs, gg G Gi S j , Gi Gk G h h H H H H H i ˆı I I I Ib Il , Ilm I0 , I 1 ℑ j ˆ J J k k k k kf kˆ K K l, m, n L L L Le L0 , Lm L m

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

LIST OF SYMBOLS

m m˙ M n n n nˆ N Nc NT Nu O{} p p P Pl , Plm Pr q, q qR qlum Q Q ′′′ Q˙ r r r R Ru R R R, Rn ℜ Re s sˆ si s j , si 1k ss, sg S S S S St Ste Si Sj , Si Gk SS, SG t t t ˆt T T, Tn

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

xix

xx u u u u uk U v v v V V w wi W W x, y, z x x x X X X Y Ylm z α α α α α, β, γ β β β∗ γ γ γ γ γE δ δ δi j δk ∆nij ǫ ǫ ε η η η ηlum θ

LIST OF SYMBOLS

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

LIST OF SYMBOLS

θ Θ Θ κ λ λm λ µ µ µ ν ν ξ ξ ρ ρ ρf σ σs σe , σdc σh σl τ τ φ φ φ Φ Φ Φ Φ ψ ψ Ψ Ψ ω ω ω Ω Subscripts 0 1, 2 ∞ a av b B c C D

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

reference value, or in vacuum, or at length = 0 in medium, or at location, “1” or “2” far from surface absorbing, or apparent average blackbody value band integrated value at band center, or at cylinder, or critical value, or denoting a complex quantity, or cold collision Doppler, or based on diameter

xxi

xxii e f 1 h i j k L m n o p p P r ref R s S sol t u v w W x, y, z, r θ, ψ η λ ν k ⊥

LIST OF SYMBOLS

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

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

CHAPTER

1

FUNDAMENTALS OF THERMAL RADIATION

1.1

INTRODUCTION

The terms radiative heat transfer and thermal radiation are commonly used to describe the science of the heat transfer caused by electromagnetic waves. Obvious everyday examples of thermal radiation include the heating effect of sunshine on a clear day, the fact that—when one is standing in front of a fire—the side of the body facing the fire feels much hotter than the back, and so on. More subtle examples of thermal radiation are that the clear sky is blue, that sunsets are red, and that, during a clear winter night, we feel more comfortable in a room whose curtains are drawn than in a room (heated to the same temperature) with open curtains. All materials continuously emit and absorb electromagnetic waves, or photons, by lowering or raising their molecular energy levels. The strength and wavelengths of emission depend on the temperature of the emitting material. As we shall see, for heat transfer applications wavelengths between 10−7 m and 10−3 m (ultraviolet, visible, and infrared) are of greatest importance and are, therefore, the only ones considered here. Before embarking on the analysis of thermal radiation we want briefly to compare the nature of this mode of heat transfer with the other two possible mechanisms of transferring energy, conduction and convection. In the case of conduction in a solid, energy is carried through the atomic lattice by free electrons or by phonon–phonon interactions (i.e., excitation of vibrational energy levels for interatomic bonds). In gases and liquids, energy is transferred from molecule to molecule through collisions (i.e., the faster molecule loses some of its kinetic energy to the slower one). Heat transfer by convection is similar, but many of the molecules with raised kinetic energy are carried away by the flow and are replaced by colder fluid (low-kinetic-energy molecules), resulting in increased energy transfer rates. Thus, both conduction and convection require the presence of a medium for the transfer of energy. Thermal radiation, on the other hand, is transferred by electromagnetic waves, or photons, which may travel over a long distance without interacting with a medium. The fact that thermal radiation does not require a medium for its transfer makes it of great importance in vacuum and space applications. This so-called “action at a distance” also manifests itself in a number of everyday thermodynamic applications. For example, on a cold winter day in a heated room we feel more comfortable when the curtains are closed: our bodies exchange heat by convection with the warm air surrounding us, but also by radiation with walls (including cold window panes if they are without curtains); we feel the heat from a fire a distance away from us, and so on. 1

2

1 FUNDAMENTALS OF THERMAL RADIATION

Another distinguishing feature between conduction and convection on the one hand and thermal radiation on the other is the difference in their temperature dependencies. For the vast majority of conduction applications heat transfer rates are well described by Fourier’s law as qx = −k

∂T , ∂x

(1.1)

where qx is conducted heat flux1 in the x-direction, T is temperature, and k is the thermal conductivity of the medium. Similarly, convective heat flux may usually be calculated from a correlation such as q = h(T − T∞ ), (1.2) where h is known as the convective heat transfer coefficient, and T∞ is a reference temperature. While k and h may depend on temperature, this dependence is usually not very strong. Thus, for most applications, conductive and convective heat transfer rates are linearly proportional to temperature differences. As we shall see, radiative heat transfer rates are generally proportional to differences in temperature to the fourth (or higher) power, i.e., 4 q ∝ T 4 − T∞ .

(1.3)

Therefore, radiative heat transfer becomes more important with rising temperature levels and may be totally dominant over conduction and convection at very high temperatures. Thus, thermal radiation is important in combustion applications (fires, furnaces, rocket nozzles, engines, etc.), in nuclear reactions (such as in the sun, in a fusion reactor, or in nuclear bombs), during atmospheric reentry of space vehicles, etc. As modern technology strives for higher efficiencies, this will require higher and higher temperatures, making thermal radiation ever more important. Other applications that are increasing in importance include solar energy collection and the greenhouse effect (both due to emission from our high-temperature sun). And, finally, one of the most pressing issues for mankind today are the effects of global warming, caused by the absorption of solar energy by man-made carbon dioxide released into the Earth’s atmosphere. The same reasons that make thermal radiation important in vacuum and high-temperature applications also make its analysis more difficult, or at least quite different from “conventional” analyses. Under normal conditions, conduction and convection are short-range phenomena: The average distance between molecular collisions (mean free path for collision) is generally very small, maybe around 10−10 m. If it takes, say, 10 collisions until a high-kinetic-energy molecule has a kinetic energy similar to that of the surrounding molecules, then any external influence is not directly felt over a distance larger than 10−9 m. Thus we are able to perform an energy balance on an “infinitesimal volume,” i.e., a volume negligibly small in comparison with overall dimensions, but very large in comparison with the mean free path for collision. The principle of conservation of energy then leads to a partial differential equation to describe the temperature field and heat fluxes for both conduction and convection. This equation may have up to four independent variables (three space coordinates and time) and is linear in temperature for the case of constant properties. Thermal radiation, on the other hand, is generally a long-range phenomenon. The mean free path for a photon (i.e., the average distance a photon travels before interacting with a molecule) may be as short as 10−10 m (e.g., absorption in a metal), but can also be as long as 10+10 m or larger (e.g., the sun’s rays hitting Earth). Thus, conservation of energy cannot be applied over an infinitesimal volume, but must be applied over the entire volume under consideration. This leads to an integral equation in up to seven independent variables (the frequency of radiation, three space coordinates, two coordinates describing the direction of travel of photons, and time). The analysis of thermal radiation is further complicated by the behavior of the radiative properties of materials. Properties relevant to conduction and convection (thermal conductivity, 1 In this book we shall use the term heat flux to denote the flow of energy per unit time and per unit area and the term heat rate for the flow of energy per unit time (i.e., not per unit area).

1.2 THE NATURE OF THERMAL RADIATION

3

kinematic viscosity, density, etc.) are fairly easily measured and are generally well behaved (isotropic throughout the medium, perhaps with weak temperature dependence). Radiative properties are usually difficult to measure and often display erratic behavior. For liquids and solids the properties normally depend only on a very thin surface layer, which may vary strongly with surface preparation and often even from day to day. All radiative properties (in particular for gases) may vary strongly with wavelength, adding another dimension to the governing equation. Rarely, if ever, may this equation be assumed to be linear. Because of these difficulties inherent in the analysis of thermal radiation, a good portion of this book has been set aside to discuss radiative properties and different approximate methods to solve the governing energy equation for radiative transport.

1.2 THE NATURE OF THERMAL RADIATION Thermal radiative energy may be viewed as consisting of electromagnetic waves (as predicted by electromagnetic wave theory) or as consisting of massless energy parcels, called photons (as predicted by quantum mechanics). Neither point of view is able to describe completely all radiative phenomena that have been observed. It is, therefore, customary to use both concepts interchangeably. In general, radiative properties of liquids and solids (including tiny particles), and of interfaces (surfaces) are more easily predicted using electromagnetic wave theory, while radiative properties of gases are more conveniently obtained from quantum mechanics. All electromagnetic waves, or photons, are known to propagate through any medium at a high velocity. Since light is a part of the electromagnetic wave spectrum, this velocity is known as the speed of light, c. The speed of light depends on the medium through which it travels, and may be related to the speed of light in vacuum, c 0 , by the formula c=

c0 , n

c 0 = 2.998 × 108 m/s,

(1.4)

where n is known as the refractive index of the medium. By definition, the refractive index of vacuum is n ≡ 1. For most gases the refractive index is very close to unity, for example, air at room temperature has n = 1.00029 over the visible spectrum. Therefore, light propagates through gases nearly as fast as through vacuum. Electromagnetic waves travel considerably slower through dielectrics (electric nonconductors), which have refractive indices between approximately 1.4 and 4, and they hardly penetrate at all into electrical conductors (metals). Each wave may be identified either by its frequency, ν wavelength, λ wavenumber, η angular frequency, ω

(measured in cycles/s = s−1 = Hz); (measured in µm = 10−6 m or nm = 10−9 m); (measured in cm−1 ); or (measured in radians/s = s−1 ).

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

ω c = = cη. 2π λ

(1.5)

Each wave or photon carries with it an amount of energy, ǫ, determined from quantum mechanics as ǫ = hν, h = 6.626 × 10−34 J s, (1.6) where h is known as Planck’s constant. The frequency of light does not change when light penetrates from one medium to another since the energy of the photon must be conserved. On the other hand, wavelength and wavenumber do, depending on the values of the refractive

4

1 FUNDAMENTALS OF THERMAL RADIATION

Violet Blue Green Yellow Red

Visible

Infrared

X rays Ultraviolet

Microwave Gamma rays

Thermal radiation

0.40 0.70 10

–5

10

10 9

–4

10

10 8 10 19

–3

10

10 7 10 18

–2

10 6 10 17

–1

10 1 Wavelength λ, µ m

10

10 2

10 3

10 4

10 5 104 Wavenumber η, c m–1

103

10 2

10

1

10 16

10 15 10 14 Frequency ν, Hz

10 13

10 12

10 11

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

index for the two media. Sometimes electromagnetic waves are characterized in terms of the energy that a photon carries, hν, using the energy unit electron volt (1 eV = 1.6022×10−19 J). Thus, light with a photon energy (or “frequency”) of a eV has a wavelength (in vacuum) of λ=

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

(1.7)

Since electromagnetic waves of vastly different wavelengths carry vastly different amounts of energy, their behavior is often quite different. Depending on their behavior or occurrence, electromagnetic waves have been grouped into a number of different categories, as shown in Fig. 1-1. Thermal radiation may be defined to be those electromagnetic waves which are emitted by a medium due solely to its temperature [1]. As indicated earlier, this definition limits the range of wavelengths of importance for heat transfer considerations to between 0.1 µm (ultraviolet) and 100 µm (midinfrared).

1.3 BASIC LAWS OF THERMAL RADIATION When an electromagnetic wave traveling through a medium (or vacuum) strikes the surface of another medium (solid or liquid surface, particle or bubble), the wave may be reflected (either partially or totally), and any nonreflected part will penetrate into the medium. While passing through the medium the wave may become continuously attenuated. If attenuation is complete so that no penetrating radiation reemerges, it is known as opaque. If a wave passes through a medium without any attenuation, it is termed transparent, while a body with partial attenuation is called semitransparent.2 Whether a medium is transparent, semitransparent or opaque depends on the material as well as on its thickness (i.e., the distance the electromagnetic wave must travel through the medium). Metals are nearly always opaque, although it is a 2 A medium that allows a fraction of light to pass through, while scattering the transmitted light into many different directions, for example, milky glass, is called translucent.

1.4 EMISSIVE POWER

5

T = const.

T

FIGURE 1-2 Kirchhoff’s law.

common high school physics experiment to show that light can penetrate through extremely thin layers of gold. Nonmetals generally require much larger thicknesses before they become opaque, and some are quite transparent over part of the spectrum (for example, window glass in the visible part of the spectrum). An opaque surface that does not reflect any radiation is called a perfect absorber or a black surface: When we “see” an object, our eyes absorb electromagnetic waves from the visible part of the spectrum, which have been emitted by the sun (or artificial light) and have been reflected by the object toward our eyes. We cannot see a surface that does not reflect radiation, and it appears “black” to our eyes.3 Since black surfaces absorb the maximum possible amount of radiative energy, they serve as a standard for the classification of all other surfaces. It is easy to show that a black surface also emits a maximum amount of radiative energy, i.e., more than any other body at the same temperature. To show this, we use one of the many variations of Kirchhoff’s law:∗ Consider two identical black-walled enclosures, thermally insulated on the outside, with each containing a small object—one black and the other one not— as shown in Fig. 1-2. After a long time, in accordance with the Second Law of Thermodynamics, both entire enclosures and the objects within them will be at a single uniform temperature. This characteristic implies that every part of the surface (of the enclosure as well as the objects) emits precisely as much energy as it absorbs. Both objects in the different enclosures receive exactly the same amount of radiative energy. But since the black object absorbs more energy (i.e., the maximum possible), it must also emit more energy than the nonblack object (i.e., also the maximum possible). By the same reasoning it is easy to show that a black surface is a perfect absorber and emitter at every wavelength and for any direction (of incoming or outgoing electromagnetic waves), and that the radiation field within an isothermal black enclosure is isotropic (i.e., the radiative energy density is the same at any point and in any direction within the enclosure).

1.4

EMISSIVE POWER

Every medium continuously emits electromagnetic radiation randomly into all directions at a rate depending on the local temperature and on the properties of the material. This is sometimes referred to as Pr´evost’s law (after Pierre Pr´evost, an early 19th century Swiss philosopher and physicist). The radiative heat flux emitted from a surface is called the emissive power, E. We distinguish between total and spectral emissive power (i.e., heat flux emitted over the entire 3 Note that a surface appearing black to our eyes is by no means a perfect absorber at nonvisible wavelengths and vice versa; indeed, many white paints are actually quite “black” at longer wavelengths.



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

6

1 FUNDAMENTALS OF THERMAL RADIATION

Spectral solar irradiation, W/ m2 µ m

2500

2000

O3

Blackbody emissive power at 5777 K, normalized to 1366 W/m 2 Extraterrestrial solar spectrum, 1366 W/m 2 "Air mass one" solar spectrum O2 H2O O2

1500

H2O

1000

H2O H2O

500

H2O

O3 UV 0

0.2

Visible 0.4

0.6

H2O

Infrared 0.8

1.0

1.2 1.4 1.6 1.8 Wavelength λ, µ m

2.0 2.2

2.4 2.6

H2O CO2 2.8 3.0

3.2

FIGURE 1-3 Solar irradiation onto Earth.

spectrum, or at a given frequency per unit frequency interval), so that spectral emissive power, Eν ≡ emitted energy/time/surface area/frequency, total emissive power, E ≡ emitted energy/time/surface area. Here and elsewhere we use the subscripts ν, λ, or η (depending on the choice of spectral variable) to express a spectral quantity whenever necessary for clarification. Thermal radiation of a single frequency or wavelength is sometimes also called monochromatic radiation (since, over the visible range, the human eye perceives electromagnetic waves to have the colors of the rainbow). It is clear from their definitions that the total and spectral emissive powers are related by Z ∞

E(T) =

0

Eν (T, ν) dν.

(1.8)

Blackbody Emissive Power Spectrum Scientists had tried for many years to theoretically predict the sun’s emission spectrum, which we know today to behave very nearly like a blackbody at approximately 5777 K [2]. The spectral solar flux falling onto Earth, or solar irradiation, is shown in Fig. 1-3 for extraterrestrial conditions (as measured by high-flying balloons and satellites) and for unity air mass (air mass is defined as the value of 1/ cos θS , where the zenith angle θS is the angle between the local vertical and a vector pointing toward the sun) [3, 4]. Solar radiation is attenuated significantly as it penetrates through the atmosphere by phenomena that will be discussed in Sections 1.12 and 1.14. Lord Rayleigh (1900) [5]∗ and Sir James Jeans (1905) [6]† independently applied the principles of ∗

John William Strutt, Lord Rayleigh (1842–1919) English physical scientist. Rayleigh obtained a mathematics degree from Cambridge, where he later served as professor of experimental physics for five years. He then became secretary, and later president, of the Royal Society. His work resulted in a number of discoveries in the fields of acoustics and optics, and he was the first to explain the blue color of the sky (cf. the Rayleigh scattering laws in Chapter 12). Rayleigh received the 1904 Nobel Prize in Physics for the isolation of argon.



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

1.4 EMISSIVE POWER

7

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

2πhν3 n2  , c20 ehν/kT − 1

(1.9)

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

Z

∞ 0

Ebν dν =

Z

∞ 0

Ebλ dλ =

Z

∞ 0

Ebη dη,

or

(1.11) (1.12)

Ebν dν = −Ebλ dλ = Ebη dη.

Here λ and η are wavelength and wavenumber for the electromagnetic waves within the medium of refractive index n (while λ0 = nλ and η0 = η/n would be wavelength and wavenumber of the same wave traveling through vacuum). Equation (1.10) shows that equation (1.9) gives convenient relations for Ebλ and Ebη only if the refractive index is independent of frequency (or wavelength, or wavenumber). This is certainly the case for vacuum (n = 1) and ordinary gases (n ≃ 1), and may be of acceptable accuracy for some semitransparent media over large parts of the spectrum (for example, for quartz 1.52 < n < 1.68 between the wavelengths of 0.2 and 2.4 µm). Thus, with the assumption of constant refractive index, Ebλ (T, λ) =

4

2πhc20  , n2 λ5 ehc 0 /nλkT − 1

(n = const),

(1.13)



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

§

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

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

8

1 FUNDAMENTALS OF THERMAL RADIATION 8

Visible part of spectrum 7

10

6

3

10

10 00 K

0K 200

5000

3000 K

K

/λ) =C 3

4

10

T =5

5

10

777 K

10

(T E bλ

Blackbody emissive power Ebλ, W/m2 µm

10

50

2

10

10-1

0K

100 Wavelength λ, µm

101

FIGURE 1-4 Blackbody emissive power spectrum.

Ebη (T, η) =

2πhc20 η3  , n2 ehc 0 η/nkT − 1

(n = const).

(1.14)

Figure 1-4 is a graphical representation of equation (1.13) for a number of blackbody temperatures. As one can see, the overall level of emission rises with rising temperature (as dictated by the Second Law of Thermodynamics), while the wavelength of maximum emission shifts toward shorter wavelengths. The blackbody emissive power is also plotted in Fig. 1-3 for an effective solar temperature of 5777 K. This plot is in good agreement with extraterrestrial solar irradiation data. It is customary to introduce the abbreviations C1 = 2πhc20 = 3.7418 × 10−16 W m2 , C2 = hc 0 /k = 14,388 µm K = 1.4388 cm K, so that equation (1.13) may be recast as C1 Ebλ = , 3 5 5 C n T (nλT) [e 2 /(nλT) − 1]

(n = const),

(1.15)

which is seen to be a function of (nλT) only. Thus, it is possible to plot this normalized emissive power as a single line vs. the product of wavelength in vacuum (nλ) and temperature (T), as shown in Fig. 1-5, and a detailed tabulation is given in Appendix C. The maximum of this curve may be determined by differentiating equation (1.15),   Ebλ d = 0, d(nλT) n3 T 5 leading to a transcendental equation that may be solved numerically as (nλT) max = C3 = 2898 µm K.

(1.16)

Equation (1.16) is known as Wien’s displacement law since it was developed independently by Wilhelm Wien [9] in 1891 (i.e., well before the publication of Planck’s emissive power law). Example 1.1. Earth?

At what wavelength has the sun its maximum emissive power? At what wavelength

1.4 EMISSIVE POWER

15.0

9

1.00

Planck’s law Rayleigh-Jeans distribution Wien’s distribution

7.5

12

0.50

5 3

1% 5.0

90% 0.25

10% 2.5 98% 0.0 0

5 10 15 3 Wavelength × temperature nλT, 10 µm K

Fractional emissive power f(nλT )

10.0

2

W/m µm K

5

0.75

Ebλ /n T , 10

Scaled blackbody emissive power

12.5

0.00 20

FIGURE 1-5 Normalized blackbody emissive power spectrum.

Solution From equation (1.16), with the sun’s surface at Tsun ≃ 5777 K and bounded by vacuum (n = 1), it follows that 2898 µm K C3 λmax,sun = = = 0.50 µm, Tsun 5777 K which is near the center of the visible region. Apparently, evolution has caused our eyes to be most sensitive in that section of the electromagnetic spectrum where the maximum daylight is available. In contrast, Earth’s average surface temperature may be in the vicinity of TEarth = 290 K, or λmax,Earth ≃

2898 µm K = 10 µm, 290 K

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

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

2πc 0 kT , nλ4

hc 0 ≪ 1. nλkT

(1.17)

The same result is obtained if one lets h → 0, i.e., if one allows photons of arbitrarily small energy content to be emitted, as postulated by classical statistics. Thus, equation (1.17) is identical to the one derived by Rayleigh and Jeans and bears their names. The Rayleigh–Jeans distribution is also included in Fig. 1-5. Obviously, this formula is accurate only for very large values of (nλT), where the energy of the emissive power spectrum is negligible. Thus, this formula is of little significance for engineering purposes. For large values of (hc 0 /nλkT) the −1 in the denominator of equation (1.13) may be neglected, leading to Wien’s distribution (or Wien’s law), Ebλ ≃

2πhc20 n2 λ5

e−hc 0 /nλkT =

C1 −C2 /nλT e , n2 λ5

hc 0 ≫ 1, nλkT

(1.18)

since it is identical to the formula first proposed by Wien, before the advent of quantum mechanics. Examination of Wien’s distribution in Fig. 1-5 shows that it is very accurate over most

10

1 FUNDAMENTALS OF THERMAL RADIATION

of the spectrum, with a total energy content of the entire spectrum approximately 8% lower than for Planck’s law. Thus, Wien’s distribution is frequently utilized in theoretical analyses in order to facilitate integration.

Total Blackbody Emissive Power The total emissive power of a blackbody may be determined from equations (1.11) and (1.13) as Z ∞ Z ∞ d(nλT) Eb (T) = Ebλ (T, λ) dλ = C1 n2 T 4  C /(nλT)  5 2 −1 0 0 (nλT) e   Z ∞  C1 ξ3 dξ  2 4 (1.19) =  4  n T , (n = const). C2 0 eξ − 1 

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

σ=

π 4 C1 W = 5.670 × 10−8 2 4 , 4 m K 15C2

(1.20)

where σ is known as the Stefan–Boltzmann constant.∗ If Wien’s distribution is to be used then the −1 is absent from the denominator of equation (1.19), and a corrected Stefan–Boltzmann constant should be employed, evaluated as σW =

6C1 W = 5.239 × 10−8 2 4 , 4 m K C2

(1.21)

indicating that Wien’s distribution underpredicts total emissive power by about 7.5%. Historically, the “T 4 radiation law,” equation (1.20), predates Planck’s law and was found through thermodynamic arguments. A short history may be found in [10]. It is often necessary to calculate the emissive power contained within a finite wavelength band, say between λ1 and λ2 . Then Z

λ2

Ebλ dλ = λ1

C1 C42

Z

C2 /nλ1 T C2 /nλ2 T

ξ3 dξ 2 4 n T . eξ − 1

(1.22)

It is not possible to evaluate the integral in equation (1.22) in simple analytical form. Therefore, it is customary to express equation (1.22) in terms of the fraction of blackbody emissive power contained between 0 and nλT, λ

R

f (nλT) = R 0∞ 0



Ebλ dλ Ebλ dλ

=

Z

0

nλT

Z  15 ∞ ξ3 dξ Ebλ d(nλT) = , 3 5 4 n σT π C2 /nλT eξ − 1

(1.23)

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

11

1.5 SOLID ANGLES

d Aj

θ dθ

dA j´´ 1

n

nj

θj

s

d Ajp

sin θ dψ dA

P



ψ

so that Z

λ2 λ1

FIGURE 1-6 Emission direction and solid angles as related to a unit hemisphere.

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

(1.24)

Equation (1.23) can be integrated only after expanding the denominator into an infinite series, resulting in f (nλT) =

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

ζ=

C2 . nλT

(1.25)

The fractional emissive power is a function in a single variable, nλT, and is therefore easily tabulated, as has been done in Appendix C. For computer calculations a little Fortran routine of equation (1.25), bbfn, is given in Appendix F, as well as a stand-alone program, planck, which, after inputting wavelength (or wavenumber) and temperature, returns Ebλ , Ebη , and f . Example 1.2. What fraction of total solar emission falls into the visible spectrum (0.4 to 0.7 µm)? Solution With n = 1 and a solar temperature of 5777 K it follows that for λ1 = 0.4 µm, nλ1 Tsun = 1 × 0.4 × 5777 = 2310.8 µm K; and for λ2 = 0.7 µm, nλ2 Tsun = 4043.9 µm K. From Appendix C we find f (nλ1 Tsun ) = 0.12220 and f (nλ2 Tsun ) = 0.48869. Thus, from equations (1.20) and (1.24) the visible fraction of sunlight is f (nλ2 Tsun ) − f (nλ1 Tsun ) = 0.48869 − 0.12220 = 0.36649. (Writing a one-line program bbfn(4043.9)-bbfn(2310.8) returns the slightly more accurate value of 0.36661.) Therefore, with a bandwidth of only 0.3 µm the human eye responds to approximately 37% of all emitted sunlight!

1.5

SOLID ANGLES

When radiative energy leaves one medium and enters another (i.e., emission from a surface into another medium), this energy flux usually has different strengths in different directions. Similarly, the electromagnetic wave, or photon, flux passing through any point inside any medium may vary with direction. It is customary to describe the direction vector in terms of a spherical or polar coordinate system. Consider a point P on an opaque surface dA radiating into another medium, say air, as shown in Fig. 1-6. It is apparent that the surface can radiate into infinitely many directions, with every ray penetrating through a hemisphere of unit radius as indicated in the figure. The total surface area of this hemisphere, 2π 12 = 2π, is known as the

12

1 FUNDAMENTALS OF THERMAL RADIATION

total solid angle above the surface. An arbitrary emission direction from the surface is specified by the unit direction vector sˆ , which may be expressed in terms of the polar angle θ (measured ˆ and the azimuthal angle ψ (measured between an arbitrary axis on the from the surface normal n) surface and the projection of sˆ onto the surface). It is seen that, for a hemisphere, 0 ≤ θ ≤ π/2 and 0 ≤ ψ ≤ 2π. The solid angle with which an infinitesimal surface dA j is seen from a point P is defined as the projection of the surface onto a plane normal to the direction vector, divided by the square of the distance S between dA j and P, as also shown in Fig. 1-6. If the surface is projected onto the unit hemisphere above the point, the solid angle is equal to the projected area itself, or dΩ =

dA jp S2

=

cos θj dA j S2

= dA′′j .

(1.26)

Thus, an infinitesimal solid angle is simply an infinitesimal area on a unit sphere, or dΩ = dA′′j = (1 × sin θ dψ)(1 × dθ) = sin θ dθ dψ. Integrating over all possible directions we obtain Z 2π Z π/2 sin θ dθ dψ = 2π, ψ=0

(1.27)

(1.28)

θ=0

for the total solid angle above the surface, as already seen earlier. The solid angle, with which a finite surface A j is seen from point P, follows immediately from equation (1.26) as Z Z Z dA jp cos θj dA j Ω= = = dA′′j = A′′j , (1.29) 2 S2 A jp S A Aj i.e., the projection of A j onto the hemisphere above P. While a little unfamiliar at first, solid angles are simply two-dimensional angular space: Similar to the way a one-dimensional angle can vary between 0 and π (measured in dimensionless radians, equivalent to length along a semicircular line), the solid angle may vary between 0 and 2π (measured in dimensionless steradians, sr, equivalent to surface area on a hemisphere). Example 1.3. Determine the solid angle with which the sun is seen from Earth. Solution The area of the sun projected onto a plane normal to the vector pointing from Earth to the sun (or, simply, the image of the sun that we see from Earth) is a disk of radius Rs ≃ 6.96 × 108 m (i.e., the radius of the sun), at a distance of approximately SES ≃ 1.496 × 1011 m (averaged over Earth’s yearly orbit). Thus the solid angle of the sun is ΩS =

(πR2S ) 2

SES

=

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

This solid angle is so small that we may generally assume that solar radiation comes from a single direction, i.e., that all the light beams are parallel. Example 1.4. What is the solid angle with which the narrow strip shown in Fig. 1-7 is seen from point “0”? Solution Since the strip is narrow we may assume that the projection angle for equation (1.29) varies only in the x-direction as indicated in Fig. 1-7, leading to Z L cos θ0 dx h , cos θ0 = , r2 = h2 + x2 , Ω=w 2 r r 0 and

1.6 RADIATIVE INTENSITY

13

x L n

θ0 r

w

0

h

FIGURE 1-7 Solid angle subtended by a narrow strip.

s n dAp = d A cos θ

θ

FIGURE 1-8 Relationship between blackbody emissive power and intensity.

dA

Z Ω=w

1.6

L 0

L Z L h dx dx w x wL = wh = . √ = √ 2 2 3/2 h h2 + x2 0 h h2 + L2 r3 0 (h + x )

RADIATIVE INTENSITY

While emissive power appears to be the natural choice to describe radiative heat flux leaving a surface, it is inadequate to describe the directional dependence of the radiation field, in particular inside an absorbing/emitting medium, where photons may not have originated from a surface. Therefore, very similar to the emissive power, we define the radiative intensity I, as radiative energy flow per unit solid angle and unit area normal to the rays (as opposed to surface area). Again, we distinguish between spectral and total intensity. Thus, spectral intensity, Iλ ≡ radiative energy flow/time/area normal to rays/solid angle/wavelength, total intensity, I ≡ radiative energy flow/time/area normal to rays/solid angle. Again, spectral and total intensity are related by Z ∞ Iλ (r, sˆ , λ) dλ. I(r, sˆ ) = 0

(1.30)

Here, r is a position vector fixing the location of a point in space, and sˆ is a unit direction vector as defined in the previous section. While emissive power depends only on position and wavelength, the radiative intensity depends, in addition, on the direction vector sˆ . The emissive power can be related to intensity by integrating over all the directions pointing away from the surface. Considering Fig. 1-8, we find that the emitted energy from dA into the direction sˆ , and contained within an infinitesimal solid angle dΩ = sin θ dθ dψ is, from the definition of intensity, I(r, sˆ ) dAp dΩ = I(r, sˆ ) dA cos θ sin θ dθ dψ, where dAp is the projected area of dA normal to the rays (i.e., the way dA is seen when viewed from the −ˆs direction). Thus, integrating this expression over all possible directions gives the

14

1 FUNDAMENTALS OF THERMAL RADIATION

n d As

θ

dA

FIGURE 1-9 Kirchhoff’s law for the directional behavior of blackbody intensity.

total energy emitted from dA, or, after dividing by dA E(r) =

Z

2π 0

Z

π/2 0

I(r, θ, ψ) cos θ sin θ dθ dψ =

Z



I(r, sˆ ) nˆ · sˆ dΩ.

(1.31)

This expression is, of course, also valid on a spectral basis. The directional behavior of the radiative intensity leaving a blackbody is easily obtained from a variation of Kirchhoff’s law: Consider a small, black surface suspended at the center of an isothermal spherical enclosure, as depicted in Fig. 1-9. Let us assume that the enclosure has a (hypothetical) surface coating that reflects all incoming radiation totally and like a mirror everywhere except over a small area dAs , which also reflects all incoming radiation except for a small wavelength interval between λ and λ + dλ. Over this small range of wavelengths dAs behaves like a blackbody. Now, all radiation leaving dA, traveling to the sphere (with the exception of light of wavelength λ traveling toward dAs ), will be reflected back toward dA where it will be absorbed (since dA is black). Thus, the net energy flow from dA to the sphere is, recalling the definitions for intensity and solid angle, ! dAs Ibλ (T, θ, ψ, λ)(dA cos θ) dΩs dλ = Ibλ (T, θ, ψ, λ)(dA cos θ) dλ, R2 where dΩ s is the solid angle with which dAs is seen from dA. On the other hand, also by Kirchhoff’s law, the sphere does not emit any radiation (since it does not absorb anything), except over dAs at wavelength λ. All energy emitted from dAs will eventually come back to itself except for the fraction intercepted by dA. Thus, the net energy flow from the sphere to dA is ! dA cos θ Ibnλ (T, λ) dAs dΩ dλ = Ibnλ (T, λ) dAs dλ, R2 where the subscript n denotes emission into the normal direction (θs = 0, ψs arbitrary), and dΩ is the solid angle with which dA is seen from dAs . Now, from the Second Law of Thermodynamics, these two fluxes must be equal for an isothermal enclosure. Therefore, Ibλ (T, θ, ψ, λ) = Ibnλ (T, λ). Since the direction (θ, ψ), with which dAs is oriented, is quite arbitrary we conclude that Ibλ is independent of direction, or Ibλ = Ibλ (T, λ) only. (1.32)

1.7 RADIATIVE HEAT FLUX

15

n n

θi

s0

θ0 si

d Ω0

d Ωi

n

FIGURE 1-10 Radiative heat flux on an arbitrary surface.

dA

Substituting this expression into equation (1.31) we obtain the following relationship between blackbody intensity and emissive power: Ebλ (r, λ) = π Ibλ (r, λ).

(1.33)

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

(1.34)

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

(1.35)

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

1.7

RADIATIVE HEAT FLUX

Consider the surface shown in Fig. 1-10. Let thermal radiation from an infinitesimal solid angle around the direction sˆ i impinge onto the surface with an intensity of Iλ (ˆsi ). Such radiation is often called a “pencil of rays” since the infinitesimal solid angle is usually drawn looking like the tip of a sharpened pencil. Recalling the definition for intensity we see that it imparts an infinitesimal heat flow rate per wavelength on the surface in the amount of dQλ = Iλ (ˆsi ) dΩ i dAp = Iλ (ˆsi ) dΩ i (dA cos θi ), where heat rate is taken as positive in the direction of the outward surface normal (going into the medium), so that the incoming flux going into the surface is negative since cos θi < 0. Integrating ∗

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

16

1 FUNDAMENTALS OF THERMAL RADIATION

over all 2π incoming directions and dividing by the surface area gives the total incoming heat flux per unit wavelength, i.e., Z  qλ in = Iλ (ˆsi ) cos θi dΩ i . (1.36) cos θi 0

If the surface is black (ǫλ = 1), there is no energy reflected from the surface and Iλ = Ibλ , leading to (qλ ) out = Ebλ . If the surface is not black, the outgoing intensity consists of contributions from emission as well as reflections. The outgoing heat flux is positive since it is going into the medium. The net heat flux from the surface may be calculated by adding both contributions, or Z    qλ net = qλ in + qλ out = Iλ (ˆs) cos θ dΩ, (1.38) 4π

where a single direction vector sˆ was used to describe the total range of solid angles, 4π. It is readily seen from Fig. 1-10 that cos θ = nˆ · sˆ and, since the net heat flux is evaluated as the flux ˆ into the positive n-direction, one gets Z  qλ net = qλ · nˆ = Iλ (ˆs) nˆ · sˆ dΩ. (1.39) 4π

In order to obtain the total radiative heat flux at the surface, equation (1.39) needs to be integrated over the spectrum, and Z ∞ Z ∞Z q · nˆ = qλ · nˆ dλ = Iλ (ˆs) nˆ · sˆ dΩ dλ. (1.40) 0

0



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

4πR2S Eb (TS ) 2

4πSES

= Ib (TS )

πR2S S2ES

= Ib (TS ) Ω S ,

where we have replaced the sun’s emissive power by intensity, Eb = πIb , and Ω S = 6.79 × 10−5 sr is the solid angle with which the sun is seen from Earth, as determined in Example 1.3. Therefore, with Ib (TS ) = σTS4 /π and TS = 5777 K, 1 qin = −(σTS4 /π)(Ω S ) = − 5.670 × 10−8 × 57774 × 6.79 × 10−5 W/m2 π = −1366 W/m2 , where we have added a minus sign to emphasize that the heat flux is going into the collector. The total incoming heat flux may, of course, also be determined from equation (1.36) as Z I(ˆsi ) cos θi dΩ i . qin = cos θi 0), the source function in equation (1.64) contains the radiative intensity at every point along the path, for all possible directions (not just sˆ ): the radiative transfer equation, equation (1.60), is an integro-differential equation (intensity appears, both, as a derivative and also inside the integral on the right-hand side) in five dimensions (three space dimensions and two directional coordinates). This makes the RTE extremely difficult to solve, and much of this book will be devoted to describing the various methods of solution that have been devised over the years (Chapters 14–21).

1.16 OUTLINE OF RADIATIVE TRANSPORT THEORY When considering heat transfer by conduction and/or convection within a medium, we require knowledge of a number of material properties, such as thermal conductivity k, thermal diffusivity α, kinematic viscosity ν, and so on. This knowledge, together with the law of conservation

28

1 FUNDAMENTALS OF THERMAL RADIATION

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

References 1. Sparrow, E. M., and R. D. Cess: Radiation Heat Transfer, Hemisphere, New York, 1978. 2. NASA web page on the solar system and the sun http://solarsystem.nasa.gov/planets, 2011. 3. Thekaekara, M. P.: “The solar constant and spectral distribution of solar radiant flux,” Solar Energy, vol. 9, no. 1, pp. 7–20, 1965. 4. Thekaekara, M. P.: “Solar energy outside the earth’s atmosphere,” Solar Energy, vol. 14, pp. 109–127, 1973. 5. Rayleigh, L.: “The law of complete radiation,” Phil. Mag., vol. 49, pp. 539–540, 1900. 6. Jeans, J. H.: “On the partition of energy between matter and the ether,” Phil. Mag., vol. 10, pp. 91–97, 1905. ¨ 7. Wien, W.: “Uber die Energieverteilung im Emissionsspektrum eines schwarzen Korpers,” Annalen der Physik, ¨ vol. 58, pp. 662–669, 1896. 8. Planck, M.: “Distribution of energy in the spectrum,” Annalen der Physik, vol. 4, no. 3, pp. 553–563, 1901. 9. Wien, W.: “Temperatur und Entropie der Strahlung,” Annalen der Physik, vol. 52, pp. 132–165, 1894. 10. Crepeau, J.: “A brief history of the T4 radiation law,” ASME paper no. HT2009-88060, 2009. 11. Frohlich, C., and J. Lean: “Solar radiative output and its variability: evidence and mechanisms,” Astron. Astrophys. ¨ Rev, no. 3, pp. 273–320, 2004. 12. World Radiation Center solar constant web page http://www.pmodwrc.ch/pmod.php?topic=tsi/composite/SolarConstant, 2010. 13. Moon, P.: Scientific Basis of Illuminating Engineering, Dover Publications, New York, 1961, (originally published by McGraw-Hill, New York, 1936). 14. Hopkinson, R. B., P. Petherbridge, and J. Longmore: Daylighting, Pitman Press, London, 1966. 15. Kaufman, J. E. (ed.): IES Lighting Handbook, Illuminating Engineering Society of North America, New York, 1981. 16. White, F. M.: Heat Transfer, Addison-Wesley, Reading, MA, 1984. 17. Edwards, D. K.: “Radiation interchange in a nongray enclosure containing an isothermal CO2 –N2 gas mixture,” ASME Journal of Heat Transfer, vol. 84C, pp. 1–11, 1962. 18. Neuroth, N.: “Der Einfluss der Temperatur auf die spektrale Absorption von Gl¨asern im Ultraroten, I (Effect of temperature on spectral absorption of glasses in the infrared, I),” Glastechnische Berichte, vol. 25, pp. 242–249, 1952. 19. Rayleigh, L.: Phil. Mag., vol. 12, 1881. 20. Mie, G. A.: “Beitr¨age zur Optik truber Medien, speziell kolloidaler Metallosungen,” Annalen der Physik, vol. 25, ¨ ¨ pp. 377–445, 1908.

PROBLEMS

29

Problems 1.1 Solar energy impinging on the outer layer of Earth’s atmosphere (usually called the “solar constant”) has been measured as 1366 W/m2 . What is the solar constant on Mars? (Distance from Earth to sun = 1.496 × 1011 m, Mars to sun = 2.28 × 1011 m.) 1.2 Assuming Earth to be a blackbody, what would be its average temperature if there was no internal heating from the core of Earth? 1.3 Assuming Earth to be a black sphere with a surface temperature of 300 K, what must Earth’s internal heat generation be in order to maintain that temperature (neglect radiation from the stars, but not the sun) (radius of the Earth RE = 6.37 × 106 m). 1.4 To estimate the diameter of the sun, one may use solar radiation data. The solar energy impinging onto the Earth’s atmosphere (called the “solar constant”) has been measured as 1366 W/m2 . Assuming that the sun may be approximated to have a black surface with an effective temperature of 5777 K, estimate the diameter of the sun (distance from sun to Earth SES ≃ 1.496 × 1011 m). 1.5 Solar energy impinging on the outer layer of Earth’s atmosphere (usually called the “solar constant”) has been measured as 1366 W/m2 . Assuming the sun may be approximated as having a surface that behaves like a blackbody, estimate its effective surface temperature (distance from sun to Earth SES ≃ 1.496 × 1011 m, radius of sun RS ≃ 6.96 × 108 m). 1.6 A rocket in space may be approximated as a black cylinder of length L = 20 m and diameter D = 2 m. It flies past the sun at a distance of 140 million km such that the cylinder axis is perpendicular to the sun’s rays. Assuming that (i) the sun is a blackbody at 5777 K and (ii) the cylinder has a high conductivity (i.e., is essentially isothermal), what is the temperature of the rocket? (Radius of sun RS = 696,000 km; neglect radiation from Earth and the stars.) 1.7 A black sphere of very high conductivity (i.e., isothermal) is orbiting Earth. What is its temperature? (Consider the sun but neglect radiation from the Earth and the stars.) What would be the temperature of the sphere if it were coated with a material that behaves like a blackbody for wavelengths between 0.4 µm and 3 µm, but does not absorb and emit at other wavelengths? 1.8 A 100 W lightbulb may be considered to be an isothermal black sphere at a certain temperature. If the light flux (i.e., visible light, 0.4 µm < λ < 0.7 µm) impinging on the floor directly (2.5 m) below the bulb is 42.6 mW/m2 , and assuming conduction/convection losses to be negligible, what is the lightbulb’s effective temperature? What is its efficiency? 1.9 When a metallic surface is irradiated with a highly concentrated laser beam, a plume of plasma (i.e., a gas consisting of ions and free electrons) is formed above the surface that absorbs the laser’s energy, often blocking it from reaching the surface. Assume that a plasma of 1 cm diameter is located 1 cm above the surface, and that the plasma behaves like a blackbody at 20,000 K. Based on these assumptions calculate the radiative heat flux and the total radiation pressure on the metal directly under the center of the plasma. 1.10 Solar energy incident on the surface of the Earth may be broken into two parts: a direct component (traveling unimpeded through the atmosphere) and a sky component (reaching the surface after being scattered by the atmosphere). On a clear day the direct solar heat flux has been determined as qsun = 1000 W/m2 (per unit area normal to the rays), while the intensity of the sky component has been found to be diffuse (i.e., the intensity of the sky radiation hitting the surface is the same for all directions) and Isky = 70 W/m2 sr. Determine the total solar irradiation onto Earth’s surface if the sun is located 60◦ above the horizon (i.e., 30◦ from the normal). 1.11 A window (consisting of a vertical sheet of glass) is exposed to direct sunshine at a strength of 1000 W/m2 . The window is pointing due south, while the sun is in the southwest, 30◦ above the horizon. Estimate the amount of solar energy that (i) penetrates into the building, (ii) is absorbed by

30

1 FUNDAMENTALS OF THERMAL RADIATION

the window, and (iii) is reflected by the window. The window is made of (a) plain glass, (b) tinted glass, whose radiative properties may be approximated by ρλ = 0.08  0.90 τλ = 0  0.90 τλ = 0

for all wavelengths (both glasses), for 0.35 µm < λ < 2.7 µm for all other wavelengths for 0.5 µm < λ < 1.4 µm for all other wavelengths

(plain glass), (tinted glass).

(c) By what fraction is the amount of visible light (0.4 µm < λ < 0.7 µm) reduced, if tinted rather than plain glass is used? How would you modify this statement in the light of Fig. 1-11? 1.12 On an overcast day the directional behavior of the intensity of solar radiation reaching the surface of the Earth after being scattered by the atmosphere may be approximated as Isky (θ) = Isky (θ = 0) cos θ, where θ is measured from the surface normal. For a day with Isky (0) = 100 W/m2 sr determine the solar irradiation hitting a solar collector, if the collector is (a) horizontal, (b) tilted from the horizontal by 30◦ . Neglect radiation from the Earth’s surface hitting the collector (by emission or reflection). 1.13 A 100 W lightbulb is rated to have a total light output of 1750 lm. Assuming the lightbulb to consist of a small, black, radiating body (the light filament) enclosed in a glass envelope (with a transmittance τ1 = 0.9 throughout the visible wavelengths), estimate the filament’s temperature. If the filament has an emittance of ǫ f = 0.7 (constant for all wavelengths and directions), how does it affect its temperature? 1.14 A pyrometer is a device with which the temperature of a surface may be determined remotely by measuring the radiative energy falling onto a detector. Consider a black detector of 1 mm × 1 mm area that is exposed to a 1 cm2 hole in a furnace located a distance of 1 m away. The inside of the furnace is at 1500 K and the intensity escaping from the hole is essentially blackbody intensity at that temperature. (a) What is the radiative heat rate hitting the detector? (b) Assuming that the pyrometer has been calibrated for the situation in (a), what temperature would the pyrometer indicate if the nonabsorbing gas between furnace and detector were replaced by one with an (average) absorption coefficient of κ = 0.1 m−1 ? 1.15 Consider a pyrometer, which also has a detector area of 1 mm × 1 mm, which is black in the wavelength range 1.0 µm ≤ λ ≤ 1.2 µm, and perfectly reflecting elsewhere. In front of the detector is a focusing lens ( f = 10 cm) of diameter D = 2 cm, and transmissivity of τl = 0.9 (around 1 µm). In order to measure the temperature inside a furnace, the pyrometer is focused onto a hot black surface inside the furnace, a distance of 1 m away from the lens. (a) How large a spot on the furnace wall does the detector see? (Remember that geometric optics dictates 1 1 1 h (detector size) v  = , = + ; M= f u v u H spot size

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

CHAPTER

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

2.1

INTRODUCTION

The basic radiative properties of surfaces forming an enclosure, i.e., emissivity, absorptivity, reflectivity, and transmissivity, must be known before any radiative heat transfer calculations can be carried out. Many of these properties vary with incoming direction, outgoing direction, and wavelength, and must usually be found through experiment. However, for pure, perfectly smooth surfaces these properties may be calculated from classical electromagnetic wave theory.1 These predictions make experimental measurements unnecessary for some cases, and help interpolating as well as extrapolating experimental data in many other situations. The first important discoveries with respect to light were made during the seventeenth century, such as the law of refraction (by Snell in 1621), the decomposition of white light into monochromatic components (by Newton in 1666), and the first determination of the speed of light (by Romer in 1675). However, the true nature of light was still unknown: The corpuscular ¨ theory (suggested by Newton) competed with a rudimentary wave theory. Not until the early nineteenth century was the wave theory finally accepted as the correct model for the description of light. Young proposed a model of purely transverse waves in 1817 (as opposed to the model prevalent until then of purely longitudinal waves), followed by Fresnel’s comprehensive treatment of diffraction and other optical phenomena. In 1845 Faraday proved experimentally that there was a connection between magnetism and light. Based on these experiments, Maxwell presented in 1861 his famous set of equations for the complete description of electromagnetic waves, i.e., the interaction between electric and magnetic fields. Their success was truly remarkable, in particular because the theories of quantum mechanics and special relativity, with 1 The National Institute of Standards and Technology (NIST, formerly NBS) has recommended to reserve the ending “-ivity” for radiative properties of pure, perfectly smooth materials (the ones discussed in this chapter), and “-ance” for rough and contaminated surfaces. Most real surfaces fall into the latter category, discussed in Chapter 3. While we will follow this convention throughout this book, the reader should be aware that many researchers in the field employ endings according to their own personal preference.

31

32

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

which electromagnetic waves are so strongly related, were not discovered until half a century later. To this day Maxwell’s equations remain the basis for the study of light.∗

2.2 THE MACROSCOPIC MAXWELL EQUATIONS The original form of Maxwell’s equations is based on electrical experiments available at the time, with their very coarse temporal and spatial resolution. Thus any of these measurements were spatial averages taken over many layers of atoms and temporal averages over many oscillations of an electromagnetic wave. For this reason the original set of equations is termed macroscopic. Today we know that electromagnetic waves interact with matter at the molecular level, with strong field fluctuations over each wave period. Therefore, more detailed treatises on optics and electromagnetic waves now generally start with a microscopic description of the wave equations, for example, the book by Stone [1]. While there is little disagreement in the literature on the microscopic equations, the macroscopic equations often differ somewhat from book to book, depending on assumptions made and constitutive relations used. Following the development of Stone [1], we may state the macroscopic Maxwell equations as ∇ · (ǫE) = ρ f , ∇ · (µH) = 0, ∂H , ∂t ∂E ∇×H=ǫ + σe E, ∂t ∇ × E = −µ

(2.1) (2.2) (2.3) (2.4)

where E and H are the electric field and magnetic field vectors, respectively, ǫ is the electrical permittivity, µ is the magnetic permeability, σe is the electrical conductivity, and ρ f is the charge density due to free electrons, which is generally assumed to be related to the electric field by the equation ∂ρ f (2.5) = −∇ · (σe E). ∂t The phenomenological coefficients σe , µ, and ǫ depend on the medium under consideration, but may be assumed independent of the fields (for a linear medium) and independent of position and direction (for a homogeneous and isotropic medium); they may, however, depend on the wavelength of the electromagnetic waves [2].

2.3 ELECTROMAGNETIC WAVE PROPAGATION IN UNBOUNDED MEDIA We seek a solution to the above set of equations in the form of a wave. The most general form of a time-harmonic field (i.e., a wave of constant frequency or wavelength) is F = A cos ωt + B sin ωt = A cos 2πνt + B sin 2πνt,

(2.6)

where ω is the angular frequency (in radians/s), and ν = ω/2π is the frequency in cycles per second. While a little less convenient, we will use the cyclical frequency ν in the following ∗

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

2.3 ELECTROMAGNETIC WAVE PROPAGATION IN UNBOUNDED MEDIA

33

development in order to limit the number of different spectral variables employed in this book. When it comes to the time-harmonic solution of linear partial differential equations, it is usually advantageous to introduce a complex representation of the real field. Thus, setting Fc = Fc e2πiνt ,

Fc = A − iB,

(2.7)

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

(2.8)

where the symbol ℜ denotes that the real part of the complex vector Fc is to be taken. Since the Maxwell equations are linear in the fields E and H, one may solve them for their complex fields, and then extract their real parts after a solution has been found. Therefore, setting E = ℜ{Ec } = ℜ{Ec e2πiνt }, 2πiνt

H = ℜ{Hc } = ℜ{Hc e

(2.9)

},

(2.10)

results in ∇ · (γEc ) = 0,

(2.11)

∇ · Hc = 0,

(2.12)

∇ × Ec = −2πiνµHc ,

(2.13)

∇ × Hc = 2πiνγEc ,

(2.14)

σe 2πν

(2.15)

where γ=ǫ−i

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

(2.16)

−2πi(w·r−νt)

(2.17)

2πiνt

H = ℜ{Hc e

} = ℜ{H0 e

},

where r is a vector pointing to an arbitrary point in space, w is known as the wave vector2 and E0 and H0 are constant vectors. In general w is a complex vector, w = w′ − iw′′ ,

(2.18)

where w′ turns out to be a vector whose magnitude is the wavenumber, and w′′ is known as the attenuation vector. Employing equation (2.18), equations (2.16) and (2.17) may be rewritten as ′′

Ec = E0 e−2πw

·r −2πi(w′ ·r−νt)

e

,

(2.19)

−2πw′′ ·r −2πi(w′ ·r−νt)

(2.20)

Hc = H0 e

e

. ′′

Thus, the complex electric and magnetic fields have local amplitude vectors E0 e−2πw ·r and ′′ ′ H0 e−2πw ·r and an oscillatory part e−2πi(w ·r−νt) with phase angle φ = 2π(w′ · r − νt). The position vector r may be considered to have two components: one parallel to w′ , and the other perpendicular to it. The vector product w′ · r is constant for all vectors r that have the same component parallel to w′ , i.e., on planes normal to the vector w′ ; these planes are known as planes of equal phase. To see how the wave travels let us look at the phase angle at two different times and locations (Fig. 2-1). First, consider the point r = 0 at time t = 0 with a zero phase 2 The present definition of the wave vector differs by a factor of 2π and in name from the definition k = 2πw in most optics texts in order to conform with our definition of wavenumber.

34

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

t = w´z / ν

0

z

w´. r

t =0

FIGURE 2-1 Phase propagation of an electromagnetic wave.

angle. Second, consider another point a distance z away into the direction of w′ ; we see that the phase angle is zero at that point when t = |w′ |z/ν. Thus, the phase velocity with which the wave travels from one point to the other is c = z/t = ν/w′ . We conclude that the wave propagates into the direction of w′ , and that the vector’s magnitude, w′ , is equal to the wavenumber η. Examining the amplitude vectors we see that w′′ · r = const are planes of equal amplitude, and that the amplitude of the fields diminishes into the direction of w′′ . If planes of equal phase and equal amplitude coincide (i.e., if w′ and w′′ are parallel) we say the wave is homogeneous, otherwise the wave is said to be inhomogeneous. Since E0 and w are independent of position, we can substitute equation (2.19) into equation (2.11) and, assuming γ to be also invariant with space, find that     ∇ · (γEc ) = γ∇ · E0 e−2πi(w·r−νt) = γE0 · ∇ e−2πi(w·r−νt) = γE0 e−2πi(w·r−νt) · ∇ (−2πiw · r) = −2πiγw · E0 e−2πi(w·r−νt) = 0.

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

= −2πiw e−2πi(w·r−νt) × E0 = −2πiνµH0 e−2πi(w·r−νt) .

(2.21)

(2.22)

Thus, the partial differential equations (2.11) through (2.14) may be replaced by a set of algebraic equations, w · E0 = 0,

(2.23)

w · H0 = 0,

(2.24)

w × E0 = νµH0 ,

(2.25)

w × H0 = −νγE0 .

(2.26)

It is clear from equations (2.23) and (2.24) that both E0 and H0 are perpendicular to w, and it follows then from equations (2.25) and (2.26) that they are also perpendicular to each other.3 If the wave is homogeneous, then w points into the direction of wave propagation, and the electric and magnetic fields lie in planes perpendicular to this direction, as indicated in Fig. 2-2. It remains to relate the complex wave vector w to the properties of the medium. Taking the vector product of equation (2.25) with w and recalling the vector identity derived, for example, in Wylie [3], A × (B × C) = B(A · C) − C(A · B), (2.27) 3 Remember that all three vectors are complex and, therefore, the interpretation of “perpendicular” is not straightforward.

2.3 ELECTROMAGNETIC WAVE PROPAGATION IN UNBOUNDED MEDIA

35

w, s S=E×H E z H

λ

FIGURE 2-2 Electric and magnetic fields of a homogeneous wave.

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

(2.28)

If the wave travels through vacuum there can be no attenuation (w′′ = 0) and µ = µ0 , γ = ǫ0 . We thus obtain the speed of light in vacuum as √ 1 c 0 = ν/w′ = ν/ w · w = √ . ǫ0 µ0

(2.29)

It is customary to introduce the complex index of refraction m = n − ik

(2.30)

into equation (2.28) such that w · w = ν2 µγ = ν2 ǫ0 µ0

! σe µ ǫµ −i = η20 m2 , ǫ 0 µ0 2πνǫ0 µ0

(2.31)

where η0 = ν/c 0 is the wavenumber of a wave with frequency ν and phase velocity c 0 , i.e., of a wave traveling through vacuum. This definition of m demands that ǫµ = ǫµc20 , ǫ 0 µ0 σe µ σe µλ0 c 0 nk = = , 4πνǫ0 µ0 4π

n2 − k 2 =

(2.32) (2.33)

where λ0 = 1/η0 = c 0 /ν is the wavelength for the wave in vacuum. Equations (2.32) and (2.33) may be solved for the refractive index n and the absorptive index4 k as 4 The absorptive index is often referred to as extinction coefficient in the literature. Since the term extinction coefficient is also employed for another, related property we will always use the term absorptive index in this book to describe the imaginary part of the index of refraction.

36

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

  s  2  2   ǫ 1 λ ǫ σ   0 e  , + n2 =  + 2  ǫ0 ǫ0 2πc 0 ǫ0    s  2  2   ǫ 1 ǫ λ σ   0 e  , k2 = − + + 2  ǫ0 ǫ0 2πc 0 ǫ0 

(2.34)

(2.35)

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

(2.36)

−2πη0 kz −2πiη0 n(z−c 0 t/n)

(2.37)

Hc = H0 e

e

,

where z = sˆ · r is distance along the direction of propagation. For a nonvacuum, the phase velocity c of an electromagnetic wave is5 c0 (2.38) c= . n Further, the field strengths decay exponentially for nonzero values of k; thus, the absorptive index gives an indication of how quickly a wave is absorbed within the medium. Inspection of equation (2.35) shows that a large absorptive index k corresponds to a large electrical conductivity σe : Electromagnetic waves tend to be attenuated rapidly in good electrical conductors, such as metals, but are often transmitted with weak attenuation in media with poor electrical conductivity, or dielectrics, such as glass. The magnitude and direction of the transfer of electromagnetic energy is given by the Poynting vector, i.e., a vector of magnitude EH pointing into the direction of propagation (cf. Fig. 2-2),6 S = E × H = ℜ{Ec } × ℜ{Hc }. (2.39) The instantaneous value for the Poynting∗ vector is a rapidly varying function of time. Of greater value to the engineer is a time-averaged value of the Poynting vector, say Z t+δt 1 S= S(t) dt, (2.40) δt t where δt is a very small amount of time, but significantly larger than the duration of a period, 1/ν; since S repeats itself after each period (if no attenuation occurs) a δt equal to any multiple of 1/ν will give the same result for S, namely S = 12 ℜ{Ec × H∗c },

(2.41)

5

Since there are materials that have n < 1 it is possible to have phase velocities (i.e., the velocity with which the amplitude of continuous waves penetrates through a medium) larger than c 0 ; these should be distinguished from the signal velocities (i.e., the velocity with which the energy contained in the waves travels), which can never exceed the speed of light in vacuum. The difference between the two may be grasped more easily by visualizing the movement of ocean waves: The wave crests move at a certain speed across the ocean surface (phase velocity), while the actual velocity of the water (signal velocity) is relatively slow. 6 Note that, since the vector cross-product is a nonlinear operation, the Poynting vector may not be calculated from S = ℜ{Ec × Hc }. ∗

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

2.4 POLARIZATION

37

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

S=

(2.42)

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

(2.43)

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

H0 =

nE0 ˆ [(−6 + 15i)ˆı + (50 + 25i)ˆ + (8 − 20i)k] √ c 0 µ0 5 154 H0 ˆ = √ [(−6+15i)ˆı + (50+25i)ˆ + (8−20i)k], 3850 =

where H0 =

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

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

2.4

n E2 sˆ = Sˆs, 2c 0 µ0 0

S=

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

= 955.6 W/m2 .

POLARIZATION

Knowledge of the frequency, direction of propagation, and the energy content [i.e., the magnitude of the Poynting vector, equation (2.42)] does not completely describe a monochromatic (or time-harmonic) electromagnetic wave. Every train of electromagnetic waves has a property known as the state of polarization. Polarization effects are generally not very important to the heat transfer engineer since emitted light generally is randomly polarized. In some applications partially or fully polarized light is employed, for example, from laser sources; and the engineer

38

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

needs to know (i) how the reflective behavior of a surface depends on the polarization of incoming light, and (ii) how reflection from a surface tends to alter the state of polarization. We shall give here only a very brief introduction to polarization, based heavily on the excellent short description in Bohren and Huffman [2]. More detailed accounts on the subject may be found in the books by van de Hulst [4], Chandrasekhar [5], and others. Consider a plane monochromatic wave with wavenumber η propagating through a nonabsorbing medium (k ≡ 0) in the z-direction. When describing polarization, it is customary to relate parameters to the electric field (keeping in mind that the magnetic field is simply perpendicular to it), which follows from equation (2.36) as E = ℜ{Ec } = ℜ{(A − iB) e−2πiηn(z−ct) } = A cos 2πηn(z − ct) − B sin 2πηn(z − ct),

(2.44)

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

(2.45)

This curve, shown in Fig. 2-3, describes an ellipse that is known as the vibration ellipse. The ellipse collapses into a straight line if either A or B vanishes, in which case the wave is said to be linearly polarized (sometimes also called plane polarized). If A and B are perpendicular to one another and are of equal magnitude, the vibration ellipse becomes a circle and the wave is known as circularly polarized. In general, the wave in equation (2.44) is elliptically polarized. At any given time, say t = 0, the curve described by the tip of the electric field vector is a helix (Fig. 2-4), or E(z, t = 0) = A cos 2πnηz − B sin 2πnηz. (2.46) Equation (2.46) describes the electric field at any one particular time. As time increases the helix moves into the direction of propagation, and its intersection with any plane z = const describes the local vibration ellipse. The state of polarization, which is characterized by its vibration ellipse, is defined by its ellipticity, b/a (the ratio of the length of its semiminor axis to that of its semimajor axis, as shown in Fig. 2-3), its azimuth γ (the angle between an arbitrary reference direction and its semimajor axis), and its handedness (i.e., the direction with which the tip of the electric field vector traverses through the vibration ellipse, clockwise or counterclockwise). These three parameters together with the magnitude of the Poynting vector are the ellipsometric parameters of a plane wave. Example 2.2. Calculate the ellipsometric parameters a, b, and γ for the wave considered in Example 2.1. Solution From equation (2.44) we find √ ˆ 154, A = E0 (6ˆı + 2ˆ − 8k)/

√ ˆ B = −E0 (3ˆı − 5ˆ − 4k)/ 154,

A a

γ B

b

FIGURE 2-3 Vibration ellipse for a monochromatic wave.

2.4 POLARIZATION

39

t =0

–B A

A –A

z

B

t= 1 4cnη A B

z =0

–B

–A

–B

1 8nη

2 8nη

3 8nη

4 8nη

5 8nη

6 8nη

7 8nη

z

1 nη

FIGURE 2-4 Space variation of electric field at fixed times.

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

E20

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

= E20 (50 − 80 cos 2πνt sin 2πνt + 54 cos2 2πνt)/154. The maximum (a) and minimum (b) of |E| may be found by differentiating the last expression with respect to t and setting the result equal to zero. This operation leads to −80(cos2 2πνt − sin2 2πνt) = 108 sin 2πνt cos 2πνt −80 cos 4πνt = 54 sin 4πνt or   80 . 2πνt = 0.5 tan−1 − 54 This function is double-valued, leading to (2πνt) 1 = −27.99◦ and (2πνt) 2 = 62.01◦ . Substituting these values into the expression for E gives ˆ |E| = a = 0.9009E0 E1 = E0 (0.5404ˆı − 0.0468ˆ − 0.7205k), and ˆ |E| = b = 0.4339E0 . E2 = E0 (0.0134ˆı + 0.4314ˆ − 0.0179k), The evaluation of the azimuth depends on the choice of a reference axis in the plane of the vibration ellipse. In the present problem the y-axis lies in this plane and is, therefore, the natural choice. Thus, cos γ =

E · ˆ 0.0468 =− = −0.0519, |E| 0.9009

γ = 92.97◦ .

While the ellipsometric parameters completely describe any monochromatic wave, they are difficult to measure directly (with the exception of the Poynting vector). In addition, when

40

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

two or more waves of the same frequency but different polarization are superposed, only their strengths are additive: The other three ellipsometric parameters must be calculated anew. For these reasons a different but equivalent description of polarized light, known as Stokes’ parameters, is usually preferred. The Stokes’ parameters are defined by separating the wave train into two perpendicular components: Ec = E0 e−2πiηn(z−ct) ;

E0 = Ek eˆ k + E⊥ eˆ ⊥ ,

(2.47)

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

E⊥ = a⊥ e−iδ⊥ ,

(2.48)

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

(2.49)

Thus, the arbitrary wave given by equation (2.44) has been decomposed into two linearly polarized waves that are perpendicular to one another. The four Stokes’ parameters I, Q, U, and V are defined by I = Ek E∗k + E⊥ E∗⊥ = a2k + a2⊥ , U=

Ek E∗k − E⊥ E∗⊥ Ek E∗⊥ + E⊥ E∗k

V=

i(Ek E∗⊥

Q=



=

a2k



a2⊥ ,

= 2ak a⊥ cos(δk − δ⊥ ),

E⊥ E∗k )

= 2ak a⊥ sin(δk − δ⊥ ),

(2.50) (2.51) (2.52) (2.53)

where the asterisks again denote complex conjugates. It can be shown that these four parameters may be determined through power measurements either directly (I ), using a linear polarizer (arranged in the parallel and perpendicular directions for Q, rotated 45◦ for U ), or a circular polarizer (V ) (see, for example, Bohren and Huffman [2]). It is clear that only three of the Stokes’ parameters are independent, since I2 = Q2 + U2 + V 2 .

(2.54)

Since the Stokes’ parameters of a wave train are expressed in terms of the energy contents of its component waves [which can be seen by comparison with equation (2.42)], it follows that the Stokes’ parameters for a collection of waves are additive. The Stokes’ parameters may also be related to the ellipsometric parameters by I = a2 + b2 ,

(2.55)

2

2

(2.56)

2

2

U = (a − b ) sin 2γ,

(2.57)

V = ±2ab,

(2.58)

Q = (a − b ) cos 2γ,

7 In the literature subscripts p and s are also commonly used, from the German words “parallel” and “senkrecht” (perpendicular).

2.4 POLARIZATION

41

TABLE 2.1

Stokes’ parameters for several cases of polarized light. Linearly Polarized 0◦ 90◦ ↔ l     1 1     −1 1      0  0     0 0 Circularly Polarized Right    1   0   0   1

+45◦ ց   1   0   1   0

−45◦ ւ   1    0    −1   0

γ    1  cos 2γ      sin 2γ    0

Left   1    0     0    −1

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

(2.59)

Such waves are called quasi-monochromatic. If, through their slow respective variations with time, Ek and E⊥ are uncorrelated, then the wave is said to be unpolarized. In such a case the vibration ellipse changes slowly with time, eventually tracing out ellipses of all shapes, orientations, and handedness. All waves discussed so far had a fixed relationship between Ek and E⊥ , and are known as (completely) polarized. If some correlation between Ek and E⊥ exists (for example, a wave of constant handedness, ellipticity, or azimuth), then the wave is called partially polarized. For quasi-monochromatic waves the Stokes’ parameters are defined in terms of time-averaged values, and equation (2.54) must be replaced by I2 ≥ Q2 + U2 + V 2 ,

(2.60)

where the equality sign holds only for polarized light. For unpolarized light one gets Q = U = V = 0, while for partially polarized light the magnitudes of Q, U, and V give the following: p degree of polarization = pQ2 + U2 + V 2 /I, degree of linear polarization = Q2 + U2 /I, degree of circular polarization = V/I. Example 2.3. Reconsider the plane wave of the last two examples. Decompose the wave into two linearly polarized waves, one in the x-z-plane, and the other perpendicular to it. What are the Stokes’ coefficients, the phase differences between the two polarizations, and the different degrees of polarization?

42

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

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

eˆ ⊥ = ˆ,

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

or

 √  Ek = 5/ 154 (2 + i)E0 =

with

r

125 E0 e−iδk , 154 r h √ i 29 E⊥ = (2 − 5i)/ 154 E0 = E0 e−iδ⊥ , 154   1 = −26.565◦ , 2   5 δ⊥ = − tan−1 − = 68.199◦ , 2

δk = − tan−1

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

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

2.5

REFLECTION AND TRANSMISSION

When an electromagnetic wave is incident on the interface between two homogeneous media, the wave will be partially reflected and partially transmitted into the second medium. We will

2.5 REFLECTION AND TRANSMISSION

A

43

δs Medium 1, m 1 = n 1 – ik 1 Medium 2, m 2 = n 2 – ik 2

δs A

FIGURE 2-5 Geometry for derivation of interface conditions.

n

limit our discussion here to plane interfaces, i.e., to cases where the local radius of curvature is much greater than the wavelength of the incoming light, λ, for which the problem may be reduced to algebraic equations. Some discussion on strongly curved surfaces in the form of small particles will be given in Chapter 12, which deals with radiative properties of particulate clouds. In the following, after first establishing the general conditions for Maxwell’s equations at the interface, we shall consider a wave traveling from one nonabsorbing medium into another nonabsorbing medium, followed by a short discussion of a wave incident from a nonabsorbing onto an absorbing medium.

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

Stokes’ theorem:

Z

Γ

∇ × F dV = − V

Z

F × dΓ,

(2.62)

Γ

where dΓ = nˆ dΓ and nˆ is a unit surface normal pointing out of the volume. Now consider a thin volume element δV = A δs containing part of the interface as shown in Fig. 2-5. Applying Gauss’ theorem to the first of Maxwell’s equations, equation (2.11) yields Z Z Z ˆ + (γEc ) 2 · n] ˆ dA = 0, ∇· (γEc ) dV = γEc · dΓ ≈ [(γEc ) 1 · (−n) (2.63) Γ

δV

A

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

(2.64)

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

(2.65)

Thus, the normal components of m2 Ec and Hc are conserved across a plane boundary. Stokes’ theorem may be applied to equations (2.13) and (2.14), again for the volume element shown in Fig. 2-5. For example, Z Z Z Z ∇ × Hc dV = − Hc × dΓ ≈ (Hc1 −Hc2 ) × nˆ dA = 2πiνγEc dV, (2.66) δV

Γ

A

V

44

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

n

ef ro nt

w´i, si

Planes of constant phase and amplitude

W

av

A

Medium 1

θ1 B θ1

n1



θ2

n 2 > n1

θ2

Medium 2



w´t , st FIGURE 2-6 Transmission and reflection of a plane wave at the interface between two nonabsorbing media.

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

(2.67)

ˆ Hc1 × nˆ = Hc2 × n.

(2.68)

and

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

The Interface between Two Nonabsorbing Media The reflection and transmission relationships become particularly simple if homogeneous plane waves reach the plane interface between two nonabsorbing media. For such a wave train the planes of equal phase and equal amplitude coincide and are normal to the direction of propagation, as shown in Fig. 2-6. This plane, also called the wavefront, moves at constant speed c 1 = c 0 /n1 through Medium 1, and at a constant but speed c 2 = c 0 /n2 through Medium 2. If n2 > n1 then, as shown in Fig. 2-6, the wavefront will move more slowly through Medium 2, lagging behind the wavefront traveling through Medium 1. This is readily put in mathematical terms by looking at points A and B on the wavefront at a certain time t. At time t + ∆t the part of the wavefront initially at A will have reached point A′ on the interface while the wavefront at point B, traveling a shorter distance through Medium 2, will have reached point B′ , where ∆t =

AA′ BB′ = . c1 c2

(2.69)

Using geometric relations for AA′ and BB′ and substituting for the phase velocities, we obtain ∆t =

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

(2.70)

2.5 REFLECTION AND TRANSMISSION

45

where the last term pertains to reflection, for which a similar relationship must exist (but which is not shown to avoid overcrowding of the figure). Thus we conclude that θr = θi = θ1 ,

(2.71)

that is, according to electromagnetic wave theory, reflection of light is always purely specular. This is a direct consequence of a “plane” interface, i.e., a surface that is not only flat (with infinite radius of curvature) but also perfectly smooth. Equation (2.70) also gives a relationship between the directions of the incoming and transmitted waves as sin θ2 n1 = , sin θ1 n2

(2.72)

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



Ec1 = E0i e−2πi(wi ·r−νt) + E0r e−2πi(wr ·r−νt) , Hc1 = H0i e

−2πi(w′i ·r−νt)

−2πi(w′r ·r−νt)

+ H0r e

(2.73) .

(2.74)

Similarly for Medium 2, ′

Ec2 = E0t e−2πi(wt ·r−νt) ,

(2.75)

−2πi(w′t ·r−νt)

(2.76)

Hc2 = H0t e

.

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

(2.77)

ˆ (H0i + H0r ) × nˆ = H0t × n.

(2.78)

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

H0 = Hk eˆ k + H⊥ eˆ ⊥ .

(2.79)

This is shown schematically in Fig. 2-7. It is readily apparent from the figure that, in the plane ˆ and tangential to the interface (ˆt) may of incidence, the unit vectors normal to the interface (n) be expressed as nˆ = sˆ i cos θ1 − eˆ ik sin θ1 = −ˆsr cos θ1 + eˆ rk sin θ1 = sˆ t cos θ2 − eˆ tk sin θ2 , tˆ = sˆ i sin θ1 + eˆ ik cos θ1 = sˆ r sin θ1 + eˆ rk cos θ1 = sˆ t sin θ2 + eˆ tk cos θ2 . ∗

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

(2.80a) (2.80b)

46

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

w´r , sr

er⊥ ei ei⊥

er n Unit vector pointing out of plane of paper

θ1 θ1 w´i , si

Medium 1, m1 = n 1

t

Medium 2, m 2 = n 2

n

θ2 et

et⊥

w´t , st FIGURE 2-7 Orientation of wave vectors at an interface.

As defined in Fig. 2-7 the unit vectors eˆ k , eˆ ⊥ and sˆ form right-handed coordinate systems for the incident and transmitted waves, i.e., eˆ k = eˆ ⊥ × sˆ ,

eˆ ⊥ = sˆ × eˆ k ,

sˆ = eˆ k × eˆ ⊥ ,

(2.81)

and a left-handed coordinate system for the reflected wave (leading to opposite signs for the above cross-products of unit vectors).8 Therefore, from equation (2.80) eˆ k × nˆ = ±ˆek × sˆ cos θ = −ˆe⊥ cos θ, eˆ ⊥ × nˆ = ±ˆe⊥ × sˆ cos θ ∓ eˆ ⊥ × eˆ k sin θ = eˆ k cos θ + sˆ sin θ = ˆt, where the top sign applies to the incident and transmitted waves, while the lower sign applies to the reflected component. The second of these relations can also be obtained directly from Fig. 2-7. Using these relations, equations (2.77) and (2.78) may be rewritten in terms of polarized components as  Eik + Erk cos θ1 = Etk cos θ2 ,

Ei⊥ + Er⊥ = Et⊥ ,  Hik + Hrk cos θ1 = Htk cos θ2 , Hi⊥ + Hr⊥ = Ht⊥ .

(2.82) (2.83) (2.84) (2.85)

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

H0 =

(2.86)

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

2.5 REFLECTION AND TRANSMISSION

47

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

(2.87) (2.88)

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

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

For an interface between two nonabsorbing media these coefficients turn out to be real, even though the electric field amplitudes are complex. The reflectivity ρ is defined as the fraction of energy in a wave that is reflected and must, therefore, be calculated from the Poynting vector, equation (2.42), so that !2 Erk Srk ρk = = = r2k (2.93) E ik Sik gives the reflectivity of that part of the wave whose electric field vector lies in the plane of incidence (with its magnetic field normal to it), and   Er⊥ 2 Sr⊥ ρ⊥ = = = r2⊥ (2.94) E i⊥ Si⊥ is the reflectivity for the part whose electric field vector is normal to the plane of incidence. In terms of these polarized components the overall reflectivity may be stated as “reflected energy for both polarizations, divided by the total incoming energy,” or ρ=

Eik E∗ik ρk + Ei⊥ E∗i⊥ ρ⊥ Eik E∗ik + Ei⊥ E∗i⊥

.

(2.95)

For unpolarized and circularly polarized light Eik = Ei⊥ , and the reflectivity for the entire wave train is "   #   1 n1 cos θ2 − n2 cos θ1 2 1 n1 cos θ1 − n2 cos θ2 2 ρ= + ρ k + ρ⊥ = . (2.96) 2 2 n1 cos θ2 + n2 cos θ1 n1 cos1 +n2 cos θ2 From this relationship the refractive indices may be eliminated through Snell’s law, giving # " 1 tan2 (θ1 − θ2 ) sin2 (θ1 − θ2 ) + , (2.97) ρ= 2 tan2 (θ1 + θ2 ) sin2 (θ1 + θ2 ) which is known as Fresnel’s relation.∗ Subroutine fresnel in Appendix F is a generalized version of Fresnel’s relation for an interface between a perfect dielectric and an absorbing medium (see following section), where n = n2 /n1 , k = k2 /n1 , and th = θ1 . ∗

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

48

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

1

1 r

0

0.5

ρ⊥ 1 ρ +ρ ( ⊥ ) 2

r⊥

–1

0

30°

60°

90°

0

0

θp Incidence angle θ

30°

ρ

60° θp

90°

Incidence angle θ

FIGURE 2-8 Reflection coefficients and reflectivities for the interface between two dielectrics (n2 /n1 = 1.5).

The overall transmissivity τ may similarly be evaluated from the Poynting vector, equation (2.42), but the different refractive indices and wave propagation directions in the transmitting and incident media must be considered, so that τ=

n2 cos θ2 2 t = 1 − ρ. n1 cos θ1

(2.98)

An example for the angular reflectivity at the interface between two dielectrics (with n2 /n1 = 1.5) is given in Fig. 2-8. It is seen that, at an angle of incidence of θ1 = θp , rk passes through zero resulting in a zero reflectivity for the parallel component of the wave. This angle is known as the polarizing angle or Brewster’s angle,∗ since light reflected from the surface—regardless of the incident polarization—will be completely polarized. Brewster’s angle follows from equations (2.72) and (2.89) as n2 tan θp = . (2.99) n1 Different behavior is observed if light travels from one dielectric into another, optically less dense medium (n1 > n2 ),9 shown in Fig. 2-9. Examination of equation (2.72) shows that θ2 reaches the value of 90◦ for an angle of incidence θc , called the critical angle, sin θc =

n2 . n1

(2.100)

It is left as an exercise for the reader to show that, for θ1 > θc , light of any polarization is reflected, and nothing is transmitted into the second medium. It is important to realize that upon reflection a wave changes its state of polarization, since Ek and E⊥ are attenuated by different amounts. If the incident wave is unpolarized (e.g., emission from a hot surface), Ek and E⊥ are unrelated and will remain so after reflection. If the incident wave is polarized (e.g., laser radiation), the relationship between Ek and E⊥ will change, causing a change in polarization. Example 2.4. The plane homogeneous wave of the previous examples encounters the flat interface with another dielectric (n2 = 8/3) that is described by the equation z = 0 (i.e., the x-y-plane at z = 0). Calculate ∗

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

9 The optical density of a medium is related to the number of atoms contained over a distance equal to the wavelength of the light and is proportional to the refractive index.

2.5 REFLECTION AND TRANSMISSION

49

1

1

r⊥ 0

0.5

r

ρ⊥ ρ –1

0

θp θc

60°

90°

0

0

Incidence angle θ

θp θc

60°

90°

Incidence angle θ

FIGURE 2-9 Reflection coefficients and reflectivities for the interface between two dielectrics (n1 /n2 = 1.5).

the angles of incidence, reflection, and refraction. What fraction of energy of the wave is reflected, and how much is transmitted? In addition, determine the state of polarization of the reflected wave. Solution ˆ Since the interface is described by z = 0, the surface normal (pointing into Medium 2) is simply nˆ = k. From sˆ = 0.8ˆı + 0.6kˆ and nˆ · sˆ = cos θ1 = 0.6, it follows that the angle of incidence is θ1 = 53.13◦ off normal, which is equal to the angle of reflection, while the angle of refraction follows from Snell’s law, equation (2.72), as n1 2 × 0.8 = 0.6, θ2 = 36.87◦ . sin θ1 = sin θ2 = n2 8/3 It follows that cos θ2 = 0.8 and the reflection coefficients are calculated from equations (2.89) and (2.90) as 2 × 0.8 − (8/3) × 0.6 1.6 − 1.6 = = 0, 2 × 0.8 + (8/3) × 0.6 3.2 2 × 0.6 − (8/3) × 0.8 3.6 − 6.4 = = −0.28, r⊥ = 2 × 0.6 + (8/3) × 0.8 10.0 rk =

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

Eik E∗ik ρk + Ei⊥ E∗i⊥ ρ⊥ Eik E∗ik + Ei⊥ E∗i⊥

=

125 × 0 + 29 × 0.0784 = 0.0148, 154

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

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

50

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

and, from equations (2.50) through (2.53), I = −Q = Er⊥ E∗r⊥ =

0.282 29 E20 = 0.01476 E20 , 154

U = V = 0. Therefore, the wave remains 100% polarized, but the polarization is not completely linear. Indeed, any polarized radiation reflecting off a surface at Brewster’s angle will become linearly polarized with only a perpendicular component.

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





′′

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

H0i × nˆ e

−2πiw′r ·r

+ H0r × nˆ e

−2πi(w′t ·r−iw′′ t ·r)

= H0t × nˆ e

,

(2.101) (2.102)

where r is left arbitrary here in order to derive formally the generalized form of Snell’s law although, for convenience, we still assume that the coordinate origin lies on the interface. We note that none of the amplitude vectors, E0i , H0i , etc., depends on location, and that r is a vector to an arbitrary point on the interface, which may be varied independently. Thus, in order for equations (2.101) and (2.102) to hold at any point on the interface, we must have w′i · r = w′r · r = w′t · r, 0=

w′′ t

· r,

(2.103) (2.104)

that is, since r is tangential to the interface, the tangential components of the wave vector w′ must be continuous across the interface, while the tangential component of the attenuation ′′ ′′ ˆ Thus, within the absorbing medium, planes of equal vector w′′ t must be zero, or wt = wt n. amplitude are parallel to the interface, as indicated in Fig. 2-10. Since w′r has the same tangential component as w′i as well as the same magnitude [cf. equation (2.31)], it follows again that the reflection must be specular, or θr = θi . The continuity of the tangential component for the transmitted wave vector indicates that w′i sin θ1 = η0 n1 sin θ1 = w′t sin θ2 .

(2.105)

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

or

(2.106a)

2.5 REFLECTION AND TRANSMISSION

w´i , si

51

Plane of equal phase and amplitude

Medium 1 (m1 = n1)

θ1 θ2

Medium 2 (m 2 = n2 – ik 2)

n , w´´t

Plane of equal amplitude

w´t , st FIGURE 2-10 Transmission and reflection at the interface between a dielectric and an absorbing medium.

Plane of equal phase

2 2 2 2 w′t 2 − w′′ t = η0 (n2 − k2 ),

(2.106b)

′ ′′ 2 w′t · w′′ t = wt wt cos θ2 = η0 n2 k2 .

(2.106c)

Thus, equations (2.105) and (2.106) constitute three equations in the three unknowns θ2 , w′t , and w′′ t . This system of equations may be solved to yield !  w′t cos θ2 2 1  q 2 2 (2.107a) = (n2 − k22 − n21 sin2 θ1 )2 + 4n22 k22 + (n22 − k22 − n21 sin2 θ1 ) , p = η0 2 !2 q  w′′ 1 t q2 = (n22 − k22 − n21 sin2 θ1 )2 + 4n22 k22 − (n22 − k22 − n21 sin2 θ1 ) , = (2.107b) η0 2 and the refraction angle θ2 may be calculated from equation (2.105) as p tan θ2 = n1 sin θ1 .

(2.108)

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

e r⊥ =

2 ′ Erk n21 (w′t cos θ2 − iw′′ t ) − m2 wi cos θ1 = 2 ′ , 2 ′′ Eik n1 (wt cos θ2 − iwt ) + m2 w′i cos θ1

(2.109a)

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

(2.109b)

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

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

(2.110)

Eliminating the wave vectors, the reflection coefficients may be written as e rk =

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

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

,

(2.111a) (2.111b)

52

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

1.00

ρ⊥

Spectral reflectivity, ρ

0.95

(ρ||+ρ⊥)/2 0.90

ρ|| 0.85

Al at 3.1µm n=4.46, k =31.5

0.80

0.75 0

15

30

45

60

75

90

Incidence angle θ, degrees

FIGURE 2-11 Directional reflectivity for a metal (aluminum at 3.1 µm with n 2 = 4.46, k 2 = 31.5) in contact with air (n1 = 1).

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

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

(2.112)

Finally, the reflectivities are again calculated as ρk = e rke r∗k =

ρ⊥ = e r⊥e r∗⊥ =

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

(2.113a)

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

(2.113b)

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

(n1 − n2 )2 + k22 (n1 + n2 )2 + k22

.

(2.114)

The directional behavior of the reflectivity for a typical metal with n2 = 4.46 and k2 = 31.5 (corresponding to the experimental values for aluminum at 3.1 µm [6]) exposed to air (n1 = 1) is shown in Fig. 2-11. Example 2.5. Redo Example 2.4 for a metallic interface, i.e., the plane homogeneous wave of the previous examples encounters the flat interface with a metal (n2 = k2 = 90), which again is described by the equation z = 0. Calculate the incidence, reflection, and refraction angles. What fraction of energy of the wave is reflected, and how much is transmitted? Solution If n2 and k2 are much larger than n1 it follows from equations (2.107) that p ≈ n2 and q ≈ k2 and, from equation (2.105), n1 sin θ1 ≈ n2 tan θ2 ≈ n2 sin θ2 (i.e., as long as n2 ≫ n1 , Snell’s law between dielectrics holds) and it follows that θ2 = 1.02◦ . With n2 = k2

2.5 REFLECTION AND TRANSMISSION

z=0

53

z=d

Reflected wave Er

E –2 Transmitted wave Et

E +2

Incident wave Ei

n m1 = n1 – ik1

m 2 = n 2 – ik 2

m3 = n3 – ik3

FIGURE 2-12 Reflection and transmission by a slab.

equations (2.113) reduce to ρ⊥ = ρk =

(n1 cos θ1 − n2 )2 + n22

=

(1.2 − 90)2 + 902

= 0.9737,

(1.2 + 90)2 + 902 (n1 cos θ1 + n2 + n22 2 (n2 −n1 sin θ1 tan θ1 ) +n22 (90 − 2×0.82 /0.6)2 +902 ρ = 2 ⊥ 2 (90+2×0.82 /0.6)2 +902 (n2 +n1 sin θ1 tan θ1 ) +n2 )2

×0.9737 = 0.9286,

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

Eik E∗ik ρk + Ei⊥ E∗i⊥ ρ⊥ Eik E∗ik + Ei⊥ E∗i⊥

=

125 × 0.9286 + 29 × 0.9737 = 0.9371. 154

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

or

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

Reflection and Transmission by a Thin Film or Slab As a final topic we shall briefly consider the reflection and transmission by a thin film or slab of thickness d and complex index of refraction m2 = n2 − ik2 , embedded between two media with indices of refraction m1 and m3 , as illustrated in Fig. 2-12. While the theory presented in this section is valid for slabs of arbitrary thickness, it is most appropriate for the study of interference wave effects in thin films or coatings. When an electromagnetic wave is reflected by a thin film, the waves reflected from both interfaces have different phases and interfere with one another (i.e., they may augment each other for small phase differences, or cancel each other for phase differences of 180◦ ). For thick slabs, such as window panes, geometric optics provides a much simpler vehicle to determine overall reflectivity and transmissivity. However, for an antireflective coating on a window, thin film optics should be considered.

54

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

Normal Incidence Since the computations become rather cumbersome, we shall limit ourselves to the simpler case of normal incidence (θ = 0). For more detailed discussions, including oblique incidence angles, the reader is referred to books on the subject such as the one by Knittl [7] or to the very readable monograph by Anders [8]. Consider the slab shown in Fig. 2-12: The wave incident at the left interface is partially reflected, and partially transmitted toward the second interface. At the second interface, again, the wave is partially reflected and partially transmitted into Medium 3. The reflected part travels back to the first interface where a part is reflected back toward the second interface, and a part is transmitted into Medium 1, i.e., it is added to the reflected wave, etc. Therefore, the reflected wave Er and the transmitted wave Et consist of many contributions, and inside ˆ respectively. Medium 2 there are two waves E+2 and E−2 traveling into the directions nˆ and −n, Thus, the boundary conditions, equations (2.67) and (2.68), may be written for the first interface, similar to equations (2.82) through (2.85), as z = r · nˆ = 0 :

Ei + Er = E+2 + E−2 ,

(2.115)

Hi + Hr = H2+ + H2− ,

(2.116)

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

(2.117)

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

z = r · nˆ = d :

(E+2 e−2πiη0 m2 d



E−2 e+2πiη0 m2 d )m2

−2πiη0 m3 d

= Et e

(2.118) m3 .

(2.119)

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

e r23 e−4πiη0 dm2 Er r12 + e = , Ei 1 +e r12e r23 e−4πiη0 dm2

(2.120)

e Et e−2πiη0 dm3 t23 e−2πiη0 dm2 t12e e tfilm = = , Ei 1 +e r12e r23 e−4πiη0 dm2

(2.121)

ti j are the complex reflection and transmission coefficients of the two interfaces, where e ri j and e m 1 − m2 , m1 + m2 2m1 e , t12 = m1 + m2

e r12 =

m 2 − m3 ; m2 + m3 2m2 e t23 = . m2 + m3 e r23 =

(2.122a)

(2.122b)

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

e ti j = ti j eiǫi j ,

ri j = |e ri j |,

tan δij =

ti j = |e ti j |,

tan ǫij =

ℑ(e rij ) ℜ(e rij ) ℑ(e tij )

ℜ(e ti j )

,

(2.123a)

,

(2.123b)

2.5 REFLECTION AND TRANSMISSION

55

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

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

,

n3 ee∗ τ12 τ23 e−κ2 d tt = , n1 1 + 2r12 r23 e−κ2 d cos(δ12 + δ23 − ζ2 ) + r212 r223 e−2κ2 d r2i j = ρi j = nj ni

t2ij = τi j =

tan δij =

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

4(n2i + ki2 ) ni , nj (ni + nj )2 + (ki + k j )2

2(ni k j − nj ki ) n2i

,

+ ki2 − (nj2 + k2j )

κi = 4πη0 ki ,

(2.124)

(2.125)

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

,

ζi = 4πη0 ni d.

(2.126d)

The correct quadrant for δi j is found by checking the sign of both the numerator and denominator in equation (2.126c) (which, while different from the real and imaginary parts of e rij , carry their signs). If both adjacent media, i and j, are dielectrics thene rij = rij is real. In that case we set δij = 0 and let ri j carry a sign. The definition of the thin film transmissivity includes the factor (n3 /n1 ), since it is the magnitude of the transmitted and incoming Poynting vector, equation (2.42), that must be compared. Example 2.6. Determine the reflectivity and transmissivity of a 5 µm thick manganese sulfide (MnS) crystal (n = 2.68, k ≪ 1), suspended in air, for the wavelength range between 1 µm and 1.25 µm. Solution Assuming n1 = n3 = 1, k1 = k2 = k3 = 0, and n2 = 2.68 and substituting these into equations (2.126) leads to n2 − 1 2 2n2 ; t12 = , t23 = ; n2 + 1 n2 + 1 n2 + 1 0 0 = = 0; tan δ23 = 2 = 0. 1 − n22 n2 − 1

r12 = r23 = tan δ12

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

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

,

(2.127)

.

(2.128)

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

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

Tfilm =

0.6005 , 1 − 0.3995 cos ζ2

56

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

Normal reflectivity of layer, R

0.75

5 µm thick layer of dielectric in air (n1 = n3 = 1, n2 = 2.68, k 1 = k 3 = 0)

k2 = 0 k 2 = 0.01

0.50

0.25

0.00 1.00

1.05

1.10 1.15 Wavelength λ, µm

1.20

1.25

FIGURE 2-13 Normal reflectivity of a thin film with interference effects.

with ζ2 = 4πn2 dη0 = 168.4 µm η0 = 168.4 µm/λ0 . Rfilm and Tfilm are periodic with a period of ∆η0 = 2π/168.4 µm = 0.0373 µm−1 . At λ0 = 1 µm this fact implies ∆λ0 = λ20 ∆η0 = 0.0373 µm. The reflectivity of the dielectric film in Fig. 2-13 shows a periodic reflectivity with maxima of 0.5709 (at ζ2 = π, 3π, . . .). For values of ζ2 = 2π, 4π, . . ., the reflectivity of the layer vanishes altogether. Also shown is the case of a slightly absorbing film, with k2 = 0.01. Maximum and minimum reflectivity (as well as transmissivity) decrease and increase somewhat, respectively. This effect is less pronounced at larger wavelengths, i.e., wherever the absorption coefficient κ2 is smaller [cf. equation (2.126d)].

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

Tslab =

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

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

(2.129)

(2.130)

where for Tslab use has been made of the fact that k1 and k2 must be very small, if an appreciable amount of energy is to reach Medium 3. The same relations for thick sheets without wave interference will be developed in the following chapter through geometric optics. Oblique Incidence Knittl [7] has shown that equations (2.124) and (2.125) remain valid for each polarization for oblique incidence if the interface reflectivities, ρij , and transmissivities, τij , are replaced by their directional values; see, for example, equations (2.113). We will state the final result here, mostly following the development of Zhang [9]. The field reflection and transmission coefficients are then expressed as e r =e r12 +

e t21e r23 e−2iβ t12e , 1 −e r21e r23 e−2iβ

(2.131a)

2.6 THEORIES FOR OPTICAL CONSTANTS

e t=

e t23 e−iβ t12e , 1 −e r21e r23 e−2iβ

57

(2.131b)

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

(2.131c)

The overall reflectivity of the film follows from Rfilm

2 e t21e r23 e−2iβ t12e =e re r = e r + , 12 1 − e r21e r23 e−2iβ ∗

and, if Media 1 and 3 are dielectrics, the film transmissivity is evaluated as t23 e−iβ n3 cos θ3 ee∗ n3 cos θ3 e t12e tt = Tfilm = . n1 cos θ1 n1 cos θ1 1 − e r21e r23 e−2iβ

(2.132)

(2.133)

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

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

σe ǫ −i = m2 ǫ0 2πνǫ0

(2.134)

[compare equations (2.31) through (2.35)] and, therefore, ǫ = n2 − k2 , ǫ0 σe = 2nk, ε′′ = 2πνǫ0 ε′ =

 1  ′ √ ′2 ε + ε + ε′′2 , 2   √ 1 −ε′ + ε′2 + ε′′2 , k2 = 2

n2 =

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

where we have again assumed the medium to be nonmagnetic (µ = µ0 ). Any material may absorb or emit radiative energy at many different wavelengths as a result of impurities (presence of foreign atoms) and imperfections in the ionic crystal lattice. However, a number of phenomena tend to dominate the optical behavior of a substance. In the frequency

58

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

NONCONDUCTORS Insulator

CONDUCTORS

Semiconductor

Electron energy

Conduction bands

Metals

Band gap

Partly filled







Band overlap



Valence bands ✁







Core electrons

FIGURE 2-14 Electron energy bands and band gaps in a solid (shading indicates amount of electrons filling the bands) [2].

range of interest to the heat transfer engineer (ultraviolet to midinfrared), electromagnetic waves are primarily absorbed by free and bound electrons or by change in the energy level of lattice vibration (converting a photon into a phonon, i.e., a quantum of lattice vibration). Since electricity is conducted by free electrons, and since free electrons are a major contributor to a solid’s ability to absorb radiative energy, there are distinct optical differences between conductors and nonconductors of electricity. Every solid has a large number of electrons, resulting in a nearcontinuum of possible energy states (and, therefore, a near-continuum of photon frequencies that can be absorbed). However, these allowed energy states occur in bands. Between the bands of allowed energy states may be band gaps, i.e., energy states that the solid cannot attain. This is schematically shown in Fig. 2-14. If a material has a band gap between completely filled and completely empty energy bands, the material is a nonconductor, i.e., an insulator (wide band gap), or a semiconductor (narrow band gap). If a band of electron energy states is incompletely filled or overlaps another, empty band, electrons can be excited into adjacent energy states resulting in an electric current, and the material is called a conductor. Electronic absorption by nonconductors is likely only for photons with energies greater than the band gap, although sometimes two or more photons may combine to bridge the band gap. An intraband transition occurs when an electron changes its energy level, but stays within the same band (which can only occur in a conductor); if an electron moves into a different band (i.e., overcomes the band gap) the movement is termed an interband transition (and can occur in both conductors and nonconductors). This difference between conductors and nonconductors causes substantially different optical behavior: Insulators tend to be transparent and weakly reflecting for photons with energies less than the band gap, while metals tend to be highly absorbing and reflecting between the visible and infrared wavelengths [2]. During the beginning of the century Lorentz [10]∗ developed a classical theory for the evaluation of the dielectric function by assuming electrons and ions are harmonic oscillators (i.e., springs) subjected to forces from interacting electromagnetic waves. His result was equivalent to ∗

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

2.6 THEORIES FOR OPTICAL CONSTANTS

ε´´

59

ρ

ε´

n k n = ε0

νi

Frequency, ν

νi

Frequency, ν

(a) (b) FIGURE 2-15 Lorentz model for (a) the dielectric function, (b) the index of refraction, and normal, spectral reflectivity.

the subsequent quantum mechanical development, and may be stated, as described by Bohren and Huffman [2], as X ν2pj ε(ν) = 1 + , (2.137) νj2 − ν2 + iγj ν j where the summation is over different types of oscillators, νpj is known as the plasma frequency (and ν2p j is proportional to the number of oscillators of type j), νj is the resonance frequency, and γj is the damping factor of the oscillators. Thus, the dielectric function may have a number of bands centered at νj , which may or may not overlap one another. Inspecting equation (2.137), we see that for ν ≫ νj the contribution of band j to ε vanishes, while for ν ≪ νj it goes to the constant value of (νp j /νj )2 . Therefore, for any nonoverlapping band i, we may rewrite equation (2.137) as ε(ν) = ε0 +

ν2pi ν2i − ν2 + iγi ν

(2.138)

,

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

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

(ν2i − ν2 )2 + γ2i ν2

.

,

(2.139a) (2.139b)

The frequency dependence of the real and imaginary parts of the dielectric function for a single oscillating band is shown qualitatively in Fig. 2-15; also shown are the corresponding curves for the real and imaginary parts of the complex index of refraction as evaluated from equation (2.136), along with the qualitative behavior of the normal, spectral reflectivity of a surface from equation (2.114). A strong band with k ≫ 0 results in a region with strong absorption around the resonance frequency and an associated region of high reflection: Incoming photons are mostly reflected, and those few that penetrate into the medium are rapidly attenuated. On either side outside the band the refractive index n increases with increasing frequency (or decreasing wavelength); this is called normal dispersion. However, close to the resonance frequency, n decreases with increasing frequency; this decrease is known as anomalous dispersion. Note that ε′ may become negative, resulting in spectral regions with n < 1. All solids and liquids may absorb photons whose energy content matches the energy difference between filled and empty electron energy levels on separate bands. Since such transitions

60

2 RADIATIVE PROPERTY PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

require a substantial amount of energy, they generally occur in the ultraviolet (i.e., at high frequency). A near-continuum of electron energy levels results in an extensive region of strong absorption (and often many overlapping bands). It takes considerably less energy to excite the vibrational modes of a crystal lattice, resulting in absorption bands in the midinfrared (around 10 µm). Since generally few different vibrational modes exist in an isotropic lattice, such transitions can often be modeled by equation (2.137) with a single band. In the case of electrical conductors photons may also be absorbed to raise the energy levels of free electrons and of bound electrons within partially filled or partially overlapping electron bands. The former, because of the nearly arbitrary energy levels that a free electron may assume, results in a single large band in the far infrared; the latter causes narrower bands in the ultraviolet to infrared.

References 1. Stone, J. M.: Radiation and Optics, McGraw-Hill, New York, 1963. 2. Bohren, C. F., and D. R. Huffman: Absorption and Scattering of Light by Small Particles, John Wiley & Sons, New York, 1983. 3. Wylie, C. R.: Advanced Engineering Mathematics, 5th ed., McGraw-Hill, New York, 1982. 4. van de Hulst, H. C.: Light Scattering by Small Particles, John Wiley & Sons, New York, 1957, (also Dover Publications, New York, 1981). 5. Chandrasekhar, S.: Radiative Transfer, Dover Publications, New York, 1960, (originally published by Oxford University Press, London, 1950). 6. Weast, R. C. (ed.): CRC Handbook of Chemistry and Physics, 68th ed., Chemical Rubber Company, Cleveland, OH, 1988. 7. Knittl, Z.: Optics of Thin Films, John Wiley & Sons, New York, 1976. 8. Anders, H.: Thin Films in Optics, The Focal Press, New York, London, 1967. 9. Zhang, Z. M.: Nano/Microscale Heat Transfer, McGraw-Hill, New York, 2007. 10. Lorentz, H. A.: Collected Papers, vol. 8, Martinus Nijhoff, The Hague, 1935.

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

CHAPTER

3

RADIATIVE PROPERTIES OF REAL SURFACES

3.1

INTRODUCTION

Ideally, electromagnetic wave theory may be used to predict all radiative properties of any material (reflectivity and transmissivity at an interface, absorption and emission within a medium). For a variety of reasons, however, the usefulness of the electromagnetic wave theory is extremely limited in practice. For one, the theory incorporates a large number of assumptions that are not necessarily good for all materials. Most importantly, electromagnetic wave theory neglects the effects of surface conditions on the radiative properties of these surfaces, instead assuming optically smooth interfaces of precisely the same (homogeneous) material as the bulk material—conditions that are very rarely met in practice. In the real world surfaces of materials are generally coated to varying degree with contaminants, oxide layers, and the like, and they usually have a certain degree of roughness (which is rarely even known on a quantitative basis). Thus, the greatest usefulness of the electromagnetic wave theory is that it provides the engineer with a tool to augment sparse experimental data through intelligent interpolation and extrapolation. Still, it is important to realize that radiative properties of opaque materials depend exclusively on the makeup of a very thin surface layer and, thus, may, for the same material, change from batch to batch and, indeed, overnight. This behavior is in contrast to most other thermophysical properties, such as thermal conductivity, which are bulk properties and as such are insensitive to surface contamination, roughness, and so on. The National Institute of Standards and Technology (NIST, formerly NBS) has recommended to reserve the ending “-ivity” for radiative properties of pure, perfectly smooth materials (the ones discussed in the previous chapter), and “-ance” for rough and contaminated surfaces. Most real surfaces fall into the latter category, discussed in the present chapter. Consequently, we will use the ending “-ance” for the definitions in the following section, and for most surface properties throughout this chapter (and the remainder of this book), unless the surface in question is optically smooth and the property is obtained from electromagnetic wave theory. Note that there will be occasions when either term could be used (“almost smooth” surfaces, comparing experimental data with electromagnetic wave theory, etc.). In the present chapter we shall first develop definitions of all radiative properties that are relevant for real opaque surfaces. We then apply electromagnetic wave theory to predict trends of radiative properties for metals and for dielectrics (electrical nonconductors). These theoretical results are compared with a limited number of experimental data. This is followed by a brief 61

62

3 RADIATIVE PROPERTIES OF REAL SURFACES

discussion of phenomena that cannot be predicted by electromagnetic wave theory, such as the effects of surface roughness, of surface oxidation and contamination, and of the preparation of “special surfaces” (i.e., surfaces whose properties are customized through surface coatings and/or controlled roughness). Most experimental data available today were taken in the 1950s and 1960s during NASA’s “Golden Age,” when considerable resources were directed toward sending a man to the moon. Interest waned, together with NASA’s funding, during the 1970s and early 1980s. More recently, because of the development of high-temperature ceramics and high-temperature applications, there has been renewed interest in the measurement of radiative surface properties. No attempt is made here to present a complete set of experimental data for radiative surface properties. Extensive data sets of such properties have been collected in a number of references, such as [1–8], although all of these surveys are somewhat outdated.

3.2

DEFINITIONS

Emittance The most basic radiative property for emission from an opaque surface is its spectral, directional emittance, defined as ǫλ′ (T, λ, sˆ o ) ≡

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

(3.1)

which compares the actual spectral, directional emissive power with that of a black surface at the same conditions. We have added a prime to the letter ǫ to distinguish the directional emittance from the hemispherical (i.e., directionally averaged) value, and the subscript λ to distinguish the spectral emittance from the total (i.e., spectrally averaged) value. The direction vector is denoted by sˆ o to emphasize that, for emission, we are considering directions away from a surface (outgoing). Finally, we have chosen wavelength λ as the spectral variable, since this is the preferred variable by most authors in the field of surface radiation phenomena. Expressions identical to equation (3.1) hold if frequency ν or wavenumber η are employed. Some typical trends for experimentally determined directional emittances for actual materials are shown in Fig. 3-1a,b, as given by Schmidt and Eckert [9] (all emittances in these figures have been averaged over the entire spectrum; see the definition of the total, directional emittance below). For nonmetals the directional emittance varies little over a large range of polar angles but decreases rapidly at grazing angles until a value of zero is reached at θ = π/2. Similar trends hold for metals, except that, at grazing angles, the emittance first increases sharply before dropping back to zero (not shown). Note that emittance levels are considerably higher for nonmetals. A spectral surface whose emittance is the same for all directions is called a diffuse emitter, or a Lambert surface [since it obeys Lambert’s law, equation (1.35)]. No real surface can be a diffuse emitter since electromagnetic wave theory predicts a zero emittance at θ = π/2 for all materials. However, little energy is emitted into grazing directions, as seen from equation (1.31), so that the assumption of diffuse emission is often a good one. The spectral, hemispherical emittance, defined as ǫλ (T, λ) ≡

Eλ (T, λ) , Ebλ (T, λ)

(3.2)

compares the actual spectral emissive power (i.e., emission into all directions above the surface) with that of a black surface. The spectral, hemispherical emittance may be related to the

3.2 DEFINITIONS

63

0 20

20

θ , degrees 40

θ , degrees 40 Glass Wood

60

Clay Copper oxide

Paper

(a)

Ice

60

Aluminum oxide 80

1.0

80

0.8

0.2

0.4

0.6

0

0.2

0.4

50

1.0

θ , degrees

θ , degrees 60

0.8

0.6 ∋



40

20

0

20

40

50

60

Cr 70

70

Ni, polished

(b)

Mn 80

80 Al

Ni, dull 0.14 0.12 0.10 0.08 0.06 0.04 0.02 ∋

0

0.02 0.04 0.06 0.08 0.10 0.12 0.14 ∋

FIGURE 3-1 Directional variation of surface emittances (a) for several nonmetals and (b) for several metals [9].

directional one through equations (1.31) and (1.33), ǫλ (T, λ) =

R

=

R

2π 0 2π 0

R R

π/2 0

Iλ (T, λ, θ, ψ) cos θ sin θ dθ dψ

π Ibλ (T, λ) π/2 ′ ǫλ (T, λ, θ, ψ)Ibλ (T, λ) cos θ 0

sin θ dθ dψ

π Ibλ (T, λ)

,

(3.3)

which may be simplified to 1 ǫλ (T, λ) = π

Z

2π 0

Z

π/2

ǫλ′ (T, λ, θ, ψ) cos θ sin θ dθ dψ,

0

(3.4)

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

Z

π/2 0

ǫλ′ (T, λ, θ) cos θ sin θ dθ.

(3.5)

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

64

3 RADIATIVE PROPERTIES OF REAL SURFACES

ǫλ (T, λ) = ǫλ′ (T, λ).

(3.6)

The total, directional emittance is a spectral average of ǫλ′ , defined by ǫ′ (T, sˆ ) =

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

or, from equations (1.30) and (1.34), Z ∞ Z ∞ Z ∞ 1 1 1 ǫλ′ (T, λ, sˆ ) Ebλ (T, λ) dλ. Iλ dλ = ǫλ′ Ibλ dλ = 2 4 ǫ′ (T, sˆ ) = Ib 0 Ib 0 n σT 0

(3.7)

(3.8)

Finally, the total, hemispherical emittance is defined as ǫ(T) =

E(T) , Eb (T)

and may be related to the spectral, hemispherical emittance through R∞ Z ∞ Eλ (T, λ) dλ 1 0 ǫ(T) = = 2 4 ǫλ (T, λ) Ebλ (T, λ) dλ. Eb (T) n σT 0

(3.9)

(3.10)

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

(3.11)

Such surfaces are termed gray. If we have the very special case of a gray, diffuse surface, this implies ǫ(T) = ǫλ = ǫ′ = ǫλ′ .

(3.12)

While no real surface is truly gray, it often happens that ǫλ is relatively constant over that part of the spectrum where Ebλ is substantial, making the simplifying assumption of a gray surface warranted. Example 3.1. A certain surface material has the following spectral, directional emittance when exposed to air:  0.9 cos θ, 0 < λ < 2 µm, ǫλ′ (λ, θ) = 0.3, 2 µm < λ < ∞. Determine the total hemispherical emittance for a surface temperature of T = 500 K. Solution We first determine the hemispherical, spectral emittance from equation (3.5) as  R π/2 2    2 × 0.9 0 cos θ sin θ dθ = 0.6, 0 < λ < 2 µm, ǫλ (λ) =  R π/2   2 × 0.3 cos θ sin θ dθ = 0.3, 2 µm < λ < ∞. 0

The total, hemispherical emittance follows from equation (3.10) as ! Z 2µm Z ∞ Z 2µm 0.6 − 0.3 1 Ebλ dλ + 0.3 Ebλ dλ = 0.3 + 2 4 Ebλ dλ ǫ(T) = 2 4 0.6 n σT n σT 0 2µm 0   = 0.3 1 + f (1×2 µm×500 K) = 0.3 × (1 + 0.00032) ≃ 0.3,

where the fractional blackbody emissive power f (nλT) is as defined in equation (1.23). For a temperature of 500 K the spectrum below 2 µm is unimportant, and the surface is essentially gray and diffuse.

3.2 DEFINITIONS

65

n Iλ ( r, s

i)

d Ωi

si

θi

dA

ψi

FIGURE 3-2 Directional irradiation onto a surface.

Absorptance Unlike emittance, absorptance (as well as reflectance and transmittance) is not truly a surface property, since it depends on the external radiation field, as seen from its definition, equation (1.51). As for emittance we distinguish between directional and hemispherical, as well as spectral and total absorptances. The radiative heat transfer rate per unit wavelength impinging onto an infinitesimal area dA, from the direction of sˆ i over a solid angle of dΩ i is, as depicted in Fig. 3-2, Iλ (r, λ, sˆ i )(cos θi dA) dΩ i , where we have used the definition of intensity as radiative heat transfer rate per unit area normal to the rays, and per unit solid angle. Iλ is the local radiative intensity at location r (just above the surface). This incoming heat transfer rate, when evaluated per unit surface area dA and per unit incoming solid angle dΩ i , is known as spectral, directional irradiation, Hλ′ (r, λ, sˆ i ) = Iλ (r, λ, sˆ i ) cos θi .

(3.13)

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

α′λ (r, λ, sˆ i ) ′



Hλ,abs Hλ′

,

(3.14)



where Hλ,abs is that part of Hλ that is absorbed by dA. If local thermodynamic equilibrium prevails, the fraction α′λ will not change if Hλ′ increases or decreases. Under this condition we find that the spectral, directional absorptance does not depend on the external radiation field and is a surface property that depends on local temperature, wavelength, and incoming direction. To determine its magnitude, we consider an isothermal spherical enclosure shown in Fig. 3-3, similar to the one used in Section 1.6 to establish the directional isotropy of blackbody intensity. The enclosure coating is again perfectly reflecting except for a small area dAs , which is also perfectly reflecting except over the wavelength interval between λ and λ + dλ, over which it is black. However, the small surface dA suspended at the center is now nonblack. Following the same arguments as for the development of equation (1.32), augmenting the emitted flux by ǫλ′ and the absorbed flux by α′λ , we find immediately α′λ (T, λ, θ, ψ) = ǫλ′ (T, λ, θ, ψ).

(3.15)

66

3 RADIATIVE PROPERTIES OF REAL SURFACES

d As

∋´λ = 1 n

θ

∋´λ = 0 ∋´λ (θ,ψ ) dA

T = const.

FIGURE 3-3 Kirchhoff’s law for the spectral, directional absorptance.

Therefore, if local thermodynamic equilibrium prevails, the spectral, directional absorptance is a true surface property and is equal to the spectral, directional emittance. The spectral radiative heat flux incident on a surface per unit wavelength from all directions, i.e., from the hemisphere above dA, is Z Z Hλ (r, λ) = Hλ′ (r, λ, sˆ i ) dΩ i = Iλ (r, λ, sˆ i ) cos θi dΩ i . (3.16) 2π



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

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

(3.17)

Since the incoming radiation, Iλ , depends on the radiation field of the surrounding enclosure, the spectral, hemispherical absorptance normally depends on the entire temperature field and is not a surface property. However, if the incoming radiation is approximately diffuse (i.e., if Iλ is independent of sˆ i ), then the Iλ may be moved outside the integrals in equation (3.17) and cancelled. Then Z 2π Z π/2 1 αλ (T, λ) = α′λ (T, λ, θi , ψi ) cos θi sin θi dθi dψi , (3.18) π 0 0 or, using equations (3.4) and (3.15), αλ (T, λ) = ǫλ (T, λ)

(diffuse irradiation).

(3.19)

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

(3.20)

3.2 DEFINITIONS

67

and αλ (T, λ) = α′λ (T, λ, sˆ i ) = ǫλ′ (T, λ, sˆ i )

(collimated irradiation).

(3.21)

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

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

(3.23)

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

(3.24)

where C(ˆs) is an otherwise arbitrary function that does not depend on wavelength, i.e., if the incoming radiation is gray (based on the local surface temperature T), then, from equations (3.8) and (3.15), α′ (T, θ, ψ) = ǫ′ (T, θ, ψ).

(3.25)

Of course, this relation also holds if the surface is gray (i.e., α′λ = ǫλ′ do not depend on wavelength). Finally, the total irradiation per unit area from all directions and over the entire spectrum is Z ∞Z H(r) = Iλ (r, λ, sˆ i ) cos θi dΩ i dλ. (3.26) 0



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

0

(3.27)



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

(3.28)

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

(3.29)

Example 3.2. Let the surface considered in the previous example be irradiated by the sun from a 30◦ off-normal direction (i.e., a vector pointing to the sun from the surface forms a 30◦ angle with the outward surface normal). Determine the relevant surface absorptance.

68

3 RADIATIVE PROPERTIES OF REAL SURFACES

Solution Since the sun irradiates the surface from only one direction, but over the entire spectrum, we need to find the total, directional absorptance. From the last example, with θi = 30◦ , we have √  (  π 0.45 3, 0 < λ < 2 µm, = α′λ λ, θi = 6 0.3, 2 µm < λ < ∞. Since we know that the sun behaves like a blackbody at a temperature of Tsun = 5777 K, we also know the spectral behavior of the sunshine falling onto our surface, or Iλ (λ, θi ) = CIbλ (Tsun , λ),

(3.30) 1

where C is a proportionality constant independent of wavelength. Substituting this into equation (3.23) leads to R∞   ǫλ′ (λ, θi )Ibλ (Tsun , λ) dλ π ′ = 0 R∞ α θi = 6 Ibλ (Tsun , λ) dλ 0 # " Z 2µm Z ∞ √ 1 Ebλ (Tsun , λ) dλ Ebλ (Tsun , λ) dλ + 0.3 = 2 4 0.45 3 n σTsun 2µm 0 √ = 0.3 + (0.45 3 − 0.3) f (1×2×5777) = 0.3 + (0.779 − 0.3) × 0.93962 = 0.750. In contrast to the previous example we find that at a temperature of 5777 K the spectrum above 2 µm is of very little importance, and the surface is again essentially gray.

We realize from this example that (i) if a surface is irradiated from a gray source at temperature Tsource , and (ii) if the spectral, directional emittance of the surface is independent of temperature (as it is for most surfaces with good degree of accuracy), then the total absorptance is equal to its total emittance evaluated at the source temperature, or α = ǫ(Tsource ).

(3.31)

This relation holds on a directional basis, and also for hemispherical values if the irradiation is diffuse.

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

(3.32)

Of this, the finite fraction α′λ will be absorbed by the surface (assuming it to be opaque), and the rest will be reflected into all possible directions (total solid angle 2π). Therefore, in general, only an infinitesimal fraction will be reflected into an infinitesimal cone of solid angle dΩ r around direction sˆ r , as shown in Fig. 3-4. Denoting this fraction by ρ′′ (r, λ, sˆ i , sˆ r ) dΩ r we obtain the λ reflected energy within the cone dΩ r as dIλ (r, λ, sˆ i , sˆ r ) dΩ r = (Hλ′ dΩ i )ρ′′ ˆ i , sˆ r ) dΩ r . λ (r, λ, s

(3.33)

2

The spectral, bidirectional reflection function ρ′′ (r, λ, sˆ i , sˆ r ) is directly proportional to the magnitude λ of reflected light that travels into the direction of sˆ r , ˆ i , sˆ r ) = ρ′′ λ (r, λ, s 1

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

(3.34)

As we have seen in Section 1.7, this constant is equal to unity. is sometimes referred to as a bidirectional reflectance; we avoid this nomenclature since the bidirectional reflectance function is not a fraction (i.e., constrained to values between 0 and 1), but may be larger than unity. 2 ′′ ρλ

3.2 DEFINITIONS

69

n I λ

(r,

si )

si (r, dI λ

d Ωi

θi

θr

s r)

sr

d Ωr

dA

ψi ψr

FIGURE 3-4 The bidirectional reflection function.

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

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

(3.35a) (3.35b)

This is done with another variation of Kirchhoff’s law by placing a surface element into an isothermal black enclosure and evaluating the net heat transfer rate—which must be zero— between two arbitrary, infinitesimal surface elements on the enclosure wall. The sign change on the right-hand side of equation (3.35) emphasizes that sˆ i points into the surface, while sˆ r points away from it. Examination of equation (3.34) shows that 0 ≤ ρ′′ < ∞. Reaching the limit of λ ρ′′ → ∞ implies that a finite fraction of Hλ′ is reflected into an infinitesimal cone of solid angle λ dΩ r . Such ideal behavior is achieved by an optically smooth surface, resulting in specular reflection (perfect mirror). For a specular reflector we have ρ′′ = 0 for all sˆ r except the specular direction λ θr = θi , ψr = ψi + π, for which ρ′′ → ∞ (see Fig. 3-4). λ Some measurements by Torrance and Sparrow [12] for the bidirectional reflection function are shown in Fig. 3-5 for magnesium oxide, a material widely used in radiation experiments because of its diffuse reflectance, as defined in equation (3.38) below, in the near infrared (discussed in the last part of this chapter). The data in Fig. 3-5 are for an average surface roughness of 1 µm and are normalized with respect to the value in the specular direction. It is apparent that the material reflects rather diffusely at shorter wavelengths, but displays strong specular peaks for λ > 2 µm. A property of greater practical importance is the spectral, directional–hemispherical reflectance, which is defined as the total reflected heat flux leaving dA into all directions due to the spectral, directional irradiation Hλ′ . With the reflected intensity (i.e., reflected energy per unit area normal to sˆ r ) given by equation (3.33), we have, after multiplying with cos θr , R dIλ (r, λ, sˆ i , sˆ r ) cos θr dΩ r ′✄ , (3.36) ρλ (r, λ, sˆ i ) ≡ 2π Hλ′ (r, λ, sˆ i ) dΩ i

3 RADIATIVE PROPERTIES OF REAL SURFACES

ρ" (θi , π, θr , ψr ) / ρ" (θi , π, θi , 0 )

70

1.2 1.0 0.8 0.6 0.4 0.2 0 1.0 0.8 0.6 0.4 0.2 0 1.0 0.8 0.6 0.4 0.2 0

λ = 3.0 µ m

λ = 0.5 µ m λ = 6.0 µ m

ψr θ i = 10° θ i = 45° 0° 45° 90° 180° 0° 180°

λ = 1.5 µ m

Sample rotated 90°

λ = 10 µ m

λ = 2.0 µ m 60

40

20

0 20 θ , degrees

40

60

60

40

20 0 20 θ , degrees

40

60

FIGURE 3-5 Normalized bidirectional reflection function for magnesium oxide [12].

or

✄ ρλ′ (r, λ, sˆ i )

=

Z



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

(3.37)

where the (Hλ′ dΩ i ) cancels out since it does not depend on outgoing direction sˆ r . Here we have temporarily added the superscript “✄ ” to distinguish the directional–hemispherical reflectance ✄ ✄ (ρ′ ) from the hemispherical–directional reflectance (ρ ′ , defined below). If the reflection function is independent of both sˆ i and sˆ r , then the surface reflects equal amounts into all directions, regardless of incoming direction, and ✄ ρλ′ (r, λ) = πρ′′ (3.38) λ (r, λ). Such a surface is called a diffuse reflector. Comparing the definition of the spectral, directional–hemispherical reflectance with that of the spectral, directional absorptance, equation (3.14), we also find, for an opaque surface, ✄ ρλ′ (r, λ, sˆ i ) = 1 − α′λ (r, λ, sˆ i ). (3.39) Sometimes it is of interest to determine the amount of energy reflected into a certain direction, coming from all possible incoming directions. Equation (3.33) gives the reflected intensity due to a single incoming direction. Integrating this expression over the entire hemisphere of incoming directions leads to Z Iλ (r, λ, sˆ r ) = ρ′′ ˆ i , sˆ r ) Hλ′ (r, λ, sˆ i ) dΩ i λ (r, λ, s Z 2π = ρ′′ ˆ i , sˆ r ) Iλ (r, λ, sˆ i ) cos θi dΩ i . (3.40) λ (r, λ, s 2π

On the other hand, the spectral, hemispherical irradiation is Z Iλ (r, λ, sˆ i ) cos θi dΩ i . Hλ (r, λ) = 2π

(3.41)

3.2 DEFINITIONS

71

If the surface were a perfect reflector, it would reflect all of Hλ , and it would reflect it equally into all outgoing directions. Thus, for the ideal case, the outgoing intensity would be, from equation (1.34), Hλ /π. Consequently, the spectral, hemispherical–directional reflectance is defined as R ρ′′ (r, λ, sˆ i , sˆ r ) Iλ (r, λ, sˆ i ) cos θi dΩ i Iλ (r, λ, sˆ r ) ✄′ λ ρλ (r, λ, sˆ r ) ≡ = 2π 1 R . (3.42) Hλ (r, λ)/π Iλ (r, λ, sˆ i ) cos θi dΩ i π



For the special case of diffuse irradiation (i.e., the incoming intensity does not depend on sˆ i ) equation (3.42) reduces to Z ✄ ρλ ′ (r, λ, sˆ r ) = ρ′′ ˆ i , sˆ r ) cos θi dΩ i , (3.43) λ (r, λ, s 2π

which is identical to equation (3.37) if the reciprocity of the bidirectional reflection function, equation (3.35), is invoked. Thus, for diffuse irradiation, ✄ ✄ ρλ ′ (r, λ, sˆ r ) = ρ′λ (r, λ, sˆ i ), sˆ i = −ˆs r , (3.44a) or

✄ ✄ ρλ ′ (r, λ, θr , ψr ) = ρ′λ (r, λ, θi = θr , ψi = ψr ),

(3.44b)

that is, reciprocity exists between the spectral directional–hemispherical and hemispherical– directional reflectances for any given irradiation/reflection direction. Use of this fact is often made in experimental measurements: While the directional–hemispherical reflectance is of great practical importance, it is very difficult to measure; the hemispherical–directional reflectance, on the other hand, is not very important but readily measured (see Section 3.10). Finally, we define a spectral, hemispherical reflectance as the fraction of the total irradiation from all directions reflected into all directions. From equation (3.36) we have the heat flux reflected into all directions for a single direction of incidence, sˆ i , as ✄ ρλ′ (r, λ, sˆ i ) Hλ′ (r, λ, sˆ i ) dΩ i . Integrating this expression as well as Hλ′ itself over all incidence angles gives R R ✄ ✄ ρλ′ (r, λ, sˆ i ) Iλ (r, λ, sˆ i ) cos θi dΩ i ρ′ (r, λ, sˆ i ) Hλ′ (r, λ, sˆ i ) dΩi 2π λ R = 2π R . ρλ (r, λ) = ′ H (r, λ, s ˆ ) dΩ I (r, λ, s ˆ ) cos θ dΩ i i λ i i i 2π λ 2π

(3.45)

If the incident intensity is independent of direction (diffuse irradiation), then equation (3.45) may be simplified again, and Z 1 ✄ ρ′λ (r, λ, sˆ i ) cos θi dΩ i . (3.46) ρλ (r, λ) = π 2π Also, comparing the definitions of spectral, hemispherical absorptance and reflectance, we obtain, for an opaque surface, ρλ (r, λ) = 1 − αλ (r, λ).

(3.47)

Finally, as for emittance and absorptance we need to introduce spectrally-integrated or “total” reflectances. This is done by integrating numerator and denominator independently over the full spectrum for each of the spectral reflectances, leading to the following relations: Total, bidirectional reflection function R∞ (r, λ, sˆ i , sˆ r ) Iλ (r, λ, sˆ i ) dλ ρ′′ λ ′′ R∞ ρ (r, sˆ i , sˆ r ) = 0 ; (3.48) I (r, λ, s ˆ ) dλ λ i 0

72

3 RADIATIVE PROPERTIES OF REAL SURFACES

TABLE 3.1

Summary of definitions for radiative properties of surfaces. Property

Symbol

Equation

Emittance Spectral, directional

ǫλ′ (T, λ, θ, ψ)

(3.1)

hemispherical

ǫλ (T, λ)

(3.4)

ǫ′ (T, θ, ψ)

(3.8)

ǫ(T)

(3.10)

Total, directional hemispherical Absorptance

Comments

directional average of ǫλ′ (over outgoing directions) spectral average of ǫλ′ (with Ibλ as weight factor) directional and spectral average of ǫλ′ depends on incoming intensity Iin

Spectral, directional

α′λ (T, λ, θ, ψ)

(3.14)

hemispherical

αλ (Iλ,in , T, λ)

(3.17)

Total, directional

α′ (Iin , T, θ, ψ)

(3.23)

α(Iin , T)

(3.27)

directional average of α′λ (over incoming directions) spectral average of α′λ (with Iλ,in as weight factor) directional and spectral average of α′λ

ρ′′ (T, λ, θi , ψi , θr , ψr ) λ

(3.34)

reflection function, 0 ≤ ρ′′ ≤∞ λ

directional–hemispherical

ρλ′ (Iλ,in , T, λ, θi , ψi )

(3.37)

hemispherical–directional

ρλ ′ (Iλ,in , T, λ, θr , ψr )

(3.42)

hemispherical

ρλ (Iλ,in , T, λ)

(3.45)

Total, bidirectional

ρ′′ (Iin , T, θi , ψi , θr , ψr )

(3.48)

directional–hemispherical

ρ′ (Iin , T, θi , ψi )

(3.49)

hemispherical–directional

ρ (Iin , T, θr , ψr )

(3.50)

ρ(Iin , T)

(3.51)

integral of ρ′′ over outgoing λ directions directional average of ρ′′ λ over incoming directions directional average of ρ′λ (incoming and outgoing directions) spectral average of ρ′′ λ (with Iλ,in as weight factor) integral of ρ′′ over outgoing directions directional average of ρ′′ over incoming directions directional and spectral ✄ average of ρλ′

hemispherical Reflectance

depends on incoming intensity Iin

Spectral, bidirectional







✄′

hemispherical

Total, directional–hemispherical reflectance R∞ ✄ ρ′λ (r, λ, sˆ i ) Iλ (r, λ, sˆ i ) dλ ′✄ R∞ ; ρ (r, sˆ i ) = 0 Iλ (r, λ, sˆ i ) dλ 0

(3.49)

Total, hemispherical–directional reflectance R∞ ✄ R ρλ ′ (r, λ, sˆ r ) 2π Iλ (r, λ, sˆ i ) cos θi dΩ i dλ ✄′ 0 R ∞R ; ρ (r, sˆ r ) = I (r, λ, s ˆ ) cos θ dΩ dλ i i i λ 2π 0 Total, hemispherical reflectance R∞ ρ(r) =

0

ρλ (r, λ) R ∞R 0

R

I (r, λ, sˆ i ) cos θi 2π λ

I (r, λ, sˆ i ) cos θi 2π λ

dΩi dλ

dΩ i dλ

.

(3.50)

(3.51)

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

73

3.3 PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY

TABLE 3.2

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

Relation

Restrictions

α′λ (T, λ, θ, ψ) = 1 − ρλ (T, λ, θ, ψ) ′✄

= ǫλ′ (T, λ, θ, ψ)

Spectral, hemispherical

Total, directional

αλ (T, λ) = 1 − ρλ (T, λ) = ǫλ (T, λ) α′ (T, θ, ψ) = 1 − ρ′ (T, θ, ψ)



= ǫ′ (T, θ, ψ) α (Ts , T, θ, ψ) = ǫ′ (Ts , θ, ψ) ′

Total, hemispherical

α(T) = 1 − ρ(T) = ǫ(T) α(Ts , T) = ǫ(Ts )

opaque surfaces (θ, ψ = incoming directions) none (θ, ψ = outgoing directions) opaque surfaces (values depend on directional distribution of source) irradiation and/or ǫλ′ independent of direction (diffuse) opaque surfaces (values depend on spectral distribution of source) ǫλ′ independent of wavelength (gray) source is gray with source temperature Ts , and ǫλ′ is independent of T, or Ts = T opaque surfaces (values depend on spectral and directional distribution of source) ǫλ′ independent of wavelength and direction (gray and diffuse) source is gray and diffuse with source temperature Ts , and ǫλ′ is independent of T, or Ts = T

3.3 PREDICTIONS FROM ELECTROMAGNETIC WAVE THEORY In Chapter 2 we developed in some detail how the spectral, directional–hemispherical reflectivity of an optically smooth interface (specular reflector) can be predicted by the electromagnetic wave and dispersion theories. Before comparing such predictions with experimental data, we shall briefly summarize the results of Chapter 2. Consider an electromagnetic wave traveling through air (refractive index = 1), hitting the surface of a conducting medium (complex index of refraction m = n − ik) at an angle of θ1 with the surface normal (cf. Fig. 3-6). Fresnel’s relations predict the reflectivities for parallel- and perpendicular-polarized light from equations (2.107) through (2.113)3 as ρk =

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

(3.52)

ρ⊥ =

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

(3.53)

where 1 2

# (n2 − k2 − sin2 θ1 )2 + 4n2 k2 + (n2 − k2 − sin2 θ1 ) , "q # 1 (n2 − k2 − sin2 θ1 )2 + 4n2 k2 − (n2 − k2 − sin2 θ1 ) . q2 = 2

p2 =

"q

(3.54) (3.55)

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

3

p tan θ2 = sin θ1 . (3.56) ′✄ For simplicity of notation we shall drop the superscripts for the directional–hemispherical reflectivity whenever

there is no possibility of confusion.

74

3 RADIATIVE PROPERTIES OF REAL SURFACES

Plane of equal phase and amplitude

si

sr

θ1 θ1

Medium 1 (m = 1)

θ2

Medium 2 (m = n – ik)

n

Plane of equal amplitude

st FIGURE 3-6 Transmission and reflection at an interface between air and an absorbing medium.

Plane of equal phase

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

(n − 1)2 + k2 . (n + 1)2 + k2

(3.57)

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

(3.58)

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

(3.59)

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

ρk = ρ⊥



(3.60a) (3.60b)

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

(3.61)

To predict radiative properties from electromagnetic wave theory, the complex index of refraction, m, must be known, either from direct measurements or from dispersion theory predictions. In the dispersion theory the complex dielectric function, ε = ε′ − iε′′ , is predicted by assuming that the surface material consists of harmonic oscillators interacting with electromagnetic waves. The complex dielectric function is related to the complex index of refraction by ε = m2 , or   √ (3.62a) n2 = 21 ε′ + ε′2 + ε′′2 ,   √ 2 ′ 1 k = 2 −ε + ε′2 + ε′′2 , (3.62b)

3.4 RADIATIVE PROPERTIES OF METALS

75

where ε′ =

ǫ , ǫ0

ε′′ =

σe ; 2πνǫ0

ǫ is the electrical permittivity, ǫ0 is its value in vacuum, and σe is the medium’s electrical conductivity. Both ǫ and σe are functions of the frequency of the electromagnetic wave ν. For an isolated oscillator (nonoverlapping band) ε is predicted by the Lorentz model, equation (2.139), as ε′ = ε0 + ′′

ε

=

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

(ν2i − ν2 )2 + γ2i ν2

,

(3.63a) (3.63b)

,

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

3.4

RADIATIVE PROPERTIES OF METALS

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

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

ν2p ν2 + γ 2

ν2p γ ν(ν2 + γ2 )

.

,

(3.64a) (3.64b)

Figure 3-7 shows the spectral, normal reflectivity of three metals—aluminum, copper, and silver. The theoretical lines are from Ehrenreich and coworkers [14] (aluminum) and Ehrenreich and Phillip [15] (copper and silver), who semiempirically determined the values of the unknowns ε0 , νp , and γ in equation (3.64). The experimental reflectance data are taken from Shiles and coworkers [16] (aluminum) and Hagemann and coworkers [17] (copper and silver). The agreement between experiment and theory in the infrared is very good. For wavelengths

76

3 RADIATIVE PROPERTIES OF REAL SURFACES

1.00

Normal, spectral reflectivity, ρnλ

Al 0.95

Ag Ag Cu Al

0.90

Experiment Hagen-Rubens Drude theory: Al

0.85

Cu

νp [1015Hz]

0.80 0.1

Cu

3.07 2.25

Ag 2.22

γ [1012Hz] 31.2

4.55

4.30

ε0

5.6

3.4

[-]

1.0

1.0 10.

Wavelength λ, µ m FIGURE 3-7 Spectral, normal reflectivity at room temperature for aluminum, copper, and silver.

λ > 1 µm the Drude theory has been shown to represent the reflectivity of many metals accurately, if samples are prepared with great care. Discrepancies are due to surface preparation methods and the limits of experimental accuracy. Aluminum has a dip in reflectivity centered at ∼ 0.8 µm; this is due to bound electron transitions that are not considered by the Drude model. Since γ ≪ νp always, there exists for each metal a frequency in the vicinity of the plasma frequency, ν ≃ νp , where ε′ = 1 and ε′′ ≪ 1 or n ≃ 1, k ≪ 1: This fact implies that many metals neither reflect nor absorb radiation in the ultraviolet near νp , but are highly transparent! For extremely long wavelengths (very small frequency ν), we find from equations (3.64) and (2.134) that ε′′ =

ν2p νγ

=

σe , 2πνǫ0

ν ≪ γ,

(3.65)

where σe is the (in general, frequency-dependent) electrical conductivity, and σe = 2πǫ0 ν2p /γ = const = σdc .

(3.66)

Note that at the long-wavelength limit the electrical conductivity becomes independent of wavelength and is known as the dc-conductivity. Since the dc-conductivity is easily measured it is advantageous to recast equation (3.64) as σdc γ/2πǫ0 , ν(ν + iγ) σdc γ/2πǫ0 ε′ = ε0 − 2 , ν + γ2

ε(ν) = ε0 −

ε′′ =

σdc γ2 /2πǫ0 . ν(ν2 + γ2 )

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

Room temperature values for electrical resistivity, 1/σdc , and for electron relaxation time, 1/2πγ, have been given by Parker and Abbott [18] for a number of metals. They have been converted and are reproduced in Table 3.3. Note that these values differ appreciably from those given in Fig. 3-7. No values for ε0 are given; however, the influence of ε0 is generally negligible in the

3.4 RADIATIVE PROPERTIES OF METALS

77

TABLE 3.3

Inverse relaxation times and dc electrical conductivities for various metals at room temperature [18]. Metal Lithium Sodium Potassium Cesium Copper Silver Gold Nickel Cobalt Iron Palladium Platinum

γ, Hz

σdc , Ω−1 cm−1

ν2p = σdc γ/2πǫ0 , Hz2

1.85 × 1013 5.13 × 1012 3.62 × 1012 7.56 × 1012 5.89 × 1012 3.88 × 1012 5.49 × 1012 1.62 × 1013 1.73 × 1013 6.63 × 1012 1.73 × 1013 1.77 × 1013

1.09 × 105 2.13 × 105 1.52 × 105 0.50 × 105 5.81 × 105 6.29 × 105 4.10 × 105 1.28 × 105 1.02 × 105 1.00 × 105 0.91 × 105 1.00 × 105

3.62 × 1030 1.96 × 1030 9.88 × 1029 6.78 × 1029 6.14 × 1030 4.38 × 1030 4.04 × 1030 3.72 × 1030 3.17 × 1030 1.19 × 1030 2.83 × 1030 3.18 × 1030

infrared. Extensive sets of spectral data for a large number of metals have been collected by Ordal and coworkers [19] (for a smaller number of metals they also give the Drude parameters, which are also conflicting somewhat with the data of Table 3.3), while a listing of spectral values of the complex index of refraction for a large numbers of metals and semiconductors has been given in a number of handbooks [20–23]. For long wavelengths equation (3.62) may be simplified considerably, since for such case, ε′′ ≫ |ε′ |, and it follows that n2 ≈ k2 ≈ ε′′ /2 =

σdc λ0 σdc = ≫ 1, 4πνǫ0 4πc 0 ǫ0

(3.68)

where λ0 is the wavelength in vacuum. Substituting values for the universal constants c 0 and ǫ0 , equation (3.68) becomes p (3.69) n ≃ k ≃ 30λ0 σdc , λ0 in cm, σdc in Ω−1 cm−1 ,

which is known as the Hagen–Rubens relation [24]. For comparison, results from equation (3.69) are also included in Fig. 3-7. It is commonly assumed that the Hagen–Rubens relation may be used for λ0 > 6 µm, although this assumption can lead to serious errors, in particular as far as evaluation of the index of refraction is concerned. While equation (3.69) is valid for the metal being adjacent to an arbitrary material, we will—for notational simplicity—assume for the rest of this discussion that the adjacent material has a refractive index of unity (vacuum or gas), that is, λ0 = λ. Substituting equation (3.69) into equation (3.57) leads to ρnλ =

2n2 − 2n + 1 , 2n2 + 2n + 1

ǫnλ = 1 − ρnλ =

(3.70)

4n . 2n2 + 2n + 1

(3.71)

Since n ≫ 1 equation (3.71) may be further simplified to ǫnλ = and, with equation (3.69), to

2 2 − 2 + ··· , n n

2 1 , − ǫnλ ≃ √ 30λ σdc 15λ σdc

λ in cm,

(3.72a)

σdc in Ω−1 cm−1 .

(3.72b)

78

3 RADIATIVE PROPERTIES OF REAL SURFACES

√ This 1/ λ dependence is not predicted by the Drude theory (except for the far infrared), nor is it observed with optically smooth surfaces. However, it often approximates the behavior of polished (i.e., not entirely smooth) surfaces. Example 3.3. Using the constants given in Fig. 3-7 calculate the complex index of refraction and the normal, spectral reflectivity of silver at λ = 6.2 µm, using (a) the Drude theory, and (b) the Hagen–Rubens relation. Solution (a) From Fig. 3-7 we have for silver ε0 = 3.4, νp = 2.22 × 1015 Hz, and γ = 4.30 × 1012 Hz. Substituting these into equation (3.64) with ν = c 0 /λ = 2.998 × 108 m/s × (106 µm/m)/6.2 µm = 4.84 × 1013 Hz, we obtain (2.22 × 1015 )2 = 3.4 − 2087 = −2084, (4.84 × 1013 )2 + (4.30 × 1012 )2 . = 2087 × 4.30 × 1012 4.84 × 1013 = 185.1.

ε′ = 3.4 − ε′′

The complex index of refraction follows from equation (3.62) as 1 2 1 k2 = 2

n2 =





−2084 +



2084 +



 20842 + 185.12 = 4.102,

 20842 + 185.12 = 2088,

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

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

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

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

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

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

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

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

Total Properties for Normal Incidence The total, normal reflectance and emittance may be evaluated from equation (3.8), with spectral, normal properties evaluated from the Drude theory or from the simple Hagen–Rubens relation. While the Hagen–Rubens relation is not very accurate, it does predict the emittance trends correctly in the infrared, and it does allow an explicit evaluation of total, normal emittance. Substituting equation (3.72) into equation (3.8) leads to an integral that may be evaluated in a similar fashion as for the total emissive power, equation (1.19), and, retaining the first three terms of the series expansion ǫn = 0.578 (T/σdc )1/2 − 0.178 (T/σdc ) + 0.0584 (T/σdc )3/2 ,

T in K, σdc in Ω−1 cm−1 .

(3.73)

3.4 RADIATIVE PROPERTIES OF METALS

79

Total normal emittance ∋ n

0.3 Theoretical (Hagen-Rubens) Tungsten Tantalum Niobium Platinum Gold

0.2

0.1

Silver Copper Zinc Tin

0 0.0

0.1

0.2

0.3

0.4

0.5

T/σdc , (Ω cm K)1/ 2 FIGURE 3-8 Total, normal emittance of various polished metals as a function of temperature [18].

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

Tref . T

(3.74)

Therefore, for low enough temperatures, the total, normal emittance of a pure metal should be approximately linearly proportional to temperature. Comparison with experiment (Fig. 38) shows that this nearly linear relationship holds for many metals up to surprisingly high temperatures; for example, for platinum (T/σdc )1/2 = 0.5 corresponds to a temperature of 2700 K. It is interesting to note that spectral integration of the Drude model results in 30% to 70% lower total emissivities for all metals and, thus, fails to follow experimental trends. Such integration was carried out by Parker and Abbott [18] in an approximate fashion. They attributed the discrepancy to imperfections in the molecular lattice induced by surface preparation and to the anomalous skin effect [25], both of which lower the electrical conductivity in the surface layer.

Directional Dependence of Radiative Properties The spectral, directional reflectivity at the interface between an absorber and a nonabsorber is given by Fresnel’s relations, (3.52) through (3.55). Since, in the infrared, n and k are generally fairly large for metals one may with little error neglect the sin2 θ1 in equations (3.54) and (3.55), leading to p ≃ n and q ≃ k. Then, from equations (3.52) and (3.53) the reflectivities for paralleland perpendicular-polarized light are evaluated from4 (n cos θ − 1)2 + (k cos θ)2 , (n cos θ + 1)2 + (k cos θ)2 (n − cos θ)2 + k2 = . (n + cos θ)2 + k2

ρk =

(3.75a)

ρ⊥

(3.75b)

The directional, spectral emissivity (unpolarized) follows as 4 The simple form for ρk used here is best obtained from the reflection coefficient given by equation (2.111) by neglecting sin2 θ1 and canceling m = n − ik from both numerator and denominator.

80

3 RADIATIVE PROPERTIES OF REAL SURFACES

1.0

Spectral Reflectance of Platinum at λ = 2µm

Spectral reflectance, ρλ′

0.9

0.8

0.7

Fresnel’s relation (n=5.29, k =6.71) ρ ρ  ρ⊥ Experiment: T=300K T=657K T=1400K

0.6

0.5

0.4

0.3

0

15

30

45

Polar angle θ , degrees

60

75

ǫλ′ = 1 − 12 (ρk + ρ⊥ ),

90

FIGURE 3-9 Spectral, directional reflectance of platinum at λ = 2 µm.

(3.76)

and is shown (as reflectance) in Fig. 3-9 for platinum at λ = 2 µm. The theoretical line for room temperature has been calculated with n = 5.29, k = 6.71 from [23]. Comparison with experimental emittances of Brandenberg [26], Brandenberg and Clausen [27], and Price [28] demonstrates the validity of Fresnel’s relations.5 Equation (3.75) may be integrated analytically over all directions to obtain the spectral, hemispherical emissivity from equation (3.5). This was done by Dunkle [29] for the two different polarizations, resulting in ! i (n2 −k2 ) h 8n n 2 2 −1 k ǫk = 2 2 1 − 2 2 ln (n+1) +k + tan , (3.77a) n+1 n +k n +k k(n2 +k2 ) ! (n+1)2 +k2 (n2 −k2 ) k tan−1 + ǫ⊥ = 8n 1 − n ln , (3.77b) 2 2 k n +k n(n+1)+k2 1 (3.77c) ǫλ = (ǫk + ǫ⊥ ). 2 Figure 3-10, from Dunkle [30], is a plot of the ratio of the hemispherical and normal emissivities, ǫλ /ǫnλ . For the case of k/n = 1 the dashed line represents results from equation (3.77), while the solid lines were obtained by numerically integrating equations (3.52) through (3.55). For k/n > 1 the two lines become indistinguishable. Hering and Smith [31] reported that equation (3.77) is accurate to within 1–2% for values of n2 + k2 larger than 40 and 3.25, respectively. In view of the large values that n and, in particular, k assume for metals, equation (3.77) is virtually always accurate to better than 2% for metals in the visible and infrared wavelengths. For the reader’s convenience the function emmet is included in Appendix F for the evaluation of equation (3.77). Example 3.4. Determine the spectral, hemispherical emissivity for room-temperature nickel at a wavelength of λ = 10 µm, using (a) the Drude theory, and (b) the Hagen–Rubens relation. Solution We first need to determine the optical constants n and k from either theory, then calculate the hemispherical emissivity from equation (3.77) or read it from Fig. 3-10. 5 In the original figure of Brandenberg and Clausen [27] older values for n and k were used that gave considerably worse agreement with experiment.

3.4 RADIATIVE PROPERTIES OF METALS

81

Hemispherical emissivity, ∋λ Normal emissivity ∋ nλ

1.40

k /n = 4

1.30

2 1.20

1

1.10

0

1.00

0.90 1

10 Refractive index, n

100

FIGURE 3-10 Ratio of hemispherical and normal spectral emissivity for electrical conductors as a function of n and k [30].

(a) Using values for nickel from Table 3.3 in equation (3.64), we find with ν = c 0 /λ = 2.998 × 108 m/s/10−5 m = 2.998 × 1013 Hz, .h i ε′ = 1.0 − 3.72 × 1030 (2.998 × 1013 )2 + (1.62 × 1013 )2 = 1 − 3204 = −3203, . ε′′ = 3204 × 1.62 × 1013 2.998 × 1013 = 1731,   √ n2 = 0.5 × −3208 + 32082 + 17312 = 219,   √ k2 = 0.5 × 3208 + 32082 + 17312 = 3422, and

n = 14.8,

k = 58.5,

k/n = 58.5/14.8 = 3.95.

To use Fig. 3-10 we first determine ρnλ as ρnλ =

13.82 + 58.52 = 0.984, 15.82 + 58.52

and ǫnλ = 1 − ρnλ = 0.016. From Fig. 3-10 ǫλ /ǫnλ ≃ 1.29 and, therefore, ǫλ ≃ 0.021. (b) Using the Hagen–Rubens relation we find, from equation (3.72), ǫnλ = √

2 30 × 10−3 × 1.28 × 105



1 = 0.032. 15 × 10−3 × 1.28 × 105

√ Further, with n ≃ k ≃ 30 × 10−3 × 1.28 × 105 = 62.0, we obtain from Fig. 3-10 ǫλ /ǫnλ ≃ 1.275 and ǫλ ≃ 0.041. The answers from both models differ by a factor of ∼2. This agrees with the trends shown in Fig. 3-7.

Theoretical values for total, directional emissivities are obtained by (numerical) integration of equations (3.75) and (3.76) over the entire spectrum. The directional behavior of total emissivities is similar to that of spectral emissivities, as shown by the early measurements of Schmidt and Eckert [9], as depicted in Fig. 3-1b in a polar diagram (as opposed to the Cartesian representation of Fig. 3-9). The emittances were determined from total radiation measurements from samples heated to a few hundred degrees Celsius.

82

3 RADIATIVE PROPERTIES OF REAL SURFACES

Total hemispherical emittance ∋

0.3

0.2

Theoretical (Hagen-Rubens) Tungsten Tantalum Niobium Molybdenum Platinum

Gold Silver Copper Zinc Tin Lead

0.1

0 0.0

0.1

0.2

0.3 0.4 T/σdc , (Ω cm K) 1/2

0.5

0.6

FIGURE 3-11 Total, hemispherical emittance of various polished metals as a function of temperature [18].

Total, Hemispherical Emittance Equation (3.77) may be integrated over the spectrum using equation (3.10), to obtain the total, hemispherical emittance of a metal. Several approximate relations, using the Hagen–Rubens limit, have been proposed, notably the ones by Davisson and Weeks [32] and by Schmidt and Eckert [9]. Expanding equation (3.77) into a series of powers of 1/n (with n = k ≫ 1), Parker and Abbott [18] were able to integrate equation (3.77) analytically, leading to ǫ(T) = 0.766(T/σdc )1/2 − [0.309 − 0.0889 ln(T/σdc )] (T/σdc ) −0.0175(T/σdc )3/2 ,

T in K,

σdc in Ω−1 cm−1 .

(3.78)

As for the total, normal emittance the total, hemispherical emittance is seen to be approximately linearly proportional to temperature (since σdc ∝ 1/T) as long as the surface temperature is relatively low (so that only long wavelengths are of importance, for which the Hagen–Rubens relation gives reasonable results). Emittances calculated from equation (3.78) are compared with experimental data in Fig. 3-11. Parker and Abbott also integrated the series expansion of equation (3.77) with n and k evaluated from the Drude theory. As for normal emissivities, the Drude model predicts values 30–70% lower than the Hagen–Rubens relations, contrary to experimental evidence shown in Fig. 3-11. Again, the discrepancy was attributed to lattice imperfections and to the anomalous skin effect.

Effects of Surface Temperature The Hagen–Rubens relation, equation (3.72), predicts that the spectral, normal emittance of √ a metal should be proportional to 1/ σdc . Since the electrical conductivity is approximately inversely proportional to temperature, the spectral emittance should, therefore, be proportional to the square root of absolute temperature for long enough wavelengths. This trend should also hold for the spectral, hemispherical emittance. Experiments have shown that this is indeed true for many metals. A typical example is given in Fig. 3-12, showing the spectral dependence of the hemispherical emittance for tungsten for a number of temperatures [33]. Note that the emittance for tungsten tends to increase with temperature beyond a crossover wavelength of approximately 1.3 µm, while the temperature dependence is reversed for shorter wavelengths. Similar trends of a single crossover wavelength have been observed for many metals.

3.5 RADIATIVE PROPERTIES OF NONCONDUCTORS

83

0.5

λ

0.4 Hemispherical emittance,



Tungsten 0.3

0.2 T =1600K T =2000K T =2400K T =2800K

0.1

0.0 0.0

0.5

1.0 1.5 Wavelength λ, µm

2.0

2.5

3.0

FIGURE 3-12 Temperature dependence of the spectral, hemispherical emittance of tungsten [33].

The total, normal or hemispherical emittances are calculated by integrating spectral values over all wavelengths, with the blackbody emissive power as weight function. Since the peak of the blackbody emissive power shifts toward shorter wavelengths with increasing temperature, we infer that hotter surfaces emit a higher fraction of energy at shorter wavelengths, where the spectral emittance is higher, resulting in an increase in total emittance as demonstrated in Figs. 3-8 and 3-11. Since the crossover wavelength is fairly short for many metals, the Hagen– Rubens temperature relation often holds for surprisingly high temperatures.

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

Wavelength Dependence of Spectral, Normal Properties The spectral behavior of pure, crystalline nonconductors is often well described by the single oscillator Lorentz model of equation (3.63). One such material is the semiconductor α-SiC (silicon carbide), a high-temperature ceramic of ever increasing importance. The spectral, normal reflectivity of pure, smooth α-SiC at room temperature is shown in Fig. 3-13, as given by Spitzer and coworkers [34]. The theoretical reflectivity in Fig. 3-13 is evaluated from equations (3.63), (3.62) and (3.57) with ε0 = 6.7, νpi = 4.327 × 1013 Hz, νi = 2.380 × 1013 Hz, γi = 1.428 × 1011 Hz. Agreement between theory and experiment is superb for the entire range between 2 µm and 22 µm. Inspection of equations (3.63) and (3.62) shows that outside the spectral range 10 µm < λ < 13 µm (or 2.5 × 1013 Hz > ν > 1.9 × 1013 Hz), α-SiC is essentially transparent

84

3 RADIATIVE PROPERTIES OF REAL SURFACES

Normal reflectivity, ρnλ

1.0 Experimental Theoretical ε 0 = 6.7 ν i = 2.380 × 1013 Hz νpi = 4.327 × 1013 Hz γ i = 1.428 × 1011 Hz

0.8

SiC 0.6

0.4 0.2

0.0 2

4

6

8

10

12

14

16

18

20

22

Wavelength λ, µ m FIGURE 3-13 Spectral, normal reflectivity of α-SiC at room temperature [34].

Normal reflectivity, ρn λ

50 1.00

Wavelength λ, µ m 16.67 12.5

25

0.80

8.33

Experimental Theoretical

ε0 = ν1 = νp1 = γ1 = ν2 = νp2 = γ2 =

0.60

0.40 MgO 0.20

0.0 200

10

400

600 800 Wavenumber η, cm–1

3.01 1.202 3.089 2.284 1.919 4.070 3.070

× 1013 Hz × 1013 Hz × 1011 Hz × 1013 Hz × 1012 Hz × 1012 Hz

1000

1200

FIGURE 3-14 Spectral, normal reflectivity of MgO at room temperature [36].

(absorptive index k ≪ 1) and weakly reflecting. Within the range of 10 µm < λ < 13 µm αSiC is not only highly reflecting but also opaque (i.e., any radiation not reflected is absorbed within a very thin surface layer, since k > 1). The reflectivity drops off sharply on both sides of the absorption band. For this reason materials such as α-SiC are sometimes used as bandpass filters: If electromagnetic radiation is reflected several times by an α-SiC mirror, the emerging light will nearly exclusively lie in the spectral band 10 µm < λ < 13 µm. This effect has led to the term Reststrahlen band (German for “remaining rays”) for absorption bands due to crystal vibrational transitions. Bao and Ruan [35] have demonstrated that the dielectric function for semiconductors can be calculated through density functional theory, resulting in good agreement with experiment for GaAs. Not all crystals are well described by the single oscillator model since two or more different vibrational transitions may be possible and can result in overlapping bands. Magnesium oxide (MgO) is an example of material that can be described by a two-oscillator model (two overlapping bands), as Jasperse and coworkers [36] have shown (Fig. 3-14). The theoretical reflectivities are obtained with the parameters for the evaluation of equation (3.63) given in the figure. Note that for the calculation of ε′ and ε′′ , equation (3.63) needs to be summed over both

3.5 RADIATIVE PROPERTIES OF NONCONDUCTORS

85

1.0

Normal reflectance ρnλ

0.9

P doped, impurity 7.5 × 1019 cm–3

0.8 Influence of heat treatment

0.7 0.6 As, 9.03 × 1019

0.5 0.4

P-type, polished

Bulk, polished

P, 16.7 × 1019

Sb, 4.47 × 1019 As P Sb

0.3 0.2

Doped, impurities 10−2 for a nonconductor outside Reststrahlen bands. At first glance it might appear, therefore, that all nonconductors must be highly transparent in the near infrared (and the visible). That this is not the case is readily seen from equation (1.55), which relates transmissivity to absorption coefficient. This, in turn, is related to the absorptive index through equation (2.42): τ = e−κs = e−4πks/λ0 .

(3.79)

For a 1 mm thick layer of a material with k = 10−3 at a wavelength (in vacuum) of λ0 = 2 µm, equation (3.79) translates into a transmissivity of τ = exp(−4π × 10−3 × 1/2 × 10−3 ) = 0.002, i.e., the layer is essentially opaque. Still, the low values of k allow us to simplify Fresnel’s relations considerably for the reflectivity of an interface. With k2 ≪ (n − 1)2 the nonconductor essentially behaves like a perfect dielectric and, from equation (3.57), the spectral, normal reflectivity may be evaluated as   n−1 2 ρnλ = , k2 ≪ n2 . (3.80) n+1 Therefore, for optically smooth nonconductors the radiative properties may be calculated from refractive index data. Refractive indices for a number of semitransparent materials at room temperature are displayed in Fig. 3-16 as a function of wavelength [20]. All these crystalline materials show similar spectral behavior: The refractive index drops rapidly in the visible

86

3 RADIATIVE PROPERTIES OF REAL SURFACES

Ge 4.0

InSb

Si

3.0 Refractive index n

TiO2 As2 S3 glass

SrTiO3

Se(As) glass KRS-6 Sapphire

Amorphous selenium

KRS-5

TiO2 AgCl

2.0 MgO

KI Calcite

CsI

Crystal quartz LiF NaF Crystal Water quartz

Fused silica 1.0 0.1

1.0

CaF2

KI KBr NaCl

CsBr

Soda lime glass 10

60

Wavelength λ, µ m FIGURE 3-16 Refractive indices for various semitransparent materials [20].

region, then is nearly constant (declining very gradually) until the midinfrared, where n again starts to drop rapidly. This behavior is explained by the fact that crystalline solids tend to have an absorption band, due to electronic transitions, near the visible, and a Reststrahlen band in the infrared: The first drop in n is due to the tail end of the electronic band, as illustrated in Fig. 2-15b;6 the second drop in the midinfrared is due to the beginning of a Reststrahlen band. Listings of refractive indices for various glasses, water, inorganic liquids, and air are also available [23].

Directional Dependence of Radiative Properties For optically smooth nonconductors experiment has been found to follow Fresnel’s relations of electromagnetic wave theory closely. Figure 3-17 shows a comparison between theory and experiment for the directional reflectivity of glass (blackened on one side to avoid multiple reflections) for polarized, monochromatic irradiation [26]. Because k2 ≪ n2 , the absorptive index may be eliminated from equations (3.52) and (3.53), and the relations for a perfect dielectric become valid. Thus, for unpolarized light incident from vacuum (or a gas), from equations (3.59) 6

Note that the abscissa in Fig. 2-15b is frequency ν, i.e., wavelength increases to the left.

3.5 RADIATIVE PROPERTIES OF NONCONDUCTORS

87

1.0 Black glass: n = 1.517, λ = 0.546 µ m

0.9

Electromagnetic theory Experimental

0.8

Reflectivity ρ´λ

0.7 0.6 0.5 0.4 0.3

ρ⊥

0.2

ρ'λ

0.1 0

ρ 0

10

20

30

40

50

60

70

80

90

Angle of incidence θ , degrees

FIGURE 3-17 Spectral, directional reflectivity of glass at room temperature, for polarized light [26].

Polar angle θ , d egrees 0 10 20 30

Directional emissivity, ∋´λ (θ )

40 50

60

n = 4.0

3.0

2.0 1.5

70

80

0

0.2

0.4

0.6 Directional emissivity, ∋´λ (θ )

0.8

l.0

90

FIGURE 3-18 Directional emissivities of nonconductors as predicted by electromagnetic wave theory.

and (3.60) √  2  1  n2 cos θ − n2 − sin2 θ  1 ′ ǫλ = 1 − ρk + ρ⊥ = 1 −   + √ 2 2 n2 cos θ + n2 − sin2 θ 

√ 2    cos θ − n2 − sin2 θ     .  √    cos θ + n2 − sin2 θ

(3.81)

Of course, the spectral, directional reflectivity for a dielectric can also be calculated from subroutine fresnel in Appendix F by setting k equal to zero. The directional variation of the spectral emissivity of dielectrics is shown in Fig. 3-18. Comparison with Fig. 3-1 demonstrates that experiment agrees well with electromagnetic wave theory for a large number of nonconductors, even for total (rather than spectral) directional emittances. The spectral, hemispherical emissivity of a nonconductor may be obtained by integrating

88

3 RADIATIVE PROPERTIES OF REAL SURFACES

1.0

∋ ∋ /



0.9





0.8



Hemispherical, and normal emissivity

λ

λ

0.7

0.6 1.0

1.5

2.0

2.5 Refractive index n

3.0

3.5

4.0

FIGURE 3-19 Normal and hemispherical emissivities for nonconductors as a function of refractive index.

equation (3.81) with equation (3.5). While tedious, such an integration is possible, as shown by Dunkle [30]: 4(2n + 1) , 3(n + 1)2   16n4 (n4 +1) ln n 4n3 (n2 +2n−1) 2n2 (n2 −1)2 n+1 − + ln , ǫ⊥ = 2 n−1 (n2 +1)3 (n +1)(n4 −1) (n2 +1)(n4 −1)2 1 ǫλ = (ǫk + ǫ⊥ ). 2 ǫk =

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

The variation of normal and hemispherical emissivities with refractive index may be calculated with functions emdiel (ǫλ ) and emdielr (ǫλ /ǫnλ ) from Appendix F and is shown in Fig. 3-19. While for metals the hemispherical emittance is generally larger than the normal emittance (cf. Fig. 3-10), the opposite is true for nonconductors. The reason for this behavior is obvious from Fig. 3-1: Metals have a relatively low emittance over most directions, but display a sharp increase for grazing angles before dropping back to zero. Nonconductors, on the other hand, have a (relatively high) emittance for most directions, which gradually drops to zero at grazing angles (without a peak). Example 3.5. The directional reflectance of silicon carbide at λ = 2 µm and an incidence angle of θ = 10◦ has been measured as ρ′λ = 0.20 (cf. Fig. 3-13). What is the hemispherical emittance of SiC at 2 µm? Solution Since at θ = 10◦ the directional reflectance does not deviate substantially from the normal reflectance (cf. Fig. 3-18), we have ǫnλ = 1 − ρnλ ≃ 1 − 0.20 = 0.80. Then, from Fig. 3-19, n ≃ 2.6 and ǫλ ≃ 0.76.

Effects of Surface Temperature The temperature dependence of the radiative properties of nonconductors is considerably more difficult to quantify than for metals. Infrared absorption bands in ionic solids due to excitation of lattice vibrations (Reststrahlen bands) generally increase in width and decrease in strength with temperature, and the wavelength of peak reflection/absorption shifts toward higher values. Figure 3-20 shows the behavior of the MgO Reststrahlen band [36]; similar results have been

3.6 EFFECTS OF SURFACE ROUGHNESS

89

Wavelength λ, µ m 1.0

50

40

30

25

20

15

12.5

0.9

Normal reflectance ρnλ

0.8 0.7 0.6 0.5 MgO

0.4 8K 295 K 950 K 1950 K

0.3 0.2

Experiment Theory

0.1 0 200

300

400

500 600 Wavenumber η, cm–1

700

800

FIGURE 3-20 Variation of the spectral, normal reflectance of MgO with temperature [36].

obtained for SiC [38]. The reflectance for shorter wavelengths largely depends on the material’s impurities. Often the behavior is similar to that of metals, i.e., the emittance increases with temperature for the near infrared, while it decreases with shorter wavelengths. As an example, Fig. 3-21 shows the normal emittance for zirconium carbide [39]. On the other hand, the emittance of amorphous solids (i.e., solids without a crystal lattice) tends to be independent of temperature [40].

3.6

EFFECTS OF SURFACE ROUGHNESS

Up to this point, our discussion of radiative properties has assumed that the material surfaces are optically smooth, i.e., that the average length scale of surface roughness is much less than the wavelength of the electromagnetic wave. Therefore, a surface that appears rough in visible light (λ ≃ 0.5 µm) may well be optically smooth in the intermediate infrared (λ ≃ 50 µm). This difference is the primary reason why the electromagnetic wave theory ceases to be valid for very short wavelengths. In this section we shall very briefly discuss some fundamental aspects of how surface roughness affects the radiative properties of opaque surfaces. Detailed discussions have been given in the books by Beckmann and Spizzichino [41] and Bass and Fuks [42], and in a review article by Ogilvy [43] The character of roughness may be very different from surface to surface, depending on the material, method of manufacture, surface preparation, and so on, and classification of this character is difficult. A common measure of surface roughness is given by the root-mean-square roughness σh , defined as (cf. Fig. 3-22) σh =

hD

2

(z − zm )

Ei1/2

1 = A "

Z

#1/2 (z − zm ) dA , 2

A

(3.83)

where A is the surface to be examined, and |z − zm | is the local height deviation from the mean. The root-mean-square roughness can be readily measured with a profilometer (a sharp

90

3 RADIATIVE PROPERTIES OF REAL SURFACES

0.7 2100 K Zirconium carbide

2270 K

Normal emittance ∋ n λ

0.6 2470 K 0.5 2670 K

2670 K 2470 K

0.4

2270 K 0.3 2100 K 0.2

1.0

2.0 Wavelength λ , µ m

3.0

4.0

FIGURE 3-21 Temperature dependence of the spectral, normal emittance of zirconium carbide [39].

λ z

zm

z (a)

zm (b)

FIGURE 3-22 Topography of a rough surface: (a) roughness with gradual slopes, (b) roughness with steep slopes. Both surfaces have similar root-mean-square roughness.

stylus that traverses the surface, recording the height fluctuations). Unfortunately, σh alone is woefully inadequate to describe the roughness of a surface as seen by comparing Fig. 3-22a and b. Surfaces of identical σh may have vastly different frequencies of roughness peaks, resulting in different average slopes along the rough surface; in addition, σh gives no information on second order (or higher) roughness superimposed onto the fundamental roughness. A first published attempt at modeling was made by Davies [44], who applied diffraction theory to a perfectly reflecting surface with roughness distributed according to a Gaussian probability distribution. The method neglects shading from adjacent peaks and, therefore, does poorly for grazing angles and for roughness with steep slopes (Fig. 3-22b). Comparison with experiments of Bennett [45] shows that, for small incidence angles, Davies’ model predicts the decay of specular peaks rather well (e.g., Fig. 3-14 for MgO). Davies’ model predicts a sharp peak in the bidirectional reflection function, ρ′′ , for the λ specular reflection direction, as has been found to be true experimentally for most cases as long

3.6 EFFECTS OF SURFACE ROUGHNESS

91

ρ λ´´(θ i , π , θ r , 0) /ρ λ´´ (θ i , π , θ i , 0)

4.0 3.6

λ = 0.5 µ m σ m = 1.9 µ m

MgO Break in scale

2.4 2.0 1.6

θ i = 10˚

45˚

1.2 0.8 60˚ 75˚

0.4 0 – 90

– 60

– 30

30

0

θr , degrees

60

90

FIGURE 3-23 Normalized bidirectional reflection function (in plane of incidence) for magnesium oxide ceramic; σh = 1.9 µm, λ = 0.5 µm [47].

as the incidence angle was not too large (e.g., Fig. 3-5). For large off-normal angles of incidence, experiment has shown that the bidirectional reflectance function has its peak at polar angles greater than the specular direction. An example is given in Fig. 3-23 for magnesium oxide with a roughness of σh = 1.9 µm, illuminated by radiation with a wavelength of λ = 0.5 µm. Shown is the bidirectional reflection function (normalized with its value in the specular direction) for the plane of incidence (the plane formed by the surface normal and the direction of the incoming radiation). We see that for small incidence angles (θi = 10◦ ) the reflection function is relatively diffuse, with a small peak in the specular direction. For comparison, diffuse reflection with a direction-independent reflection function is indicated by the dashed line. For larger incidence angles the reflection function displays stronger and stronger off-specular peaks. For example, for an incidence angle of θi = 45◦ , the off-specular peak lies in the region of θ = 80◦ to 85◦ . Apparently, these off-specular peaks are due to shadowing of parts of the surface by adjacent peaks. The effects of shadowing have been incorporated into the model by Beckmann [46] and Torrance and Sparrow [47]. With the appropriate choice for two unknown constants, Torrance and Sparrow found their model agreed very well with their experimental data (Fig. 3-23). The above models assumed that the surfaces have a certain root-mean-square roughness, but that they were otherwise random—no attempt was made to classify roughness slopes, secondary roughness, etc. Berry and coworkers [48, 49] considered diffraction of radiation from fractal surfaces. The behavior of fractal surfaces is such that the enlarged images appear very similar to the original surface when the surface roughness is repeatedly magnified (Fig. 3-22b). Majumdar and colleagues [50, 51] carried out roughness measurements on a variety of surfaces and found that both processed and unprocessed surfaces are generally fractal. Majumdar and Tien [52] extended Davies’ theory to include fractal surfaces, resulting in good agreement for experiments with different types of metallic surfaces [53, 54]. However, since shadowing effects have not been considered, the model is again limited to near-normal incidence. Buckius and coworkers [55–58] have investigated various one-dimensionally rough surfaces (i.e., where surface height is a function of one coordinate only, z = z(x) in Fig. 3-23), including the effects of roughness peak frequency (or slopes). For a randomly rough surface peak-to-peak spacing is usually characterized by a correlation length σl in a Gaussian correlation function C(L), where L is the length over which the correlation diminishes by a factor of e, or C(L) =

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

(3.84)

92

3 RADIATIVE PROPERTIES OF REAL SURFACES

h

l

101

100

10-1

10-2 0.1

Geometric Optics Region

Statistical Model Region

1

h cos

10

FIGURE 3-24 Domains of validity for the geometric optics and the statistical rough surface reflection models, constructed for incidence angles between −45◦ and +45◦ from the surface normal.

They first considered triangular grooves with roughnesses σh , σl and wavelength λ all of the same order, finding the bidirectional reflectance by solving an integral form of Maxwell’s equations. They found that these exact solutions predict the same scattering peaks as found from optical grating theory. They then applied their model to randomly rough surfaces described by equations (3.83) and (3.84), and compared their electromagnetic wave theory results with those from the simple Kirchhoff approximation [41]. In the Kirchhoff approximation a simplified set of electromagnetic wave equations is considered, assuming that at every point on the surface the electromagnetic field is equal to the field that would exist on a local tangent plane, and multiple reflections between local peaks are neglected. This approximation has been applied by a number of researchers to one- and two-dimensionally rough surfaces, and domains of validity have been constructed [56, 59–61]. It is generally understood that the Kirchhoff approximation gives satisfactory results when surface geometric parameters (σh , σl ) are less than or comparable to the wavelength and the slope of the roughness is small (σh /σl . 0.3). In more recent work Buckius and coworkers have concentrated on geometric optics (i.e., assuming Fresnel’s relations to hold at every point on the surface), noting that Kirchhoff’s approximation results in considerably larger numerical effort without significant improvement over the specular approximation. They considered one- and two-dimensionally uncoated rough surfaces [58, 62, 63], and surfaces coated with a thin film [64] (together with thin film theory). A map was constructed, shown in Fig. 3-24, depicting under what conditions geometric optics gives satisfactory results as compared to exact electromagnetic wave theory calculations, using the criterion , Z π/2 Z π/2 Ed = Ie cos θ dθ < 0.2, (3.85) (I − I ) cos θ dθ −π/2 e a −π/2 where Ie and Ia are exact and approximate reflected intensities, respectively. In general, geometric optics requires generation of statistical surfaces together with ray tracing, a relatively timeconsuming task. Along the same line Zhang and coworkers investigated scattering from rough silicon surfaces and wafers [65–67]. Surface topographic data obtained with an atomic force microscope showed the surface roughness to be significantly non-Gaussian and anisotropic. Nevertheless, the use of two-dimensional slope distributions and statistical ray tracing recovered experimental bidirectional reflection very accurately. Tang and Buckius [68] also introduced a statistical geometric optics model that does not require ray tracing. The resulting closedform expressions were found to be satisfactory for σh /σl . 1, as also indicated in Fig. 3-24. Comparison of geometric optics calculations with experiment (Al2 O3 film on aluminum) showed good agreement, corroborating the applicability of their model [64]. Figure 3-24 was further confirmed (and augmented somewhat) by Fu and Hsu [69], who compared statistical ray tracing results with numerical solutions of Maxwell’s equations.

3.7 EFFECTS OF SURFACE DAMAGE AND OXIDE FILMS

93

Carminati and colleagues [70] used Kirchhoff’s approximation to provide an expression for the spectral, directional emittance (polarized or unpolarized) of a one-dimensionally randomly rough surface as Z ∞h i ′ ǫλ (θ) = 1 − ρλ (θ − tan−1 p) 1 − p tan θ P(p) dp, (3.86) −∞

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

σl 2 e−(pσl /2σh ) . √ σh 4π

(3.87)

Calling this a “small slope emission model” (since, similar to the conclusions of Fig. 3-24, its validity—in particular for parallel polarization—is limited to σh /λ . 0.3), they extended this formula to a “large slope emission model,” using Ishimaru and Chen’s [71] shadowing function and assuming secondary reflection fields to be isotropic.

3.7 EFFECTS OF SURFACE DAMAGE AND OXIDE FILMS Even optically smooth surfaces have a surface structure that is different from the bulk material, due to either surface damage or the presence of thin layers of foreign materials. Surface damage is usually caused by the machining process, particularly for metals and semiconductors, which distorts or damages the crystal lattice near the surface. Thin foreign coats may be formed by chemical reaction (mostly oxidation), adsorption (e.g., coats of grease or water), or electrostatics (e.g., dust particles). All of these effects may have a severe impact on the radiation properties of metals, and may cause considerable changes in the properties of semiconductors. Other materials are usually less affected, because metals have large absorptive indices, k, and thus high reflectances. A thin, nonmetallic layer with small k can significantly decrease the composite’s reflectance (and raise its emittance). Dielectric materials, on the other hand, have small k’s and their relatively strong emission and absorption take place over a very thick surface layer. The addition of a thin, different dielectric layer cannot significantly alter their radiative properties. A minimum amount of surface damage is introduced during sample preparation if (i) the technique of electropolishing is used [45], (ii) the surface is evaporated onto a substrate within an ultra-high vacuum environment [72], or (iii) the metal is evaporated onto a smooth sheet of transparent material and the reflectance is measured at the transparent medium–metal interface [73]. Figure 3-25 shows the spectral, normal emittance of aluminum for a surface prepared by the ultra-high vacuum method [72], and for several other aluminum surface finishes [74]. While ultra-high vacuum aluminum follows the Drude theory for λ > 1 µm (cf. Fig. 3-7), polished aluminum (clean and optically smooth for large wavelengths) has a much higher emittance over the entire spectrum. Still, the overall level of emittance remains very low, and the reflectance remains rather specular. Similar results have been obtained by Bennett [45], who compared electropolished and mechanically polished copper samples. As Fig. 3-25 shows, the emittance is much larger still when off-the-shelf commercial aluminum is tested, probably due to a combination of roughness, contamination, and slight atmospheric oxidation. Bennett and colleagues [75] have shown that deposition of a thin oxide layer on aluminum (up to 100 Å) appreciably increases the emittance only for wavelengths less than 1.5 µm. This statement clearly is not true for thick oxide layers, as evidenced by Fig. 3-25: Anodized aluminum (i.e., electrolytically oxidized material with a thick layer of alumina, Al2 O3 ) no longer displays the typical trends of a metal, but rather shows the behavior of the dielectric alumina. The effects of thin and thick oxide layers have been measured for many metals, with similar results. A good collection of such measurements has been given by Wood and coworkers [3]. As a rule of thumb, clean metal exposed to air at room temperature grows oxide films so thin that infrared

94

3 RADIATIVE PROPERTIES OF REAL SURFACES

Spectral, normal emittance ∋nλ

1.0 Aluminum, anodized

0.8

0.6

0.4

Aluminum, polished

0.2

0

2

Aluminum, Aluminum, ultra-high vacuum commercial finish evaporated

4 6 Wavelength λ, µ m

8

10

FIGURE 3-25 Spectral, normal emittance for aluminum with different surface finishes [72, 74].

Spectral, normal absorptance α nλ

1.0 Ultraviolet irradiated (300 ESH)

0.8

Titanium dioxide/epoxy (Skyspar) Ultraviolet intensity = 3 × solar Pressure = 1 × 10 –7 torr

0.6 Gamma irradiated (385 mrad)

0.4

0.2 Unirradiated 0 0.2

0.4

0.6

0.8

1.0

1.2

Wavelength λ, µ m

1.4

1.6

1.8

2.0

FIGURE 3-26 Effects of ultraviolet and gamma ray irradiation on a titanium dioxide/epoxy coating [78].

emittances are not affected appreciably. On the other hand, metal surfaces exposed to hightemperature oxidizing environments (furnaces, etc.) generally have radiative properties similar to those of their oxide layer. While most severe for metallic surfaces the problem of surface modification is not unknown for nonmetals. For example, it is well known that silicon carbide (SiC), when exposed to air at high temperature, forms a silica (SiO2 ) layer on its surface, resulting in a reflection band around 9 µm [76]. Nonoxidizing chemical reactions can also significantly change the radiative properties of dielectrics. For example, the strong ultraviolet radiation in outer space (from the sun) as well as gamma rays (from inside the Earth’s van Allen belt) can damage the surface of spacecraft protective coatings like white acrylic paint [77] or titanium dioxide/epoxy coating [78], as shown in Fig. 3-26. In summary, radiative properties for opaque surfaces, when obtained from figures in this chapter, from the tables given in Appendix B, or from other tabulations and figures of [1–8,79,80], should be taken with a grain of salt. Unless detailed descriptions of surface purity, preparation, treatment, etc., are available, the data may not give any more than an order-of-magnitude estimate. One should also keep in mind that the properties of a surface may change during a process or overnight (by oxidation and/or contamination).

3.8 RADIATIVE PROPERTIES OF SEMITRANSPARENT SHEETS

ρ12

Ii (θ 1) = 1

θ1 θ1

(1 – ρ12)2 ρ23 τ 2

(1 – ρ12)2 ρ12 ρ223 τ 4

θ1

1 – ρ12

n1 (1 – ρ 12)ρ23 τ 2 (1 – ρ12 ) ρ12 ρ223 τ 4

θ2 θ2 (1 – ρ12) τ ρ23

(1 – ρ12) τ

θ3

95

m2 = n2 – ik2

(1 – ρ12) ρ12 ρ23 τ 3

n3 (1 – ρ12 ) (1 – ρ23 ) ρ12 ρ23τ 3 (1 – ρ12 ) (1 – ρ23 )τ

FIGURE 3-27 Reflectivity and transmissivity of a thick semitransparent sheet.

3.8 RADIATIVE PROPERTIES OF SEMITRANSPARENT SHEETS The properties of radiatively participating media will be discussed in Chapters 11 through 13; i.e., semitransparent media that absorb and emit in depth and whose temperature distribution is, thus, strongly affected by thermal radiation. There are, however, important applications where thermal radiation enters an enclosure through semitransparent sheets, and where the temperature distribution within the sheet is unimportant or not significantly affected by thermal radiation. Applications include solar collector cover plates, windows in connection with light level calculations within interior spaces, and so forth. We shall, therefore, briefly present here the radiative properties of window glass, for single and multiple pane windows with and without surface coatings. Glass and other amorphous solids tend to have extremely smooth surfaces, allowing for accurate predictions of interface reflectivities from electromagnetic wave theory.

Properties of Single Pane Glasses For an optically smooth window pane of a thickness d substantially larger than the wavelength of incident light, d ≫ λ, the radiative properties are readily determined through geometric optics and ray tracing. Consider the sheet of semitransparent material depicted in Fig. 3-27. The sheet has a complex index of refraction m2 = n2 − ik2 with k2 ≪ 1, so that the transmission through the sheet (not counting surface reflections), τ = e−κ2 d/cos θ2 = e−4πk2 d/λ0 cos θ2 ,

(3.88)

is appreciable [cf. equation (2.42)]. Here κ2 = 4πk2 /λ0 is the absorption coefficient, λ0 is the wavelength of the incident light in vacuum, and d/cos θ2 is the distance a light beam of oblique incidence travels through Medium 2 in a single pass. The semitransparent sheet is surrounded by two dielectric materials with refractive indices n1 and n3 . To calculate the reflectivity at the interfaces 1–2 and 2–3 it is sufficient to use Fresnel’s relations for dielectric media, since k2 ≪ 1. Interchanging n1 and n2 , as well as θ1 and θ2 , in equation (2.96) shows that the reflectivity at the 1–2 interface is the same, regardless of whether radiation is incident from Medium 1 or Medium 2, i.e., ρ12 = ρ21 and ρ23 = ρ32 . Now consider radiation of unit strength to be incident upon the sheet from Medium 1 in the direction of θ1 . As indicated in Fig. 3-27 the fraction ρ12 is reflected at the first interface, while the fraction (1 − ρ12 ) is refracted into Medium 2, according to Snell’s law. After traveling a distance d/cos θ2 through Medium 2 the attenuated fraction

96

3 RADIATIVE PROPERTIES OF REAL SURFACES

(1 − ρ12 )τ arrives at the 2–3 interface. Here the amount (1 − ρ12 )τρ23 is reflected back to the 1–2 interface, while the fraction (1 − ρ12 )τ(1 − ρ23 ) leaves the sheet and penetrates into Medium 3 in a direction of θ3 . The internally reflected fraction keeps bouncing back and forth between the interfaces, as indicated in the figure, until all energy is depleted by reflection back into Medium 1, by absorption within Medium 2, and by transmission into Medium 3. Therefore, the slab reflectivity, Rslab , may be calculated by summing over all contributions, or h i Rslab = ρ12 + ρ23 (1 − ρ12 )2 τ2 1 + ρ12 ρ23 τ2 + (ρ12 ρ23 τ2 )2 + · · · . Since ρ12 ρ23 τ2 < 1 the series is readily evaluated [81], and Rslab = ρ12 +

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

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

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

(3.89)

(3.90)

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

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

(3.91)

If Media 1 and 3 are identical (say, air), then ρ12 = ρ23 = ρ and equations (3.89) through (3.91) reduce to " # (1 − ρ)2 τ2 Rslab = ρ 1 + , (3.92) 1 − ρ2 τ2 Tslab =

(1 − ρ)2 τ , 1 − ρ 2 τ2

(3.93)

Aslab =

(1 − ρ)(1 − τ) . 1 − ρτ

(3.94)

Figure 3-28 shows typical slab transmissivities and reflectivities of several different types of glasses for normal incidence and for a pane thickness of 12.7 mm. Most glasses have fairly constant and low slab reflectivity in the spectral range from 0.1 µm up to about 9 µm (relatively constant refractive index n, small absorptive index k). Beyond 9 µm the reflectivity increases because of two Reststrahlen bands [82] (not shown). Glass transmissivity tends to be very high between 0.4 µm and 2.5 µm. Beyond 2.5 µm the transmissivity of window glass diminishes rapidly, making windows opaque to infrared radiation. This gives rise to the so-called “greenhouse” effect: Since the sun behaves much like a blackbody at 5777 K, most of its energy (≈ 95%) falling onto Earth lies in the spectral range of high glass transmissivities. Therefore, solar energy falling onto a window passes readily into the space behind it. The spectral variation of solar irradiation, for extraterrestrial and unity air mass conditions, was given in Fig. 1-3. On the other hand, if the space behind the window is at low to moderate temperatures (300 to 400 K), emission from such surfaces is at fairly long wavelengths, which is absorbed by the glass and, thus, cannot escape.

3.8 RADIATIVE PROPERTIES OF SEMITRANSPARENT SHEETS

Normal slab reflectivity, R slab Normal slab transmissivity, T slab

1.00

2

3 Reflectivity or transmissivity

97

3 1

0.75

5 4

2 4

5 1

4

0.50

1 2 3 4 5

5 0.25

2

Aluminum silicate glass Borosilicate glass Fused silica Pyrex Soda−lime glass

1 0.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Wavelength λ, µm FIGURE 3-28 Spectral, normal slab transmissivity and reflectivity for panes of five different types of glasses at room temperature; data from [7]. Normal slab reflectivity, R slab

Normal slab transmissivity, T slab

Reflectivity or transmissivity

1.00

1 0.75

2 3 4

0.50 Glass sheet thickness 1 1.588 mm 2 6.350 mm 3 9.525 mm 4 12.7 mm

0.25

0.00 0.0

0.5

1.0 1.5 Wavelength λ, µm

2.0

2.5

3.0

FIGURE 3-29 Spectral, normal slab transmissivity and reflectivity of soda–lime glass at room temperature, for a number of pane thicknesses; data from [7].

The influence of pane thickness on reflectivity and transmissivity is shown in Fig. 3-29 for the case of soda–lime glass (i.e., ordinary window glass). As the pane thickness increases, transmissivity decreases due to the increasing absorption. Since the absorption coefficient is small for λ < 2.7 µm (see Fig. 1-17), the effect is rather minor (and even less so for the other glasses shown in Fig. 3-28). In some high-temperature applications the emission from hot glass surfaces becomes important (e.g., in the manufacture of glass). Gardon [83] has calculated the spectral, hemispherical and total, hemispherical emissivity of soda–lime glass sheets at 1000◦ C based on the data of Neuroth [84]. Spectral emissivities beyond 2.7 µm do not depend strongly on temperature since the absorption coefficient is relatively temperature-independent (see Fig. 1-17). For all but the thinnest glass sheets the material becomes totally opaque, and the hemispherical emissivity is

98

3 RADIATIVE PROPERTIES OF REAL SURFACES

evaluated as ǫλ = 1 − ρλ ≃ 0.91.7

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



n1 − n2 n1 + n2

2

and ρ23 =



n2 − n3 n2 + n3

2

,

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

Rcoat =

which is readily simplified to Rcoat = 1 −

(n22

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

(3.95)

If the aim is to minimize the overall reflectivity of the semitransparent sheet, then a value for the refractive index of the coating must be chosen to make Rcoat a minimum. Thus, setting dRcoat /dn2 = 0 leads to √ n2,min = n1 n3 . (3.96) Substituting equation (3.96) into (3.95) results in a minimum coated-surface reflectivity of √ 2 n1 n3 . (3.97) Rcoat,min = 1 − n1 + n3 The slab reflectivity for a thin dielectric coating on a dielectric substrate, d = O(λ), is subject to wave interference effects and has been evaluated in Chapter 2, from equation (2.124), with δ12 = π and δ23 = 0 (cf. Example 2.6), as Rcoat = r12 =

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

r23 =

(3.98a)

,

n 2 − n3 , n2 + n3

ζ=

4πn2 d . λ

(3.98b)

Equation (3.98) has an interference minimum when ζ = π (i.e., if the film thickness is a quarter of the wavelength inside the film, d = 0.25λ/n2 ). For this interference minimum the reflectivity of the coated surface becomes   r12 − r23 2 Rcoat = . (3.99) 1 − r12 r23 7 The hemispherical emissivity is evaluated by first evaluating ρnλ : With n ≃ 1.5 (for λ > 2.7 µm), from Fig. 3-16 ρnλ = 0.04 and ǫnλ = 0.96; finally, from Fig. 3-19 ǫλ ≃ 0.91.

3.8 RADIATIVE PROPERTIES OF SEMITRANSPARENT SHEETS

99

1.0

Transmissivity Tλ , and reflectivity Rλ

0.8



0.6

0.4

0.2

0



0.3 0.5

0.7 1.0 Wavelength λ, µ m

2.0

3.0

FIGURE 3-30 Spectral, normal reflectivity and transmissivity of a 0.35 µm thick Sn-doped In2 O3 film deposited on Corning 7059 glass [86].

√ Clearly, this equation results in a minimum (or zero) reflectivity if r12 = r23 , or n2,min = n1 n3 , which is the same as for thick films, equation (3.96). To obtain minimum reflectivities for glass (n3 ≃ 1.5) facing air (n1 ≃ 1) would require a dielectric film with n2 ≃ 1.22. Dielectric films of such low refractive index do not appear possible. However, Yoldas and Partlow [85] showed that a porous film (pore size ≪ λ) can effectively lower the refractive index, and they obtained glass transmissivities greater than 99% throughout the visible. In other applications a strong reflectivity is desired. An example of experimentally determined reflectivity and transmissivity of a coated dielectric is given in Fig. 3-30 for a 0.35 µm thick layer of Sn-doped In 2 O 3 film on glass [86]. The oscillating properties clearly demonstrate the effects of wave interference at shorter wavelengths. At wavelengths λ > 1.5 µm the material has a strong absorption band, making it highly reflective and opaque. Thus, this coated glass makes a better solar collector cover plate than ordinary glass, since internally emitted infrared radiation is reflected back into the collector (rather than being absorbed), keeping the cover glass cool and reducing losses. Similar behavior was obtained by Yoldas and O’Keefe [87], who deposited thin (20 to 50 nm) triple-layer films (titanium dioxide–silver–titanium dioxide) on soda–lime glass. It is also possible to tailor the directional reflection behavior using special, obliquely deposited films [88].

Multiple Parallel Sheets To minimize convection losses, two or more parallel sheets of windows are often employed, as illustrated in Fig. 3-31a. To find the total reflectivity and transmissivity of n layers, we break the system up into a single layer and the remaining (n − 1) layers. Then ray tracing (see Fig. 3-31b) results in h i T12 Rn−1 Rn = R1 + T12 Rn−1 1 + R1 Rn−1 + (R1 Rn−1 )2 + · · · = R1 + , (3.100) 1 − R1 Rn−1 and, similarly, T1 Tn−1 Tn = , (3.101) 1 − R1 Rn−1 where Rn−1 and Tn−1 are the net reflectivity and transmissivity of (n − 1) layers. The net absorptivity of the n layers can be calculated directly either from An = A1 + A1 T1 Rn−1 (1 + R1 Rn−1 + · · · ) + An−1 T1 (1 + R1 Rn−1 + · · · ) T1 (A1 Rn−1 + An−1 ) = A1 + , 1 − R1 Rn−1

(3.102)

100

3 RADIATIVE PROPERTIES OF REAL SURFACES

T 21Rn –1

1

T 21R 2n –1R1

n

Sheet n

n –1 R1, T1

R1, T 1 2

T1

T1Rn–1

Sheets 1 through n–1

Sheet 1 Rn –1, Tn –1 R1, T1

T1Tn –1

T1Rn –1R1Tn –1

(a) (b) FIGURE 3-31 Reflectivity and transmissivity of multiple sheets: (a) geometric arrangement, (b) ray tracing for interaction between a single layer and the remainder of the sheets.

or from conservation of energy, i.e., An + Rn + Tn = 1. In the development of equation (3.100) we have assumed that R1 is the same for light shining onto the top or the bottom of the sheet (ρ12 = ρ23 ), in other words, that equation (3.92) is valid. The above recursion formulae were first derived by Edwards [89] without the restriction of ρ12 = ρ23 . In a later paper Edwards [90] expanded the method to include wave interference effects for stacked thin films. Multiple sheets subject to mixed diffuse and collimated irradiation, but without interference effects, were analyzed by Mitts and Smith [91]. Example 3.6. Determine the normal transmissivity of a triple-glazed window for visible wavelengths. The window panes are thin sheets of soda–lime glass, separated by layers of air. Solution The reflectivity R1 and transmissivity T1 of a single sheet are readily calculated from equations (3.92) and (3.93). For thin sheets (e.g., curve 1 in Fig. 3-29) we have τ ≃ 1, and with n ≃ 1.5 (cf. Fig. 3-16), ρ = [(1.5 − 1)/(1.5 + 1)]2 = 0.04. Therefore, " # (1 − ρ)2 2ρ 2 × 0.04 R1 = ρ 1 + = = 0.0769, = 1+ρ 1 + 0.04 1 − ρ2 T1 =

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

(and A1 = 0, since we assumed τ ≃ 1). For two panes, from equations (3.100) and (3.101) with n = 2, ! T2 R1 0.92312 = 0.1429, R2 = R1 + 1 2 = 0.0769 1 + 1 − 0.07692 1 − R1 T2 =

T12 1 − R21

= 0.8571

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

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

R3 = R1 +

= 0.0769 +

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

Although they are valid, equations (3.89) and (3.90) are quite cumbersome for oblique incidence, in particular, if absorption cannot be neglected. Some calculations for nonabsorbing

3.9 SPECIAL SURFACES

1.0

κ d = 0 (per plate)

101

1 2

0.75

Number of covers

4

3

1.0 0.25

Transmissivity T

0.75

κ d = 0.0125 (per plate) 1.0

0

κ d = 0.0370 (per plate)

0.75

0.25

50

1.0

0

0.75

0.25 0

0.50

κ d = 0.0524 (per plate) 0.25 0

0.25 0

0

20 40 60 Angle of incidence, degrees

80

FIGURE 3-32 Transmissivities of 1, 2, 3, and 4 sheets of glass (n = 1.526) for different optical thicknesses per sheet, κd [93].

(for n = 1.5 [92] and for n = 1.526 [93]) and absorbing [93] (n = 1.526) multiple sheets of window glass have been carried out. Note that, for oblique incidence, the overall reflectivity and transmissivity are different for parallel- and perpendicular-polarized light. Even for unpolarized light the polarized components must be determined before averaging, as Rn =

1 (Rn⊥ + Rnk ), 2

Tn =

1 (Tn⊥ + Tnk ). 2

(3.103)

The results of the calculations by Duffie and Beckman [93] are given in graphical form in Fig. 3-32.

3.9

SPECIAL SURFACES

For many engineering applications it would be desirable to have a surface material available with very specific radiative property characteristics. For example, the net radiative heat gain of a solar collector is the difference between absorbed solar energy and radiation losses due to emission by the collector surface. While a black absorber plate would absorb all solar irradiation, it unfortunately would also lose a maximum amount of energy due to surface emission. An ideal solar collector surface has a maximum emittance for those wavelengths and directions over which solar energy falls onto the surface, and a minimum emittance for all other wavelengths and directions. On the other hand, a radiative heat rejector, such as the ones used by the U.S. Space Shuttle to reject excess heat into outer space, should have a high emittance at longer wavelengths, and a high reflectance for those wavelengths and directions with which sunshine falls onto the heat rejector. To a certain degree the radiative properties of a surface can be tailored toward desired characteristics. Surfaces that absorb and emit strongly over one wavelength range, and reflect strongly over the rest of the spectrum are called spectrally selective, while surfaces with tailored directional properties are known as directionally selective.

102

3 RADIATIVE PROPERTIES OF REAL SURFACES

1.0

White epoxy paint on aluminum

Hemispherical emittance ∋λ

0.8 Black nickel 0.6 Black chrome 0.4

301 stainless steel 0.2 Scale changes 0

0.5

1.0

3

5 7 Wavelength λ, µ m

11

15

19

23

FIGURE 3-33 Spectral, hemispherical reflectances of several spectrally selective surfaces [99].

q sun

θs ∋

Eb

Solar collector

Tcoll , ∋λ´ FIGURE 3-34 Solar irradiation on and emission from a solar collector plate.

An ideal, spectrally selective surface would be black (αλ = ǫλ = 1) over the wavelength range over which maximum absorption (or emission) is desired, and would be totally reflective (αλ = ǫλ = 0) beyond a certain cutoff wavelength λc , where undesirable emission (or absorption) would occur. Of course, in practice such behavior can only be approximated. Such an ideal surface is indicated by the long-dash line in Fig. 3-33. The performance of a selective surface is usually measured by the “α/ǫ-ratio,” where α is the total, directional absorptance of the material for solar irradiation, while ǫ is the total, hemispherical emittance for infrared surface emission. Consider a solar collector plate (Fig. 334), irradiated by the sun at an off-normal angle of θs . Making an energy balance (per unit area of the collector), we find 4 qnet = ǫσTcoll − αqsun cos θs ,

(3.104)

where the factor cos θs appears since qsun is solar heat flux per unit area normal to the sun’s rays. The total, hemispherical emittance may be related to spectral, hemispherical values through equation (3.10), while the total, directional absorptance is found from equation (3.23). Thus Z ∞ 1 ǫ= (3.105a) ǫλ (Tcoll , λ) Ebλ (Tcoll , λ) dλ, 4 σTcoll 0 Z ∞ Z ∞ 1 1 α= αλ (Tcoll , λ, θs ) qsun,λ dλ = αλ (Tcoll , λ, θs ) Ebλ (Tsun , λ) dλ, (3.105b) 4 qsun 0 σTsun 0

3.9 SPECIAL SURFACES

103

where we have made use of the fact that the spectral distribution of qsun is the same as the blackbody emission from the sun’s surface. Clearly, for optimum performance of a collector the solar absorptance should be maximum, while the infrared emittance should be minimum. Therefore, a large α/ǫ-ratio indicates a better performance for a solar collector. On the other hand, for radiative heat rejectors a minimum value for α/ǫ is desirable. Most selective absorbers are manufactured by coating a thin nonmetallic film onto a metal. Over most wavelengths the nonmetallic film is very transmissive and incoming radiation passes straight through to the metal interface with its very high reflectance. However, many nonconductors have spectral regions over which they do absorb appreciably without being strongly reflective (usually due to lattice defects or contaminants). The result is a material that acts like a strongly reflecting metal over most of the spectrum, but like a strongly absorbing nonconductor for selected wavelength ranges. A few examples of such selective surfaces are also given in Fig. 3-33. Black chrome (chrome-oxide coating) and black nickel (nickel-oxide coating) are popular solar collector materials, while epoxy paint may be used as an efficient solar energy rejector. If the coatings are extremely thin, interference effects can also be exploited to improve selectivity. For example, Martin and Bell [94] showed that a three-layer coating of SiO2 –Al–SiO2 on metallic substrates has a solar absorptance greater than 90%, but an infrared emittance of < 10%. Fan and Bachner [86] produced a coating for glass that raised its reflectance to > 80% for infrared wavelengths, without appreciably affecting solar transmittance (Fig. 3-30). The advantages of spectrally selective surface properties were first recognized by Hottel and Woertz [95]. With the growing interest in solar energy collection during the 1950s and 1960s, a number of selective coatings were developed, and the subject was discussed by Gier and Dunkle [96], and Tabor and coworkers [97, 98]. There are several compilations for radiative properties of selective absorbers [3,8,99]. A somewhat more detailed discussion about spectrally selective surface properties has been given by Duffie and Beckman [93]. Example 3.7. Let us assume that it is possible to manufacture a diffusely absorbing/emitting selective absorber with a spectral emittance ǫλ = ǫs = 0.05 for 0 < λ < λc and ǫλ = ǫc = 0.95 for λ > λc , where the cutoff wavelength can be varied through manufacturing methods. Determine the optimum cutoff wavelength for a solar collector with an absorber plate at 350 K that is exposed to solar irradiation of qsun = 1000 W/m2 at an angle of θs = 30◦ off-normal. What is the net radiative energy gain for such a collector? Solution A simple energy balance on the surface, using equations (3.9) and (3.41) leads to ′ (θs ) = ǫ Eb − α′ (θs ) H′ (θs ) qnet = E − Habs

where qnet > 0 if a net amount of energy leaves the surface, and qnet < 0 if energy is collected. Total, hemispherical emittance follows from equation (3.10) while total, directional absorptance is determined from equation (3.23). For our diffuse absorber we have α′λ (λ, θ) = ǫλ (λ) and # " Z λc Z ∞ Z ∞ (ǫc − ǫs ) 1 E (T , λ) dλ + ǫ E (T , λ) dλ = ǫ + Ebλ (Tcoll , λ) dλ, ǫ ǫ = s c s coll coll bλ bλ 4 4 σTcoll σTcoll 0 λc λc # " Z λc Z ∞ Z ∞ (ǫc − ǫs ) 1 α = Ebλ (Tsun , λ) dλ. E (T , λ) dλ + ǫ E (T , λ) dλ = ǫ + ǫ s sun c sun s bλ bλ 4 4 σTsun σTsun 0 λc λc Substituting these expressions into our energy balance leads to # Z ∞" qsun cos θs 4 Ebλ (Tsun , λ) dλ. Ebλ (Tcoll , λ) − qnet = ǫs (σTcoll − qsun cos θs ) + (ǫc − ǫs ) 4 σTsun λc Optimizing the value of λc implies finding a maximum for qnet . Therefore, from Leibniz’s rule (see, e.g., [81]), which states that Z b Z b(x) df d da db f (x, b) − f (x, a) + (x, y) dy, (3.106) f (x, y) dy = dx a(x) dx dx a dx

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3 RADIATIVE PROPERTIES OF REAL SURFACES

we find # " dqnet qsun cos θs E (T , λ ) = 0, = −(ǫc − ǫs ) Ebλ (Tcoll , λc ) − sun c bλ 4 dλc σTsun or Ebλ (Tcoll , λc ) =

qsun cos θs 4 σTsun

Ebλ (Tsun , λc ).

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

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

This transcendental equation needs to be solved by iteration. As a first guess one may employ Wien’s distribution, equation (1.18) (dropping two ‘−1’ terms), exp(C2 /λc Tcoll ) ≃ or exp



4 σTsun exp(C2 /λc Tsun ) qsun cos θs

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

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

λc ≃ C2



1



1

.

ln

Iterating the full Planck’s law leads to a cutoff wavelength of λc = 3.69 µm. Substituting these values into the expressions for emittance and absorptance,   ǫ = ǫs + (ǫc − ǫs ) 1 − f (λc Tcoll ) = 0.95 − 0.90 + 0.90 f (3.69 × 350) = 0.05 + 0.90 × 0.00413 = 0.054,

  α = ǫs + (ǫc − ǫs ) 1 − f (λc Tsun ) = 0.05 + 0.90 × f (3.69 × 5777) = 0.05 + 0.90 × 0.98785 = 0.939.

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

Surfaces can be made directionally selective by mechanically altering the surface finish on a microscale (microgrooves) or macroscale. For example, large V-grooves (large compared with the wavelengths of radiation) tend to reflect incoming radiation several times for near-normal incidence, as indicated in Fig. 3-35 (from Trombe and coworkers [101]) for an opening angle of γ = 30◦ , each time absorbing a fraction of the beam. The number of reflections decreases with increasing incidence angle, down to a single reflection for incidence angles θ > 90◦ − γ (or 60◦ in the case of Fig. 3-35). Hollands [102] has shown that this type of surface has a significantly higher normal emittance, which is important for collection of solar irradiation, than hemispherical emittance, which governs emission losses. A similarly shaped material, with flat black bottoms, was theoretically analyzed by Perlmutter and Howell [103]. Their analytical values for directional emittance were experimentally confirmed by Brandenberg and Clausen [27], as illustrated in Fig. 3-36. 8 Based on a blackbody at 5777 K. This number remains essentially unchanged for true, extraterrestrial solar irradiation [100], while the 95% fraction moves to even shorter wavelengths if atmospheric absorption is taken into account (cf. Fig. 1-3).

3.10 EXPERIMENTAL METHODS

105

Angle of incidence of radiation

30°

45°

60°

30°

Folded reflecting metal sheet

Angle of emission θ , degrees 0 15

FIGURE 3-35 Directional absorption and reflection of irradiation by a V-grooved surface [101].

Highly 18.2° reflecting

30 d

Black Theory Experiment

D

45

60

75

0

3.10

0.2

0.4 0.8 0.6 Directional emittance ∋ ´(θ )

90 l.0

FIGURE 3-36 Directional emittance of a grooved surface with highly reflective, specular sidewalls and near-black base. Results are for plane perpendicular to groove length. Theory (ρsides = ǫbase = 1) from [103], experiment [taken at λ = 8 µm with aluminum sidewalls and black paint base with ǫλ (8 µm) = 0.95] from [27].

EXPERIMENTAL METHODS

It is quite apparent from the discussion in the preceding sections that, although electromagnetic wave theory can be used to augment experimental data, it cannot replace them. While the spectral, bidirectional reflection function, equation (3.34), is the most basic radiation property of an opaque surface, to which all other properties can be related, it is rarely measured. Obtaining the bidirectional reflection function is difficult because of the low achievable signal strength. It is also impractical since it is a function of both incoming and outgoing directions and of wavelength and temperature. A complete description of the surface requires enormous amounts of data. In addition, the use of the bidirectional reflection function complicates the analysis to such a point that it is rarely attempted. If bidirectional data are not required it is sufficient, for an opaque material, to measure one of the following, from which all other ones may be inferred: absorptance, emittance, directional– hemispherical reflectance, and hemispherical–directional reflectance. Various different measurement techniques have been developed, which may be separated into three loosely-defined groups: calorimetric emission measurements, radiometric emission measurements, and reflection measurements. The interest in experimental methods was at its peak during the 1960s as a result of the advent of the space age. Compilations covering the literature of that period have been

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3 RADIATIVE PROPERTIES OF REAL SURFACES

given in two NASA publications [104, 105]. Interest waned during the 1970s and 1980s but has recently picked up again because of the development of better and newer materials operating at higher temperatures. Sacadura [106] has given an updated review of experimental methods. While measurement techniques vary widely from method to method, most of them employ similar optical components, such as light sources, monochromators, and detectors. Therefore, we shall begin our discussion of experimental methods with a short description of important optical components.

Instrumentation Radiative property measurements generally require a light source, a monochromator, a detector, and the components of the optical path, such as mirrors, lenses, beam splitters, optical windows, and so on. Depending on the nature of the experiment and/or detector, other accessories, such as optical choppers, may also be necessary. LIGHT SOURCES. Light sources are required for the measurement of absorption by, or reflection from, an opaque surface, as well as for the alignment of optical components in any spectroscopic system. In addition, light sources are needed for transmission and scattering measurements of absorbing/scattering media, such as gases, particles, and semitransparent solids, and liquids (to be discussed in later chapters). We distinguish between monochromatic and polychromatic light sources. Monochromatic sources. These types of sources operate through stimulated emission, producing light over an extremely narrow wavelength range. Their monochromaticity, low beam divergence, coherence, and high power concentration make lasers particularly attractive as light sources. While only invented some 30 years ago, there are today literally dozens of solid-state and gas lasers covering the spectrum between the ultraviolet and the far infrared. Although lasers are generally monochromatic, there are a number of gas lasers that can be tuned over a part of the spectrum by stimulating different transitions. For example, dye lasers (using large organic dye molecules as the lasing medium) may be operated at a large number of wavelengths in the range 0.2 µm < λ < 1 µm, while the common CO2 laser (usually operating at 10.6 µm) may be equipped with a movable grating, allowing it to lase at a large number of wavelengths in the range 9 µm < λ < 11 µm. Even solid-state lasers can be operated at several wavelengths through frequency-doubling. For example, the Nd-YAG laser, the most common solid-state laser, can be used at 1.064 µm, 0.532 µm, 0.355 µm, and 0.266 µm. Of particular importance for radiative property measurements is the helium–neon laser because of its low price and small size and because it operates in the visible at 0.633 µm (making it useful for optical alignment). A different kind of monochromatic source is the low-pressure gas discharge lamp, in which a low-density electric current passes through a low-pressure gas. Gas atoms and molecules become ionized and conduct the current. Electrons bound to the gas atoms become excited to higher energy levels, from which they fall again, emitting radiation over a number of narrow spectral lines whose wavelengths are characteristic of the gas used, such as zinc, mercury, and so on. Polychromatic sources. These usually incandescent light sources emit radiation by spontaneous emission due to the thermal excitation of source atoms and molecules, resulting in a continuous spectrum. The spectral distribution and total radiated power depend on the temperature, area, and emittance of the surface. Incandescent sources may be of the filament type (similar to an ordinary light bulb) or of the bare-element type. The quartz–tungsten–halogen lamp has a doped tungsten filament inside a quartz envelope, which is filled with a rare gas and a small amount of a halogen. Operating at a filament temperature greater than 3000 K, this lamp produces a near-blackbody spectrum with maximum emission below 1 µm. However, because of the transmission characteristics of quartz (which is the same as fused silica, Fig. 3-28), there is no appreciable emission beyond 3 µm. Bare-element sources are either rods

3.10 EXPERIMENTAL METHODS

107

100

Spectral irradiance mW/cm2 µm at 50 cm from source

150W Xe Hg

10 1.0

Xe

200W Hg

Quartz tungsten halogen lamp 100 W 3400K

10–1

Nernst glower 22 W 1550K

10–2 Globar 100 W 1000 K

10–3 10– 4 0.2

0.3

0.4

0.5

0.6

0.7

0.9 1 2 3 4 5 6 7 8 9 10 11 12 13 0.8 Wavelength λ, µ m

FIGURE 3-37 Spectral irradiation on a distant surface from various incandescent light sources.

of silicon carbide, called globars, or heating wires embedded in refractory oxides, called Nernst glowers. Globars operate at a temperature of 1000 K and produce an almost-gray spectrum with a maximum around 2.9 µm. Nernst glowers operate at temperatures up to 1500 K, with a somewhat less ideal spectral distribution. The irradiation onto a distant surface from different incandescent sources is shown in Fig. 3-37. None of the light sources shown in Fig. 3-37 has a truly “black” spectral distribution, since their output is influenced by their spectral emittance. In most experiments this is of little importance since, in general, sample and reference signals (coming from the same spectral source) are compared. If a true blackbody source is required (primarily for calibration of instruments) blackbody cavity sources are available from a number of manufacturers. In these sources a cylindrical and/or conical cavity, made of a high-temperature, high-emittance material (such as silicon carbide) is heated to a desired temperature. Radiation leaving the cavity, also commonly called Hohlraum (German for “hollow space”), is essentially black (cf. Table 5.1). The brightest conventional source of optical radiation is the high-pressure gas discharge lamp, which combines the characteristics of spontaneous and stimulated emission. The lamp is similar to a low-pressure gas discharge source, but with high current density and gas pressure. This configuration results in an arc with highly excited atoms and molecules forming a plasma. While the hot plasma emits as an incandescent source, ionized atoms emit over substantially broadened spectral lines, resulting in a mixed spectrum (Fig. 3-37). Commonly used gases for such arc sources are xenon, mercury, and deuterium. SPECTRAL SEPARATORS. Spectral radiative properties can be measured over part of the spectrum in one of two ways: (i) Measurements are made using a variety of monochromatic light sources, which adequately represent the desired part of the spectrum, or (ii) a polychromatic source is used together with a device that allows light of only a few select wavelengths to reach the detector. Such devices may consist of simple optical filters, manually driven or motorized monochromators, or highly sophisticated FTIR (Fourier Transform InfraRed) spectrometers. Optical filters. These are multilayer thin-film devices that selectively transmit radiation only over desired ranges of wavelengths. Bandpass filters transmit light only over a finite, usually narrow, wavelength region, while edge filters transmit only above or below certain cutoff or edge wavelengths. Bandpass filters consist of a series of thin dielectric films that, at each interface, partially reflect and partially transmit radiation (cf. Fig. 2-13). The spacing between layers is such that beams of the desired wavelength are, after multiple reflections within the layers, in phase with the transmitted beam (constructive interference). Other wavelengths are rejected

108

3 RADIATIVE PROPERTIES OF REAL SURFACES

(a)

(b)

FIGURE 3-38 Schematic of spectral separation with (a) a transparent prism, (b) a diffraction grating.

because they destructively interfere with one another. Bandpass filters for any conceivable wavelength between the ultraviolet and the midinfrared are routinely manufactured. Edge filters operate on the same principle, but are more complex in design. Monochromators. These devices separate an incoming polychromatic beam into its spectral components. They generally consist of an entrance slit, a prism or grating that spreads the incoming light according to its wavelengths, and an exit slit, which allows only light of desired wavelengths to escape. If a prism is used, it is made of a highly-transparent material with a refractive index that varies slightly across the spectrum (cf. Fig. 3-16). As shown in Fig. 3-38a, the incoming radiant energy is separated into its constituent wavelengths since, by Snell’s law, the prism bends different wavelengths (with different refractive index) by different amounts. Rotating the prism around an axis allows different wavelengths to escape through the exit slit. Instead of a prism one can use a diffraction grating to separate the wavelengths of incoming light, employing the principle of constructive and destructive interference [107], as schematically indicated in Fig. 3-38b. Until a few years ago all monochromators employed salt prisms, while today almost all systems employ diffraction gratings, since they are considerably cheaper and simpler to handle (salt prisms tend to be hygroscopic, i.e., they are attacked by the water vapor in the surrounding air). However, diffraction gratings have the disadvantages that their spectral range is more limited (necessitating devices with multiple gratings), and they may give erroneous readings due to higher-order signals (frequency-doubling). FTIR spectrometers. These instruments collect the entire radiant energy (i.e., comprising all FTIR spectrometer wavelengths) after reflection from a moving mirror. The measured intensity depends on the position of the moving mirror owing to constructive and destructive interference. This signal is converted by a computer through an inverse Fast Fourier Transform into a power vs. wavelength plot. The spectral range of FTIRs is limited only by the choice of beam splitters and detectors, and is comparable to that of prism monochromators. However, while monochromators generally require several minutes to collect data over their entire spectral range, the FTIR is able to do this in a fraction of a second. Detailed descriptions of the operation of FTIRs may be found in books on the subject, such as the one by Griffiths and de Haseth [108]. DETECTORS. In a typical spectroscopic experiment the detector measures the intensity of incoming radiation due to transmission through, emission from, or reflection by, a sample. This irradiation may be relatively monochromatic (i.e., covers a very narrow wavelength range after having passed through a filter or monochromator), or may be polychromatic (for total emittance measurements, or if an FTIR is used). In either case, the detector converts the beam’s power into an electrical signal, which is amplified and recorded. The performance of detectors is measured by certain criteria, which are generally functions of several operating conditions, such as wavelength, temperature, modulating frequency, bias voltage, and gain of any internal amplifier. The response time (τ) is the time for a detector’s output to reach 1 − 1/e = 63% of its final value, after suddenly being subjected to constant irradiation. The linearity range of a detector is the range of input power over which the output signal is a linear function of the input. The noise equivalent power (NEP) is the radiant energy rate in watts that is necessary to give an output signal equal to the rms noise output from the detector. More widely used is the

109

3.10 EXPERIMENTAL METHODS

Black coating

Deposited photoconductor

Pyroelectric material

Electrodes Substrate

(a) FIGURE 3-39 Schematic of (a) a pyroelectric detector, (b) a photoconductive detector.

(b)

reciprocal of NEP, the detectivity (D). The detectivity is known to vary inversely with the square root of the detector area, AD , while the signal noise is proportional to the square root of the amplifier’s noise-equivalent bandwidth ∆ f (in Hz). Thus, a normalized detectivity (D*) is defined to allow comparison between different types of detectors regardless of their detector areas and amplifier bandwidths as D∗ = (AD ∆ f )1/2 D.

(3.107)

Depending on how the incoming radiation interacts with the detector material, detectors are grouped into thermal and photon (or quantum) detectors. Thermal detectors. These devices convert incident radiation into a temperature rise. This temperature change is measured either through one or more thermocouples, or by using the pyroelectric effect. A single, usually blackened (to increase absorptance) thermocouple is the simplest and cheapest of all thermal detectors. However, it suffers from high amplifier noise and, therefore, limited detectivity. One way to increase output voltage and detectivity is to connect a number of thermocouples in series (typically 20 to 120), constituting a thermopile. Thermopiles can be manufactured economically through thin-film processes. Pyroelectric detectors are made of crystalline materials that have permanent electric polarization. When heated by irradiation, the material expands and changes its polarization, which causes a current to flow in a circuit that connects the detector’s top and bottom surfaces, as shown in the schematic of Fig. 3-39a. Since the change in temperature produces the current, pyroelectric detectors respond only to pulsed or chopped irradiation. They respond to changes in irradiation much more rapidly than thermocouples and thermopiles, and are not affected by steady background radiation. Photon detectors. These absorb the energy of incident radiation with their electrons, producing free charge carriers (photoconductive and photovoltaic detectors) or even ejecting electrons from the material (photoemissive detectors). In photoconductive and photovoltaic detectors the production of free electrons increases the electrical conductivity of the material. In the photoconductive mode an applied voltage, or reverse bias, causes a current that is proportional to the strength of irradiation to flow, as schematically shown in Fig. 3-39b. In the photovoltaic mode no bias is applied and, closing the electric circuit, a current flows as a result of the excitation of electrons (as in the operation of photovoltaic, or solar, cells). Photovoltaic detectors have greater detectivity, while photoconductive detectors exhibit extremely fast response times. For optimum performance each mode requires slightly different design, although a single device may be operated in either mode. Typical semiconductor materials used for photovoltaic and photoconductive detectors are silicon (Si), germanium (Ge), indium antimonide (InSb), mercury cadmium telluride (HgCdTe),9 lead sulfide and selenide (PbS and PbSe), and cadmium sulfide (CdS). While most semiconductor detectors have a single detector element, many of them today 9

Mercury–Cadmium–Telluride detectors are also commonly referred to as MCT detectors.

110

3 RADIATIVE PROPERTIES OF REAL SURFACES

Photomultipliers : D* ~ 1015–1016 λ ~ 0.2–1.0 µ m

Normalized detectivity D*, cm Hz1/2/ W

1013

1012

Si (PV) 295 K InAs (PV) 77 K

Ge (PV) 295 K 1011 PbS (PC ) 295 K

InSb (PV) 77 K

1010 HgCdTe (PC) 77 K 109 Thermopile and pyroelectric 295 K 108

0.2

0.5

1

2 5 Wavelength λ, µ m

10

20

50

FIGURE 3-40 Typical spectral ranges and normalized detectivities for various detectors.

are also available as linear arrays and surface arrays (up to 512 × 512 elements), which—when combined with a monochromator—allows for ultra-fast data acquisition at many wavelengths. The most basic photoemissive device is a photodiode, in which high-energy photons (ultraviolet to near infrared) cause emission of electrons from photocathode surfaces placed in a vacuum. Applying a voltage causes a current that is proportional to the intensity of incident radiation to flow. The signal of a vacuum photodiode is amplified in a photomultiplier by fitting it with a series of anodes (called dynodes), which produce secondary emission electrons and a current. The latter is an order-of-magnitude higher than the original photocurrent. Thermal detectors generally respond evenly across the entire spectrum, while photon detectors have limited spectral response but higher detectivity and faster response times. The normalized detectivity of several detectors is compared in Fig. 3-40. The spectral response of photon detectors can be tailored to a degree by varying the relative amounts of detector material components. The response time of thermal detectors is relatively slow, normally in the order of milliseconds, while the response time of photon detectors ranges from microseconds to a few nanoseconds. The detectivity is often increased by cooling the detector thermoelectrically (to −30◦ C), with dry ice (195 K), or by attaching it to a liquid-nitrogen Dewar flask (77 K). OTHER COMPONENTS. In a spectroscopic experiment light from a source and/or sample is guided toward the detector by a number of mirrors and lenses. Plane mirrors are employed to bend the beam path while curved mirrors are used to focus an otherwise diverging beam onto a sample, the monochromator entrance slit, or the detector. Today’s optical mirrors provide extremely high reflectivities (> 99.5%) over the entire spectrum of interest. While focusing mirrors are generally preferable for a number of reasons, sometimes lenses need to be used for focusing. The most important drawbacks of lenses are that they tend to have relatively large reflection losses and their spectral range (with high transmissivity) is limited. While antireflection coatings can be applied, these coatings are generally only effective over narrow spectral ranges as a result of interference effects. Common lens materials for the infrared are zinc selenide (ZnSe), calcium fluoride (CaF2 ), germanium (Ge), and others. Sometimes it is necessary to split a beam into two portions (e.g., to create a reference beam that does not pass over the sample) using a beam splitter. Beam splitters are made of the same material as lenses,

3.10 EXPERIMENTAL METHODS

111

1 2

2 3

11 5

6

7

8

9 10

4 1. Vacuum feed-through flange 2. Coolant fill and vent tubes 3. Stainless steel vacuum jacket 4. Copper chamber walls 5. Vacuum inlet 6. Power leads 7. Thermocouple leads 8. Sample and heating element 9. Magnet-operated shutter 10. Sample viewing port 11. Radiation shields

FIGURE 3-41 Typical setup for calorimetric emission measurements [110].

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

Calorimetric Emission Measurement Methods If only knowledge of the total, hemispherical emittance of a surface is required, this is most commonly determined by measuring the net radiative heat loss or gain of an isolated specimen [109–125]. Figure 3-41 shows a typical experimental setup, which was used by Funai [110]. The specimen is suspended inside an evacuated test chamber, the walls of which are coated with a near-black material. The chamber walls are cooled, while the specimen is heated electrically, directly (metallic samples), through a metal substrate (nonconducting samples), or by some other means. Temperatures of the specimen and chamber wall are monitored by thermocouples. The emittance of the sample can be determined from steady-state [109–117] or transient measurements [111, 118–125]. In the steady-state method the sample is heated to, and kept at, a desired temperature by passing the appropriate current through the heating element. The total, hemispherical emittance may then be calculated by equating electric heat input to the specimen with the radiative heat loss from the specimen to the surroundings, or ǫ(T) =

I2 R , As σ(Ts4 − Tw4 )

(3.108)

where I2 R is the dissipated electrical power, As is the exposed surface area of the specimen, and Ts and Tw are the temperatures of specimen and chamber walls, respectively. As will be discussed in Chapter 5, equation (3.108) assumes that the surface area of the chamber is much larger than As and/or that the emittance of the chamber wall is near unity [cf. equation (5.36)]. In the transient calorimetric technique the current is switched off when the desired temperature has been reached, and the rates of loss of internal energy and radiative heat loss are equated, or ǫ(T) = −

ms cs dTs /dt , As σ(Ts4 − Tw4 )

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

(3.109)

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3 RADIATIVE PROPERTIES OF REAL SURFACES

RCA #7102 Photomultiplier detector

Quartz lens

Thermocouple detector

Exit slit

Blackbody furnace

Ellipsoidal mirror Parabolic mirror 90° fold

Littrow mirror

Chopper

Specimen furnace

Prism

Entrance slit

FIGURE 3-42 Emissometer with separate reference blackbody and two optical paths [126].

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

Iλ (T, λ, θ, ψ) . Ibλ (T, λ)

(3.110)

The comparison blackbody may be a separate blackbody kept at the same temperature, or it may be an integral part of the sample chamber. The latter is generally preferred at high temperatures, where temperature control is difficult, and for short wavelengths, where small deviations in temperatures can cause large inaccuracies. Separate reference blackbody. In this method a blackbody, usually a long, cylindrical, isothermal cavity with an L/D-ratio larger than 4, is kept separate from the sample chamber, while both are heated to the same temperature. Radiation coming from this Hohlraum is essentially black (cf. Table 5.1). The control system keeps the sample and blackbody at the same temperature by monitoring temperature differences with a differential thermocouple and taking corrective action whenever necessary. To monitor sample and blackbody emission via an identical optical path, either two identical paths have to be constructed, or sample and blackbody must be alternately placed into the single optical path. In the former method, identical paths are formed either through two sets of optics [126], or by moving optical components back and forth [39]. Figure 3-42 shows an example of a system with two different optical paths [126], while Fig. 3-43 is an example of a linearly actuated blackbody/sample arrangement [127]. It is also possible to combine blackbody and sample, and the device is rotated or moved back-and-forth inside a single furnace [128]. Markham and coworkers [129] mounted sample/reference blackbody individually on a turntable, heated them with a torch, and measured the directional, spectral emittance of sandblasted aluminum (up to 750 K), alumina (1300 to 2200 K), fused quartz (900 K), and sapphire (1000 K) with an FTIR spectrometer. Other materials measured with the separate reference blackbody technique include the normal, spectral emittance of solid and liquid silicon just below and above the melting point [130], and of a collection of 30 metals and alloys at temperatures up to 1200◦ C [131, 132].

3.10 EXPERIMENTAL METHODS

113

Blackbody oven B Flat mirror Perkin-Elmer Model 112 infrared spectrophotometer

A ✕

OA

φ OA

Concave mirror Support table

A´´ S Sample oven Oven motion

OA - Optic axis φ - Field of view of spectrometer (about 12° ) A - Entrance slit A´´ - Image of "A" in concave mirror (sample) B - Radiant aperture of blackbody S - Sample FIGURE 3-43 Emissometer with separate reference blackbody and linearly actuated sample/blackbody arrangement [127].

Integrated reference blackbody. At high temperatures it is preferable to incorporate the reference blackbody into the design of the sample furnace. If the sample rests at the bottom of a deep isothermal, cylindrical cavity, the radiation leaving the sample (by emission and reflection) corresponds to that of a black surface. If the hot side wall is removed or replaced by a cold one, radiation leaving the sample is due to emission only. Taking the ratio of the two signals then allows the determination of the spectral, directional emittance from equation (3.110). Removing the reflection component from the signal may be achieved in one of two ways. Several researchers have used a tubular furnace with the sample mounted on a movable rod [133–135]. When the sample is deep inside the furnace the signal corresponds to a blackbody. The sample is then rapidly moved to the exit of the furnace and the signal is due to emission alone. Disadvantages of the method are (i) maintaining isothermal conditions up to close to the end of the tube, (ii) keeping the sample at the same temperature after displacement, and (iii) stress on the high-temperature sample due to the rapid movement. In the approach of Vader and coworkers [136] and Postlethwait et al. [137], reflection from the sample is suppressed by freely dropping a cold tube into the blackbody cavity. A schematic of the apparatus of Postlethwait et al. is shown in Fig. 3-44. Once the cold tube has been dropped, measurements must be taken rapidly (in a few seconds’ time), before substantial heating of the drop tube (and cooling of the sample). Vader and coworkers obtained spectral measurements by placing various filters in front of their detector, performing a number of drops for each sample temperature. Postlethwait employed an FTIR spectrometer, allowing them to measure the entire spectral range from 1 µm to 9 µm in a single drop. In a method more akin to the separate blackbody technique, Havstad and colleagues [128] incorporated a small blackbody cavity into a tungsten crucible (holding liquid metal samples). The entire assembly is then moved to have the optics focus on sample or blackbody, respectively.

Reflection Measurements Reflection measurements are carried out to determine the bidirectional reflection function, the directional–hemispherical reflectance, and the hemispherical–directional reflectance. The latter two provide indirect means to determine the directional absorptance and emittance of opaque specimens, in particular, if sample temperatures are too low for emission measurements.

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3 RADIATIVE PROPERTIES OF REAL SURFACES

Optical path M

M

L W

S - Sample L - Lens M - Mirror W - Window

Drop tube

Furnace W M

FTIR spectrometer

W

S

SiC tube

Temperature controller FIGURE 3-44 Schematic of a drop-tube emissometer [137].

T

M

SM

M ES

SM

SM M M S SM

P

A - Globar source SM - Spherical mirror S - Sample ES - Entrance slit P - Salt prism M - Mirror T - Thermopile

Monochromator

A

FIGURE 3-45 Schematic of the bidirectional reflection measurement apparatus of Birkebak and Eckert [138].

BIDIRECTIONAL REFLECTION MEASUREMENTS. If the bidirectional reflection behavior of a surface is of interest, the bidirectional reflection function, ρ′′ , must be measured directly, λ by irradiating the sample with a collimated beam from one direction and collecting the reflected intensity over various small solid angles. A sketch of an early apparatus used by Birkebak and Eckert [138] and Torrance and Sparrow [139] is shown in Fig. 3-45. Radiation from a globar A travels through a diaphragm to a spherical mirror SM, which focuses it onto the test sample S. A pencil of radiation reflected from the sample into the desired direction is collected by another spherical mirror and focused onto the entrance slit of the monochromator, in which the wavelengths are separated by the rock salt prism P, and the signal is recorded by the thermopile T. The test sample is mounted on a multiple-yoke apparatus, which allows independent rotation around three perpendicular axes. The resulting measurements are relative (i.e., abso-

3.10 EXPERIMENTAL METHODS

115

Sample temperature control

Resistance heaters Sample

Nickel cavity

Insulation

Cooling coils

Exit port

FIGURE 3-46 Schematic of a heated cavity reflectometer [146].

lute values can only be obtained by calibrating the apparatus with a known standard in place of the test sample). Example measurements for magnesium oxide are shown in Fig. 3-5 [12]. More recently built devices use sophisticated, multiple-degree-of-freedom sample mounts as well as FTIR spectrometers, such as the one of Ford and coworkers [140], who measured the bidirectional reflectances of diffuse gold and grooved nickel. The main problem with bidirectional reflection measurements is the low level of reflected radiation that must be detected (particularly in off-specular directions), even with the advent of FTIR spectrometers and highly sensitive detectors. Consequently, a number of designs have employed strong monochromatic laser sources to overcome this problem, for example, [141–145]. An overview of the different methods to determine directional–hemispherical and hemispherical–directional reflectances has been given by Touloukian and DeWitt [6]. The different types of experiments may be grouped into three categories, heated cavity reflectometers, integrating sphere reflectometers, and integrating mirror reflectometers, each having their own ranges of applicability, advantages, and shortcomings. HEATED CAVITY REFLECTOMETERS. The heated cavity reflectometer [6, 146–148] (sometimes known as the Gier–Dunkle reflectometer after its inventors [148]) consists of a uniformly heated enclosure fitted with a water-cooled sample holder and a viewport, as schematically shown in Fig. 3-46. Since the sample is situated within a more or less closed isothermal enclosure, the intensity striking it from any direction is essentially equal to the blackbody intensity Ibλ (Tw ) (evaluated at the cavity-wall temperature, Tw ). Images of the sample and a spot on the cavity wall are alternately focused onto the entrance slit of a monochromator. The signal from the specimen corresponds to emission (at the sample’s temperature, Ts ) plus reflection of the cavity-wall’s blackbody intensity, Ibλ (Tw ). Since the signal from the cavity wall is proportional to Ibλ (Tw ), the ratio of the two signals corresponds to

✄ ρλ ′ (ˆs) Ibλ (Tw ) + ǫλ′ (ˆs) Ibλ (Ts ) Is . = Iw Ibλ (Tw )

(3.111)

If the sample is relatively cold (Ts ≪ Tw ), emission may be neglected and the device simply measures the hemispherical–directional reflectance. For higher specimen temperatures, and for

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3 RADIATIVE PROPERTIES OF REAL SURFACES

Sample

Sphere Reference area

MgO surface Detector Beam splitter

Monochromatic beam

Lamp Screen Sample

To detector

Reference Diffuse plate (a) FIGURE 3-47 Typical integrating sphere reflectometers: (a) direct mode, (b) indirect mode.

Sample holder rotatable about vertical (b)

an opaque surface with diffuse irradiation, from equations (3.42), (3.39) and (3.44), ✄ ✄ ρ ′λ (ˆs) = ρ′λ (ˆs) = 1 − α′λ (ˆs) = 1 − ǫλ′ (ˆs), and

" # Ibλ (Ts ) Is = 1 − ǫλ′ (ˆs) 1 − . Iw Ibλ (Tw )

(3.112) (3.113)

The principal source of error in this method is the difficulty in making the entire cavity reasonably isothermal and (as a consequence) making the reference signal proportional to a blackbody at the cavity-wall temperature. To make these errors less severe the method is generally only used for low sample temperatures. INTEGRATING SPHERE REFLECTOMETERS. These devices are most commonly employed for reflectance measurements [147,149–159] and are available commercially in a variety of forms, either as separate instruments or already incorporated into spectrophotometers. A good early discussion of different designs was given by Edwards and coworkers [154]. The integrating sphere may be used to measure hemispherical–directional or directional–hemispherical reflectance, depending on whether it is used in indirect or direct mode. Schematics of integrating spheres operating in the two modes are shown in Fig. 3-47. The ideal device is coated on its inside with a material of high and perfectly diffuse reflectance. The most common material in use is smoked magnesium oxide, which reflects strongly and very diffusely up to λ ≃ 2.6 µm (cf. Fig. 3-5). Other materials, such as “diffuse gold” [155–158], have been used to overcome the wavelength limitations. The strong, diffuse reflectance, together with the spherical geometry, assures that any external radiation hitting the surface of the sphere is converted into a perfectly diffuse intensity field due to many diffuse reflections. In the direct method the sample is illuminated directly by an external source, as shown in Fig. 3-47a. All of the reflected radiation is collected by the sphere and converted into a diffuse intensity field, which is measured by a detector. Similar readings are then taken on a comparison standard of known reflectance, under the same conditions. The sample may be removed and replaced by the standard (substitution method); or there may be separate sample and standard holders, which are alternately irradiated by the external source (comparison method), the latter being generally preferred. In the indirect method a spot on the sphere surface is irradiated while the detector measures the intensity reflected by the sample (or the comparison standard) directly. Errors in integrating sphere measurements are primarily caused by imperfections of the surface coating (imperfectly diffuse reflectance), losses out of apertures, and unwanted irradiation onto the detector (direct reflection from the sample in the direct mode, direct reflection from the externally-irradiated spot on the sphere in the indirect mode). Because of temperature sensitivity of the diffuse coatings, integrating-sphere measurements have mostly been limited

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117

From chopper and monochromatic source Entrance port

Specular reflecting hemisphere

Aluminum first surfaces Collimated monochromatic beam Plane mirror

First conjugate focal point (sample)

Second conjugate focal point (detector) (a)

Sample Detector

Incident energy Reflected energy (b)

Spherical mirror

Ellipsoidal mirror Monochromator Specimen Detector at second focal point (c)

FIGURE 3-48 Design schematics of several integrating mirror reflectometers, using (a) a hemispherical, (b) a paraboloidal, and (c) an ellipsoidal mirror.

to moderate temperature levels. However, for monochromatic and high-speed FTIR measurements it is possible to rapidly heat up only the sample by a high-power source, such as a laser, as was done by Zhang and Modest [160]. INTEGRATING MIRROR REFLECTOMETERS. An alternative to the integrating sphere is a similar design utilizing an integrating mirror. Mirrors in general have high reflectivities in the infrared and are much more efficient than integrating spheres and, hence, are highly desirable in the infrared where the energy of the light source is low. On the other hand, it is difficult to collect the radiant energy, reflected by the sample into the hemisphere above it, into a parallel beam of small cross-section. For this reason, an integrating mirror reflectometer requires a large detector area. There are three types of integrating mirrors: hemispherical [161], paraboloidal [146, 162] and ellipsoidal [163–169]. Schematics of the three different types are shown in Fig. 3-48. The principle of operation of all three is the same, only the shape of the mirror is different. Each of these mirrors has two conjugate focal points, i.e., if a point source of light is placed at one focal point, all radiation will, after reflection off the mirror, fall onto the second focal point. Thus, in the integrating mirror technique an external beam is focused onto the sample, which is located at one of the focal points, through a small opening in the mirror. Radiation reflected from the sample into any direction will be reflected by the integrating mirror and is then collected by the detector located at the other focal point. This technique yields the directional–hemispherical reflectance of the sample, after comparison with a reference signal. Alternatively, one of the focal points can hold a blackbody source, with the ellipsoidal mirror focusing the energy onto the sample at the second focal point. Radiation leaving the sample is then probed through a small hole in the

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mirror and spectrally resolved and detected by a monochromator or FTIR spectrometer, yielding the hemispherical–directional reflectance of the sample [169]. Sources for error in the integrating mirror method are absorption by the mirror, energy lost through the entrance port, nonuniform angular response of detectors, and energy missing the detector owing to mirror aberrations. To minimize aberrations, ellipsoids are preferable over hemispheres. The method has generally been limited to relatively large wavelengths, > 2.5 µm (because of mirror limitations), and to moderate temperatures. Designs allowing sample temperatures up to about 1000◦ C have been reported by Battuello and coworkers [167], Ravindra and colleagues [170], and by Freeman et al. [171], while the torch-heated sample of Markham and coworkers’ design [169] allows sample temperatures up to 2000◦ C. In general, integrating mirrors are somewhat less popular than integrating spheres because mirrors are more sensitive to flux losses and misalignment errors.

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Woertz: “The performance of flat-plate solar-heat collectors,” Transactions of ASME, Journal of Heat Transfer, vol. 64, pp. 91–104, 1942. 96. Gier, J. T., and R. V. Dunkle: “Selective spectral characteristics as an important factor in the efficiency of solar collectors,” in Transactions of the Conference on the Use of Solar Energy, vol. 2, University of Arizona Press, Tucson, AZ, p. 41, 1958. 97. Tabor, H., J. A. Harris, H. Weinberger, and B. Doron: “Further studies on selective black coatings,” Proceedings of the UN Conference on New Sources of Energy, vol. 4, p. 618, 1964. 98. Tabor, H.: “Selective surfaces for solar collectors,” in Low Temperature Engineering Applications of Solar Energy, ASHRAE, 1967. 99. Edwards, D. K., K. E. Nelson, R. D. Roddick, and J. T. Gier: “Basic studies on the use and control of solar energy,” Technical Report 60-93, The University of California, Los Angeles, CA, 1960. 100. Thekaekara, M. P.: “Solar energy outside the earth’s atmosphere,” Solar Energy, vol. 14, pp. 109–127, 1973. 101. Trombe, F., M. Foex, and V. LePhat: “Research on selective surfaces for air conditioning dwellings,” Proceedings of the UN Conference on New Sources of Energy, vol. 4, pp. 625–638, 1964. 102. Hollands, K. G. T.: “Directional selectivity, emittance, and absorptance properties of vee corrugated specular surfaces,” Solar Energy, vol. 7, no. 3, pp. 108–116, 1963. 103. Perlmutter, M., and J. R. Howell: “A strongly directional emitting and absorbing surface,” ASME Journal of Heat Transfer, vol. 85, no. 3, pp. 282–283, 1963. 104. Richmond, J. C. (ed.): Measurement of Thermal Radiation Properties of Solids, NASA SP-31, 1963. 105. Katzoff, S. (ed.): Symposium on Thermal Radiation Properties of Solids, NASA SP-55, 1964. 106. Sacadura, J.-F.: “Measurement techniques for thermal radiation properties,” in Proceedings of the Ninth International Heat Transfer Conference, Hemisphere, Washington, D.C., pp. 207–222, 1990. 107. Hutley, M. C.: Diffraction Gratings, Academic Press, New York, 1982. 108. Griffiths, P. R., and J. A. de Haseth: Fourier Transform Infrared Spectrometry, vol. 83 of Chemical Analysis, John Wiley & Sons, New York, 1986. 109. Sadler, R., L. Hemmerdinger, and I. Rando: “A device for measuring total hemispherical emittance,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 217–223, 1963. 110. Funai, A. I.: “A multichamber calorimeter for high-temperature emittance studies,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 317–327, 1963. 111. McElroy, D. L., and T. G. Kollie: “The total hemispherical emittance of platinum, columbium-1%, zirconium, and polished and oxidized iron-8 in the range 100◦ C to 1200◦ C,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 365–379, 1963. 112. Moore, V. S., A. R. Stetson, and A. G. Metcalfe: “Emittance measurements of refractory oxide coatings up to 2900◦ C,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 527–533, 1963. 113. Nyland, T. W.: “Apparatus for the measurement of hemispherical emittance and solar absorptance from 270◦ C to 650◦ C,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 393–401, 1963. 114. Zerlaut, G. A.: “An apparatus for the measurement of the total normal emittance of surfaces at satellite temperatures,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 275–285, 1963. 115. Chen, S. H. P., and S. C. Saxena: “Experimental determination of hemispherical total emittance of metals as a function of temperature,” Ind. Eng. Chem. Fundam., vol. 12, no. 2, pp. 220–224, 1973. 116. Jody, B. J., and S. C. Saxena: “Radiative heat transfer from metal wires: Hemispherical total emittance of platinum,” Journal of Physics E: Scientific Instruments, vol. 9, pp. 359–362, 1976. 117. Taylor, R. E.: “Determination of thermophysical properties by direct electrical heating,” High Temperatures High Pressures, vol. 13, pp. 9–22, 1981. 118. Gordon, G. D., and A. London: “Emittance measurements at satellite temperatures,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 147–151, 1963. 119. Rudkin, R. L.: “Measurement of thermal properties of metals at elevated temperatures,” in Temperature, Its Measurement and Control in Science and Industry, vol. 3, part 2, Reinhold Publishing Corp., New York, pp. 523–534, 1962. 120. Gaumer, R. E., and J. V. Stewart: “Calorimetric determination of infrared emittance and the α/ǫ ratio,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 127–133, 1963. 121. Butler, C. P., and R. J. Jenkins: “Space chamber emittance measurements,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 39–43, 1963.

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122. Butler, C. P., and E. C. Y. Inn: “A method for measuring total hemispherical and emissivity of metals,” in First Symposium - Surface Effects on Spacecraft Materials, John Wiley & Sons, New York, pp. 117–137, 1960. 123. Smalley, R., and A. J. Sievers: “The total hemispherical emissivity of copper,” Journal of the Optical Society of America, vol. 68, pp. 1516–1518, 1978. 124. Ramanathan, K. G., and S. H. Yen: “High-temperature emissivities of copper, aluminum and silver,” Journal of the Optical Society of America, vol. 67, pp. 32–38, 1977. 125. Masuda, H., and M. Higano: “Measurement of total, hemispherical emissivities of metal wires by using transient calorimetric techniques,” ASME Journal of Heat Transfer, vol. 110, pp. 166–172, 1988. 126. Limperis, T., D. M. Szeles, and W. L. Wolfe: “The measurement of total normal emittance of three nuclear reactor materials,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 357–364, 1963. 127. Fussell, W. B., and F. Stair: “Preliminary studies toward the determination of spectral absorption coefficients of homogeneous dielectric material in the infrared at elevated temperatures,” in Symposium on Thermal Radiation of Solids, ed. S. Katzoff, NASA SP-55, pp. 287–292, 1965. 128. Havstad, M. A., W. I. McLean, and S. A. Self: “Apparatus for the measurement of the optical constants and thermal radiative properties of pure liquid metals from 0.4 to 10 µm,” Review of Scientific Instruments, vol. 64, pp. 1971–1978, 1993. 129. Markham, J. R., P. R. Solomon, and P. E. Best: “An FT-IR based instrument for measuring spectral emittance of material at high temperature,” Review of Scientific Instruments, vol. 61, no. 12, pp. 3700–3708, 1990. 130. Takasuka, E., E. Tokizaki, K. Terashima, and S. O. Kazutaka: “Emissivity of liquid silicon in visible and infrared regions,” Journal of Applied Physics, vol. 81, pp. 6384–6389, 1997. 131. Kobayashi, M., M. Otsuki, H. Sakate, F. Sakuma, and A. Ono: “System for measuring the spectral distribution of normal emissivity of metals with direct current heating,” International Journal of Thermophysics, vol. 20, no. 1, 1999. 132. Kobayashi, M., A. Ono, M. Otsuki, H. Sakate, and F. Sakuma: “Database of normal spectral emissivities of metals at high temperatures,” International Journal of Thermophysics, vol. 20, no. 1, pp. 299–308, 1999. 133. Knopken, S., and R. Klemm: “Evaluation of thermal radiation at high temperatures,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 505–514, 1963. 134. Bennethum, W. H.: “Thin film sensors and radiation sensing techniques for measurement of surface temperature of ceramic components,” in HITEMP Review, Advanced High Temperature Engine Materials Technology Program, NASA CP-10039, 1989. 135. Atkinson, W. H., and M. A. Cyr: “Sensors for temperature measurement for ceramic materials,” in HITEMP Review, Advanced High Temperature Engine Materials Technology Program, NASA CP-10039, pp. 287–292, 1989. 136. Vader, D. T., R. Viskanta, and F. P. Incropera: “Design and testing of a high-temperature emissometer for porous and particulate dielectrics,” Review of Scientific Instruments, vol. 57, no. 1, pp. 87–93, 1986. 137. Postlethwait, M. A., K. K. Sikka, M. F. Modest, and J. R. Hellmann: “High temperature normal spectral emittance of silicon carbide based materials,” Journal of Thermophysics and Heat Transfer, vol. 8, no. 3, pp. 412–418, 1994. 138. Birkebak, R. C., and E. R. G. Eckert: “Effect of roughness of metal surfaces on angular distribution of monochromatic reflected radiation,” ASME Journal of Heat Transfer, vol. 87, pp. 85–94, 1965. 139. Torrance, K. E., and E. M. Sparrow: “Off-specular peaks in the directional distribution of reflected thermal radiation,” ASME Journal of Heat Transfer, vol. 88, pp. 223–230, 1966. 140. Ford, J. N., K. Tang, and R. O. Buckius: “Fourier transform infrared system measurement of the bidirectional reflectivity of diffuse and grooved surfaces,” ASME Journal of Heat Transfer, vol. 117, no. 4, pp. 955–962, 1995. 141. Hsia, J. J., and J. C. Richmond: “A high resolution laser bidirectional reflectometer,” Journal Research of N.B.S., vol. 80A, no. 2, pp. 189–220, 1976. 142. De Silva, A. A., and B. W. Jones: “Bidirectional spectral reflectance and directional-hemispherical spectral reflectance of six materials used as absorbers of solar energy,” Solar Energy Materials, vol. 15, pp. 391–401, 1987. 143. Greffet, J.-J.: “Design of a fully automated bidirectional laser reflectometer; applications to emissivity measurement,” in Proceedings of SPIE on Stray Light and Contamination in Optical Systems, ed. R. P. Breault, vol. 967, pp. 184–191, 1989. 144. Al Hamwi, M., and J.-F. Sacadura: “M´ethode de d´etermination des propri´etes radiatives spectrales et directionnelles, dans le proche et moyen i.r., de surfaces opaques m´etalliques et non-m´etalliques,” Proceedings of JITH ’89, pp. 126–136, November 1989. 145. Zaworski, J. R., J. R. Welty, and M. K. Drost: “Measurement and use of bi-directional reflectance,” International Journal of Heat and Mass Transfer, vol. 39, pp. 1149–1156, 1996. 146. Dunkle, R. V.: “Spectral reflection measurements,” in First Symposium - Surface Effects on Spacecraft Materials, John Wiley & Sons, New York, pp. 117–137, 1960. 147. Hembach, R. J., L. Hemmerdinger, and A. J. Katz: “Heated cavity reflectometer modifications,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 153–167, 1963. 148. Gier, J. T., R. V. Dunkle, and J. T. Bevans: “Measurement of absolute spectral reflectivity from 1.0 to 15 microns,” Journal of the Optical Society of America, vol. 44, pp. 558–562, 1954. 149. Fussell, W. B., J. J. Triolo, and F. A. Jerozal: “Portable integrating sphere for monitoring reflectance of spacecraft coatings,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 103–116, 1963.

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150. Drummeter, L. F., and E. Goldstein: “Vanguard emittance studies at NRL,” in First Symposium - Surface Effects on Spacecraft Materials, John Wiley & Sons, New York, pp. 152–163, 1960. 151. Snail, K. A., and L. M. Hangsen: “Integrating sphere designs with isotropic throughput,” Applied Optics, vol. 28, pp. 1793–1799, May 1989. 152. Egan, W. G., and T. Hilgeman: “Integrating spheres for measurements between 0.185 µm and 12 µm,” Applied Optics, vol. 14, pp. 1137–1142, May 1975. 153. Kneissl, G. J., and J. C. Richmond: “A laser source integrating sphere reflectometer,” Technical Report NBS-TN439, National Bureau of Standards, 1968. 154. Edwards, D. K., J. T. Gier, K. E. Nelson, and R. D. Roddick: “Integrating sphere for imperfectly diffuse samples,” Journal of the Optical Society of America, vol. 51, pp. 1279–1288, 1961. 155. Willey, R. R.: “Fourier transform infrared spectrophotometer for transmittance and diffuse reflectance measurements,” Applied Spectroscopy, vol. 30, pp. 593–601, 1976. 156. Richter, W.: “Fourier transform reflectance spectrometry between 8000 cm−1 (1.25 µm) and 800 cm−1 (12.5 µm) using an integrating sphere,” Applied Spectroscopy, vol. 37, pp. 32–38, 1983. 157. Gindele, K., M. Kohl, and M. Mast: “Spectral reflectance measurements using an integrating sphere in the ¨ infrared,” Applied Optics, vol. 24, pp. 1757–1760, 1985. 158. Richter, W., and W. Erb: “Accurate diffuse reflection measurements in the infrared spectral range,” Applied Optics, vol. 26, no. 21, pp. 4620–4624, November 1987. 159. Sheffer, D., U. P. Oppenheim, D. Clement, and A. D. Devir: “Absolute reflectometer for the 0.8–2.5 µm region,” Applied Optics, vol. 26, no. 3, pp. 583–586, 1987. 160. Zhang, Z., and M. F. Modest: “Temperature-dependent absorptances of ceramics for Nd:YAG and CO2 laser processing applications,” ASME Journal of Heat Transfer, vol. 120, no. 2, pp. 322–327, 1998. 161. Janssen, J. E., and R. H. Torborg: “Measurement of spectral reflectance using an integrating hemisphere,” in Measurement of Thermal Radiation Properties of Solids, ed. J. C. Richmond, NASA SP-31, pp. 169–182, 1963. 162. Neher, R. T., and D. K. Edwards: “Far infrared reflectometer for imperfectly diffuse specimens,” Applied Optics, vol. 4, pp. 775–780, 1965. 163. Neu, J. T.: “Design, fabrication and performance of an ellipsoidal spectroreflectometer,” NASA CR 73193, 1968. 164. Dunn, S. T., J. C. Richmond, and J. F. Panner: “Survey of infrared measurement techniques and computational methods in radiant heat transfer,” Journal of Spacecraft and Rockets, vol. 3, pp. 961–975, July 1966. 165. Heinisch, R. P., F. J. Bradar, and D. B. Perlick: “On the fabrication and evaluation of an integrating hemiellipsoid,” Applied Optics, vol. 9, no. 2, pp. 483–489, 1970. 166. Wood, B. E., P. G. Pipes, A. M. Smith, and J. A. Roux: “Hemi-ellipsoidal mirror infrared reflectometer: Development and operation,” Applied Optics, vol. 15, no. 4, pp. 940–950, 1976. 167. Battuello, M., F. Lanza, and T. Ricolfi: “Infrared ellipsoidal mirror reflectometer for measurements between room temperature and 1000◦ C,” High Temperature, vol. 18, pp. 683–688, 1986. 168. Snail, K. A.: “Reflectometer design using nonimaging optics,” Applied Optics, vol. 26, no. 24, pp. 5326–5332, 1987. 169. Markham, J. R., K. Kinsella, R. M. Carangelo, C. R. Brouillette, M. D. Carangelo, P. E. Best, and P. R. Solomon: “Bench top Fourier transform infrared based instrument for simultaneously measuring surface spectral emittance and temperature,” Review of Scientific Instruments, vol. 64, no. 9, pp. 2515–2522, 1993. 170. Ravindra, N. M., S. Abedrabbo, W. Chen, F. M. Tong, A. K. Nanda, and A. C. Speranza: “Temperature-dependent emissivity of silicon-related materials and structures,” IEEE Transactions on Semiconductor Manufacturing, vol. 11, no. 1, pp. 30–39, 1998. 171. Freeman, R. K., F. A. Rigby, and N. Morley: “Temperature-dependent reflectance of plated metals and composite materials under laser irradiation,” Journal of Thermophysics and Heat Transfer, vol. 14, no. 3, pp. 305–312, 2000. 172. Hale, G. M., and M. R. Querry: “Optical constants of water in the 200 nm to 200 µm wavelength region,” Applied Optics, vol. 12, pp. 555–563, 1973.

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



0.9 cos θ, 0.2,

λ < 1 µm, λ > 1 µm.

(a) What is the total, hemispherical emittance of the material? (b) If the sun irradiates this surface at an angle of θ = 60◦ off-normal, what is the relevant total absorptance? (c) What is the net radiative energy gain or loss of this surface (per unit time and area)?

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3.3 For optimum performance a solar collector surface has been treated so that, for the spectral, directional emittance ( ) 0.9 cos 2θ, θ < 45◦ ǫλ′ = , λ < 2 µm, 0.0 θ > 45◦ =

0.1,

all θ,

λ > 2 µm.

For solar incidence of 15◦ off-normal and a collector temperature of 400 K, what is the relevant ratio of absorptance to emittance? 3.4 A long, cylindrical antenna of 1 cm radius on an Earth-orbiting satellite is coated with a material whose emittance is  0, λ < 1 µm, ǫλ′ = cos θ, λ ≥ 1 µm. Find the absorbed energy per meter length. (Assume irradiation is from the sun only, and in a direction normal to the antenna’s axis; neglect the Earth and stars.) 3.5 The spectral, hemispherical emittance of a (hypothetical) metal may be approximated by the relationship  0.5, λ < λc = 0.5 µm, ǫλ = 0.5λc /λ, λ > λc (independent of temperature). Determine the total, hemispherical emittance of this material using (a) Planck’s law, and (b) Wien’s distribution, for a surface temperature of (i) 300 K, and (ii) 1000 K. How accurate is the prediction using Wien’s distribution? 3.6 A treated metallic surface is used as a solar collector material; its spectral, directional emittance may be approximated by  0.5 µm/λ, θ < 45◦ , ǫλ′ = 0, θ > 45◦ . What is the relevant α/ǫ-ratio for near normal solar incidence if Tcoll ≃ 600 K? 3.7 A surface sample with ǫλ′ =



0.9 cos θ, λ < 2 µm, 0.2, λ > 2 µm,

is irradiated by three tungsten lights as shown. The tungsten lights may be approximated by black spheres at T = 2000 K fitted with mirrors to produce parallel light beams aimed at the sample. Neglecting background radiation, determine the absorptance of the sample.

Tungsten lamps

45° 45° Sample

3.8 An antenna of a satellite may be approximated by a long half cylinder, which is exposed to sunshine as shown in the sketch. The antenna has a high conductivity (i.e., is isothermal), and is coated with the material of Fig. 3-36, i.e., the material may be assumed to be gray with the following directional characteristics: 2R  ◦ 0.9, 0 ≤ θ < 40 , ǫλ′ = 0, θ > 40◦ . Determine the equilibrium temperature of the antenna, assuming it exchanges heat only with the sun (and cold outer space).

qsun=1367 W/m2

PROBLEMS

3.9 A large isothermal plate (temperature T = 400 K) is exposed to a long monochromatic (λ = 1 µm) line source as shown. The strength of the line source is Q′ (W/m length of source) = hσT 4 , spreading equally into all directions. The plate has a spectral, directional emittance of ( π 0.9 cos2 θ, λ < 2.5 µm, 0≤θ< . ǫλ′ = 0.1, λ > 2.5 µm, 2

125

Q'

θ h dx

For a general location, x, determine relevant absorptance, emittance, and the net local heat flux qnet (x), which must be supplied to/removed from the plate to keep it isothermal at T. 3.10 A large isothermal plate (temperature T = 400 K) is exposed to a long tungsten–halogen line source as shown in the sketch next to Problem 3.9. The strength of the line source is Q′ = 1000 W/m length of source, spreading equally into all directions, and it has the spectral distribution of a blackbody at 4000K. The plate has a spectral, directional emittance of ( π 0.8 cos θ, λ < 3µm, 0≤θ< ǫ′λ = 0.2, λ > 3µm. 2 For a general location, x, give an expression for local irradiation H, determine the relevant absorptance and emittance, and give an expression for the net local heat flux qnet (x) that must be supplied to/removed from the plate to keep it isothermal at T. 3.11 An isothermal disk (temperature T = 400 K) is exposed to a small black spherical source (temperature Ts = 4000 K) as shown. The strength of the source is Q (W), spreading equally into all directions. The plate has a spectral, directional emittance of ( π 0.9 cos θ, λ < 4µm, 0≤θ< ǫ′λ = 0.3, λ > 4µm. 2

Q

θ H

DR

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



QL

= 10 3 W/cm2

0.15 cos θ, λ < 6 µm, 0.8 cos2 θ, λ > 6 µm,

determine the relevant total absorptance and emittance.

2D 2D

3.13 A metal (m2 = 50 − 50 i) is coated with a dielectric (m1 = 2 − 0 i), which is exposed to vacuum. (a) What is the range of possible directions from which radiation can impinge on the metal? (b) What is the normal reflectance of the dielectric–metal interface? (c) What is the (approximate) relevant hemispherical reflectance for the dielectric–metal interface?

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3.14 For a certain material, temperature, and wavelength the spectral, hemispherical emittance has been measured as ǫλ . Estimate the refractive index of the material under these conditions, assuming the material to be (a) a dielectric with ǫλ = 0.8, (b) a metal in the infrared with ǫλ = 0.2 (the Hagen–Rubens relation being valid). 3.15 It can be derived from electromagnetic wave theory that ǫλ 4 1 ≃ − ǫnλ ǫnλ 3 4

for

ǫnλ ≪ 1.

Determine ǫλ for metals with ǫnλ ≪ 1 as a function of wavelength and temperature. 3.16 A solar collector surface with emittance ǫλ′ =



0.9 cos θ, 0.2,

λ < 2 µm, λ > 2 µm,

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

ǫnλ =



0.9, 0.1,

λ < 2.2 µm, λ > 2.2 µm,

where ǫnλ is the normal, spectral emittance. Determine the net radiative flux on the collector. 3.18 A small plate, insulated at the bottom, is heated by irradiation from a defocused CO2 laser beam (wavelength 10.6 µm) with an incidence angle of 30◦ off-normal. The radiative properties of the surface are ( 0.2 cos2 θ, λ < 3 µm, ǫλ′ = 0.8 cos θ, λ > 3 µm. The strength of the laser beam is 1300 W/m2 . Neglecting losses due to natural convection, determine the temperature of the plate. Note: For such weak laser irradiation levels the heating effect is relatively small. 3.19 A thin disk, insulated at the bottom, is irradiated by a CO2 laser (λ = 10.6 µm) as shown. The top surface is exposed to a low temperature (300K) environment. Assume that the entire disk surface is uniformly irradiated with qL = 5 MW/m2 and that the specific heat/area of the disk is ρcp δ = 2 kJ/m2 K. The disk is at ambient temperature when the laser is turned on. The emittance of the disk surface is ǫ′λ =



0.2, 0.9 cos θ,

qL qL =60°

θL

d

λ < 6 µm, λ > 6 µm.

(a) Indicate how to calculate the temperature history of the disk. (b) Determine the initial heating rate (in K/s) at t = 0. (c) What is the steady state temperature of the disk? (This is expected to be very high, say > 3000 K.) 3.20 Determine the total, normal emittance of copper, silver, and gold for a temperature of 1500 K. Check your results by comparing with Fig. 3-8.

PROBLEMS

127

3.21 Determine the total, hemispherical emittance of copper, silver, and gold for a temperature of 1500 K. Check your results by comparing with Fig. 3-11. 3.22 A polished platinum sphere is heated until it is glowing red. An observer is stationed a distance away, from where the sphere appears as a red disk. Using the various aspects of electromagnetic wave theory and/or Fig. 3-9 and Table 3.3, explain how the brightness of emitted radiation would vary across the disk, if observed with (a) the human eye, (b) an infrared camera. 3.23 Two aluminum plates, one covered with a layer of white enamel paint, the other polished, are directly facing the sun, which is irradiating the plates with 1000 W/m2 . Assuming that convection/conduction losses of the plates to the environment at 300 K can be calculated by using a heat transfer coefficient of 10 W/m2 K, and that the back sides of the plates are insulated, estimate the equilibrium temperature of each plate. 3.24 Consider a metallic surface coated with a dielectric layer. (a) Show that the fraction of energy reflected at the vacuum–dielectric interface is negligible (n1 = 1.2; k1 = 0). (b) Develop an expression for the normal, spectral emittance for the metal substrate, similar to the Hagen–Rubens relationship. (c) Develop an approximate relation for the directional, spectral emittance of the metal substrate for large wavelengths and moderate incidence angles, say θ < 75◦ . 3.25 A plate of metal with n2 = k2 = 100 is covered with a dielectric as shown. The dielectric has an absorption band such that n1 = 2, and k1 = 1 for 0.2 µm < λ < 2 µm and k1 = 0 elsewhere. The dielectric is thick enough, such that any light traveling through it of wavelengths 0.2 µm < λ < 2 µm is entirely absorbed before it reaches the metal.

vacuum, n0 = 1 dielectric, n1, k1

metal, n2, k2

(a) What is the total, normal emittance of the composite if its temperature is 400 K? (b) What is the total, normal absorptance if the sun shines perpendicularly onto the composite? 3.26 Estimate the total, normal emittance of α-SiC for a temperature of (i) 300 K, (ii) 1000 K. You may assume the spectral, normal emittance to be independent of temperature. 3.27 Estimate the total, hemispherical emittance of a thick slab of pure silicon at room temperature. 3.28 Estimate and compare the total, normal emittance of room temperature aluminum for the surface finishes given in Fig. 3-25. 3.29 A satellite orbiting Earth has part of its (flat) surface coated with spectrally selective “black nickel,” which is a diffuse emitter and whose spectral emittance may be approximated by ǫλ =



0.9, 0.25,

λ < 2 µm, λ > 2 µm.

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



0.1, 0.9,

λ < 2 µm, λ > 2 µm.

3.31 Estimate the spectral, hemispherical emittance of the grooved materials shown in Fig. 3-36. Repeat Problem 3.29 for these materials, assuming them to be gray.

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3 RADIATIVE PROPERTIES OF REAL SURFACES

3.32 Repeat Problem 1.7 for a sphere covered with the grooved material of Fig. 3-36, whose directional, spectral emittance may be approximated by  0.9, 0 ≤ θ < 40◦ , ǫλ′ = 0.0, 40◦ < θ < 90◦ . Assume the material to be gray. 3.33 A solar collector consists of a metal plate coated with “black nickel.” The collector is irradiated by the sun with a strength of qsol = 1000 W/m2 from a direction that is θ = 30◦ from the surface normal. On its top the surface loses heat by radiation and by free convection (heat transfer coefficient h1 = 10 W/m2 K), both to an atmosphere at Tamb = 20◦ C. The bottom surface delivers heat to the collector fluid (h2 = 50 W/m2 K), which flows past the surface at Tfluid = 20◦ C. What is the equilibrium temperature of the collector plate? How much energy (per unit area) is collected (i.e., carried away by the fluid)? Discuss the performance of this collector. Assume black nickel to be a diffuse emitter. 3.34 Make a qualitative plot of temperature vs. the total hemispherical emittance of: (a) a 3 mm thick sheet of window glass, (b) polished aluminum, and (c) an ideal metal that obeys the Hagen–Rubens relation. 3.35 A horizontal sheet of 5 mm thick glass is covered with a 2 mm thick layer of water. If solar radiation is incident normal to the sheet, what are the transmissivity and reflectivity of the water/glass layer at λ1 = 0.6 µm and λ2 = 2 µm? For water mH2 O (0.6 µm) = 1.332 − 1.09 × 10−8 i, mH2 O (2 µm) = 1.306 − 1.1 × 10−3 i [172]; for glass mglass (0.6 µm) = 1.517 − 6.04 × 10−7 i, mglass (2 µm) = 1.497 − 5.89 × 10−5 i [82]. 3.36 A solar collector plate of spectral absorptivity αcoll = 0.90 is fitted with two sheets of 5 mm thick glass as shown in the adjacent sketch. What fraction of normally incident solar radiation 5mm is absorbed by the collector plate at a wavelength of 0.6 µm? 5mm At 0.6 µm mglass = 1.517 − 6.04 × 10−7 i [82].

Sunshine Glass Glass Collector

CHAPTER

4 VIEW FACTORS

4.1

INTRODUCTION

In many engineering applications the exchange of radiative energy between surfaces is virtually unaffected by the medium that separates them. Such (radiatively) nonparticipating media include vacuum as well as monatomic and most diatomic gases (including air) at low to moderate temperature levels (i.e., before ionization and dissociation occurs). Examples include spacecraft heat rejection systems, solar collector systems, radiative space heaters, illumination problems, and so on. In the following four chapters we shall consider the analysis of surface radiation transport, i.e., radiative heat transfer in the absence of a participating medium, for different levels of complexity. It is common practice to simplify the analysis by making the assumption of an idealized enclosure and/or of ideal surface properties. The greatest simplification arises if all surfaces are black: for such a situation no reflected radiation needs to be accounted for, and all emitted radiation is diffuse (i.e., the intensity leaving a surface does not depend on direction). The next level of difficulty arises if surfaces are assumed to be gray, diffuse emitters (and, thus, absorbers) as well as gray, diffuse reflectors. The vast majority of engineering calculations are limited to such ideal surfaces, which are the topic of Chapter 5. If the reflective behavior of a surface deviates strongly from a diffuse reflector (e.g., a polished metal, which reflects almost like a mirror) one may often approximate the reflectance to consist of a purely diffuse and a purely specular component. This situation is discussed in Chapter 6. However, if greater accuracy is desired, if the reflectance cannot be approximated by purely diffuse and specular components, or if the assumption of a gray surface is not acceptable, a more general approach must be taken. A few such methods are briefly outlined in Chapter 7. As discussed in Chapter 1 thermal radiation is generally a long-range phenomenon. This is always the case in the absence of a participating medium, since photons will travel unimpeded from surface to surface. Therefore, performing a thermal radiation analysis for one surface implies that all surfaces, no matter how far removed, that can exchange radiative energy with one another must be considered simultaneously. How much energy any two surfaces exchange depends in part on their size, separation distance, and orientation, leading to geometric functions known as view factors. In the present chapter these view factors are developed for gray, diffusely radiating (i.e., emitting and reflecting) surfaces. However, the view factor is a very basic function that will also be employed in the analysis of specular reflectors as well as for the analysis for surfaces with arbitrary emission and reflection properties. Making an energy balance on a surface element, as shown in Fig. 4-1, we find

129

130

4 VIEW FACTORS

Reflection Irradiation H

ρH Emission E

q FIGURE 4-1 Surface energy balance.

q = qemission − qabsorption = E − αH.

(4.1)

In this relation qemission and qabsorption are absolute values with directions as given by Fig. 4-1, while q is the net heat flux supplied to the surface, as defined in Chapter 1 by equation (1.38). According to this definition q is positive if the heat is coming from inside the wall material, by conduction or other means (q > 0), and negative if going from the enclosure into the wall (q < 0). Alternatively, the heat flux may be expressed as q = qout − qin = (qemission + qreflection ) − qirradiation = (E + ρH) − H,

(4.2)

which is, of course, the same as equation (4.1) since, for opaque surfaces, ρ = 1−α. The irradiation H depends, in general, on the level of emission from surfaces far removed from the point under consideration, as schematically indicated in Fig. 4-2a. Thus, in order to make a radiative energy balance we always need to consider an entire enclosure rather than an infinitesimal control volume (as is normally done for other modes of heat transfer, i.e., conduction or convection). The enclosure must be closed so that irradiation from all possible directions can be accounted for, and the enclosure surfaces must be opaque so that all irradiation is accounted for, for each direction. In practice, an incomplete enclosure may be closed by introducing artificial surfaces. An enclosure may be idealized in two ways, as indicated in Fig. 4-2b: by replacing a complex geometrical shape with a few simple surfaces, and by assuming surfaces to be isothermal with constant (i.e., average) heat flux values across them. Obviously, the idealized enclosure approaches the real enclosure for sufficiently small isothermal subsurfaces.

Tk , ∋ k

Tn , ∋ n T1 , ∋ 1 dA

T2 , ∋ 2

T(r), ∋ (r ) r 0 (a) (b) FIGURE 4-2 (a) Irradiation from different locations in an enclosure, (b) real and ideal enclosures for radiative transfer calculations.

4.2 DEFINITION OF VIEW FACTORS

4.2

131

DEFINITION OF VIEW FACTORS

To make an energy balance on a surface element, equation (4.1), the irradiation H must be evaluated. In a general enclosure the irradiation will have contributions from all visible parts of the enclosure surface. Therefore, we need to determine how much energy leaves an arbitrary surface element dA′ that travels toward dA. The geometric relations governing this process for “diffuse” surfaces (for surfaces that absorb and emit diffusely, and also reflect radiative energy diffusely) are known as view factors. Other names used in the literature are configuration factor, angle factor, and shape factor, and sometimes the term diffuse view factor is used (to distinguish from specular view factors for specularly reflecting surfaces; see Chapter 6). The view factor between two infinitesimal surface elements dAi and dA j , as shown in Fig. 4-3, is defined as

dFdAi −dA j ≡

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

,

(4.3)

where the word “directly” is meant to imply “on a straight path, without intervening reflections.” This view factor is infinitesimal since only an infinitesimal fraction can be intercepted by an infinitesimal area. From the definition of intensity and Fig. 4-3 we may determine the heat transfer rate from dAi to dA j as I(ri )(dAi cos θi ) dΩ j = I(ri ) cos θi cos θj dAi dA j /S2 ,

(4.4)

where θi (or θj ) is the angle between the surface normal nˆ i (or nˆ j ) and the line connecting dAi and dA j (of length S). The total radiative energy leaving dAi into the hemisphere above it is J = E + ρH, where J is called the radiosity. Since the surface emits and reflects diffusely both E and ρH obey equation (1.33), and the outgoing flux may be related to intensity by   J(ri ) dAi = E(ri ) + ρ(ri ) H(ri ) dAi = πI(ri ) dAi .

Note that the radiative intensity away from dAi , due to emission and/or reflection, does not depend on direction. Therefore, the view factor between two infinitesimal areas is dFdAi −dA j =

cos θi cos θj πS2

dA j .

(4.5)

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

(nˆ i · si j )(nˆ j · s ji ) πS4

dA j .

(4.6)

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

(4.7)

Often, enclosures are idealized to consist of a number of finite isothermal subsurfaces, as indicated in Fig. 4-2b. Therefore, we should like to expand the definition of the view factor to include radiative exchange between one infinitesimal and one finite area, and between two finite areas. Consider first the exchange between an infinitesimal dAi and a finite A j , as shown in Fig. 4-4. The total energy leaving dAi toward all of A j is, from equation (4.4),

I(ri ) dAi

Z

cos θi cos θj Aj

S2

dA j ,

132

4 VIEW FACTORS

rj

rj

nj θ j

nj θ j dAj

dAj Aj

S

S

ni θ i

ni θ i

ri

ri dAi

dAi

FIGURE 4-4 Radiative exchange between one infinitesimal and one finite surface element.

FIGURE 4-3 Radiative exchange between two infinitesimal surface elements.

while the total energy leaving the dAi into all directions remains unchanged. Thus, we find Z cos θi cos θj FdAi −A j = dA j , (4.8) πS2 Aj which is now finite since the intercepting surface, A j , is finite. Next we consider the view factor from A j to the infinitesimal dAi . The amount of radiation leaving all of A j toward dAi is, from equation (4.4) (after switching subscripts i and j), Z cos θi cos θj dAi dA j , I(r j ) S2 Aj and the total amount leaving A j into all directions is Z π I(r j ) dA j . Aj

Thus, we find the view factor between surfaces A j and dAi is , Z Z cos θi cos θj dFA j −dAi = dA j dAi π I(r j ) I(r j ) dA j , S2 Aj Aj

(4.9)

which is infinitesimal since the intercepting surface, dAi , is infinitesimal. The view factor in equation (4.9)—unlike equations (4.5) and (4.8)—is not a purely geometric parameter since it depends on the radiation field I(r j ). However, for an ideal enclosure as shown in Fig. 4-2b, it is usually assumed that the intensity leaving any surface is not only diffuse but also does not vary across the surface, i.e., I(r j ) = I j = const. With this assumption equation (4.9) becomes Z cos θi cos θj 1 dFA j −dAi = dA j dAi . (4.10) A j Aj πS2 Comparing this with equation (4.8) we find another law of reciprocity, with A j dFA j −dAi = dAi FdAi −A j ,

(4.11)

4.2 DEFINITION OF VIEW FACTORS

133

rj nj θ j

dAj Aj

S

ni θ i

ri

Ai dAi

FIGURE 4-5 Radiative exchange between two finite surfaces.

subject to the restriction that the intensity leaving A j does not vary across the surface. Finally, we consider radiative exchange between two finite areas Ai and A j as depicted in Fig. 4-5. The total energy leaving Ai toward A j is, from equation (4.4), Z

Ai

Z

I(ri )

cos θi cos θj S2

Aj

dA j dAi ,

and the view factor follows as , Z Z Z cos θi cos θj I(ri ) I(ri ) dAi . dA dA FAi −A j = j i π S2 Ai Aj Ai

(4.12)

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

(4.14)

which is now subject to the condition that the radiation intensities leaving Ai and A j must both be constant across their respective surfaces. In a somewhat more compact notation, the law of reciprocity may be summarized as dAi dFdi−d j = dA j dFd j−di ,

(4.15a)

dAi Fdi− j = A j dF j−di ,

(I j = const),

(4.15b)

Ai Fi− j = A j F j−i ,

(Ii , I j = const).

(4.15c)

The different levels of view factors may be related to one another by Z Fdi− j = dFdi−dj ,

(4.16a)

Aj

Fi− j =

1 Ai

Z

Fdi− j dAi . Ai

(4.16b)

134

4 VIEW FACTORS

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

K X

Fi−( j,k) , with A j =

k=1

K X

A(j,k) .

(4.17)

k=1

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

Fdi− j =

N X

Fi− j = 1.

(4.18)

j=1

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

4.3 METHODS FOR THE EVALUATION OF VIEW FACTORS The calculation of a radiative view factor between any two finite surfaces requires the solution to a double area integral, or a fourth-order integration. Such integrals are exceedingly difficult to evaluate analytically except for very simple geometries. Even numerical quadrature may often be problematic because of singularities in the integrand, and because of excessive CPU time requirements. Therefore, considerable effort has been directed toward tabulation and the development of evaluation methods for view factors. Early tables and charts for simple configurations were given by Hamilton and Morgan [1], Leuenberger and Pearson [2], and Kreith [3]. Fairly extensive tabulations were given in the books by Sparrow and Cess [4] and Siegel and Howell [5]. Siegel and Howell also give an exhaustive listing of sources for more involved view factors. The most complete tabulation is given in a catalogue by Howell [6, 7], the latest version of which can also be accessed on the Internet via http://www.engr.uky.edu/rtl/Catalog/. A number of commercial and noncommercial computer programs for their evaluation are also available [8–18], and a review of available numerical methods has been given by Emery and coworkers [19]. Some experimental methods have been discussed by Jakob [20] and Liu and Howell [21]. Within the present book Appendix D gives view factor formulae for an extensive set of geometries. Self-contained Fortran/C++/Matlabr programs viewfactors are included in Appendix F for the evaluation of all view factors listed in Appendix D [these programs call a function view, which may also be used from within other programs]. Radiation view factors may be determined by a variety of methods. One possible grouping of different approaches could be: 1. Direct integration: (i) analytical or numerical integration of the relations given in the previous section (surface integration); (ii) conversion of the relations to contour integrals, followed by analytical or numerical integration (contour integration). 2. Statistical determination: View factors may be determined through statistical sampling with the Monte Carlo method. 3. Special methods: For many simple shapes integration can be avoided by employing one of the following special methods: (i) view factor algebra, i.e., repeated application of the rules of reciprocity and the summation relationship;

4.4 AREA INTEGRATION

135

z n θz

y

θy θx

x

r

FIGURE 4-6 Unit normal and direction cosines for a surface element.

0

(ii) crossed-strings method: a simple method for evaluation of view factors in twodimensional geometries; (iii) unit sphere method: a powerful method for view factors between one infinitesimal and one finite area; (iv) inside sphere method: a simple method for a few special shapes. All of the above methods will be discussed in the following pages, except for the Monte Carlo method, which is treated in considerable detail in Chapter 8.

4.4

AREA INTEGRATION

To evaluate equation (4.5) or to carry out the integrations in equations (4.8) and (4.13) the integrand (i.e., cos θi , cos θj , and S) must be known in terms of a local coordinate system that describes the geometry of the two surfaces. While the evaluation of the integrand may be straightforward for some simple configurations, it is desirable to have a more generally applicable formula at one’s disposal. Using an arbitrary coordinate origin, a vector pointing from the origin to a point on a surface may be written as ˆ r = xˆı + yˆ + zk, (4.19) where ˆı, ˆ, and kˆ are unit vectors pointing into the x-, y-, and z-directions, respectively. Thus the vector from dAi going to dA j is determined (see Fig. 4-5) as ˆ si j = −s ji = r j − ri = (x j − xi )ˆı + (y j − yi )ˆ + (z j − zi )k.

(4.20)

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

(4.21)

We will now assume that the local surface normals are also known in terms of the unit vectors ˆ or, from Fig. 4-6, ˆı, ˆ, and k, ˆ nˆ = l ˆı + mˆ + nk,

(4.22)

where l, m, and n are the direction cosines for the unit vector n, ˆ i.e., l = nˆ · ˆı = cos θx is the cosine of the angle θx between nˆ and the x-axis, etc. We may now evaluate cos θi and cos θj as nˆ i · si j

i 1h (x j − xi )li + (y j − yi )mi + (z j − zi )ni , S S i nˆ j · s ji 1h cos θj = = (xi − x j )l j + (yi − y j )m j + (zi − z j )n j . S S

cos θi =

=

(4.23a) (4.23b)

136

4 VIEW FACTORS

d A strip2 dA2 z n2

y

θ2 S

θ1

n2 u2

b

d A strip1

n1

α

dA1

x1

x

a

FIGURE 4-7 View factor for strips on an infinitely long groove.

Example 4.1. Consider the infinitely long (−∞ < y < +∞) wedge-shaped groove as shown in Fig. 4-7. The groove has sides of widths a and b and an opening angle α. Determine the view factor between the narrow strips shown in the figure. Solution After placing the coordinate system as shown in the figure, we find z1 = 0, x2 = u2 cos α, and z2 = u2 sin α, leading to S2 = (x1 − u2 cos α)2 + (y1 − y2 )2 + u22 sin2 α = (x21 − 2x1 u2 cos α + u22 ) + (y1 − y2 )2 = S20 + (y1 − y2 )2 , where S0 is the projection of S in the x-z-plane and is constant in the present problem. The two surface normals are readily determined as ˆ nˆ 1 = k,

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

leading to cos θ1 = u2 sin α/S, cos θ2 = [(x1 −u2 cos α) sin α + u2 sin α cos α] /S = x1 sin α/S. For illustrative purposes we will first calculate dFd1−strip 2 from equation (4.8), and then dFstrip 1−strip 2 from equation (4.16). Thus dFd1−strip 2 =

Z

du2 cos θ1 cos θ2 dA2 = π πS2

dAstrip 2

x1 u2 sin2 α du2 = π

=

Z

+∞ −∞

x1 u2 sin2 α dy2 h i2 S20 + (y1 − y2 )2

 +∞   y2 − y1 1  −1 y2 − y1   h i + 3 tan   2S2 S2 +(y − y )2 S0  2S0 1 2 0 0 −∞

1 u2 sin α x1 sin α du2 1 du2 x1 u2 sin2 α du2 = = cos θ10 cos θ20 , 3 2 S S S 2 S0 2S0 0 0 0

where θ10 and θ20 are the projections of θ1 and θ2 in the x-z-plane. Looking at Fig. 4-8 this may be rewritten as dFd1−strip 2 =

1 2

cos φ dφ,

137

4.4 AREA INTEGRATION

du2 cosθ 20

b

du 2

θ 20 n2

u

2

dφ n1

φ = θ10 α x1

FIGURE 4-8 Two-dimensional wedge-shaped groove with projected distances.

dx1 a

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

1 2

cos φ dφ =

x1 sin2 α u2 du2 . 2S30

Example 4.2. Determine the view factor F1−2 for the infinitely long groove shown in Fig. 4-8. Solution Since we already know the view factor between two infinite strips, we can write Fstrip 1−2 = F1−2 =

b

Z

0

1 a

Z

dFstrip 1−strip 2 , a

Fstrip 1−2 dx1 . 0

Therefore, from Example 4.1, 2

Fstrip 1−2 =

x1 sin α 2

Z

b 0

u2 du2 (x21 − 2x1 u2 cos α + u22 )3/2

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

0

      b cos α − x1 1   . 1 + q =   2   2 2 x1 − 2bx1 cos α + b 

Finally, carrying out the second integration we obtain

F1−2

1 = a

Z

a 0

  s !2  a ! q    1 1 1 b b b   . 1− x21 − 2bx1 cos α + b2 = 1 + − 1 − 2 cos α + Fstrip 1−2 dx1 =  2 a 2 a a a   0

Example 4.3. As a final example for area integration we shall consider the view factor between two parallel, coaxial disks of radius R1 and R2 , respectively, as shown in Fig. 4-9. Solution Placing x-, y-, and z-axes as shown in the figure, and making a coordinate transformation to cylindrical

138

4 VIEW FACTORS

dA2

ψ2

r2

R2

n2

θ2 S

h

ψ2

θ1

y

z n1

r2

ψ1

R1

x

r1 dA1

FIGURE 4-9 Coordinate systems for the view factor between parallel, coaxial disks.

A1

coordinates, we find x1 = r1 cos ψ1 , y1 = r1 sin ψ1 , z1 = 0;

dA1 = r1 dr1 dψ1 ;

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

dA2 = r2 dr2 dψ2 ;

2

2

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

1 (πR21 )π

Z

R1 r1 =0

Z

R2 r2 =0

Z

2π ψ1 =0

Z

2π ψ2 =0

h2 r1 r2 dψ2 dψ1 dr2 dr1 h i2 . h2 +r21 +r22 −2r1 r2 cos(ψ1 −ψ2 )

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

2h2 πR21

Z

R1 r1 =0

Z

R2 r2 =0

Z

2π ψ=0

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

.

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

4.5

CONTOUR INTEGRATION

According to Stokes’ theorem, as developed in standard mathematics texts such as Wylie [22], a surface integral may be converted to an equivalent contour integral (see Fig. 4-10) through I Z f · ds = (∇ × f) · nˆ dA, (4.24) Γ

A

where f is a vector function defined everywhere on the surface A, including its boundary Γ, nˆ is the unit surface normal, and s is the position vector for a point on the boundary of A (ds, therefore, is the vector describing the boundary contour of A). By convention the contour integration in

4.5 CONTOUR INTEGRATION

139

n

A

ds

Γ

s

FIGURE 4-10 Conversion between surface and contour integral; Stokes’ theorem.

0

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

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

1 s12 × nˆ 1 , 2π S2

(4.25)

I

(4.26)

leading to Fd1−2 =

1 2π

Γ2

(s12 × nˆ 1 ) · ds2 , S2

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

(4.27)

    1 1 s12 × nˆ 1 = 2 ∇2 ×(s12 × nˆ 1 )−(s12 × nˆ 1 )×∇2 2 . 2 S S S

(4.28)

we may write1 2π∇2 ×f = ∇2 ×

From equations (4.20) and (4.21) it follows that ∇2



 2 2 s12 2s12 1 = − 3 ∇2 S = − 3 =− 4 . S2 S S S S

1 We add the subscript 2 to all operators to make clear that differentiation is with respect to position coordinates on A2 , for example, x 2 , y 2 , and z 2 if a Cartesian coordinate system is employed.

140

4 VIEW FACTORS

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

(4.29a)

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

(4.29b)

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

∇2 s12 = ˆıˆı + ˆˆ + kˆ kˆ = δ,

(4.30)

where δ is the unit tensor whose diagonal elements are unity and whose nondiagonal elements are zero:     1 0 0       (4.31) δ = 0 1 0 .     0 0 1 With nˆ 1 · δ = nˆ 1 equation (4.29b) reduces to

∇2 × (s12 × nˆ 1 ) = nˆ 1 − 3nˆ 1 = −2nˆ 1 . Substituting all this into equation (4.28), we obtain 2π∇2 × f = − and

i 2nˆ 1 2 h 2 + 4 S2 nˆ 1 − s12 (s12 · nˆ 1 ) = − 4 s12 (s12 · nˆ 1 ), 2 S S S

(∇2 × f) · nˆ 2 = −

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

(4.32)

Together with Stokes’ theorem this completes the proof that equation (4.26) is equivalent to an area integral over the function given by equation (4.32). For a Cartesian coordinate system, using equations (4.19) through (4.22), we have ˆ ds2 = dx2 ˆı + dy2 ˆ + dz2 k, and equation (4.26) becomes I I (z2 −z1 ) dy2 − (y2 − y1 ) dz2 m1 (x2 −x1 ) dz2 − (z2 −z1 ) dx2 l1 Fd1−2 = + 2π Γ2 2π Γ2 S2 S2 I (y2 − y1 ) dx2 − (x2 −x1 ) dy2 n1 + . 2π Γ2 S2 Example 4.4. Determine the view factor Fd1−2 for the configuration shown in Fig. 4-11. Solution With the coordinate system as shown in the figure we have S=

q x2 + y2 + c2 ,

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

(4.33)

4.5 CONTOUR INTEGRATION

141

x a A2

b

y z

S

c

FIGURE 4-11 View factor to a rectangular plate from a parallel infinitesimal area element located opposite a corner.

dA1

y dx − x dy S2 Γ2 "Z # "Z x=0 #  # "Z y=a # "Z y=0 x=b    y y (−x) (−x) 1    dx + dy + dx + dy =−     2 2 2 2  2π  S S S S x=b y=0 y=a x=0 y=0 y=a x=b x=0 ! Z a Z b b dy a dx 1 + = 2 2 2 2 2 2 2π y=0 b + y +c x=0 x +a +c

Fd1−2 = −

1 2π

1 = 2π Fd1−2

1 = 2π

I

 a b   y  a x −1 −1  √ b tan √ tan √ + √   b2 +c2 b2 +c2 0 a2 +c2 a2 +c2 0 b

−1

tan √ b2 +c2

a

a

+ √ tan √ b2 +c2 b2 +c2

−1



b

a2 +c2

!

.

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

(4.36)

142

4 VIEW FACTORS

where differentiation is with respect to the coordinates of surface A1 (for which Stokes’ theorem has been applied). Forming the dot product with nˆ 1 then results in I I 1 nˆ 1 × ∇1 S nˆ 1 · (∇1 × f) = nˆ 1 · (∇1 S × ds2 ) = · ds2 , (4.37) 2πS 2πS Γ2 Γ2 where use has been made of the vector relationship u · (v × w) = (u × v) · w.

(4.38)

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

(4.39)

which is, of course, identical to equation (4.13). For Cartesian coordinates, with s1 and s2 from equation (4.19), equation (4.34) becomes I I 1 A1 F1−2 = ln S (dx2 dx1 + dy2 dy1 + dz2 dz1 ). (4.40) 2π Γ1 Γ2 Example 4.5. Determine the view factor between two parallel, coaxial disks, Example 4.3, by contour integration. Solution With ds = dx ˆı + dy ˆ + dz kˆ it follows immediately from the coordinates given in Example 4.3 that ds1 = R1 dψ1 (− sin ψ1 ˆı + cos ψ1 ˆ), ds2 = R2 dψ2 (− sin ψ2 ˆı + cos ψ2 ˆ), ds1 · ds2 = R1 R2 dψ1 dψ2 (sin ψ1 sin ψ2 + cos ψ1 cos ψ2 ) = R1 R2 cos(ψ1 − ψ2 ) dψ1 dψ2 , where, it should be remembered, ds is along the periphery of a disk, i.e., at r = R. Substituting the last expression into equation (4.34) leads to Z 2π Z −2π h i1/2 R1 R2 ln h2 +R21 +R22 −2R1 R2 cos(ψ1 −ψ2 ) cos(ψ1 −ψ2 ) dψ2 dψ1 , F1−2 = 2 2π(πR1 ) ψ1 =0 ψ2 =0 where the integration for ψ2 is from 0 to −2π since, for an observer standing on top of A2 , the integration must be in a counterclockwise sense. Just like in Example 4.3, we can eliminate one of the integrations immediately since the angles appear only as differences, i.e., ψ1 − ψ2 : Z 2π  1/2 1 R2 F1−2 = − cos ψ dψ. ln h2 +R21 +R22 −2R1 R2 cos ψ π R1 0 Integrating by parts we obtain:   Z 2π  1/2 2π  sin2 ψ dψ 1 R2   F1−2 = −  sin ψ ln h2 +R21 +R22 −2R1 R2 cos ψ − R1 R2 2 +R2 +R2 −2R R cos ψ  π R1 h 1 2 0 2 1 0 Z 2π sin2 ψ dψ R2 /R1 = , 2π X − cos ψ 0

143

4.6 VIEW FACTOR ALGEBRA

d

c

A5

A6

a

A3

A4

b

e

dA1

FIGURE 4-12 View factor configuration for Example 4.6.

where we have introduced the abbreviation X=

h2 + R21 + R22 2R1 R2

.

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

4.6

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

VIEW FACTOR ALGEBRA

Many view factors for fairly complex configurations may be calculated without any integration by simply using the rules of reciprocity and summation, and perhaps the known view factor for a more basic geometry. That is, besides one (or more) known view factor we will only use the following three basic equations: Reciprocity Rule:

Ai Fi− j = A j F j−i , N X

Summation Relation:

Fi− j = 1,

(4.15c) (4.18)

j=1

Subsurface Summation A j =

K X

A(j,k) :

Fi− j =

k=1

K X

Fi−( j,k)

(4.17)

k=1

We shall illustrate the usefulness of this view factor algebra through a few simple examples. Example 4.6. Suppose we have been given the view factor for the configuration shown in Fig. 4-11, that is, Fd1−2 = F(a, b, c) as determined in Example 4.4. Determine the view factor Fd1−3 for the configuration shown in Fig. 4-12. Solution To express Fd1−3 in terms of known view factors F(a, b, c) (with the differential area opposite one of the corners of the large plate), we fill the plane of A3 with hypothetical surfaces A4 , A5 , and A6 as indicated in Fig. 4-12. From the definition of view factors, or equation (4.13), it follows that Fd1−(3+4+5+6) = Fd1−3 + Fd1−4 + Fd1−(5+6) , Fd1−4 = Fd1−(4+6) − Fd1−6 . Thus, Fd1−3 = Fd1−(3+4+5+6) − Fd1−(4+6) + Fd1−6 − Fd1−(5+6) .

144

4 VIEW FACTORS

A4

A2

A2 A1

c

A1

A3

b a (a)

(b) FIGURE 4-13 Configuration for Example 4.7: (a) full corner piece, (b) strips on a corner piece.

All four of these are of the type discussed in Example 4.4. Therefore, Fd1−3 = F(a+b, c+d, e) − F(a, c+d, e) + F(a, c, e) − F(a+b, c, e). We have successfully converted the present complex view factor to a summation of four known, more basic ones. Example 4.7. Assuming the view factor for a finite corner, as shown in Fig. 4-13a, is known as F1−2 = f (a, b, c), where f is a known function of the dimensions of the corner pieces (as given in Appendix D), determine the view factor F3−4 , between the two perpendicular strips as shown in Fig. 413b. Solution From the definition of the view factor, and since the energy traveling to A4 is the energy going to A2 plus A4 minus the energy going to A2 , it follows that F3−4 = F3−(2+4) − F3−2 , and, using reciprocity, F3−4 = Similarly, we find F3−4 =

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

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

All view factors on the right-hand side are corner pieces and are, thus, known by evaluating the function f with appropriate dimensions. Example 4.8. Again, assuming the view factor is known for the configuration in Fig. 4-13a, determine F1−6 as shown in Fig. 4-14. Solution Examining Fig. 4-14, and employing reciprocity, we find   (A5 + A6 )F(5+6)−(1+2) = (A5 + A6 ) F(5+6)−1 + F(5+6)−2

= A1 (F1−5 + F1−6 ) + A2 (F2−5 + F2−6 )     = A1 F1−(3+5) − F1−3 + A2 F2−(4+6) − F2−4 + A1 F1−6 + A2 F2−5 .

On the other hand, we also have

  (A5 + A6 ) F(5+6)−(1+2) = (A1 + A2 ) F(1+2)−(3+4+5+6) − F(1+2)−(3+4) .

4.6 VIEW FACTOR ALGEBRA

A6

z d

A5

A3

y

b

A4

c

0

145

A2

a A1 e

FIGURE 4-14 Configuration for Example 4.8.

x

In both expressions all view factors, with the exceptions of F1−6 and F2−5 , are of the type given in Fig. 4-13a. These last two view factors may be related to one another, as is easily seen from their integral forms. From equation (4.13) we have Z Z cos θ2 cos θ5 dA5 dA2 . A2 F2−5 = πS2 A2 A5 With a coordinate system as shown in Fig. 4-14, we get from equations (4.21) and (4.23) S2 = x22 + (y2 − y5 )2 + z25 , cos θ2 = z5 /S, cos θ5 = x2 /S, or A2 F2−5 =

Z

e x2 =0

Z

b y2 =a

Z

a y5 =0

Z

d z5 =c

x2 z5 dz5 dy5 dy2 dx2 i2 . h π x22 +(y2 − y5 )2 +z25

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

Z

e x1 =0

Z

a y1 =0

Z

b y6 =a

Z

d z6 =c

x1 z6 dz6 dy6 dy1 dx1 i2 . h π x21 +(y1 − y6 )2 +z26

Switching the names for dummy integration variables, it is obvious that A2 F2−5 = A1 F1−6 , which may be called the law of reciprocity for diagonally opposed pairs of perpendicular rectangular plates. Finally, solving for F1−6 we obtain F1−6 =

 1   A1 + A2  A2  F(1+2)−(3+4+5+6) − F(1+2)−(3+4) − F2−(4+6) − F2−4 . F1−(3+5) − F1−3 − 2A1 2 2A1

Using similar arguments, one may also determine the view factor between two arbitrarily orientated rectangular plates lying in perpendicular planes (Fig. 4-15a) or in parallel planes (Fig. 4-15b). After considerable algebra, one finds [1]: Perpendicular plates (Fig. 4-15a): 2A1 F1−2 = f (x2 , y2 , z3 ) − f (x2 , y1 , z3 ) − f (x1 , y2 , z3 ) + f (x1 , y1 , z3 ) + f (x1 , y2 , z2 ) − f (x1 , y1 , z2 ) − f (x2 , y2 , z2 ) + f (x2 , y1 , z2 ) − f (x2 , y2 , z3 −z1 ) + f (x2 , y1 , z3 −z1 ) + f (x1 , y2 , z3 −z1 ) − f (x1 , y1 , z3 −z1 ) + f (x2 , y2 , z2 −z1 ) − f (x2 , y1 , z2 −z1 ) − f (x1 , y2 , z2 −z1 ) + f (x1 , y1 , z2 −z1 ),

(4.41)

146

4 VIEW FACTORS

y y2 A2 y1 A2

c x1

x2

x

y3 y1

z2 z3 z

y

y2

A1

z1

A1

0

0

x1

x2

x3

x

(a) (b) FIGURE 4-15 View factors between generalized rectangles: (a) surfaces are on perpendicular planes, (b) surfaces are on parallel planes.

where f (w, h, l) = A1 F1−2 is the product of area and view factor between two perpendicular rectangles with a common edge as given by Configuration 39 in Appendix D. Parallel plates (Fig. 4-15b): 4A1 F1−2 = f (x3 , y3 ) − f (x3 , y2 ) − f (x3 , y3 − y1 ) + f (x3 , y2 − y1 )   − f (x2 , y3 ) − f (x2 , y2 ) − f (x2 , y3 − y1 ) + f (x2 , y2 − y1 )   − f (x3 −x1 , y3 ) − f (x3 −x1 , y2 ) − f (x3 −x1 , y3 − y1 ) + f (x3 −x1 , y2 − y1 ) + f (x2 −x1 , y3 ) − f (x2 −x1 , y2 ) − f (x2 −x1 , y3 − y1 ) + f (x2 −x1 , y2 − y1 ),

(4.42)

where f (a, b) = A1 F1−2 is the product of area and view factor between two directly opposed, parallel rectangles, as given by Configuration 38 in Appendix D. Equations (4.41) and (4.42) are not restricted to x3 > x2 > x1 , and so on, but hold for arbitrary values, for example, they are valid for partially overlapping surfaces. Fortran functions perpplates and parlplates are included in Appendix F for the evaluation of these view factors, based on calls to Fortran function view (i.e., calls to function view to evaluate the various view factors for Configurations 39 and 38, respectively). Example 4.9. Show that equation (4.42) reduces to the correct expression for directly opposing rectangles. Solution For directly opposing rectangles, we have x1 = x3 = a, y1 = y3 = b, and x2 = y2 = 0. We note that the formula for A1 F1−2 for Configuration 38 in Appendix D is such that f (a, b) = f (−a, b) = f (a, −b) = f (−a, −b), i.e., the view factor and area are both “negative” for a single negative dimension, making their product positive, and similarly if both a and b are negative. Also, if either a or b is zero (zero area), then f (a, b) = 0. Thus, 4A1 F1−2 = f (a, b) − 0 − 0 + f (a, −b) − [0 − 0 − 0 + 0] − [0 − 0 − 0 + 0] + f (−a, b) − 0 − 0 + f (−a, −b) =4 f (a, b).

Many other view factors for a multitude of configurations may be obtained through view factor algebra. A few more examples will be given in this and the following chapters (when radiative exchange between black, gray-diffuse, and gray-specular surfaces is discussed).

4.7 THE CROSSED-STRINGS METHOD

A2

a

b

d

c A1

4.7

147

FIGURE 4-16 The crossed-strings method for arbitrary two-dimensional configurations.

THE CROSSED-STRINGS METHOD

View factor algebra may be used to determine all view factors in long enclosures with constant cross-section. The method is credited to Hottel [26],∗ and is called the crossed-strings method since the view factors can be determined experimentally by a person armed with four pins, a roll of string, and a yardstick. Consider the configuration in Fig. 4-16, which shows the cross-section of an infinitely long enclosure, continuing into and out of the plane of the figure: We would like to determine F1−2 . Obviously, the surfaces shown are rather irregular (partly convex, partly concave), and the view between them may be obstructed. We shudder at the thought of having to carry out the view factor determination by integration, and plant our four pins at the two ends of each surface, as indicated by the labels a, b, c, and d. We now connect points a and c and b and d with tight strings, making sure that no visual obstruction remains between the two strings. Similarly, we place tight strings ab and cd across the surfaces, and ad and bc diagonally between them, as shown in Fig. 4-16. Now assuming the strings to be imaginary surfaces Aab , Aac , and Abc , we apply the summation rule to the “triangle” abc: Aab Fab−ac + Aab Fab−bc = Aab ,

(4.43a)

Aac Fac−ab + Aac Fac−bc = Aac ,

(4.43b)

Abc Fbc−ac + Abc Fbc−ab = Abc ,

(4.43c)

where Fab−ab = Fac−ac = Fbc−bc = 0 since a tightened string will always form a convex surface. Equations (4.43) are three equations in six unknown view factors, which may be solved by applying reciprocity to three of them:



Aab Fab−ac + Aab Fab−bc = Aab ,

(4.44a)

Aab Fab−ac + Aac Fac−bc = Aac ,

(4.44b)

Aac Fac−bc + Aab Fab−bc = Abc .

(4.44c)

Hoyte Clark Hottel (1903–1998) American engineer. Obtained his M.S. from the Massachusetts Institute of Technology in 1924, and was on the Chemical Engineering faculty at M.I.T. from 1927 until his death. While Hottel is credited with the method’s discovery, he has stated that he found it in a publication while in the M.I.T. library; but, by the time he first published it, he was unable to rediscover its source. Hottel’s major contributions have been his pioneering work on radiative heat transfer in furnaces, particularly his study of the radiative properties of molecular gases (Chapter 11) and his development of the zonal method (Chapter 18).

148

4 VIEW FACTORS

Adding the first two equations and subtracting the last leads to the view factor for an arbitrarily shaped triangle with convex surfaces, Fab−ac =

Aab + Aac − Abc , 2Aab

(4.45)

which states that the view factor between two surfaces in an arbitrary “triangle” is equal to the area of the originating surface, plus the area of the receiving surface, minus the area of the third surface, divided by twice the originating surface. Applying equation (4.45) to triangle abd we find immediately Fab−bd =

Aab + Abd − Aad . 2Aab

(4.46)

But, from the summation rule, Fab−ac + Fab−bd + Fab−cd = 1. Thus

Aab + Aac − Abc Aab + Abd − Aad − 2Aab 2Aab (Abc + Aad ) − (Aac + Abd ) = . 2Aab

(4.47)

Fab−cd = 1 −

(4.48)

Inspection of Fig. 4-16 shows that all radiation leaving Aab traveling to Acd will hit surface A1 . At the same time all radiation from Aab going to A1 must pass through Acd . Therefore, Fab−cd = Fab−1 . Using reciprocity and repeating the argument for surfaces Aab and A2 , we find Fab−cd = Fab−1 =

A1 A1 F1−ab = F1−2 , Aab Aab

and, finally, F1−2 =

(Abc + Aad ) − (Aac + Abd ) . 2A1

(4.49)

This formula is easily memorized by looking at the configuration between any two surfaces as a generalized “rectangle,” consisting of A1 , A2 , and the two sides Aac and Abd . Then F1−2 =

diagonals − sides . 2 × originating area

Example 4.10. Calculate F1−2 for the configuration shown in Fig. 4-17. Solution From the figure it is obvious that s21 = (c − d cos α)2 + d2 sin2 α = c2 + d2 − 2cd cos α. Similarly, we have s22 = (a + c)2 + (b + d)2 − 2(a + c)(b + d) cos α, d21 = (a + c)2 + d2 − 2(a + c)d cos α, d22 = c2 + (b + d)2 − 2c(b + d) cos α, and F1−2 =

d1 + d2 − (s1 + s2 ) . 2a

(4.50)

4.7 THE CROSSED-STRINGS METHOD

149

b A2 d2 d

s1

s2 d1

α

c

a

A1 FIGURE 4-17 Infinitely long wedge-shaped groove for Examples 4.10 and 4.11.

For c = d = 0, this reduces to the result of Example 4.2, or √ a + b − a2 + b2 − 2ab cos α F1−2 = . 2a Example 4.11. Find the view factor Fd1−2 of Fig. 4-17 for the case that A1 is an infinitesimal strip of width dx. Use the crossed-strings method. Solution We can obtain the result right away by replacing a by dx in the previous example. Throwing out differentials of second and higher order, we find that s1 and d2 remain unchanged, and p d1 = (c + dx)2 + d2 − 2(c + dx) d cos α p c2 + d2 − 2cd cos α + 2(c − d cos α) dx ≃ # " √ (c − d cos α) dx dx = s1 + (c−d cos α) ≃ c2 +d2 −2cd cos α 1+ 2 s1 c + d2 −2cd cos α p

(c + dx)2 + (b + d)2 − 2(c + dx)(b + d) cos α dx [c − (b + d) cos α] . ≃ d2 + d2

s2 =

Substituting this into equation (4.50), we obtain s1 + (c−d cos α) dx/s1 + d2 − s1 − d2 − [c−(b+d) cos α] dx/d2 2 dx     c − (b+d) cos α 1  c − d cos α . = − p  √  2 2 2 2 2 c + (b+d) − 2c(b+d) cos α c + d − 2cd cos α

Fd1−2 =

The same result could also have been obtained by letting

Fd1−2 = lim F1−2 , a→0

where F1−2 is the view factor from the previous example. Using de l’Hopital’s rule to determine the value of the resulting expression leads to ! 1 ∂d1 ∂s2 Fd1−2 = − , 2 ∂a ∂a a=0

and the above result.

Thus, the crossed-strings method may also be applied to strips. Example 4.1 could also have been solved this way; since the result is infinitesimal this computation would require retaining differentials up to second order. However, integration becomes simpler for strips of differential widths, while application of the crossed-strings method becomes more involved.

150

4 VIEW FACTORS

A1

A1

a

b

a

b

h

A

l

α

l

F

H D B

β

δ

β

C

γ

G

H

E

α

δ

α 0

π –2 β

cd (a)

A2

A2

I J

0 π/ 2–δ –γ

cd (b)

FIGURE 4-18 Configuration for view factor calculation of Example 4.12; string placement (a) for Fl1−2 , (b) for Fr1−2 .

We shall present one final example to show how view factors for curved surfaces and for configurations with floating obstructions can be determined by the crossed-strings method. Example 4.12. Determine the view factor F1−2 for the configuration shown in Fig. 4-18. Solution In the figure the end points of A1 and A2 (pin points) have been labeled a, b, c, and d, and other strategic points have been labeled with capital letters. A closed-contour surface such as a cylinder may be modeled by placing two pins right next to each other, with surface A2 being a strongly bulging convex surface between the pins. While the location of the two pins on the cylinder is arbitrary, it is usually more convenient to pick a location out of sight of A1 . Since A1 can see A2 from both sides of the obstruction, F1−2 cannot be determined with a single set of strings. Using view factor algebra, we can state that F1−2 = Fl1−2 + Fr1−2 , where Fl1−2 and Fr1−2 are the view factors between A1 and A2 when considering only light paths on the left or right of the obstruction, respectively. The placement of strings for Fl1−2 is given in Fig. 4-18a, and for Fr1−2 in Fig. 4-18b. Considering first Fl1−2 , the diagonals and sides may be determined from d1 = aD + DE + Ed,

d2 = bA + AB + BC + Cc,

s1 = aC + Cc,

s2 = bA + AE + Ed.

Substituting these expressions into equation (4.50) and canceling those terms that appear in a diagonal as well as in a side (Ed, bA, and Cc), we obtain Fl1−2 =

aD + DE + AB + BC − (aC+AE) . 2ab

Looking at Fig. 4-18a we also notice that aC = aD and AB = AE, so that   αR + (π−2β−α)R 1 π BC + DE = = −β . Fl1−2 = 2ab 2 × 2R 2 2

4.8 THE INSIDE SPHERE METHOD

151

A2 dA2

θ2

R

S R

θ1

dA1

A1

FIGURE 4-19 The inside sphere method.

 But cot β = tan π/2 − β = R/(h + H). Thus,

Fl1−2 =

R 1 tan−1 . 2 h+H

Similarly, we find from Fig. 4-18b for Fr1−2 , d1 = aF + FI + IJ + Jd,

d2 = bG + GH + Hc,

s1 = aF + FH + Hc,

s2 = bJ + Jd,

Fr1−2 =

FI + IJ + bG + GH − (FH+bJ) . 2ab

By inspection bG = bJ and FI = FH, leading to     π π IJ + GH 2 −δ−γ R + π−2β+δ− 2 −γ R r = F1−2 = 2ab 2 × 2R !   1 π 1 R l = −β−γ = tan−1 − tan−1 . 2 2 2 h+H h Note that this formula only holds as long as GH > 0 (i.e., as long as the cylinder is seen without obstruction from point b). Finally, adding the left and right contributions to the view factor, F1−2 = tan−1

R 1 l − tan−1 . h+H 2 h

4.8 THE INSIDE SPHERE METHOD Consider two surfaces A1 and A2 that are both parts of the surface of one and the same sphere, as shown in Fig. 4-19. We note that, for this type of configuration, θ1 = θ2 = θ and S = 2R cos θ. Therefore, Z Z Z 1 A2 cos θ1 cos θ2 cos2 θ Fd1−2 = dA = dA = dA2 = , (4.51) 2 2 2 2 2 As πS 4πR A2 A2 A2 π(2R cos θ)

152

4 VIEW FACTORS

A2´

A2

R2

R h β2

β1 R1

A1

FIGURE 4-20 View factor between coaxial parallel disks.

A1´

where As = 4πR2 is the surface area of the entire sphere. Similarly, from equation (4.16), F1−2 = Fd1−2 =

A2 , As

(4.52)

since Fd1−2 does not depend on the position of dA1 . Therefore, because of the unique geometry of a sphere, the view factor between two surfaces on the same sphere only depends on the size of the receiving surface, and not on the location of either one. The inside sphere method is primarily used in conjunction with view factor algebra, to determine the view factor between two surfaces that may not necessarily lie on a sphere. Example 4.13. Find the view factor between two parallel, coaxial disks of radius R1 and R2 using the inside sphere method. Solution Inspecting Fig. 4-20 we see that it is possible to place the parallel disks inside a sphere of radius R in such a way that the entire peripheries of both disks lie on the surface of the sphere. Since all radiation from A1 to A2 travels on to the spherical cap A2′ (in the absence of A2 ), and since all radiation from A1 to A2′ must pass through A2 , we have F1−2 = F1−2′ . Using reciprocity and applying a similar argument for A1 and spherical cap A1′ , we find F1−2 = F1−2′ =

A 2′ A1′ A2′ A 2′ F2′ −1 = F2′ −1′ = . A1 A1 A1 As

The areas of the spherical caps are readily calculated as Z βi sin β dβ = 2πR2 (1 − cos βi ), Ai′ = 2πR2

i = 1, 2.

0

Thus, with A1 = πR21 and As = 4πR2 , this results in F1−2 =

(2πR2 )2 (1 − cos β1 )(1 − cos β2 )

. πR21 4πR2 q From Fig. 4-20 one finds (assuming βi ≤ π/2) cos βi = R2 − R2i /R, and F1−2 =

q q    1 2 − R2 R − R − R R2 − R22 . 1 2 R1

4.9 THE UNIT SPHERE METHOD

153

A2 dA2

θ2 n1

n2 S

dA´2 A´2

θ1 dA´´2 dA1

R

A´´2

FIGURE 4-21 Surface projection for the unit sphere method.

It remains to find the radius of the sphere R, since only the distance between disks, h, is known. From Fig. 4-20 q q h = R2 − R21 + R2 − R22 , which may be solved (by squaring twice), to give R2 = (X2 − 1)



R1 R2 h

2

,

X=

h2 + R21 + R22 2R1 R2

.

This result is, of course, identical to the one given in Example 4.5, although it is not trivial to show this.

4.9

THE UNIT SPHERE METHOD

The unit sphere method is a powerful tool to calculate view factors between one infinitesimal and one finite area. It is particularly useful for the experimental determination of such view factors, as first stated by Nusselt [27]. An experimental implementation of the method through optical projection has been discussed by Farrell [28]. To determine the view factor Fd1−2 between dA1 and A2 we place a hemisphere2 of radius R on top of A1 , centered over dA1 , as shown in Fig. 4-21. From equations (4.4) and (4.8) we may write Z Z cos θ1 cos θ1 cos θ2 dA2 = dΩ 2 . (4.53) Fd1−2 = 2 π πS Ω2 A2 The solid angle dΩ 2 may also be expressed in terms of area dA′2 (dA2 projected onto the hemisphere) as dΩ 2 = dA′2 /R2 . Further, the area dA′2 may be projected along the z-axis onto the plane of A1 as dA′′ = cos θ1 dA′2 . Thus, 2 Z Z ′ dA′′ A′′ cos θ1 dA2 2 2 = = , (4.54) Fd1−2 = 2 2 π R2 πR πR A′′ A′2 2 that is, Fd1−2 is the fraction of the disk πR2 that is occupied by the double projection of A2 . Experimentally this can be measured, for example, by placing an opaque area A2 within a 2 The name unit sphere method originated with Nusselt, who used a sphere of unit radius; however, a sphere of arbitrary radius may be used.

154

4 VIEW FACTORS

a

d R

FIGURE 4-22 Geometry for the view factor in Example 4.14.

dA1

hemisphere, made of a translucent material, and which has a light source at the center (at dA1 ). Looking down onto the translucent hemisphere in the negative z-direction, A2′ will appear as a shadow. A photograph of the shadow (and the bright disk) can be taken, showing the double projection of A2 , and Fd1−2 can be measured. Example 4.14. Determine the view factor for Fd1−2 between an infinitesimal area and a parallel disk as shown in Fig. 4-22. Solution While a hemisphere of arbitrary radius could be employed, we shall choose here for convenience a √ radius of R = a2 + d2 , i.e., a hemisphere that includes the periphery of the disk on its surface. Then = A2 = πa2 , and the view factor follows as A′′ 2 Fd1−2 =

πa2 a2 = 2 . 2 πR a + d2

Obviously, only a few configurations will allow such simple calculation of view factors. For a more general case it would be desirable to have some “cookbook formula” for the application of the method. This is readily achieved by looking at the vector representation of the surfaces. Any point on the periphery of A2 may be expressed as a vector ˆ s12 = xˆı + yˆ + zk.

(4.55)

The corresponding point on A′2 may be expressed as

as and on A′′ 2

s′12 = x′ ˆı + y′ ˆ + z′ kˆ = p

R x2

+ y2 + z2

s12 ,

′′ ′′ ′ ′ s′′ 12 = x ˆı + y ˆ = x ˆı + y ˆ.

(4.56)

(4.57)

as Thus, any point (x, y, z) on A2 is double-projected onto A′′ 2 x x′′ R, 2 = p 2 x + y2 + z2

y y′′ R. 2 = p 2 x + y 2 + z2

(4.58)

Only the area formed by the projection of the periphery of A2 through equation (4.58) needs to be found. This integration is generally considerably less involved than the one in equation (4.8).

References 1. Hamilton, D. C., and W. R. Morgan: “Radiant interchange configuration factors,” NACA TN 2836, 1952. 2. Leuenberger, H., and R. A. Pearson: “Compilation of radiant shape factors for cylindrical assemblies,” ASME paper no. 56-A-144, 1956.

PROBLEMS

155

3. Kreith, F.: Radiation Heat Transfer for Spacecraft and Solar Power Design, International Textbook Company, Scranton, PA, 1962. 4. Sparrow, E. M., and R. D. Cess: Radiation Heat Transfer, Hemisphere, New York, 1978. 5. Siegel, R., and J. R. Howell: Thermal Radiation Heat Transfer, 4th ed., Taylor and Francis-Hemisphere, Washington, 2002. 6. Howell, J. R.: A Catalog of Radiation Configuration Factors, McGraw-Hill, New York, 1982. 7. Howell, J. R., and M. P. Menguc ¨ ¸ : “Radiative transfer configuration factor catalog: A listing of relations for common geometries,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 112, pp. 910–912, 2011. 8. Wong, R. L.: “User’s manual for CNVUFAC–the General Dynamics heat transfer radiation view factor program,” Technical report, University of California, Lawrence Livermore National Laboratory, 1976. 9. Shapiro, A. B.: “FACET–a computer view factor computer code for axisymmetric, 2D planar, and 3D geometries with shadowing,” Technical report, University of California, Lawrence Livermore National Laboratory, August 1983, (maintained by Nuclear Energy Agency under http://www.oecd-nea.org/tools/abstract/detail/nesc9578/). 10. Burns, P. J.: “MONTE–a two-dimensional radiative exchange factor code,” Technical report, Colorado State University, Fort Collins, 1983. 11. Emery, A. F.: “VIEW–a radiation view factor program with interactive graphics for geometry definition (version 5.5.3),” Technical report, NASA computer software management and information center, Atlanta, 1986, (available from http://www.openchannelfoundation.org/projects/VIEW). 12. Ikushima, T.: “MCVIEW: A radiation view factor computer program or three-dimensional geometries using Monte Carlo method,” Technical report, Japan Atomic Energy Research Institute (JAERI), 1986, (maintained by Nuclear Energy Agency under http://www.oecd-nea.org/tools/abstract/detail/nea-1166). 13. Jensen, C. L.: “TRASYS-II user’s manual–thermal radiation analysis system,” Technical report, Martin Marietta Aerospace Corp., Denver, 1987. 14. Walton, G. N.: “Algorithms for calculating radiation view factors between plane convex polygons with obstructions,” in Fundamentals and Applications of Radiation Heat Transfer, vol. HTD-72, ASME, pp. 45–52, 1987. 15. Chin, J. H., T. D. Panczak, and L. Fried: “Spacecraft thermal modeling,” Int. J. Numer. Methods Eng., vol. 35, pp. 641–653, 1992. 16. Zeeb, C. N., P. J. Burns, K. Branner, and J. S. Dolaghan: “User’s manual for Mont3d – Version 2.4,” Colorado State University, Fort Collins, CO, 1999. 17. Walton, G. N.: “Calculation of obstructed view factors by adaptive integration,” Technical Report NISTIR–6925, National Institute of Standards and Technology (NIST), Gaithersburg, MD, 2002. 18. MacFarlane, J. J.: “VISRAD-a 3D view factor code and design tool for high-energy density physics experiments,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 81, pp. 287–300, 2003. 19. Emery, A. F., O. Johansson, M. Lobo, and A. Abrous: “A comparative study of methods for computing the diffuse radiation viewfactors for complex structures,” ASME Journal of Heat Transfer, vol. 113, no. 2, pp. 413–422, 1991. 20. Jakob, M.: Heat Transfer, vol. 2, John Wiley & Sons, New York, 1957. 21. Liu, H. P., and J. R. Howell: “Measurement of radiation exchange factors,” ASME Journal of Heat Transfer, vol. 109, no. 2, pp. 470–477, 1956. 22. Wylie, C. R.: Advanced Engineering Mathematics, 5th ed., McGraw-Hill, New York, 1982. 23. Moon, P.: Scientific Basis of Illuminating Engineering, Dover Publications, New York, 1961, (originally published by McGraw-Hill, New York, 1936). 24. de Bastos, R.: “Computation of radiation configuration factors by contour integration,” M.S. thesis, Oklahoma State University, 1961. 25. Sparrow, E. M.: “A new and simpler formulation for radiative angle factors,” ASME Journal of Heat Transfer, vol. 85, pp. 73–81, 1963. 26. Hottel, H. C.: “Radiant heat transmission,” in Heat Transmission, ed. W. H. McAdams, 3rd ed., ch. 4, McGraw-Hill, New York, 1954. 27. Nusselt, W.: “Graphische Bestimming des Winkelverh¨altnisses bei der W¨armestrahlung,” VDI Zeitschrift, vol. 72, p. 673, 1928. 28. Farrell, R.: “Determination of configuration factors of irregular shape,” ASME Journal of Heat Transfer, vol. 98, no. 2, pp. 311–313, 1976.

Problems 4.1 For Configuration 11 in Appendix D, find Fd1−2 by (a) area integration, and (b) contour integration. Compare the effort involved. 4.2 Using the results of Problem 4.1, find F1−2 for Configuration 33 in Appendix D. 4.3 Find F1−2 for Configuration 32 in Appendix D, by area integration. 4.4 Evaluate Fd1−2 for Configuration 13 in Appendix D by (a) area integration, and (b) contour integration. Compare the effort involved.

156

4 VIEW FACTORS

4.5 Using the result from Problem 4.4, calculate F1−2 for Configuration 40 in Appendix D. 4.6 Find the view factor Fd1−2 for Configuration 11 in Appendix D, with dA1 tilted toward A2 by an angle φ. 4.7 Find Fd1−2 for the surfaces shown in the figure, using (a) area integration, (b) view factor algebra, and Configuration 11 in Appendix D. A2 a b c d e dA1

4.8 For the infinite half-cylinder depicted in the figure, find F1−2 .

r1 A2

r2

R A1

4.9 Find Fd1−2 for the surfaces shown in the figure.

a c A2

d

b

e

dA1

4.10 Find the view factor of the spherical ring shown in the figure to itself, F1−1 , using the inside sphere method.

R

α α

A1

4.11 Determine the view factor for Configuration 51 in Appendix D, using (a) other, more basic view factors given in Appendix D, (b) the crossed-strings rule.

157

PROBLEMS

4.12 To reduce heat transfer between two infinite concentric cylinders a third cylinder is placed between them as shown in the figure. The center cylinder has an opening of half-angle θ. Calculate F4−2 .



r1

r2 = r3 r4

A2

A4

A3 A1

4.13 Consider the two long concentric cylinders as shown in the figure. Between the two cylinders is a long, thin flat plate as also indicated. Determine F4−2 .

A3

2R A2

R 3R

R A4

A1

4.14 Calculate the view factor F1−2 for surfaces on a cone as shown in the figure.

ϕ

b

a

A2 A1

4.15 Determine the view factor F1−2 for the configuration shown in the figure, if (a) the bodies are two-dimensional (i.e., infinitely long perpendicular to the paper); (b) the bodies are axisymmetric (cones).

D

A2 h = 2D A1

158

4 VIEW FACTORS

4.16 Consider the configuration shown; determine the view factor F1−2 assuming the configuration is a) axisymmetric (1 is conical, 2 is a disk with a hole), or b) two-dimensional Cartesian (1 is a V-groove, 2 is comprised of two infinitely long strips).

A2

D/2

D/2

2D

A1 2D

2D

4.17 Find F1−2 for the configuration shown in the figure (infinitely long perpendicular to paper).

A2 r r/2

r

r

r

r/2 r

r

A1

4.18 Calculate the view factor between two infinitely long cylinders as shown in the figure. If a radiation shield is placed between them to obstruct partially the view (dashed line), how does the view factor change?

L R R

R

l

4.19 Find the view factor between spherical caps as shown in the figure, for the case of H≥ q

R21

R2 + q 2 , R21 − a21 R22 − a22

A1

R1 a1

a2

where H = distance between sphere centers, R = sphere radius, and a = radius of cap base. Why is this restriction necessary? H

4.20 Determine the view factor for Configuration 18 in Appendix D, using the unit sphere method.

R2

159

PROBLEMS

4.21 Consider the axisymmetric configuration shown in the figure. Calculate the view factor F1−3 .

4 cm

A3 5cm A2

A2 1cm

A1

4.22 Find Fd1−2 from the infinitesimal area to the disk as shown in the figure, with 0 ≤ β ≤ π.

dA1 β

n h r

4.23 Consider the configuration shown (this could be a long cylindrical BBQ with a center shelf/hole; or an integrating sphere). Determine the view factors F2−2 and F2−3 assuming the configuration is (a) axisymmetric (sphere), (b) two-dimensional Cartesian (cylinder), using view factor algebra, (c) two-dimensional Cartesian (cylinder), using the string rule (F2−3 only).

A2

A2

A1T A1B A3 2R 2R

pR

4.24 In the solar energy laboratory at UC Merced parabolic concentrators are employed to enhance the absorption of tubular solar collectors as shown in the sketch. Calculate the view factor from the parabolic concentrator A1 to collecting cylinder A2 , using (a) view factor algebra, (b) Hottel’s string rule.

concentrator

4R 6R oil tube

R

4.25 The interior of a right-circular cylinder of length L = 4R, where R is its radius, is to be broken up into 4 ring elements of equal width. Determine the view factors between all the ring elements, using (a) view factor algebra and the view factors of Configuration 40, (b) Configuration 9 with the assumption that this formula can be used for rings of finite widths. Assess the accuracy of the approximate view factors. What would be the maximum allowable value for ∆X to ensure that all view factors within a distance of 4R are accurate to at least 5%? (Exclude the view factor from a ring to itself, which is best evaluated last, applying the summation rule.) Use the program viewfactors or the function view in your calculations. 4.26 The inside surfaces of a furnace in the shape of a parallelepiped with dimensions 1 m × 2 m × 4 m are to be broken up into 28 1 m × 1 m subareas. Determine all necessary view factors using the functions parlplates and perpplates in Appendix F.

CHAPTER

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

5.1

INTRODUCTION

In this chapter we shall begin our analysis of radiative heat transfer rates within enclosures without a participating medium, making use of the view factors developed in the preceding chapter. We shall first deal with the simplest case of a black enclosure, that is, an enclosure where all surfaces are black. Such simple analysis may often be sufficient, for example, for furnace applications with soot-covered walls. This will be followed by expanding the analysis to enclosures with gray, diffuse surfaces, whose radiative properties do not depend on wavelength, and which emit as well as reflect energy diffusely. Considerable experimental evidence demonstrates that most surfaces emit (and, therefore, absorb) diffusely except for grazing angles (θ > 60◦ ), which are unimportant for heat transfer calculations (for example, Fig. 3-1). Most surfaces tend to be fairly rough and, therefore, reflect in a relatively diffuse fashion. Finally, if the surface properties vary little across that part of the spectrum over which the blackbody emissive powers of the surfaces are appreciable, then the simplification of gray properties may be acceptable. In both cases—black enclosures as well as enclosures with gray, diffuse surfaces—we shall first derive the governing integral equation for arbitrary enclosures, which is then reduced to a set of algebraic equations by applying it to idealized enclosures. At the end of the chapter solution methods to the general integral equations are briefly discussed.

5.2 RADIATIVE EXCHANGE BETWEEN BLACK SURFACES Consider a black-walled enclosure of arbitrary geometry and with arbitrary temperature distribution as shown in Fig. 5-1. An energy balance for dA yields, from equation (4.1), q(r) = Eb (r) − H(r),

(5.1)

where H is the irradiation onto dA. From the definition of the view factor, the rate with which energy leaves dA′ and is intercepted by dA is (Eb (r′ ) dA′ ) dFdA′ −dA . Therefore, the total rate of

160

5.2 RADIATIVE EXCHANGE BETWEEN BLACK SURFACES

161

Ho

dA´



T(r)

dA r

FIGURE 5-1 A black enclosure of arbitrary geometry.

0

incoming heat transfer onto dA from the entire enclosure and from outside (for enclosures with some semitransparent surfaces and/or holes) is Z H(r) dA = Eb (r′ ) dFdA′ −dA dA′ + Ho (r) dA, (5.2) A

where Ho (r) is the external contribution to the irradiation, i.e., any part not due to emission from the enclosure surface. Using reciprocity, this may be stated as Z H(r) = Eb (r′ ) dFdA−dA′ + Ho (r) A Z cos θ cos θ′ Eb (r′ ) = (r, r′ ) dA′ + Ho (r), (5.3) 2 πS A where θ and θ′ are angles at the surface elements dA and dA′ , respectively, and S is the distance between them, as defined in Section 4.2. For an enclosure with known surface temperature distribution, the local heat flux is readily calculated as1 Z q(r) = Eb (r) − Eb (r′ ) dFdA−dA′ − Ho (r). (5.4) A

To simplify the problem it is customary to break up the enclosure into N isothermal subsurfaces, as shown in Fig. 4-2b. Then equation (5.4) becomes qi (ri ) = Ebi −

N X

Eb j

j=1

Z

dFdAi −dA j − Hoi (ri ),

(5.5)

Aj

or, from equation (4.16), qi (ri ) = Ebi −

N X

Eb j Fdi− j (ri ) − Hoi (ri ).

(5.6)

j=1

1 When looking at equation (5.4) one is often tempted by intuition to replace dFdA−dA′ by dFdA′ −dA . It should always be remembered that we have used reciprocity, since dFdA′ −dA is per unit area at r′ , while equation (5.4) is per unit area at r.

162

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

40 cm A 3: T 1

30 cm

A2: T2 A4: T2 A1: T1

FIGURE 5-2 Two-dimensional black duct for Example 5.1.

Even though the temperature may be constant across Ai , the heat flux is usually not since (i) the local view factor Fdi−j nearly always varies across Ai , and (ii) the external irradiation Hoi may Rnot be uniform. We may calculate an average heat flux by averaging equation (5.6) over Ai . With F dAi = Ai Fi− j this leads to A di− j i

qi =

1 Ai

Z

qi (ri ) dAi = Ebi − Ai

N X

Eb j Fi− j − Hoi ,

i = 1, 2, . . . , N,

(5.7)

j=1

where qi and Hoi are now understood to be average values. P Employing equation (4.18) we rewrite Ebi as Nj=1 Ebi Fi− j , or qi =

N X

Fi− j (Ebi − Eb j ) − Hoi ,

i = 1, 2, . . . , N.

(5.8)

j=1

In this equation the heat flux is expressed in terms of the net radiative energy exchange between surfaces Ai and A j , Qi− j = qi− j Ai = Ai Fi− j (Ebi − Ebj ) = −Q j−i .

(5.9)

Example 5.1. Consider a very long duct as shown in Fig. 5-2. The duct is 30 cm × 40 cm in cross-section, and all surfaces are black. The top and bottom walls are at temperature T1 = 1000 K, while the side walls are at temperature T2 = 600 K. Determine the net radiative heat transfer rate (per unit duct length) on each surface. Solution We may use either equation (5.7) or (5.8). We shall use the latter here since it takes better advantage of the symmetry of the problem (i.e., it uses the fact that the net radiative exchange between two surfaces at the same temperature must be zero). Thus, with no external irradiation, and using symmetry (e.g., Eb1 = Eb3 , F1−2 = F1−4 , etc.), q1 = F1−2 (Eb1 − Eb2 ) + F1−3 (Eb1 − Eb3 ) + F1−4 (Eb1 − Eb4 ) = 2F1−2 (Eb1 − Eb2 ) = q3 , q2 = q4 = 2F2−1 (Eb2 − Eb1 ). Only the view factors F1−2 and F2−1 are required, which are readily determined from the crossed-strings method as √ 30 + 40 − ( 302 + 402 + 0) 1 = , F1−2 = 2 × 40 4 A1 40 1 1 F1−2 = × = . F2−1 = A2 30 4 3

5.2 RADIATIVE EXCHANGE BETWEEN BLACK SURFACES

R1

163

R2

A1: T1 A2 : T 2 FIGURE 5-3 Concentric black spheres for Example 5.2.

Therefore (using a prime to indicate “per unit duct length”), Q′1 = Q′3 = 2A′1 F1−2 σ(T14 − T24 ) W (10004 −6004 ) K4 = 9870 W/m m2 K4 Q′2 = Q′4 = 2A′2 F2−1 σ(T24 − T14 ) = −9870 W/m = 2×0.4 m×0.25×5.670×10−8

It is apparent from this example that the sum of all surface heat transfer rates must vanish. This follows immediately from conservation of energy: The total heat transfer rate into the enclosure (i.e., the heat transfer rates summed over all surfaces) must be equal to the rate of change of radiative energy within the enclosure. Since radiation travels at the speed of light, steady state is reached almost instantaneously, so that the rate of change of radiative energy may nearly always be neglected. Mathematically, we may multiply equation (5.7) by Ai and sum over all areas: N N N N N N N X X X X X X X Ai Ebi − A j Ebj F j−i = 0. Ai Eb j Fi− j = (Qi + Ai Hoi ) = Ai Ebi − i=1

i=1

i=1

i=1

j=1

j=1

(5.10)

i=1

This relationship is most useful to check the correctness of one’s calculations, or their accuracy (for computer calculations). Example 5.2. Consider two concentric, isothermal, black spheres with radii R1 and R2 , and temperatures T1 and T2 , respectively, as shown in Fig. 5-3. Show how the temperature of the inner sphere can be deduced, if temperature and heat flux of the outer sphere are measured. Solution We have only two surfaces, and equation (5.8) becomes q1 = F1−2 (Eb1 − Eb2 );

q2 = F2−1 (Eb2 − Eb1 ).

Since all radiation from Sphere 1 travels to 2, we have F1−2 = 1 and, by reciprocity, F2−1 = A1 /A2 . Thus, Q1 = −Q2 = A1 σ(T14 − T24 ). Solving this for T1 we get, with Ai = 4πR2i , T14 = T24 −



R2 R1

2 q 2 . σ

Whenever T1 is larger than T2 , q2 is negative, and vice versa.

164

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

y

α

a

qsol

A2 A1

a

x

FIGURE 5-4 Right-angled groove exposed to solar irradiation, Example 5.3.

Example 5.3. A right-angled groove, consisting of two long black surfaces of width a, is exposed to solar radiation qsol (Fig. 5-4). The entire groove surface is kept isothermal at temperature T. Determine the net radiative heat transfer rate from the groove. Solution Again, we may employ either equation (5.7) or (5.8). However, this time the enclosure is not closed; and we must close it artificially. We note that any radiation leaving the cavity will not come back (barring any reflection from other surfaces nearby). Thus, our artificial surface should be black. We also assume that, with the exception of the (parallel) solar irradiation, no external radiation enters the cavity. Since the solar irradiation is best treated separately through the external irradiation term Ho , our artificial surface is nonemitting. Both criteria are satisfied by covering the groove with a black surface at 0 K. Even though we now have three surfaces, the last one does not really appear in equation (5.7) (since Eb3 = 0), but it does appear in equation (5.8). Using equation (5.7) we find q1 = Eb1 − F1−2 Eb2 − Ho1 = σT 4 (1 − F1−2 ) − qsol cos α, q2 = Eb2 − F2−1 Eb1 − Ho2 = σT 4 (1 − F2−1 ) − qsol sin α. From Configuration 33 in Appendix D we find, with H = 1,  √  F1−2 = 21 2 − 2 = 0.293 = F2−1 ,

and

Q′ = a(q1 + q2 ) = a

i h√ 2σT 4 − qsol (cos α + sin α) .

These examples demonstrate that equation (5.8) is generally more convenient to use for closed configurations, since it takes advantage of the fact that the net exchange between two surfaces at the same temperature (or with itself) is zero. Equation (5.7), on the other hand, is more convenient for open configurations, since the hypothetical surfaces employed to close the configuration do not contribute (because of their zero emissive power): With this equation the hypothetical closing surfaces may be completely ignored! Equation (5.7) may be written in a third form that is most convenient for computer calculations. Using Kronecker’s delta function, defined as  1, i = j, δi j = (5.11) 0, i , j, we find

N X j=1

δi j = 1 and

N X

Eb j δi j = Ebi . Thus,

j=1

qi =

N X (δi j − Fi− j )Eb j − Hoi , j=1

i = 1, 2, . . . , N.

(5.12)

5.3 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

165

Let us suppose that for surfaces i = 1, 2, . . . , n the heat fluxes are prescribed (and temperatures are unknown), while for surfaces i = n + 1, . . . , N the temperatures are prescribed (heat fluxes unknown). Unlike for the heat fluxes, no explicit relations for the unknown temperatures exist. Placing all unknown temperatures on one side of equation (5.12), we may write n N X X (δi j − Fi− j )Eb j = qi + Hoi + Fi− j Ebj , j=1

i = 1, 2, . . . , n,

(5.13)

j=n+1

where everything on the right-hand side of the equation is known. In matrix form this is written2 as A · eb = b,

(5.14)

where    1 − F1−1     −F2−1  A =  ..   .     −Fn−1

   Eb1     E  b2 eb =   ..  .     Ebn

−F1−2

···

−F1−n

1 − F2−2

···

−F2−n

..

.. .

−Fn−2

.

···

1 − Fn−n

         ,       

  P     q1 +Ho1 + Nj=n+1 F1− j Ebj         q2 +Ho2 + PN   j=n+1 F2− j Ebj  , b =    ..     .         qn +Hon + PN j=n+1 Fn− j Ebj

(5.15)

         .       

(5.16)

The n × n matrix A is readily inverted on a computer (generally with the aid of a software library subroutine), and the unknown temperatures are calculated as eb = A−1 · b.

(5.17)

5.3 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES We shall now assume that all surfaces are gray, that they are diffuse emitters, absorbers, and reflectors. Under these conditions ǫ = ǫλ′ = α′λ = α = 1 − ρ. The total heat flux leaving a surface at location r is, from Fig. 4-1, J(r) = ǫ(r)Eb (r) + ρ(r)H(r),

(5.18)

which is called the surface radiosity J at location r. Since both emission and reflection are diffuse, so is the resulting intensity leaving the surface: I(r, sˆ ) = I(r) = J(r)/π.

(5.19)

Therefore, an observer at a different location is unable to distinguish emitted and reflected 2 For easy readability of matrix manipulations we shall follow here the convention that a two-dimensional matrix is denoted by a bold capitalized letter, while a vector is written as a bold lowercase letter.

166

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

J J

const × Ebλ(T2)

∋ Ebλ (T1) FIGURE 5-5 Qualitative spectral behavior of radiosity for irradiation from an isothermal source.

λ

radiation on the basis of directional behavior. However, the observer may be able to distinguish the two as a result of their different spectral behavior. Consider Example 5.2 for the case of a black outer sphere but a gray, diffuse inner sphere. On the inner sphere the emitted radiation has the spectral distribution of a blackbody at temperature T1 , while the reflected radiation— which was originally emitted at the outer sphere—has the spectral distribution of a blackbody at temperature T2 . Thus, the spectral radiosity will behave as shown qualitatively in Fig. 5-5. An observer will be able to distinguish between emitted and reflected radiation if he has the ability to distinguish between radiation at different wavelengths. A gray surface does not have this ability, since it behaves in the same fashion toward all incoming radiation at any wavelength, i.e., it is “color blind.” Consequently, a gray surface does not “know” whether its irradiation comes from a gray, diffuse surface or from a black surface with an effective emissive power J. This fact simplifies the analysis considerably since it allows us to calculate radiative heat transfer rates between surfaces by balancing the net outgoing radiation (i.e., emission and reflection) traveling directly from surface to surface (as opposed to emitted radiation traveling to another surface directly or after any number of reflections). For this reason the following analysis is often referred to as the net radiation method. Making an energy balance on a surface dA in the enclosure shown in Fig. 5-6 we obtain from equation (4.2) q(r) = ǫ(r)Eb (r) − α(r)H(r) = J(r) − H(r).

(5.20)

The irradiation H(r) is again found by determining the contribution from a differential area dA′ (r′ ), followed by integrating over the entire surface. From the definition of the view factor the heat transfer rate leaving dA′ intercepted by dA is (J(r′ ) dA′ ) dFdA′ −dA . Thus, similar to the black-surfaces case, Z H(r) dA = J(r′ ) dFdA′ −dA dA′ + Ho (r) dA, (5.21) A

where Ho (r) is again any external radiation arriving at dA. Using reciprocity this equation reduces to Z J(r′ ) dFdA−dA′ + Ho (r). (5.22) H(r) = A

Substitution into equation (5.20) yields q(r) = ǫ(r)Eb (r) − α(r)

"Z



J(r ) dFdA−dA′ A

# + Ho (r) .

(5.23)

Thus, the unknown heat flux (or temperature) could be calculated if the radiosity field had been known. A governing integral equation for radiosity is readily established by solving

5.3 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

167

Ho

dA´



T(r), ∋ (r)

dA r

FIGURE 5-6 Radiative exchange in a gray, diffuse enclosure.

0

equation (5.20) for J: J(r) = ǫ(r)Eb (r) + ρ(r)

"Z



#

J(r ) dFdA−dA′ + Ho (r) , A

for those surface locations where the temperature is known, or Z J(r′ ) dFdA−dA′ + Ho (r), J(r) = q(r) +

(5.24)

(5.25)

A

for those parts of the surface where the local heat flux is specified. However, in problems without participating media there is rarely a need to determine radiosity, and it is usually best to eliminate radiosity from equation (5.23). Expressing radiosity in terms of local temperature and heat flux and eliminating irradiation H from equation (5.20) we have q − αq = (ǫEb − αH) − α(J − H) = ǫEb − αJ. Up to this point we have differentiated between emittance and absorptance, to keep the relations as general as possible (i.e., to accommodate nongray surface properties if necessary). We shall now invoke the assumption of gray, diffuse surfaces, or α = ǫ. Then ǫ(r) [Eb (r) − J(r)]. 1 − ǫ(r)

(5.26)

! 1 J(r) = Eb (r) − − 1 q(r). ǫ(r)

(5.27)

q(r) = Solving for radiosity, we get

Substituting this into equation (5.23), we obtain an integral equation relating temperature T and heat flux q: ! Z Z q(r) 1 ′ Eb (r′ ) dFdA−dA′ . (5.28) − − 1 q(r ) dFdA−dA′ + Ho (r) = Eb (r) − ′ ǫ(r) A ǫ(r ) A

168

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

Note that equation (5.28) reduces to equation (5.4) for a black enclosure. However, for a black enclosure with known temperature field the local heat flux can be determined with a simple integration over emissive power. For a gray enclosure an integral equation must be solved, i.e., an equation where the unknown dependent variable q(r) appears inside an integral. This requirement makes the solution considerably more difficult. As for a black enclosure it is customary to break up a gray enclosure into N subsurfaces, over each of which the radiosity is assumed constant. Then equation (5.23) becomes N

X qi (ri ) = Ebi (ri ) − Jj Fdi−j (ri ) − Hoi (ri ), ǫi (ri )

i = 1, 2, . . . , N,

(5.29)

j=1

and, taking an average over subsurface Ai , N

X qi = Ebi − Jj Fi− j − Hoi , ǫi

i = 1, 2, . . . , N.

(5.30)

j=1

Taking a similar average for equation (5.26) gives qi =

ǫi [Ebi − Ji ] . 1 − ǫi

(5.31)

Solving for J and substituting into equation (5.30) then leads to ! N N X qi X 1 − − 1 Fi− j q j + Hoi = Ebi − Fi− j Eb j , ǫi ǫj j=1

i = 1, 2, . . . , N.

(5.32)

j=1

This relation also follows directly from equation (5.28) if both (1/ǫ − 1)q and Eb (the components P of J) are assumed constant across the subsurfaces. Recalling the summation rule, Nj=1 Fi− j = 1, we may also write equation (5.32) as an interchange between surfaces, ! N N X qi X 1 − − 1 Fi− j q j + Hoi = Fi− j (Ebi − Ebj ), ǫi ǫj j=1

i = 1, 2, . . . , N.

(5.33)

j=1

Either one of these equations, of course, reduces to equation (5.8) for a black enclosure. Equation (5.32) is preferred for open configurations, since it allows one to ignore hypothetical closing surfaces; and equation (5.33) is preferred for closed enclosures, because it eliminates transfer between surfaces at the same temperature. Sometimes one wishes to determine the radiosity of a surface, for example, in the field of pyrometry (relating surface temperature to radiative intensity leaving a surface). Depending on which of the two is unknown, elimination of qi or Ebi from equation (5.30) with the help of equation (5.31) leads to   N  X   (5.34a) Ji = ǫi Ebi + (1−ǫi )  Jj Fi− j + Hoi    j=1

= qi +

N X

Jj Fi− j + Hoi ,

i = 1, 2, . . . , N.

(5.34b)

j=1

These two relations simply repeat the definition of radiosity, the first stating that radiosity consists of emitted and reflected heat fluxes and the second that radiosity, or outgoing heat flux, is equal to net heat flux (with negative qin ) plus the absolute value of qin .

5.3 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

169

40 cm A3: T1, ∋ 1

30 cm

A2: T2 , ∋ 2 A4: T2 , ∋ 2 A1: T1, ∋ 1

FIGURE 5-7 Two-dimensional gray, diffuse duct for Example 5.4.

Example 5.4. Reconsider Example 5.1 for a gray, diffuse surface material. Top and bottom walls are at T1 = T3 = 1000 K with ǫ1 = ǫ3 = 0.3, while the side walls are at T2 = T4 = 600 K with ǫ2 = ǫ4 = 0.8 as shown in Fig. 5-7. Determine the net radiative heat transfer rates for each surface. Solution Using equation (5.33) for i = 1 and i = 2, and recalling that F1−2 = F1−4 and F2−1 = F2−3 ,     q1 1 1 −2 − 1 F1−2 q2 − − 1 F1−3 q1 = 2F1−2 (Eb1 − Eb2 ), i=1: ǫ1 ǫ2 ǫ1     q2 1 1 i=2: −2 − 1 F2−1 q1 − − 1 F2−4 q2 = 2F2−1 (Eb2 − Eb1 ). ǫ2 ǫ1 ǫ2 We have already evaluated F1−2 = 14 and F2−1 = 31 in Example 5.1. From the summation rule F1−3 = 1 − 2F1−2 = 21 and F2−4 = 1 − 2F2−1 = 13 . Substituting these, as well as emittance values, into the relations reduces them to the simpler form of       1 1 1 1 1 − −1 − 1 q2 = 2 × 14 (Eb1 − Eb2 ), q1 − 2 0.3 0.3 2 0.8 4       1 1 1 1 1 q2 = 2 × 13 (Eb2 − Eb1 ), − 1 q1 + − −1 −2 0.3 3 0.8 0.8 3 or 13 q1 − 6 14 − q1 + 9

1 1 q2 = (Eb1 − Eb2 ), 8 2 7 2 q2 = − (Eb1 − Eb2 ). 6 3

Thus, 

   13 7 14 1 1 7 2 1 q1 = (Eb1 − Eb2 ), × − × × − × 6 6 9 8 2 6 3 8 3 3 1 σ(T14 − T24 ), q1 = × (Eb1 − Eb2 ) = 7 2 14

and 



   1 14 7 13 1 14 2 13 × + × q2 = × − × (Eb1 − Eb2 ), 8 9 6 6 2 9 3 6 3 2 2 q2 = − × (Eb1 − Eb2 ) = − σ(T14 − T24 ). 7 3 7

Finally, substituting values for temperatures, W (10004 −6004 ) K4 = 4230 W/m, m2 K4 W Q′2 = −0.3 m× 72 ×5.670×10−8 2 4 (10004 −6004 ) K4 = −4230 W/m. m K 3 ×5.670×10−8 Q′1 = 0.4 m× 14

Of course, both heat transfer rates must again add up to zero. We observe that these rates are less than half the ones for the black duct.

170

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

T2, ∋ 2

T1, ∋1 R1

A2: T2, ∋ 2

R2

A1: T1, ∋ 1

(a) (b) FIGURE 5-8 Radiative transfer between (a) two concentric spheres, (b) a convex surface and a large isothermal enclosure.

Example 5.5. Determine the radiative heat flux between two isothermal gray concentric spheres with radii R1 and R2 , temperatures T1 and T2 , and emittances ǫ1 and ǫ2 , respectively, as shown in Fig. 5-8a. Solution Again applying equation (5.33) for i = 1 (inner sphere) and i = 2 (outer sphere), we obtain: i=1: i=2:

   q1  1 1 − − 1 F1−1 q1 − − 1 F1−2 q2 = F1−2 (Eb1 − Eb2 ), ǫ1 ǫ1 ǫ2    q2  1 1 − − 1 F2−1 q1 − − 1 F2−2 q2 = F2−1 (Eb2 − Eb1 ). ǫ2 ǫ1 ǫ2

With F1−1 = 0, F1−2 = 1, F2−1 = A1 /A2 , and F2−2 = 1 − F2−1 = 1 − A1 /A2 , these two equations reduce to   1 1 q1 − − 1 q2 = σ(T14 − T24 ), ǫ1 ǫ2       A1 A1 1 A1 1 1 q2 = − σ(T14 − T24 ). −1 q1 + − −1 1− ǫ1 A2 ǫ2 ǫ2 A2 A2 This may be solved for q1 by eliminating q2 (or using conservation of energy, i.e., A1 q1 + A2 q2 = 0), or q1 =

σ(T14 − T24 )  . 1 A1 1 + −1 ǫ1 A2 ǫ2

(5.35)

We note that equation (5.35) is not just limited to concentric spheres, but holds for any convex surface A1 (i.e., with F1−1 = 0) that radiates only to A2 (i.e., F1−2 = 1) as indicated in Fig. 5-8b. This is often convenient for a convex surface Ai placed into a large, isothermal environment (Aa ≫ Ai ) at temperature Ta , leading to qi = ǫi σ(Ti4 − Ta4 ).

(5.36)

Surface Ai may also be a hypothetical one, closing an open configuration contained within a large environment. Example 5.6. Repeat Example 5.3 for a groove whose surface is gray and diffuse, with emittance ǫ, rather than black.

5.3 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

171

d

L T,∋

D

FIGURE 5-9 Cylindrical cavity with partial cover plate, Example 5.7.

Solution Using equation (5.32) for the open configuration we obtain  q1  1 − − 1 F1−2 q2 + Ho1 = σT 4 (1 − F1−2 ), i=1: ǫ ǫ  q2  1 − − 1 F2−1 q1 + Ho2 = σT 4 (1 − F2−1 ), i=2: ǫ ǫ where we have made use of the fact that Eb1 = Eb2 = σT 4 and ǫ1 = ǫ2 = ǫ. As in Example 5.3 we have √ F1−2 = F2−1 = 1 − 2/2 and Ho1 = qsol cos α, Ho2 = qsol sin α. Since we are only interested in the total heat loss we add the two equations, leading to     √ 1 1 − − 1 F1−2 (q1 + q2 ) = 2σT 4 − qsol (cos α + sin α), ǫ ǫ and h√ i a 2σT 4 − qsol (cos α + sin α) Q′ = a(q1 + q2 ) = .   √ 1 1+ 2 −1 ǫ Comparing this result with that of Example 5.3, we see that the heat loss due to emission is decreased (less emission, but more effective heat loss of emitted energy due to reflection from the opposing surface), as is the solar heat gain (since some of the irradiation is reflected back out of the cavity). Example 5.7. Consider the cavity shown in Fig. 5-9, which consists of a cylindrical hole of diameter D and length L. The top of the cavity is covered with a disk, which has a hole of diameter d. The entire inside of the cavity is isothermal at temperature T, and is covered with a gray, diffuse material of emittance ǫ. Determine the amount of radiation escaping from the cavity. Solution For simplicity, since the entire surface is isothermal and has the same emittance, we use a single zone A1 , which comprises the entire groove surface (sides, bottom, and top). Therefore, equation (5.32) reduces to     1 1 − − 1 F1−1 q1 = (1 − F1−1 )Eb1 . ǫ1 ǫ1 Since the total radiative energy rate leaving the cavity is Q1 = A1 q1 , we get Q1 =

1 − F1−1 A1 Eb1 .   1 1 − − 1 F1−1 ǫ1 ǫ1

172

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

The view factor F1−1 is easily determined by recognizing that Fo−1 = 1 (and Ao is the opening at the top) and, by reciprocity, F1−1 = 1 − F1−o = 1 −

Ao Ao Fo−1 = 1 − . A1 A1

Therefore, the radiative heat flux leaving the cavity, per unit area of opening, is   A o A1 1−1+ Eb1 Q1 Eb1 A1 Ao = .  =     1 1 Ao 1 Ao Ao − −1 1− 1+ −1 ǫ1 ǫ1 A1 ǫ1 A1 Thus, if Ao /A1 ≪ 1, the opening of the cavity behaves like a blackbody with emissive power Eb1 . Such cavities are commonly used in experimental methods in which blackbodies are needed for comparison. For example, a cavity with d/D = 1/2 and L/D = 2 has d2 πd2 /4 Ao = = A1 2πD2 /4−πd2 /4+πDL 2D2 −d2 +4DL (d/D)2 1/4 1 = = . = 2−1/4+4×2 39 2−(d/D)2 +4(L/D) For ǫ1 = 0.5 this results in an apparent emittance of ǫa =

Q1 = Ao Eb1

1 1 39 = = 0.975. =    1 1 1 Ao 40 −1 1+ 1+ −1 0.5 39 ǫ1 A1 

For computer calculations the Kronecker delta is introduced into equation (5.32), as was done for a black enclosure, leading to N " X δi j j=1

ǫj



! # N h X i 1 δi j − Fi− j Ebj − Hoi . − 1 Fi− j q j = ǫj

(5.37)

j=1

If all the temperatures are known and the radiative heat fluxes are to be determined, equation (5.37) may be cast in matrix form as C · q = A · eb − ho ,

(5.38)

where C and A are matrices with elements ! 1 − 1 Fi− j , ǫj ǫj = δi j − Fi− j ,

Ci j = Ai j

δi j



and q, eb , and ho are vectors of the unknown heat fluxes q j and the known emissive powers Ebj and external irradiations Ho j . Equation (5.38) is solved by matrix inversion as q = C−1 · [A · eb − ho ] .

(5.39)

If the emissive power is known over only some of the surfaces, and the heat fluxes are specified elsewhere, equation (5.38) may be rearranged into a similar equation for the vector containing all the unknowns. Subroutine graydiff is provided in Appendix F for the solution of the simultaneous equations (5.38), requiring surface information and a partial view factor matrix as input. The solution to a three-dimensional version of Example 5.4 is also given in the form of a program graydiffxch, which may be used as a starting point for the solution to other problems. Fortran90, C++ as well as Matlabr versions are provided. Several commercial solvers are also available, usually including software for view factor evaluation, such as TRASYS [1] and TSS [2].

5.4 ELECTRICAL NETWORK ANALOGY

1 A1F1–2

J1

J2

Q1

1– ∋1 A1 ∋1

Eb1

Q1

(a)

Eb1

J1

Q1

Q2

(b)

1– ∋ 1 A1∋ 1

J1

1 A1F1–2

J2

1– ∋ 2 A2 ∋ 2

Eb2

Q1

Q2 (c)

5.4

173

FIGURE 5-10 Electrical network analogy for infinite parallel plates: (a) space resistance, (b) surface resistance, (c) total resistance.

ELECTRICAL NETWORK ANALOGY

While equation (5.37) represents the most convenient set of governing equations for numerical calculations on today’s digital computers, some people prefer to get a physical feeling for the radiative exchange problem by representing it through an analogous electrical network, a method more suitable for analog computers—now nearly extinct. For completeness, we shall briefly present this electrical network method, which was first introduced by Oppenheim [3]. From equation (5.20) we have qi = Ji − Hi ,

i = 1, 2, . . . , N,

(5.40)

or, with equations (5.30) and (5.31), qi = Ji −

N X

Jj Fi− j − Hoi ,

(5.41)

j=1

=

N X

(Ji − Jj )Fi− j − Hoi,

i = 1, 2, . . . , N.

(5.42)

j=1

We shall first consider the simple case of two infinite parallel plates without external irradiation. Thus, N = 2, Hoi = 0, and Q1 = A1 q1 =

J1 − J2 = −Q2 . 1 A1 F1−2

(5.43)

As written, equation (5.43) may be interpreted as follows: If the radiosities are considered potentials, 1/A1 F1−2 is a radiative resistance between surfaces, or a space resistance, and Q is a radiative heat flow “current,” then equation (5.43) is identical to the one governing an electrical current flowing across a resistor due to a voltage potential, as indicated in Fig. 5-10a. The space resistance is a measure of how easily a radiative heat flux flows from one surface to another: The larger F1−2 , the more easily heat can travel from A1 to A2 , resulting in a smaller resistance. The same heat flux is also given by equation (5.31) as Q1 =

J2 − Eb2 Eb1 − J1 = = −Q2 , 1 − ǫ1 1 − ǫ2 A1 ǫ1 A2 ǫ2

(5.44)

where (1 − ǫi )/Ai ǫi are radiative surface resistances. This situation is shown in Fig. 5-10b. The surface resistance describes a surface’s ability to radiate. For the maximum radiator, a black surface, the resistance is zero. This fact implies that, for a finite heat flux, the potential drop

174

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

1 /A

i

Fi

–N

Q

iN

JN

E bi Qi

1– ∋ i A i∋ i

1/A i F i– j

Ji

1/A

i

1

1/A i F i –

Ai Hoi

Jj

Q ij F

i–

2

Q

i2

J2

Q i1

FIGURE 5-11 Network representation for radiative heat flux between surface Ai and all other surfaces.

J1

across a zero resistance must be zero, i.e., Ji = Ebi . Of course, the radiosities may be eliminated from equations (5.43) and (5.44), and Q1 =

Eb1 − Eb2 = −Q2 , 1 1 − ǫ2 1 − ǫ1 + + A1 ǫ1 A1 F1−2 A2 ǫ2

(5.45)

where the denominator is the total radiative resistance between surfaces A1 and A2 . Since the three resistances are in series they simply add up as electrical resistances do; see Fig. 5-10c. This network analogy is readily extended to more complicated situations by rewriting equation (5.42) as Qi =

N N X Ebi − Ji X Ji − Jj = − Ai Hoi = Qi− j − Ai Hoi . 1 − ǫi 1 j=1 j=1 Ai ǫi Ai Fi− j

(5.46)

Thus, the total heat flux at surface i is the net radiative exchange between Ai and all the other surfaces in the enclosure. The electrical analog is shown in Fig. 5-11, where the current flowing from Ebi to Ji is divided into N parallel lines, each with a different potential difference and with different resistors. Example 5.8. Consider a solar collector shown in Fig. 5-12a. The collector consists of a glass cover plate, a collector plate, and side walls. We shall assume that the glass is totally transparent to solar irradiation, which penetrates through the glass and hits the absorber plate with a strength of 1000 W/m2 . The absorber plate is black and is kept at a constant temperature T1 = 77◦ C by heating water flowing underneath it. The side walls are insulated and made of a material with emittance ǫ2 = 0.5. The glass cover may be considered opaque to thermal (i.e., infrared) radiation with an emittance ǫ3 = 0.9. The collector is 1 m × 1 m × 10 cm in dimension and is reasonably evacuated to suppress free convection between absorber plate and glass cover. The convective heat transfer coefficient at the top of the glass cover is known to be h = 5.0 W/m2 K, and the temperature of the ambient is Ta = 17◦ C. Estimate the collected energy for normal solar incidence. Solution We may construct an equivalent network (Fig. 5-12b), leading to Q1 =

σ(T14 − Ta4 ) − A 1 qs , 1 − ǫ3 R13 + + R3a A3 ǫ3

where R13 is the total resistance between surfaces A1 and A3 , and R3a is the resistance, by radiation as well as free convection, between glass cover and environment. We note that, since A2 is insulated, there

5.4 ELECTRICAL NETWORK ANALOGY

Ta

=

17 °C

qs = 1000 W/m

175

2

h = 5 W/m2 K

∋ 3 = 0.9 10 cm

q2 = 0, ∋ 2 = 0.5

T1 = 77°C, ∋ 1 = 1 1m (a) J2 = Eb2

1

F 1– /A 1

1/

2

A

3F 3–

2

Eb1 Q1 A1qs

Eb3

J3 1–∋3 A3∋ 3

1 /A1F1–3 (b)

Q 3a R3a

Eba FIGURE 5-12 Schematics for Example 5.8: (a) geometry, (b) network.

is no heat flux entering/leaving at Eb2 and, from equation (5.44), J2 = Eb2 . Thus, the total resistance between A1 and A3 comes from two parallel circuits, one with resistance 1/(A1 F1−3 ) and the other with two resistances in series, 1/(A1 F1−2 ) and 1/(A3 F3−2 ), or 1 1 1 + = R13 1/(A1 F1−3 ) 1/(A1 F1−2 ) + 1/(A3 F3−2 )   = A1 F1−3 + 12 A1 F1−2 = A1 F1−3 + 21 F1−2 ,

where we have used the fact that A1 F1−2 = A3 F3−2 by symmetry. From Configuration 38 in Appendix D we obtain, with X = Y = 10, F1−3 = 0.827 and F1−2 = 1 − F1−3 = 0.173, and .h i R13 = 1 1 m2 × (0.827 + 0.5 × 0.173) = 1.095 m−2 .

The resistance between glass cover and ambient is a little more complicated. The total heat loss from the cover plate, by free convection and radiation, is Q3a = ǫ3 A3 σ(T34 − Ta4 ) + hA3 (T3 − Ta ), where we have assumed that the environment (sky) radiates to the collector with the ambient temperature Ta . To convert this to the correct form we rewrite it as    h(T3 − Ta )  4 4  Q3a = σ(T3 − Ta )A3 ǫ3 + , σ(T34 − Ta4 )  or        1 1 h T3 − Ta  h  = A ǫ = A3 ǫ3 + +  .   3 3  3 3 2 2 4 4 R3a σ T3 − Ta σ T3 + T3 Ta + T3 Ta + Ta  As a first approximation, if T3 is not too different from Ta , ! ! 1 h 1 5 W/m2 K 2 ≃ A3 ǫ3 + m2. = 1 m 0.9+ = R3a 4σTa3 4×5.670×10−8 W/m2 K4 ×(273+17)3 K3 0.554 Finally, substituting the resistances into the expression for Q1 we get h i 5.670×10−8 W/m2 K4 (273+77)4 −(273+17)4 K4 − 1 m2 × 1000 W/m2 Q1 = 1−0.9 −2 −2 +0.554 m 1.095 m + 0.9 m2 = −744 W.

176

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

∋ N–1,o ∋ N–1, i

∋ N,o

∋ N,i

∋o

Shield 1 Di

∋ 3i

∋ 1i

∋ 3o

∋1o

2 3

∋ 2i

∋i

∋4i

∋4o

∋2o

4

N –1

N

FIGURE 5-13 Concentric cylinders (or spheres) with N radiation shields between them.

Since the system could collect a theoretical maximum of −1000 W, the collector efficiency is ηcollector =

Q1 744 = = 0.744 = 74.4%. A1 qs 1000

This efficiency should be compared with an uncovered black collector plate, whose net heat flux would be h i Q1 = A1 σ(T14 − Ta4 ) + h(T1 − Ta ) − qs i h = 1 m2 5.670×10−8 ×(3504 −2904 )+5×(350−290) − 1000 W/m2 = −250 W.

Thus, an unprotected collector at that temperature would have an efficiency of only 25%.

The electrical network analogy is a very simple and physically appealing approach for simple two- and three-surface enclosures, such as the one of the previous example. However, in more complicated enclosures with multiple surfaces the method quickly becomes tedious and intractable.

5.5

RADIATION SHIELDS

In high-performance insulating materials it is common to suppress conductive and convective heat transfer by evacuating the space between two surfaces. This leaves thermal radiation as the dominant heat loss mode even for low-temperature applications such as insulation in cryogenic storage tanks. The radiation loss may be minimized by placing a multitude of closely spaced, parallel, highly reflective radiation shields between the surfaces. The radiation shields are generally made of thin metallic foils or, to reduce conductive losses further, of dielectric foils coated with metallic films. In either case radiation shields tend to be very specular reflectors. However, for closely spaced shields the directional behavior of the reflectance tends to be irrelevant and assuming diffuse reflectances gives excellent accuracy (see also Example 6.9 in the following chapter). A typical arrangement for N radiation shields between two concentric cylinders (or concentric spheres) is shown in Fig. 5-13. This geometry includes the case of parallel plates for

5.5 RADIATION SHIELDS

177

large (and nearly equal) radii. Let the inner cylinder have temperature Ti , surface area Ai , and emittance ǫi . Similarly, each shield has temperature Tn (unknown), An , ǫni (on its inner surface), and ǫno (on its outer surface). The last shield, AN , faces the outer cylinder with To , Ao and ǫo . The net radiative heat rate leaving Ai is, of course, equal to the heat rate going through each shield and to the one arriving at Ao . This net heat rate may be readily determined from the electrical network analogy, or by repeated application of the enclosure relations, equation (5.32). However, this is the type of problem for which the network analogy truly shines and we will use this method here. The case of concentric surfaces was already evaluated in Example 5.5, so that the net heat rate between any two of the concentric cylinders is then Q=

Eb j − Ebk R j−k

,

R j−k =

  1 1 1 −1 . + ǫ j A j Ak ǫk

(5.47)

Therefore, we may write QRi−1i = Ebi − Eb1 , QR1o−2i = Eb1 − Eb2 , .. . QRNo−o = EbN − Ebo . Adding all these equations eliminates all the unknown shield temperatures, and, after solving for the heat flux, we obtain Q=

Ebi − Ebo . PN−1 Ri−1i + n=1 Rno−n+1,i + RNo−o

(5.48)

Example 5.9. A Dewar holding 4 liters of liquid helium at 4.2 K consists essentially of two concentric stainless steel (ǫ = 0.3) cylinders of 50 cm length, and inner and outer diameters of Di = 10 cm and Do = 20 cm, respectively. The space between the cylinders is evacuated to a high vacuum to eliminate conductive/convective heat losses. Radiation shields are to be placed between the Dewar walls to reduce radiative losses to the point that it takes 24 hours for the 4-liter filling to evaporate if the Dewar is placed into an environment at 298 K. For the purpose of this example the following may be assumed: (i) End losses as well as conduction/convection losses are negligible, (ii) the wall temperatures are at Ti = 4.2 K and To = 298 K, respectively, and (iii) radiation is one-dimensional. Thin plastic sheets coated on both sides with aluminum (ǫ = 0.05) are available as shield material. Estimate the number of shields required. The heat of evaporation for helium at atmospheric pressure is hfg,He = 20.94 J/g (which is a very low value compared with other liquids), and the liquid density is ρHe = 0.125 g/cm3 [4]. Solution The total heat required to evaporate 4 liters of liquid helium is Q = ρHe VHe hfg,He = 0.125

g 103 cm3 J × 4 liters × × 20.94 = 10.47 kJ. 3 liter g cm

If all of this energy is supplied through radial radiation over a time period of 24 hours, one infers that the heat flux in equation (5.48) must be held at or below Q˙ = Q/24 h = 10,470 J/24 h×(1 h/3600 s) = 0.1212 W, or qi = Q˙ /Ai = 0.1212 W/(π × 10 cm × 50 cm) = 7.71 × 10−5 W/cm2 . Therefore, the total resistance must, from equation (5.48), be a minimum of Ai Rtot = |Ebi − Ebo |/qi = 5.670 × 10−12 × |4.24 − 2984 |/7.71 × 10−5 = 580.0. We note from equation (5.47) that the resistances are inversely proportional to shield area. Therefore, it is best to place the shields as close to the inner cylinder as possible. We will assume that the shields

178

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

can be so closely spaced that Ai ≃ A2 ≃ . . . ≃ AN = As = πDs L, with Ds = 11 cm. Evaluating the total resistance from equations (5.47) and (5.48), we find Ai Rtot =

N−1       1 Ai 1 1 Ai X 2 Ai 1 Ai + −1 + −1 + + −1 , ǫw ǫs As ǫs As ǫs As ǫw Ao n=1

where ǫw = 0.3 is the emittance of the (stainless steel) walls, and ǫs = 0.05 is the emittance of the (aluminized) shields. Since the elements of the series in the last equation do not depend on n, we may solve for N as   1 1 Ai − −1 Ai Rtot − ǫw ǫw Ao N =   2 Ai −1 ǫs As   1 1 10 580.0 − 0.3 − 0.3 − 1 20   = 2 10 0.05 − 1 11 = 16.23. Therefore, a minimum of 17 radiation shields would be required. Note from equation (5.35) that, without radiation shields, qi =

|Ebi − Ebo | 5.670 × 10−12 |4.24 − 2984 | =     1 1 1 1 1 Ai + −1 × + −1 0.3 0.3 2 ǫw ǫw Ao

= 9.94 × 10−3 W/cm2 , that is, the heat loss is approximately 100 times larger!

5.6 SOLUTION METHODS FOR THE GOVERNING INTEGRAL EQUATIONS The usefulness of the method described in the previous sections is limited by the fact that it requires the radiosity to be constant over each subsurface. This is rarely the case if the subsurfaces of the enclosure are relatively large (as compared with typical distances between surfaces). Today, with the advent of powerful digital computers, more accurate solutions are usually obtained by increasing the number of subsurfaces, N, in equation (5.37), which then become simply a finite-difference solution to the integral equation (5.28). Still, there are times when more accurate methods for the solution of equation (5.28) are desired (for computational efficiency), or when exact or approximate solutions are sought in explicit form. Therefore, we shall give here a very brief outline of such solution methods. If radiosity J is to be determined, the governing equation that needs to be solved is either equation (5.24), if the surface temperature is given, or equation (5.25), if surface heat flux is specified. If unknown temperatures or heat fluxes are to be determined directly, equation (5.28) must be solved. In all cases the governing equation may be written as a Fredholm integral equation of the second kind, Z φ(r) = f (r) + K(r, r′ ) φ(r′ ) dA′ , (5.49) A

where K(r, r′ ) is called the kernel of the integral equation, f (r) is a known function, and φ(r) is the function to be determined (e.g., radiosity or heat flux). Comprehensive discussions for the treatment of such integral equations are given in mathematical texts such as Courant and Hilbert [5] or Hildebrand [6]. A number of radiative heat transfer examples have been discussed ¨ ¸ ik [7]. by Ozis

5.6 SOLUTION METHODS FOR THE GOVERNING INTEGRAL EQUATIONS

179

Numerical solutions to equation (5.49) may be found in a number of ways. In the method of successive approximation a first guess of φ(r) = f (r) is made with which the integral in equation (5.49) is evaluated (analytically in some simple situations, but more often through numerical quadrature). This leads to an improved value for φ(r), which is substituted back into the integral, and so on. This scheme is known to converge for all surface radiation problems. Another possible solution method is reduction to algebraic equations by using numerical quadrature for the integral, i.e., replacing it by a series of quadrature coefficients and nodal values. This leads to a set of equations similar to equation (5.37), but of higher accuracy. This type of solution method is most easily extended to arbitrary, three-dimensional geometries, for example, as recently demonstrated by Daun and Hollands [8], who employed nonuniform rational B-splines (NURBS) to express the surfaces. A third method of solution has been given by Sparrow and Haji-Sheikh [9], who demonstrated that the method of variational calculus may be applied to general problems governed by a Fredholm integral equation. Most early numerical solutions in the literature dealt with two very basic systems. The problem of two-dimensional parallel plates of finite width was studied in some detail by Sparrow and coworkers [9–11], using the variational method. The majority of studies have concentrated on radiation from cylindrical holes because of the importance of this geometry for cylindrical tube flow, as well as for the preparation of a blackbody for calibrating radiative property measurements. The problem of an infinitely long isothermal hole radiating from its opening was first studied by Buckley [12] and by Eckert [13]. Buckley’s work appears to be the first employing the kernel approximation method. Much later, the same problem was solved exactly through the method of successive approximation (with numerical quadrature) by Sparrow and Albers [14]. A finite hole, but with both ends open, was studied by a number of investigators. Usiskin and Siegel [15] considered the constant wall heat flux case, using the kernel approximation as well as a variational approach. The constant wall temperature case was studied by Lin and Sparrow [16], and combined convection/surface radiation was investigated by Perlmutter and Siegel [17, 18]. Of greater importance for the manufacture of a blackbody is the isothermal cylindrical cavity of finite depth, which was studied by Sparrow and coworkers [19, 20] using successive approximations. If part of the opening is covered by a flat ring with a smaller hole, such a cavity behaves like a blackbody for very small L/R ratios. This problem was studied by Alfano [21] and Alfano and Sarno [22]. Because of their importance for the manufacture of blackbody cavities these results are summarized in Table 5.1. A detector removed from the cavity will sense a signal proportional to the intensity leaving the bottom center of the cavity in the normal direction. Thus the effectiveness of the blackbody is measured by how close to unity the ratio In /Ib (T) is. For perfectly diffuse reflectors, In = J/π, and with Ib = σT 4 /π an apparent emittance is defined as ǫa = In /Ib (T) = J/σT 4 .

(5.50)

To give an outline of how the different methods may be applied we shall, over the following few pages, solve the same simple example by three different methods, the first two being “exact,” and the third being the kernel approximation. Example 5.10. Consider two long parallel plates of width w as shown in Fig. 5-14. Both plates are isothermal at the (same) temperature T, and both have a gray, diffuse emittance of ǫ. The plates are separated by a distance h and are placed in a large, cold environment. Determine the local radiative heat fluxes along the plate using the method of successive approximation. Solution From equation (5.24) we find, with dFdi−di = 0, J1 (x1 ) = ǫσT 4 + (1 − ǫ) J2 (x2 ) = ǫσT 4 + (1 − ǫ)

Z Z

w

J2 (x2 ) dFd1−d2 , 0 w

J1 (x1 ) dFd2−d1 , 0

180

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

TABLE 5.1

Apparent emittance, ǫa = J/σT 4 , at the bottom center of an isothermal partially covered cylindrical cavity [21, 22]. ǫa

2Ri

ǫ

Ri /R

(L/R = 2)

(L/R = 4)

(L/R = 8)

0.25

0.4 0.6 0.8 1.0

0.916 0.829 0.732 0.640

0.968 0.931 0.888 0.844

0.990 0.981 0.969 0.965

0.50

0.4 0.6 0.8 1.0

0.968 0.932 0.887 0.839

0.990 0.979 0.964 0.946

0.998 0.995 0.992 0.989

0.75

0.4 0.6 0.8 1.0

0.988 0.975 0.958 0.939

0.997 0.997 0.988 0.982

0.999 0.998 0.997 0.996

L T,∋ 2R

x2, ξ dx2

A2 : T, ∋

s12

h

φ

A1 : T, ∋

dx1

x1, ξ

FIGURE 5-14 Radiative exchange between two long isothermal parallel plates.

w

and, from Configuration 1 in Appendix D, with s12 = h/cos φ, s12 dφ = dx2 cos φ, and cos φ = p h/ h2 + (x2 − x1 )2 , dx1 dFd1−d2 = dx2 dFd2−d1 =

cos3 φ h2 dx1 dx2 1 1 cos φ dφ dx1 = dx1 dx2 = . 2 2 2h 2 [h + (x1 − x2 )2 ]3/2

Introducing nondimensional variables W = w/h, ξ = x/h, and J(x) = J(x)/σT 4 , and realizing that, as a result of symmetry, J1 = J2 (and q1 = q2 ), we may simplify the governing integral equation to 1 2

J(ξ) = ǫ + (1 − ǫ) Making a first guess of J

(1)

Z

W

J(ξ′ ) 0

dξ′ . [1 + (ξ′ − ξ)2 ]3/2

= ǫ we obtain a second guess by substitution, (

1 2

J (2) (ξ) = ǫ 1 + (1 − ǫ)

Z

W 0

dξ′ [1 + (ξ′ − ξ)2 ]3/2

)

        W − ξ ξ 1   = ǫ + p 1 + (1 − ǫ)  p   .   2 2 1 + (W − ξ) 1 + ξ2 

(5.51)

5.6 SOLUTION METHODS FOR THE GOVERNING INTEGRAL EQUATIONS

181

1.00

4





Nondimensional heat flux, Ψ/ = q/ σT

= 0.1

∋ ∋



0.75 = 0.5

= 0.9

0.50

1st successive approx. Kernel approximation Exact 5-point quadrature

0.00

0.25

0.50 0.75 Location along plates, ξ = x/w

1.00

FIGURE 5-15 Local radiative heat flux on long, isothermal parallel plates, determined by various methods.

Repeating the procedure we get        1 ξ W−ξ (3)  J (ξ) = ǫ 1 + (1 − ǫ)  p + p   2 2 2 1 + (W − ǫ) 1+ξ    Z W     dξ′ 1 W − ξ′ ξ′  2   + (1 − ǫ) + p   p , 3/2  ′ 2 ′ 2 4 ′2 0 1 + (W − ξ ) 1 + ξ [1 + (ξ − ξ) ] 

where the last integral becomes quite involved. We shall stop at this point since further successive integrations would have to be carried out numerically. It is clear from the above expression that the terms in the series diminish as ǫ [(1 − ǫ)W]n , i.e., few successive iterations are necessary for surfaces with low reflectances and/or w/h ratios. Once the radiosity has been determined the local heat flux follows (2) from equation (5.26). Limiting ourselves to J (single successive approximation), this yields q(ξ) ǫ = [1 − J(ξ)] 1−ǫ σT 4      W−ξ ξ ǫ2   − O ǫ2 (1 − ǫ)W 2 , + p = ǫ− p  2  1 + (W − ξ)2 2 1+ξ

Ψ(ξ) =

where O(z) is shorthand for “order of magnitude z.” Some results are shown in Fig. 5-15 and compared with other solution methods for the case of W = w/h = 1 and three values of the emittance. Observe that the heat loss is a minimum at the center of the plate, since this location receives maximum irradiation from the other plate (i.e., the view factor from this location to the opposing plate is maximum). For decreasing ǫ the heat loss increases, of course, since more is emitted; however, this increase is less than linear since also more energy is coming in, of which a larger fraction is absorbed. The first successive approximation does very well for small and large ǫ as expected from the order of magnitude of the neglected terms. Example 5.11. Repeat Example 5.10 using numerical quadrature. Solution The governing equation is, of course, again equation (5.51). We shall approximate the integral on the right-hand side by a series obtained through numerical integration, or quadrature. In this method an integral is approximated by a weighted series of the integrand evaluated at a number of nodal points; or

182

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

Z

b

f (ξ, ξ′ ) dξ′ ≃ (b − a) a

J X

J X

c j f (ξ, ξ j ),

j=1

c j = 1.

(5.52)

j=1

Here the ξ j represent J locations between a and b, and the c j are weight coefficients. The nodal points ξ j may be equally spaced for easy presentation of results (Newton–Cotes quadrature), or their location may be optimized for increased accuracy (Gaussian quadrature); for a detailed treatment of quadrature see, for example, the book by Froberg [23]. ¨ Using equation (5.52) in equation (5.51) we obtain

Ji = ǫ + (1 − ǫ)W

J X

c j Jj fij ,

i = 1, 2, . . . , J,

j=1

where fi j =

i3/2 1.h 1 + (ξ j − ξi )2 . 2

This system of equations may be further simplified by utilizing the symmetry of the problem, i.e., J(ξ) = J(W − ξ). Assuming that nodes are placed symmetrically about the centerline, ξJ+1−j = ξ j , leads to c J+1−j = c j and JJ+1−j = Jj , or     (J−1)/2     J+1  X  , c J [ f + f ] + c J f J odd: Ji = ǫ + (1 − ǫ)W  , i = 1, 2, . . . , j ij i,J+1−j (J+1)/2 i,(J+1)/2    j (J+1)/2   2  j=1 

J even:

Ji = ǫ + (1 − ǫ)W

J/2 X

c j Jj ( fij + fi,J+1−j ),

j=1

J i = 1, 2, . . . , . 2

The values of the radiosities may be determined by successive approximation, or by direct matrix inversion. In Fig. 5-15 the simple case of J = 5 (resulting in three simultaneous equations) is included, using Newton–Cotes quadrature with ξ j = W( j − 1)/4 and c1 = c5 = 7/90, c2 = c4 = 32/90, and c3 = 12/90 [23].

Exact analytical solutions that yield explicit relations for the unknown radiosity are rare and limited to a few special geometries. However, approximate analytical solutions may be found for many geometries through the kernel approximation method. In this method the kernel K(x, x′ ) ′ is approximated by a linear series of special functions such as e−ax , cos ax′ , cosh ax′ , and so on (i.e., functions that, after one or two differentiations with respect to x′ , turn back into the original function except for a constant factor). It is then often possible to convert integral equation (5.49) into a differential equation that may be solved explicitly. The method is best illustrated through an example. Example 5.12. Repeat Example 5.11 using the kernel approximation method. Solution We again need to solve equation (5.51), this time by approximating the kernel. For convenience we shall choose a simple exponential form, K(ξ, ξ′ ) =

1 [1 + (ξ′ − ξ)2 ]3/2



≃ a e−b|ξ −ξ| .

We shall determine “optimum” parameters a and b by letting the approximation satisfy the 0th and 1st moments. This implies multiplying the expression by |ξ′ − ξ| raised to the 0th and 1st powers, followed by integration over the entire domain for |ξ′ − ξ|, i.e., from 0 to ∞ (since W could be arbitrarily large).3 3 Using the actual W at hand will result in a better approximation, but new values for a and b must be determined if W is changed; in addition, the mathematics become considerably more involved.

5.6 SOLUTION METHODS FOR THE GOVERNING INTEGRAL EQUATIONS

183

Thus, Z

0th moment:

Z

1st moment:

∞ 0 ∞ 0

dx =1= (1 + x2 )3/2

Z

x dx =1= (1 + x2 )3/2

Z



a e−bx dx =

a , b

a e−bx x dx =

a , b2

0 x 0

leading to a = b = 1 and ′

K(ξ, ξ′ ) ≃ e−|ξ −ξ| . Substituting this expression into equation (5.51) leads to "Z ξ # Z W 1 ′ ′ J(ξ) ≃ ǫ + (1 − ǫ) J(ξ′ ) e−(ξ−ξ ) dξ′ + J(ξ′ ) e−(ξ −ξ) dξ′ . 2 0 ξ We shall now differentiate this expression twice with respect to ξ, for which we need to employ Leibniz’s rule, equation (3.106). Therefore, " # Z ξ Z W dJ 1 ′ ′ = (1 − ǫ) J(ξ) − J(ξ′ ) e−(ξ−ξ ) dξ′ − J(ξ) + J(ξ′ ) e−(ξ −ξ) dξ′ , dξ 2 0 ξ " # Z ξ Z W d2 J 1 ′ −(ξ−ξ′ ) ′ ′ −(ξ′ −ξ) ′ = (1 − ǫ) − J (ξ) + J (ξ ) e dξ − J (ξ) + J (ξ ) e dξ , 2 dξ2 0 ξ or, by comparison with the expression for J(ξ), d2 J dξ2

= J − ǫ − (1 − ǫ) J = ǫ( J − 1).

Thus, the governing integral equation has been converted into a second-order ordinary differential equation, which is readily solved as

J(ξ) = 1 + C1 e−

√ ǫξ

+ C2 e+

√ ǫξ

.

While an integral equation does not require any boundary conditions, we have converted the governing equation into a differential equation that requires two boundary conditions in order to determine C1 and C2 . The dilemma is overcome by substituting the general solution back into the governing integral equation (with approximated kernel). This calculation can be done for variable values of ξ by comparing coefficients of independent functions of ξ, or simply for two arbitrarily selected values for ξ. The first method gives the engineer proof that his analysis is without mistake, but is usually considerably more tedious. Often it is also possible to employ symmetry, as is the case here, since J(ξ) = J(W − ξ) or h √ i i h √ i h √ √ √ √ √ C1 e− ǫξ − e− ǫ(W−ξ) = −C2 e ǫξ − e ǫ(W−ξ) = C2 e ǫW e− ǫξ − e− ǫ(W−ξ) , or

C1 = C2 e

√ ǫW

.

Consequently, h

J(ξ) = 1 + C1 e−

√ ǫξ

+ e−

i √ ǫ(W−ξ)

,

and substituting this expression into the governing equation at ξ = 0 gives   √ J(0) = 1 + C1 1+ e− ǫW Z Wn h √ ′ io ′ √ 1 ′ 1+C1 e− ǫξ + e− ǫ(W−ξ ) e−ξ dξ′ = ǫ + (1−ǫ) 2 0 Z Wn io h √ ′ √ 1 ′ ′ ′ = ǫ + (1−ǫ) e−ξ + C1 e−(1+ ǫ)ξ + e−ξ − ǫ(W−ξ ) dξ′ 2 0 √ ( " −(1+√ǫ)ξ′ ′ #) W 1 e e−ξ− ǫ(W−ξ ) −ξ′ = ǫ − (1−ǫ) e +C1 √ + √ 2 1+ ǫ 1− ǫ 0 = ǫ+

√ √ ( " #) 1− e−(1+ ǫ)W e− ǫW − e−W 1 (1−ǫ) 1−e−W +C1 + . √ √ 2 1+ ǫ 1− ǫ

184

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

qsun

R

ϕ ϕ´

R d A´

ϕ dA

∋ , T = const FIGURE 5-16 Isothermal hemispherical cavity irradiated normally by the sun, Example 5.13.

Solving this for C1 gives

or

h  √    √  √   i √ √ 1−ǫ− 21 (1−ǫ)(1−e−W ) = C1 21 1− ǫ 1−e−(1+ ǫ)W + 12 1+ ǫ e− ǫW −e−W − 1−e− ǫW n  √  o   √  √  √ i  h  √  √ √ 1 −W = C1 21 1− ǫ + 21 1+ ǫ e− ǫW −1−e− ǫW − 12 1− ǫ e− ǫW + 21 1+ ǫ e−W , 2 (1−ǫ) 1+e C1 = −

(1 +



1−ǫ ǫ) + (1 −



ǫ) e−

√ ǫW

and √

J(ξ) = 1 − (1 − ǫ)



e− ǫξ + e− ǫ(W−ξ) . √ √ √ (1 + ǫ) + (1 − ǫ) e− ǫW

Finally, the nondimensional heat flux follows as i h √ √ ǫ e− ǫξ + e− ǫ(W−ξ) ǫ [1 − J(ξ)] = , Ψ(ξ) = √ √ √ 1−ǫ (1 + ǫ) + (1 − ǫ) e− ǫW which is also included in Fig. 5-15. ′ Note that e−|ξ −ξ| is not a particularly good approximation for the kernel, since the actual kernel has a zero first derivative at ξ′ = ξ. A better approximation can be obtained by using ′



K(ξ, ξ′ ) ≃ a1 e−b1 |ξ −ξ| + a2 e−b2 |ξ −ξ| (with a1 > 1 and a2 < 0). If W is relatively small, say < 12 , a good approximation may be obtained using K(ξ, ξ′ ) ≃ cos a(ξ′ − ξ) (since the kernel has an inflection point at |ξ′ − ξ| = 21 ).

We shall conclude this chapter with two examples that demonstrate that exact analytical solutions are possible for a few simple geometries for which the view factors between area elements attain certain special forms. Example 5.13. Consider a hemispherical cavity irradiated by the sun as shown in Fig. 5-16. The surface of the cavity is kept isothermal at temperature T and is coated with a gray, diffuse material with emittance ǫ. Assuming that the cavity is, aside from the solar irradiation, exposed to cold surroundings, determine the local heat flux rates that are necessary to maintain the cavity surface at constant temperature.

5.6 SOLUTION METHODS FOR THE GOVERNING INTEGRAL EQUATIONS

185

Solution From equation (5.24) the local radiosity at position (ϕ, ψ) is determined as J(ϕ) = ǫσT 4 + (1 − ǫ)H(ϕ) "Z # J(ϕ′ ) dFdA−dA′ + Ho (ϕ) , = ǫσT 4 + (1 − ǫ) A

where we have already stated that radiosity is a function of ϕ only, i.e., there is no dependence on azimuthal angle ψ. The view factor between infinitesimal areas on a sphere is known from the inside sphere method, equation (4.33), as dFdA−dA′ =

R2 sin ϕ′ dϕ′ dψ′ dA′ = . 4πR2 4πR2

The external irradiation at dA is readily determined as Ho (ϕ) = qsun cos ϕ, and the expression for radiosity becomes # "Z 2π Z π/2 sin ϕ′ dϕ′ dψ′ + qsun cos ϕ J(ϕ′ ) J(ϕ) = ǫσT 4 + (1 − ǫ) 4π 0 0 Z π/2 1−ǫ = ǫσT 4 + J(ϕ′ ) sin ϕ′ dϕ′ + (1 − ǫ)qsun cos ϕ. 2 0 Because of the unique behavior of view factors between sphere surface elements we note that the irradiation at location ϕ that arrives from other parts of the sphere, Hs , does not depend on ϕ. Thus, Hs =

1 2

Z

π/2

J(ϕ′ ) sin ϕ′ dϕ′ = const, 0

and J(ϕ) = ǫσT 4 + (1 − ǫ)Hs + (1 − ǫ)qsun cos ϕ. Substituting this equation into the expression for Hs leads to Hs = = or Hs =

1 2

Z

π/2

0 1 4 2 ǫσT

h i ǫσT 4 + (1 − ǫ)Hs + (1 − ǫ)qsun cos ϕ′ sin ϕ′ dϕ′

+ 12 (1 − ǫ)Hs + 14 (1 − ǫ)qsun ,

1−ǫ ǫ σT 4 + qsun . 1+ǫ 2(1 + ǫ)

An energy balance at dA gives q(ϕ) = ǫσT 4 − ǫH(ϕ) = ǫ(σT 4 − Hs − qsun cos ϕ) or

! # 1−ǫ σT 4 − + cos ϕ qsun . q(ϕ) = ǫ 1+ǫ 2(1 + ǫ) "

We observe from this example that in problems where all radiating surfaces are part of a sphere, none of the view factors involved depend on the location of the originating surface, and an exact analytical solution can always be found in a similar fashion. Apparently, this was first recognized by Jensen [24] and reported in the book by Jakob [25]. Exact analytical solutions are also possible for such configurations where all relevant view factors have repeating derivatives (as in the kernel approximation). Example 5.14. A long thin radiating wire is to be employed as an infrared light source. To maximize the output of infrared energy into the desired direction, the wire is fitted with an insulated, highly reflective sheath as shown in Fig. 5-17. The sheath is cylindrical with radius R (which is much larger than the diameter of the wire), and has a cutout of half-angle ϕ to let the concentrated infrared light escape. Assuming that the wire is heated with a power of Q′ W/m length of wire, and that the sheath

186

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

d A´

ϕ ϕ

β θ´ Wire

θ

R

β

dA

FIGURE 5-17 Thin radiating wire with radiating sheath, Example 5.14.

can lose heat only by radiation and only from its inside surface, determine the temperature distribution across the sheath. Solution From an energy balance on a surface element dA it follows from equation (5.20) that, with q(θ) = 0, σT 4 (θ) = J(θ) = H(θ), and H(θ) =

Z

J(θ′ ) dFdA−dA′ + Ho (θ). A

We may treat the energy emitted from the wire as external radiation (neglecting absorption by the wire since it is so small). Since the total released energy will spread equally into all directions, we find Ho (θ) = Q′ /2πR = const. The view factor dFdA−dA′ between two infinitely long strips on the cylinder surface is given by Configuration 1 in Appendix D as FdA−dA′ =

1 2

cos β dβ,

where the angle β is indicated in Fig. 5-17 and may be related to θ through 2β + |θ′ − θ| = π. Differentiating β with respect to θ′ we obtain dβ = ±dθ′ /2, depending on whether θ′ is larger or less than θ. Substituting for β in the view factor, this becomes  ′  π θ′ − θ 1 ′ 1 1 θ − θ ′ − dθ = sin dθ , FdA−dA′ = cos 2 2 2 2 4 2

where the ± has been omitted since the view factor is always positive (i.e., |dβ| is to be used). Substituting this into the above relationship for radiosity we obtain Z π−ϕ ′ 1 θ − θ ′ dθ + Ho J(θ′ ) sin J(θ) = 4 −π+ϕ 2 Z θ Z π−ϕ 1 θ − θ′ ′ 1 θ′ − θ ′ = dθ + dθ + Ho . J(θ′ ) sin J(θ′ ) sin 4 −π+ϕ 2 4 θ 2

Since the view factor in the integrand has repetitive derivatives we may convert this integral equation into a second-order differential equation, as was done in the kernel approximation method. Differentiating

5.6 SOLUTION METHODS FOR THE GOVERNING INTEGRAL EQUATIONS

187

twice, we have dJ 1 = dθ 8

Z

θ

J(θ′ ) cos −π+ϕ

d2 J 1 1 = J(θ) − 8 16 dθ2

Z

θ − θ′ ′ 1 dθ − 2 8

θ

J(θ′ ) sin −π+ϕ

Z

π−ϕ

J(θ′ ) cos θ

θ′ − θ ′ dθ , 2

θ − θ′ ′ 1 1 dθ + J(θ) − 2 8 16

Z

π−ϕ

J(θ′ ) sin θ

θ′ − θ ′ dθ . 2

Comparing this result with the above integral equation for J(θ) we find d2 J = 14 J(θ) − 14 [J(θ) − Ho ] = dθ2

1 4

Ho .

This equation is readily solved as J(θ) =

1 8

Ho θ 2 + C 1 θ + C 2 .

The two integration constants must now be determined by substituting the solution back into the governing integral equation. However, C1 may be determined from symmetry since, for this problem, J(θ) = J(−θ) and C1 = 0. To determine C2 we evaluate J at θ = 0: J(0) = C2 =

1 4

=

1 2

=

1 2

0

Z  ′ 1 π−ϕ ′ θ θ′ ′ dθ′ + dθ + Ho J(θ′ ) sin − J(θ ) sin 2 4 2 −π+ϕ 0 Z π−ϕ θ′ ′ dθ + Ho J(θ′ ) sin 2 0 Z π−ϕ   θ′ ′ Ho ′2 C2 + θ sin dθ + Ho . 8 2 0

Z

Integrating twice by parts we obtain Z π−ϕ   θ′ π−ϕ Ho Ho ′2 θ′ ′ + θ cos dθ θ′ cos C2 = Ho − C2 + 8 2 0 4 0 2 ! Z π−ϕ     π ϕ θ′ π−ϕ Ho Ho ′ θ′ ′ 2 θ sin (π − ϕ) cos − dθ + C2 + = H o − C2 + − sin 8 2 2 2 2 0 2 0 !     ϕ Ho θ′ π−ϕ Ho π ϕ + 2 cos = H o + C2 − C 2 + (π − ϕ) sin (π − ϕ)2 sin + − 8 2 2 2 2 2 0   ϕ ϕ ϕ Ho Ho = Ho + C 2 − C 2 + (π − ϕ)2 sin + (π − ϕ) cos + Ho sin − Ho . 8 2 2 2 2 Solving this equation for C2 we get   π−ϕ ϕ 1 C2 = Ho 1 + cos − (π − ϕ)2 . 2 2 8

Therefore, T 4 (θ) =

 i ϕ 1h π−ϕ Q′ J = cos − 1+ (π − ϕ)2 − θ2 . σ 2πRσ 2 2 8

We find that the temperature has a minimum at θ = 0, since around that location the view factor to the opening is maximum, resulting in a maximum of escaping energy. The temperature level increases as ϕ decreases (since less energy can escape) and reaches T → ∞ as ϕ = 0 (since this produces an insulated closed enclosure with internal heat production).

The fact that long cylindrical surfaces lend themselves to exact analysis was apparently first recognized by Sparrow [26]. The preceding two examples have shown that exact solutions may be found for a number of special geometries, namely, (i) enclosures whose surfaces all lie on a single sphere, and (ii) enclosures for which view factors between surface elements have repetitive derivatives. For other still fairly simple geometries an approximate analytical solution may be determined from the kernel approximation method. However, the vast majority of radiative heat

188

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

transfer problems in enclosures without a participating medium must be solved by numerical methods. A large majority of these are solved using the net radiation method described in the first few sections of this chapter. If greater accuracy or better numerical efficiency is desired, one of the numerical methods briefly described in this section needs to be used, such as numerical quadrature leading to a set of linear algebraic equations (as in the net radiation method).

References 1. Jensen, C. L.: “TRASYS-II user’s manual–thermal radiation analysis system,” Technical report, Martin Marietta Aerospace Corp., Denver, 1987. 2. Chin, J. H., T. D. Panczak, and L. Fried: “Spacecraft thermal modeling,” Int. J. Numer. Methods Eng., vol. 35, pp. 641–653, 1992. 3. Oppenheim, A. K.: “Radiation analysis by the network method,” Transactions of ASME, Journal of Heat Transfer, vol. 78, pp. 725–735, 1956. 4. Kropschot, R. H., B. W. Birmingham, and D. B. Mann (eds.): Technology of Liquid Helium, National Bureau of Standards, Monograph 111, Washington, D.C., 1968. 5. Courant, R., and D. Hilbert: Methods of Mathematical Physics, Interscience Publishers, New York, 1953. 6. Hildebrand, F. B.: Methods of Applied Mathematics, Prentice Hall, Englewood Cliffs, NJ, 1952. ¨ ¸ ik, M. N.: Radiative Transfer and Interactions With Conduction and Convection, John Wiley & Sons, New York, 7. Ozis 1973. 8. Daun, K. J., and K. G. T. Hollands: “Infinitesimal-area radiative analysis using parametric surface representation, through NURBS,” ASME Journal of Heat Transfer, vol. 123, no. 2, pp. 249–256, 2001. 9. Sparrow, E. M., and A. Haji-Sheikh: “A generalized variational method for calculating radiant interchange between surfaces,” ASME Journal of Heat Transfer, vol. 87, pp. 103–109, 1965. 10. Sparrow, E. M.: “Application of variational methods to radiation heat transfer calculations,” ASME Journal of Heat Transfer, vol. 82, pp. 375–380, 1960. 11. Sparrow, E. M., J. L. Gregg, J. V. Szel, and P. Manos: “Analysis, results, and interpretation for radiation between simply arranged gray surfaces,” ASME Journal of Heat Transfer, vol. 83, pp. 207–214, 1961. 12. Buckley, H.: “On the radiation from the inside of a circular cylinder,” Phil. Mag., vol. 4, no. 23, pp. 753–762, 1927. 13. Eckert, E. R. G.: “Das Strahlungsverh¨altnis von Fl¨achen mit Einbuchtungen und von zylindrischen Bohrungen,” Arch. W¨armewirtschaft, vol. 16, pp. 135–138, 1935. 14. Sparrow, E. M., and L. U. Albers: “Apparent emissivity and heat transfer in a long cylindrical hole,” ASME Journal of Heat Transfer, vol. 82, pp. 253–255, 1960. 15. Usiskin, C. M., and R. Siegel: “Thermal radiation from a cylindrical enclosure with specified wall heat flux,” ASME Journal of Heat Transfer, vol. 82, pp. 369–374, 1960. 16. Lin, S. H., and E. M. Sparrow: “Radiant interchange among curved specularly reflecting surfaces, application to cylindrical and conical cavities,” ASME Journal of Heat Transfer, vol. 87, pp. 299–307, 1965. 17. Perlmutter, M., and R. Siegel: “Effect of specularly reflecting gray surface on thermal radiation through a tube and from its heated wall,” ASME Journal of Heat Transfer, vol. 85, pp. 55–62, 1963. 18. Siegel, R., and M. Perlmutter: “Convective and radiant heat transfer for flow of a transparent gas in a tube with a gray wall,” International Journal of Heat and Mass Transfer, vol. 5, pp. 639–660, 1962. 19. Sparrow, E. M., L. U. Albers, and E. R. G. Eckert: “Thermal radiation characteristics of cylindrical enclosures,” ASME Journal of Heat Transfer, vol. 84, pp. 73–81, 1962. 20. Sparrow, E. M., and R. P. Heinisch: “The normal emittance of circular cylindrical cavities,” Applied Optics, vol. 9, pp. 2569–2572, 1970. 21. Alfano, G.: “Apparent thermal emittance of cylindrical enclosures with and without diaphragms,” International Journal of Heat and Mass Transfer, vol. 15, no. 12, pp. 2671–2674, 1972. 22. Alfano, G., and A. Sarno: “Normal and hemispherical thermal emittances of cylindrical cavities,” ASME Journal of Heat Transfer, vol. 97, no. 3, pp. 387–390, 1975. 23. Froberg, C. E.: Introduction to Numerical Analysis, Addison-Wesley, Reading, MA, 1969. ¨ 24. Jensen, H. H.: “Some notes on heat transfer by radiation,” Kgl. Danske Videnskab. Selskab. Mat.-Fys. Medd., vol. 24, no. 8, pp. 1–26, 1948. 25. Jakob, M.: Heat Transfer, vol. 2, John Wiley & Sons, New York, 1957. 26. Sparrow, E. M.: “Radiant absorption characteristics of concave cylindrical surfaces,” ASME Journal of Heat Transfer, vol. 84, pp. 283–293, 1962.

Problems 5.1 A firefighter (approximated by a two-sided black surface at 310 K 180 cm long and 40 cm wide) is facing a large fire at a distance of 10 m (approximated by a semi-infinite black surface at 1500 K). Ground and sky are at 0◦ C (and may also be approximated as black). What are the net radiative

189

PROBLEMS

heat fluxes on the front and back of the firefighter? Compare these with heat rates by free convection (h = 10 W/m2 K, Tamb = 0◦ C). T3 = 500 K

5.2 A small furnace consists of a cylindrical, black-walled enclosure, 20 cm long and with a diameter of 10 cm. The bottom surface is electrically heated to 1500 K, while the cylindrical sidewall is insulated. The top plate is exposed to the environment, such that its temperature is 500 K. Estimate the heating requirements for the bottom wall, and the temperature of the cylindrical sidewall, by treating the sidewall as (a) a single zone, (b) two equal rings of 10 cm height each.

q2 = 0

5.3 Repeat Problem 5.2 for a 20 cm high furnace of quadratic (10 cm × 10 cm) cross-section.

T1 = 1500 K

5.4 A small star has a radius of 100,000 km. Suppose that the star is originally at a uniform temperature of 1,000,000 K before it “dies,” i.e., before nuclear fusion stops supplying heat. If it is assumed that the star has a constant heat capacity of ρcp = 1 kJ/m3 K, and that it remains isothermal during cool-down, estimate the time required until the star has cooled to 10,000 K. Note: A body of such proportions radiates like a blackbody (Why?). 5.5 A collimated light beam of q0 = 10 W/cm2 originating from a blackbody source at 1250 K is aimed at a small target A1 = 1 cm2 as shown. The target is coated with a diffusely reflecting material, whose emittance is ǫλ′ =



0.9 cos θ, 0.2,

λ < 4 µm, λ > 4 µm.

A1 q0=10 W/cm2

h=10 cm A2

w=20 cm

Light reflected from A1 travels on to a detector A2 = 1 cm2 , coated with the same material as A1 . How much of the collimated energy q0 is absorbed by detector A2 ?

5.6 Repeat Problem 5.2 for the case that the top surface of the furnace is coated with a gray, diffuse material with emittance ǫ3 = 0.5 (other surfaces remain black). 5.7 A long half-cylindrical rod is enclosed by a long diffuse, gray isothermal cylinder as shown. Both rod and cylinder may be considered isothermal (T1 = T2 , ǫ1 = ǫ2 , T3 , ǫ3 ) and gray, diffuse reflectors. Give an expression for the heat lost from the rod (per unit length).

A3 A2 Ri

Ro

A1

5.8 Consider a 90◦ pipe elbow as shown in the figure (pipe diameter = D = 1 m; inner elbow radius = 0, outer elbow radius = D). The elbow is isothermal at temperature T = 1000 K, has a gray diffuse emittance ǫ = 0.4, and is placed in a cool environment. What is the total heat loss from the isothermal elbow (inside and outside)?

π A1 = 2 D

2

( )

D

D

190

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

5.9 For the configuration shown in the figure, determine the temperature of Surface 2 with the following data: Surface 1 :

2 cm

A2

T1 = 1000 K, q1 = −1 W/cm2 , ǫ1 = 0.6;

Surface 2 :

ǫ2 = 0.2;

Surface 3 :

ǫ3 = 0.3, perfectly insulated.

5 cm A3 4 cm

All configurations are gray and diffuse. A1

5.10 Two pipes carrying hot combustion gases are enclosed in a cylindrical duct as shown. Assuming both pipes to be isothermal at 2000 K and diffusely emitting and reflecting (ǫ = 0.5), and the duct wall to be isothermal at 500 K and diffusely emitting and reflecting (ǫ = 0.2), determine the radiative heat loss from the pipes.

r 2r A1 2r A3

A2 2r r

5.11 A cubical enclosure has gray, diffuse walls which interchange energy. Four of the walls are isothermal at Ts with emittance ǫs , the other two are isothermal at Tt with emittance ǫt . Calculate the heat flux rates per unit time and area. 5.12 During launch the heat rejector radiative panels of the Space Shuttle are folded against the inside of the Shuttle doors. During orbit the doors are opened and the panels are rotated out by an angle ϕ as shown in the figure. Assuming door and panel can be approximated by infinitely long, isothermal quarter-cylinders of radius a and emittance ǫ = 0.8, calculate the necessary rotation angle ϕ so that half the total energy emitted by panel (2) and door (1) escapes through the opening. At what opening angle will a maximum amount of energy be rejected? How much and why?

a ϕ

a

a Panel

ϕ

T, ∋ A1

Door

5.13 Consider two 1 × 1 m2 , thin, gray, diffuse plates located a distance h = 1 m apart. The temperature of the top plate is maintained at T1 = 1200 K, whereas the bottom plate is initially at T2 = 300 K and insulated on the outside. In case 1, the surface of the top plate is flat, whereas in case 2 grooves, whose dimensions are indicated below, have been machined in the plate’s surface. In either case the surfaces are gray and diffuse, and the surroundings may be considered as black and having a temperature T∞ = 500 K; convective heat transfer effects may be neglected.

a

A2

1m 1m A1

1m

(flat or grooved)

A2

(a) Estimate the effect of the surface preparation of the top surface (1mm thick, insulated) on the initial temperature change of the bottom plate (dT2 /dt at 1cm t = 0). A1 for case 2 (b) Justify, then use, a lumped-capacity analysis for the bottom plate 1cm (100 grooves) to predict the history of temperature and heating rates of the bottom plate until steady state is reached. The following properties are known: top plate: ǫ1 = 0.6, T1 = 1200 K; bottom plate: T2 (t = 0) = 300 K, ǫ2 = 0.5, ρ2 = 800 kg/m3 (density), cp2 = 440 J/kg K, k2 = 200 W/m K.

191

PROBLEMS

5.14 A row of equally spaced, cylindrical heating elements (s = 2d) is used to heat the inside of a furnace as shown. Assuming that the outer wall is made of firebrick with ǫ3 = 0.3 and is perfectly insulated, that the heating rods are made of silicon carbide (ǫ1 = 0.8), and that the inner wall has an emittance of ǫ2 = 0.6, what must the operating temperature of the rods be to supply a net heat flux of 300 kW/m2 to the furnace, if the inner wall is at a temperature of 1300 K?

Outer wall, ∋3 = 0.3, q3 = 0 Silicon carbide rods (s/d = 2) ∋ 1 = 0.8

d

s

∋ 2 = 0.6 Inner wall q = 300 kW/m2

Tamb 5.15 A thermocouple used to measure the temperature of cold, low∋ss, ho pressure helium flowing through a long duct shows a tempera∋ ss vacuum ture reading of 10 K. To minimize heat losses from the duct to the ∋ ss, hi surroundings the duct is made of two concentric thin layers of TC stainless steel with an evacuated space in between (inner diamhelium ∋ TC, hTC eter di = 2 cm, outer diameter do = 2.5 cm; stainless layers very thin and of high conductivity). The emittance of the thermocouple is ǫTC = 0.6, the convection heat transfer coefficient between helium and tube wall is hi = 5 W/m2 K, and between thermocouple and helium is hTC = 2 W/m2 K, and the emittance of the stainless steel is ǫss = 0.2 (gray and diffuse, all four surfaces). The free convection heat transfer coefficient between the outer tube and the surroundings at Tamb = 300 K is ho = 5 W/m2 K. To determine the actual temperature of the helium,

(a) Prepare an energy balance for the thermocouple. (b) Prepare an energy balance for the heat loss through the duct wall (the only unknowns here should be THe , Ti , and To ). (c) Outline how to solve for the temperature of the helium (no need to carry out solution). (d) Do you expect the thermocouple to be accurate? (Hint: Check the magnitudes of the terms in (a).) 5.16 During a materials processing experiment on the Space Shuttle (under microgravity conditions), a platinum sphere of 3 mm diameter is levitated in a large, cold black vacuum chamber. A spherical aluminum shield (with a circular cutout) is placed around the sphere as shown, to reduce heat loss from the sphere. Initially, the sphere is at 200 K and is suddenly irradiated with a laser providing an irradiation of 100 W (normal to beam) to raise its temperature rapidly to its melting point (2741 K). Determine the time required to reach the melting point. You may assume the platinum and aluminum to be gray and diffuse (ǫPt = 0.25, ǫAl = 0.1), the sphere to be essentially isothermal at all times, and the shield to have zero heat capacity.

0K Pt sphere 1 cm

laser

Al shield 10 cm

5.17 Two identical circular disks are connected at one point of their periphery by a hinge. The configuration is then opened by an angle φ as shown in the figure. Assuming the opening angle to be φ = 60◦ , d = 1 m, calculate the average equilibrium temperature for each of the two disks, with solar radiation entering the configuration parallel to Disk 2 with a strength of qsun = 1000 W/m2 . Disk 1 is gray and diffuse with α = ǫ = 0.5, Disk 2 is black. Both disks are insulated.

qsun A1 d

A2

φ d

192

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

qsun

5.18 A long greenhouse has the cross-section of an equilateral triangle as shown. The side exposed to the sun consists of a thin sheet of glass (A1 ) with reflectivity ρ1 = 0.1. The glass may be assumed perfectly transparent to solar radiation, and totally opaque to radiation emitted inside the greenhouse. The other side wall (A2 ) is opaque with emittance ǫ2 = 0.2, while the floor (A3 ) has ǫ3 = 0.8. All surfaces reflect diffusely. For simplicity, you may assume surfaces A1 and A2 to be perfectly insulated, while the floor loses heat to the ground according to

A1 A2 A3 ground

60°

q3,conduction = U(T3 − T∞ )

60°

L = 1m

where T∞ = 280 K is the temperature of the ground, and U = 19.5 W/m2 K is an overall heat transfer coefficient. Determine the temperatures of all three surfaces for the case that the sun shines onto the greenhouse with strength qsun = 1000 W/m2 in a direction parallel to surface A2 . 5.19 A long, black V-groove is irradiated by the sun as shown. Assuming the groove to be perfectly insulated, and radiation to be the only mode of heat transfer, determine the average groove temperature as a function of solar incidence angle θ (give values for θ = 0◦ , 15◦ , 30◦ , 60◦ , 90◦ ). For simplicity the V-groove wall may be taken as a single zone.

qsol = 1000 W/m2

0K

θ 30° L

5.20 Consider the conical cavity shown (radius of opening R, opening angle γ = 30◦ ), which has a gray, diffusely reflective coating (ǫ = 0.6) and is perfectly insulated. The cavity is irradiated by a collimated beam of strength H0 and radius Rb = 0.5R).

Ho

γ

2R

2Rb

T,∋

(a) Using a single node analysis, develop an expression relating H0 to the average cavity temperature T. (b) For a more accurate analysis a two-node analysis is to be performed. What nodes would you choose? Develop expressions for the necessary view factors in terms of known ones (including those given in App. D) and surface areas, then relate the two temperatures to H0 . (c) Qualitatively, what happens to the cavity’s overall average temperature, if the beam is turned away by an angle α? 5.21 A (simplified) radiation heat flux meter consists of a conical cavity coated with a gray, diffuse material, as shown in the figure. To measure the radiative heat flux, the cavity is perfectly insulated. (a) Develop an expression that relates the flux, Ho , to the cavity temperature, T. (b) If the cavity is turned away from the incoming flux by an angle α, what happens to the cavity temperature?

Ho T,∋

30° qsun = 1000 W/m2 l2 = 60 cm Reflector

5.22 A very long solar collector plate is to collect energy at a temperature of T1 = 350 K. To improve its performance for off-normal solar incidence, a highly reflective surface is placed next to the collector as shown in the adjacent figure. How much energy (per unit length) does the collector plate collect for a solar incidence angle of 30◦ ? For simplicity you may make the following assumptions: The collector is isothermal and gray-diffuse with emittance ǫ1 = 0.8; the reflector is gray-diffuse with ǫ2 = 0.1, and heat losses from the reflector by convection as well as all losses from the collector ends may be neglected.

γ

q2 = 0, ∋2

Collector plate l1 = 80cm

T1, ∋1

PROBLEMS

5.23 A thermocouple (approximated by a 1 mm diameter sphere A3 with gray-diffuse emittance ǫ1 = 0.5) is suspended inside a tube through which a hot, nonparticipating gas at T1 = 2000 K is flowing. In the vicinity of the thermocouple the tube temD=10 cm perature is known to be T2 = 1000 K (wall emittance ǫ2 = 0.5). For the purpose of this problem you may assume both ends of the tube to be closed with a black surface at the temperature of the gas, T3 = 2000 K. Again, for the purpose of this problem, you may assume that the thermocouple gains a heat flux of 104 W/m2 of thermocouple surface area, which it must reject again in the form of radiation. Estimate the temperature of the thermocouple. Hints:

193

A2 A3

TC(A1)

L=10 cm

(a) Treat the tube ends together as a single surface A3 . (b) Note that the thermocouple is small, i.e., Fx−1 ≪ 1. 5.24 A small spherical heat source outputting Qs = 10 kW power, spreading equally into all directions, is encased in a reflector as shown, consisting of a hemisphere of radius R = 40 cm, plus a ring of radius R and height h = 30 cm. The arrangement is used to heat a disk of radius = 25 cm a distance of L = 30 cm below the reflector. All surfaces are gray and diffuse, with emittances of ǫ1 = 0.8 and ǫ2 = 0.1. Reflector A2 is insulated.

A2: ∋ 2 = 0.1, q2 = 0

R = 40 cm

Qs=10 kW R h=30 cm

(a) Determine (per unit area of receiving surface) the irradiation from heat source to reflector and to disk; (b) all relevant view factors; and (c) the temperature of the disk, if 0.4 kW of power is extracted from the disk.

L=30 cm A1: ∋ 1 = 0.8, Q1 = -0.4 kW r=25 cm

5.25 A long thin black heating wire radiates 300 W per cm length of wire and is used to heat a flat surface by thermal radiation. To increase its efficiency the wire is surrounded by an insulated half-cylinder as shown in the figure. Both surfaces are gray and diffuse with emittances ǫ2 and ǫ3 , respectively. What is the net heat flux at Surface 3? How does this compare with the case without cylinder? Hint: You may either treat the heating wire as a thin cylinder whose radius you eventually shrink to zero, or treat radiation from the wire as external radiation (the second approach being somewhat simpler). 5.26 Consider the configuration shown, consisting of a cylindrical cavity A2 , a circular disk A1 at the bottom, and a small spherical radiation source (blackbody at 4000K) of strength Q = 10, 000 W as shown (R = 10 cm, h = 10 cm). The cylinder wall A2 is covered with a gray, diffuse material with ǫ2 = 0.1, and is perfectly insulated. Surface A1 is kept at a constant temperature of 400 K. No other external surfaces or sources affect the heat transfer. Assuming surface A1 to be gray and diffuse with ǫ1 = 0.3 determine the amount of heat that needs to be removed from A1 (Q1 ).

A3: ∋ 3 = 0.5, T3 = 300K A1: wire R = 2 cm

h = 3 cm

R

A2: ∋ 2 = 0.2, q2 = 0

A2

Q

H

A1 2R

2H

194

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

5.27 Determine F1−2 for the rotationally symmetric configuration shown in the figure (i.e., a big sphere, R = 13 cm, with a circular hole, r = 5 cm, and a hemispherical cavity, r = 5 cm). Assuming Surface 2 to be gray and diffuse (ǫ = 0.5) and insulated and Surface 1 to be black and also insulated, what is the average temperature of the black cavity if collimated irradiation of 1000 W/m2 is penetrating through the hole as shown?

qs = 1000W/m2

A2

R

r

r A1

5.28 An integrating sphere (a device to measure surface properties) is 10 cm in radius. It contains on its inside wall a 1 cm2 Collimated beam black detector, a 1 × 2 cm entrance port, and a 1 × 1 cm sample as shown. The remaining portion of the sphere is smoked with magnesium oxide having a short-wavelength reflectance of 0.98, which is almost perfectly diffuse. A collimated beam of radiant energy (i.e., all energy is contained within a very small cone of solid angles) enters the sphere through the entrance port, falls onto the sample, and then is reflected and interreflected, giving rise to a sphere wall radiosity and irradiation. Entrance Radiation emitted from the walls is not detected because the port source radiation is chopped, and the detector–amplifier system responds only to the chopped radiation. Find the fractions of the chopped incoming radiation that are

MgO Sample

Detector

(a) lost out the entrance port, (b) absorbed by the MgO-smoked wall, and (c) absorbed by the detector. [Item (c) is called the “sphere efficiency.”] 5.29 The side wall of a flask holding liquid helium may be approximated as a long double-walled cylinder as shown in the adjacent sketch. The container walls are made of 1 mm thick stainless steel (k = 15 W/m K, ǫ = 0.2), and have outer radii of R2 = 10 cm and R4 = 11 cm. The space between walls is evacuated, and the outside is exposed to free convection with the ambient at Tamb = 20◦ C and a heat transfer coefficient of ho = 10 W/m2 K (for the combined effects of free convection and radiation). It is reasonable to assume that the temperature of the inner wall is at liquid helium temperature, or T(R2 ) = 4 K.

Foil

Liquid helium

ho

Tamb

R4 R1 R2

R3

(a) Determine the heat gain by the helium, per unit length of flask. (b) To reduce the heat gain a thin silver foil (ǫ = 0.02) is placed midway between the two walls. How does this affect the heat flux? For the sake of the problem, you may assume both steel and silver to be diffuse reflectors. 5.30 Repeat Problem 5.6, breaking up the sidewall into four equal ring elements. Use the view factors calculated in Problem 4.25 together with program graydiffxch of Appendix F. 5.31 The inside surfaces of a furnace in the shape of a parallelepiped with dimensions 1 m × 2 m × 4 m are to be broken up into 28 1 m × 1 m subareas. The gray-diffuse side walls (of dimension 1 m × 2 m and 1 m × 4 m) have emittances of ǫs = 0.7 and are perfectly insulated, the bottom surface has an emittance of ǫb = 0.9 and a temperature Tb = 1600 K, while the top’s emittance is ǫt = 0.2 and its temperature is Tt = 500 K. Using the view factors calculated in Problem 4.26 and program graydiffxch of Appendix

195

PROBLEMS

F, calculate the heating/cooling requirements for bottom and top surfaces, as well as the temperature distribution along the side walls. 5.32 For your Memorial Day barbecue you would like to broil a steak on your backyard BBQ, which consists of a base unit in the shape of a hemisphere (D= 60 cm), fitted with a disk-shaped coal rack, and a disk-shaped grill, as shown in the sketch. Hot coal may be assumed to cover the entire floor of the unit, with uniform temperature Tc = 1200 K, and an emittance of ǫc = 1. The side wall is soot-covered and black on the inside, but has an outside emittance of ǫo = 0.5. The steak (modeled as a ds = 15 cm disk, 1 cm thick, emittance ǫs = 0.8, initially at Ts = 280 K) is now placed on the grill (assumed to be so lightweight as to be totally transparent and not participating in the heat transfer). The environment is at 300 K, and free convection may be neglected.

lid

steak grill

BBQ base

coal rack

30 cm 60 cm

(a) Assuming that the lid is not placed on top of the unit, estimate the initial heating rates on the two surfaces of the steak. (b) How would the heating rates change, if the lid (also a hemisphere) is put on (ǫi = ǫo = 0.5)? Could one achieve a more even heating rate (top and bottom) if the emittance of the inside surface is increased or decreased? Note: Part (b) will be quite tedious, unless program graydiffxch of Appendix F is used (which, in turn, will require iteration or a little trickery). 5.33 Consider Configuration 33 in Appendix D with h = w. The bottom wall is at constant temperature T1 and has emittance ǫ1 ; the side wall is at T2 = const and ǫ2 . Find the exact expression for q1 (x) if ǫ2 = 1. 5.34 An infinitely long half-cylinder is irradiated by the sun as shown in the figure, with qsun = 1000 W/m2 . The inside of the cylinder is gray and diffuse, the outside is insulated. There is no radiation from the background. Determine the equilibrium temperature distribution along the cylinder periphery, (a) using four isothermal zones of 45◦ each, (b) using the exact relations. Hint: Use differentiation as in the kernel approximation method.

5.35 To calculate the net heat loss from a part of a spacecraft, 0 K this part may be approximated by an infinitely long black plate at temperature T2 = 600 K, as shown. Parallel to this plate is another (infinitely long) thin plate that is gray and emits/reflects diffusely with the same emittance ǫ1 on both sides. You may assume the surroundings to be black at 0 K. Calculate the net heat loss from the black plate.

q sun

r ϕ ∋, T(ϕ)

0K ∋ 1 = 0.1

w1 =1 m h = 1m w2 = 2 m T2, ∋ 2 = 1

5.36 A large isothermal surface (exposed to vacuum, temperature Tw , diffuse-gray emittance ǫw ) is irradiated by the sun. To reduce the heat gain/loss from the surface, a thin copper shield (emittance ǫc and initially at temperature Tc0 ) is placed between surface and sun as shown in the figure. (a) Determine the relationship between Tc and time t (it is sufficient to leave the answer in implicit form with an unsolved integral). (b) Give the steady state temperature for Tc (i.e., for t → ∞). (c) Briefly discuss qualitatively the following effects:

q sun

ϕ Tc , ∋ c

Tw , ∋w

196

5 RADIATIVE EXCHANGE BETWEEN GRAY, DIFFUSE SURFACES

(i) The shield is replaced by a moderately thick slab of styrofoam coated on both sides with a very thin layer of copper. (ii) The surfaces are finite in size. 5.37 Consider two infinitely long, parallel, black plates of width L Insulation L as shown. The bottom plate is uniformly heated electrically with a heat flux of q1 = const, while the top plate is insulated. The entire configuration is placed into a large cold A2, ∋ 2 environment. 0K y h (a) Determine the governing equations for the temperature variation across the plates. A1, ∋ 1 x (b) Find the solution by the kernel substitution method. To avoid tedious algebra, you may leave the final result in q1 terms of two constants to be determined, as long as you outline carefully how these constants may be found. (c) If the plates are gray and diffuse with emittances ǫ1 and ǫ2 , how can the temperature distribution be determined, using the solution from part (b)? 5.38 To reduce heat transfer between two infinite concentric cylinders a third cylinder is placed between them as shown in the figure. The center cylinder has an opening of half-angle θ. The inner cylinder is black and at temperature T1 = 1000 K, while the outer cylinder is at T4 = 300 K. The outer cylinder and both sides of the shield are coated with a reflective material, such that ǫc = ǫ2 = ǫ3 = ǫ4 . Determine the heat loss from the inner cylinder as function of coating emittance ǫc , using



r1

r2 = r3 r4

A3 A2

(a) the net radiation method, (b) the network analogy. 5.39 Consider the two long concentric cylinders as shown in the figure. Between the two cylinders is a long, thin flat plate as also indicated. The inner cylinder is black and generating heat on its inside in the amount of Q′1 = 1 kW/m length of the cylinder, which must be removed by radiation. The plate is gray and diffuse with emittance ǫ2 = ǫ3 = 0.5, while the outer cylinder is black and cold (T4 = 0 K). Determine the temperature of the inner cylinder, using (a) the net radiation method, (b) the network analogy.

5.40 An isothermal black disk at T1 = 500K is flush with the outer 0 K surface of a spacecraft and is thus exposed to outer space. To minimize heat loss from the disk a disk-shaped radiation shield is placed coaxially and parallel to the disk as shown; the shield radius is R2 (which may be smaller or larger than R1 ), and its distance from the black disk is a variable h. Determine an expression for the heat loss from the black disk as a function of shield radius and distance, using (a) the net radiation method, (b) the network analogy.

A4

A1

A3

2R A2

R 3R

R A4

A1

0K ∋ 2 = 0.1

R2 =? H

=?

R1 = 0.1m T1, ∋1 = 1

CHAPTER

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

6.1

INTRODUCTION

In the previous two chapters it was assumed that all surfaces constituting the enclosure are— besides being gray—diffuse emitters as well as diffuse reflectors of radiant energy. Diffuse emission is nearly always an acceptable simplification. The assumption of diffuse reflection, on the other hand, often leads to considerable error, since many surfaces deviate substantially from this behavior. Electromagnetic wave theory predicts reflection to be specular for optically smooth surfaces, i.e., to reflect light like a mirror. All clean metals, many nonmetals such as glassy materials, and most polished materials display strong specular reflection peaks. Nevertheless, they all, to some extent, reflect somewhat into other directions as a result of their surface roughness. Surfaces may appear dull (i.e., diffusely reflecting) to the eye, but are rather specular in the infrared, since the ratio of every surface’s root-mean-square roughness to wavelength decreases with increasing wavelength. For a surface with diffuse reflectance the reflected radiation has the same (diffuse) directional distribution as the emitted energy, as discussed in the beginning of Section 5.3. Therefore, the radiation field within the enclosure is completely specified in terms of the radiosity, which is a function of location along the enclosure walls (but not a function of direction as well). If reflection is nondiffuse, then the radiation intensities leaving any surface are functions of direction as well as surface location, and the analysis becomes immensely more complicated.1 To make the analysis tractable, one may make the idealization that the reflectance, while not diffuse, can be adequately represented by a combination of a diffuse and a specular component, as illustrated in Fig. 6-1 for oxidized brass [1]. Thus, for the present chapter, we assume the radiative properties to be of the form ρ = ρs + ρ d = 1 − α = 1 − ǫ = 1 − ǫλ′ ,

(6.1)

where ρs and ρ d are the specular and diffuse components of the reflectance, respectively. Since the surfaces are assumed to be gray, diffuse emitters (ǫ = ǫλ′ ), it follows that neither α nor ρ 1 In addition, if the irradiation is polarized (e.g., owing to irradiation from a laser source), specular reflections will change the state of polarization (because of the different values for ρk and ρ⊥ , as discussed in Chapter 2). We shall only consider unpolarized radiation.

197

198

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

θr

15°



θi =0°

θr

15°

30°

15°

15°

θi =0°

45°

θi =15°



θi =30° θi = 45°

30°

45°

θi =15° θi =30° θi = 45°

60°

75°

0.2

0 0.4 0.2 ρ ´´(θi , ψi , θr , ψi + π) cos θr

60°

75°

0.2

0.2 0.4 0 ρ ´´(θi , ψi , θr , ψi + π) cos θr

(a) (b) FIGURE 6-1 (a) Subdivision of the reflectance of oxidized brass (shown for plane of incidence) into specular (shaded) and diffuse components (unshaded), from [1]; (b) equivalent idealized reflectance.

may depend on wavelength or on incoming direction (i.e., the magnitude of ρ does not depend on incoming direction); how ρ is distributed over outgoing directions depends on incoming direction through ρs . With this approximation, the separate reflection components may be found analytically by splitting the bidirectional reflection function into two parts, ρ′′ (r, sˆ i , sˆ r ) = ρ′′s (r, sˆ i , sˆ r ) + ρ′′d (r, sˆ i , sˆ r ).

(6.2)

Substituting this expression into equation (3.43) and equation (3.46) then leads to ρs and ρ d . Values of ρs and ρ d may also be determined directly from experiment, as reported by Birkebak and coworkers [2], making detailed measurements of the bidirectional reflection function unnecessary. Within an enclosure consisting of surfaces with purely diffuse and purely specular reflection components, the complexity of the problem may be reduced considerably by realizing that any specularly reflected beam may be traced back to a point on the enclosure surface from which it emanated diffusely (i.e., any beam was part of an energy stream leaving the surface after emission or diffuse reflection), as illustrated in Fig. 6-2. Therefore, by redefining the view factors to include specular reflection paths in addition to direct view, the radiation field may again be described by a diffuse energy function that is a function of surface location but not of direction.

6.2

SPECULAR VIEW FACTORS

To accommodate surfaces with reflectances described by equation (6.1), we define a specular view factor as diffuse energy leaving dAi intercepted by dA j , by direct travel or any number of specular reflections s . (6.3) dFdA ≡ i −dA j total diffuse energy leaving dAi The concept of the specular view factor is illustrated in Figs. 6-2 and 6-3. Diffuse radiation leaving dAi (by emission or diffuse reflection) can reach dA j either directly or after one or more reflections. Usually only a finite number of specular reflection paths such as dAi − a − dA j or dAi − b − c − dA j (and others not indicated in the figure) will be possible. The surface at points

199

6.2 SPECULAR VIEW FACTORS

d Ai a c

FIGURE 6-2 Radiative exchange in an enclosure with specular reflectors.

b

d Aj

c

b

b(c)

dAi(bc)

mirror c

d Aj

mirr

or b

d Ai(b)

d Ai FIGURE 6-3 Specular view factor between infinitesimal surface elements; formation of images.

a, b, and c behaves like a perfect mirror as far as the specular part of the reflection is concerned. Therefore, if an observer stood on top of dA j looking toward c, it would appear as if point b as well as dAi were situated behind point c as indicated in Fig. 6-3; the point labeled b(c) is the image of point b as mirrored by the surface at c, and dAi (cb) is the image of dAi as mirrored by the surfaces at c and b. Therefore, as we examine Figs. 6-2 and 6-3, we may formally evaluate the specular view factor between two infinitesimal areas as s dFdA = dFdAi −dA j + ρsa dFdAi (a)−dA j + ρsb ρsc dFdAi (cb)−dA j i −dA j

+ other possible reflection paths.

(6.4)

Thus, the specular view factor may be expressed as a sum of diffuse view factors, with one contribution for each possible direct or reflection path. Note that, for images, the diffuse view factors must be multiplied by the specular reflectances of the mirroring surfaces, since radiation traveling from dAi to dA j is attenuated by every reflection. If all specularly reflecting parts of the enclosure are flat, then all images of dAi have the same shape and size as dAi itself. However, curved surfaces tend to distort the images (focusing and defocusing effects). In the case of only flat, specularly reflecting surfaces we may multiply equation (6.4) by dAi and, invoking the law of reciprocity for diffuse view factors, equation (4.7), we obtain s dAi dFdA = dA j dFdA j −dAi + ρas dA j dFdA j −dAi (a) + ρsb ρsc dA j dFdA j −dAi (bc) i −dA j

= dA j dFdA j −dAi + ρsa dA j dFdA j (a)−dAi + ρsb ρsc dA j dFdA j (bc)−dAi + . . . s , = dA j dFdA j −dAi

(6.5)

200

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

A2

A2

∋ 2 , ρ 2s , ρ d2

ρ 1s ( ρ 2s )2

ρ 2s

1

∋ 1, ρ s1, ρ1d

ρ 1s ρ 2s A1

A1 d A1

(a)

(b)

FIGURE 6-4 s s (a) Geometry for Example 6.1, (b) ray tracing for the evaluation of F1−1 and F1−2 .

that is, the law of reciprocity holds for specular view factors as long as all specularly reflecting surfaces are flat. Although considerably more complicated, it is possible to show that the law of reciprocity also holds for curved specular reflectors. If we also assume that the diffuse energy leaving Ai and A j is constant across each respective area, we have the equivalent to equation (4.15), s s dAi dFdi−d j = dA j dFd j−di , s dAi Fdi−j s Ai Fi−j

= =

s , A j dFj−di s A j Fj−i ,

(6.6a) (Jj = const),

(6.6b)

(Ji , Jj = const),

(6.6c)

where we have adopted the compact notation first introduced in Chapter 4, and Ji is the total diffuse energy (per unit area) leaving surface Ai (again called the radiosity). s s Example 6.1. Evaluate the specular view factors F1−1 and F1−2 for the parallel plate geometry shown in Fig. 6-4a.

Solution s We note that, because of the one-dimensionality of the problem, Fd1−2 must be the same for any dA1 s s s s on surface A1 . Since F1−2 is nothing but a surface average of Fd1−2 , we conclude that Fd1−2 = F1−2 . It is sufficient to consider energy leaving from an infinitesimal area (rather than all of A1 ). Examining Fig. 64b we see that every beam (assumed to have unity strength) leaving dA1 , regardless of its direction, must travel to surface A2 (a beam of strength “1” is intercepted). After reflection at A2 a beam of strength ρs2 returns to A1 specularly, where it is reflected again and a beam of strength ρs2 ρs1 returns to A2 specularly. After one more reflection a beam of strength (ρs2 ρ1s )ρs2 returns to A1 , and so on. Thus, the specular view factor may be evaluated as s s = 1 + ρs1 ρs2 + (ρs1 ρs2 )2 + (ρs1 ρs2 )3 + . . . . Fd1−2 = F1−2

(6.7)

Since ρs1 ρs2 < 1 the sum in this equation is readily evaluated by the methods given in Wylie [3], and s = F1−2

1 s . = F2−1 1 − ρs1 ρs2

(6.8)

The last part of this relation is found by switching subscripts or by invoking reciprocity (and A1 = A2 ). We notice that specular view factors are not limited to values between zero and one, but are often greater than unity because much of the radiative energy leaving a surface is accounted for more than once. All energy from A1 is intercepted by A2 after direct travel, but only the fraction (1 − ρs2 ) is removed (by absorption and/or diffuse reflection) from the specular reflection path. The fraction ρs2 travels on s that must have a value specularly and is, therefore, counted a second time, etc. Thus, it is (1 − ρs2 )F1−2 between zero and one, and the summation relation, equation (4.18), must be replaced by N X s (1 − ρsj )Fi−j = 1. j=1

(6.9)

6.2 SPECULAR VIEW FACTORS

∋ 2 , ρ 2s , ρ d2

201

( ρ 2s )3

1 ∋ 1, ρ 1s, ρ 1d

ρ 2s ρ 2s ρ 1s ρ 2s ρ 1s ρ 2s

( ρ 2s )2 1

( ρ 2s ρ 1s )2

ρ 2s

(a) FIGURE 6-5 (a) Geometry for Example 6.2, (b) repeated reflections along outer surface.

(b)

Equation (6.9), formed here through intuition, will be developed rigorously in the next section. s may be found similarly as F1−1 s F1−1 = ρs2 + (ρs1 ρs2 )ρs2 + (ρs1 ρs2 )2 ρs2 + . . . =

We note in passing that s s (1 − ρs1 )F1−1 = + (1 − ρs2 )F1−2

ρs2

1 − ρs1 ρs2

(1 − ρs1 )ρs2 + 1 − ρs2

1 − ρs1 ρs2

.

= 1,

as postulated by equation (6.9). Example 6.2. Evaluate all specular view factors for two concentric cylinders or spheres. Solution Possible beam paths with specular reflections from inner to outer cylinders (or spheres) and vice versa are shown in Fig. 6-5a. As in the previous example a beam leaving A1 in any direction must hit surface A2 (with strength “1”). Because of the circular geometry, after specular reflection the beam (now of strength ρs2 ) must return to A1 (i.e., it cannot hit A2 again before hitting A1 ). After renewed reflections the beam keeps bouncing back and forth between A1 and A2 . Thus, as for parallel plates, s F1−2 = 1 + ρs1 ρs2 + (ρs1 ρs2 )2 + . . . =

Similarly, we have s F1−1 = ρs2 + (ρs1 ρs2 )ρs2 + . . . =

1 . 1 − ρs1 ρs2

ρs2 1 − ρs1 ρs2

.

A beam emanating from A2 will first hit either A1 , and then keep bouncing back and forth between A1 and A2 (cf. Fig. 6-5a), or A2 , and then keep bouncing along A2 without ever hitting A1 (cf. Fig. 6-5b). Thus, since the fraction F2−1 of the diffuse energy leaving A2 hits A1 after direct travel, we have h i A1 /A2 s , F2−1 = F2−1 1 + ρs1 ρs2 + (ρs1 ρs2 )2 + . . . = 1 − ρs1 ρs2 h h i i s F2−2 = F2−2 1 + ρs2 + (ρs2 )2 + (ρ2s )3 + . . . + F2−1 ρs1 + ρs1 (ρs1 ρs2 ) + . . .

=

s 1 − A1 /A2 ρ1 A1 /A2 + , s 1 − ρ2 1 − ρs1 ρs2

where the simple diffuse view factors F2−1 and F2−2 have been evaluated in terms of A1 and A2 . Of s s s could have been found from F1−2 by reciprocity, and F2−2 with the aid of equation (6.9). course, F2−1

202

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

ρsH ρdH

J

∋ Eb

H

q

FIGURE 6-6 Energy balance for surfaces with partially specular reflection.

A few more examples of specular view factor determinations will be given once the appropriate heat transfer relations have been developed.

6.3 ENCLOSURES WITH PARTIALLY SPECULAR SURFACES Consider an enclosure of arbitrary geometry as shown in Fig. 6-2. All surfaces are gray, diffuse emitters and gray reflectors with purely diffuse and purely specular components, i.e., their radiative properties obey equation (6.1). Under these conditions the net heat flux at a surface at location r is, from Fig. 6-6, q(r) = qemission − qabsorption = ǫ(r)[Eb (r) − H(r)] = qout − qin = ǫ(r)Eb (r) + ρ d (r)H(r) + ρs (r)H(r) − H(r).

(6.10)

The first two terms on the last right-hand side of equation (6.10), or the part of the outgoing heat flux that leaves diffusely, we will again call the surface radiosity, J(r) = ǫ(r)Eb (r) + ρ d (r)H(r),

(6.11)

q(r) = J(r) − [1 − ρs (r)]H(r).

(6.12)

so that Eliminating the irradiation H(r) from equations (6.10) and (6.12) leads to q(r) =

i ǫ(r) h [1 − ρs (r)]Eb (r) − J(r) , d ρ (r)

(6.13)

which, of course, reduces to equation (5.26) for a diffusely reflecting surface if ρs = 0 and ρ d = 1 − ǫ. For a purely specular reflecting surface (ρ d = 0) equation (6.13) is indeterminate since the radiosity consists only of emission, or J = ǫ Eb . As in Chapter 5 the irradiation H(r) is found by determining the contribution to H from a differential area dA′ (r′ ), followed by integration over the entire enclosure surface. A subtle difference is that we do not track the total energy leaving dA′ (multiplied by a suitable directtravel view factor); rather, the contribution from specular reflections is subtracted and attributed to the surface from which it leaves diffusely. The more complicated path of such energy is then accounted for by the definition of the specular view factor. Thus, similar to equation (5.21), Z s ′ s H(r) dA = J(r′ ) dFdA (6.14) ′ −dA dA + Ho (r) dA, A

Hos (r)

where is any external irradiation arriving at dA (through openings or semitransparent walls). Similar to the specular view factors, the Hos includes external radiation hitting dA directly

203

6.3 ENCLOSURES WITH PARTIALLY SPECULAR SURFACES

or after any number of specular reflections. Using reciprocity, equation (6.14) becomes Z s s H(r) = J(r′ ) dFdA−dA ′ + Ho (r),

(6.15)

A

and, after substitution into equation (6.11), an integral equation for the unknown radiosity is obtained as "Z # s s J(r) = ǫ(r)Eb (r) + ρ d (r) J(r′ ) dFdA−dA + H (r) . (6.16) ′ o A

For surface locations for which heat flux q(r) is given rather than Eb (r), equation (6.12) should be used rather than equation (6.11). It is usually more desirable to eliminate the radiosity, to obtain a single relationship between surface blackbody emissive powers and heat fluxes. Solving equation (6.13) for J gives ρ d (r)   q(r), J(r) = 1 − ρs (r) Eb (r) − ǫ(r)

(6.17)

and substituting this expression into equation (6.16) leads to ρd (1 − ρ )Eb − q = (1 − ρs − ρ d )Eb + ρ d ǫ s

or Eb (r) −

Z

A



"Z

(1 − ρ

s

A

q(r)  s 1 − ρs (r′ ) Eb (r′ ) dFdA−dA − ′ = ǫ(r)

Z

s )Eb dFdA−dA ′

A



Z

A

# ρd s s q dFdA−dA′ + Ho , ǫ

ρ d (r′ ) ′ s s q(r ) dFdA−dA ′ + Ho (r). ǫ(r′ )

(6.18)

s We note that, for diffusely reflecting surfaces with ρs = 0, ρ d = 1 − ǫ, Fi−j = Fi−j , and Hos = Ho , equation (6.18) reduces to equation (5.28). If the specular view factors can be calculated (and that is often a big “if”), then equation (6.18) is not any more difficult to solve than equation (5.28). Indeed, if part or all of the surface is purely specular (ρ d = 0), equation (6.18) becomes considerably simpler. As for black and gray-diffuse enclosures, it is customary to simplify the analysis by using an idealized enclosure, consisting of N relatively simple subsurfaces, over each of which the radiosity is assumed constant. Then

Z

A

s J(r′ ) dFdA−dA ′ ≃

N X

Jj

j=1

Z

Aj

s dFdA−dA = j

N X

s Jj FdA−A , j

j=1

and, after averaging over a subsurface Ai on which dA is situated, equation (6.16) simplifies to   N X   d s s (6.19) Ji = ǫi Ebi + ρi  Jj Fi−j + Hoi  , i = 1, 2, . . . , N.   j=1

Eliminating radiosity through equation (6.17) then simplifies equation (6.18) to Ebi −

d N N X qi X ρ j s s F q j + Hois , (1 − ρsj )Fi−j Eb j = − ǫi ǫ j i−j j=1

i = 1, 2, . . . , N.

(6.20)

j=1

The summation relation, equation (6.9), is easily obtained from equation (6.20) by considering a special case: In an isothermal enclosure (Eb1 = Eb2 = · · · = EbN ) without external irradiation

204

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

s s = · · · = 0), according to the Second Law of Thermodynamics, all heat fluxes must = Ho2 (Ho1 vanish (q1 = q2 = · · · = 0). Thus, canceling emissive powers, N X s (1 − ρsj )Fi−j = 1,

i = 1, 2, . . . , N.

(6.21)

j=1

s are geometric factors and do not depend on temperature distribution, equation (6.21) Since the Fi−j is valid for arbitrary emissive power values. Finally, for computer calculations it may be advantageous to write the emissive power and heat fluxes in matrix form. Introducing Kronecker’s delta equation (6.20) becomes   d N h N  X X i  δi j ρ j s  s s  − Fi−j  q j + Hois , i = 1, 2, . . . , N, (6.22) δi j − (1 − ρ j )Fi−j Eb j =   ǫj ǫj j=1

j=1

2

or

A · eb = C · q + hso ,

(6.23)

where C and A are matrices with elements s , Ai j = δi j − (1 − ρsj )Fi−j

Ci j =

δi j ǫj



ρ dj ǫj

s , Fi−j

and q, eb , and hso are vectors for the surface heat fluxes, emissive powers, and external irradiations, respectively. If all temperatures and external irradiations are known, the unknown heat fluxes are readily found by matrix inversion as   q = C−1 · A · eb − hso . (6.24)

If the emissive power is only known over some of the surfaces, and the heat fluxes are specified elsewhere, equation (6.23) may be rearranged into a similar equation for the vector containing all the unknowns. Subroutine graydifspec is provided in Appendix F for the solution of the simultaneous equations (6.23), requiring surface information and a partial view factor matrix as input. The solution to a sample problem is also given in the form of a program grspecxch, which may be used as a starting point for the solution to other problems. Fortran90, C++ as well as Matlabr versions are provided. Example 6.3. Two large parallel plates are separated by a nonparticipating medium as shown in Fig. 64a. The bottom surface is isothermal at T1 , with emittance ǫ1 and a partially specular, partially diffuse reflectance ρ1 = ρ1d + ρs1 . Similarly, the top surface is isothermal at T2 with ǫ2 and ρ2 = ρ2d + ρs2 . Determine the radiative heat flux between the surfaces. Solution s = 0, From equation (6.20) we have, for i = 1, with Ho1 s s Eb1 − (1 − ρs2 )F1−2 Eb2 = Eb1 − (1 − ρs1 )F1−1

ρd s q1 ρ1d s F1−1 q1 − 2 F1−2 q2 . − ǫ1 ǫ2 ǫ1

While we could apply i = 2 to equation (6.20) to obtain a second equation for q1 and q2 , it is simpler here to use overall conservation of energy, or q2 = −q1 . Thus, h i s s 1 − (1 − ρs1 )F1−1 Eb1 − (1 − ρs2 )F1−2 Eb2 . q1 = s s − ρ 1 − ǫ 1 − ǫ2 − ρ 2 s 1 1 1 s F1−1 + F1−2 − ǫ1 ǫ2 ǫ1 2 Again, for easy readability of matrix manipulations we shall follow here the convention that a two-dimensional matrix is denoted by a bold capitalized letter, while a vector is written as a bold lowercase letter.

6.3 ENCLOSURES WITH PARTIALLY SPECULAR SURFACES

205

s Using the results from Example 6.1 and dividing both numerator and denominator by F1−2 , we obtain

q1 = 

s (1 − ρs2 )(1)(Eb1 − Eb2 ) (Eb1 − Eb2 ) (1 − ρs2 )F1−2 Eb1 − Eb2 = =  ,   1 1 1 1 1 1 s s s s s s (1 − ρ2 )(1) + ρ2 − 1 (1 − ρ2 )F1−2 + F1−1 − F1−2 −1 + + + ǫ1 ǫ2 ǫ1 ǫ2 ǫ1 ǫ2

(6.25)

which produces the same result whether we have diffusely or specularly reflecting surfaces. Indeed, equation (6.25) is valid for the radiative transfer between two isothermal parallel plates, regardless of the directional behavior of the reflectance (i.e., it is not limited to the idealized reflectances considered in this chapter). Any beam leaving A1 must hit surface A2 and vice versa, regardless of whether the reflectance is diffuse, specular, or neither of the two; the surface locations will be different but the directional variation of reflectance has no influence on the heat transfer rate since the surfaces are isothermal. Example 6.4. Repeat the previous example for concentric spheres and cylinders. Solution Again, from equation (6.20) with i = 1 and Hois = 0, we obtain s s Eb1 − (1 − ρs2 )F1−2 Eb2 = Eb1 − (1 − ρs1 )F1−1

ρd s q1 ρ1d s − F1−1 q1 − 2 F1−2 q2 . ǫ1 ǫ1 ǫ2

In this case conservation of energy demands q2 A2 = −q1 A1 , and h i s s s 1 − (1 − ρs1 )F1−1 Eb1 − (1 − ρ2s )F1−2 Eb2 (Eb1 − Eb2 ) (1 − ρs2 )F1−2 =  . q1 =  s s 1 − ǫ1 − ρ1 s 1 A1 s 1 A1 1 − ǫ2 − ρ2 A1 s 1 s s F1−2 (1 − ρs2 )F1−2 + F1−1 − + − F1−2 F1−1 + A2 ǫ1 ǫ2 A2 ǫ1 A2 ǫ2 ǫ1 s s are the same as in the previous example (cf. Example 6.2), leading The specular view factors F1−1 and F1−2 to

q1 =

Eb1 − Eb2 s . 1 A1 A1 /A2 − ρ2 1 − + 1 − ρ2s ǫ1 ǫ2 A2

(6.26)

We note that equation (6.26) does not depend on ρs1 : Again, any radiation reflected off surface A1 must return to surface A2 , regardless of the directional behavior of its reflectance. If surface A2 is purely specular (ρs2 = 1 − ǫ2 ), all radiation from A1 bounces back and forth between A1 and A2 , and equation (6.26) reduces to equation (6.25), i.e., the heat flux between these concentric spheres or cylinders is the same as between parallel plates. On the other hand, if A2 is diffuse (ρs2 = 0) equation (6.26) reduces to the purely diffuse case since the directional behavior of ρ1 is irrelevant. Example 6.5. A very long solar collector plate is to collect energy at a temperature of T1 = 350 K. To improve its performance for off-normal solar incidence, a highly reflective surface is placed next to the collector as shown in Fig. 6-7. For simplicity you may make the following assumptions: The collector is isothermal and gray-diffuse with emittance ǫ1 = 1 − ρ1d = 0.8; the mirror is gray and specular with ǫ2 = 1 − ρs2 = 0.1, and heat losses from the mirror by convection as well as all losses from the collector ends may be neglected. How much energy (per unit length) does the collector plate collect for solar irradiation of qsun = 1000 W/m2 at an incidence angle of 30◦ ? Solution Applying equation (6.22) to the absorber plate (i = 1) as well as the mirror (i = 2) we obtain   h i ρ1d s  ρd s  1 s s s s s  F1−1  q1 − 2 F1−2 q2 + Ho1 , 1 − (1 − ρ1 )F1−1 Eb1 − (1 − ρ2 )F1−2 Eb2 =  − ǫ1 ǫ2 ǫ1   i h ρd s ρ d s   1 s s  q2 + H s . q1 +  − 2 F2−2 Eb2 = − 1 F2−1 −(1 − ρs1 )F2−1 Eb1 + 1 − (1 − ρs2 )F2−2  o2 ǫ1 ǫ2 ǫ2

206

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

ϕ

Mirror

l 2 = 60 cm

q sun = 1000 W/m2

q 2 = 0, ∋2

T1, ∋ 1

Collector plate l 2 tan ϕ l 1 = 80 cm

FIGURE 6-7 Geometry for Example 6.5.

s s s s Since ρs1 = 0, it follows that F1−1 = F2−2 = 0 and also F1−2 = F1−2 , F2−1 = F2−1 . For this configuration no specular reflections from one surface to another surface are possible (radiation leaving the absorber plate, after specular reflection from the mirror, always leaves the open enclosure). Thus, with q2 = 0,

q1 s , + Ho1 ǫ1   1 s . − 1 F2−1 q1 + Ho2 = − ǫ1

Eb1 − ǫ2 F1−2 Eb2 = −F2−1 Eb1 + Eb2

Eliminating Eb2 , by multiplying the second equation by ǫ2 F1−2 and adding, leads to     1 1 s s − −1 ǫ2 F1−2 F2−1 q1 +Ho1 . +ǫ2 F1−2 Ho2 (1 − ǫ2 F1−2 F2−1 )Eb1 = ǫ1 ǫ1 The external fluxes are evaluated as follows: The mirror receives solar flux only directly (no specular s = qsun sin ϕ. The absorber plate receives a direct reflection off the absorber plate is possible), i.e., Ho2 contribution, qsun cos ϕ, and a second contribution after specular reflection off the mirror. This second contribution has the strength of ρs2 qsun cos ϕ per unit area. However, only part of the collector plate (l2 tan ϕ) receives this secondary contribution, which, for our crude two-node description, must be averaged over l1 . Thus, # " l2 tan ϕ l2 s Ho1 = qsun cos ϕ + ρs2 qsun cos ϕ = qsun cos ϕ + (1 − ǫ2 ) sin ϕ . l1 l1

Therefore, q1 =

  (1−ǫ2 F1−2 F2−1 )Eb1 − cos ϕ+(1−ǫ2 ) sin ϕ(l2 /l1 )+ǫ2 F1−2 sin ϕ qsun .   1 1 − 1 F1−2 F2−1 − ǫ2 ǫ1 ǫ1

The view factors are readily evaluated by the crossed-strings method as F1−2 = (80+60−100)/(2×80) = and F2−1 = 80 × 41 /60 = 13 . Substituting numbers, we obtain  √   60 1−0.1× 41 × 13 5.670×10−8 ×3504 − 23 +0.9× 21 × 80 +0.1× 41 × 12 1000   q1 = = −298 W/m2 . 1 1 1 1 × − 1 × − 0.1 4 0.8 3 0.8

1 4

Under these conditions, therefore, the collector is about 30% efficient. This result should be compared with a collector without a mirror (l2 = 0 and F1−2 = 0), for which we get √ ! Eb1 − qsun cos ϕ 3 q1,no mirror = = 0.8 × 5.670 × 10−8 × 3504 − 1000 × = −12 W/m2 . 1/ǫ1 2

This absorber plate collects hardly any energy at all (indeed, after accounting for convection losses, it would experience a net energy loss). If the mirror had been a diffuse reflector the heat gain would have

6.3 ENCLOSURES WITH PARTIALLY SPECULAR SURFACES

207

A1(3) A2

A1 A2(3)

A3

A1 A3

A2 (3)

A1(3) A2 (a)

(b) FIGURE 6-8 Triangular enclosure with a single specularly reflecting surface, with a few possible beam paths indicated, (a) without obstructions, (b) with partial obstructions.

been q1,diffuse mirror = −172 W/m2 , which is significantly less than for the specular mirror (cf. Problem 5.22). We conclude from this example that (i) mirrors can significantly improve collector performance, and (ii) infrared reradiation losses from near-black collectors are very substantial. Of course, reradiation losses may be significantly reduced by using selective surfaces or glass-covered collectors (cf. Chapter 3).

We shall conclude this section with three more examples designed to clarify certain aspects of evaluating the specular view factors in enclosures comprised of only simple planar elements. Example 6.6. Consider the triangular enclosures shown in Figs. 6-8a and b. Surfaces A1 and A2 are isothermal at T1 and T2 , respectively, and are purely diffuse reflectors with ǫ1 = 1 − ρ1d and ǫ2 = 1 − ρ2d . Surface A3 is isothermal at T3 and is a purely specular reflector with ǫ3 = 1 − ρs3 . Set up the system of equations for the unknown surface heat fluxes. Solution Since there is only a single (and flat) specular surface, no multiple specular reflections are possible. s s s and F2−2 are nonzero, it is clear that F3−3 = 0. Thus, from equation (6.22), with Hois = 0, While F1−1

     1 1 1 s s q2 , − 1 F1−2 q1 − − 1 F1−1 − ǫ2 ǫ1 ǫ1       1 1 1 s s q2 , − 1 F2−2 =− − − 1 F2−1 q1 + ǫ2 ǫ1 ǫ2     q3 1 1 s s =− − 1 F3−2 q2 + . − 1 F3−1 q1 − ǫ3 ǫ2 ǫ1

s s s Eb3 = Eb2 − ǫ3 F1−3 )Eb1 − F1−2 (1 − F1−1

s s s −F2−1 Eb1 + (1 − F2−2 )Eb2 − ǫ3 F2−3 Eb3 s s −F3−1 Eb1 − F3−2 Eb2 + Eb3



We note that q3 only enters the last equation, so we only have two simultaneous equations to solve (i.e., as many as we have surfaces with diffuse reflection components). We shall need to determine the s s s , F1−2 , and F2−2 , while the rest can be evaluated through reciprocity and the specular view factors F1−1 summation rule. Considering the first case of Fig. 6-8a, we find s F1−1 = ρs3 F1(3)−1 , s = F1−2 + ρs3 F1(3)−2 , F1−2

s s s , − F1−2 = 1 − F1−1 ǫ3 F1−3

s s = A1 F1−2 /A2 , F2−1 s = ρs3 F2(3)−2 , F2−2 s F3−1

=

s A1 F1−3 /A3 ,

s s s ǫ3 F2−3 = 1 − F2−1 − F2−2 , s s F3−2 = A2 F2−3 /A3 ,

where all view factors on the right-hand sides are readily evaluated through standard diffuse view factor analysis. The problem becomes slightly more difficult in the configuration shown in Fig. 6-8b, where the specular surface is attached to another surface with an opening angle of > 90◦ . Standing in the left corner on surface A2 , one obviously cannot see all of the image A2(3) from there by looking through

208

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

30 cm A4

40 cm

A4

A4(1) A3(1)

A3

A1

A4

A3

A1

A1

A3 A2

A2(1) A3(21)

A2

A2 A1(2)

A3(2)

A2(1)

A3(12)

A3(21)

A3(12) A4(2)

A4(12), A4(21) (a)

(b)

(c)

FIGURE 6-9 Rectangular enclosure with two adjacent specular reflectors, with some possible beam paths indicated: (a) evaluation s s s of F3−4 , (b) evaluation of A3(12) contribution to F3−3 , (c) evaluation of A3(21) contribution to F3−3 .

“mirror” A3 . Care must be taken that these visual obstructions are not overlooked. If the enclosure is two-dimensional, such partially obstructed view factors are no problem for the crossed-strings method, but may pose great difficulty for an analytical solution otherwise.

The effects of partial shading become somewhat more obvious when configurations with two or more adjacent specular surfaces are considered. Example 6.7. Consider the rectangular enclosure shown in Fig. 6-9. Surfaces A1 and A2 are purely specular, and surfaces A3 and A4 are purely diffuse reflectors. Top and bottom walls are at T1 = T3 = 1000 K, with ǫ1 = 1 − ρs1 = ǫ3 = 1 − ρ3d = 0.3; the side walls are at T2 = T4 = 600 K with emittances ǫ2 = 1 − ρs2 = ǫ4 = 1 − ρ4d = 0.8. Determine the net radiative heat flux for each surface. Solution s s = F2−2 = 0, while all other specular view factors are nonzero. Looking at Fig. 6-9a, one sees that F1−1 Again, with Hois = 0, we have from equation (6.22)    q1  1 1 s s s s s − 1 F1−4 q4 , − 1 F1−3 q3 − − Eb4 = Eb3 − F1−4 Eb2 − F1−3 Eb1 − ǫ2 F1−2 ǫ4 ǫ3 ǫ1    q2  1 1 s s s s s Eb1 + Eb2 − F2−3 Eb3 − F2−4 Eb4 = − 1 F2−4 q4 , −ǫ1 F2−1 − − 1 F2−3 q3 − ǫ4 ǫ2 ǫ3       1 1 1 s s s s s s q4 , − 1 F3−4 q3 − − 1 F3−3 Eb1 − ǫ2 F3−2 Eb2 + (1 − F3−3 )Eb3 − F3−4 Eb4 = −ǫ1 F3−1 − ǫ4 ǫ3 ǫ3       1 1 1 s s s s s s q4 . − 1 F4−4 − 1 F4−3 q3 + − Eb1 − ǫ2 F4−2 Eb2 − F4−3 Eb3 + (1 − F4−4 )Eb4 = − −ǫ1 F4−1 ǫ4 ǫ3 ǫ4

Again, we have only two simultaneous equations to solve for the two (diffuse) heat fluxes q3 and q4 : The first two equations are explicit expressions for q1 and q2 , respectively (once q3 and q4 have been determined). Checking the various images in Fig. 6-9a, we find that the specular view factors for surface A1 are s = 0, F1−1 s = F1−2 , F1−2 s = F1−3 + ρs2 F1(2)−3 , F1−3 s = F1−4 + ρs2 F1(2)−4 . F1−4

Checking the summation rule, we find s s s s + F1−3 + F1−4 = 0 + F1−2 +F1−3 +F1−4 − ρs2 (F1−2 −F1(2)−3 −F1(2)−4 ) = 1 + (1 − ρs2 )F1−2 (1 − ρs1 )F1−1

or F1(2)−3 + F1(2)−4 = F1−2 .

6.3 ENCLOSURES WITH PARTIALLY SPECULAR SURFACES

209

Indeed, by checking Fig. 6-9a, we find F1(2)−3 + F1(2)−4 = F1(2)−(3+4) = F1(2)−2 = F1−2 . Similarly, we have s = 0, F2−2 s = F2−1 , F2−1 s = F2−3 + ρs1 F2(1)−3 , F2−3 s = F2−4 + ρs1 F2(1)−4 . F2−4

For surfaces A3 and A4 dual specular reflections are possible: s = F3−1 + ρs2 F3(2)−1 , F3−1 s = F3−2 + ρs1 F3(1)−2 , F3−2 s = ρs1 F3(1)−3 + ρs1 ρs2 F3(12)−3 + ρs2 ρs1 F3(21)−3 , F3−3 s = F3−4 + ρs1 F3(1)−4 + ρs2 F3(2)−4 + ρs1 ρs2 F3(12)−4 + ρs2 ρs1 F3(21)−4 , F3−4 s = F4−1 + ρs2 F4(2)−1 , F4−1 s = F4−2 + ρs1 F4(1)−2 , F4−2 s = F4−3 + ρs1 F4(1)−3 + ρs2 F4(2)−3 + ρs1 ρs2 F4(12)−3 + ρs2 ρs1 F4(21)−3 , F4−3 s = ρs2 F4(2)−4 + ρs1 ρs2 F4(12)−4 + ρs2 ρs1 F4(21)−4 . F4−4

It is tempting to assume that F4(12)−4 = F4(21)−4 , etc. Closer inspection of Figs. 6-9b and c reveals, however, that these view factors are partially obstructed: For example, for F4(21)−4 all rays from A4(21) to A4 must pass through the image A2(1) as well as A1 , i.e., all rays must stay below the corner between A1 and A2 (center point of Fig. 6-9b). On the other hand, for F4(12)−4 all rays from A4(12) must stay above the corner between A1 and A2 , and both together add up to the unobstructed view factor from the image to A4 . The same is true for F3(12)−3 + F3(21)−3 . However, the geometry is such that F4(21)−3 = 0, while F4(12)−3 is unobstructed (thus, still adding up to the unobstructed view factor). Similarly, F3(12)−4 = 0, while F3(21)−4 is unobstructed. Simplifications for partially obstructed view factor were found for this particular simple geometry. Care must be taken before extrapolating these results to other configurations. Before actually evaluating view factors one should take advantage of the fact that there are only two different surface temperatures, i.e., Eb3 = Eb1 and Eb4 = Eb2 , and only two emittances, ǫ3 = ǫ1 and ǫ4 = ǫ2 :    q1  1 1 s s s s s − 1 F1−4 q4 , − 1 F1−3 q3 − − )Eb1 − (ǫ2 F1−2 + F1−4 )Eb2 = (1 − F1−3 ǫ2 ǫ1 ǫ1    q2  1 1 s s s s s − 1 F2−4 q4 , −(ǫ1 F2−1 + F2−3 )Eb1 + (1 − F2−4 )Eb2 = − 1 F2−3 q3 − − ǫ2 ǫ2 ǫ1       1 1 1 s s s s s s (1 − ǫ1 F3−1 − F3−3 )Eb1 − (ǫ2 F3−2 + F3−4 )Eb2 = − 1 F3−4 q4 , q3 − − 1 F3−3 − ǫ2 ǫ1 ǫ1       1 1 1 s s s s s s − 1 F4−3 q3 + −(ǫ1 F4−1 + F4−3 )Eb1 + (1 − ǫ2 F4−2 − F4−4 )Eb2 = − − q4 . − 1 F4−4 ǫ1 ǫ2 ǫ2 The necessary view factors are readily found from the crossed-strings method [equation (4.50)], reciprocity, and the summation rule [equation (6.21)], as well as from Example 5.1 for the diffuse view factors: s = F1−2 = 0.25; F1−2

F1−3 = 0.5,

√ F1(2)−3 = ( 64 + 9 + 3 − 2 × 5)/2 × 4 = 0.1930 :

s = 0.5 + 0.2 × 0.1930 = 0.5386; F1−3

F1−4 = 0.25,

F1(2)−4 = (5 + 8 − 4 −

s F1−4

= 0.25 + 0.2 × 0.0570 = 0.2614;

s F2−1

= F2−1 = 0.3333;



73)/8 = 0.0570 :

210

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

F2−3 = 0.3333,

F2(1)−3 = (5 + 6 −



52 − 3)/6 = 0.1315 :

s = 0.3333 + 0.7 × 0.1315 = 0.4254; F2−3 √ F2−4 = 0.3333, F2(1)−4 = ( 52 + 4 − 2 × 5)/6 = 0.2019 : s = 0.3333 + 0.7 × 0.2019 = 0.4746; F2−4 s s F3−1 = F1−3 = 0.5386; s s = A2 F2−3 /A3 = 0.75 × 0.4254 = 0.3191; F3−2 √ F3(1)−3 = ( 52−6)/4 = 0.3028, √ F3(12)−3 + F3(21)−3 = (10+6−2 52)/8 = 0.1972 : s = 0.7 × 0.3028 + 0.2 × 0.7 × 0.1972 = 0.2396; F3−3 s s s s − ǫ2 F3−2 − F3−3 = 1 − ǫ1 F3−1 F3−4

= 1 − 0.3 × 0.5386 − 0.8 × 0.3191 − 0.2396 = 0.3436; s F4−1 s F4−2 s F4−3 s F4−4

s = A1 F1−4 /A4 = 0.2614/0.75 = 0.3485; s = 0.4746; = F2−4 s /A4 = 0.3436/0.75 = 0.4581; = A3 F3−4 s s s = 1 − ǫ1 F4−1 − ǫ2 F4−2 − F4−3

= 1 − 0.3 × 0.3485 − 0.8 × 0.4746 − 0.4581 = 0.0576. Substituting these values into the heat flux equations and realizing, from the summation rule, that the two coefficients in front of Eb1 and Eb2 are the same for each equation, we obtain

   q1  1 1 −1 0.5386q3 − −1 0.2614q4 , − 0.8 0.3 0.3    q2  1 1 −1 0.4746q4 , −1 0.4254q3 − − −(1 − 0.4746)(Eb1 −Eb2 ) = 0.8 0.8 0.3       1 1 1 −1 0.2396 q3 − (0.8× 0.3191+0.3436)(Eb1 −Eb2 ) = − −1 0.3436q4 , 0.3 0.3 0.8       1 1 1 −1 0.0576 q4 . − −1 0.4581q3 + −(0.3× 0.3485+0.4581)(Eb1 −Eb2 ) = − 0.8 0.8 0.3 (1 − 0.5386)(Eb1 −Eb2 ) =

After a little cleaning up these equations become 2.7743q3 − 0.0859q4 = 0.5989(Eb1 − Eb2 ), −1.0689q3 + 1.2356q4 = −0.5627(Eb1 − Eb2 ), q1 = 0.3770q3 + 0.0196q4 + 0.1384(Eb1 − Eb2 ), q2 = 0.7941q3 + 0.0949q4 − 0.4203(Eb1 − Eb2 ). Solving the first two equations leads to 0.5989×1.2356−0.5627×0.0859 (Eb1 −Eb2 ) = 0.2073(Eb1 −Eb2 ), 2.7743×1.2356−1.0689×0.0859 0.5989×1.0689−0.5627×2.7743 q4 = (Eb1 −Eb2 ) = −0.2761(Eb1 −Eb2 ), 2.7743×1.2356−1.0689×0.0859

q3 =

and q1 = [0.3770 × 0.2073 + 0.0196 × (−0.2761) + 0.1384](Eb1 − Eb2 ) = 0.2111(Eb1 − Eb2 ), q2 = [0.7941 × 0.2073 + 0.0949 × (−0.2761) − 0.4203](Eb1 − Eb2 ) = −0.2819(Eb1 − Eb2 ). To determine the net surface heat fluxes we evaluate Eb1 − Eb2 = σ(T14 − T24 ) = 5.670×10−8 (10004 −6004 ) W/m2 = 4.935 W/cm2

6.3 ENCLOSURES WITH PARTIALLY SPECULAR SURFACES

A2

211

A1

A1(2) A2(12) A1(212)



12)

A 2(12

A2

A

L

(121 212)



1( 21 21 2)

d

(a) (b) FIGURE 6-10 s Geometry for Example 6.8: (a) V-corrugated surface, (b) images for a single V for the evaluation of F1−1 .

and multiply by the respective surface areas. Thus, Q′1 = 40 cm × 0.2111 × 4.935 W/cm2 = 41.7 W/cm, Q′2 = 30 cm × (−0.2819) × 4.935 W/cm2 = −41.7 W/cm, Q′3 = 40 cm × 0.2073 × 4.935 W/cm2 = 40.9 W/cm, Q′4 = 30 cm × (−0.2761) × 4.935 W/cm2 = −40.9 W/cm. Checking our results, we note that the four heat fluxes add up to zero as they should. The results of the present example—an enclosure with two adjacent specular reflectors—should be compared with those of Example 5.4, dealing with the identical problem except that all four surfaces were perfectly diffuse reflectors. For Example 5.4, we had found Q′1 = −Q′2 = Q′3 = −Q′4 = 42.3 W/cm. For the present configuration the heat fluxes of the specular surfaces are reduced by 1%, while the heat fluxes of the diffuse surfaces are reduced a little more, by approximately 3%. Overall, the effects of specularity are found to be rather minor.

In the last two examples only two simultaneous equations had to be solved, even though there were three and four unknown surface heat fluxes, respectively, because for any purely specular surface with known temperature the radiosity is not unknown, but is given as J = ǫEb . Thus, for an enclosure consisting of N surfaces, of which n are purely specular with known temperature, only N − n simultaneous equations need to be solved. While this fact simplifies specular enclosure analysis as compared with diffuse enclosures, one should remember that, in general, specular view factors are considerably more difficult to evaluate. As a final example for configurations with flat surfaces we shall consider a case where many specular reflections are possible. Example 6.8. Since solar energy strikes the absorbing plate of a strategically oriented solar collector only over a narrow band of incidence directions (varying somewhat during the day, as well as during the year), the ideal collector material would be directionally selective: The emittance should be high for directions of solar incidence (to maximize energy collection), and low for all other directions (to minimize reradiation losses). One such material is a V-corrugated specular surface shown in Fig. 610a. Assuming that the V-corrugated groove, with opening angle 2γ, is coated with a purely specular

212

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

reflecting material, with emittance ǫ = 1 − ρs , what is the apparent hemispherical emittance of such a surface (i.e., what is its heat loss compared with a flat black plate at the same temperature)? Solution Calling the two surfaces in a single “V” A1 and A2 , as indicated in Fig. 6-10b, with Eb1 = Eb2 = Eb , s s = Ho2 = 0 we obtain from equation (6.22) (for i = 1) ǫ1 = ǫ2 = ǫ, and Ho1 h

 i q s s 1 − ǫ F1−1 Eb = . + F1−2 ǫ

Total heat lost from both surfaces of the groove is Q = q × 2L = qd/ sin γ; on the other hand, heat lost from a black surface covering the opening would be Qb = Eb d. Thus, the apparent emittance is i h  s s ǫ 1 − ǫ F1−1 + F1−2 q Q = . = ǫa = Eb sin γ sin γ Qb s This expression could be further simplified, using summation rule and reciprocity, to ǫa = ǫ F1−3 / sin γ = s s s , where A3 is the open top of the V (and of width d). However, F1−1 and F1−2 are somewhat simpler 2ǫ F3−1 to evaluate, and we shall do so here: A beam leaving surface A1 can return to A1 (i) after a single reflection off surface A2 [appearing to come from the image A1(2) , as indicated in Fig. 6-10b], or (ii) after hitting A2 , traveling back to A1 , returning one more time to A2 , and hitting A1 a second time [i.e., a beam that appears to come from image A1(212) ], and so on. Thus, s F1−1 = ρF1(2)−1 + ρ3 F1(212)−1 + ρ5 F1(21212)−1 + . . . . s s s F1−2 may be similarly evaluated. We shall here determine F2−1 = F1−2 instead, since this expression allows us to employ the images shown in Fig. 6-10b: Energy may travel directly from A2 to A1 , or go from A2 to A1 , get reflected back to A2 , and reflected back to A1 again [appearing to come from image A2(12) ], and so forth. Therefore, s s = F2−1 + ρ2 F2(12)−1 + ρ4 F2(1212)−1 + . . . . = F2−1 F1−2

Adding both together and using reciprocity (with all areas being the same), we obtain s s F1−1 = F1−2 + ρF1−1(2) + ρ2 F1−2(12) + ρ3 F1−1(212) + . . . . + F1−2

Each one of these view factors Fi−j is subject to the restriction that all beams from A1 to the image A j must pass through all the images between A1 and A j ; however, in this geometry no partial obstruction occurs as seen from Fig. 6-10b. The series above ends as soon as the image can no longer be seen from A1 , i.e., when the opening angle between A1 and the image exceeds 180◦ . The view factor for a V-groove with opening angle 2φ is, from Configuration 34 in Appendix D, F2φ = 1 − sin φ. Thus, s s F1−1 + F1−2 = 1 − sin γ + ρ(1 − sin 2γ) + ρ2 (1 − sin 3γ) + . . . + ρn−1 (1 − sin nγ),

Finally, the apparent hemispherical emittance of the V-corrugated surface is   n X  ǫ  k−1 1 − ǫ  , n < π/2γ. ρ (1 − sin kγ) ǫa =   sin γ

nγ < π/2.

k=1

Figure 6-11 shows the apparent hemispherical emittance of V-corrugated surfaces as a function of opening angle for a number of flat-surface emittances. Also shown in the figure is the normal emittance (or absorptance), which may also be calculated from equation (6.22) (left as an exercise). For example, for ǫ = 0.5 and a groove opening angle of γ = 30◦ , the apparent hemispherical emittance (important for reradiation losses) is 0.72, and the normal emittance (important for solar energy collection) is 0.88. While the difference between these two values is not huge, the corrugated groove (i) helps to make the absorber plate more black, and (ii) substantially reduces the reradiation losses (by ≃ 20% for the ǫ = 0.5, γ = 30◦ surface). More detail about the radiative properties of V-corrugated grooves may be found in the papers by Eckert and Sparrow [4], Sparrow and Lin [5], and Hollands [6], and the book by Sparrow and Cess [7].

6.3 ENCLOSURES WITH PARTIALLY SPECULAR SURFACES

1.0

213

∋ = 0.90

Apparent hemispherical emittance, ∋ a ; normal absorptance, αan

0.9

0.8

∋a αan

0.7

∋ = 0.50

0.6

0.5

0.4

∋ = 0.25 0.3

∋ = 0.15 0.2

∋ = 0.10 0.1

∋ = 0.05 ∋ = 0.01

0 0

10°

20° 30° 40° Opening angle, γ

50°

60°

FIGURE 6-11 Apparent normal and hemispherical emittances for specularly reflecting V-corrugated surfaces [6].

Curved Surfaces with Specular Reflection Components In all our examples we have only considered idealized enclosures consisting of flat surfaces, for which the mirror images necessary for specular view factor calculations are relatively easily determined. If some or all of the reflecting surfaces are curved then equations (6.18) and (6.20) remain valid, but the specular view factors tend to be much more difficult to obtain. Analytical solutions can be found only for relatively simple geometries, such as axisymmetric surfaces, but even then they tend to get very involved. The very simple case of cylindrical cavities (with and without specularly reflecting end plate) has been studied by Sparrow and coworkers [8–10] and by Perlmutter and Siegel [11]. The more involved case of conical cavities has been treated by Sparrow and colleagues [9,10,12] as well as Polgar and Howell [13], while spherical cavities have been addressed by Tsai and coworkers [14,15] and Sparrow and Jonsson [16,17]. Somewhat more generalized discussions on the determination of specular view factors for curved surfaces have been given by Plamondon and Horton [18] and by Burkhard and coworkers [19]. In view of the complexity involved in these evaluations, specular view factors for curved surfaces are probably most conveniently calculated by a statistical method, such as the Monte Carlo method, which will be discussed in detail in Chapter 8. A considerably more detailed discussion of thermal radiation from and within grooves and cavities is given in the book by Sparrow and Cess [7].

214

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

1 ∋1 A1 F s1–2

Eb1

1– ∋ 2 ∋ 2 A2

J2

Eb2

1

– (1

2–

Fs

1

4

∋ s –3

(1 –

s 4)

A

2

F1 1 A1 s ) ρ3

1 ∋1 (1– ρ s4) A1 F s1–4

ρ

1 (1– ρ s3) A2 F s2–3

Eb3 J4 (1– ρ s4)

6.4

1 (1– ρs3) (1– ρs4) A3 F s3– 4

J3 (1– ρ s3)

ρ d3 (1– ρ s3)∋ 3 A3

FIGURE 6-12 Electrical network equivalent for a four-surface enclosure (A1 = specular, A2 = diffuse, A3 = partially diffuse and specular, A4 = insulated, partially specular).

ELECTRICAL NETWORK ANALOGY

The electrical network analogy, first introduced in Section 5.4, may be readily extended to allow for partially specular reflectors. This possibility was first demonstrated by Ziering and Sarofim [20]. Expressing equations (6.12) and (6.15) for an idealized enclosure [i.e., an enclosure with finite surfaces of constant radiosity, exactly as was done in equation (6.19)], we can evaluate the nodal heat fluxes as   N  X  s  s s qi = Ji − (1 − ρi )  (6.27) Jj Fi−j + Hoi  , i = 1, 2, . . . , N.   j=1

Using the summation rule, equation (6.21), this relation may also be written as the sum of net radiative interchange between any two surfaces, qi =

N h X j=1

i s (1 − ρsj )Ji − (1 − ρsi )Jj Fi−j − (1 − ρsi )Hois

 N  X Jj   Ji  (1 − ρs )(1 − ρs )F s − (1 − ρs )H s .  − =  i j i−j i oi 1 − ρsi 1 − ρsj 

(6.28)

j=1

Similarly, from equation (6.13),

qi =

(1 − ρsi )ǫi ρid

Ebi −

! Ji . 1 − ρsi

(6.29)

After multiplication with Ai these relations may be combined and written in terms of potentials [Ebi and Ji /(1 − ρsi )] and resistances as Ebi − Qi =

Ji 1 − ρsi

ρid (1 − ρsi )ǫi Ai

=

N X j=1

Jj Ji s − 1 − ρi 1 − ρjs 1 (1 − ρsi )(1 − ρjs )Ai Fi−s j

− (1 − ρsi ) Ai Hois .

(6.30)

Of course, this relation reduces to equation (5.46) for the case of purely diffuse surfaces (ρsi = 0, i = 1, 2, . . . , N). As an example, Fig. 6-12 shows the equivalent electrical network for an

6.5 RADIATION SHIELDS

215

enclosure consisting of four surfaces: Surface A1 is a specular reflector (ρ1d = 0), surface A2 is a diffuse reflector (ρs2 = 0), surface A3 has specular and diffuse reflectance components, and surface A4 (also partially specular) is insulated. Note that, unlike diffuse reflectance, the specular reflectance is not irrelevant for insulated surfaces.

6.5

RADIATION SHIELDS

As noted in Section 5.5 radiation shields tend to be made of specularly reflecting materials, such as polished metals or dielectric sheets coated with a metallic film. We would like, therefore, to extend the analysis to partly specular surfaces, i.e., (referring to Fig. 5-13) ǫk = 1 − ρsk − ρkd for all surfaces (inside and outside wall, all shield surfaces). Again, the analysis is most easily carried out using the electrical network analogy, and the resistance between any two layers has already been evaluated in Example 6.4, equation (6.26), as ! ρsk 1 1 1 1 . (6.31) − − + R j−k = ǫ j A j ǫk Ak 1 − ρks Ak A j

The resistances given in equation (6.31) may be simplified somewhat if surface Ak is either a purely diffuse reflector (ρsk = 0), or a purely specular reflector (1 − ρsk = ǫk ):   1 1 1 + , (6.32a) −1 R j−k = Ak diffuse : ǫ jA j Ak ǫk ! 1 1 1 −1 + . (6.32b) R j−k = Ak specular : ǫ j ǫk Aj Following the procedure of Section 5.5, equation (5.48) still holds, i.e., Q=

Ri−1i +

Ebi − Ebo . PN−1 n=1 Rno−n+1,i + RNo−o

(6.33)

Example 6.9. Repeat Example 5.9 for purely specularly reflecting shields. The wall material (steel) may be diffusely or specularly reflecting. Solution As before we note from equation (6.32) that the resistances are inversely proportional to shield area, and will again assume A1 ≃ A2 ≃ . . . ≃ AN = As = πDs L, with Ds = 11 cm. Evaluating the total resistance from equations (6.33) and (6.32), we find Ai Rtot =

N−1   A∗ X     1 Ai Ai 1 1 2 Ai 1 −1 i + + −1 + + −1 ∗, ǫs As ǫs As ǫs ǫw Ao As ǫw n=1

where, if the steel is specular A∗i = Ai , A∗o = As , and if it is diffuse A∗i = As , A∗o = Ao . We shall investigate both possibilities to see whether specularity of the steel is an important factor in this arrangement. Again, we may solve for N as !  A∗    Ai 1 Ai 1 1 i − −1 −1 ∗ + − Ai Rtot − As As ǫs Ao ǫw ǫw N=   2 Ai −1 ǫs As      1 10 1 1 − 0.3 − 1 11 580.0 − 0.3 − 1 1 − 10 − 0.05 11   = 16.16, steel specular, = 2 10 0.05 − 1 11      10 1 1 10 1 580.0 − 0.3 − 0.3 − 1 10 11 − 11 20 − 0.05 − 1   = = 16.23, steel diffuse. 2 10 0.05 − 1 11

216

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

Therefore, the same minimum of 17 radiation shields would be required. We note that the specularity of the shields has no impact whatsoever (because we assumed them to be infinitely close together in this analysis), while specular inner and outer cylinder walls marginally improve performance. Without radiation shields we obtain qi =

=

5.670 × 10−12 |4.24 − 2984 | |Ebi − Ebo |      = 1 Ai 1 1 1 + − 1 × [1 or 12 ] − 1 × 1 or + 0.3 0.3 ǫw Ao ǫw



(

9.94 × 10−3 W/cm2 , steel diffuse, 7.89 × 10−3 W/cm2 , steel specular,

i.e, without shields the aspect ratio Ai /Ao = 1/2 deviates considerably from unity, making the differences between specular and diffuse cylinders more apparent.

6.6 SEMITRANSPARENT SHEETS (WINDOWS) When we developed the governing relations for radiative heat transfer in an enclosure bounded by diffusely reflecting surfaces (Chapter 5) or by partially diffuse/partially specular reflectors (this chapter), we made allowance for external radiation to penetrate into the enclosure through holes and/or semitransparent surfaces (windows). While we have investigated some examples with external radiation entering through holes, only one (Example 5.8) has dealt with a simple semitransparent surface. Radiative heat transfer in enclosures with semitransparent windows occurs in a number of important applications, such as solar collectors, externally irradiated specimens kept in a controlled atmosphere, furnaces with sight windows, and so on. We shall briefly outline in this section how such enclosures may be analyzed with equation (6.18) or (6.22). To this purpose we shall assume that properties of the semitransparent window are wavelength-independent (gray), that equation (6.1) describes the reflectance (facing the inside of the enclosure), and that the transmittance of the window also has specular (light is transmitted without change of direction) and diffuse (light leaving the window is perfectly diffuse) components.3 Thus, ρ + τ + α = ρs + ρ d + τs + τ d + α = 1,

ǫ = α.

(6.34)

Further, we shall assume that radiation hitting the outside of the window has a collimated component qoc (i.e., parallel rays coming from a single direction, such as sunshine) and a diffuse component qod (such as sky radiation coming in from all directions with equal intensity). Making an energy balance for the net radiative heat flux from the semitransparent window into the enclosure leads to (cf. Fig. 6-13): q(r) = qem + qtr,in − qabs − qtr,out = ǫ(r)Eb (r) + τ d (r)qoc (r) + τ(r)qod (r) − α(r)H(r) − τ(r)H(r),

(6.35)

where the specularly transmitted fraction of the collimated external radiation, τs qoc , has not been accounted for since it enters the enclosure in a nondiffuse fashion; it is accounted for in Hos (r′ ) as part of the irradiation at another enclosure location r′ (traveling there directly, or after any number of specular reflections). Using equation (6.34), equation (6.35) may also be written as   q(r) = qout − qin = ǫEb + τ d qoc + τqod + ρ d H + ρs H − H, (6.36)

3 It is unlikely that a realistic window has both specular and diffuse transmittance components; rather its transmittance will either be specular (clear windows) or diffuse (milky windows, glass blocks, etc.). We simply use the more general expression to make it valid for all types of windows.

6.6 SEMITRANSPARENT SHEETS (WINDOWS)

217

τ sqoc τ dqoc qoc qod

τ qod H

τ sH

τ dH

ρ dH

∋ Eb

ρ sH

FIGURE 6-13 Energy balance for a semitransparent window.

where qin is the energy falling onto the inside of the window coming from within the enclosure. The first four terms of qout are diffuse and may be combined to form the radiosity J(r) = ǫEb + τ d qoc + τqod + ρ d H.

(6.37)

Examination of equations (6.34) through (6.37) shows that they may be reduced to equations (6.10) through (6.12) if we introduce an apparent emittance ǫa and an apparent blackbody emissive power Eb,a as ǫa (r) = ǫ + τ = 1 − ρ, d

ǫa Eb,a (r) = ǫEb + τ qoc + τqod .

(6.38a) (6.38b)

Thus, the semitransparent window is equivalent to an opaque surface with apparent emittance ǫa and apparent emissive power Eb,a (if the radiative properties are gray). Therefore, equations (6.18) and (6.22) remain valid as long as the emittance and blackbody emissive powers of semitransparent surfaces are understood to be apparent values. Example 6.10. A long hallway 3 m wide by 4 m high is lighted with a skylight that covers the entire ceiling. The skylight is double-glazed with an optical thickness of κd = 0.037 per window plate. The floor and sides of the hallway may be assumed to be gray and diffuse with ǫ = 0.2. The outside of the skylight is exposed to a clear sky, so that diffuse visible light in the amount of qsky = 20,000 lm/m2 is incident on the skylight. Direct sunshine also falls on the skylight in the amount of qsun = 80,000 lm/m2 (normal to the rays). For simplicity assume that the sun angle is θs = 36.87◦ as indicated in Fig. 6-14. Determine the amount of light incident on a point in the lower right-hand corner (also indicated in the figure) if (a) the skylight is clear, (b) the skylight is diffusing (with the same transmittance and reflectance). Solution From Fig. 3-32 for double glazing and κd = 0.037 we find a hemispherical transmittance (i.e., directionally averaged) of τ ≃ 0.70, while for solar incidence with θ = 36.87◦ we have τθ ≃ 0.75. The hemispherical reflectance of the skylight may be estimated by assuming that the reflectance is the same as the one of a nonabsorbing glass. Then, from Fig. 3-31 ρ1 = ρs1 = 1 − τ(κd = 0) ≃ 1 − 0.75 = 0.25. From equation (6.38) we find ǫ1,a = 1 − ρ1 = 0.75 and, for a clear skylight, ǫ1,a Eb1,a = 0 + 0 + τqsky since τ d = 0, and since there is no luminous emission from the window (or from any of the other walls, for that matter). Because of the s = τθ qsun sin θs . special sun angle, direct sunshine falls only onto surface A2 , filling the entire wall, i.e., Ho2 To determine the illumination at the point in the corner, we need to calculate the local irradiation H (in terms of lumens). This calculation, in turn, requires knowledge of the radiosity for all the surfaces of the hallway (for the skylight it is already known as J1 = ǫ1,a Eb1,a = τqsky , since ρ1d = 0). To this purpose we shall approximate the hallway as a four-surface enclosure for which we shall calculate the

218

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

A3(1) A4(1)

A2(1) qsun qsky

Skylight, A1

θs

A2

A4

θs

A3

p FIGURE 6-14 Geometry for a skylit hallway (Example 6.10).

average radiosities. Based on these radiosities we may then calculate the local irradiation for a point from equation (6.15). While equation (6.22) is most suitable for heat transfer calculations, we shall use equation (6.19) for this example since radiosities are more useful in lighting calculations.4 Therefore, for i = 2, 3, and 4,   s s s s s + Ho2 , + J2 F2−2 + J3 F2−3 + J4 F2−4 J2 = ρ2 J1 F2−1   s s s s J3 = ρ3 J1 F3−1 + J2 F3−2 + J3 F3−3 + J4 F3−4 ,   s s s s J4 = ρ4 J1 F4−1 . + J2 F4−2 + J3 F4−3 + J4 F4−4 The necessary view factors are readily calculated from the crossed-strings method: s F2−1 = F2−1 =

3+4−5 = 0.25, 2×4

s F2−2 = 0,

√ 8+5−(4+ 73) = 0.25(1+0.05700) = 0.26425, 2×4 √ 3+ 73−2 × 5 = 0.5 + 0.25 × = 0.5 + 0.25 × 0.19300 = 0.54825, 2×4

s F2−3 = F2−3 + ρ1 F2(1)−3 = 0.25 + 0.25 × s = F2−4 + ρ1 F2(1)−4 F2−4

2×5 − 2×4 = 0.33333, 2×3 A2 s 4 = F = × 0.26425 = 0.35233, A3 2−3 3 √ 2× 73−2×8 = 0.25 × 0.18133 = 0.04533, = ρ1 F3(1)−3 = 0.25 × 2×3 s = F3−2 = 0.35233,

s = F3−1 = F3−1 s F3−2 s F3−3

s F3−4

s s = 0.2500, = F2−1 F4−1 s F4−3

=

s F2−3

= 0.26425,

s s = 0.54825, = F2−4 F4−2 s F4−4 = 0.

4 If equation (6.22) is used the resulting heat fluxes are converted to radiosities using equation (6.13), or J = −ρ d q/ǫ (since Eb = 0).

6.6 SEMITRANSPARENT SHEETS (WINDOWS)

219

s Therefore, after normalization with Ji = Ji /J1 and H = Ho2 /J1 , and with ρ2 = ρ3 = ρ4 = 1 − 0.2 = 0.8,

J2 = 0.8(0.25 + 0 + 0.26425 J3 + 0.54825 J4 ) + H, J3 = 0.8(0.33333 + 0.35233 J2 + 0.04533 J3 + 0.35233 J4 ), J4 = 0.8(0.25 + 0.54825 J2 + 0.26425 J3 + 0), or

J2 − 0.21140 J3 − 0.43860 J4 = H + 0.2, −0.28186 J2 + 0.96374 J3 − 0.28186 J4 = 0.26667, −0.43860 J2 − 0.21140 J3 + J4 = 0.2. Omitting the details of solving these three simultaneous equations, we find

J2 = 1.48978H + 0.59051, J3 = 0.66812H + 0.62211, J4 = 0.79466H + 0.59051. The irradiation onto the corner point is, from equation (6.15) Hp =

4 X j=1

  s s s s s Jj Fp− j = J1 Fp−1 + J2 Fp−2 + J3 Fp−3 + J4 Fp−4 ,

where the view factors may be determined from Configurations 10 and 11 in Appendix D (with b → ∞, and multiplying by 2 since the strip tends to infinity in both directions):

1 3 a 1 = × = 0.3, √ 2 5 2 a2 + c 2 h i = Fp−2 + ρ1 Fp(1)−2 = Fp−2 + ρ1 Fp(1)−2+2(1) − Fp(1)−2(1) , !   c 3 1 1 = 0.2, 1− √ 1− Fp−2 = = 5 2 2 a2 + c2 ! 1 3 Fp(1)−2(1) = Fp−2 = 0.2, Fp(1)−2+2(1) = 1− √ = 0.32444, 2 73

s Fp−1 = Fp−1 =

s Fp−2

s = 0.2 + 0.25 × (0.32444 − 0.2) = 0.23111, Fp−2 s = ρ1 Fp(1)−3 = 0.25 × Fp−3

3 1 × √ = 0.04389, 2 73

s = 0.5. Fp−4

Therefore,

Hp =

Hp J1

= 0.3+0.23111×(1.48978H +0.59051) +0.04389×(0.66812H +0.62211)+0.5×(0.79466H +0.59051) = 0.77096H + 0.75903.

s = τθ qsun sin 36.87◦ = 0.75 × Finally, for a clear window, J1 = τ1 qsky = 0.7 × 20,000 = 14,000 lx, and Ho2 80,000 × 0.6 = 36,000 lx, and

Hp = 0.77096 × 36,000 + 0.75903 × 14,000 = 38,381 lx. s On the other hand, if the window has a diffusing transmittance τ = τ d = 0.7, then Ho2 = 0 and, from equation (6.37), J1 = τ(qsky + qsun cos 36.87◦ ) = 0.7 × (20,000 + 80,000 × 0.8) = 58,800 lx. This results in

Hp = 0.75903 × 58,800 = 44,631 lx. For a diffusing window the light is more evenly distributed throughout the hallway, resulting in higher illumination at point p.

220

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

dx´2 x2

h

x1 0

dx´1

dx1

FIGURE 6-15 Radiative exchange between two long isothermal plates with specular reflection components.

w

6.7 SOLUTION OF THE GOVERNING INTEGRAL EQUATION As in the case for diffusely reflecting surfaces the methods of the previous sections require the radiosity to be constant over each subsurface, a condition rarely met in practice. More accurate results may be obtained by solving the governing integral equation, either equation (6.16) (to determine radiosity J) or equation (6.18) (to determine the unknown heat flux and/or surface temperature directly), by any of the methods outlined in Chapter 5. This is best illustrated by repeating Examples 5.10 to 5.12. Example 6.11. Consider two long parallel plates of width w as shown in Fig. 6-15. Both plates are isothermal at the (same) temperature T, and both have a gray, diffuse emittance of ǫ. The reflectance of the material is partly diffuse, partly specular, so that ǫ = 1 − ρs − ρ d . The plates are separated by a distance h and are placed in a large, cold environment. Determine the local radiative heat fluxes along the plate using numerical quadrature. Solution From equation (6.18) we find, for location x1 on the lower plate, Eb − (1 − ρs )Eb

"Z

w 0

s dFdx ′ + 1 −dx 1

Z

"Z w # # Z w q(x1 ) ρ d ′ s ′ s s q(x ) dF + q(x ) dF − dFdx = ′ ′ ′ 2 1 dx1 −dx2 . dx1 −dx1 1 −dx2 ǫ ǫ 0 0

w 0

The necessary specular view factors are readily found from s = ρs dFdx1 (2)−dx′1 + (ρs )3 dFdx1 (212)−dx′1 + . . . , dFdx ′ 1 −dx 1

s dFdx = dFdx1 −dx′2 + (ρs )2 dFdx1 (21)−dx′2 + . . . . ′ 1 −dx 2

The view factor between two infinitely long parallel strips of infinitesimal width and separated by a distance kh (k = 1, 2, . . .) is given by Example 5.10 as (kh)2 dx′ 1 . 2 [(kh)2 + (x − x′ )2 ]3/2 Thus, s 2 s s s dFdx ′ + dFdx −dx′ = dFdx1 −dx′ + ρ dFdx1 (2)−dx′ + (ρ ) dFdx1 (21)−dx′ + . . . 2 2 1 1 −dx 1 1

2

=

∞ (kh)2 dx′ 1 X s k−1 , (ρ ) 2 [(kh)2 + (x1 − x′ )2 ]3/2 k=1

6.7 SOLUTION OF THE GOVERNING INTEGRAL EQUATION

221

ρs = 0.9

ρs = 0

ρs = 0.6

ρs = 0

ρs = 0.25



Nondimensional heat flux Ψ/ = q/ σT 4

1.00

= 0.1



∋ 0.75

∋ = 0.9

ρs = 0.5

= 0.5

ρs = 0 s

ρ = 0.1

∋ 0.50

Exact 5-point quadrature

0.00

0.25

0.50 0.75 Location along plates ξ = x/w

1.00

FIGURE 6-16 Local radiative heat flux on isothermal, parallel plates with diffuse and specular reflection components.

where we have made use of x′1 = x′2 = x′ . This expression may be substituted into the governing integral equation. Realizing that, by symmetry, q(x′1 ) = q(x′2 ) = q(x′ ) and nondimensionalizing with ξ = x/h, W = w/h, and Ψ = q(ξ)/Eb , lead to  Z Z W X ∞ ∞ X (ρs )k−1 k2 dξ′  (ρs )k−1 k2 dξ′ Ψ(ξ) ρ d  W 1 ′ 1 s  .  − Ψ(ξ ) = 1 − (1 − ρ )  2 ′ 2 3/2 ǫ 2 ǫ  0 2 [k2 + (ξ − ξ′ )2 ]3/2  0 k=1 k=1 [k + (ξ − ξ ) ]

As in Example 5.11 this equation may be solved by numerical quadrature as   J J X X     s d c j fij  , Ψi − ρ W c j Ψj fij = ǫ 1 − (1 − ρ )W   j=1

j=1

where Ψi is evaluated at J nodal positions ξi , i = 1, 2, . . . , J, and the c j are weight coefficients for the numerical integration. The fij are an abbreviation for the integration kernel, fij =

∞ 1X h 2 k=1

k2 (ρs )k−1

k2 + (ξi − ξ j )2

i3/2 .

They must be evaluated by summing as many terms as necessary (decreasing as (ρs )k−1 /k for large k). Results for the same simple J = 5 quadrature of Example 5.11 are given in Fig. 6-16, together with “exact” solutions (high-order quadrature). The results show that, for W = w/h = 1, the heat loss from the plates decreases if reflection is specular: Specular reflection traps emitted radiation somewhat more through repeated reflections between the plates.

Note that, if both surfaces are purely specular, the heat flux may be calculated directly (i.e., no solution of an integral equation is necessary). This calculation was first done for the parallelplate case by Eckert and Sparrow [4]. In general, equation (6.18) is actually easier to solve than its diffuse-reflection counterpart if some or all of the surfaces are purely specular. However, the necessary specular view factors are generally much more difficult—if not impossible—to evaluate. Such a case arises, for example, for curved surfaces with multiple specular reflections. Since the specular view factors for such problems are most easily found from statistical methods, such as the Monte Carlo method (Chapter 8), it is usually best to solve the entire heat transfer problem using the Monte Carlo method.

222

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

∋ = 0.1, 2 γ = 90° 0.9

L

Nondimensional radiative flux, q/∋ σ T 4

1.0

Diffuse Specular ρ = ρ s = const Specular ρ = ρ s (θ ) Bidirectional (σo/ λ = 2/3; a/λ =10) Bidirectional (σo / λ = 1/3; a /λ =5) Bidirectional (σo / λ = 1/15; a/ λ =1) 0.8

6.8

x

0

0.2

0.4 0.6 Distance from vertex, x/L

0.8

1.0

FIGURE 6-17 Local radiative heat flux from the surface of an isothermal V-groove for different reflection behavior; for all surfaces 2γ = 90◦ and ǫ = 0.1; σo is root-mean-square optical roughness, a is a measure [21] for average distance between roughness peaks [22].

CONCLUDING REMARKS

Before leaving the topic of specularly reflecting surfaces we want to discuss briefly under what circumstances the assumption of a partly diffuse, partly specular reflector is appropriate. The analysis for such surfaces is generally considerably more involved than for diffusely reflecting surfaces, as a result of the more difficult evaluation of specular view factors. On the other hand, the analysis is substantially less involved than for surfaces with more irregular reflection behavior (as will be discussed in the following chapter). Examples 6.3 and 6.7 have shown that in fully closed configurations (without external irradiation) the heat fluxes show very little dependence on specularity. This is true for all closed configurations as long as there are no long and narrow channels separating surfaces of widely different temperatures (cf. Problems 6.3 and 6.4). Therefore, for most practical enclosures it should be sufficient to evaluate heat fluxes assuming purely diffuse reflectors—even though a number of surfaces may be decidedly specular. On the other hand, in open configurations, in long and narrow channels, in configurations with collimated irradiation—whenever there is a possibility of beam channeling—the influence of specularity can be very substantial and must be accounted for. It is tempting to think of diffuse and specular reflection as not only extreme but also limiting cases: This leads to the thought that—if heat fluxes have been determined for purely diffuse reflection, and again for purely specular reflection—the heat flux for a surface with more irregular reflection behavior must always lie between these two limiting values. This consideration is true in most cases, in particular since most real surfaces tend to have a reflectance maximum near the specular direction. However, there are cases when the actual heat flux is not bracketed by the diffuse and specular reflection models, particularly for directionally selective surfaces. As an example consider the local radiative heat flux from an isothermal groove, such as the one given by Fig. 6-10. Toor [22] has investigated this problem for diffuse reflectors, for specular reflectors, and for three different types of surface roughnesses analyzed with the Monte Carlo method, and his results are shown in Fig. 6-17. It is quite apparent that, near the vertex of the groove, diffuse and specular reflectors both seriously overpredict the heat loss. The reason is that, at grazing angles, rough surfaces tend to reflect strongly back into the direction of incidence.

References 1. Sarofim, A. F., and H. C. Hottel: “Radiation exchange among non-Lambert surfaces,” ASME Journal of Heat Transfer, vol. 88, pp. 37–44, 1966. 2. Birkebak, R. C., E. M. Sparrow, E. R. G. Eckert, and J. W. Ramsey: “Effect of surface roughness on the total and specular reflectance of metallic surfaces,” ASME Journal of Heat Transfer, vol. 86, pp. 193–199, 1964.

223

PROBLEMS

3. Wylie, C. R.: Advanced Engineering Mathematics, 5th ed., McGraw-Hill, New York, 1982. 4. Eckert, E. R. G., and E. M. Sparrow: “Radiative heat exchange between surfaces with specular reflection,” International Journal of Heat and Mass Transfer, vol. 3, pp. 42–54, 1961. 5. Sparrow, E. M., and S. L. Lin: “Absorption of thermal radiation in v-groove cavities,” International Journal of Heat and Mass Transfer, vol. 5, pp. 1111–1115, 1962. 6. Hollands, K. G. T.: “Directional selectivity, emittance, and absorptance properties of vee corrugated specular surfaces,” Solar Energy, vol. 7, no. 3, pp. 108–116, 1963. 7. Sparrow, E. M., and R. D. Cess: Radiation Heat Transfer, Hemisphere, New York, 1978. 8. Sparrow, E. M., L. U. Albers, and E. R. G. Eckert: “Thermal radiation characteristics of cylindrical enclosures,” ASME Journal of Heat Transfer, vol. 84, pp. 73–81, 1962. 9. Lin, S. H., and E. M. Sparrow: “Radiant interchange among curved specularly reflecting surfaces, application to cylindrical and conical cavities,” ASME Journal of Heat Transfer, vol. 87, pp. 299–307, 1965. 10. Sparrow, E. M., and S. L. Lin: “Radiation heat transfer at a surface having both specular and diffuse reflectance components,” International Journal of Heat and Mass Transfer, vol. 8, pp. 769–779, 1965. 11. Perlmutter, M., and R. Siegel: “Effect of specularly reflecting gray surface on thermal radiation through a tube and from its heated wall,” ASME Journal of Heat Transfer, vol. 85, pp. 55–62, 1963. 12. Sparrow, E. M., and V. K. Jonsson: “Radiant emission characteristics of diffuse conical cavities,” Journal of the Optical Society of America, vol. 53, pp. 816–821, 1963. 13. Polgar, L. G., and J. R. Howell: “Directional thermal-radiative properties of conical cavities,” NASA TN D-2904, 1965. 14. Tsai, D. S., F. G. Ho, and W. Strieder: “Specular reflection in radiant heat transport across a spherical void,” Chemical Engineering Science–Genie Chimique, vol. 39, pp. 775–779, 1984. 15. Tsai, D. S., and W. Strieder: “Radiation across a spherical cavity having both specular and diffuse reflectance components,” Chemical Engineering and Science, vol. 40, no. 1, p. 170, 1985. 16. Sparrow, E. M., and V. K. Jonsson: “Absorption and emission characteristics of diffuse spherical enclosures,” NASA TN D-1289, 1962. 17. Sparrow, E. M., and V. K. Jonsson: “Absorption and emission characteristics of diffuse spherical enclosures,” ASME Journal of Heat Transfer, vol. 84, pp. 188–189, 1962. 18. Plamondon, J. A., and T. E. Horton: “On the determination of the view function to the images of a surface in a nonplanar specular reflector,” International Journal of Heat and Mass Transfer, vol. 10, no. 5, pp. 665–679, 1967. 19. Burkhard, D. G., D. L. Shealy, and R. U. Sexl: “Specular reflection of heat radiation from an arbitrary reflector surface to an arbitrary receiver surface,” International Journal of Heat and Mass Transfer, vol. 16, pp. 271–280, 1973. 20. Ziering, M. B., and A. F. Sarofim: “The electrical network analog to radiative transfer: Allowance for specular reflection,” ASME Journal of Heat Transfer, vol. 88, pp. 341–342, 1966. 21. Beckmann, P., and A. Spizzichino: The Scattering of Electromagnetic Waves from Rough Surfaces, Macmillan, New York, 1963. 22. Toor, J. S.: “Radiant heat transfer analysis among surfaces having direction dependent properties by the Monte Carlo method,” M.S. thesis, Purdue University, Lafayette, IN, 1967.

Problems 6.1 An infinitely long, diffusely reflecting cylinder is opposite a large, infinitely long plate of semiinfinite width (in plane of paper) as shown in the adjacent sketch. The plate is specularly reflecting with ρs2 = 0.5. As the center of the cylinder moves from x = +∞ to x = −∞ plot Fs1−1 vs. position h (your plot should include at least three precise values).

R A2

2R

A1 h

6.2 Consider two identical conical cavities (such as the ones depicted next to Problems 6.7 and 6.8), which are identical except for their surface treatment, making one surface a diffuse and the other a specular reflector. If both cones are isothermal, and both lose the same total amount of heat by radiation, which one has the higher temperature? 6.3 Two infinitely long black plates of width D are separated by a long, narrow channel, as indicated in the adjacent sketch. A T1 A2 ∋ 1 D T2 One plate is isothermal at T1 , the other is isothermal at T2 . The emittance of the insulated channel wall is ǫ. Determine L the radiative heat flux between the plates if the channel wall is (a) specular, (b) diffuse. For simplicity you may treat the channel wall as a single node. The diffuse case approximates the behavior of a light guide, a device used to pipe daylight into interior, windowless spaces.

224

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

6.4 Two circular black plates of diameter D are separated by a long, narrow tubular channel, as indicated in the sketch next to Problem 6.3. One disk is isothermal at T1 , the other is isothermal at T2 . The channel wall is a perfect reflector, i.e., ǫ = 0. Determine the radiative heat flux between the disks if the channel wall is (a) specular, (b) diffuse. For simplicity, you may treat the channel wall as a single node. If the channel is made of a transparent material, the specular arrangement approximates the behavior of an optical fiber; if the channel is filled with air, the diffuse case approximates the behavior of a light guide, a device used to pipe daylight into interior, windowless spaces.

6.7 Consider the infinite groove cavity shown. The entire surface of the groove is isothermal at T and coated with a gray, diffusely emitting material with emittance ǫ. (a) Assuming the coating is a diffuse reflector, what is the total heat loss (per unit length) of the cavity? (b) If the coating is a specular reflector, what is the total heat loss for the cavity?

A2

L=

6.6 A long duct has the cross-section of an equilateral triangle with side lengths L = 1 m. Surface 1 is a diffuse reflector to which an external heat flux at the rate of Q′1 = 1 kW/m length of duct is supplied. Surfaces 2 and 3 are isothermal at T2 = 1000 K and T3 = 500 K, respectively, and are purely specular reflectors with ǫ1 = ǫ2 = ǫ3 = 0.5. (a) Determine the average temperature of Surface 1, and the heat fluxes for Surfaces 2 and 3. (b) How would the results change if Surfaces 2 and 3 were also diffusely reflecting?

1m

6.5 Two infinitely long parallel plates of width w are spaced h = 2w apart. Surface 1 has ǫ1 = 0.2 and T1 = 1000 K, Surface 2 has ǫ2 = 0.5 and T2 = 2000 K. Calculate the heat transfer on these plates if (a) the surfaces are diffuse reflectors, (b) the surfaces are specular.

A3 A1

60°

0K T, ∋

L

A1 :

0K T, ∋

L

L

total flux leaving cavity ? area of groove opening × Eb

6.9 Determine the temperature of surface A2 in the axisymmetric configuration shown in the adjacent sketch, with the following data:

L

90o

6.8 Consider the infinite groove cavity shown in the adjacent sketch. The entire surface (L = 2 cm) is isothermal at T = 1000 K and is coated with a gray material whose reflectance may be idealized to consist of purely diffuse and specular components such that ǫ = ρ d = ρs = 13 . What is the total heat loss from the cavity? What is its apparent emittance, defined by ǫa =

o

60

20 cm

A2

10 cm

T1 = 1000 K, q1 = −1 W/cm2 , ǫ1 = 0.6

(diffuse reflector);

A2 :

ǫ2 = 0.2

(specular reflector);

A3 :

q3 = 0.0

(perfectly insulated),

ǫ3 = 0.3

(diffuse reflector).

60°

A1

A3

40 cm

All surfaces are gray and emit diffusely. Note: Some view factors may have to be approximated if integration is to be avoided.

10 cm

225

PROBLEMS

6.10 To calculate the net heat loss from a part of a spacecraft, this 0 K part may be approximated by an infinitely long black plate at temperature T2 = 600 K, as shown. Parallel to this plate is an (infinitely long) thin shield that is gray and reflects specularly with the same emittance ǫ1 on both sides. You may assume the surroundings to be black at 0 K. Calculate the net heat loss from the black plate.

0K ∋ 1 = 0.1

w1 = 1 m h = 1m w2 = 2 m T2, ∋ 2 = 1

6.11 A long isothermal plate (at T1 ) is a gray, diffuse emitter (ǫ1 ) and purely specular reflector, and is used to reject heat into space. To regulate the heat flux the plate is shielded by another (black) plate, which is perfectly insulated as illustrated in the adjacent sketch. Give an expression for heat loss as a function of shield opening angle (neglect variations along plates). At what opening angle 0 ≤ φ ≤ 180◦ does maximum heat loss occur?

L A2, black and insulated T1, ∋ 1

φ L

6.12 Reconsider Problem 6.11, but assume the entire configuration to be isothermal at temperature T, and covered with a partially diffuse, partially specular material, ǫ = 1 − ρs − ρ d . Determine an expression for the heat lost from the cavity. 6.13 An infinitely long cylinder with a gray, diffuse surface (ǫ1 = 0.8) at T1 = 2000 K is situated with its axis parallel to an infinite plane with ǫ2 = 0.2 at T2 = 1000 K in a vacuum environment with a background temperature of 0 K. The axis of the cylinder is two diameters from the plane. Specify the heat loss from the cylinder when the plate surface is (a) gray and diffuse, or (b) gray and specular.

T1, ∋ 1 D

2D T2, ∋ 2

6.14 A pipe carrying hot combustion gases is in radiative contact with a thin plate as shown. Assuming (a) the pipe to be isothermal at 2000 K and black, (b) the thin plate to be coated on both sides with a gray, diffusely emitting/specularly reflecting material (ǫ = 0.1), determine the radiative heat loss from the pipe. The surroundings are at 0 K and convection may be neglected.

A1 2r

r

A2 2r

6.15 Repeat Problem 5.7 for the case that the flat part of the rod (A1 ) is a purely specular reflector. 6.16 A long furnace may, in a simplified scenario, be considered to consist of a strip plate (the material to be heated, A1 : ǫ1 = 0.2, T1 = 500 K, specular reflector), unheated refractory brick (flat sides and bottom, A2 : ǫ2 = 0.1, diffuse reflector), and a cylindrical dome of heated refractory brick (A3 : ǫ3 = 1, T3 = 1000 K). Heat release inside the heated brick is qh (W/m2 ). The total heat release is radiated into the furnace cavity and is removed by convection, such that the convective heat loss is uniform everywhere (at qc W/m2 on all three surfaces).

A3

R A2

(a) Express the net radiative fluxes on all three surfaces in terms of qh . (b) Determine the qh necessary to maintain the indicated temperatures.

R=1m A1

A2

h=0.1m

2w=0.1m

A2

226

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

6.17 A typical space radiator may have a shape as shown in the adjacent sketch, i.e., a small tube to which are attached a number of flat plate fins, spaced at equal angle intervals. Assume that the central tube is negligibly small, and that a fixed amount of specularly-reflecting fin material is available (ǫ = ρs = 0.5), to give (per unit length of tube) a total, one-sided fin area of A′ = N × L. Also assume the whole structure to be isothermal. Develop an expression for the total heat loss from the radiator as a function of the number of fins (each fin having length L = A′ /N). Does an optimum exist? Qualitatively discuss the more realistic case of supplying a fixed amount of heat to the bases of the fins (rather than assuming isothermal fins).

∋, T

L

∋, T

α

6.18 Repeat Problem 5.15 for the case that the stainless steel, while being a gray and diffuse emitter, is a purely specular reflector (all four surfaces). 6.19 Repeat Problem 5.16 for the case that both the platinum sphere as well as the aluminum shield, while being gray and diffuse emitters, are purely specular reflectors. 6.20 Repeat Problem 5.29, but assume steel and silver to be specular reflectors. 6.21 Reconsider the spacecraft of Problem 6.10. To decrease the 0 K heat loss from Surface 2 a specularly reflecting shield, of the same dimensions as the black surface and with emittance ǫ = 0.1, is placed between the two plates. Determine the net heat loss from the black plate as a function of shield location. Where would you place the shield?

0K ∋ = 0.1

w1 = 1 m ∋ = 0.1

ht w2 = 2 m

hb T2, ∋ 2 = 1

6.22 Evaluate the normal emittance for the V-corrugated surface shown in Fig. 6-10a. Hint: This is most easily calculated by determining the normal absorptance, or the net heat flux on a cold groove irradiated by parallel light from the normal direction; see Problem 6.8 for the definition of “apparent emittance.” 6.23 Redo Problem 6.22 for an arbitrary off-normal direction 0 < θ < π/2 in a two-dimensional sense (i.e., determine the off-normal absorptance for parallel incoming light whose propagation vector is in the same plane as all the surface normal, namely the plane of the paper in Fig. 6-10). 6.24 A long, thin heating wire, radiating energy in the amount of S′ = 300 W/cm (per cm length of wire), is located between two long, parallel plates as shown in the adjacent sketch. The bottom plate is insulated and specularly reflecting with ǫ2 = 1−ρs2 = 0.2, while the top plate is isothermal at T1 = 300 K and diffusely reflecting with ǫ1 = 1 − ρ1d = 0.5. Determine the net radiative heat flux on the top plate.

q 1, ∋ 1

A1 h1 = 3 cm Wire

A2

T2, ∋ 2

h2 = 1 cm b/2 b = 10 cm

6.25 A long groove has diffuse walls that are insulated. All surfaces are gray with ǫ = 0.5. A parallel beam of radiation, q0 = 1 W/cm2 enters the open end of the cavity in the center line direction, flooding the cavity opening completely. (a) What is the apparent reflectance of the groove (i.e., how much radiative energy is leaving it), and what is the temperature of surface A1 ? (b) What are these values if surface A1 is a specular reflector instead of diffuse?

4 cm

A3 3 cm

q0

A1 A2

227

PROBLEMS

6.26 An infinitely long corner of characteristic length w = 1 m is a gray, diffuse emitter and purely specular reflector with ǫ = ρs = 12 . The entire corner is kept at a constant temperature T = 500 K, and is irradiated externally by a line source of strength S′ = 20 kW/m, located a distance w away from both sides of the corner, as shown in the sketch. What is the total heat flux Q′ (per m length) to be supplied or extracted from the corner to keep the temperature at 500 K?

w S´

w

w T, ∋ w

qsun

6.27 A long greenhouse has the cross-section of an equilateral triangle as shown. The side exposed to the sun consists of a thin sheet of glass (A1 ) with reflectance ρ1 = 0.1. The glass may be assumed perfectly transparent to solar radiation, and totally opaque to radiation emitted inside the greenhouse. The other side wall (A2 ) is opaque with emittance ǫ2 = 0.2, while the floor (A3 ) has ǫ3 = 0.8. Both walls (A1 and A2 ) are specular reflectors, while the floor reflects diffusely. For simplicity, you may assume surfaces A1 and A2 to be perfectly insulated, while the floor loses heat to the ground according to

A1 A2 60°

q3,conduction = U(T3 − T∞ )

A3 ground

60°

L = 1m

where T∞ = 280 K is the temperature of the ground, and U = 19.5 W/m2 K is an overall heat transfer coefficient. Determine the temperatures of all three surfaces for the case that the sun shines onto the greenhouse with strength qsun = 1000 W/m2 in a direction parallel to surface A2 . 6.28 Two long plates, parallel to each other and of width w, are √ spaced a distance L = 3w/2 apart, and are facing each other as shown. The bottom plate is a gray, diffuse emitter and specularly reflecting with emittance ǫ1 and temperature T1 . The top plate is a gray, diffuse emitter and diffusely reflecting with emittance ǫ2 and temperature T2 . The bottom plate is irradiated by the sun as shown (strength qsol [W/m2 ], angle θ). Determine the net heat fluxes on the two plates. How accurate do you expect your answer to be? What would be a first step to achieve better accuracy?

θ = 30o qsol

A2, ∋ 2 (diffuse) L

A1, ∋ 1 (specular)

w

6.29 Consider the solar collector shown. The collector plate is gray and diffuse, while the insulated guard plates are gray and specularly reflecting. Sun strikes the cavity at an angle α (α < 45◦ ). How much heat is collected? Compare with a collector without guard plates. For what values of α is your theory valid?

q sun = 1250 W/m2 Insulated

∋ 3 = 0.2

3m

∋ 1 = 0.9

T1 = 400 K

∋ 2 = 0.2

α 4m

228

6 RADIATIVE EXCHANGE BETWEEN PARTIALLY SPECULAR GRAY SURFACES

6.30 A rectangular cavity as shown is irradiated by a parallel-light source of strength qs = 1000 W/m2 . The entire cavity is held at constant temperature T = 300 K and is coated with a gray material whose reflectance may be idealized to consist of purely w2 = 3 cm diffuse and specular components, such that ǫ = ρ d = ρs = 13 . How must the cavity be oriented toward the light source (i.e., what is φ?) so that there is no net heat flux on surface A1 ?

qs

A2

φ A1

w1 = 4 cm

6.31 Reconsider the spacecraft of Problem 6.10. To decrease the 0 K heat loss from Surface 2 the specularly reflecting shield 1 is replaced by an array of N shields (parallel to each other and very closely spaced), of the same dimensions as the black surface and made of the original, specularly reflecting shield material with emittance ǫ = 0.1. Determine the net heat loss from the black plate as a function of shield number N.

0K

N shields, ∋ = 0.1 h = 1m w = 2m Tw, ∋ w = 1

6.32 Repeat Problem 6.26 using subroutine graydifspec of Appendix F (or modifying the sample program grspecxch). Break up each surface into N subsurfaces of equal width (n = 1, 2, 4, 8). 6.33 Repeat Problem 6.24 using subroutine graydifspec of Appendix F (or modifying the sample program grspecxch). Break up each surface into N subsurfaces of equal width (n = 1, 2, 4, 8). 6.34 An infinitely long corner piece as shown is coated with a material of (diffuse and gray) emittance ǫ, and purely specular reflectance. Calculate the variation of heat flux along the surfaces per unit area. Both surfaces are isothermal at T1 and T2 , respectively.

0K A2: T2 , ∋

h

A1: T1, ∋ w

6.35 An infinitely long cavity as shown is coated with gray, specular materials ǫ1 and ǫ2 (but the materials are diffuse emitters). The vertical surface is insulated, while the horizontal surface is at constant temperature T1 . The surroundings may be assumed to be black at 0 K. Specify the variation of the temperature along the vertical plate.

∋2

0K

L

∋ 1, T1

L

6.36 Consider the corner for Problem 6.30, which is irradiated by sunshine at an angle φ. Both plates are gray and specularly reflecting (emittance ǫ = 1 − ρs ) and isothermal at T. Develop an expression for the local heat fluxes as a function of ǫ, T, x, y, qs , and φ.

CHAPTER

7 RADIATIVE EXCHANGE BETWEEN NONIDEAL SURFACES

7.1

INTRODUCTION

In Chapter 6 we saw that, in certain situations, the directional nature of the reflectance of surfaces can strongly influence radiative heat transfer rates. This effect occurs particularly in open configurations, in enclosures with long channels, or in applications with collimated irradiation. Since real surfaces are neither diffuse nor specular reflectors, the actual directional behavior may have substantial impact, as we saw from the data in Fig. 6-17. We also noted that solar collectors did not appear to perform very well because, in our gray analysis, the reradiation losses were rather large. However, experience has shown that reradiation losses can be reduced substantially if selective surfaces (i.e., strongly nongray surfaces) are used for the collector plates. Apparently, there are a substantial number of applications for which our idealized treatment (gray, diffuse—i.e., direction-independent—absorptance and emittance, gray and diffuse or specular reflectance) is not sufficiently accurate. Actual surface properties deviate from our idealized treatment in a number of ways: 1. As seen from the discussion in Chapter 3, radiative properties can vary appreciably across the spectrum. 2. Spectral properties and, in particular, spectrally averaged properties may depend on the local surface temperature. 3. Absorptance and reflectance of a surface may depend on the direction of the incoming radiation. 4. Emittance and reflectance of a surface may depend on the direction of the outgoing radiation. 5. The components of polarization of incident radiation are reflected differently by a surface. Even for unpolarized radiation this difference can cause errors if many consecutive specular reflections take place. In the case of polarized laser irradiation this effect will always be important.

229

230

7 RADIATIVE EXCHANGE BETWEEN NONIDEAL SURFACES

In this chapter we shall briefly discuss how nongray effects may be incorporated into the analyses of the previous chapters. We shall also develop the governing equation for the intensity leaving the surface of an enclosure with arbitrary radiative properties (spectrally and directionally), from which heat transfer rates may be calculated. This expression will be applied to a simple geometry to show how directionally irregular surface properties may be incorporated in the analysis.

7.2 RADIATIVE EXCHANGE BETWEEN NONGRAY SURFACES In this section we shall consider radiative exchange between nongray surfaces that are directionally ideal: Their absorptances and emittances are independent of direction, while their reflectance is idealized to consist of purely diffuse and/or specular components. For such a situation equation (6.22) becomes, on a spectral basis, N h X j=1

i

s s )Fλ,i− δi j − (1−ρλj j Ebλj

  d N  X  δi j ρλj s  s   =  ǫλj − ǫλj Fλ,i− j  qλj + Hoλi ,

i = 1, 2, . . . , N.

(7.1)

j=1

While diffuse view factors are purely geometric quantities and, therefore, never depend on wavelength, the specular view factors depend on the spectral variation of specular reflectances. In principle, equation (7.1) may be solved for all the unknown qλj and/or Ebλj . This operation is followed by integrating the results over the entire spectrum, leading to Z ∞ Z ∞ qj = qλj dλ, Ebj = Ebλj dλ. (7.2) 0

0

In matrix form this may be written, similar to equation (6.23), as s , Aλ · ebλ = Cλ · qλ + hoλ

(7.3)

s are defined as in Chapter 6, but on a spectral basis. Assuming where Aλ , ebλ , Cλ , qλ , and hoλ that all the q j are unknown (and all temperatures are known), equation (7.3) may be solved and integrated as Z ∞ Z ∞ s q= qλ dλ = C−1 (7.4) λ · [Aλ · ebλ − hoλ ] dλ. 0

0

A similar expression may be found if the heat flux is specified over some of the surfaces (with temperatures unknown). Branstetter [1] carried out integration of equation (7.4) for two infinite, parallel plates with platinum surfaces. In practice, accurate numerical evaluation of equation (7.4) is considered too complicated for most applications: For every wavelength used in the numerical integration (or quadrature) the matrix C needs to be inverted, which—for large numbers of nodes—is generally done by iteration. In addition, if one or more of the surfaces are specular reflectors, the specular view factors need to be recalculated for each wavelength (though not the diffuse view factors of which they are composed). Therefore, nongray effects are usually addressed by simplified models such as the semigray approximation or the band approximation.

Semigray Approximation In some applications there is a natural division of the radiative energy within an enclosure into two or more distinct spectral regions. For example, in a solar collector the incoming energy comes from a high-temperature source with most of its energy below 3 µm, while radiation losses

7.2 RADIATIVE EXCHANGE BETWEEN NONGRAY SURFACES

231

q sun = 1000 W/m2

A2

l 2 = 60cm 30° A1 FIGURE 7-1 Solar collector geometry for Example 7.1.

l 1 = 80 cm

for typical collector temperatures are at wavelengths above 3 µm. In the case of laser heating and processing the incoming energy is monochromatic (at the laser wavelength), while reradiation takes place over the entire near- to midinfrared (depending on the workpiece temperature), etc. In such a situation equation (6.22) may be split into two sets of N equations each, one set for each spectral range, and with different radiative properties for each set. For example, consider an enclosure subject to external irradiation, which is confined to a certain spectral range “(1)”. The surfaces in the enclosure, owing to their temperature, emit over spectral range “(2)”.1 Then from equation (6.22),   d(1) N  X   δi j ρ j  s (1) + Hois = 0, (7.5a)  (1) − (1) Fi− j  q(1)  j ǫ ǫ j j j=1   d(2) N h N  X X  i  δi j ρ j s(2)  s(2)  q(2) =  δi j − (1−ρ sj (2) )Fi− Ebj , − F (7.5b) (2)  ǫ(2) i− j  j j  ǫj j j=1 j=1 + q(2) , qi = q(1) i i

i = 1, 2, . . . , N,

(7.5c)

where ǫ(1) is the average emittance for surface j over spectral interval (1), and so on. j Example 7.1. A very long solar collector plate is to collect energy at a temperature of T1 = 350 K. To improve its performance for off-normal solar incidence, a surface, which is highly reflective at short wavelengths, is placed next to the collector as shown in Fig. 7-1. For simplicity you may make the following assumptions: (i) The collector A1 is isothermal and a diffuse reflector; (ii) the mirror A2 is a specular reflector; (iii) the spectral properties of the collector and mirror may be approximated as  0.8, λ < λc = 4 µm, ǫ1 = 1 − ρ1d = 0.1, λ > λc ,  0.1, λ < λc , ǫ2 = 1 − ρ2s = 0.8, λ > λc , and (iv) heat losses from the mirror by convection as well as all losses from the collector ends may be neglected. How much energy (per unit length) does the collector plate collect for a solar incidence angle of 30◦ ? Solution s s s s = F1−2 , F2−1 , = F2−1 , and F1−1 = F2−2 = 0, for range (1), From equation (7.5) we find, with F1−2 q(1) 1 ǫ(1) 1 −

s + Ho1 = 0,

! q(1) 1 2 (1) s − 1 F q + + Ho2 = 0, 2−1 1 (1) ǫ(1) ǫ 2 1

and for range (2), 1

Note that spectral ranges “(1)” and “(2)” do not need to cover the entire spectrum and, indeed, they may overlap.

232

7 RADIATIVE EXCHANGE BETWEEN NONIDEAL SURFACES

q(2) 1 ǫ(2) 1 −

= Eb1 − ǫ(2) F E , 2 1−2 b2

! q(2) 1 2 (2) − 1 F q + = −F2−1 Eb1 + Eb2 . 2−1 (2) 1 ǫ(2) ǫ 2 1

Eliminating Eb2 from the last two equations, we find ! " # 1 1 (2) + F1−2 q(2) = (1 − ǫ(2) F F )E . − − 1 ǫ F F q(2) (2) 2 2 1−2 2−1 b1 2 1−2 2−1 1 ǫ(2) ǫ 1 1 Multiplying the second equation for range (1) by ǫ(1) F results in 2 1−2 ! 1 − (1) − 1 ǫ(1) F F q(1) + F1−2 q(1) = −ǫ(1) F Hs . 2 1−2 2−1 1 2 2 1−2 o2 ǫ1 Adding the last two equations and using q2 = q(1) + q(2) = 0 then leads to 2 2 " ! ! #   1 1 1 Eb1 , − (2) − 1 ǫ(2) F F F F q(1) − ǫ(1) F H s + 1 − ǫ(2) F F = (1) − 1 ǫ(1) q(2) 2 1−2 2−1 2 1−2 2−1 1 2 1−2 o2 2 1−2 2−1 1 ǫ(2) ǫ ǫ 1 1 1 s or, with q(1) = −ǫ(1) Ho1 , 1 1

q1 = q(1) + q(2) = 1 1

F F H s −ǫ(1) F Hs F F )E −(1−ǫ(1) )ǫ(1) (1−ǫ(2) 2 1−2 2−1 b1 2 1−2 2−1 o1 2 1−2 o2 1 s   − ǫ(1) Ho1 . 1 (2) (2) 1/ǫ(2) − 1/ǫ − 1 ǫ F F 1−2 2−1 2 1 1

From Example 6.5 we have s Ho2 = qsun sin ϕ = 1000 × sin 30◦ = 500 W/m2 , h i s Ho1 = qsun cos ϕ+ρ2s(1) sin ϕ (l2 /l1 ) = 1000 [cos 30◦ +0.9×sin 30◦ (60/80)] = 1203.5 W/m2 .

1 , and Eb1 = 5.670 × 10−8 × 3504 = 850.9 W/m2 , q1 may now be With F1−2 = 14 , F2−1 = 13 , F1−2 F2−1 = 12 evaluated as   0.2×0.1 1− 0.8 × 1203.5− 0.1 12 × 850.9− 12 4 × 500   − 0.8 × 1203.5 = 82.9 − 962.8 = −880.1 W/m2 , q1 = 1 1 0.8 − − 1 × 0.1 0.1 12

or a collection efficiency of 88%! In addition, surface A2 remains much cooler than for the gray case (Example 6.5); from the first equation for region (2) Eb2 = or

Eb1 −

q(2) 1 (2)

ǫ1

!,

  82.9 0.8 F = 850.9 − = 109.5 W/m2 , ǫ(2) 1−2 2 0.1 4

i1/4 h T2 = (Eb2 /σ)1/4 = 109.5/5.670 × 10−8 = 209 K.

Obviously, surface A2 would heat up by convection from the surroundings. Surface emission from A2 would then further improve the collection efficiency.

Thus, selective surfaces can have enormous impact on radiative heat fluxes in configurations with irradiation from high-temperature sources. Subroutine semigray is provided in Appendix F for the solution of the simultaneous equations (7.5), requiring surface information and a partial view factor matrix as input (i.e., the code is limited to two spectral ranges, separating external irradiation from surface emission). The solution to Example 7.1 is also given in the form of program semigrxch, which may be used as a starting point for the solution to other problems. Fortran90, C++ as well as Matlabr versions are provided. The semigray approximation is not limited to two distinct spectral regions. Each surface of the enclosure may be given a set of absorptances and reflectances, one value for each different surface temperature (with its different emission spectra). Armaly and Tien [2] have indicated

233

7.2 RADIATIVE EXCHANGE BETWEEN NONGRAY SURFACES

how such absorptances may be determined. However, while simple and straightforward, the method can never become “exact,” no matter how many different values of absorptance and reflectance are chosen for each surface. Bobco and coworkers [3] have given a general discussion of the semigray approximation. The method has been applied to solar irradiation falling into a V-groove cavity with a spectrally selective, diffusely reflecting surface by Plamondon and Landram [4]. Comparison with exact (i.e., spectrally integrated) results proved the method to be very accurate. Shimoji [5] used the semigray approximation to model solar irradiation onto conical and V-groove cavities whose reflectances had purely diffuse and specular components.

Band Approximation Another commonly used method of solving equation (7.1) is the band approximation. In this method the spectrum is broken up into M bands, over which the radiative properties of all surfaces in the enclosure are constant. Therefore,   d(m) N h N  X X  ρj i δ  ij  (m)  s(m)  s(m) s(m) s(m) (m)  δi j − (1 − ρ j )Fi− j Eb j =  (m) − (m) Fi− j  q j + Hoi , ǫ ǫ j=1

j=1

Eb j =

M X

(m)

Eb j ,

m=1

j

j

i = 1, 2, . . . , N, m = 1, 2, . . . , M; M M X X (m) s(m) qj = q j , Hois = Hoi . m=1

(7.6a)

(7.6b)

m=1

Equation (7.6) is, of course, nothing but a simple numerical integration of equation (7.1), using the trapezoidal rule with varying steps. This method has the advantage that the widths of the bands can be tailored to the spectral variation of properties, resulting in good accuracy with relatively few bands. For very few bands the accuracy of this method is similar to that of the semigray approximation, but is a little more cumbersome to apply, and requires an iterative approach if some surfaces have prescribed radiative flux rather than temperature. On the other hand, the band approximation can achieve any desired accuracy by using many bands. Example 7.2. Repeat Example 7.1 using the band approximation. Solution Since the emittances in this example have been idealized to have constant values across the spectrum with the exception of a step at λ = 4 µm, a two-band approximation (λ < λc = 4 µm and λ > 4 µm) will produce the “exact” solution (within the framework of the net radiation method). From equation (7.6) (m)

(m)

(m)

(m)

Eb1 − ǫ2 F1−2 Eb2 = (m)

(m)

−F2−1 Eb1 +Eb2 = where E(1) bi unknowns

R

λc

0 (m) q1 ,

q1

s(m)

+ Ho1 , (m) ǫ1   (m)  1  s(m) (m) q  = −  (m) − 1 F2−1 q1 + 2(m) +Ho2 , ǫ1 ǫ2

m = 1, 2,

  Ebλi dλ = f (λc Ti )Ebi , E(2) = 1 − f (λc Ti ) Ebi , etc. These are four equations in the six bi (m)

(m)

q2 , Eb2 , m = 1, 2. Two more conditions are obtained from q2 = q(1) + q(2) = 0 and 2 2 (m)

Eb2 + Eb2 = Eb2 = σT24 . The problem is that Eb2 are nonlinear relations in T2 , making it impossible to find (1)

(2)

(m)

explicit relations for the desired q1 = q(1) + q(2) . The system is solved by iteration, by solving for qi : 1 1 (m)

 (m) (m) (m) s(m) Eb1 − ǫ2 F1−2 Eb2 − Ho1 ,     1   (m)  (m) (m) (m) s(m)     = ǫ2  (m) − 1 F2−1 q1 − F2−1 Eb1 + Eb2 − Ho2  , ǫ1 (m)

q1 = ǫ1 (m)

q2



m = 1, 2.

234

7 RADIATIVE EXCHANGE BETWEEN NONIDEAL SURFACES (m)

First, T2 is guessed, from which the Eb2 may be evaluated. This computation is followed by determining (m) q1 ,

(m) q2

the after which the can be calculated. If q2 > 0, surface A2 is too hot and its temperature is reduced and vice versa until the correct temperature is obtained. This calculation may be done by writing a simple computer code, resulting in T2 = 212 K and q1 = −867 W/m2 . As expected, for the present example the band approximation offers little improvement while complicating the analysis. However, the band approximation is the method of choice if no distinct spectral regions are obvious and/or the spectral behavior of properties is more involved.

Subroutine bandapp is provided in Appendix F for the solution of the simultaneous equations (7.6), requiring surface information and a partial view factor matrix as input. The solution to Example 7.2 is also given in the form of a program bandmxch, which may be used as a starting point for the solution to other problems. Fortran90, C++ as well as Matlabr versions are provided. Dunkle and Bevans [6] applied the band approximation to the same problem as Branstetter [1] (infinite, parallel, tungsten plates) as well as to some other configurations, showing that the band approximation generally achieves accuracies of 2% and better with very few bands, while a gray analysis may result in errors of 30% or more.

7.3 DIRECTIONALLY NONIDEAL SURFACES In the vast majority of applications the assumption of “directionally ideal” surfaces gives results of sufficient accuracy, i.e., surfaces may be assumed to be diffusely emitting and absorbing and to be diffusely and/or specularly reflecting (with the magnitude of reflectance independent of incoming direction). However, that these results are not always accurate and that heat fluxes are not necessarily bracketed by the diffuse- and specular-reflection cases have been shown in Fig. 6-17 for V-grooves. There will be situations where (i) the directional properties, (ii) the geometrical considerations, and/or (iii) the accuracy requirements are such that the directional behavior of radiation properties must be addressed. If radiative properties with arbitrary directional behavior are to be accounted for, it is no longer possible to reduce the governing equation to an integral equation in a single quantity (the radiosity) that is a function of surface location only (but not of direction). Rather, applying conservation of energy to this problem produces an equation governing the directional intensity leaving a surface that is a function of both location on the enclosure surface and direction.

The Governing Equation for Intensity Consider the arbitrary enclosure shown in Fig. 7-2. The spectral radiative heat flux leaving an infinitesimal surface element dA′ into the direction of sˆ ′ and arriving at surface element dA is Iλ (r′ , λ, sˆ ′ ) dA′p dΩ = Iλ (r′ , λ, sˆ ′ )(dA′ cos θ′ )

dA cos θi , S2

(7.7)

where S = |r′ − r| is the distance between dA′ and dA, cos θ′ = sˆ ′ · nˆ ′ is the cosine of the angle between the unit direction vector sˆ ′ = (r − r′ )/S and the outward surface normal nˆ ′ at dA′ and, similarly, cos θi = (−ˆs′ ) · nˆ at dA. This irradiation at dA coming from dA′ may also be expressed, from equation (3.32), as Hλ′ (r, λ, sˆ ′ ) dA dΩ i = Iλ (r, λ, sˆ ′ ) dA cos θi

dA′ cos θ′ . S2

(7.8)

Equating these two expressions, we find Iλ (r, λ, sˆ ′ ) = Iλ (r′ , λ, sˆ ′ ), that is, the radiative intensity remains unchanged as it travels from dA′ to dA.

(7.9)

7.3 DIRECTIONALLY NONIDEAL SURFACES

235

dA´

s´ S

s

θ´ n´

n

θ

θi r´

dA r

0 FIGURE 7-2 Radiative exchange in an enclosure with arbitrary surface properties.

The outgoing intensity at dA into the direction of sˆ consists of two contributions: locally emitted intensity and reflected intensity. The locally emitted intensity is, from equation (3.1), ǫ′λ (r, λ, sˆ )Ibλ (r, λ). The amount of irradiation at dA coming from dA′ [equation (7.8)] that is reflected into a solid angle dΩ o around the direction sˆ is, from the definition of the bidirectional reflection function, equation (3.33),   dIλ (r, λ, sˆ ) dΩ o = ρ′′ ˆ ′ , sˆ ) Hλ′ (r, λ, sˆ ′ ) dΩ i dΩ o , λ (r, λ, s or

dIλ (r, λ, sˆ ) = ρ′′ ˆ ′ , sˆ )Iλ (r, λ, sˆ ′ ) cos θi dΩ i λ (r, λ, s cos θi cos θ′ dA′ . = ρ′′ ˆ ′ , sˆ )Iλ (r, λ, sˆ ′ ) λ (r, λ, s S2

Integrating the reflected intensity over all incoming directions (or over the entire enclosure surface), and adding the locally emitted intensity, we find an expression for the outgoing intensity at dA as Z Iλ (r, λ, sˆ ) = ǫλ′ (r, λ, sˆ )Ibλ (r, λ) + ρ′′ ˆ ′ , sˆ )Iλ (r′ , λ, sˆ ′ ) cos θi dΩ i λ (r, λ, s 2π Z ′ ′ ′ cos θi cos θ ′ = ǫλ′ (r, λ, sˆ )Ibλ (r, λ) + ρ′′ (r, λ, s ˆ , s ˆ )I (r , λ, s ˆ ) dA′ . (7.10) λ λ S2 A Equation (7.10) is an integral equation for outgoing intensity (nˆ · sˆ > 0) anywhere on the surface enclosure. Once a solution to equation (7.10) has been obtained (analytically, numerically, or statistically; approximately or “exactly”), the net radiative heat flux is determined from qλ (r, λ) = qout − qin Z Z = Iλ (r, λ, sˆ ) cos θ dΩ − Iλ (r, λ, sˆ ′ ) cos θi dΩ i n·ˆ ˆ s>0 n·ˆ ˆ s0 A

(7.11)

236

7 RADIATIVE EXCHANGE BETWEEN NONIDEAL SURFACES

or, equivalently, from qλ (r, λ) = qem − qabs = ǫλ Ebλ − αλ Hλ Z Z cos θi cos θ′ ′ α′λ (r, λ, sˆ ′ )Iλ (r′ , λ, sˆ ′ ) = ǫλ (r, λ, sˆ ) cos θ dΩ Ibλ (r, λ) − dA′ . S2 A n·ˆ ˆ s>0

(7.12)

Both forms of equation (7.10) (solid angle and area integration) may be employed, depending on the problem at hand. For example, if dA is a diffuse emitter and reflector then, from equation (3.38), ρ′′ (r, λ, sˆ ′ , sˆ ) = ρ′λ (r, λ)/π and, from equation (5.19), Iλ (r, λ, sˆ ) = Jλ (r, λ)/π. If dA′ λ is also diffuse, we obtain from the second form of equation (7.10) Z Jλ (r′ , λ) dFdA−dA′ , (7.13) Jλ (r, λ) = ǫλ (r, λ)Ebλ (r, λ) + ρλ (r, λ) A

which is nothing but the spectral form of equation (5.24) without external irradiation.2 Similarly, equation (7.11) reduces to Z qλ (r, λ) = Jλ (r, λ) − Jλ (r′ , λ) dFdA−dA′ , (7.14) A

the spectral form of equation (5.25). On the other hand, if dA is a specular reflector the first form of equation (7.10) becomes more convenient: For a specular surface we have ρ′′ = 0 for all sˆ ′ except for sˆ ′ = sˆ s , where sˆ s is the λ “specular direction” from which a beam must originate in order to travel on into the direction of sˆ after specular reflection. For that direction ρ′′ → ∞, and it is clear that the integrand of λ the integral in equation (7.10) will be nonzero only in the immediate vicinity of sˆ ′ = sˆ s . In that vicinity Iλ (r′ , λ, sˆ ′ ) varies very little and we may remove it from the integral. From the definition of the spectral, directional–hemispherical reflectance, equation (3.37), and the law of reciprocity for the bidirectional reflectance function, equation (3.35), we obtain Z Z ′ ′ ′ ρ′′ (r, λ, s ˆ , s ˆ )I (r, λ, s ˆ ) cos θ dΩ = I (r , λ, s ˆ ) ρ′′ ˆ ′ , sˆ ) cos θi dΩ i λ i i λ s λ λ (r, λ, s 2π 2π Z ′ = Iλ (r , λ, sˆ s ) ρ′′ s, −ˆs′ ) cos θi dΩ i λ (r, λ, −ˆ 2π

= Iλ (r′ , λ, sˆ s )ρ′λ (r, λ, −ˆs), where −ˆs denotes an incoming direction, pointing toward dA, and ρ′λ (r, λ, −ˆs) is the directional– hemispherical reflectance. From the same Kirchhoff’s law used to establish equation (3.35), it follows that ρ′λ (r, λ, −ˆs) = ρ′λ (r, λ, sˆ s ) and Iλ (r, λ, sˆ ) = ǫλ′ (r, λ, sˆ )Ibλ (r, λ) + ρ′λ (r, λ, sˆ s )Iλ (r′ , λ, sˆ s ).

(7.15)

Example 7.3. Consider a very long V-groove with an opening angle of 2γ = 90◦ and with optically smooth metallic surfaces with index of refraction m = n−ik = 23.452(1−i), i.e., the surfaces are specularly reflecting and their directional dependence obeys Fresnel’s equations. The groove is isothermal at temperature T, and no external irradiation is entering the configuration. Calculate the local net radiative heat loss as a function of the distance from the vertex of the groove. Solution This is one of the problems studied by Toor [7], using the Monte Carlo method (the solid line in Fig. 6-17). The directional emittance may be calculated from Fresnel’s equations for a metal, equations (3.75) and (3.76), as ǫ′ (θ) = 1 − ρ′ (θ) = 2

2n cos θ 2n cos θ + , (n + cos θ)2 + k2 (n cos θ + 1)2 + (k cos θ)2

External irradiation is readily included in equations (7.10) and (7.11) by replacing Iλ with Iλ +Ioλ inside the integrals.

7.3 DIRECTIONALLY NONIDEAL SURFACES

237

A2

y x

ψ2

A1

θ1

z

θ2

ψ1 S

FIGURE 7-3 Isothermal V-groove with specularly reflecting, directionally dependent reflectance (Example 7.3).

while the hemispherical emittance follows from equation (3.77) or Fig. 3-10 as ǫ = 0.1. The present problem is particularly simple since the surfaces are specular reflectors and since the opening angle of the groove is 90◦ (cf. Fig. 7-3). Any radiation leaving surface A1 traveling toward A2 will be absorbed by A2 or reflected out of the groove; none can be reflected back to A1 . This fact implies that all radiation arriving at A1 is due to emission from A2 , which is a known quantity. Therefore, for those azimuthal angles ψ2 pointing toward A1 we have −

π π < ψ2 < : 2 2

I2 (θ2 ) = ǫ′ (θ2 )Ib ,

and the local heat flux follows from equation (7.12) as Z ǫ′ (θ1 )I2 (θ2 ) cos θ1 dΩ 1 q(x) = ǫEb − 2π

= ǫEb − 2

Z

π/2 ψ1 =0

or q(x) 2 = 1− ǫEb πǫ

Z

π/2 ψ1 =0

Z Z

π/2

ǫ′ (θ1 )ǫ′ (θ2 )Ib cos θ1 sin θ1 dθ1 dψ1 , θ1 =θ1min (ψ1 ) π/2

ǫ′ (θ1 )ǫ′ (θ2 ) cos θ1 sin θ1 dθ1 dψ1 . θ1 =θ1min (ψ1 )

Here the limits on the integral express the fact that the solid angle, with which A2 is seen from A1 , is limited. It remains to express θ1min as well as θ2 in terms of θ1 and ψ1 . From Fig. 7-3 it follows that cos θ1 =

y , S

cos θ2 =

x , S

S sin θ1 =

x . cos ψ1

From these three relations and the fact that the minimum value of θ1 occurs when y = L, we find cos θ2 = sin θ1 cos ψ1

and θ1min (ψ1 ) = tan−1

x . L cos ψ1

Using Fresnel’s equation for the directional emittance, the nondimensional local heat flux q(x)/ǫEb may now be calculated using numerical integration. The resulting heat flux is shown as the solid line in Fig. 6-17. This result should be compared with the simpler case of diffuse emission, or ǫ′ (θ) = ǫ = 0.1 = const. For that case the integral above is readily integrated analytically, resulting in the dash-dotted line of Fig. 6-17. The two results are very close, with a maximum error of ≃ 2% near the vertex of the groove.

While the evaluation of the “exact” heat flux, using Fresnel’s equations, was quite straightforward in this very simple problem, these calculations are normally much, much more involved than the diffuse-emission approximation. Before embarking on such extensive calculations it is important to ask oneself whether employing Fresnel’s equations will lead to substantially different results for the problem at hand.

238

7 RADIATIVE EXCHANGE BETWEEN NONIDEAL SURFACES

Few numerical solutions of the exact integral equations have appeared in the literature. For example, Hering and Smith [8] considered the same problem as Example 7.3, but for varying opening angles and for rough surface materials (with the bidirectional reflection function as given in an earlier paper [9]). Lack of detailed knowledge of bidirectional reflection distributions, as well as the enormous complexity involved in the solution of the integral equation (7.10), makes it necessary in practice to make additional simplifying assumptions or to employ a different approach, such as the Monte Carlo method (to be discussed in Chapter 8).

Net Radiation Method It is possible to apply the net radiation method to surfaces with directionally nonideal properties, although its application is considerably more difficult and restrictive. Breaking up the enclosure into N subsurfaces we may write equation (7.10), for r pointing to a location on subsurface Ai , as I(r, λ, sˆ ) = ǫ′ (r, λ, sˆ )Ib (r, λ) + π

N X

ρ′′j (r, λ, sˆ )I j (r, λ)Fdi−j (r),

(7.16)

j=1

where we have dropped the subscript λ for simplicity of notation, and where ρ′′j and I j are “suitable” average values between point r and surface A j . Averaging equation (7.16) over Ai leads to Ii (λ, sˆ ) = ǫ′i (λ, sˆ )Ibi (λ) + π

N X

ρ′ji (λ, sˆ )I ji (λ)Fi− j ,

i = 1, 2, . . . , N.

(7.17)

j=1

Here I ji is an average value of the intensity leaving surface A j traveling toward Ai , and ρ′ji is a corresponding value for the bidirectional reflection function. If we assume that the enclosure temperature and surface properties are known everywhere, then equation (7.17) has N unknown intensities I ji (j = 1, 2, . . . , N) for each subsurface Ai . Thus, if equation (7.17) is averaged over all the solid angles with which subsurface Ak is seen from Ai , it becomes a set of N × N equations in the N2 unknown Iik : Iik (λ) = ǫik (λ)Ibi (λ) + π

N X

ρ jik (λ)I ji (λ)Fi− j ,

i, k = 1, 2, . . . , N.

(7.18)

j=1

Here ρ jik is an average value of the bidirectional reflection function for radiation traveling from A j to Ak via reflection at Ai . For a diffusely emitting, absorbing, and reflecting enclosure we have ǫik = ǫi , πρ jik = ρi , and equation (7.18) becomes, with I ji = I j = Jj /π, Ji = ǫi Ebi + ρi

N X

Jj Fi− j ,

i = 1, 2, . . . , N,

(7.19)

j=1

which is identical to equations (5.30) and (5.31) (without external irradiation). If the N subsurfaces are relatively small (as compared with the distance-squared between them), average properties ǫik and ρ jik may be obtained simply by evaluating ǫ′ and ρ′′ at the directions given by connecting the centerpoints of surface Ai with A j and Ak . For larger subsurfaces a more elaborate averaging may be desirable. A discussion on that subject has been given by Bevans and Edwards [10].

239

7.3 DIRECTIONALLY NONIDEAL SURFACES

Polar angle, θr

n=0 n=4 n=8

30° ( jo , ko )

y

so 60° z

( ji , ki ) w

si n (i, k)

t2 t1

L x

w

1

0

1

2

3

4

π ρ ´´( θi , ψi , θr , ψr = ψ i + π ) (b) (a) FIGURE 7-4 (a) Geometry for Example 7.4, (b) bidirectional reflection function in plane of incidence for θi = 0◦ and θi = 45◦ , for the material of Example 7.4.

Once the N2 unknown Iik have been determined, the average heat flux on Ai may be calculated from equations (7.18) and (7.11) or (7.12) as qi (λ) = π

N X

Iik (λ)Fi−k − π

N X

I ji (λ)Fi− j = π

j=1

k=1

= ǫi (λ)Ebi (λ) − π

N X

αi j (λ)I ji (λ)Fi− j ,

N X (Iij −I ji )Fi− j

(7.20a)

i = 1, 2, . . . , N,

(7.20b)

j=1

j=1

where ǫi is the hemispherical emittance of Ai and αij is the average absorptance of subsurface Ai for radiation coming from A j . It is apparent from equations (7.10) and (7.18) that the net radiation method for directionally nonideal surfaces is valid (i) if each Ibi varies little over each subsurface Ai , (ii) if each Iik varies little between any two positions on Ai and Ak , and (iii) if similar restrictions apply to ǫik , αij , and ρ jik . Restrictions (ii) and (iii) are likely to be easily violated unless the surfaces are near-diffuse reflectors or are very small (as compared with the distance between them). Equations (7.10) and (7.18) are valid for an enclosure with gray surface properties, or on a spectral basis. For nongray surface properties the governing equations are readily integrated over the spectrum using the methods outlined in the previous section. To illustrate the difficulties associated with directionally nonideal surfaces, we shall consider one particularly simple example. Example 7.4. Consider the isothermal corner of finite length as depicted in Fig. 7-4a. The surface material is similar to the one of the infinitely long corner of the previous example, i.e., the absorptance and emittance obey Fresnel’s equations with m = n − ik = 23.452(1 − i), and a hemispherical emittance of ǫ = 0.1. However, in the present example we assume that the material is reflecting in a nonspecular fashion with a bidirectional reflection function of ρ′′ (ˆs i , sˆ r ) =

ρ′ (ˆs i ) (1 + sˆ s · sˆ r )n , πCn (ˆs i )

where sˆ i is the direction of incoming radiation, sˆ s is the specular reflection direction (i.e., θs = θi , ψs = ψi + π), and sˆ r is the actual direction of reflection. This form of the bidirectional reflection function describes a surface that has a reflectance maximum in the specular direction, and whose reflectance

240

7 RADIATIVE EXCHANGE BETWEEN NONIDEAL SURFACES

drops off equally in all directions away from the specular direction (i.e., with changing polar angle and/or azimuthal angle). Since the directional–hemispherical reflectance must obey ρ′ (ˆs i ) = 1 − ǫ′ (ˆs i ), the function Cn (ˆs i ) follows from equation (3.37) as Z 1 (1 + sˆ s · sˆ r )n cos θr dΩ r . Cn (ˆs i ) = π 2π Determine the local radiative heat loss rates from the plates for the case that both plates are isothermal at the same temperature. Solution The direction vectors sˆ may be expressed in terms of polar angle θ and azimuthal angle ψ, or sˆ = ˆ where nˆ is the unit surface normal and ˆt1 and ˆt2 are two perpendicular sin θ(cos ψˆt1 + sin ψˆt2 ) + cos θn, unit vectors tangential to the surface. Therefore, the bidirectional reflection function may be written as ρ′ (θi , ψi )  n 1 + cos θi cos θr − sin θi sin θr cos(ψi − ψr ) , πCn (θi ) Z 2π Z π/2 n 1 1 + cos θi cos θ + sin θi sin θ cos ψ cos θ sin θ dθ dψ. Cn (θi ) = π 0 0

ρ′′ (θi , ψi , θr , ψr ) =

(7.21a) (7.21b)

The bidirectional reflection function within the plane of incidence (ψr = ψi or ψi + π) is shown in Fig. 7-4b for two different incidence directions and three different values of n. Obviously, for n = 0 the surface reflects diffusely (but the amount of reflection, as well as absorption and emission, depends on direction through Fresnel’s equation). As n grows, the surface becomes more specular, and purely specular reflection would be reached with n → ∞. For this configuration and surface material we should like to determine the heat lost from the plates using the net radiation method. As indicated in Fig. 7-4a we shall apply the net radiation method, equations (7.18) and (7.20), by breaking up each surface into M × N subsurfaces (M divisions in the x- and y-directions, N in the z-direction). Considering the intensity at node (i, k) on the bottom surface directed toward node (jo , ko ) on the vertical wall, we find that equation (7.18) becomes, after division by Ib , Φi,k→jo ,ko =

Ii,k→ jo ,ko Ib

= ǫi,k→ jo ,ko +

M X N X

πρ ji ,ki →i,k→jo ,ko Fi,k→ji ,ki Φ ji ,ki →i,k .

(7.22)

ji =1 ki =1

In this relation we have made use of the fact that a node on the bottom surface can only see nodes on the side wall and vice versa. Also, by symmetry we have Φi,k→jo ,ko = Φ j,k→io ,ko

if

j=i

and

io = jo ,

and Φi,k→ jo ,ko = Φi,N+1−k→ jo ,N+1−ko , that is, the intensity must be symmetric to the two planes x = y and z = L/2. We, therefore, have a total of M × (N/2) unknowns (assuming N to be even) and need to apply equation (7.22) for i = 1, 2, . . . , M and k = 1, 2, . . . , N/2. To calculate the necessary ǫ′ and ρ′′ values, one must establish a number of polar and azimuthal angles. From Fig. 7-4a it follows that y ji

(cos θi ) i,k→ji ,ki = q

x2i + y2ji + (zk − zki )2

(cos θr ) i,k→ jo ,ko = q

x2i

y jo +

y2jo

,

.

+ (zk − zko )2

Using the values for (cos θr ) i,k→jo ,ko one can readily calculate the directional emittances ǫi,k→ jo ,ko = 1 − ρ′ (cos θr ) from Fresnel’s equation as given in Example 7.3. Similarly, ρ′ (cos θi ) and Cn (cos θi ) are determined from Fresnel’s equation and equation (7.21),3 respectively; and all values of ρ ji ,ki →i,k→ jo ,ko follow from equation (7.21). All necessary view factors may be calculated from equation (4.41), for 3 For integer values of n the integration may be carried out analytically, either by hand or on a computer using a symbolic mathematics analyzer (the latter having been used here).

7.3 DIRECTIONALLY NONIDEAL SURFACES

241

1.00



Nondimensional heat flux Ψ = q/ σT

4

w/L = 1

0.95

diffuse Fresnel, n = 0 Fresnel, n = 4 Fresnel, n = 8

0.90

0.00

0.25 0.50 0.75 Nondimensional distance from vertex x/w

1.00

FIGURE 7-5 Nondimensional, local heat fluxes for the corner geometry of Example 7.4, for w/L = 1. Solid symbols: Surfaces are broken up into 2 × 2 subsurfaces; open symbols: 4 × 4 subsurfaces; lines: 20 × 20 subsurfaces.

arbitrarily oriented perpendicular plates. For all view factors the opposing surfaces are of identical and constant size with x2 −x1 = y2 −y1 = w/M and z1 = z3 −z2 = L/N. Offsets x1 and y1 may vary between 0 and (M−1)w/M and z2 between 0 and (N −1)L/N. Thus, using symmetry and reciprocity, one must evaluate a total of (M/2)×M×N view factors. In many of today’s workstations and computers all different values of directional emittance, the factor ρ′ /Cn in the bidirectional reflection function, and all view factors may be calculated—once and for all—and stored (requiring memory allocation for often millions of numbers). The bidirectional reflection function itself depends on surface locations and on all possible incoming as well as all possible outgoing directions. Even after employing symmetry and reciprocity (for the bidirectional reflection function), this would require storing [M×(N/2)]×[M×N]2 /2 = (MN)3 /4 numbers. Unless relatively few subdivisions are used (say M, N < 10), it will be impossible to precalculate and store values of the bidirectional reflection function; rather, part of it must be recalculated every time it is required. The nondimensional intensities are now easily found from equation (7.22) by successive approximation: A first guess for the intensity field is made by setting Φi,k→ jo ,ko = ǫi,k→jo ,ko . Improved values for Φi,k→jo ,ko are found by evaluating equation (7.22) again and again until the intensities have converged to within specified error bounds. The local net radiative heat flux may then be determined from equation (7.20b) as Ψi,k =

M N qi,k 1 XX =1− ǫi,k→ji ,ki Fi,k→ ji ,ki Φ ji ,ki →i,k . ǫEb ǫ ji =1 ki =1

Some representative results for the local radiative heat flux near z = L/2 (i.e., for k = N/2) are shown in Fig. 7-5 for the case of w = L (square plates). Clearly, taking into consideration substantially different reflective properties has rather small effects on the local heat transfer rates. Obviously, as the surface becomes more specular (increasing n) the heat loss rates increase (since less radiation will be reflected back to the emitting surface), but the increases are very minor except for the region close to the vertex (and even there, they are less than 4%). The directional distribution of the emittance is just as important as that of the bidirectional reflection function: The curve labeled “diffuse” shows the case of diffuse emission and reflection, i.e., ǫ′ (ˆs) = α′ (ˆs) = ǫ = 0.1 and πρ′′ (ˆs i , sˆ r ) = ρ′ = 1 − ǫ = 0.9. In contrast, the curve labeled “Fresnel, n = 0” corresponds to the case of ǫ′ (ˆs) = α′ (ˆs) = 1 − ρ′ (ˆs) evaluated from Fresnel’s equation and πρ(ˆsi , sˆ r ) = ρ′ (ˆs i ). All lines in Fig. 7-5 have been calculated by breaking up each surface into 20 × 20 subsurfaces. Also included are the data points for results obtained by breaking up each surface into only 2 × 2 (solid symbols) and 4 × 4 surfaces (open symbols). Local heat fluxes are predicted accurately with few subsurfaces, even for

242

7 RADIATIVE EXCHANGE BETWEEN NONIDEAL SURFACES

strongly nondiffuse reflection. Total heat loss is predicted even more accurately, with maximum errors of < 0.6% (2 × 2 subsurfaces) and < 0.3% (4 × 4 subsurfaces), respectively. The results should be compared with those of Toor [7] for w/L → 0, as shown in Fig. 6-17: The “diffuse” case of Fig. 7-5 virtually coincides with the corresponding case in Fig. 6-17, while the n = 8 case falls very close to the specular case with Fresnel-varying reflectance of Toor (solid line in Fig. 6-17).

For the present example at least, taking into account the directional behavior of emittance and reflectance is rarely justifiable in view of the additional complexity and computational effort required. Only if the radiative properties are known with great accuracy, and if heat fluxes need to be determined with similar accuracy, should this type of analysis be attempted. Similar statements may be made for most other configurations. For example, if Example 7.4 is recalculated for directly opposed parallel quadratic plates, the effects of Fresnel’s equation and the bidirectional reflection function are even less: Heat fluxes for diffuse reflection—whether Fresnel’s equation is used or not—differ by less than 0.6%, while differences due to the value of n in the bidirectional reflection function never exceed 0.2%. Only in configurations with collimated irradiation and/or strong beam-channeling possibilities should one expect substantial impact as a result of the directional variations of surface properties.

7.4 ANALYSIS FOR ARBITRARY SURFACE CHARACTERISTICS The discussion in the previous two sections has demonstrated that the evaluation of radiative transfer rates in enclosures with nonideal surface properties, while relatively straightforward to formulate, is considerably more complex and time-consuming. If one considers nongray surface properties, the computational effort increases roughly by a factor of M if M spectral bands (band approximation) or M sets of property values (semigray approximation) are employed. In an analysis with directional properties for an enclosure with N subsurfaces, the computational effort is increased roughly by a factor of N (an enormous increase if a substantial number of subdivisions are made). If the radiative properties are both nongray and directionally varying, the problem becomes even more difficult. While it is relatively simple to combine the methods of the previous two sections for the analysis of an enclosure with such surface properties, to the author’s knowledge this has not yet been done in any reported work. Few analytical solutions for such problems can be found (for the very simplest of geometries), and even standard numerical techniques may fail for nontrivial geometries; because of the four-dimensional character, huge matrices would have to be inverted. Therefore, such calculations are normally carried out with statistical methods such as the Monte Carlo method (to be discussed in detail in Chapter 8). For example, Toor [7] has studied the radiative interchange between simply arranged flat surfaces having theoretically determined directional surface properties; Modest and Poon [11] and Modest [12] evaluated the heat rejection and solar absorption rates of the U.S. Space Shuttle’s heat rejector panels, using nongray and directional properties determined from experimental data. The validity and accuracy of several directional models have been tested and verified experimentally by Toor and Viskanta [13, 14]. They studied radiative transfer among three simply arranged parallel rectangles, comparing experimental results with a simple analysis employing (i) the semigray model, (ii) Fresnel’s equation for the evaluation of directional properties, and (iii) reflectances consisting of purely diffuse and specular parts. They found good agreement with experiment and concluded that, for the gold surfaces studied, (i) directional effects are more pronounced than nongray effects, and (ii) in the presence of one or more diffusely reflecting surfaces the effects of specularity of other surfaces become unimportant. Employing a combination of band approximation and the net radiation method has the disadvantage that (i) either large amounts of directional properties and/or view factors must be calculated repeatedly in the iterative solution process (making the method numerically inefficient), or (ii) large amounts of precalculated properties and/or view factors must be stored

PROBLEMS

243

(requiring enormous amounts of computer storage). In addition, this method tends to have a voracious appetite for computer CPU time. On the other hand, it avoids the statistical scatter that is always present in Monte Carlo solutions. In light of today’s rapid development in the computer field, with many small workstations and personal computers boasting internal storage capacities of several gigabytes, as well as enormous number-crunching capabilities, it appears that the methods discussed in this chapter may become attractive alternatives to the Monte Carlo method.

References 1. Branstetter, J. R.: “Radiant heat transfer between nongray parallel plates of tungsten,” NASA TN D-1088, 1961. 2. Armaly, B. F., and C. L. Tien: “A note on the radiative interchange among nongray surfaces,” ASME Journal of Heat Transfer, vol. 92, pp. 178–179, 1970. 3. Bobco, R. P., G. E. Allen, and P. W. Othmer: “Local radiation equilibrium temperatures in semigray enclosures,” Journal of Spacecraft and Rockets, vol. 4, no. 8, pp. 1076–1082, 1967. 4. Plamondon, J. A., and C. S. Landram: “Radiant heat transfer from nongray surfaces with external radiation. Thermophysics and temperature control of spacecraft and entry vehicles,” Progress in Astronautics and Aeronautics, vol. 18, pp. 173–197, 1966. 5. Shimoji, S.: “Local temperatures in semigray nondiffuse cones and v-grooves,” AIAA Journal, vol. 15, no. 3, pp. 289–290, 1977. 6. Dunkle, R. V., and J. T. Bevans: “Part 3, a method for solving multinode networks and a comparison of the band energy and gray radiation approximations,” ASME Journal of Heat Transfer, vol. 82, no. 1, pp. 14–19, 1960. 7. Toor, J. S.: “Radiant heat transfer analysis among surfaces having direction dependent properties by the Monte Carlo method,” M.S. thesis, Purdue University, Lafayette, IN, 1967. 8. Hering, R. G., and T. F. Smith: “Surface roughness effects on radiant energy interchange,” ASME Journal of Heat Transfer, vol. 93, no. 1, pp. 88–96, 1971. 9. Hering, R. G., and T. F. Smith: “Apparent radiation properties of a rough surface,” AIAA paper no. 69-622, 1969. 10. Bevans, J. T., and D. K. Edwards: “Radiation exchange in an enclosure with directional wall properties,” ASME Journal of Heat Transfer, vol. 87, no. 3, pp. 388–396, 1965. 11. Modest, M. F., and S. C. Poon: “Determination of three-dimensional radiative exchange factors for the space shuttle by Monte Carlo,” ASME paper no. 77-HT-49, 1977. 12. Modest, M. F.: “Determination of radiative exchange factors for three dimensional geometries with nonideal surface properties,” Numerical Heat Transfer, vol. 1, pp. 403–416, 1978. 13. Toor, J. S., and R. Viskanta: “A critical examination of the validity of simplified models for radiant heat transfer analysis,” International Journal of Heat and Mass Transfer, vol. 15, pp. 1553–1567, 1972. 14. Toor, J. S., and R. Viskanta: “Experiment and analysis of directional effects on radiant heat transfer,” ASME Journal of Heat Transfer, vol. 94, pp. 459–466, November 1972.

Problems 7.1 Two identical circular disks of diameter D = 1 m are connected at one point of their periphery by a hinge. The configuration is then opened by an angle φ. Disk 1 is a diffuse reflector, but emits and absorbs according to ǫλ′ =



0.95 cos θ, 0.5,

qsun A1 d

λ ≤ 3 µm, λ > 3 µm.

Disk 2 is black. Both disks are insulated. Assuming the opening angle to be φ = 60◦ , calculate the average equilibrium temperature for each of the two disks, with solar radiation entering the configuration parallel to Disk 2 with a strength of qsun = 1000 W/m2 .

A2

φ d

7.2 Reconsider Problem 7.1 for the case that surfaces A1 and A2 are long, rectangular plates. 7.3 Repeat Problem 5.17 using the semigray approximation. Disk 1 is covered with a diffuse coating of black chrome (Fig. 3-33). 7.4 Repeat Example 5.8 for an absorber plate made of black chrome (Fig. 3-33) and a glass cover made of soda–lime glass (Fig. 3-28). Use the semigray or the band approximation.

244

7 RADIATIVE EXCHANGE BETWEEN NONIDEAL SURFACES

7.5 Repeat Problem 5.36 for the case that the top of the copper shield is coated with white epoxy paint (Fig. 3-33). 7.6 A cubical enclosure has five of its surfaces maintained at 300 K, while the sixth is isothermal at 1200 K. The entire enclosure is coated with a material that emits and reflects diffusely with ǫλ =



0.2, 0 ≤ λ < 4 µm, 0.8, 4 µm < λ < ∞.

Determine the net radiative heat fluxes on the surfaces. 7.7 Repeat Problem 6.13 for the case that Surface 1 is coated with the material described in Problem 7.6. 7.8 Repeat Problem 6.26 for the case that the corner is coated with a diffusely emitting, specularly reflecting layer whose spectral behavior may be approximated by ǫλ =



0.8, 0 ≤ λ < 3 µm, 0.2, 3 µm < λ < ∞.

The line source consists of a long filament at 2500 K inside a quartz tube, i.e., the source behaves like a gray body for λ < 2.5 µm but has no emission beyond 2.5 µm. 7.9 Repeat Problem 6.27 for the case that the side wall A2 is coated with a diffusely emitting, specularly reflecting layer whose spectral behavior may be approximated by ǫλ =



0.1, 0 ≤ λ < 3 µm, 0.8, 3 µm < λ < ∞.

7.10 Repeat Problem 5.26 for the case that A1 is coated with a material that has a spectral, directional emittance of ( π 0.9 cos θ, λ < 4µm, ′ 0≤θ< . ǫλ = 0.3, λ > 4µm. 2 7.11 Consider the configuration shown, consisting of a conical cavity A1 and an opposing circular disk with a hole at the center, as shown (d = 1 cm). Defocused laser radiation at 10.6 µm enters the configuration through the hole in the disk as shown, the beam having a strength of qL = 103 W/cm2 . The down-facing disk A2 is a gray, diffuse material with ǫ2 = 0.1, and is perfectly insulated (toward top). Surface A1 is kept at a constant temperature of 500 K. No other external surfaces or sources affect the heat transfer.

qL = 10 W/cm2 3

A2

d/2

d/2

2d

(a) Assuming surface A1 to be gray and diffuse with ǫ1 = 0.3 determine the amount of heat that needs to be removed from A1 (Q1 ). (b) If A1 were coated with the material of Problem 3.12, how would you determine Q1 ? Set up any necessary equations and indicate how you would solve them (no actual solution necessary). Would you expect Q1 to increase/decrease/stay the same (and why)? (c) What other simple measures can you suggest to improve the accuracy of the solution (to either (a) or (b))?

A1 2d

2d

PROBLEMS

7.12 During a materials processing experiment on the Space Shuttle (under microgravity conditions) a platinum sphere of 3 mm diameter is levitated in a large, cold black vacuum chamber. A spherical aluminum shield (with a circular cutout) is placed around the sphere as shown, to reduce heat loss from the sphere. Initially, the sphere is at 200 K and is suddenly irradiated with a laser providing an irradiation of 100 W (normal to beam) to raise its temperature rapidly to its melting point (2741 K). Determine the time required to reach the melting point. You may assume the sphere to be essentially isothermal at all times, and the shield to have zero heat capacity. The platinum and aluminum may be taken as diffuse emitters and reflectors with r λ0 ǫPt,0 = 0.25, ǫPt = ǫPt,0 λ r λ0 ǫAl,0 = 0.1, ǫAl = ǫAl,0 λ

245

0K Pt sphere 1 cm

laser

Al shield 10 cm

λ0 = 2 µm, λ0 = 2 µm.

(a) Use the semigray approximation, using gray values for reemission from sphere and shield. (b) Use the band approximation, splitting the spectrum into three appropriate bands. 7.13 In the solar energy laboratory at UC Merced parabolic concentrators are employed to enhance the absorption of tubular solar collectors as shown in the sketch. Assume that solar energy enters the cavity normal to the opening, with a strength of qsun = 1000 W/m2 (per unit area normal to the rays). The parabolic receiver is coated with a highly reflective gray, diffuse material with ǫ1 = 0.05, and is kept cold by convection (i.e., emission from it is negligible). Calculate the collected solar energy as a function of tube outer temperature (say, for 300 K, 400 K, 500 K), (a) assuming the tube to be gray with emittance ǫ2 = 0.90, (b) assuming the tube to be covered with black nickel, using the 2-band approach.

qsun=103W/m2

pR concentrator

4R 6R oil tube

R

It is sufficient to treat tube and concentrator each as single zones. 7.14 A small spherical heat source outputting Qs = 10 kW power, spreading equally into all directions, is encased in a reflector as shown, consisting of a hemisphere of radius R = 40 cm, plus a ring of radius R and height h = 30 cm. The arrangement is used to heat a disk of radius = 25 cm a distance of L = 30 cm below the reflector. Reflector A2 is gray and diffuse with emittance of ǫ2 = 0.1 and is insulated. Disk A1 is diffuse and coated with a selective absorber, i.e.,  0.8, 0 ≤ λ < 3 µm, ǫ1λ = 0.2, 3 µm < λ < ∞. The source is of the tungsten–halogen type, i.e., the spectral variation of its emissive power follows that of a blackbody at 4000 K.

A2: ∋ 2 = 0.1, q2 = 0 QS =10 kW R

R = 40 cm

h30 cm

H

L=30 cm A1: ∋ 1 = 0.8, Q1 = -0.4 kW R=25

cm

(a) Determine (per unit area of receiving surface) the irradiation from heat source to reflector and to disk. (b) Determine all relevant view factors.

246

7 RADIATIVE EXCHANGE BETWEEN NONIDEAL SURFACES

(c) Outline how you would obtain the temperature of the disk, if 0.4 kW of power is extracted from it. (“Outline” implies setting up all the necessary equations, plus a sentence on how you would solve them.) 7.15 Repeat Problem 7.8 using subroutine bandapp of Appendix F (or modifying the sample program bandmxch). Break up each surface into N subsurfaces of equal width (n = 1, 2, 4, 8). 7.16 Repeat Problem 5.25 for the case that the insulated cylinder is coated with a material that has ǫ2λ =



0.2, 0 ≤ λ < 4 µm, 0.8, 4 µm < λ < ∞

(the flat surface remains gray with ǫ3 = 0.5). Note that the wire heater is gray and diffuse and at a temperature of T1 = 3000 K. (a) Find the solution using the semigray method; also set up the same problem and find the solution by using program semigrayxchdf. (b) Set up the solution using the band approximation, i.e., to the point of having a set of simultaneous equations and an outline of how to solve them. Also find the solution using program bandmxchdf. 7.17 Repeat Problem 5.2 assuming that the furnace walls are made of alumina ceramic (aluminum oxide, Fig. 1-14). Use subroutine bandapp of Appendix F (or modifying the sample program bandmxch). Break up the spectrum into several parts, and compare your results for N = 1, 2, 3, and 5. 7.18 Repeat Problem 5.19 assuming that the furnace walls are made of alumina ceramic (aluminum oxide, Fig. 1-14). Use subroutine semigray of Appendix F (or modifying the sample program semigrxch). Break up the groove surface into N subsurfaces of equal size (N = 2 and 4), but only consider incidence angles of θ = 0◦ and 60◦ . 7.19 Repeat Problem 6.26 for the case that the corner is cold (i.e., has negligible emission), and that the surface is gray and specularly reflecting with ǫ = ρ s = 0.5, but has a directional emittance/absorptance of ǫ′ (θ) = ǫn cos θ. Determine local and total absorbed radiative heat fluxes. 7.20 Consider two infinitely long, parallel plates of width w = 1 m, spaced a distance h = 0.5 m apart (see Configuration 32 in Appendix D). Both plates are isothermal at 1000 K and are coated with a gray material with a directional emittance of ǫ′ (θi ) = α′ (θi ) = 1 − ρ′ (θi ) = ǫn cos θi and a hemispherical emittance of ǫ = 0.5. Reflection is neither diffuse nor specular, but the bidirectional reflection function of the material is ρ′′ (θi , θr ) =

3 ′ ρ (θi ) cos θr . 2π

Write a small computer program to determine the total heat lost (per unit length) from each plate. Compare with the case for a diffusely emitting/reflecting surface.

CHAPTER

8 THE MONTE CARLO METHOD FOR SURFACE EXCHANGE

8.1

INTRODUCTION

Very few exact, closed-form solutions to thermal radiation problems exist, even in the absence of a participating medium. Under most circumstances the solution has to be found by numerical means. For most engineers, who are used to dealing with partial differential equations, this implies use of finite difference and finite element techniques. These methods are, of course, applicable to thermal radiation problems whenever a solution method is chosen that transforms the governing equations into sets of partial differential equations. For surface exchange, however, radiative transfer is governed by integral equations, which may be solved numerically by employing numerical quadrature for the evaluation of integrals, or more approximately using the “net radiation method” of the previous three chapters. With these techniques the solutions to relatively simple problems are readily found. However, if the geometry is involved, and/or if radiative properties vary with direction, then a solution by conventional numerical techniques may quickly become extremely involved if not impossible. Many mathematical problems may also be solved by statistical methods, through sampling techniques, to any degree of accuracy. For example, consider predicting the outcome of the next presidential elections. Establishing a mathematical model that would predict voter turnout and voting behavior is, of course, impossible, let alone finding the analytical solution to such a model. However, if an appropriate sampling technique is chosen, the outcome can be predicted by conducting a poll. The accuracy of its prediction depends primarily on the sample size, i.e., how many people have been polled. Solving mathematical problems statistically always involves the use of random numbers, which may be picked, e.g., by placing a ball into a spinning roulette wheel. For this reason these sampling methods are called Monte Carlo methods (named after the principality of Monte Carlo in the south of France, famous for its casino). There is no single scheme to which the name Monte Carlo applies. Rather, any method of solving a mathematical problem with an appropriate statistical sampling technique is commonly referred to as a Monte Carlo method. Problems in thermal radiation are particularly well suited to solution by a Monte Carlo technique, since energy travels in discrete parcels (photons) over (usually) relatively long distances along a (usually) straight path before interaction with matter. Thus, solving a thermal radiation problem by Monte Carlo implies tracing the history of a statistically meaningful random sample of photons from their points of emission to their points of absorption. The advantage of the 247

8 THE MONTE CARLO METHOD FOR SURFACE EXCHANGE

Conventional

Monte Carlo

Complexity of problem

Solution effort (CPU time)

Complexity of formulation

248

FIGURE 8-1 Comparison of Monte Carlo and conventional solution techniques.

Monte Carlo method is that even the most complicated problem may be solved with relative ease, as schematically indicated in Fig. 8-1. For a trivial problem, setting up the appropriate photon sampling technique alone may require more effort than finding the analytical solution. As the complexity of the problem increases, however, the complexity of formulation and the solution effort increase much more rapidly for conventional techniques. For problems beyond a certain complexity, the Monte Carlo solution will be preferable. Unfortunately, there is no way to determine a priori precisely where this crossover point in complexity lies. The disadvantage of Monte Carlo methods is that, as statistical methods, they are subject to statistical error (very similar to the unavoidable error associated with experimental measurements). The name and the systematic development of Monte Carlo methods dates from about 1944 [1], although some crude mathematical sampling techniques were used off and on during previous centuries. Their first use as a research tool stems from the attempt to model neutron diffusion in fission material, for the development of the atomic bomb during World War II. The method was first applied to thermal radiation problems in the early 1960s by Fleck [2, 3] and Howell and Perlmutter [4–6]. For a thorough understanding of Monte Carlo methods, a good background in statistical methods is necessary, which goes beyond the scope of this book. In this chapter the method as applied to thermal radiation is outlined, and statistical considerations are presented in an intuitive way rather than in a rigorous mathematical fashion. For a more detailed description, the reader may want to consult the books by Hammersley and Handscomb [1], Cashwell and Everett [7], and Schreider [8], or the monographs by Kahn [9], Brown [10], Halton [11], and Hajji-Sheikh [12]. A first monograph dealing specifically with Monte Carlo methods as applied to thermal radiation has been given by Howell [13]. Another more recent one by Walters and Buckius [14] emphasizes the treatment of scattering. An exhaustive review of the literature up until 1997, that uses some form of radiative Monte Carlo analysis, has been given also by Howell [15]. Since then a large number of researchers have applied Monte Carlo simulations to a vast array of problems, ranging from nanoscale radiation properties to large-scale tomography, surface radiation, participating media, transient radiation, combined modes heat transfer, etc., too numerous to review in this book.

Probability Distributions When a political poll is conducted, people are not selected at random from a telephone directory. Rather, people are randomly selected from different groups according to probability distributions, to ensure that representative numbers of barbers, housewives, doctors, smokers, gun owners, bald people, heat transfer engineers, etc. are included in the poll. Similarly, in order to follow the history of radiative energy bundles in a statistically meaningful way, the points, directions

8.1 INTRODUCTION

249

and wavelengths of emission, reflective behavior, etc. must be chosen according to probability distributions. As an example, consider the total radiative heat flux being emitted from a surface, i.e., the total emissive power, Z ∞ Z ∞ E= Eλ dλ = ǫλ Ebλ dλ. (8.1) 0

0

Between the wavelengths of λ and λ + dλ the emitted heat flux is Eλ dλ = ǫλ Ebλ dλ, and the fraction of energy emitted over this wavelength range is P(λ) dλ = R

Eλ dλ ∞ 0

=

Eλ dλ

Eλ dλ. E

(8.2)

We may think of all the photons leaving the surface as belonging to a set of N energy bundles of equal energy (each consisting of many photons of a single wavelength). Then each bundle carries the amount of energy (E/N) with it, and the probability that any particular bundle has a wavelength between λ and λ + dλ is given by the probability density function P(λ). The fraction of energy emitted over all wavelengths between 0 and λ is then R(λ) =

Z

λ 0

R

P(λ) dλ = R

λ 0 ∞ 0

Eλ dλ Eλ dλ

.

(8.3)

It is immediately obvious that R(λ) is also the probability that any given energy bundle has a wavelength between 0 and λ, and it is known as the cumulative distribution function. The probability that a bundle has a wavelength between 0 and ∞ is, of course, R(λ → ∞) = 1, a certainty. Equation (8.3) implies that if we want to simulate emission from a surface with N energy bundles of equal energy, then the fraction R(λ) of these bundles must have wavelengths smaller than λ. Now consider a pool of random numbers equally distributed between the values 0 and 1. Since they are equally distributed, this implies that a fraction R of these random numbers have values less than R itself. Let us now pick a single random number, say R0 . Inverting equation (8.3), we find λ(R0 ), i.e., the wavelength corresponding to a cumulative distribution function of value R0 , and we assign this wavelength to one energy bundle. If we repeat this process many times, then the fraction R0 of all energy bundles will have wavelengths below λ(R0 ), since the fraction R0 of all our random numbers will be below this value. Thus, in order to model correctly the spectral variation of surface emission, using N bundles of equal energy, their wavelengths may be determined by picking N random numbers between 0 and 1, and inverting equation (8.3).

Random Numbers If we throw a ball onto a spinning roulette wheel, the ball will eventually settle on any one of the wheel’s numbers (between 0 and 36). If we let the roulette wheel decide on another number again and again, we will obtain a set of random numbers between 0 and 36 (or between 0 and 1, if we divide each number by 36). Unless the croupier throws in the ball and spins the wheel in a regular (nonrandom) fashion,1 any number may be chosen each time with equal probability, regardless of what numbers have been picked previously. However, if sufficiently many numbers are picked, we may expect that roughly half (i.e., 18/37) of all the picked numbers will be between 0 and 17, for example. 1 This is, of course, the reason casinos tend to employ a number of croupiers, each of whom works only for a very short period each day.

250

8 THE MONTE CARLO METHOD FOR SURFACE EXCHANGE

0.75 10 cm × 10 cm parallel black plates spaced 2.5 cm 0.70

F1–2

0.65

0.60 Starting value = 12,345 Starting value = 1 Analytical (F1–2 = 0.632)

0.55

0.50 5

10

15 20

25

30

35

40

Number of bundles

45 50

55

60 ×102

FIGURE 8-2 Convergence of Monte Carlo method for different sets of random numbers.

During the course of a Monte Carlo simulation, generally somewhere between 105 and 10 random numbers need to be drawn, and they need to be drawn very rapidly. Obviously, spinning a roulette wheel would be impractical. One solution to this problem is to store an (externally determined) set of random numbers. However, such a table would require a prohibitive amount of computer storage, unless it were a relatively small table, that would be used repeatedly (thus destroying the true randomness of the set). The only practical answer is to generate the random numbers within the computer itself. This appears to be a contradiction, since a digital computer is the incarnation of logic (nonrandomness). Substantial research has been carried out on how to generate sets of sufficiently random numbers using what are called pseudorandom number generators. A number of such generators exist that, after making the choice of a starting point, generate a new pseudorandom number from the previous one. The randomness of such a set of numbers depends on the quality of the generator as well as the choice of the starting point and should be tested by different “randomness tests.” For a more detailed discussion of pseudorandom number generators, the reader is referred to Hammersley and Handscomb [1], Schreider [8], or Taussky and Todd [16]. 7

Accuracy Considerations Since Monte Carlo methods are statistical methods, the results, when plotted against number of samples, will generally fluctuate randomly around the correct answer. If a set of truly random numbers is used for the sampling, then these fluctuations will decrease as the number of samples increases. Let the answer obtained from the Monte Carlo method after tracing N energy bundles be S(N), and the exact solution obtained after sampling infinitely many energy bundles S(∞). For some simple problems it is possible to calculate directly the probability that the obtained answer, S(N), differs by less than a certain amount from the correct answer, S(∞). Even if it were possible to directly calculate the confidence level for more complicated situations, this would not take into account the pseudorandomness of the computer-generated random number set. That this effect can be rather substantial is seen from Fig. 8-2, which depicts the Monte Carlo evaluation of the view factor between two parallel black plates [17]. Both sets of data use the same computer code and the same random number generator (on a UNIVAC 1110). If a starting value of 1 is used, the results are still fairly inaccurate after 5000 bundles; if a starting value of 12,345 is used (this number gave the fastest-converging results of the ones tested for the random number generator used here), good convergence is achieved after only 4000 bundles. Obviously, careful investigation of the random number generator can increase

8.2 NUMERICAL QUADRATURE BY MONTE CARLO

251

convergence and accuracy and thus decrease computer time considerably. Randomness tests performed on sets of generated numbers showed that a starting value of 12,345 performs well in all tests and indeed results in a “better” set of random numbers than the starting value of 1 for the random number generator employed by Modest and Poon [17]. For radiative heat transfer calculations the most straightforward way of estimating the error associated with the sampling result S(N) is to break up the result into a number of I subsamples S(Ni ), such that I X N = N1 + N2 + . . . + NI = Ni , (8.4) i=1

I   1 1 X N1 S(N1 ) + . . . + NI S(NI ) = S(N) = Ni S(Ni ). N N

(8.5)

i=1

Normally, each subsample would include identical amounts of bundles, leading to Ni = N/I;

i = 1, 2, . . . , I, I X 1 S(N) = S(Ni ). I

(8.6) (8.7)

i=1

The I subsamples may then be treated as if they were independent experimental measurements of the same quantity. We may then calculate the variance or adjusted mean square deviation of the mean I X 1 σ2m = [S(Ni ) − S(N)]2 . (8.8) I(I − 1) i=1

The central limit theorem states that the mean S(N) of I measurements S(Ni ) follows a Gaussian distribution, whatever the distribution of the individual measurements. This implies that we can say with 68.3% confidence that the correct answer S(∞) lies within the limits of S(N) ± σm , with 95.5% confidence within S(N) ± 2σm , or with 99% confidence within S(N) ± 2.58σm . Details on statistical analysis of errors may be found in any standard book on experimentation, for example, the one by Barford [18].

8.2 NUMERICAL QUADRATURE BY MONTE CARLO Before discussing how statistical methods can be used to solve complicated radiative transfer problems, we will quickly demonstrate that the Monte Carlo method can also be employed to evaluate integrals numerically (known as numerical quadrature). Consider the integral Rb f (x) dx. The most primitive form of numerical quadrature is the trapezoidal rule, in which f (x) a is assumed constant over a small interval ∆x, i.e. [19, 20], Z

b a

N X i h f xi = (i − 12 )∆x ∆x; f (x) dx ≃

∆x =

i=1

b−a . N

(8.9)

For large enough values of N equation (8.9) converges to the correct result. Note that the values of xi are equally distributed across the interval between a and b. If we were to draw N random locations equally distributed between a and b, we would achieve the same result in a statistical sense. Therefore, we can evaluate any integral via the Monte Carlo method as Z

b

f (x) dx ≃ a

N X i=1

f [xi = a + (b − a)Ri ] ∆x; ∆x =

b−a , N

(8.10)

252

8 THE MONTE CARLO METHOD FOR SURFACE EXCHANGE

where Ri is a set of random numbers equally distributed between 0 and 1. Equation (8.10) is an efficient means of integration if the integrand f (x) is poorly behaved as, e.g., in the evaluation of k-distributions in Chapter 11 (integration over spectral variations of the absorption coefficient of molecular gases). However, if f (x) varies by orders of magnitude (but in a predictable manner) across a ≤ x ≤ b, picking equally distributed xi results in putting equal emphasis on important as well as unimportant regions. The stochastic integration can be made more efficient by determining the xi from a probability density function (PDF) p(x). We may write Z

b

f (x) dx = a

Z

where ξ(x) =

Z

b a

f (x) p(x) dx = p(x)

x

p(x) dx, a

Z

Z

1 0

f (x(ξ)) dξ p(x(ξ))

(8.11)

b

p(x) dx ≡ 1.

(8.12)

a

The PDF is chosen in such a way that f /p remains relatively constant across a ≤ x ≤ b, assuring that each stochastic sample makes roughly the same contribution to the result. The integral may then be evaluated as Z b N b − a X f (xi ) , xi = ξ−1 (Ri ). (8.13) f (x) dx ≃ N p(xi ) a i=1

Equations (8.10) and (8.13) are also useful if integration is an integral part of a Monte Carlo simulation, such as the Backward Monte Carlo scheme described at the end of this chapter. Finally, extension to two- and higher-dimensional integrals is obvious and trivial.

8.3 HEAT TRANSFER RELATIONS FOR RADIATIVE EXCHANGE BETWEEN SURFACES In the absence of a participating medium and assuming a refractive index of unity, the radiative heat flux leaving or going into a certain surface, using the Monte Carlo technique, is governed by the following basic equation: Z dFdA′ →dA 4 q(r) = ǫ(r)σT (r) − dA′ , (8.14) ǫ(r′ ) σT 4 (r′ ) dA A where q(r) = local surface heat flux at location r, T(r) = surface temperature at location r, ǫ(r) = total hemispherical emittance of the surface at r, A = surface area of the enclosure, and dFdA′ →dA = generalized radiation exchange factor between surface elements dA′ and dA. In equation (8.14) the first term on the right-hand side describes the emission from the surface, and the integrand of the second term is the fraction of energy, originally emitted from the surface at r′ , which eventually gets absorbed at location r. Therefore, the definition for the generalized exchange factor must be: dFdA′ →dA ≡ fraction of the total energy emitted by dA′ that is absorbed by dA, either directly or after any number and type of reflections.

(8.15)

This definition appears to be the most compatible one for solution by ray-tracing techniques and is therefore usually employed for calculations by the Monte Carlo method. Figure 8-3 shows a schematic of an arbitrary enclosure with energy bundles emitted at dA′ and absorbed at dA.

8.3 HEAT TRANSFER RELATIONS FOR RADIATIVE EXCHANGE BETWEEN SURFACES

253

dA´

dA



r

FIGURE 8-3 Possible energy bundle paths in an arbitrary enclosure.

0

If the enclosure is not closed, i.e., has openings into space, some artificial closing surfaces must be introduced. For example, an opening directed into outer space without irradiation from the sun or Earth can be replaced by a black surface at a temperature of 0 K. If the opening is irradiated by the sun, it is replaced by a nonreflecting surface with zero emittance for all angles but the solar angle, etc. The enclosure surface is now divided into J subsurfaces, and equation (8.14) reduces to Qi =

Z

qi dAi = Ai

ǫi σTi4 Ai



J X

ǫ j σT4j A j Fj→i − q ext As Fs→i ,

1 ≤ i ≤ J,

(8.16)

j=1

where q ext = external energy entering through any opening in the enclosure, As = area of the opening irradiated from external sources, and the ǫ j and T j are suitable average values for each subsurface, i.e., Z 1 ǫσT 4 dA. ǫ j σT4j = A j Aj

(8.17)

Although heat flow rates Qi can be calculated directly by the Monte Carlo method, it is of advantage to instead determine the exchange factors: Although the Qi ’s depend on all surface temperatures in the enclosure, the Fi→ j ’s either do not (gray surfaces) or depend only on the temperature of the emitting surface (nongray surfaces), provided that surface reflectances (and absorptances) are independent of temperature (as they are to a very good degree of accuracy). Since all emitted energy must go somewhere, and, by the Second Law of Thermodynamics the net exchange between two equal temperature surfaces must be zero, the summation rule and reciprocity also hold for exchange factors, i.e., J X Fi→ j = 1, (8.18) j=1

ǫi Ai Fi→ j = ǫ j A j F j→i ,

(8.19)

(the former, of course, only for enclosures without openings). A large statistical sample of energy bundles Ni is emitted from surface Ai , each of them carrying the amount of radiative energy ∆Ei = ǫi σTi4 Ai /Ni .

(8.20)

254

8 THE MONTE CARLO METHOD FOR SURFACE EXCHANGE

If Ni j of these bundles become absorbed by surface A j either after direct travel or after any number of reflections, the exchange factor may be calculated from ! ! Ni j Ni j . (8.21) Fi→ j = lim ≃ Ni →∞ Ni Ni Ni ≫1 MONT3D is a publicly available Fortran code [21–24], given in Appendix F, that calculates general exchange factors for complicated three-dimensional geometries. Monte Carlo calculations of exchange factors, by their nature, automatically obey the summation rule, equation (8.18), but—due to the inherent statistical scatter—reciprocity, equation (8.19), is not fulfilled. Several smoothing schemes have been given in the literature that assure that both equations (8.18) and (8.19) are satisfied [25–29].

8.4 RANDOM NUMBER RELATIONS FOR SURFACE EXCHANGE In order to calculate the exchange factor by tracing the history of a large number of energy bundles, we need to know how to pick statistically meaningful energy bundles as explained in Section 8.1: for each emitted bundle we need to determine a point of emission, a direction of emission, and a wavelength of emission. Upon impact of the bundle onto another point of the enclosure surface, we need to decide whether the bundle is reflected and, if so, into what direction.

Points of Emission Similar to equation (8.1) we may write for the total emission from a surface A j : Z Ej = ǫσT 4 dA.

(8.22)

Aj

Since integration over an area is a double integral, we may rewrite this equation, without loss of generality, as Z X Z Y Z X 4 Ej = ǫσT dy dx = E′j dx, (8.23) x=0

y=0

0

where E′j (x)

=

Z

Y

ǫσT 4 dy.

(8.24)

0

Thus, we may apply equation (8.3) and find Z x 1 E′ dx. Rx = Ej 0 j

(8.25)

This relationship may be inverted to find the x-location of the emission point as a function of a random number Rx : x = x(Rx ).

(8.26)

Once the x-location has been determined, equation (8.3) may also be applied to equation (8.24), leading to an expression for the y-location of emission: Z y 1 Ry = ′ (8.27) ǫσT 4 dy, E j (x) 0 and

8.4 RANDOM NUMBER RELATIONS FOR SURFACE EXCHANGE

255

dA = rdφ dr

ro

ri

r

φ

FIGURE 8-4 Geometry for Example 8.1.

y = y(R y , x).

(8.28)

Note that the choice for the y-location depends not only on the random number R y , but also on the location of x. If the emissive power may be separated in x and y, i.e., if E = ǫσT 4 = Ex (x)E y (y),

(8.29)

then equation (8.25) reduces to Rx =

Z

x

Ex (x) dx 0

,Z

X

Ex (x) dx,

(8.30)

E y (y) dy,

(8.31)

0

and equation (8.27) simplifies to Ry =

Z

y 0

,Z E y (y) dy

Y 0

that is, choices for x- and y-locations become independent of one another. In the simplest case of an isothermal surface with constant emittance, these relations reduce to x = Rx X,

y = R y Y.

(8.32)

Example 8.1. Given a ring surface element on the bottom of a black isothermal cylinder with inner radius ri = 10 cm and outer radius ro = 20 cm, as indicated in Fig. 8-4, calculate the location of emission for a pair of random numbers Rr = 0.5 and Rφ = 0.25. Solution We find E=

Z

Eb dA = Eb A

Z

2π 0

Z

ro

r dr dφ. ri

Since this expression is separable in r and φ, this leads to Z φ ,Z 2π φ Rφ = dφ dφ = , 2π 0 0 and Rr = or

Z

r

r dr ri

,Z

ro

r dr = ri

or

φ = 2πRφ ,

r2 − r2i r2o − r2i

,

256

8 THE MONTE CARLO METHOD FOR SURFACE EXCHANGE

r=

q

r2i + (r2o − r2i )Rr .

p Therefore, φ = 2π × 0.25 = π/2 and r = 100 + (400 − 100)0.5 = 15.8 cm. While, as expected for a random number of 0.25, the emission point angle is 90◦ away from the φ = 0 axis, the r-location does not fall onto the midpoint. This is because the cylindrical ring has more surface area at larger radii, resulting in larger total emission. This implies that more energy bundles must be emitted from the outer part of the ring.

Wavelengths of Emission Once an emission location has been chosen, the wavelength of the emitted bundle needs to be determined (unless all surfaces in the enclosure are gray; in that case the wavelength of the bundle does not enter the calculations, and its determination may be omitted). The process of finding the wavelength has already been outlined in Section 8.1, leading to equation (8.3), i.e., Z λ 1 Rλ = ǫλ Ebλ dλ, (8.33) ǫσT 4 0 and, after inversion, λ = λ(Rλ , x, y). (8.34) We note that the choice of wavelength, in general, depends on the choice for the emission location (x, y), unless the surface is isothermal with constant emittance. If the surface is black or gray, equation (8.33) reduces to the simple case of Z λ 1 Rλ = Ebλ dλ = f (λT). (8.35) σT 4 0

Directions of Emission The spectral emissive power (for a given position and wavelength) is Z 2π Z π/2 Z 1 ′ ǫλ′ cos θ sin θ dθ dψ. ǫλ Ibλ cos θ dΩ = Ebλ Eλ = π 0 0 2π As we did for choosing the (two-dimensional) point of emission, we write Z ψ Z π/2 Z ψ Z π/2 ′ ǫλ Ebλ 1 ǫλ′ cos θ sin θ dθ dψ = cos θ sin θ dθ dψ, Rψ = πEλ 0 π 0 ǫλ 0 0 or ψ = ψ(Rψ , x, y, λ).

(8.36)

(8.37) (8.38)

We note from equation (8.37) that ψ does not usually depend on emission location, unless the emittance changes across the surface. However, ψ does depend on the chosen wavelength, unless spectral and directional dependence of the emittance are separable. Once the azimuthal angle ψ is found, the polar angle θ is determined from ,Z π/2 Z θ ′ Rθ = (8.39) ǫλ′ cos θ sin θ dθ, ǫλ cos θ sin θ dθ 0

0

or θ = θ(Rθ , x, y, λ, ψ).

(8.40)

Most surfaces tend to be isotropic so that the directional emittance does not depend on azimuthal R π/2 angle ψ. In that case ǫλ = 2 0 ǫλ′ cos θ sin θ dθ, and equation (8.37) reduces to Rψ =

ψ , 2π

or

ψ = 2πRψ ,

(8.41)

8.4 RANDOM NUMBER RELATIONS FOR SURFACE EXCHANGE

257

and the choice of polar angle becomes independent of azimuthal angle. For a diffuse emitter, equation (8.39) simplifies to Rθ = sin2 θ,

or

θ = sin−1

p

Rθ .

(8.42)

Order of Evaluation In the foregoing we have chosen to first determine an emission location, followed by an emission wavelength and, finally, the direction of emission, as is most customary. However, the only constraint that we need to satisfy in a statistical manner is the total emitted energy from a surface, given by Z Z Z Z ∞

ǫσT 4 dA =

E=

A

A

0



ǫλ′ Ibλ cos θ dΩ dλ dA.

(8.43)

While we have obtained the random number relationships by peeling the integrals in equation (8.43) in the order shown, integration may be carried out in arbitrary order (e.g., first evaluating emission wavelength, etc.).

Absorption and Reflection When radiative energy impinges on a surface, the fraction α′λ will be absorbed, which may depend on the wavelength of irradiation, the direction of the incoming rays, and, perhaps, the local temperature. Of many incoming bundles the fraction α′λ will therefore be absorbed while the rest, 1 − α′λ , will be reflected. This can clearly be simulated by picking a random number, Rα , and comparing it with α′λ : If Rα ≤ α′λ , the bundle is absorbed, while if Rα > α′λ , it is reflected. The direction of reflection depends on the bidirectional reflection function of the material. The fraction of energy reflected into all possible directions is equal to the directional– hemispherical spectral reflectance, or Z ′✄ ρλ (λ, θi , ψi ) = ρ′′ λ (λ, θi , ψi , θr , ψr ) cos θr dΩ r Z

=

2π 2π

Z

0

π/2 0

ρ′′ λ (λ, θi , ψi , θr , ψr ) cos θr sin θr dθr dψr .

(8.44)

As before, the direction of reflection may then be determined from Rψr =

1 ✄ ρλ′

Z

ψr

Z

0

and Z

Rθr = Z

π/2

ρ′′ λ (λ, θi , ψi , θr , ψr ) cos θr sin θr dθr dψr ,

0

θr

ρ′′ λ (λ, θi , ψi , θr , ψr ) cos θr sin θr dθr

0 π/2

0

(8.45)

. ρ′′ λ (λ, θi , ψi , θr , ψr ) cos θr

(8.46)

sin θr dθr

✄ If the surface is a diffuse reflector, i.e., ρ′′ (λ, θi , ψi , θr , ψr ) = ρ′′ (λ) = ρλ′ (λ)/π, then equaλ λ tions (8.45) and (8.46) reduce to Rψr =

ψr , 2π

or

and Rθr = sin2 θr ,

or

ψr = 2πRψr , θr = sin−1

p Rθr ,

(8.47) (8.48)

258

8 THE MONTE CARLO METHOD FOR SURFACE EXCHANGE

n t1

t2

v1

v2 r

FIGURE 8-5 Surface description in terms of a position vector.

0

which are the same as for diffuse emission. For a purely specular reflector, the reflection direction follows from the law of optics as ψr = ψi + π,

θr = θi ,

(8.49)

that is, no random numbers are needed.2

8.5

SURFACE DESCRIPTION

When Monte Carlo simulations are applied to very simple configurations such as flat plates, e.g., Toor and Viskanta [30], the surface description, bundle intersection points, intersection angles, reflection angles, etc. are relatively obvious and straightforward. If more complicated surfaces are considered, such as the second-order polynomial description by Weiner and coworkers [31] or the arbitrary-order polynomial description by Modest and Poon [17] and Modest [32], a systematic way to describe surfaces is preferable. It appears most logical to describe surfaces in vectorial form, as indicated in Fig. 8-5, r=

3 X

xi (v1 , v2 ) eˆ i ,

v1min ≤ v1 ≤ v1max ,

v2min (v1 ) ≤ v2max (v1 ),

(8.50)

i=1

that is, r is the vector pointing from the origin to a point on the surface, v1 and v2 are two surface parameters, the xi are the (x, y, z) coordinates of the surface point, and the eˆ i are unit vectors ˆ into the x, y, z directions, respectively. We may define two unit tangents to the surface at (ˆı, ˆ, k) any point as , , ∂r ∂r ∂r ∂r ˆt1 = , ˆt2 = . (8.51) ∂v1 ∂v1 ∂v2 ∂v2 While it is usually a good idea to choose the surface parameters v1 and v2 perpendicular to one another (making ˆt1 and ˆt2 perpendicular to each other), this is not necessary. In either case, one can evaluate the unit surface normal as nˆ =

ˆt1 × ˆt2 , |tˆ1 × ˆt2 |

(8.52)

where it has been assumed that v1 and v2 have been ordered such that nˆ is the outward surface normal. 2 by an Mathematically, equation (8.49) may also be obtained from equations (8.45) and (8.46) by replacing ρ′′ λ appropriate Dirac-delta function.

8.6 RAY TRACING

259

z

sr t1

n t2 r

n

z

si φ

x r0

8.6

FIGURE 8-6 Rocket nozzle diffuser geometry for Example 8.2.

RAY TRACING

Points of emission may be found by establishing a relationship such as equation (8.22) for the general vectorial surface description given by equation (8.50). The infinitesimal area element on the surface may be described by ∂r ∂r ∂r ∂r dv1 dv2 = |ˆt1 × ˆt2 | dv1 dv2 . dA = (8.53) × ∂v1 ∂v2 ∂v1 ∂v2 Thus, if we replace x by v1 and y by v2 , emission points (v1 , v2 ) are readily found from equations (8.25) and (8.27).

Example 8.2. Consider the axisymmetric rocket nozzle diffuser shown in Fig. 8-6. Assuming that the diffuser is gray and isothermal, establish the appropriate random number relationships for the determination of emission points. Solution The diffuser surface is described by the formula z = a(r2 − r20 ),

0 ≤ z ≤ L,

r0 ≤ r ≤ rL ,

a=

1 , 2r0

where L is the length of the diffuser and r0 and rL are its radius at z = 0 and L, respectively. In vectorial form, we may write ˆ r = r cos φˆı + r sin φˆ + a(r2 − r20 )k, where φ is the azimuthal angle in the x-y-plane, measured from the x-axis. This suggests the choice v1 = r and v2 = φ. The two surface tangents are now readily calculated as cos φˆı + sin φˆ + 2arkˆ , √ 1 + 4a2 r2 ˆt2 = − sin φˆı + cos φˆ. ˆt1 =

It is seen that ˆt1 · ˆt2 = 0, i.e., the tangents are perpendicular to one another. The surface normal is then   ˆ kˆ  −2ar(cos φˆı + sin φˆ) + kˆ  ˆı 1     cos φ nˆ = tˆ1 × tˆ2 = √ , sin φ 2ar = √   1 + 4a2 r2 − sin φ cos φ 1 + 4a2 r2 0

and, finally, an infinitesimal surface area is

ˆ − r sin φˆı + r cos φˆ| dr dφ = dA = | cos φˆı + sin φˆ + 2ark||

√ 1 + 4a2 r2 r dr dφ.

260

8 THE MONTE CARLO METHOD FOR SURFACE EXCHANGE

sr n si

D

n t2 s cos θ

θ

sin

α

θ

ψ

α t1 FIGURE 8-7 Vector description of emission direction and point of impact.

Since there is no dependence on azimuthal angle φ in either dA or the emissive power, we find immediately φ Rφ = , or φ = 2πRφ , 2π and for the radial position parameter r Rr √ (1 + 4a2 r2 )3/2 r 1 + 4a2 r2 r dr (1 + 4a2 r2 )3/2 − (1 + 4a2 r20 )3/2 r0 rL0 = R r = R rL √ = . (1 + 4a2 r2L )3/2 − (1 + 4a2 r20 )3/2 (1 + 4a2 r2 )3/2 r 1 + 4a2 r2 r dr r 0

0

The above expression is readily solved to give an explicit expression for r = r(Rr ).

Once a point of emission has been found, a wavelength and a direction are calculated from equations (8.33), (8.37), and (8.39). As shown in Fig. 8-7, the direction may be specified as a unit direction vector with polar angle θ measured from the surface normal, and azimuthal angle ψ measured from ˆt1 , leading to sˆ = and

i sin θ h sin(α − ψ)ˆt1 + sin ψˆt2 + cos θn, ˆ sin α sin α = ˆt1 × ˆt2 ,

(8.54) (8.55)

where α is the angle between ˆt1 and ˆt2 . If tˆ1 and tˆ2 are perpendicular (α = π/2), equation (8.54) reduces to h i sˆ = sin θ cos ψˆt1 + sin ψˆt2 + cos θn. ˆ (8.56)

As also indicated in Fig. 8-7, the intersection point of an energy bundle emitted at location re , traveling into the direction sˆ , with a surface described in vectorial form may be determined as re + Dˆs = r,

(8.57)

where r is the vector describing the intersection point, and D is the distance traveled by the energy bundle. Equation (8.57) may be written in terms of its x, y, z components and solved for

8.7 EFFICIENCY CONSIDERATIONS

261

ˆ D by forming the dot products with unit vectors ˆı, ˆ, and k: y(v1 , v2 ) − ye x(v1 , v2 ) − xe z(v1 , v2 ) − ze . (8.58) = = sˆ · ıˆ sˆ · ˆ sˆ · kˆ Equation (8.58) is a set of three equations in the three unknowns v1 , v2 , and D: First v1 and v2 are calculated, and it is determined whether the intersection occurs within the confines of the surface under scrutiny. If so, and if more than one intersection is a possibility (in the presence of convex surfaces, protruding corners, etc.), then the path length D is also determined; if more than one intersection is found, the correct one is the one after the shortest positive path. If the bundle is reflected, and if reflection is nonspecular, a reflection direction is chosen from equations (8.45) and (8.46). This direction is then expressed in vector form using equation (8.54). If the surface is a specular reflector, the direction of reflection is determined from equation (8.49), or in vector form as sˆ r = sˆ i + 2|ˆs i · n| ˆ n. ˆ (8.59) D=

Once the intersection point and the direction of reflection have been determined, a new intersection may be found from equation (8.58), etc., until the bundle is absorbed. Example 8.3. Consider again the geometry of Example 8.2. An energy bundle is emitted from the ˆ Determine the intersection point on the diffuser origin (x = y = z = 0) into the direction sˆ = 0.8ˆı + 0.6k. and the direction of reflection, assuming the diffuser to be a specular reflector. Solution With re = 0 and equations (8.54) and (8.58), we find D=

a(r2 − r20 ) r sin φ r cos φ = = . 0.8 0 0.6

Obviously, φ = 0,3 and solving the quadratic equation for r, r2 − r20 =

3r 1 = 32 rr0 , or r = 2r0 and z = (4r2 − r20 ) = 32 r0 . 4a 2r0 0

At that location we form the unit vectors as given in Example 8.2, 1 ˆ ˆt2 = ˆ, and nˆ = √1 (−2ˆı + k). ˆ ˆt1 = √ (ˆı + 2k), 5 5 Therefore, the direction of reflection is determined from equation (8.59) as −2 × 0.8 + 0.6 −2ˆı + kˆ ˆ ˆ sˆ r = 0.8ˆı + 0.6k + 2 = k, √ √ 5 5 as is easily verified from Fig. 8-6.

8.7

EFFICIENCY CONSIDERATIONS

The accuracy of results for generalized radiation exchange factors or wall heat fluxes, as characterized by the standard deviation, equation (8.8), is determined by the statistical scatter of the results. The scatter may be expected to be inversely proportional to the number of bundles absorbed by a subsurface. This number of bundles, on the other hand, is directly proportional to both total number of bundles and size of subsurface. Thus, in order to achieve good spatial resolution (small element sizes), very large numbers of bundles—often several million or even billions—must be emitted and traced. Consequently, even with the availability of today’s fast digital computers, it is imperative that the Monte Carlo implementation and its ray tracings be as numerically efficient as possible, if many hours of CPU time for each computer run are to be avoided. Today’s trend toward massively parallel computing brings new efficiency challenges with it that—while beyond the scope of this book—have been discussed in some detail by several investigators [15]. 3

In computer calculations care must be taken here and elsewhere to avoid division by zero.

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8 THE MONTE CARLO METHOD FOR SURFACE EXCHANGE

Inversion of Random Number Relations Many of the random number relationships governing emission location, wavelength, direction, etc., cannot be inverted explicitly. For example, to determine the wavelength of emission, even for a simple black surface, for a given random number Rλ requires the solution of the transcendental equation (8.33), Z λ 1 Rλ = Ebλ dλ = f (λT). (8.60) σT 4 0 In principle, this requires guessing a λ, calculating Rλ , etc. until the correct wavelength is found; this would then be repeated for each emitted photon bundle. It would be much more efficient to invert equation (8.60) once and for all before the first energy bundle is traced as λT = f −1 (Rλ ).

(8.61)

This is done by first calculating Rλ,j corresponding to a (λT) j for a sufficient number of points j = 0, 1, . . . , J. These data points may then be used to obtain a polynomial description λT = A + BRλ + CR2λ + · · · ,

(8.62)

as proposed by Howell [13]. With the math libraries available today on most digital computers it would, however, be preferable to invert equation (8.61) using a (cubic) spline. Even more efficient is the method employed by Modest and Poon [17] and Modest [32], who used a cubic spline to determine values of (λT) j for (J + 1) equally spaced random numbers ! j (8.63) , j = 0, 1, 2, . . . , J. (λT) j = f −1 Rλ = J If, for example, a random number Rλ = 0.6789 is picked, it is immediately known that (λT) lies between (λT)m and (λT)m+1 , where m is the largest integer less than J × Rλ (= 67 if J = 100). The actual value for (λT) may then be found by (linear) interpolation. The quantity to be determined may depend on more than a single random number. For example, to fix an emission wavelength on a surface with nonseparable emissive power (say a surface in the x-y-plane with locally varying, nongray emittance) requires the determination of x = x(Rx ),

y = y(R y , x),

λ = λ(Rλ , x, y).

(8.64)

That is, first the x-location is chosen, requiring the interpolation between and storage of J data points x j (R j ); next the y-location is determined, requiring a double interpolation and storage of a J × K array for y jk (Rk , x j ); and finally λ is found from a triple interpolation from a J × K × L array for λ jkl (Rl , x j , yk ). This may lead to excessive computer storage requirements if J, K, L are chosen too large: If J = K = L = 100, an array with one million numbers needs to be stored for the determination of emission wavelengths alone! The problem may be alleviated by choosing a better interpolation scheme together with smaller values for J, K, L (for example, a choice of J = K = L = 40 reduces storage requirements to 64,000 numbers).

Energy Partitioning In the general Monte Carlo method, a ray of fixed energy content is traced until it is absorbed. In the absence of a participating medium, the decision whether the bundle is absorbed or reflected is made after every impact on a surface. Thus, on the average it will take 1/α tracings until the bundle is absorbed. Therefore, it takes 1/α tracings to add one statistical sample to the calculation of one of the Fi→ j ’s. If the configuration has openings, a number of bundles may be reflected a few times before they escape into space without adding a statistical sample to any of the Fi→ j ’s. Thus, the ordinary Monte Carlo method becomes extremely inefficient for open configurations

REFERENCES

263

and/or highly reflective surfaces. The former problem may be alleviated by partitioning the energy of emitted bundles. This was first applied by Sparrow and coworkers [33, 34], who, before determining a direction of emission, split the energy of the bundle into two parts: the part leaving the enclosure through the opening (equal to the view factor from the emission point to the opening) and the rest (which will strike a surface). A direction is then determined, limited to those that make the bundle hit an enclosure surface. The procedure is repeated after every reflection. This method guarantees that each bundle will contribute to the statistical sample for exchange factor evaluation. A somewhat more general and more easily implemented energy partitioning scheme was applied by Modest and Poon [17, 32]: Rather than drawing a random number Rα to decide whether a bundle is (fully) absorbed or not, they partition the energy of a bundle at each reflection into the fraction α, which is absorbed, and the fraction ρ = 1 − α, which is reflected. The bundle is then traced until it either leaves the enclosure or until its energy is depleted (below a certain fraction of original energy content). This method adds to the statistical sample of a Fi→ j after every tracing and thus leads to vastly faster convergence for highly reflective surfaces.

Data Smoothing Virtually all Monte Carlo implementation to date have been of 0th order, i.e., all properties within a given cell are considered constant throughout the cell, without connectivity to surrounding cells. This makes the estimation of local gradients difficult, if not impossible. Several smoothing schemes have been proposed for the exchange factors of equation (8.21), the simpler ones without restrictions on the size of corrections [26, 27], and others that find the smallest corrections that make the exchange factors satisfy, both, the summation and reciprocity relationships [28, 29].

Other Efficiency Improvements Other improvements are often connected to the particular geometry under scrutiny. For instance, large amounts of computer time may be wasted because it is not immediately known, which of the many subsurfaces the traveling bundle will hit. In general, an intersection between every surface and the bundle must be calculated. Only then can it be determined whether this intersection is legitimate, i.e., whether it occurs within the bounds of the surface. Often the overall enclosure can be broken up into a (relatively small) number of basic surfaces (dictated by geometry), which in turn are broken up into a number of smaller, isothermal subsurfaces. Furthermore, a bundle emitted or reflected from some subsurface may not be able to hit some basic surface by any path. In other cases, if a possible point of impact on some surface has been determined, it may not be necessary to check the remaining surfaces, etc. There are no fixed rules for the computational structure of a Monte Carlo code. In these applications the proverbial “dash of ingenuity” can go a long way in making a computation efficient.

References 1. Hammersley, J. M., and D. C. Handscomb: Monte Carlo Methods, John Wiley & Sons, New York, 1964. 2. Fleck, J. A.: “The calculation of nonlinear radiation transport by a Monte Carlo method,” Technical Report UCRL-7838, Lawrence Radiation Laboratory, 1961. 3. Fleck, J. A.: “The calculation of nonlinear radiation transport by a Monte Carlo method: Statistical physics,” Methods in Computational Physics, vol. 1, pp. 43–65, 1961. 4. Howell, J. R., and M. Perlmutter: “Monte Carlo solution of thermal transfer through radiant media between gray walls,” ASME Journal of Heat Transfer, vol. 86, no. 1, pp. 116–122, 1964. 5. Howell, J. R., and M. Perlmutter: “Monte Carlo solution of thermal transfer in a nongrey nonisothermal gas with temperature dependent properties,” AIChE Journal, vol. 10, no. 4, pp. 562–567, 1964. 6. Perlmutter, M., and J. R. Howell: “Radiant transfer through a gray gas between concentric cylinders using Monte Carlo,” ASME Journal of Heat Transfer, vol. 86, no. 2, pp. 169–179, 1964. 7. Cashwell, E. D., and C. J. Everett: A Practical Manual on the Monte Carlo Method for Random Walk Problems, Pergamon Press, New York, 1959.

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8. Schreider, Y. A.: Method of Statistical Testing – Monte Carlo Method, Elsevier, New York, 1964. 9. Kahn, H.: “Applications of Monte Carlo,” Report for Rand Corp., vol. Rept. No. RM-1237-AEC (AEC No. AECU3259), 1956. 10. Brown, G. W.: “Monte Carlo methods,” in Modern Mathematics for the Engineer, McGraw-Hill, New York, pp. 279–307, 1956. 11. Halton, J. H.: “A retrospective and prospective survey of the Monte Carlo method,” SIAM Rev., vol. 12, no. 1, pp. 1–63, 1970. 12. Haji-Sheikh, A.: “Monte Carlo methods,” in Handbook of Numerical Heat Transfer, John Wiley & Sons, New York, pp. 673–722, 1988. 13. Howell, J. R.: “Application of Monte Carlo to heat transfer problems,” in Advances in Heat Transfer, eds. J. P. Hartnett and T. F. Irvine, vol. 5, Academic Press, New York, 1968. 14. Walters, D. V., and R. O. Buckius: “Monte Carlo methods for radiative heat transfer in scattering media,” in Annual Review of Heat Transfer, vol. 5, Hemisphere, New York, pp. 131–176, 1992. 15. Howell, J. R.: “The Monte Carlo method in radiative heat transfer,” ASME Journal of Heat Transfer, vol. 120, no. 3, pp. 547–560, 1998. 16. Taussky, O., and J. Todd: “Generating and testing of pseudo-random numbers,” in Symposium on Monte Carlo Methods, John Wiley & Sons, New York, pp. 15–28, 1956. 17. Modest, M. F., and S. C. Poon: “Determination of three-dimensional radiative exchange factors for the space shuttle by Monte Carlo,” ASME paper no. 77-HT-49, 1977. 18. Barford, N. C.: Experimental Measurements: Precision, Error and Truth, Addison-Wesley, London, 1967. 19. Froberg, C. E.: Introduction to Numerical Analysis, Addison-Wesley, Reading, MA, 1969. ¨ 20. Abramowitz, M., and I. A. Stegun (eds.): Handbook of Mathematical Functions, Dover Publications, New York, 1965. 21. Maltby, J. D.: “Three-dimensional simulation of radiative heat transfer by the Monte Carlo method,” M.S. thesis, Colorado State University, Fort Collins, CO, 1987. 22. Burns, P. J., and J. D. Maltby: “Large-scale surface to surface transport for photons and electrons via Monte Carlo,” Computing Systems in Engineering, vol. 1, no. 1, pp. 75–99, 1990. 23. Maltby, J. D., and P. J. Burns: “Performance, accuracy and convergence in a three-dimensional Monte Carlo radiative heat transfer simulation,” Numerical Heat Transfer – Part B: Fundamentals, vol. 16, pp. 191–209, 1991. 24. Zeeb, C. N., P. J. Burns, K. Branner, and J. S. Dolaghan: “User’s manual for Mont3d – Version 2.4,” Colorado State University, Fort Collins, CO, 1999. 25. Larsen, M. E., and J. R. Howell: “Least-squares smoothing of direct-exchange areas in zonal analysis,” ASME Journal of Heat Transfer, vol. 108, no. 1, pp. 239–242, 1986. 26. van Leersum, J.: “A method for determining a consistent set of radiation view factors from a set generated by a nonexact method,” International Journal of Heat and Fluid Flow, vol. 10, no. 1, p. 83, 1989. 27. Lawson, D. A.: “An improved method for smoothing approximate exchange areas,” International Journal of Heat and Mass Transfer, vol. 38, no. 16, pp. 3109–3110, 1995. 28. Loehrke, R. I., J. S. Dolaghan, and P. J. Burns: “Smoothing Monte Carlo exchange factors,” ASME Journal of Heat Transfer, vol. 117, no. 2, pp. 524–526, 1995. 29. Daun, K. J., D. P. Morton, and J. R. Howell: “Smoothing Monte Carlo exchange factors through constrained maximum likelihood estimation,” ASME Journal of Heat Transfer, vol. 127, no. 10, pp. 1124–1128, 2005. 30. Toor, J. S., and R. Viskanta: “A numerical experiment of radiant heat exchange by the Monte Carlo method,” International Journal of Heat and Mass Transfer, vol. 11, no. 5, pp. 883–887, 1968. 31. Weiner, M. M., J. W. Tindall, and L. M. Candell: “Radiative interchange factors by Monte Carlo,” ASME paper no. 65-WA/HT-51, 1965. 32. Modest, M. F.: “Determination of radiative exchange factors for three dimensional geometries with nonideal surface properties,” Numerical Heat Transfer, vol. 1, pp. 403–416, 1978. 33. Heinisch, R. P., E. M. Sparrow, and N. Shamsundar: “Radiant emission from baffled conical cavities,” Journal of the Optical Society of America, vol. 63, no. 2, pp. 152–158, 1973. 34. Shamsundar, N., E. M. Sparrow, and R. P. Heinisch: “Monte Carlo solutions — effect of energy partitioning and number of rays,” International Journal of Heat and Mass Transfer, vol. 16, pp. 690–694, 1973.

Problems Because of the nature of the Monte Carlo technique, most of the following problems require the development of a small computer code. However, all problem solutions can be outlined by giving relevant relations, equations, and a detailed flow chart. Rb 8.1 Prepare a little Monte Carlo code that integrates I(z) = a f (z, x) dx. Apply your code to a few simple integrals, plus Z π/2 π e−z cos x cos(z sin x) dx = Si(z) − . si(z) = − 2 0

265

PROBLEMS

Note: Si(1) = 0.94608. 8.2 In a Monte Carlo simulation involving the plate of Problem 3.9 but of finite width w, a photon bundle is to be emitted from the plate with a wavelength of λ = 2 µm. Find the emission point and direction of this photon bundle in terms of random numbers. 8.3 A triangular, isothermal surface as shown has the following spectral emittance:  0.1, λ < 2µm; θ ≤ 60◦    ǫλ =  0.6, λ > 2µm; θ ≤ 60◦   0.0, all λ; θ > 60◦ For a Monte Carlo simulation (a) find a point of bundle emission in terms of random numbers, (b) find a wavelength of bundle emission in terms of random numbers, (c) find a direction of bundle emission in terms of random numbers.

w

w

8.4 A semicircular disk as shown has a temperature distribution given by  T(r) = T0 / 1 + (r/R)2 , and its emittance is gray and nondiffuse with ǫ′ = ǫ′λ (λ, θ, ψ) =

(

0.6, 0,

r

0 ≤ θ ≤ 30◦ , θ > 30◦ .

f For a Monte Carlo simulation R (a) find a point of emission in terms of random numbers, (b) find a direction of emission in terms of random numbers. You may leave your answer in simple implicit form.

8.5 A light pipe with direct solar irradiation is to be investigated via a Monte Carlo method. Such a device consists of a straight or curved tube covered with a highly-reflective material to pipe light into a room. At visible wavelengths the reflectance from the pipe wall is ρ′λ (θout ) = 1.5ρλ cos θout , with reflection angle θout measured from the local surface normal, and visible light intensity due to direct sunshine may be approximated by Lλ = Kλ Iλ,sun = C exp[−A2 (λ − λ0 )2 ], λ0 = 0.56µm, A = 20/µm. (a) Find the pertinent relationship to determine wavelengths of emission as a function of random number. (b) Find an expression for reflection angle vs. random number. 8.6 At the Aaronsburg (Pennsylvania) Apple Fest you have won a large piece of elderberry pie (yumh!) as shown. The wheels in the oven must have been spinning, because it appears that the number of elderberries per unit area increases linearly proportional with radius! If there are 1000 elderberries otherwise randomly distributed on the slice, make a scatter plot of elderberries on the pie slice.

60

o

20cm 2

8.7 Consider a black disk 0 ≤ r ≤ R with temperature distribution T4 (r) = T04 e−C(r/R) . Develop the random number relations for points of emission; draw random numbers for 1000 emission points and draw them in a scattergram for the cases of C = 0 and C = 5. Use R = 10 cm. 8.8 A disk of radius R is opposed by a square plate (sides of length R) parallel to it, and a distance R away. Find the view factor from disk to square plate. Use 100,000 bundles, plotting updated results after every 5,000 bundles. 8.9 Consider two infinitely long parallel plates of width w spaced a distance h apart. (see Configuration 32 in Appendix D). (a) Calculate F1−2 via Monte Carlo for the case that the top plate is horizontally displaced by a distance L. Use L = h = w.

266

8 THE MONTE CARLO METHOD FOR SURFACE EXCHANGE

s (b) Calculate F1−2 via Monte Carlo for the case that both plates are specular (with identical reflectances ρs1 = ρs2 = 0.5), but not horizontally displaced. Use L = 0, h = w. Prepare a figure similar to Fig. 8-2, also including analytical results for comparison.

8.10 Two directly opposed quadratic plates of width w = 10 cm are spaced a distance L = 10 cm apart, with a third centered quadratic plate of dimension b × b (b = 5 cm) in between at a distance l = 5 cm from the bottom. Determine the view factor F1−2 via Monte Carlo. In order to verify your code (and to have a more flexible tool) it may be best to allow for arbitrary and different top and bottom w as well as b.

W W

A2

L

B

L

B

A1

8.11 Consider two concentric parallel disks of radius R, spaced a distance H apart. Both plates are isothermal (at T1 and T2 , respectively), are gray diffuse emitters with emittance ǫ, and are gray reflectors with diffuse reflectance component ρd and purely specular component ρs . Write a computer code that calculates the generalized exchange factor F1→2 and, taking advantage of the fact that F1→2 = F2→1 , calculate the total heat loss from each plate. Compare with the analytical solution treating each surface as a single node. 8.12 Repeat Problem 8.11, but calculate heat fluxes directly, i.e., without first calculating exchange factors. 8.13 Determine the view factor for Configuration 39 of Appendix D, for h = w = l. Compare with exact results. 8.14 Consider the conical geometry of Problem 5.9: breaking up the sidewall into strips (say 4), calculate all relevant view factors (base-to-rings, ring-to-rings) via Monte Carlo.

2 cm

A2

5 cm A3 4 cm A1

8.15 Reconsider Problem 5.30: (a) find the solution by writing a small Monte Carlo program, and (b) augment this program to allow for nongray, temperature-dependent emittances. 8.16 Repeat Problem 5.33 for T1 = T2 = 1000 K, ǫ1 = ǫ2 = 0.5. Use the Monte Carlo method, employing the energy partitioning of Sparrow and coworkers [33, 34]. 8.17 Repeat Problem 5.34. Compare with the exact solutions for several values of ǫ. 8.18 Repeat Problem 6.3, using the Monte Carlo method. Compare with the solution from Chapter 6 for a few values of D/L and ǫ, and T1 = 1000 K, T2 = 2000 K. How can the problem be done by emitting bundles from only one surface? 8.19 Repeat Problem 6.10 using the Monte Carlo method. 8.20 Repeat Problem 6.23 using the Monte Carlo method. 8.21 Repeat Example 7.3 using the Monte Carlo method. 8.22 Repeat Example 7.4 using the Monte Carlo method. 8.23 Repeat Problem 7.20 using the Monte Carlo method.

CHAPTER

9 SURFACE RADIATIVE EXCHANGE IN THE PRESENCE OF CONDUCTION AND CONVECTION

9.1

INTRODUCTION

In the previous few chapters we have considered only the analysis of radiative exchange in enclosures with specified wall temperatures or fluxes, i.e., we have neglected interaction with other modes of heat transfer. In practical systems, of course, it is nearly always the case that radiation from a boundary is affected by conduction into the solid and/or by convection from the surface. Then, two or three modes of heat transfer must be accounted for simultaneously. The interaction may be quite simple, or it may be rather involved. For example, heat loss from an isothermal surface of known temperature, adjacent to a radiatively nonparticipating medium, may occur by convection as well as radiation; however, convective and radiative heat fluxes are independent of one another, can be calculated independently, and may simply be added. If boundary conditions are more complex (i.e., surface temperatures are not specified), then radiation enters the remaining conduction/convection problem as a nonlinear boundary condition. In a number of important applications, a conduction analysis needs to be performed on an opaque medium, which loses (or gains) heat from its surfaces by radiation (and, possibly, convection). In such cases radiation enters the conduction problem as a nonlinear boundary condition; however, the radiative flux in this boundary condition may depend on the radiative exchange in the surrounding enclosure. In other applications, conduction and/or convection in a transparent gas or liquid needs to be evaluated, bounded by opaque, radiating walls. Again, radiation enters only as a boundary condition, with the transparent medium itself occupying the enclosure governing the radiative transfer. In both types of applications radiation and conduction–convection are interdependent, i.e., a change in radiative heat flux disturbs the overall energy balance at the surface, causing a change in temperature as well as conductive– convective fluxes, and vice versa. Many important applications of interactions between surface radiation and other modes of heat transfer have been reported in the literature. We will limit ourselves here to the discussion 267

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9 SURFACE RADIATIVE EXCHANGE IN THE PRESENCE OF CONDUCTION AND CONVECTION

dx 2

L

2t

x2 , ξ ´

x2

2t S0

L

α α 2t T = Tb x1

x1 , ξ

dx1 L

(a)

(b)

FIGURE 9-1 Schematic of a space radiator tube with longitudinal fins.

of a few very basic cases (i) to show the basic trends of how the different modes of heat transfer interact with one another, and (ii) to outline some of the numerical schemes that have been used to solve such problems. At the end of each section a short description of more advanced problems is given, as well as a list of references.

9.2 CONDUCTION AND SURFACE RADIATION—FINS The vast majority of combined conduction–surface radiation applications involve heat transfer through vacuum, e.g., heat loss from space vehicles or vacuum insulations. As a single example we will discuss here the performance of a simple rectangular-fin radiator used to reject heat from a spacecraft. Consider a tube with a set of radial fins, as schematically shown in Fig. 9-1. In order to facilitate the analysis, we will make the following assumptions: 1. The thickness of each fin, 2t, is much less than its length in the radial direction, L, which in turn is much less than the fin extent in the direction of the tube axis. This implies that heat conduction within the fin may be calculated by assuming that the fin temperature is a function of radial distance, x, only. 2. End losses from the fin tips (by convection and radiation) are negligible, i.e., ∂Ti /∂xi (L) ≃ 0. 3. The thermal conductivity of the fin material, k, is constant. 4. The base temperatures of all fins are the same, i.e., T1 (0) = T2 (0) = Tb , and the fin arrangement is symmetrical, i.e., T1 (x1 ) = T2 (x2 = x1 ), etc. 5. The surfaces are coated with an opaque, gray, diffusely emitting and reflecting material of uniform emittance ǫ. 6. There is no external irradiation falling into the fin cavities (Ho = 0, T∞ = 0). The first three assumptions are standard simplifications made for the analysis of thin fins (see, e.g., Holman [1]), and the other three have been made to make the radiation part of the

9.2 CONDUCTION AND SURFACE RADIATION—FINS

269

problem more tractable. Performing an energy balance on an infinitesimal volume element (of unit length in the axial direction) dV = 2t dx, one finds: conduction going in at x across cross-sectional area (2t) = conduction going out at x+dx + net radiative loss from top and bottom surfaces (2 dx) or −2tk

dT dT = −2tk + 2qR dx. dx x dx x+dx

Expanding the outgoing conduction term into a truncated Taylor series, dT d2 T dT + dx 2 + · · · , = dx x+dx dx x dx x then leads to

1 d2 T = qR . tk dx2

(9.1)

Here qR (x) is the net radiative heat flux leaving a surface element of the fin, which may be determined in terms of surface radiosity, J, from equations (5.24) and (5.25) as1 qR (x1 ) = J(x1 ) −

Z

L

J(x2 ) dFd1−d2 ,

(9.2)

x2 =0

J(x1 ) = ǫσT 4 (x1 ) + (1 − ǫ)

Z

L

J(x2 ) dFd1−d2 .

(9.3)

x2 =0

The expression for radiative heat flux may be simplified by eliminating the integral, equation (5.26), qR (x1 ) =

i ǫ h 4 σT1 (x1 ) − J1 (x1 ) . 1−ǫ

(9.4)

The view factor between two infinitely long strips may be found from Appendix D, Configuration 5, or from Example 4.1 as Fd1−d2 =

x1 sin2 α x2 dx2 sin2 α x1 x2 dx2 . = 2(x21 − 2x1 x2 cos α + x22 )3/2 2S30

(9.5)

Equation (9.1) requires two boundary conditions, namely,

T(x = 0) = Tb ,

dT (x = L) = 0. dx

(9.6)

Before we attempt a numerical solution, it is a good idea to summarize the mathematical problem in terms of nondimensional variables and parameters, θ(ξ) =

T(x) , Tb

J(ξ) =

J(x) , σTb4

Nc =

kt , σTb3 L2

ξ=

x , L

(9.7)

1 For the radiative exchange it is advantageous to attach subscripts 1 and 2 to the x-coordinates to distinguish contributions from different plates, even though T(x), J(x), qR (x), etc., are the same along each of the fins.

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9 SURFACE RADIATIVE EXCHANGE IN THE PRESENCE OF CONDUCTION AND CONVECTION

where θ and J are nondimensional temperature and radiosity, and Nc is usually called the conduction-to-radiation parameter, sometimes also known as the Planck number. With these definitions, i d2 θ 1 ǫ h 4 = θ (ξ) − J(ξ) , 2 Nc 1 − ǫ dξ Z 1 J(ξ′ ) K(ξ, ξ′ ) dξ′ , J(ξ) = ǫ θ 4 (ξ) + (1 − ǫ)

(9.8a) (9.8b)

ξ′ =0

K(ξ, ξ′ ) =

1 ξ ξ′ sin2 α 2 , ′ 2 (ξ − 2ξξ cos α + ξ′2 )3/2

(9.8c)

subject to θ(ξ = 0) = 1,

dθ (ξ = 1) = 0. dξ

(9.8d)

As for convection-cooled fins, a fin efficiency, η f , is defined, comparing the heat loss from the actual fin to that of an ideal fin (a black fin, which is isothermal at Tb ). The total heat loss from an ideal fin (ǫ = 1, J = σTb4 ) is readily determined from equation (9.2) and Appendix D, Configuration 34, as Qideal = 2L qR,ideal = 2L σTb4 (1 − F1−2 ) = 2L sin

α 4 σT , 2 b

(9.9)

while the actual heat loss follows from Fourier’s law applied to the base, or by integrating over the length of the fin, as Qactual = −2tk Thus, ηf =

Z L dT = 2 qR (x) dx. dx x=0 0

Z 1 Qactual Nc dθ ǫ 1 =− (θ 4 − J ) dξ, = Qideal sin α2 dξ 0 sin α2 1 − ǫ 0

(9.10)

(9.11)

where the last expression is obtained by integrating equation (9.8a) along the length L of the fin. The set of equations (9.8) is readily solved by a host of different methods, including the net radiation method [finite-differencing equation (9.8b) into finite-width isothermal strips, to which equation (5.34) can be applied] or any of the solution methods for Fredholm equations discussed in Section 5.6. Because of the nonlinear nature of the equations it is always advisable to employ the method of successive approximations, i.e., a temperature field is guessed, a radiosity distribution is calculated, an updated temperature field is determined by solving the differential equation (for a known right-hand side), etc. Sample results for the efficiency, as obtained by Sparrow and coworkers [2], are shown in Fig. 9-2. The variation of the fin efficiency is similar to that for a convectively cooled fin (with the heat transfer coefficient replaced by a “radiative heat transfer coefficient,” hR = 4ǫσTb3 ). Maximum efficiency is obtained for Nc → ∞, i.e., when conduction dominates and the fin is essentially isothermal. For ǫ < 1 the efficiency is limited to values η f < 1 since a black configuration will always lose more heat. It is also observed that the fin efficiency (but not the actual heat lost) increases as the opening angle α decreases: For small opening angles irradiation from adjacent fins reduces the net radiative heat loss by a large fraction, but not as much as for the “ideal” fin (with irradiation from adjacent fins, which are black and at Tb ). Many other studies discussing the interaction of surface radiation and one-dimensional conduction may be found in the literature. For example, Hering [3] and Tien [4] considered the fins of Fig. 9-1 with specularly reflecting surfaces, and Sparrow and coworkers [2] investigated the influence of external irradiation. Fins connecting parallel tubes were studied by Bartas and Sellers [5], Sparrow and coworkers [6, 7], and Lieblein [8]. Single annular fins (i.e., annular

9.3 CONVECTION AND SURFACE RADIATION

271

1.00 ∋ = 1.0 ∋ = 0.5

0.90

Radiative fin efficiency η f

0.80 0.70 0.60

α = 45°

0.50 120°

0.40 0.30 0.20

α = 120° 0

1.0

60°

45°

60°

2.0

3.0

1 = L2 σ Tb3 Nc kt

4.0 FIGURE 9-2 Radiative fin efficiency for longitudinal plate fins [2].

disks attached to the outside of tubes) were studied by Chambers and Sommers [9] (rectangular cross-section), Keller and Holdredge [10] (variable cross-section), and Mackay [11] (with external irradiation), while Sparrow and colleagues [12] investigated the interaction between adjacent fins. Various other publications have appeared dealing with different geometries, surface properties (including nongrayness effects), irradiation conditions, etc. A partial listing is given with [13–35]. More recently, some researchers have considered combined conduction–surface radiation in media with cavities, such as porous media [36, 37], packed beds of spheres [38], mirror furnaces [39], and honeycomb panels [40–42].

9.3 CONVECTION AND SURFACE RADIATION As in the case of pure convection heat transfer, it is common to distinguish between external flow and internal flow applications. If the flowing medium is air or some other relatively inert gas, the assumption of a transparent, or radiatively nonparticipating, medium is often justified. As an example we will consider here the case of a transparent gas flowing through a cylindrical tube of diameter D = 2R and length L, which is heated uniformly at a rate of qw (per unit surface area). As schematically shown in Fig. 9-3, the fluid enters the tube at x = 0 with a mean, or bulk, temperature Tm1 . Over the length of the tube the supplied heat flux qw is dissipated from the inner surface by convection (to the fluid) and radiation (to the openings and to other parts of the tube wall), while the outer surface of the tube is insulated. The two open ends of the tube are exposed to radiation environments at temperatures T1 and T2 , respectively. The inner surface of the tube is assumed to be gray, diffusely emitting and diffusely reflecting, with a uniform emittance ǫ. Finally, for a simplified analysis, we will assume that the convective heat transfer coefficient, h, between tube wall and fluid is constant, independent of the radiative heat transfer, and known. With these simplifications an energy balance on a control volume dV = πR2 × dx yields: enthalpy flux in at x + convective flux in over dx = enthalpy flux out at x+dx,

272

9 SURFACE RADIATIVE EXCHANGE IN THE PRESENCE OF CONDUCTION AND CONVECTION

qw

T1

Tm1

R

Tm2

R

x

T2

dx x´

dx´

L FIGURE 9-3 Forced convection and radiation of a transparent medium flowing through a circular tube, subject to constant wall heat flux.

or # " dTm (x) dx , m˙ cp Tm (x) + h [Tw (x) − Tm (x)] 2πR dx = m˙ cp Tm (x+dx) = m˙ cp Tm (x) + dx

(9.12)

or 2h dTm = [Tw (x)−Tm (x)] , dx ρcp um R

(9.13)

where axial conduction has been neglected, and the mass flow rate has been expressed in terms of mean velocity as m˙ = ρum πR2 . Equation (9.13) is a single equation for the unknown wall and bulk temperatures Tw (x) and Tm (x) and is subject to the inlet condition Tm (x = 0) = Tm1 .

(9.14)

An energy balance for the tube surface states that the prescribed heat flux qw is dissipated by convection and radiation or, applying equation (5.26) for the radiative heat flux, qw = h [Tw (x) − Tm (x)] +

i ǫ h 4 σTw (x) − J(x) . 1−ǫ

(9.15)

The radiosity J(x) is found from equation (5.24) as ( ) Z L J(x) = ǫσTw4 (x) + (1−ǫ) σT14 Fdx−1 + σT24 Fdx−2 + J(x′ ) dFdx−dx′ ,

(9.16)

0

where Fdx−1 is the view factor from the circular strip of width dx at x to the opening at x = 0, Fdx−2 is the one to the opening at x = L, and dFdx−dx′ is the view factor between two circular strips located at x and x′ , as indicated in Fig. 9-3. All view factors are readily determined from Appendix D, Configurations 9 and 31, and will not be repeated here. Equations (9.13), (9.15), and (9.16) are a set of three simultaneous equations in the unknown Tw (x), Tm (x), and J(x), which must be solved numerically. Before we attempt such a solution, it is best to recast the equations in nondimensional form. Defining the following variables and parameters, ξ = St =

x , D

θ(ξ) =

h , ρcp um

H=

σT 4 qw

!1/4

,

J(ξ) =

  h qw 1/4 , qw σ

J , qw

(9.17a) (9.17b)

9.3 CONVECTION AND SURFACE RADIATION

273

transforms equations (9.13) through (9.16) to dθm = 4 St [θw (ξ) − θm (ξ)] , dξ

θm (ξ = 0) = θm1 , i ǫ h 4 1 = H [θw (ξ) − θm (ξ)] + θw (ξ) − J(ξ) , 1−ǫ ( ) Z L/D 4 4 4 ′ J(ξ) = ǫ θw (ξ) + (1 − ǫ) θ1 Fdξ−1 + θ2 Fdξ−2 + J(ξ ) dFdξ−dξ′ .

(9.18) (9.19) (9.20)

0

Equation (9.19) becomes indeterminate for ǫ = 1. For the case of a black tube J = θw4 , and equations (9.19) and (9.20) may be combined as Z L/D 4 4 4 1 = H [θw (ξ) − θm (ξ)] + θw (ξ) − θ1 Fdξ−1 − θ2 Fdξ−2 − θw4 (ξ′ ) dFdξ−dξ′ . (9.21) 0

Example 9.1. A transparent gas flows through a black tube subject to a constant heat flux. The convective heat transfer coefficient is known to be constant such that Stanton numbers and the nondimensional heat transfer coefficient are evaluated as St = 2.5 × 10−3 and H = 0.8. The environmental temperatures at both ends are equal to the local gas temperatures, i.e., θ1 = θm1 and θ2 = θm2 = θm (ξ = L/D), and the nondimensional inlet temperature is given as θm1 = 1.5. Determine the (nondimensional) wall temperature variation as a function of relative tube length, L/D, using the numerical quadrature approach of Example 5.11. Solution Since the tube wall is black we have only two simultaneous equations, (9.18) and (9.21), in the two unknowns θm and θw . However, the equations are nonlinear; therefore, an iterative procedure is necessary. For simplicity, we will adopt a simple backward finite-difference approach for the solution of equation (9.18), and the numerical quadrature scheme of equation (5.52) for the integral in equation (9.21). Evaluating temperatures at N + 1 nodal points ξi = i∆ξ (i = 0, 1, . . . , N) where ∆ξ = L/(ND), this implies ! θm (ξi ) − θm (ξi−1 ) dθm ≃ , i = 1, 2, . . . , N, dξ ξi ∆ξ Z L/D N dFdξ−dξ′ ′ L X θw4 (ξ′ ) dξ ≃ c j θw4 (ξ j ) K(ξi , ξ j ), i = 0, 1, . . . , N, dξ′ D 0 j=0

where the c j are quadrature weights and, from Configuration 9 in Appendix D,2 K(ξi , ξ j ) = 1 −

Xi j (2Xij2 +3) 2(Xi2j +1)

;

Xij = |ξi − ξ j |.

Similarly, the two view factors to the openings are evaluated from Configuration 31 in Appendix D as Xi2j + 12 − Xij , Fdξi −k = q Xi2j + 1

where

j = 0

if

k=1

(opening at ξ = ξ0 = 0),

j = N

if

k=2

(opening at ξ = ξN = L/D).

To solve for the unknown θm (ξi ) and θw (ξi ), we adopt the following iterative procedure: 1. A wall temperature is guessed for all wall nodes, say, θw (ξi ) = θ1 , 2





i = 0, 1, . . . , N.

Note that K(ξ, ξ ) has a sharp peak at ξ = ξ. Therefore, and also in light of the truncation error in the finitedifferencing of dθm /dξ, it is best to limit the quadrature scheme to Simpson’s rule [43].

274

9 SURFACE RADIATIVE EXCHANGE IN THE PRESENCE OF CONDUCTION AND CONVECTION

Nondimensional temperature θ = (σ T 4/qw )1/4

3.5

θw θm θw (no radiation)

L/D = 50 50

3.0 10 1 2.5 10

2.0 50 1 10 1.5 0.00

0.25

0.50 Nondimensional axial distance x/L

0.75

1.00

FIGURE 9-4 Axial surface temperature development for combined convection and surface radiation in a black tube subjected to constant wall heat flux.

2. A temperature difference is calculated from equation (9.21), i.e., φi = H [θw (ξi ) − θm (ξi )] = 1 − θw4 (ξi ) + θ14 Fdξi −1 + θ24 Fdξi −2 +

N L X c j θw4 (ξ j ) K(ξi , ξ j ). D j=0

3. The gas bulk temperature is calculated from equation (9.18) as θm (ξi ) = θm (ξi−1 ) +

4 St ∆ξ φi ; H

θm (ξ0 ) = θ1 .

4. An updated value for the wall temperatures is then determined from the definition for φi , that is,   1 θwnew (ξi ) = ω θm (ξi ) + φi + (1 − ω) θwold (ξi ), H where ω is known as the relaxation parameter. The iteration scheme is called underrelaxed if ω < 1, and overrelaxed if ω > 1. If ω is chosen too large, the iteration will become unstable and not converge at all. A good or optimal value for the relaxation parameter must usually be found by trial and error. Detailed discussions on relaxation may be found in standard numerical analysis texts such as [44,45]. Some representative results are shown in Fig. 9-4 for several values of L/D. Because of the strong nonlinearity of the problem, and the crude numerical scheme employed here, large numbers of nodes are necessary to achieve good accuracy (N ≃ 40L/D), together with strong underrelaxation (ω < 0.02). For the case of pure convection (ǫ = 0, or φi ≡ 1) the tube wall temperature rises linearly with axial distance, since constant wall heat flux implies a linear increase in bulk temperature and, therefore, (assuming a constant heat transfer coefficient) in surface temperature. This is not the case if radiation is present, in particular for short tubes (small L/D). Near both ends of the tube, much of the radiative energy leaves through the openings, causing a distinct drop in surface temperature. For long tubes (L/D > 50) the surface temperature rises almost linearly over the central parts of the tube, although the temperature stays below the convection-only case: Due to the higher temperatures downstream, some net radiative heat flux travels upstream, making overall heat transfer a little more efficient. It should be noted here that the assumption of a constant heat transfer coefficient is not particularly realistic, since it implies a fully developed thermal profile. It is well known that for pure convection h → ∞ at the inlet and, thus, θw (ξ = 0) = 1 [1]. Near the inlet of a tube the actual temperature distribution for pure convection is very similar to the one depicted in Fig. 9-3, which is driven by radiation losses. Although for pure convection a fully developed thermal profile and constant h are eventually reached

REFERENCES

275

(at L/D > 20 for turbulent flow), in the presence of radiation a constant heat transfer coefficient is never reached (because the radiation term makes the governing equations nonlinear).

A number of researchers have investigated combined convection and radiation for a transparent flowing medium. Flow through circular tubes was considered by Siegel and coworkers [46–48] for a number of situations, but always assuming a constant and known heat transfer coefficient. Dussan and Irvine [49] and Chen [50] calculated the local convection rate by solving the two-dimensional energy equation for the flowing medium, but they made severe simplifications in the evaluation of radiative heat fluxes. The most general tube flow analysis has been carried out by Thorsen and Kanchanagom [51, 52]. Similar problems for parallel-plate channel flow were investigated by Keshock and Siegel [53] (for a constant heat transfer coefficient) and Lin and Thorsen [54] (for two-dimensional convection calculations). Combined radiation and forced convection of external flow across a flat plate has been addressed by Cess [55,56], Sparrow and Lin [57], and Sohal and Howell [58]. Fluidized bed heat transfer has also been investigated by a number of researchers [59–61] and, finally, the interaction between surface radiation and free convection has been studied, both numerically and experimentally [62–71].

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22. Hrycak, P.: “Influence of conduction on spacecraft skin temperatures,” AIAA Journal, vol. 1, pp. 2619–2621, 1963. 23. Karlekar, B. V., and B. T. Chao: “Mass minimization of radiating trapezoidal fins with negligible base cylinder interaction,” International Journal of Heat and Mass Transfer, vol. 6, pp. 33–48, 1963. 24. Stockman, N. O., and J. L. Kramer: “Effect of variable thermal properties on one-dimensional heat transfer in radiating fins,” NASA TN D-1878, 1963. 25. Kotan, K., and O. A. Arnas: “On the optimization of the design parameters of parabolic radiating fins,” ASME Paper No. 65-HT-42, August 1965. 26. Mueller, H. F., and N. D. Malmuth: “Temperature distribution in radiating heat shields by the method of singular perturbations,” International Journal of Heat and Mass Transfer, vol. 8, pp. 915–920, 1965. 27. Russell, L. D., and A. J. 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Qu: “Effective thermal conductivity of gas–solid composite materials and the temperature difference effect at high temperature,” International Journal of Heat and Mass Transfer, vol. 42, no. 10, pp. 1885–1893, 1999. 38. Singh, B. P., and M. Kaviany: “Effect of solid conductivity on radiative heat transfer in packed beds,” International Journal of Heat and Mass Transfer, vol. 37, no. 16, pp. 2579–2583, 1994. 39. Haya, R., D. Rivas, and J. Sanz: “Radiative exchange between a cylindrical crystal and a monoellipsoidal mirror furnace,” International Journal of Heat and Mass Transfer, vol. 40, pp. 323–332, 1997. 40. Hollands, K. G. T., and K. Iynkaran: “Analytical model for the thermal conductance of compound honeycomb transparent insulation, with experimental validation,” Solar Energy, vol. 51, pp. 223–227, 1993. 41. Jones, P. D.: “Correlation of combined radiation and conduction in evacuated honeycomb-cored panels,” Journal of Solar Energy Engineering, vol. 118, pp. 97–100, 1996. 42. Schweiger, H., A. Oliva, M. Costa, and C. D. Segarra: “Monte Carlo method for the simulation of transient radiation heat transfer: Application to compound honeycomb transparent insulation,” Numerical Heat Transfer – Part B: Fundamentals, vol. 35, pp. 113–136, 1999. 43. Froberg, C. E.: Introduction to Numerical Analysis, Addison-Wesley, Reading, MA, 1969. ¨ 44. Hornbeck, R. W.: Numerical Methods, Quantum Publishers, Inc., New York, 1975. 45. Ferziger, J. H.: Numerical Methods for Engineering Application, John Wiley & Sons, New York, 1981. 46. Siegel, R., and M. Perlmutter: “Convective and radiant heat transfer for flow of a transparent gas in a tube with gray wall,” International Journal of Heat and Mass Transfer, vol. 5, pp. 639–660, 1962. 47. Perlmutter, M., and R. Siegel: “Heat transfer by combined forced convection and thermal radiation in a heated tube,” ASME Journal of Heat Transfer, vol. C84, pp. 301–311, 1962. 48. Siegel, R., and E. G. Keshock: “Wall temperature in a tube with forced convection, internal radiation exchange and axial wall conduction,” NASA TN D-2116, 1964. 49. Dussan, B. I., and T. F. Irvine: “Laminar heat transfer in a round tube with radiating flux at the outer wall,” in Proceedings of the Third International Heat Transfer Conference, vol. 5, Hemisphere, Washington, D.C., pp. 184–189, 1966. 50. Chen, J. C.: “Laminar heat transfer in a tube with nonlinear radiant heat-flux boundary conditions,” International Journal of Heat and Mass Transfer, vol. 9, pp. 433–440, 1966. 51. Thorsen, R. S.: “Heat transfer in a tube with forced convection, internal radiation exchange, axial wall heat conduction and arbitrary wall heat generation,” International Journal of Heat and Mass Transfer, vol. 12, pp. 1182– 1187, 1969. 52. Thorsen, R. S., and D. Kanchanagom: “The influence of internal radiation exchange, arbitrary wall heat generation and wall heat conduction on heat transfer in laminar and turbulent flows,” in Proceedings of the Fourth International Heat Transfer Conference, vol. 3, Elsevier, New York, pp. 1–10, 1970. 53. Keshock, E. G., and R. Siegel: “Combined radiation and convection in asymmetrically heated parallel plate flow channel,” ASME Journal of Heat Transfer, vol. 86C, pp. 341–350, 1964.

PROBLEMS

277

54. Lin, S. T., and R. S. Thorsen: “Combined forced convection and radiation heat transfer in asymmetrically heated parallel plates,” in Proceedings of the Heat Transfer and Fluid Mechanics Institute, Stanford University Press, pp. 32–44, 1970. 55. Cess, R. D.: “The effect of radiation upon forced-convection heat transfer,” Applied Scientific Research Part A, vol. 10, pp. 430–438, 1962. 56. Cess, R. D.: “The interaction of thermal radiation with conduction and convection heat transfer,” in Advances in Heat Transfer, vol. 1, Academic Press, New York, pp. 1–50, 1964. 57. Sparrow, E. M., and S. H. Lin: “Boundary layers with prescribed heat flux–application to simultaneous convection and radiation,” International Journal of Heat and Mass Transfer, vol. 8, pp. 437–448, 1965. 58. Sohal, M., and J. R. Howell: “Determination of plate temperature in case of combined conduction, convection and radiation heat exchange,” International Journal of Heat and Mass Transfer, vol. 16, pp. 2055–2066, 1973. 59. Flamant, G., J. D. Lu, and B. Variot: “Radiation heat transfer in fluidized beds: A comparison of exact and simplified approaches,” ASME Journal of Heat Transfer, vol. 116, no. 3, pp. 652–659, 1994. 60. Fang, Z. H., J. R. Grace, and C. J. Lim: “Radiative heat transfer in circulating fluidized beds,” ASME Journal of Heat Transfer, vol. 117, no. 4, pp. 963–968, 1995. 61. Luan, W., C. J. Lim, C. M. H. Brereton, B. D. Bowen, and J. R. Grace: “Experimental and theoretical study of total and radiative heat transfer in circulating fluidized beds,” Chemical Engineering and Science, vol. 54, no. 17, pp. 3749–3764, 1999. 62. Gianoulakis, S., and D. E. Klein: “Combined natural convection and surface radiation in the annular region between volumetrically heated inner tube and a finite conducting outer tube,” Nuclear Technology, vol. 104, pp. 241–251, 1993. 63. Balaji, C., and S. P. Venkateshan: “Natural convection in L-corners with surface radiation and conduction,” ASME Journal of Heat Transfer, vol. 118, pp. 222–225, 1996. 64. Rao, V. R., and S. P. Venkateshan: “Experimental study of free convection and radiation in horizontal fin arrays,” International Journal of Heat and Mass Transfer, vol. 39, pp. 779–789, 1996. 65. Rao, V. R., C. Balaji, and S. P. Venkateshan: “Interferometric study of interaction of free convection with surface radiation in an l corner,” International Journal of Heat and Mass Transfer, vol. 40, pp. 2941–2947, 1997. 66. Jayaram, K. S., C. Balaji, and S. P. Venkateshan: “Interaction of surface radiation and free convection in an enclosure with a vertical partition,” ASME Journal of Heat Transfer, vol. 119, pp. 641–645, 1997. 67. Cheng, X., and U. Muller: “Turbulent natural convection coupled with thermal radiation in large vertical channels ¨ with asymmetric heating,” International Journal of Heat and Mass Transfer, vol. 41, no. 12, pp. 1681–1692, 1998. 68. Ramesh, N., and S. P. Venkateshan: “Effect of surface radiation on natural convection in a square enclosure,” Journal of Thermophysics and Heat Transfer, vol. 13, no. 3, pp. 299–301, 1999. 69. Yu, E., and Y. K. Joshi: “Heat transfer in discretely heated side-vented compact enclosures by combined conduction, natural convection, and radiation,” ASME Journal of Heat Transfer, vol. 121, no. 4, pp. 1002–1010, 1999. 70. Adams, V. H., Y. K. Joshi, and D. L. Blackburn: “Three-dimensional study of combined conduction, radiation, and natural convection from discrete heat sources in a horizontal narrow-aspect-ratio enclosure,” ASME Journal of Heat Transfer, vol. 121, no. 4, pp. 992–1001, 1999. 71. Velusamy, K., T. Sundararajan, and K. N. Seetharamu: “Interaction effects between surface radiation and turbulent natural convection in square and rectangular enclosures,” ASME Journal of Heat Transfer, vol. 123, no. 6, pp. 1062– 1070, 2001. 72. Vader, D. T., R. Viskanta, and F. P. Incropera: “Design and testing of a high-temperature emissometer for porous and particulate dielectrics,” Review of Scientific Instruments, vol. 57, no. 1, pp. 87–93, 1986. 73. Sikka, K. K.: “High temperature normal spectral emittance of silicon carbide based materials,” M.S. thesis, The Pennsylvania State University, University Park, PA, 1991.

Problems 9.1 A satellite shaped like a sphere (R = 1 m) has a gray-diffuse surface qsol coating with ǫs = 0.3 and is fitted with a long, thin, cylindrical antenna, as shown in the adjacent sketch. The antenna is a specular reflector with ǫa = 0.1, ka = 100 W/m K, and d = 1 cm. Satellite and antenna are R exposed to solar radiation of strength qsol = 1300 W/m2 from a direction normal to the antenna. Assuming that the satellite produces heat at ∋ s ∋a d a rate of 4 kW and—due to a high-conductivity shell—is essentially isothermal, determine the equilibrium temperature distribution along the antenna. (Hint: Use the fact that d ≪ R not only for conduction calculations, but also for the calculation of view factors.) 9.2 A long, thin, cylindrical needle (L ≫ D) is attached perpendicularly to a large, isothermal base plate at T = Tb = const. The base plate is gray and diffuse (ǫb = αb ), while the needle is nongray and diffuse (ǫ , α). The needle exchanges heat by convection and radiation with a large, isothermal environment at T∞ .

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(a) Neglecting heat losses from the free tip of the needle, formulate the problem for the calculation of needle temperature distribution, total heat loss, and fin efficiency. (b) Implement the solution numerically for L = 1 m, D = 1 cm, k = 10 W/m K, h = 40 W/m2 K, ǫ = 0.8, α = 0.4, ǫb = 0.8, Tb = 1000 K, T∞ = 300 K. 9.3 In the emissometer of Vader and coworkers [72] and Sikka [73], Refractory the sample is kept inside a long silicon carbide tube that in turn, is brick inside a furnace, as shown in the sketch. The furnace is heated with a number of SiC heating elements, providing a uniform flux over a 120 cm Heating 45 cm length as shown. Assume that there is no heat loss through element the refractory brick or the bottom of the furnace, that the inside heat transfer coefficient for free convection (with air at 600◦ C) is Sample 30 cm 10 W/m2 K, that the silicon carbide tube is gray-diffuse (ǫ = 0.9, k = 100 W/m K), and that the sample temperature is equal to the SiC 15 cm tube temperature at the same height. What must be the steadyRefractory SiC tube state power load on the furnace to maintain a sample temperature 15 cm of 1000◦ C? In this configuration a detector receiving radiation from a small center spot of the sample is supposedly getting the same 0.5 cm 5 cm amount as from a blackbody at 1000◦ C (cf. Table 5.1). What is the actual emittance sensed by the detector, i.e., what systematic error is caused by this near-blackbody, if the sample is gray and diffuse with ǫs = 0.5? 9.4 A thermocouple with a 0.5 mm diameter bead is used to measure the local temperature of a hot, radiatively nonparticipating gas flowing through an isothermal, gray-diffuse tube (Tw = 300 K, ǫw = 0.8). The thermocouple is a diffuse emitter/specular reflector with ǫb = 0.5, and the heat transfer coefficient between bead and gas is 30 W/m2 K. (a) Determine the thermocouple error as a function of gas temperature (i.e., |Tb − T1 | vs. T1 ). (b) In order to reduce the error, a radiation shield in the form of a thin, stainless-steel cylinder (ǫ = 0.1, R = 2 mm, L = 20 mm) is placed over the thermocouple. This also reduces the heat transfer coefficient between bead and gas to 15 W/m2 K, which is equal to the heat transfer coefficient on the inside of the shield. On the outside of the cylinder the heat transfer coefficient is 30 W/m2 K. Determine error vs. gas temperature for this case. To simplify the problem, you may make the following assumptions: (i) the leads of the thermocouple may be neglected, (ii) the shield is very long as far as the radiation analysis is concerned, and (iii) the shield reflects diffusely. 9.5 Repeat Problem 5.36 for the case in which a radiatively nonparticipating, stationary gas (k = 0.04 W/m K) is filling the 1 cm thick gap between surface and shield.

CHAPTER

10 THE RADIATIVE TRANSFER EQUATION IN PARTICIPATING MEDIA (RTE)

10.1

INTRODUCTION

In previous chapters we have looked at radiative transfer between surfaces that were separated by vacuum or by a transparent (“radiatively nonparticipating”) medium. However, in many engineering applications the interaction of thermal radiation with an absorbing, emitting, and scattering (“radiatively participating”) medium must be accounted for. Examples in the heat transfer area are the burning of any fuel (be it gaseous, liquid,or solid; be it for power production, within fires, within explosions, etc.), rocket propulsion, hypersonic shock layers, ablation systems on reentry vehicles, nuclear explosions, plasmas in fusion reactors, and many more. In the present chapter we shall develop the general relationships that govern the behavior of radiative heat transfer in the presence of an absorbing, emitting, and/or scattering medium. We shall begin by making a radiative energy balance, known as the radiative transfer equation, or RTE, which describes the radiative intensity field within the enclosure as a function of location (fixed by location vector r), direction (fixed by unit direction vector sˆ ) and spectral variable (wavenumber η).1 To obtain the net radiative heat flux crossing a surface element, we must sum the contributions of radiative energy irradiating the surface from all possible directions and for all possible wavenumbers. Therefore, integrating the radiative transfer equation over all directions and wavenumbers leads to a conservation of radiative energy statement applied to an infinitesimal volume. Finally, this will be combined with a balance for all types of energy (including conduction and convection), leading to the Overall Conservation of Energy equation. In the following three chapters we shall deal with the radiation properties of participating media, i.e., with how a substance can absorb, emit, and scatter thermal radiation. In Chapter 11 we discuss how a molecular gas can absorb and emit photons by changing its energy states, how to predict the radiation properties, and how to measure them experimentally. Chapter 12 is concerned with how small particles interact with electromagnetic waves—how they absorb, 1 In our discussion of surface radiative transport we have used wavelength λ as the spectral variable throughout, largely to conform with the majority of other publications. However, for gases, frequency ν or wavenumber η are considerably more convenient to use. Again, to conform with the majority of the literature, we shall use wavenumber throughout this part.

279

280

10 THE RADIATIVE TRANSFER EQUATION IN PARTICIPATING MEDIA (RTE)

Particles Scattered photons

dA Photons

Absorbed photons

s

Transmitted photons ds

FIGURE 10-1 Attenuation of radiative intensity by absorption and scattering.

emit, and scatter radiative energy. Again, theoretical as well as experimental methods are covered. Finally, in Chapter 13 a very brief account is given of the radiation properties of solids and liquids that allow electromagnetic waves of certain wavelengths to penetrate into them for appreciable distances, known as semitransparent media.

10.2 ATTENUATION BY ABSORPTION AND SCATTERING If the medium through which radiative energy travels is “participating,” then any incident beam will be attenuated by absorption and scattering while it travels through the medium, as schematically shown in Fig. 10-1. In the following we shall develop expressions for this attenuation for a light beam which travels within a pencil of rays into the direction sˆ . The present discussion will be limited to media with constant refractive index, i.e., media through which electromagnetic waves travel along straight lines [while a varying refractive index will bend the ray, as shown by Snell’s law, equation (2.72), for an abrupt change]. It is further assumed that the medium is stationary (as compared to the speed of light), that it is nonpolarizing,and that it is (for most of the discussion) at local thermodynamic equilibrium (LTE).

Absorption The absolute amount of absorption has been observed to be directly proportional to the magnitude of the incident energy as well as the distance the beam travels through the medium. Thus, we may write, (dIη ) abs = −κη Iη ds, (10.1) where the proportionality constant κη is known as the (linear) absorption coefficient, and the negative sign has been introduced since the intensity decreases. As will be discussed in the following chapter, the absorption of radiation in molecular gases depends also on the number of receptive molecules per unit volume, so that some researchers use a mass absorption coefficient or a pressure absorption coefficient, defined by (dIη ) abs = −κρη Iη ρ ds = −κpη Iη p ds.

(10.2)

10.3 AUGMENTATION BY EMISSION AND SCATTERING

281

The subscripts ρ and p are used here only to demonstrate the differences between the coefficients. The reader of scientific literature often must rely on the physical units to determine the coefficient used. Integration of equation (10.1) over a geometric path s results in  Rs  Iη (s) = Iη (0) exp − 0 κη ds = Iη (0) e−τη , (10.3)

where

τη =

Z

s

κη ds

(10.4)

0

is the optical thickness (for absorption) through which the beam has traveled and Iη (0) is the intensity entering the medium at s = 0. Note that the (linear) absorption coefficient is the inverse of the mean free path for a photon until it undergoes absorption. One may also define an absorptivity for the participating medium (for a given path within the medium) as αη ≡

Iη (0) − Iη (s) Iη (0)

= 1 − e−τη .

(10.5)

Scattering Attenuation by scattering, or “out-scattering” (away from the direction under consideration), is very similar to absorption, i.e., a part of the incoming intensity is removed from the direction of propagation, sˆ . The only difference between the two phenomena is that absorbed energy is converted into internal energy, while scattered energy is simply redirected and appears as augmentation along another direction (discussed in the next section), also known as “inscattering.” Thus, we may write (dIη ) sca = −σsη Iη ds, (10.6) where the proportionality constant σsη is the (linear) scattering coefficient for scattering from the pencil of rays under consideration into all other directions. Again, scattering coefficients based on density or pressure may be defined. It is also possible to define an optical thickness for scattering, where the scattering coefficient is the inverse of the mean free path for scattering.

Total Attenuation The total attenuation of the intensity in a pencil of rays by both absorption and scattering is known as extinction. Thus, an extinction coefficient is defined2 as βη = κη + σsη . The optical distance based on extinction is defined as Z s βη ds. τη =

(10.7)

(10.8)

0

As for absorption and scattering, the extinction coefficient is sometimes based on density or pressure.

10.3 AUGMENTATION BY EMISSION AND SCATTERING A light beam traveling through a participating medium in the direction of sˆ loses energy by absorption and by scattering away from the direction of travel. But at the same time it also gains energy by emission, as well as by scattering from other directions into the direction of travel sˆ . 2 Care must be taken to distinguish the dimensional extinction coefficient βη from the absorptive index, i.e., the imaginary part of the index of refraction complex k (sometimes referred to in the literature as the “extinction coefficient”).

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10 THE RADIATIVE TRANSFER EQUATION IN PARTICIPATING MEDIA (RTE)

ds

si

d Ωi dA dΩ

s FIGURE 10-2 Redirection of radiative intensity by scattering.

Emission The rate of emission from a volume element will be proportional to the magnitude of the volume. Therefore, the emitted intensity (which is the rate of emitted energy per unit area) along any path again must be proportional to the length of the path, and it must be proportional to the local energy content in the medium. Thus, (dIη ) em = jη ds,

(10.9)

where jη is termed the emission coefficient. Since, at local thermodynamic equilibrium (LTE), the intensity everywhere must be equal to the blackbody intensity, it will be shown in Chapter 11, equation (11.22), that jη = κη Ibη and (dIη ) em = κη Ibη ds, (10.10) that is, at LTE the proportionality constant for emission is the same as for absorption. Similar to absorptivity, one may also define an emissivity of an isothermal medium as the amount of energy emitted over a certain path s that escapes into a given direction (without having been absorbed between point of emission and point of exit), as compared to the maximum possible. Combining equations (10.1) and (10.10) gives the complete radiative transfer equation for an absorbing–emitting (but not scattering) medium as dIη ds

= κη (Ibη − Iη ),

(10.11)

where the first term of the right-hand side is augmentation due to emission and the second term is attenuation due to absorption. The solution to the radiative transfer equation for an isothermal gas layer of thickness s is  Iη (s) = Iη (0) e−τη + Ibη 1 − e−τη , (10.12)

where the optical distance has been defined in equation (10.4). If only emission is considered, Iη (0) = 0, and the emissivity is defined as ǫη = Iη (s)/Ibη = 1 − e−τη ,

(10.13)

which, as is the case with surface radiation, is identical to the expression for absorptivity.

Scattering Augmentation due to scattering, or “in-scattering,” has contributions from all directions and, therefore, must be calculated by integration over all solid angles. Consider the radiative heat flux impinging on a volume element dV = dA ds, from an infinitesimal pencil of rays in the direction sˆ i as depicted in Fig. 10-2. Recalling the definition for radiative intensity as energy flux per unit area normal to the rays, per unit solid angle, and per unit wavenumber interval,

283

10.4 THE RADIATIVE TRANSFER EQUATION

dA

s

s s s + ds

FIGURE 10-3 Pencil of rays for radiative energy balance.

one may calculate the spectral radiative heat flux impinging on dA from within the solid angle dΩ i as Iη (ˆs i )(dA sˆ i · sˆ ) dΩ i dη. This flux travels through dV for a distance ds/ˆs i · sˆ . Therefore, the total amount of energy scattered away from sˆ i is, according to equation (10.6), !   ds σsη Iη (ˆs i )(dA sˆ i · sˆ ) dΩi dη (10.14) = σsη Iη (ˆs i ) dA dΩ i dη ds. sˆ i · sˆ Of this amount, the fraction Φη (ˆs i , sˆ ) dΩ/4π is scattered into the cone dΩ around the direction sˆ . The function Φη is called the scattering phase function and describes the probability that a ray from one direction, sˆ i , will be scattered into a certain other direction, sˆ . The constant 4π is arbitrary and is included for convenience [see equation (10.17) below]. The amount of energy flux from the cone dΩ i scattered into the cone dΩ is then Φη (ˆs i , sˆ )

dΩ. (10.15) 4π We can now calculate the energy flux scattered into the direction sˆ from all incoming directions sˆ i by integrating: Z   dΩ σsη Iη (ˆs i ) dA dΩ i dη ds Φη (ˆs i , sˆ ) , dIη (ˆs) dA dΩ dη = sca 4π 4π or Z   σsη dIη (ˆs) = ds (10.16) Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i . sca 4π 4π σsη Iη (ˆs i ) dA dΩ i dη ds

Returning to equation (10.15), we find that the amount of energy flux scattered from dΩ i into all directions is Z 1 Φη (ˆs i , sˆ ) dΩ, σsη Iη (ˆs i ) dA dΩ i dη ds 4π 4π which must be equal to the amount in equation (10.14). We conclude that Z 1 Φη (ˆs i , sˆ ) dΩ ≡ 1. (10.17) 4π 4π Therefore, if Φη = const, i.e., if equal amounts of energy are scattered into all directions (called isotropic scattering), then Φη ≡ 1. This is the reason for the inclusion of the factor 4π.

10.4 THE RADIATIVE TRANSFER EQUATION We can now make an energy balance on the radiative energy traveling in the direction of sˆ within a small pencil of rays as shown in Fig. 10-3. The change in intensity is found by summing the contributions from emission, absorption, scattering away from the direction sˆ , and scattering into the direction of sˆ , from equations (10.1), (10.6), (10.9), and (10.16) as Iη (s+ds, sˆ , t+dt) − Iη (s, sˆ , t) = jη (s, t) ds − κη Iη (s, sˆ , t) ds − σsη Iη (s, sˆ , t) ds +

σsη 4π

Z

Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i ds. 4π

(10.18)

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10 THE RADIATIVE TRANSFER EQUATION IN PARTICIPATING MEDIA (RTE)

Iη (s, s)

s

s, τη

s´, τ ´η Iη 0 ( s) 0

FIGURE 10-4 Enclosure for derivation of radiative transfer equation.

This equation is Lagrangian in nature, i.e., we are following a ray from s to s+ds; since the ray travels at the speed of light c, ds and dt are related through ds = c dt. The outgoing intensity may be developed into a truncated Taylor series, or Iη (s+ds, sˆ , t+dt) = Iη (s, sˆ , t) + dt

∂Iη ∂t

+ ds

∂Iη ∂s

,

(10.19)

so that equation (10.18) may be simplified to σsη 1 ∂Iη ∂Iη + = jη − κη Iη − σsη Iη + c ∂t 4π ∂s

Z

Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i .

(10.20)



In this radiative transfer equation (commonly abbreviated as RTE), or equation of transfer, all quantities may vary with location in space, time, and wavenumber, while the intensity and the phase function also depend on direction sˆ (and sˆ i ). Only the directional dependence, and only whenever necessary, has been explicitly indicated in this and the following equations, to simplify notation. As indicated earlier, the development of this equation is subject to a number of simplifying assumptions, viz., the medium is homogeneous and at rest (as compared to the speed of light), the medium is nonpolarizing and the state of polarization is neglected, and the medium has a constant index of refraction. An elaborate discussion of these limitations has been given by Viskanta and Menguc ¨ ¸ [1]. The RTE for a medium with varying refractive index has been given, e.g., by Pomraning [2], and some recent developments have been reported by Ben-Abdallah [3]. Equation (10.20) is valid anywhere inside an arbitrary enclosure. Its solution requires knowledge of the intensity for each direction at some location s, usually the intensity entering the medium through or from the enclosure boundary into the direction of sˆ , as indicated in Fig. 104. We have not yet brought the radiative transfer equation into its most compact form so that the four different contributions to the change of intensity may be clearly identified. Equation (10.20) is the transient form of the radiative transfer equation, valid at local thermodynamic equilibrium as well as nonequilibrium. Over the last few years, primarily due to the development of short-pulsed lasers, with pulse durations in the ps or fs range, transient radiation phenomena have been becoming of increasing importance [4]. However, for the vast majority of engineering applications, the speed of light is so large compared to local time and length scales that the first term in equation (10.20) may

10.5 FORMAL SOLUTION TO THE RADIATIVE TRANSFER EQUATION

285

be neglected. There are also several important applications that take place at thermodynamic nonequilibrium, such as the strong nonequlibrium radiation hitting a hypersonic spacecraft entering Earth’s atmosphere [5] (creating a high-temperature plasma ahead of it; cf. Fig. 11-7). Nevertheless, most engineering applications are at local thermodynamic equilibrium. We have presented here the full equation for completeness, but will omit the transient and nonequilibrium terms during the remainder of this book (with the exception of a very brief discussion of nonequlibrium properties in Chapter 11, and a somewhat more detailed consideration of transient radiation in Chapter 19). After introducing the extinction coefficient defined in equation (10.7), one may restate equation (10.20) in its equilibrium, quasi-steady form as dIη ds

= sˆ · ∇Iη = κη Ibη − βη Iη +

σsη 4π

Z

Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i ,

(10.21)



where the intensity gradient has been converted into a total derivative since we assume the process to be quasi-steady. The radiative transfer equation is often rewritten in terms of nondimensional optical coordinates (see Fig. 10-4), Z s Z s τη = (κη + σsη ) ds = βη ds, (10.22) 0

0

and the single scattering albedo, first defined in equation (1.58) as ωη ≡ leading to

dIη dτη

σsη κη + σsη

= −Iη + (1 − ωη )Ibη +

ωη 4π

= Z

σsη βη

,

Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i .

(10.23)

(10.24)



The last two terms in equation (10.24) are often combined and are then known as the source function for radiative intensity, Z ωη Sη (τη , sˆ ) = (1 − ωη )Ibη + Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩi . (10.25) 4π 4π Equation (10.24) then assumes the deceptively simple form of dIη dτη

+ Iη = Sη (τη , sˆ ),

(10.26)

which is, of course, an integro-differential equation (in space, and in two directional coordinates with local origin). Furthermore, the Planck function Ibη is generally not known and must be found by considering the overall energy equation (adding derivatives in the three space coordinates and integrations over two more directional coordinates and the wavenumber spectrum).

10.5 FORMAL SOLUTION TO THE RADIATIVE TRANSFER EQUATION If the source function is known (or assumed known), equation (10.26) can be formally integrated by the use of an integrating factor. Thus, multiplying through by eτη results in d  τη  Iη e = Sη (τη , sˆ ) eτη , dτη

(10.27)

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10 THE RADIATIVE TRANSFER EQUATION IN PARTICIPATING MEDIA (RTE)

τs τ ´s

0

θ

θ

s

θ R, τR

FIGURE 10-5 Isothermal sphere for Example 10.1.

which may be integrated from a point s′ = 0 at the wall to a point s′ = s inside the medium (see Fig. 10-4), so that Z τη

Iη (τη ) = Iη (0) e−τη +

0



Sη (τ′η , sˆ ) e−(τη −τη ) dτ′η ,

(10.28)

where τ′η is the optical coordinate at s = s′ . Physically, one can readily appreciate that the first term on the right-hand side of equation (10.28) is the contribution to the local intensity by the intensity entering the enclosure at s = 0, which decays exponentially due to extinction over the optical distance τη . The integrand of the second term, Sη (τ′η ) dτ′η , on the other hand, is the contribution from the local emission at τ′η , attenuated exponentially by self-extinction over the optical distance between the emission point and the point under consideration, τη − τ′η . The integral, finally, sums all the contributions over the entire emission path. Equation (10.28) is a third-order integral equation in intensity Iη . The integral over the source function must be carried out over the optical coordinate (for all directions), while the source function itself is also an integral over a set of direction coordinates (with varying local origin) containing the unknown intensity. Furthermore, usually the temperature and, therefore, the blackbody intensity are not known and must be found in conjunction with overall conservation of energy. There are, however, a few cases for which the radiative transfer equation becomes considerably simplified.

Nonscattering Medium If the medium only absorbs and emits, the source function reduces to the local blackbody intensity, and Z τη ′ (10.29) Iη (τη ) = Iη (0) e−τη + Ibη (τ′η ) e−(τη −τη ) dτ′η . 0

This equation is an explicit expression for the radiation intensity if the temperature field is known. However, generally the temperature is not known and must be found in conjunction with overall conservation of energy. Example 10.1. What is the spectral intensity emanating from an isothermal sphere bounded by vacuum or a cold black wall? Solution Because of the symmetry in this problem, the intensity emanating from the sphere surface is only a function of the exit angle. Examining Fig. 10-5, we see that equation (10.29) reduces to Z τs ′ Ibη (τ′s ) e−(τs −τs ) dτ′s . Iη (τR , θ) = 0

10.5 FORMAL SOLUTION TO THE RADIATIVE TRANSFER EQUATION

287

But for a sphere τs = 2τR cos θ, regardless of the azimuthal angle. Therefore, with Ibη (τ′s ) = Ibη = const, the desired intensity turns out to be   ′ 2τR cos θ = Ibη 1 − e−2τR cos θ . Iη (τR , θ) = Ibη e−(2τR cos θ−τs ) 0

Thus, for τR ≫ 1 the isothermal sphere emits equally into all directions, like a black surface at the same temperature.

The Cold Medium If the temperature of the medium is so low that the blackbody intensity at that temperature is small as compared with incident intensity, then the radiative transfer equation is decoupled from other modes of heat transfer. However, the governing equation remains a third-order integral equation, namely, Z τη Z ωη ′ −τη Iη (τη , sˆ ) = Iη (0) e + Iη (τ′η , sˆ i ) Φη (ˆs i , sˆ ) dΩ i e−(τη −τη ) dτ′η . (10.30) 4π 4π 0 If the scattering is isotropic, or Φ ≡ 1, the directional integration in equation (10.30) may be carried out, so that Z τη 1 ′ Iη (τη , sˆ ) = Iη (0) e−τη + ωη Gη (τ′η ) e−(τη −τη ) dτ′η , (10.31) 4π 0 Z where Gη (τ) ≡ (10.32) Iη (τ′η , sˆ i ) dΩ i 4π

is known as the incident radiation function (since it is the total intensity impinging on a point from all sides). The problem is then much simplified since it is only necessary to find a solution for G [by direction-integrating equation (10.31)] rather than determining the direction-dependent intensity.

Purely Scattering Medium If the medium scatters radiation, but does not absorb or emit, then the radiative transfer is again decoupled from other heat transfer modes. In this case ωη ≡ 1, and the radiative transfer equation reduces to a form essentially identical to equation (10.30), i.e., Z τη Z 1 ′ Iη (τη , sˆ ) = Iη (0) e−τη + Iη (τ′η , sˆ i ) Φη (ˆs i , sˆ ) dΩ i e−(τη −τη ) dτ′η . (10.33) 4π 0 4π Again, for isotropic scattering, this equation may be simplified by introducing the incident radiation, so that Z τη 1 ′ −τη Iη (τη , sˆ ) = Iη (0) e + Gη (τ′η , sˆ ) e−(τη −τη ) dτ′η . (10.34) 4π 0 Example 10.2. A large isothermal black plate is covered with a thin layer of isotropically scattering, nonabsorbing (and, therefore, nonemitting) material with unity index of refraction. Assuming that the layer is so thin that any ray emitted from the plate is scattered at most once before leaving the scattering layer, estimate the radiative intensity above the layer in the direction normal to the plate. Solution The exiting intensity in the normal direction (see Fig. 10-6) may be calculated from equation (10.34) by retaining only terms of order τη or higher (since τη ≪ 1). This process leads to e−τη = 1 − τη + O(τ2η ),

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10 THE RADIATIVE TRANSFER EQUATION IN PARTICIPATING MEDIA (RTE)

Inη (τη)

Ibη

τη´

τη FIGURE 10-6 Geometry for Example 10.2. ′

G(τ′η ) = G(τη ) + O(τη ) (radiation to be scattered arrives unattenuated at a point), and e−(τη −τη ) = 1 − O(τη ) (scattered radiation will leave the medium without further attenuation), so that Inη = Ibη (1 − τη ) +

1 Gη τη + O(τ2η ), 4π

where the intensity emanating from the plate is known since the plate is black. The incident radiation at any point is due to unattenuated emission from the bottom plate arriving from the lower 2π solid angles, and nothing coming from the top 2π solid angles, i.e., Gη ≈ 2πIbη and  τη  1 + O(τ2η ). Inη = Ibη (1 − τη ) + Ibη τη + O(τ2η ) = Ibη 1 − 2 2 Physically this result tells us that the emission into the normal direction is attenuated by the fraction τη (scattered away from the normal direction), and augmented by the fraction τη /2 (scattered into the normal direction): Since scattering is isotropic, exactly half of the attenuation is scattered upward and half downward; the latter is then absorbed by the emitting plate. Thus, the scattering layer acts as a heat shield for the hot plate.

10.6 BOUNDARY CONDITIONS FOR THE RADIATIVE TRANSFER EQUATION The radiative transfer equation in its quasi-steady form, equation (10.21), is a first-order differential equation in intensity (for a fixed direction sˆ ). As such, the equation requires knowledge of the radiative intensity at a single point in space, into the direction of sˆ . Generally, the point where the intensity can be specified independently lies on the surface of an enclosure surrounding the participating medium, as indicated by the formal solution in equation (10.28). This intensity, leaving a wall into a specified direction, may be determined by the methods given in Chapter 5 (diffusely emitting and reflecting surfaces), Chapter 6 (diffusely emitting and specularly reflecting surfaces) and Chapter 7 (surfaces with arbitrary characteristics).

Diffusely Emitting and Reflecting Opaque Surfaces For a surface that emits and reflects diffusely, the exiting intensity is independent of direction. Therefore, at a point rw on the surface, from equations (5.18) and (5.19), I(rw , sˆ ) = I(rw ) = J(rw )/π = ǫ(rw ) Ib (rw ) + ρ(rw ) H(rw )/π,

(10.35)

where H(rw ) is the hemispherical irradiation (i.e., incoming radiative heat flux) defined by equation (3.41), leading to Z ρ(rw ) I(rw , sˆ ′ ) |nˆ · sˆ ′ | dΩ′ , (10.36) I(rw , sˆ ) = ǫ(rw ) Ib (rw ) + π n·ˆ ˆ s′ n2 , refraction in Medium 2 is away from the surface normal, i.e., θ2 > θ1 , and there is a critical angle θ1 = θc , as given by equation (2.100), at which θ2 = 90◦ and for

10.7 RADIATION ENERGY DENSITY

Medium 1 , M 1 ; N 1

291

Medium 2 , M 2 ; N 2 < N 1 s2I

s1I

d

D

q1

qc

q1

q2 q2

s1R ,s2T

s2R ,s1T FIGURE 10-9 Intensities leaving an interface between two semitransparent media with different refractive indices (shown for n1 > n2 ).

larger θ1 there will be total internal reflection, and nothing is transmitted into Medium 2:   n2 : ρ12 = 1; Iν2 (θ2 ) = 0. (10.45) θ1 > θc = sin−1 n1 This is indicated in Fig. 10-9 by showing several additional incident directions (with thin dashed lines and open arrows), together with their transmitted (for θ1 < θc only) and reflected directions. Employing equations (10.44) and (10.45), we can now make a full energy balance for the interface, comprising intensity coming in from inside Medium 1, Iν1i (θ1 ), the fraction of it that is reflected, Iν1r (θ1 ) (with specular reflection angle θr = θ1 ), and the fraction transmitted into Medium 2, Iν1t (θ2 ), along with similar contributions from intensity striking the interface from inside Medium 2, as depicted in Fig. 10-9:  n2 2 Iν1i (θ1 ), n1  2 n1 Iν1 (θ1 ) = ρ12 Iν1i (θ1 ) + Iν2t (θ1 ) = ρ12 Iν1i (θ1 ) + (1 − ρ21 ) Iν2i (θ2 ), n2

Iν2 (θ2 ) = ρ21 Iν2i (θ2 ) + Iν1t (θ2 ) = ρ21 Iν2i (θ2 ) + (1 − ρ12 )



where, from equation (2.96),  "   #   n1 cos θ1 − n2 cos θ2 2 1 n1 cos θ2 − n2 cos θ1 2    + ,  2 n1 cos θ2 + n2 cos θ1 n1 cos1 +n2 cos θ2 ρ12 = ρ21 =      1

θ1 < θc ,

(10.46a) (10.46b)

(10.47)

θ1 ≥ θc .

The intensity entering the optically less dense Medium 2 from the interface, Iν2 (θ2 ), will have a transmitted contribution from Medium 1 for all values of θ2 (but coming from within a cone with opening angle θc ). Intensity entering Medium 1, Iν1 (θ1 ), on the other hand, will have a transmitted component from Medium 2 only if θ1 < θc .

10.7

RADIATION ENERGY DENSITY

A volume element inside an enclosure is irradiated from all directions and, at any instant in time t, contains a certain amount of radiative energy in the form of photons. Consider, for

292

10 THE RADIATIVE TRANSFER EQUATION IN PARTICIPATING MEDIA (RTE)

example, an element dV = dA ds irradiated perpendicularly to dA with intensity Iη (ˆs) as shown in Fig. 10-3. Therefore, per unit time radiative energy in the amount of Iη (ˆs) dΩ dA enters dV. From the development in Chapter 1, equation (1.48), we see that this energy remains inside dV for a duration of dt = ds/c, before exiting at the other side. Thus, due to irradiation from a single direction, the volume contains the amount of radiative energy Iη (ˆs) dΩ dA ds/c = Iη (ˆs) dΩ dV/c at any instant in time. Adding the contributions from all possible directions, we find the total radiative energy stored within dV is uη dV, where uη is the spectral radiation energy density Z 1 (10.48) Iη (ˆs) dΩ. uη ≡ c 4π Integration over the spectrum gives the total radiation energy density, Z Z ∞ Z Z ∞ 1 1 I(ˆs) dΩ. Iη (ˆs) dη dΩ = uη dη = u= c 4π 0 c 4π 0

(10.49)

Although the radiation energy density is a very basic quantity akin to internal energy for energy stored within matter, it is not widely used by heat transfer engineers. Instead, it is common practice to employ the incident radiation Gη , which is related to the energy density through Z Iη (ˆs) dΩ = cuη ; G = cu. (10.50) Gη ≡ 4π

10.8

RADIATIVE HEAT FLUX

The spectral radiative heat flux onto a surface element has been expressed in terms of incident and outgoing intensity in equation (1.39) as Z qη · nˆ = Iη nˆ · sˆ dΩ. (10.51) 4π

This relationship also holds, of course, for a hypothetical (i.e., totally transmissive) surface element placed arbitrarily inside an enclosure. Removing the surface normal from equation (1.39), we obtain the definition for the spectral, radiative heat flux vector inside a participating medium. To obtain the total radiative heat flux, equation (10.51) needs to be integrated over the spectrum, and Z ∞ Z ∞Z q= qη dη = Iη (ˆs) sˆ dΩ dη. (10.52) 0

0



Depending on the coordinate system used, or the surface being described, the radiative heat flux vector may be separated into its coordinate components, for example qx , q y , and qz (for a Cartesian coordinate system), or into components normal and tangential to a surface, and so on. Example 10.3. Evaluate the total heat loss from an isothermal spherical medium bounded by vacuum, assuming that κη = const (i.e., does not vary with location, temperature, or wavenumber). Solution Here we are dealing with a spherical coordinate system, and we are interested in the radial component of the radiative heat flux (the other two being equal to zero by symmetry). We saw in Example 10.1 that the intensity emanating from the sphere is   π Iη (τR , θ) = Ibη 1 − e−2τR cos θ , 0 ≤ θ ≤ , 2 where θ is measured from the surface normal pointing away from the sphere (Fig. 10-5). Since the sphere is bounded by vacuum, there is no incoming radiation and Iη (τR , θ) = 0,

π ≤ θ ≤ π. 2

10.9 DIVERGENCE OF THE RADIATIVE HEAT FLUX

293

Therefore, from equation (10.52), Z ∞ Z 2π Z π q(τR ) = Iη (τR , θ) cos θ sin θ dθ dψ dη 0

Z

0 0 ∞ Z π/2

  Ibη 1 − e−2τR cos θ cos θ sin θ dθ dη 0 0 ( ( ) ) i i 1 h 1 h = πIb 1 − 2 1 − (1 + 2τR ) e−2τR = n2 σT 4 1 − 2 1 − (1 + 2τR ) e−2τR , 2τR 2τR = 2π

where n is the refractive index of the medium (usually n ≈ 1 for gases, but n > 1 for semitransparent liquids and solids). As discussed in the previous example, if τR → ∞ the heat flux approaches the same value as the one from a black surface.

If the sphere in the last example is optically thin τR ≪ 1 (i.e., the medium emits radiative energy, but does not absorb any of the emitted energy), then the total heat loss (total emission) from the sphere is Q = 4πR2 q = 4πR2 × 43 τR n2 σT 4 = 4κn2 σT 4 V. (10.53) This result may be generalized to govern emission from any isothermal volume V without self-absorption, or Qemission = 4κn2 σT 4 V. (10.54)

10.9 DIVERGENCE OF THE RADIATIVE HEAT FLUX While the heat transfer engineer is interested in the radiative heat flux, this interest usually holds true only for fluxes at physical boundaries. Inside the medium, on the other hand, we need to know how much net radiative energy is deposited into (or withdrawn from) each volume element. Thus, making a radiative energy balance on an infinitesimal volume dV = dx dy dz as shown in Fig. 10-10, we have       radiative energy rad. energy generated rad. energy destroyed  stored in dV   (emitted) by dV   (absorbed) by dV    +   −         per unit time per unit time per unit time

   flux in at x − flux out at x + dx  + flux in at y − flux out at y + dy =   .   + flux in at z − flux out at z + dz

The right-hand side may be written in mathematical form as  ! q(x) dy dz − q(x + dx) dy dz    ∂q ∂q ∂q  + q(y) dx dz − q(y + dy) dx dz  =− + + dx dy dz = −∇ · q dV.   ∂x ∂y ∂z + q(z) dx dy − q(z + dz) dx dy 

Thus, within the overall energy equation, it is the divergence of the radiative heat flux that is of interest inside the participating medium.4 We have already established an energy balance for thermal radiation, the radiative transfer equation [for example, equation (10.21)], Z dIη σsη = sˆ · ∇Iη = κη Ibη − βη Iη (ˆs) + Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i , (10.55) ds 4π 4π 4 For simplicity, this equation was derived for a Cartesian coordinate system but the result holds, of course, for any arbitrary coordinate system.

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10 THE RADIATIVE TRANSFER EQUATION IN PARTICIPATING MEDIA (RTE)

qz+dz

qy+dy

qx

qx+dx dy

dz

dx

FIGURE 10-10 Control volume for derivation of divergence of radiative heat flux.

qy qz

which is a radiation balance for an infinitesimal pencil of rays. Thus, in order to get a volume balance, we integrate this equation over all solid angles, or Z Z Z Z Z σsη sˆ · ∇Iη dΩ = κη Ibη dΩ − βη Iη (ˆs) dΩ + Iη (ˆs i ) Φη (ˆs i , sˆ ) dΩ i dΩ, (10.56) 4π 4π 4π 4π 4π 4π and ∇·

Z

Iη sˆ dΩ = 4πκη Ibη − 4π

Z

βη Iη (ˆs) dΩ + 4π

σsη 4π

Z

Iη (ˆs i ) 4π

Z



! Φη (ˆs i , sˆ ) dΩ dΩ i .

(10.57)

On the left side of equation (10.57) the integral and the direction vector were taken into the gradient since direction and space coordinates are all independent from one another.5 The expression inside the operator is now, of course, the spectral radiative heat flux. On the right side of equation (10.57) the order of integration has been changed, applying the Ω-integration to the only part depending on it, the scattering phase function Φη . This last integration can be carried out using equation (10.17), leading to Z Z ∇ · qη = 4πκη Ibη − βη Iη (ˆs) dΩ + σsη Iη (ˆsi ) dΩ i . (10.58) 4π



Since Ω and Ω i are dummy arguments for integration over all solid angles, the last two terms can be pulled together, using κη = βη − σsη : ! Z   (10.59) Iη dΩ = κη 4πIbη − Gη . ∇ · qη = κη 4πIbη − 4π

Equation (10.59) states that physically the net loss of radiative energy from a control volume is equal to emitted energy minus absorbed irradiation. This direction-integrated form of the radiative transfer equation no longer contains the scattering coefficient. This fact is not surprising since scattering only redirects the stream of photons; it does not affect the energy content of any given unit volume. Equation (10.59) is a spectral relationship, i.e., it gives the heat flux per unit wavenumber at a certain spectral position. If the divergence of the total heat flux is desired, the integration over the spectrum is carried out to give ! Z ∞ Z ∞ Z Z ∞   ∇·q=∇· (10.60) qη dη = κη 4πIbη − Iη dΩ dη = κη 4πIbη − Gη dη. 0

5

0



0

While this statement is always true, care must be taken in non-Cartesian coordinate systems: Although the direction vector is independent from space coordinates, the three components may be tied to locally defined unit vectors. For example, in a cylindrical coordinate system the direction vector is usually defined in terms of eˆ r and eˆ θ , which vary with r and θ.

10.10 INTEGRAL FORMULATION OF THE RADIATIVE TRANSFER EQUATION

Iη (r, s)

295

s

0

s´´

r

Iηw (rw ,s) r´ rw FIGURE 10-11 Enclosure for the derivation of the integral form of the radiative transfer equation.

0

Equation (10.60) is a statement of the conservation of radiative energy. For the special case of a gray medium (κη = κ = constant) this may be simplified to ! Z   4 ∇ · q = κ 4σT − I dΩ = κ 4σT 4 − G . (10.61) 4π

Example 10.4. Calculate the divergence of the total radiative heat flux at the center and at the surface of the gray, isothermal spherical medium in the previous example. Solution We already know the intensity at the surface of the sphere and, therefore, Z π/2  Z π  1 − e−2τR cos θ sin θ dθ sin θIη dθ = 2πIbη Gη (τR ) = 2π 0 0   −2τR cos θ π/2     πI e  = bη 2τR − 1 + e−2τR , = 2πIbη  1 −  2τR 0 τR and

 σT 4  2τR + 1 − e−2τR . R At the center of the sphere the intensity is easily evaluated as  Iη (0) = Ibη 1 − e−τR , ∇ · q(τR ) = κ (4πIb − G) =

(10.62)

and

so that

 Gη (0) = 4πIbη 1 − e−τR , ∇ · q(0) = κ4σT 4 e−τR .

(10.63)

The right-hand sides of equations (10.62) and (10.63) are radiative heat losses per unit time and volume, which must be made up for by a volumetric heat source if the sphere is to stay isothermal.

10.10 INTEGRAL FORMULATION OF THE RADIATIVE TRANSFER EQUATION In order to obtain incident radiation, radiative heat flux, or its divergence, it is sometimes desirable to use an integral formulation of the radiative transfer equation. We start with the formal solution, equation (10.28), but rewritten in terms of the vectors shown in Fig. 10-11,   R ′′ i Z s h R s s Iη (r, sˆ ) = Iwη (rw , sˆ ) exp − 0 βη ds′′ + Sη (r′ , sˆ ) exp − 0 βη ds′′ βη ds′′ , (10.64) 0

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10 THE RADIATIVE TRANSFER EQUATION IN PARTICIPATING MEDIA (RTE)

where s = |r − rw | and the direction of integration has been switched to go along s′′ (from point r toward the wall). From the definition of the incident radiation, equation (10.32), we have Z Z s Z   R ′′ i h Rs s ′′ Gη (r) = Sη (r′ , sˆ ) exp − 0 βη ds′′ βη ds′′ dΩ, (10.65) Iwη (rw , sˆ ) exp − 0 βη ds dΩ + 4π



0

with, from equation (1.26),  dA′′    ,    − r′ |2   |r dΩ =     nˆ · sˆ dAw    ,  |r − rw |2

inside volume, (10.66) at the wall,

where dA′′ is an infinitesimal area perpendicular to the integration path (and ds′′ ), such that dV = ds′′ dA′′ is an infinitesimal volume. Therefore, equation (10.65) may be rewritten as Z  β dV  R ′′ i ˆ · sˆ dAw Z h Rs η s ′ ′′ ′′ n Gη (r) = + S (r , s ˆ ) exp − β ds , Iwη (rw , sˆ ) exp − 0 βη ds η η 2 0 |r − rw | |r − r′ |2 Aw V (10.67) with the local unit direction vector found from sˆ =

r − r′ . |r − r′ |

(10.68)

The radiative flux (and any higher moment) can be determined similarly, after first multiplying equation (10.64) by sˆ , as Z  β sˆ dV  R ′′ h R s i (nˆ · sˆ )ˆs dAw Z η s ′ ′′ qη (r) = Iwη (rw , sˆ ) exp − 0 βη ds′′ + S (r , s ˆ ) exp − β ds . η η 0 |r − rw |2 |r − r′ |2 Aw V (10.69) For a nonscattering medium Sη = Ibη , and equation (10.67) is the explicit solution for incident radiation Gη , provided the temperature field is known, and if the walls are black. For isotropic scattering the source function depends only on Ibη (or temperature) and incident radiation. For such a case (and if the walls are black) equation (10.67) is a single, independent integral equation for the incident radiation; once Gη has been determined qη is found from equation (10.69). For reflecting walls and anisotropic scattering, equations (10.67) and (10.69) (and, perhaps, higher-order moments) must be solved simultaneously. Also, for a nonparticipating medium (βη = 0) with diffusely reflecting surfaces (Iw = J/π), equation (5.25) is readily recovered from equation (10.69); this is left as an exercise (Problem 10.15). Example 10.5. Repeat Example 10.3 using the integral formulation of the RTE. Solution In this simple problem with a cold, black (i.e., nonreflecting) wall with Iwη = 0, and in the absence of scattering with Sη = Ibη = const we can determine qη directly from equation (10.69) as Z ˆ · sˆ dV ′′ n , e−κη s qη (R) = −qη (rw ) · nˆ = −Ibη κη (s′′ )2 V where s′′ is the distance between any point inside the medium (at r′ ) and the chosen point on the wall, r = rw . It is tempting at this point to introduce a spherical coordinate system at the center of the sphere to evaluate the volume integral for qη ; however, this would lead to a very difficult integral. Instead, we introduce a spherical coordinate system at the chosen point at the wall, i.e., rw = 0 (point τs in Fig. 10-5). An arbitrary location inside the sphere can then be specified as ˆ r′ = −ˆss′′ = s′′ (cos ψ sin θ ˆı + sin ψ sin θ ˆ + cos θ k),

10.11 OVERALL ENERGY CONSERVATION

297

where kˆ = nˆ is pointing toward the center of the sphere and ˆı and ˆ are arbitrary (as long as they form a right-handed coordinate system). Then, with a maximum value for s′′ max = 2R cos θ, as given in Example 10.1, qη (R) = −Ibη κη = 2πIbη

Z

Z

2π ψ=0

Z

π/2 θ=0

Z

2R cos θ

e−κη s s′′ =0

′′

(− cos θ) sin θ dθ dψ(s′′ )2 ds′′ (s′′ )2

π/2

(1 − e−2τR cos θ ) cos θ sin θ dθ, θ=0

exactly as in Example 10.3.

10.11 OVERALL ENERGY CONSERVATION Thermal radiation is only one mode of transferring heat which, in general, must compete with conductive and convective heat transfer. Therefore, the temperature field must be determined through an energy conservation equation that incorporates all three modes of heat transfer. The radiation intensity, through emission and temperature-dependent properties, depends on the temperature field and, therefore, cannot be decoupled from the overall energy equation. The general form of the energy conservation equation for a moving compressible fluid may be stated as ! Du ∂u ′′′ ρ + v · ∇u = −∇ · q − p∇ · v + µΦ + Q˙ , (10.70) =ρ Dt ∂t where u is internal energy, v is the velocity vector, q is the total heat flux vector, Φ is the ′′′ dissipation function, and Q˙ is heat generated within the medium (such as energy release due to chemical reactions). For a detailed derivation of equation (10.70), the reader is referred to standard textbooks such as [6, 7]. If the medium is radiatively participating through emission, absorption, and scattering, then the conservation equations for momentum and energy are altered by three effects [8]: 1. The heat flux term in equation (10.70), which without radiation is in most applications due only to molecular diffusion (heat conduction), now has a second component, the radiative heat flux, due to radiative energy interacting with the medium within the control volume. 2. The internal energy now contains a radiative contribution [the incident radiation G, due to the first term in equation (10.20)]. 3. The radiation pressure tensor must be added to the traditional fluid dynamics pressure tensor. We have already seen that the second effect is almost always negligible, and the same is true for the augmentation of the pressure tensor. Under these conditions the energy conservation equation can be simplified. If we assume that du = cv dT, and that Fourier’s law for heat conduction holds, q = qC + qR = −k∇T + qR , (10.71) equation (10.70) becomes DT ∂T + v · ∇T = ρcv ρcv Dt ∂t

!

′′′ = ∇ · (k∇T) − p∇ · v + µΦ + Q˙ − ∇ · qR .

(10.72)

This is an integro-differential equation for the calculation of the temperature field, since the evaluation of the divergence of the radiative heat flux must come from (10.59), which is an

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10 THE RADIATIVE TRANSFER EQUATION IN PARTICIPATING MEDIA (RTE)

integral equation in temperature. Obviously, a complete solution of this equation, even with the recent advent of supercomputers, is a truly formidable task. Example 10.6. State the radiative transfer equation and its boundary conditions for the case of combined steady-state conduction and radiation within a one-dimensional, planar, gray, and nonscattering medium, bounded by isothermal black walls. Solution Since the problem is steady state and there is no movement in the medium, the left side of equation (10.72) vanishes, and only the first (conduction) and last (radiation) terms on the right side remain. For a onedimensional planar medium this reduces to6 ! d dT k − qR = 0, (10.73) dz dz and the divergence of radiative heat flux is related to temperature and incident radiation through equation (10.59), dqR = κ(4σT 4 − G), dz where the spectral integration for the gray medium has been carried out by simply dropping the subscript η. Finally, the incident radiation is found from direction-integrating equation (10.29) (not a trivial task). The necessary boundary conditions are T = Ti , i = 1, 2 at the two walls (for conduction) and I(0, sˆ ) = σTi4 /π (for radiation) needed in equation (10.29). Solution of this seemingly simple problem is by no means trivial, and can only be achieved through relatively involved numerical analysis.

Radiative Equilibrium Much attention in the following chapters will be given to the situation in which radiation is the dominant mode of heat transfer, meaning that when conduction and convection are negligible. This situation is referred to as radiative equilibrium, meaning that thermodynamic equilibrium within the medium is achieved by virtue of thermal radiation alone. As is commonly done in the discussion of “pure” conduction or convection, we allow volumetric heat sources throughout the medium. Thus, we may write ρcv

∂T ′′′ + ∇ · qR = Q˙ , ∂t

(10.74)

which is identical in form to the basic transient heat conduction equation (before substitution of Fourier’s law). In the vast majority of cases radiative transfer occurs so fast that radiative equilibrium is achieved before a noticeable change in temperature occurs [i.e., when the unsteady term in equation (10.20) can be dropped]. Then the statement of radiative equilibrium reduces to its steady-state form ′′′ ∇ · qR = Q˙ . (10.75) Radiative equilibrium is often a good assumption in applications with extremely high temperatures, such as plasmas, nuclear explosions, and such. The inclusion of a volumetric heat source allows the treatment of conduction and convection “through the back door:” A guess is made for the temperature field and the nonradiation terms in equation (10.72) are calculated ′′′ to give Q˙ for the radiation calculations. This process is then repeated until a convergence criterion is met. 6 While in the science of conduction the variable x is usually employed for one-dimensional planar problems, for thermal radiation problems the variable z is more convenient. The reason for this is that, by convention, the polar angle for the direction vector is measured from the z-axis.

10.12 SOLUTION METHODS FOR THE RADIATIVE TRANSFER EQUATION

299

10.12 SOLUTION METHODS FOR THE RADIATIVE TRANSFER EQUATION Exact analytical solutions to the radiative transfer equation [equation (10.21)] are exceedingly difficult, and explicit solutions are impossible for all but the very simplest situations. Therefore, research on radiative heat transfer in participating media has generally proceeded in two directions: (i) exact (analytical and numerical) solutions of highly idealized situations, and (ii) approximate solution methods for more involved scenarios. Phenomena that make a radiative heat transfer problem difficult may be placed into four different categories: Geometry: The problem may be one-dimensional, two-dimensional, or three-dimensional. Most investigations to date have dealt with one-dimensional geometries, and the vast majority of these dealt with the simplest case of a one-dimensional plane-parallel slab. Temperature Field: The least difficult situation arises if the temperature profile within the medium is known, making equation (10.21) a relatively “simple” integral equation. Consequently, the most basic case of an isothermal medium has been studied extensively. Alternatively, if radiative equilibrium prevails, the temperature field is unknown but uncoupled from conduction and convection, and must be found from directional and spectral integration of the radiative transfer equation. In the most complicated scenario, radiative heat transfer is combined with conduction and/or convection, resulting in a highly nonlinear integro-differential equation. Scattering: The solution to a radiation problem is greatly simplified if the medium does not scatter. In that case the radiative transfer equation reduces to a simple first-order differential equation if the temperature field is known, and a relatively simpler integral equation if radiative equilibrium prevails. If scattering must be considered, isotropic scattering is often assumed. Relatively few investigations have dealt with the case of anisotropic scattering, and most of those are limited to the case of linear-anisotropic scattering (see Section 12.9). Properties: Although most participating media display strong nongray character, as discussed in the following three chapters, the vast majority of investigations to date have centered on the study of gray media. In addition, while radiative properties also generally depend strongly on temperature, concentration, etc., most calculations are limited to situations with constant properties. Most “exact” solutions are limited to gray media with constant properties in one-dimensional, mainly plane-parallel geometries. The media are isothermal or at radiative equilibrium, and if they scatter, the scattering is usually isotropic. Since the usefulness of such one-dimensional solutions in heat transfer applications is limited, they are only briefly discussed in Chapter 14. Several chapters are devoted to the various approximate methods that have been devised for the solution of the radiative transfer equation. Still, these seven chapters by no means cover all the different methods that have been and still are used by investigators in the field. A number of approximate methods for one-dimensional problems are discussed in Chapter 15. The optically thin and diffusion (or optically thick) approximations have historically been developed for a one-dimensional plane-parallel medium, but can readily be applied to more complicated geometries. Similarly, the Schuster–Schwarzschild or two-flux approximation [9, 10] is a forerunner to the multidimensional discrete ordinates method. In this method the intensity is assumed to be constant over discrete parts of the total solid angle of 4π. Several other flux methods exist, but they are usually tailored toward special geometries, and cannot easily be applied to other scenarios, for example, the six-flux methods of Chu and Churchill [11] and Shih and coworkers [12, 13]. Another early one-dimensional model was the moment method or Eddington approximation [14]. In this model the directional dependence is expressed by a truncated series representation (rather than discretized). In general geometries this expansion

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10 THE RADIATIVE TRANSFER EQUATION IN PARTICIPATING MEDIA (RTE)

is usually achieved through the use of spherical harmonics, leading to the spherical harmonics method. Several variations to the moment method that are tailored toward specific geometries have been proposed [15, 16], but these are of limited general utility. Finally, the exponential kernel approximation, already discussed in Chapter 5 for surface radiation problems, may be used as a tool for many one-dimensional problems. However, its extension to multidimensional geometries is problematic. A survey of the literature over the past forty years demonstrates that some solution methods have been used frequently, while others that appeared promising at one time are no longer employed on a regular basis. Apparently, some methods have been found to be more readily adapted to more difficult situations than others (such as multidimensionality, variable properties, anisotropic scattering, and/or nongray effects). The majority of radiative heat transfer analyses today appear to use one of four methods: (i) the spherical harmonics method or a variation of it, (ii) the discrete ordinates method or its more modern form, the finite volume method, (iii) the zonal method, and (iv) the Monte Carlo method. The first two of these have already been discussed briefly above with the one-dimensional approximations. The zonal method was developed by Hottel [17] in his pioneering work on furnace heat transfer. Unlike the spherical harmonics and discrete ordinates methods, the zonal method approximates spatial, rather than directional, behavior by breaking up an enclosure into finite, isothermal subvolumes. On the other hand, the Monte Carlo method [18] is a statistical method, in which the history of bundles of photons is traced as they travel through the enclosure. While the statistical nature of the Monte Carlo method makes it difficult to match it with other calculations, it is the only method that can satisfactorily deal with effects of irregular radiative properties (nonideal directional and/or nongray behavior). Because of their importance, an entire chapter is devoted to each of these four solution methods. Several other methods that can be found in the literature are not covered in this book (except for brief descriptions in appropriate places). For example, the discrete transfer method, proposed by Shah [19] and Lockwood and Shah [20], combines features of the discrete ordinates, zonal, and Monte Carlo methods. Another hybrid proposed by Edwards [21] combines elements of the Monte Carlo and zonal methods.

References 1. Viskanta, R., and M. P. Menguc ¨ ¸ : “Radiation heat transfer in combustion systems,” Progress in Energy and Combustion Science, vol. 13, pp. 97–160, 1987. 2. Pomraning, G. C.: The Equations of Radiation Hydrodynamics, Pergamon Press, New York, 1973. 3. Ben-Abdallah, P., V. Le Dez, D. Lemonnier, S. Fumeron, and A. Charette: “Inhomogeneous radiative model of refractive and dispersive semi-transparent stellar atmospheres,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 69, pp. 61–80, 2001. 4. Kumar, S., and K. Mitra: “Microscale aspects of thermal radiation transport and laser applications,” in Advances in Heat Transfer, vol. 33, Academic Press, New York, pp. 187–294, 1999. 5. Hartung, L., R. Mitcheltree, and P. Gnoffo: “Stagnation point nonequilibrium radiative heating and influence of energy exchange models,” Journal of Thermophysics and Heat Transfer, vol. 6, no. 3, pp. 412–418, 1992. 6. Rohsenow, W. M., and H. Y. Choi: Heat, Mass and Momentum Transfer, Prentice Hall, Englewood Cliffs, NJ, 1961. 7. Kays, W. M., and M. E. Crawford: Convective Heat and Mass Transfer, McGraw-Hill, 1980. 8. Sparrow, E. M., and R. D. Cess: Radiation Heat Transfer, Hemisphere, New York, 1978. 9. Schuster, A.: “Radiation through a foggy atmosphere,” Astrophysical Journal, vol. 21, pp. 1–22, 1905. ¨ 10. Schwarzschild, K.: “Uber das Gleichgewicht der Sonnenatmosph¨aren (Equilibrium of the sun’s atmosphere),” Akad. Wiss. G¨ottingen, Math.-Phys. Kl. Nachr., vol. 195, pp. 41–53, 1906. 11. Chu, C. M., and S. W. Churchill: “Numerical solution of problems in multiple scattering of electromagnetic radiation,” Journal of Physical Chemistry, vol. 59, pp. 855–863, 1960. 12. Shih, T. M., and Y. N. Chen: “A discretized-intensity method proposed for two-dimensional systems enclosing radiative and conductive media,” Numerical Heat Transfer, vol. 6, pp. 117–134, 1983. 13. Shih, T. M., and A. L. Ren: “Combined radiative and convective recirculating flows in enclosures,” Numerical Heat Transfer, vol. 8, no. 2, pp. 149–167, 1985. 14. Eddington, A. S.: The Internal Constitution of the Stars, Dover Publications, New York, 1959. 15. Chou, Y. S., and C. L. Tien: “A modified moment method for radiative transfer in non-planar systems,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 8, pp. 719–733, 1968.

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16. Hunt, G. E.: “The transport equation of radiative transfer with axial symmetry,” SIAM J. Appl. Math., vol. 16, no. 1, pp. 228–237, 1968. 17. Hottel, H. C., and E. S. Cohen: “Radiant heat exchange in a gas-filled enclosure: Allowance for nonuniformity of gas temperature,” AIChE Journal, vol. 4, pp. 3–14, 1958. 18. Howell, J. R.: “Application of Monte Carlo to heat transfer problems,” in Advances in Heat Transfer, eds. J. P. Hartnett and T. F. Irvine, vol. 5, Academic Press, New York, 1968. 19. Shah, N. G.: “New method of computation of radiation heat transfer in combustion chambers,” Ph.D. thesis, Imperial College of Science and Technology, London, England, 1979. 20. Lockwood, F. C., and N. G. Shah: “A new radiation solution method for incorporation in general combustion prediction procedures,” in Eighteenth Symposium (International) on Combustion, The Combustion Institute, pp. 1405–1409, 1981. 21. Edwards, D. K.: “Hybrid Monte-Carlo matrix-inversion formulation of radiation heat transfer with volume scattering,” in Heat Transfer in Fire and Combustion Systems, vol. HTD-45, ASME, pp. 273–278, 1985.

Problems 10.1 A semi-infinite medium 0 ≤ z < ∞ consists of a gray, absorbing–emitting gas that does not scatter, bounded by vacuum at the interface z = 0. The gas is isothermal at 1000 K, and the absorption coefficient is κ = 1 m−1 . The interface is nonreflecting; conduction and convection may be neglected. (a) What is the local heat generation that is necessary to keep the gas at 1000 K? (b) What is the intensity distribution at the interface, that is, I(z = 0, θ, ψ), for all θ and ψ? (c) What is the total heat flux leaving the semi-infinite medium? 10.2 Reconsider the semi-infinite medium of Problem 10.1 for a temperature distribution of T = T0 e−z/L , T0 = 1000 K, L = 1 m. What are the exiting intensity and heat flux for this case? Discuss how the answer would change if κ varied between 0 and ∞. 10.3 Repeat Problem 10.1 for a medium of thickness L = 1 m. Discuss how the answer would change if κ varied between 0 and ∞. 10.4 A semi-infinite, gray, nonscattering medium (n = 2, κ = 1 m−1 ) is irradiated by the sun normal to its surface at a rate of qsun = 1000 W/m2 . Neglecting emission from the relatively cold medium, determine the local heat generation rate due to absorption of solar energy. Hint: The solar radiation may be thought of as being due to a radiative intensity which has a large value Io over a very small cone of solid angles δΩ, and is zero elsewhere, i.e.,  ˆ I over δΩ along n, I(ˆs) = o 0 elsewhere, and Z I(ˆs)nˆ · sˆ dΩ = Io δΩ.

qsun =



10.5 A 1 m thick slab of an absorbing–emitting gas has an approximately linear temperature distribution as shown in the sketch. On both sides the medium is bounded by vacuum with nonreflecting boundaries. (a) If the medium has a constant and gray absorption coefficient of κ = 1 m−1 , what is the intensity (as a function of direction) leaving the hot side of the slab? (b) Give an expression for the radiative heat flux leaving the hot side.

1m T2 = 2000 K

T1 = 1000 K

10.6 A semitransparent sphere of radius R = 10 cm has a parabolic temperature profile T = Tc (1 − r2 /R2 ), Tc = 2000 K. The sphere is gray with κ = 0.1 cm−1 , n = 1.0, does not scatter, and has nonreflective boundaries. Outline how to calculate the total heat loss from the sphere (i.e., there is no need actually to carry out cumbersome integrations). 10.7 Repeat Problem 10.6, but assume that the temperature is uniform at 2000 K. What must the local production of heat be if the sphere is to remain at 2000 K everywhere? Note: The answer may be left in integral form (which must be solved numerically). Carry out the integration for r = 0 and r = R.

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10.8 Repeat Problem 10.6, but assume that the temperature is uniform at 2000 K. Also, there is no heat production, meaning that the sphere cools down. How long will it take for the sphere to cool down to 500 K (the heat capacity of the medium is ρc = 1000 kJ/m3 K and the conductivity is very large, i.e., the sphere is isothermal at all times)? 10.9 A relatively cold sphere with a radius of Ro = 1 m consists of a nonscattering gray medium that absorbs with an absorption coefficient of κ = 0.1 cm−1 and has a refractive index n = 2. At the center of the sphere is a small black sphere with radius Ri = 1 cm at a temperature of 1000 K. On the outside, the sphere is bounded by vacuum. What is the total heat flux leaving the sphere? Explain what happens as κ is increased from zero to a large value. 10.10 A laser beam is directed onto the atmosphere of a (hypothetical) planet. The planet’s atmosphere contains 0.01% by volume of an absorbing gas. The absorbing gas has a molecular weight of 20 and, at the laser wavelength, an absorption coefficient κη = 10−4 cm−1 /(g/m3 ). It is known that the pressure and temperature distributions of the atmosphere can be approximated by p = p0 e−2z/L and T = T0 e−z/L , where p0 = 0.75 atm, T0 = 400 K are values at the planet surface z = 0, and L = 2 km is a characteristic length. What fraction of the laser energy arrives at the planet’s surface? 10.11 A CO2 laser with a total power output of Q = 10 W is directed (at right angle) onto a 10 cm thick, isothermal, absorbing/emitting (but not scattering) medium at 1000 K. It is known that the laser beam is essentially monochromatic at a wavelength of 10.6 µm with a Gaussian power distribution. Thus, the intensity falling onto the medium is 2

I(0) ∝ eZ−(r/R) /(δΩ δη), Q=

0 ≤ r ≤ ∞;

10 cm Laser irradiation r z

I(0) dA δΩ δη,

A

where r is distance from beam center, R = 100 µm is the “effective radius” of the laser beam, δΩ = 5 × 10−3 sr is the range of solid angles over which the laser beam outputs intensity (assumed uniform over δΩ), and δη is the range of wavenumbers over which the intensity is distributed (also assumed uniform). At 10.6 µm the medium is known to have an absorption coefficient κη = 0.15 cm−1 . Assuming that the medium has nonreflecting boundaries, determine the exiting total intensity in the normal direction (transmitted laser radiation plus emission, assuming the medium to be gray). Is the emission contribution important? How thick would the medium have to be to make transmission and emission equally important? 10.12 Repeat Problem 10.11 for a medium with refractive index n = 2, bounded by vacuum (i.e., a slab with reflecting surfaces). Hint: (1) Part of the laser beam will be reflected when first hitting the slab, part will penetrate into the slab. Part of this energy will be absorbed by the layer, part will hit the rear face, where a fraction will be reflected back into the slab, and the rest will emerge from the slab, etc. Similar multiple internal reflections will take place with the emitted energy before emerging from the slab. (2) To calculate the slab–surroundings reflectance, show that the value of the absorptive index is negligible. 10.13 A thin column of gas of cross-section δA and length L contains a uniform suspension of small particles that absorb and scatter radiation. The scattering is according to the phase function (a) Φ = 1 (isotropic scattering), (b) Φ = 1+A1 cos Θ (linear anisotropic scattering, A1 is a constant), and (c) Φ = 43 (1+cos2 Θ) (Rayleigh scattering), where Θ is the angle between incoming and scattered directions. A laser beam hits the column normal to δA. What is the transmitted fraction of the laser power? What fraction of the laser flux goes through an infinite plane at L normal to the gas column? What fraction goes back through a plane at 0? What happens to the rest? 10.14 Repeat Example 10.2 for (a) Φ = 1 + A1 cos Θ (linear anisotropic scattering, A1 = const), and (b) Φ = 34 (1 + cos2 Θ) (Rayleigh scattering), and Θ is the angle between incoming and scattered directions. 10.15 Show that, by setting βη = 0 and Iw = J/π, the radiosity integral equation (5.25) can be recovered from equation (10.69) for a nonparticipating medium surrounded by diffusely reflecting walls. Hint: Break up the heat flux in equation (10.69) into two parts, incoming radiation H and exiting radiation J. For the latter assume r to be an infinitesimal distance above the surface and evaluate the integral in equation (10.69).

CHAPTER

11

RADIATIVE PROPERTIES OF MOLECULAR GASES 11.1

FUNDAMENTAL PRINCIPLES

Radiative transfer characteristics of an opaque wall can often be described with good accuracy by the very simple model of gray and diffuse emission, absorption, and reflection. The radiative properties of a molecular gas, on the other hand, vary so strongly and rapidly across the spectrum that the assumption of a “gray” gas is almost never a good one [1]. In the present chapter a short development of the radiative properties of molecular gases is given. Other elaborate discussions can be found, for example, in the book by Goody and Yung [2], in the monograph by Tien [3], and in the very recent treatise of Taine and Soufiani [4]. Most of the earlier work was not in the area of heat transfer but rather was carried out by astronomers, who had to deal with light absorption within Earth’s atmosphere, and by astrophysicists, who studied the spectra of stars. The study of atmospheric radiation was apparently initiated by Lord Rayleigh [5] and Langley [6] in the late nineteenth century. The radiation spectra of stars started to receive attention in the early twentieth century, for example by Eddington [7] and Chandrasekhar [8, 9]. The earliest measurements of radiation from hot gases were reported by Paschen, a physicist, in 1894 [10], but his work was apparently ignored by heat transfer engineers for many years [11]. The last few decades have seen much progress in the understanding of molecular gas radiation, in particular the radiation from water vapor and carbon dioxide, which is of great importance in the combustion of hydrocarbon fuels, and which also dominates atmospheric radiation with its thermodynamic implications on Earth’s atmosphere. The combination of the two, i.e., the man-made strong increases in the atmosphere’s CO2 content, giving rise to “global warming,” is perhaps the most pressing problem facing mankind today. Much of the pioneering work since the late 1920s was done by Hottel and coworkers [12–19] (measurements and practical calculations) and by Penner [20] and Plass [21, 22] (theoretical basis). When a photon (or an electromagnetic wave) interacts with a gas molecule, it may be either absorbed, raising the molecule’s energy level, or scattered, changing the direction of travel of the photon. Conversely, a gas molecule may spontaneously lower its energy level by the emission of an appropriate photon. As will be seen in the next chapter on particle properties (since every molecule is, of course, a very small particle), the scattering of photons by molecules is always negligible for heat transfer applications. There are three different types of radiative transitions that lead to a change of molecular energy level by emission or absorption of a photon: (i) transitions between nondissociated (“bound”) atomic or molecular states, called bound–bound transitions, (ii) transitions from a “bound” state to a “free” (dissociated) one (absorption) or 303

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from “free” to “bound” (emission), called bound–free transitions, and (iii) transitions between two different “free” states, free–free transitions. The internal energy of every atom and molecule depends on a number of factors, primarily on the energies associated with electrons spinning at varying distances around the nucleus, atoms within a molecule spinning around one another, and atoms within a molecule vibrating against each other. Quantum mechanics postulates that the energy levels for atomic or molecular electron orbit as well as the energy levels for molecular rotation and vibration are quantized; i.e., electron orbits and rotational and vibrational frequencies can only change by certain discrete amounts. Since the energy contained in a photon or electromagnetic wave is directly proportional to frequency, quantization means that, in bound–bound transitions, photons must have a certain frequency (or wavelength) in order to be captured or released, resulting in discrete spectral lines for absorption and emission. Since, according to Heisenberg’s uncertainty principle, the energy level of an atom or molecule cannot be fixed precisely, this phenomenon (and, as we shall see, some others as well) results in a slight broadening of these spectral lines. Changing the orbit of an electron requires a relatively large amount of energy, or a highfrequency photon, resulting in absorption–emission lines at short wavelengths between the ultraviolet and the near-infrared (between 10−2 µm and 1.5 µm). Vibrational energy level changes require somewhat less energy, so that their spectral lines are found in the infrared (between 1.5 µm and 10 µm), while changes in rotational energy levels call for the least amount of energy and, thus, rotational lines are found in the far infrared (beyond 10 µm). Changes in vibrational energy levels may (and often must) be accompanied by rotational transitions, leading to closely spaced groups of spectral lines that, as a result of line broadening, may partly overlap and lead to so-called vibration–rotation bands in the infrared. Similarly, electronic transitions in molecules (as opposed to atoms) are always accompanied by vibrational and rotational energy changes, generally in the ultraviolet to the near-infrared. If the initial energy level of a molecule is very high (e.g., in very high-temperature gases), then the absorption of a photon may cause the breaking-away of an electron or the breakup of the entire molecule because of too strong vibration, i.e., a bound–free transition. The postabsorption energy level of the molecule depends on the kinetic energy of the separated part, which is essentially not quantized. Therefore, bound–free transitions result in a continuous absorption spectrum over all wavelengths or frequencies for which the photon energy exceeds the required ionization or dissociation energy. The same is true for the reverse process, emission of a photon in a free–bound transition (often called radiative combination). In an ionized gas free electrons can interact with the electric field of ions resulting in a free–free transition (also known as Bremsstrahlung, which is German for brake radiation); i.e., the release of a photon lowers the kinetic energy of the electron (decelerates it), or the capture of a photon accelerates it (inverse Bremsstrahlung). Since kinetic energy levels of electrons are essentially not quantized, these photons may have any frequency or wavelength. Bound–free and free–free transitions generally occur at very high temperatures (when dissociation and ionization become substantial). The continuum radiation associated with them is usually found at short wavelengths (ultraviolet to visible). Therefore, these effects are of importance only in extremely high-temperature situations. Most engineering applications occur at moderate temperature levels, with little ionization and dissociation, making bound–bound transitions most important. At combustion temperatures the emissive power has its maximum in the infrared (between 1 µm and 6 µm), giving special importance to vibration–rotation bands. In this book we will focus our discussion on the most important case of bound–bound transitions.

11.2 EMISSION AND ABSORPTION PROBABILITIES There are three different processes leading to the release or capture of a photon, namely, spontaneous emission, induced or stimulated emission (also called negative absorption), and absorption. The

11.2 EMISSION AND ABSORPTION PROBABILITIES

305

absorption and emission coefficients associated with these transitions may, at least theoretically, be calculated from quantum mechanics. Complete descriptions of the microscopic phenomena may be found in books on statistical mechanics [23, 24] or spectroscopy [25, 26]. An informative (rather than precise) synopsis has been given by Tien [3] that we shall essentially follow here. Let there be nu atoms or molecules (per unit volume) at a nondegenerate higher energy state u and nl at a lower energy state l. “Nondegenerate” means that, if there are several states with identical energies (degeneracy), each state is counted separately. The difference of energy between the two states is hν. The number of transitions from state u to state l by release of a photon with energy hν (spontaneous emission) must be proportional to the number of atoms or molecules at that level. Thus ! dnu = −Aul nu , (11.1) dt u→l where the proportionality constant Aul is known as the Einstein coefficient for spontaneous emission. Spontaneous emission is isotropic, meaning that the direction of the emitted photon is random, resulting in equal emission intensity in all directions. Quantum mechanics postulates that, in addition to spontaneous emission, incoming radiative intensity (or photon streams) with the appropriate frequency may induce the molecule to emit photons into the same direction as the incoming intensity (stimulated emission). Therefore, the total number of transitions from state u to state l may be written as ! ! Z dnu Iν dΩ , (11.2) = −nu Aul + Bul dt u→l 4π where Iν is the incoming intensity, which must be integrated over all directions to account for all possible transitions, and Bul is the Einstein coefficient for stimulated emission. Finally, part of the incoming radiative intensity may be absorbed by molecules at energy state l. Obviously, the absorption rate will be proportional to the strength of incoming radiation as well as the number of molecules that are at energy state l, leading to ! Z dnl = nl Blu Iν dΩ, (11.3) dt l→u 4π where Blu is the Einstein coefficient for absorption. The three Einstein coefficients may be related to one another by considering the special case of equilibrium radiation. Equilibrium radiation occurs in an isothermal black enclosure, where the radiative intensity is everywhere equal to the blackbody intensity Ibν and where the average number of molecules at any given energy level is constant at any given time, i.e., the number of transitions from all upper energy levels u to all lower states l is equal to the ones from l to u, or ! ! ! Z Z dnu dnl 1u (11.4) + 1l = −1u nu Aul + Bul Ibν dΩ + 1l nl Blu Ibν dΩ = 0, dt u→l dt l→u 4π 4π where 1u and 1l are the degeneracies of the upper and lower energy state, respectively, i.e., the number of different arrangements with which a molecule can obtain this energy level. At local thermodynamic equilibrium the number of particles at any energy level is governed by Boltzmann’s distribution law [23], leading to . nl /nu = e−El /kT e−Eu /kT = ehν/kT , (11.5)

where Eu and El are the energy levels associated with states u and l, respectively. Thus, the blackbody intensity may be evaluated from equation (11.4) as Ibν =

Aul /Bul 1 .  4π 1l Blu /1u Bul ehν/kT − 1

(11.6)

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11 RADIATIVE PROPERTIES OF MOLECULAR GASES

Comparison with Planck’s law, equation (1.9), shows that all three Einstein coefficients are dependent upon another, namely, Aul =

8πhν3 Bul , c20

1u Bul = 1l Blu .

(11.7)

The Einstein coefficients are universal functions for a given transition and, therefore, the relationships between them hold also if local thermodynamic equilibrium does not prevail (i.e., the energy level populations do not obey Boltzmann’s distribution, equation (11.5). The one remaining independent Einstein coefficient is clearly an indicator of how strongly a gas is able to emit and absorb radiation. This is most easily seen by examining the number of induced transitions (by absorption and emission) in a single direction (or within a thin pencil of rays). If ! d dn 1 = (1l nl Blu − 1u nu Bul )Iν (11.8) dΩ dt l↔u is the net number of photons removed from the pencil of rays per unit time and per unit volume, then—since each photon carries the energy hν—the change of radiative energy per unit time, per unit area and distance, and per unit solid angle is ! d dn −hν 1 = −(1l nl Blu − 1u nu Bul )hνIν . (11.9) dΩ dt l↔u This relation is equivalent to equation (10.1), except that in reality the spectral line associated with a transition between an upper energy state u and a lower energy state l is “broadened,” i.e., transitions occur across a (very small) range of frequencies, and equation (11.9) captures all of these transitions. Accounting for this slight spread in frequencies (and recalling the definition of intensity, Section 1.6), we have Z Z d Iν dν = −(1l nl Blu − 1u nu Bul )hνIν = − (1l nl B′lu − 1u nu B′ul )hνIν dν, (11.10) ds ∆ν ∆ν i.e., the Einstein probabilities are not defined for a single transition frequency, but rather are spread over a small but finite frequency range ∆ν due to broadening, with [27] A′ul = Aul φν ,

B′ul = Bul φν ,

B′lu = Blu φν ,

(11.11)

and φν (ν) is a normalized line shape function (assumed here to be equal for all three probabilities), Z φν (ν) dν = 1. (11.12) ∆ν

The exact shape of line broadening will be discussed in detail in Section 11.4. Using equation (11.11) we can rewrite equation (11.10) as Z Z d Iν dν = −(1l nl Blu − 1u nu Bul ) hνφν Iν dν. (11.13) ds ∆ν ∆ν This relation gives the absorption of an entire line, and we define the line strength or line intensity as Z Sν = (1l nl Blu − 1u nu Bul )

hνφν dν = (1l nl Blu − 1u nu Bul )hν.

(11.14)

∆ν

In the last expression of equation (11.14) the (line-center) frequency has been taken out of the integral, since ν varies very little across a narrow spectral line. By the definition of the absorption

11.2 EMISSION AND ABSORPTION PROBABILITIES

307

coefficient the line strength is the (linear) absorption coefficient integrated across a line. On a spectral basis across ∆ν, this becomes Z Sν = κν dν, and κν = Sν φν , (11.15) ∆ν

so that

dIν = −κν Iν , ds

(11.16)

which is, of course, identical to equation (10.1). The absorption coefficient as defined here is often termed the effective absorption coefficient since it incorporates stimulated emission (or negative absorption). Sometimes a true absorption coefficient is defined from Z κν dν = 1l nl Blu hν. (11.17) ∆ν

Since stimulated emission and absorption always occur together and cannot be separated, it is general practice to incorporate stimulated emission into the absorption coefficient, so that only the effective absorption coefficient needs to be considered.1 Examination of equation (11.14) shows that the absorption coefficient is proportional to molecular number density. Therefore, as mentioned earlier, a number of researchers take the number density out of the definition for κν either in the form of density or pressure, by defining a mass absorption coefficient or a pressure absorption coefficient, respectively, as κρν ≡

κν , ρ

κpν ≡

κν , p

(11.18)

and similarly for Sν . If a mass or pressure absorption coefficient is used, then a ρ or p must, of course, be added to equation (11.16).2 The negative of equation (11.1) gives the rate at which molecules emit photons of strength hν randomly into all directions (into a solid angle of 4π) and per unit volume. Thus, multiplying this equation by −hν and dividing by 4π gives isotropic energy emitted per unit time, per unit solid angle, per unit area and distance along a pencil of rays or, in short, the change of intensity per unit distance due to spontaneous emission: ! Z d d dn Iν dν = −hν = 1u nu Aul hν/4π. (11.19) ds ∆ν dΩ dt u→l This is the emission of an entire line and, on a spectral basis across ∆ν this becomes dIν = 1u nu A′ul hν/4π = jν , ds

(11.20)

and jν is called the emission coefficient, which is related to the absorption coefficient through equations (11.7), (11.14), and (11.15), leading to jν = κν

2hν3 nu , c20 nl − nu

(11.21)

1 Since it is experimentally impossible to distinguish stimulated emission from absorption, its existence had initially been questioned. Equation (11.6) is generally accepted as proof that stimulated emission does indeed exist: Without it Bul → 0 and the blackbody intensity would be governed by Wien’s distribution, equation (1.18), which is known to be incorrect. 2 Thus, depending on what spectral variable is employed (wavelength λ, wavenumber η, or frequency ν), a spectrally integrated absorption coefficient may appear in nine different variations. Often the only way to determine which definition has been used is to carefully check the units given.

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At local thermodynamic equilibrium energy levels are populated according to Boltzmann’s distribution, equation (11.5), and the emission coefficient and equation (11.20) reduce to dIν = jν = κν Ibν , ds

(11.22)

which represents the augmentation of directional intensity due to spontaneous emission, as given by equation (10.10).

11.3 ATOMIC AND MOLECULAR SPECTRA We have already seen that the emission or absorption of a photon goes hand in hand with the change of rotational and/or vibrational energy levels in molecules, or with the change of electron orbits (in atoms and molecules). This change, in turn, causes a change in radiative intensity resulting in spectral lines. In this section we discuss briefly how the position of spectral lines within a vibration–rotation band can be calculated, since it is these bands that are of great importance to the heat transfer engineer. More detailed information as well as discussion of electronic spectra, and bound–free and free–free transitions may be found in more specialized books on quantum mechanics [24,25,28] or spectroscopy [26,29–31], in the book on atmospheric radiation by Goody and Yung [2], or in the monographs on gas radiation properties by Tien [3] and Taine and Soufiani [4]. Since every particle moves in three-dimensional space, it has three degrees of freedom: It can move in the forward–backward, left–right, and/or upward–downward directions. If two or more particles are connected with each other (diatomic and polyatomic molecules), then each of the atoms making up the molecule has three degrees of freedom. However, it is more convenient to say that a molecule consisting of N atoms has three degrees of freedom for translation, and 3N − 3 degrees of freedom for relative motion between atoms. These 3N − 3 degrees of internal freedom may be further separated into rotational and vibrational degrees of freedom. This fact is illustrated in Fig. 11-1 for a diatomic molecule and for linear and nonlinear triatomic molecules. The diatomic molecule has three internal degrees of freedom. Obviously, it can rotate around its center of gravity within the plane of the paper or, similarly, perpendicularly to the paper (with the rotation axis lying in the paper). It could also rotate around its own axis; however, neither one of the atoms would move (except for rotating around itself). Thus, the last degree of freedom must be used for vibrational motion between the two atoms as indicated in the figure. The situation gets rapidly more complicated for molecules with increasing number of atoms. For linear triatomic molecules (e.g., CO2 , N2 O, HCN) there are, again, only two rotational modes. Since there are six internal degrees of freedom, there are four vibrational modes, as indicated in Fig. 11-1. However, two of these vibrational modes are identical, or degenerate (except for taking place in perpendicular planes). In contrast, a nonlinear triatomic molecule has three rotational modes: In this case rotation around the horizontal axis in the plane of the paper is legitimate, so that there are only three vibrational degrees of freedom. Depending on the axis of rotation, a polyatomic molecule may have different moments of inertia for each of the three rotational modes. If symmetry is such that all three moments of inertia are the same, the molecule is classified as a spherical top (e.g., CH4 ). It is called a symmetric top, if two are the same (e.g., NH3 , CH3 Cl, C2 H6 , SF6 ), and an asymmetric top, if all three are different (e.g., H2 O, O3 , SO2 , NO2 , H2 S, H2 O2 ).

Rotational Transitions To calculate the allowed rotational energy level from quantum mechanics using Schr¨odinger’s wave equation (see, for example, [23,24]), we generally assume that the molecule consists of point

11.3 ATOMIC AND MOLECULAR SPECTRA

309

Rotational degrees of freedom

Vibrational degrees of freedom

(a) (b) (c) FIGURE 11-1 Rotational and vibrational degrees of freedom for (a) diatomic, (b) linear triatomic, and (c) nonlinear triatomic molecules.

masses connected by rigid massless rods, the so-called rigid rotator model. The solution to this wave equation dictates that possible energy levels for a linear molecule are limited to Ej =

~2 j(j + 1) = hc 0 Bj(j + 1), 2I

j = 0, 1, 2, . . . (j integer),

(11.23)

where ~ = h/2π is the modified Planck’s constant, I is the moment of inertia of the molecule, j is the rotational quantum number, and the abbreviation B has been introduced for later convenience. Allowed transitions are ∆ j = ±1 and 0 (the latter being of importance for a simultaneous vibrational transition); this expression is known as the selection rule. In the case of the absorption of a photon ( j → j + 1 transition) the wavenumbers of the resulting spectral lines can then be determined3 as η = (E j+1 − E j )/hc 0 = B(j + 1)( j + 2) − B j(j + 1) = 2B(j + 1),

j = 0, 1, 2, . . . .

(11.24)

The results of this equation produce a number of equidistant spectral lines (in units of wavenumber or frequency), as shown in the sketch of Fig. 11-2. The rigid rotator model turns out to be surprisingly accurate, although for high rotation rates (j ≫ 0) a small correction factor due to the centrifugal contribution (stretching of the “rod”) may be considered. Not all linear molecules exhibit rotational lines, since an electric dipole moment is required for a transition to occur. Thus, diatomic molecules such as O2 and N2 never undergo rotational transitions, while symmetric molecules such as CO2 show a rotational spectrum only if accompanied by a vibrational transition [3]. Evaluation of the spectral lines of nonlinear polyatomic molecules is always rather complicated and the reader is referred to specialized treatises such as the one by Herzberg [30]. 3 In our discussion of surface radiative transport we have used wavelength λ as the spectral variable throughout, largely to conform with the majority of other publications. However, for gases frequency ν or wavenumber η are considerably more convenient to use [see, for example, equation (11.24)]. Again, to conform with the majority of the literature, we shall use wavenumber throughout this part.

310

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

4

20

Absorption Emission Ej h c0 B 12

3

6

2

2

1

0

j = 0 2B

FIGURE 11-2 Spectral position and energy levels for a rigid rotator.

4B 6B 8B Wavenumber η

Vibrational Transitions The simplest model of a vibrating diatomic molecule assumes two point masses connected by a perfectly elastic massless spring. Such a model leads to a harmonic oscillation and is, therefore, called the harmonic oscillator. For this case the solution to Schrodinger’s wave equation for the ¨ determination of possible vibrational energy levels is readily found to be Ev = hνe (v + 21 ),

v = 0, 1, 2, . . . (v integer),

(11.25)

where νe is the equilibrium frequency of harmonic oscillation or eigenfrequency, and v is the vibrational quantum number. The selection rule for a harmonic oscillator is ∆v = ±1 and, thus, one would expect a single spectral line at the same frequency as the harmonic oscillation, or at a wavenumber η = (Ev+1 − Ev )/hc 0 = (νe /c 0 )(v + 1 − v) = νe /c 0 , (11.26) as indicated in Fig. 11-3. Unfortunately, the assumption of a harmonic oscillator leads to considerably less accurate results than the one of a rigid rotator. This fact is easily appreciated by looking at Fig. 11-4, which depicts the molecular energy level of a diatomic molecule vs. interatomic distance: When atoms move toward each other repulsive forces grow more and more rapidly, while the opposite is true when the atoms move apart. The heavy line in Fig. 11-4 shows the minimum and maximum distances between atoms for any given vibrational energy state

Absorption

Ev hνe

Emission 7 2

3

5 2

2

3 2

1

1 2

v = 0

νe /c0 Wavenumber η

FIGURE 11-3 Spectral position and energy levels for a harmonic oscillator.

11.3 ATOMIC AND MOLECULAR SPECTRA

311

Energy level

Harmonic oscillator

Dissociation energy 3 2 1 v=0 Equilibrium position Interatomic distance

FIGURE 11-4 Energy level vs. interatomic distance.

(showing also that the molecule may dissociate if the energy level becomes too high). In a perfectly elastic spring, force increases linearly with displacement, leading to a symmetric quadratic polynomial for the displacement limits as also indicated in the figure. If a more complicated spring constant is included in the analysis, this results in additional terms in equation (11.25); and the selection rule changes to ∆v = ±1, ±2, ±3, . . ., producing several approximately equally spaced spectral lines. The transition corresponding to ∆v = ±1 is called the fundamental, or the first harmonic, and usually is by far the strongest one. The transition corresponding to ∆v = ±2 is called the first overtone or second harmonic, and so on. For example, CO has a strong fundamental band at η0 = 2143 cm−1 and a much weaker first overtone band at η0 = 4260 cm−1 (see the data in Table 11.3 in Section 11.10). In the literature the vibrational state of a molecule is identified by the values of the vibrational quantum numbers. For example, the vibrational state of a nonlinear, triatomic molecule, such as H2 O, with its three different vibrational modes, is identified as (v1 v2 v3 ). The case is a little bit more complicated for molecules with degeneracies. For example, the linear CO2 molecule has three different vibrational modes, thesecond one being doubly degenerate (see Fig. 11-1); its  vibrational state is defined by v1 v2 l2 v3 or (v1 v2 l2 v3 ), where 0 ≤ l2 ≤ v2 is an angular momentum quantum number, describing the rotation of the molecule caused by different vibrations in perpendicular planes. More details on these issues are given by Taine and Soufiani [4] and by Herzberg [30].

Combined Vibrational–Rotational Transitions Since the energy required to change the vibrational state is so much larger than that needed for rotational changes, and since both transitions can (and indeed often must) occur simultaneously, this requirement leads to many closely spaced lines, also called a vibration–rotation band, centered around the wavenumber η = νe /c 0 , which is known as the band origin or band center. For the simplest model of a rigid rotator combined with a harmonic oscillator, assuming both modes to be independent, the combined energy level at quantum numbers j, v is given by Ev j = hνe (v + 12 ) + Bv j(j + 1),

v, j = 0, 1, 2, . . . .

(11.27)

Since the small error due to the assumption of a totally rigid rotator can result in appreciable total error when a large collection of simultaneous vibration–rotation transition is considered, allowance has been made in the above expression for the fact that Bv (or the molecular moment of inertia) may depend on the vibrational energy level. The allowed transitions (∆v = ±1 combined with ∆ j = ±1, 0) lead to three separate branches of the band, namely P (∆j = −1), Q

312

11 RADIATIVE PROPERTIES OF MOLECULAR GASES 6 5

j

v+1 4 3 2 1 0 7

6

v

j 5 4 3 2 1 0

η

P

Q η0

R

FIGURE 11-5 Typical spectrum of vibration–rotation bands.

(∆j = 0) and R (∆j = +1) branches, with spectral lines at wavenumbers ηP = η0 − (Bv+1 + Bv )j + (Bv+1 − Bv )j2 , 2

ηQ = η0 + (Bv+1 − Bv )j + (Bv+1 − Bv )j , 2

ηR = η0 + 2Bv+1 + (3Bv+1 − Bv )j + (Bv+1 − Bv )j ,

j = 1, 2, 3, . . .

(11.28a)

j = 1, 2, 3, . . .

(11.28b)

j = 0, 1, 2, . . .

(11.28c)

where j is the rotational state before the transition. It is seen that there is no line at the band origin. If Bv+1 = Bv = const, then the Q-branch vanishes and the two remaining branches yield equally spaced lines on both sides of the band center. If Bv+1 < Bv (larger moment of inertia I at higher vibrational level), then the R-branch will, for sufficiently large j, fold back toward and beyond the band origin. In that case all lines within the band are on one side of a limiting wavenumber. Those bands, where this occurs close to the band center (i.e., for small j where the line strength is strong), are known as bands with a head. A sketch of a typical vibration–rotation band spectrum is shown in Fig. 11-5. Note that in linear molecules the Q-branch often does not occur as a result of forbidden transitions [3]. Many more complicated combined transitions are possible, since every molecule has a number of rotational and vibrational energy modes, any number of which could undergo a transition simultaneously. An example is given in Fig. 11-6, which shows a calculated spectrum of the 4.3 µm CO2 band (a collection of many different vibrational transitions together with their rotational lines), generated from the HITRAN database [32]. It is apparent that this band has no Q-branch.

Electronic Transitions Electronic energy transitions, i.e., changing the orbital radius of an electron, requires a substantially larger amount of energy than vibrational and rotational transitions, with resulting photons in the ultraviolet and visible parts of the spectrum. Transitions of interest in heat transfer applications (i.e., at wavelengths above 0.25 µm) generally occur only at very high temperatures (above several thousand degrees Kelvin) and/or in the presence of large numbers of free electrons (such as fluorescent lights). At extreme temperatures atoms and molecules may also become ionized through a bound–free absorption event, or an ion and electron can recombine (free–bound emission). In addition, a free electron colliding with a molecule may absorb or emit a photon (free–free transition). If the gas is monatomic, radiation can alter only electronic energy states. Still, this results in some 914 lines for monatomic nitrogen and 682 for

313

κpη, cm 1bar

1

11.3 ATOMIC AND MOLECULAR SPECTRA

200

100

0 2300

2325

2350

2375

η, cm FIGURE 11-6 Pressure-based spectral absorption coefficient for small amounts of CO2 in nitrogen; 4.3 µm band at p = 1.0 bar, T = 296 K.

monatomic oxygen [33], contributing to heat transfer in high-temperature applications, such as the air plasma in front of a hypersonic spacecraft entering Earth’s atmosphere. As an example Fig. 11-7 shows the absorption coefficient of atomic nitrogen at T = 10,860 K, as encountered in the shock layer of the Stardust spacecraft [34]. Many of the monatomic lines are extremely strong (with absorption coefficients near 106 m−1 ), and continuum radiation (bound–free and free–free transitions) is substantial. In this part of the spectrum otherwise radiatively inert molecules, e.g., diatomic nitrogen, also emit and absorb photons, leading to simultaneous electronic–vibration– rotation bands. For comparison, the absorption coefficient for N2 is also included in Fig. 11-7, consisting of 5 electronic bands, each containing many vibration–rotation subbands. At temperatures above 10,000 K N2 is nearly completely dissociated, making its absorption coefficient small in comparison to that of monatomic N. At lower temperatures, nearly all molecules are at the lowest electronic energy level, and only the bands with η > 50,000 cm−1 , or λ < 0.2 µm remain (of no importance in most engineering applications).

Strength of Spectral Lines within a Band In equation (11.14) we related the spectral absorption coefficient to the Einstein coefficients Blu and Bul before knowing how such a transition takes place. We now want to develop equation (11.14) a little further to learn how the strength of individual lines (and, through it, the absorption coefficient) varies across vibration–rotation bands, and how they are affected by variations in temperature and pressure. For a combined vibrational (from vibrational quantum number v to v ± 1) and rotational (from rotational quantum number j to j or j ± 1) transition, the line intensity or line strength may be rewritten in terms of wavenumber (i.e., after division by c 0 ) as Sη = (nl 1l Blu − nu 1u Bul )hη,

(11.29)

where η is the associated transition wavenumber from equations (11.28). Using equations (11.5)

314

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

6

10

   

4

Absorption coefficient κ λ, m

-1

10











T = 10860 K, P = 24 kPa 19 -3 N N2 = 3.08×10 m 23 N N = 1.01×10 m

N, N



+

2

10

0

10

10

-2

10-4

N2

10-6 10-8

20000

40000

60000 80000 Wavenumber η , cm-1

100000

120000

FIGURE 11-7 Linear spectral absorption coefficient of monatomic and diatomic nitrogen in a hypersonic boundary air plasma.

and (11.7) this becomes Sη =

 nl 1u Aul  1 − e−hc 0 η/kT . 2 8πc 0 η

(11.30)

The number of molecules at the lower energy state, nl , may be related to the total number of particles per unit volume, n, through [23] nl e−El /kT = , n Q(T)

n=

p , kT

(11.31)

where Q(T) is the rovibrational partition function (a summation over all the possible rotational and vibrational energy levels of the molecule). Substituting this into equation (11.30) and relating the Einstein coefficient to matrix elements of the molecule’s electric dipole moment [20], ℜul , leads to  p  8π3 η Sη = (11.32) |ℜul |2 1 − e−hc 0 η/kT e−El /kT . 3hc 0 k Q(T)T

The rovibrational partition function Q(T) and dipole elements |ℜul |2 can, at least in principle, be calculated from quantum mechanics through very lengthy and complex calculations. For example, much of Penner’s book [20] is devoted to this subject. To gain some insight into the relative strengths of lines within a vibration–rotation band, we will look at the case of a rigid rotator–harmonic oscillator, with the additional assumptions that the bandwidth is small compared with the wavenumber at the band center and that only the P and R branches are important. For such a case the evaluation of the |ℜul |2 is relatively straightforward [20], and equation (11.32) may be restated as SP j = Cj e−hc 0 Bv j( j+1)/kT , −hc 0 Bv j( j+1)/kT

SR j = C(j + 1) e

,

j = 1, 2, 3, . . .

(11.33a)

j = 0, 1, 2, . . .

(11.33b)

where Er j = hc 0 Bv j(j + 1) is the rotational contribution to the lower energy state from equation (11.23) (i.e., before transition for absorption of a photon; after transition for emission), and C collects the coefficients in equation (11.32), as well as the vibrational contribution to the

11.4 LINE RADIATION

315

lower energy state. Examination of equations (11.33) shows that line strength first √ increases linearly with increasing j (as long as hc 0 Bv j(j + 1)/kT ≪ 1), levels off around j ≃ kT/hc 0 Bv , then drops off exponentially with large values of j. It is apparent that the band widens with temperature, and lines farther away from the band center become most important. An example is given in Fig. 11-6 for the calculated spectrum of the 4.3 µm CO2 band, generated from the HITRAN database [32]. At room temperature the 4.3 µm band is dominated by the 000 0 → 000 1 vibrational transition, centered at 2349 cm−1 . It is clear that this band has no Q-branch, and that the line strengths of the P- and R-branches closely follow equation (11.33). Temperature and pressure dependence As seen from equation (11.32) the linear line strength Sη is directly proportional to the pressure of the absorbing/emitting gas; therefore, pressurebased line strength Spη and density-based line strength Sρη are functions of temperature only. The temperature dependence comes from three contributions: (i) from the partition function Q(T), (ii) from the stimulated emission term, exp(−hc 0 η/kT), and (iii) from the lower energy state El . Evaluation of the partition function is extremely difficult, and approximations need to be made. To a good degree of accuracy rotational and vibrational contributions can be separated, i.e., Q(T) ≃ Qv (T)Qr (T). The vibrational partition function can then be determined, assuming a harmonic oscillator, as [30] Y −1k Qv (T) = 1 − e−hc 0 ηk /kT , (11.34) k

where the product is over all the different vibrational modes with their harmonic oscillation wavenumbers ηk [= νe /c 0 in equation (11.25)], and 1k is the degeneracy of the vibrational mode. The rotational partition function depends on the symmetry of the molecule and on the moments of inertia for rotation around two (linear molecule) or three (nonlinear molecule) axes. For moderate to high temperatures, i.e., when 2IkT/~2 ≫ 1 [23, 30], Linear molecules (Ix = I y = I): Nonlinear molecules:

1 2IkT ∝ T, σ ~2 !1/2 1 Y 2Ii kT Qr (T) = ∝ T3/2 , 2 σ ~ i=x,y,z Qr (T) =

(11.35a) (11.35b)

where σ is a symmetry number, or the number of distinguishable rotational modes. Examining the separate contributions to the temperature dependence we note that, at moderate temperatures, the rotational partition function causes the line strength to decrease with temperature as 1/T or 1/T3/2 , while the influences of the vibrational partition function and of stimulated emission are very minor (but may become important for T > 1000 K). The influence of the lower energy state El can be negligible or dramatic, depending on the size of El : for small values of El (low vibrational levels) exp(−El /kT) ≃ 1 and further raising the temperature will not change this value. On the other hand, large values of El (associated with high vibrational levels) make line strengths very small at low temperatures, but produce sharply increasing line strengths at elevated temperatures (when more molecules populate the higher vibrational levels), giving rise to so-called “hot lines” and “hot bands.” An example of the temperature dependence of the spectral absorption coefficient (including effects of line broadening and spacing) will be given in the next section, in Fig. 11-11.

11.4

LINE RADIATION

In the previous two sections we have seen that quantum mechanics postulates that a molecular gas can emit or absorb photons at an infinite set of distinct wavenumbers or frequencies. We already observed that no spectral line can be truly monochromatic; rather, absorption or emission occurs over a tiny but finite range of wavenumbers. The results are broadened spectral lines that have their maxima at the wavenumber predicted by quantum mechanics. In this

316

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

section we will briefly look at line strengths, the causes of line broadening and at line shapes, i.e., the variation of line strength with wavenumber for an isolated line. More detailed accounts may be found in more specialized works [2, 3, 20, 26]. The effects of line overlap, which usually occurs in vibration–rotation bands in the infrared, will be discussed in Section 11.8, “Narrow Band Models.” Numerous phenomena cause broadening of spectral lines. The four most important ones are natural line broadening, collision broadening, Stark broadening, and Doppler broadening, with collision and, to a lesser extent, Doppler broadening dominating in most engineering applications. These models have been developed for isolated lines, i.e., interaction between overlapping lines is not considered, and was found to be accurate for low-to-moderate pressures. However, at elevated pressures (roughly 10 bar) collisional interference (or line mixing) effects should be accounted for [35, 36].

Natural Line Broadening Every excited molecule will have its energy levels decay spontaneously to a lower state by emitting a photon, even if the molecule is completely undisturbed. According to Heisenberg’s uncertainty principle no energy transition can occur with precisely the same amount of energy, thus causing the energy of emitted photons to vary slightly and the spectral lines to be broadened. The mechanism of decay for that of spontaneous emission is the same as that for collision broadening as discussed in the next section, resulting in identical line shapes. However, the average time for spontaneous decay is much larger than the average time between molecular collisions. Therefore, natural line broadening is generally not important from an engineering point of view, and its effect is invariably small compared to collision broadening. Its small effect may be accounted for by adding a line half-width γN to the collision line half-width γC discussed below.

Collision Broadening As the name indicates, collision broadening of spectral lines is attributable to the frequency of collisions between gas molecules. The shape of such a line can be calculated from the electron theory of Lorentz∗ or from quantum mechanics [2, 37] as Z γC S = Sφ S ≡ κη dη, (11.36) κη = Lη (γC , η − η0 ), π (η − η0 )2 + γ2C ∆η where S is the line-integrated absorption coefficient or line strength, γC is the so-called line halfwidth in units of wavenumber (half the line width at half the maximum absorption coefficient), and η0 is the wavenumber at the line center. The line shape function is a normalized Lorentz profile, such that Z φLη (η) dη = 1.

(11.37)

∆η

The line shape function is not dimensionless, but has the units of reciprocal spectral variable. In equation (11.37) this is reciprocal wavenumber (or cm), since κη is expressed in terms of wavenumber. The shape of a collision-broadened line is identical to that of natural line broadening, and the combined effect is generally termed Lorentz broadening with a line half-width γL . The spectral distribution of a Lorentz line is shown in Fig. 11-8 (together with the shape of Doppler- and Voigt-broadened lines). Since molecular collisions are proportional to the ∗

A biographical footnote for Hendrik A. Lorentz may be found in Section 2.6.

11.4 LINE RADIATION

κη

317

(π ln2)1/2 Doppler

(S/π g)

g /g = 0.1 L D  0.3  0.5  Voigt 1   2

1 Lorentz

gD

g

0.5

L

-

4

-

3

-

2

-

1

0 (η - η 0)/g

1

2

3

4

FIGURE 11-8 Spectral line shape for Lorentz (collision), Doppler, and Voigt broadening (for equal line strength and half-width).

√ number density of molecules (n ∝ ρ ∝ p/T) and to the average molecular speed (vav ∝ T), it is not surprising that the half-width for a pure gas can be calculated from kinetic theory [2] as ! p  T0 n D2 p 2 γC = √ = γC0 , √ p0 T π c 0 mkT

(11.38)

where D is the effective diameter of the molecule, m is its mass, p is total gas pressure, T is absolute temperature, and the subscript “0” denotes a reference state. The collisional diameter depends on the temperature of the gas and the value for the exponent n must, in general, be found from experiment. If the absorbing–emitting gas is part of a mixture, the fact that collisions involving only nonradiating gases do not cause broadening, and that the nonradiating gases have different molecular diameters, must be accounted for, and equation (11.38) must be generalized to γC =

r

!   2 pi  T0 ni 1 1/2 X 2 X σi pi 1 + = γC0,i , √ π p0 T c 0 kT m mi i

(11.39)

i

where pi and mi are partial pressure and molecular mass of the various broadening gases (including the radiating gas), respectively, and σi is the effective collisional diameter with species i. Temperature-dependent broadening coefficients for some absorbing gases have been tabulated by Rosenmann et al. [38] (CO2 ), Delaye et al. [39] (H2 O), and Hartmann et al. [40], all for mixtures containing N2 , O2 , CO2 , and H2 O.

Stark Broadening Stark broadening occurs if the radiative transition occurs in the presence of a strong electric field. The electrical field may be externally applied, but it is most often due to an internal field, such as the presence of ions and free electrons in a high-temperature plasma. At low-enough pressures Stark broadened lines are symmetric and have Lorentzian shape, equation (11.36). Line widths depend strongly on free electron number density, ne , and free electron temperature,

318

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

Te , and may be calculated as [26, 41] γS = γS0



Te T0

n 

 ne , n0

(11.40)

where again the subscript “0” denotes a reference state. The Stark effect can also result in a shift in the line’s spectral position.

Doppler Broadening According to the Doppler effect a wave traveling toward an observer appears slightly compressed (shorter wavelength or higher frequency) if the emitter is also moving toward the observer, and slightly expanded (longer wavelength or lower frequency) if the emitter is moving away. This is true whether the wave is a sound wave (for example, the pitch of a whistle of a train passing an observer) or an electromagnetic wave. Thus,   v · sˆ ηobs = ηem 1 + , (11.41) c where v is the velocity of the emitter and sˆ is a unit vector pointing from the emitter to the observer. Assuming local thermodynamic equilibrium, so that Maxwell’s velocity distribution applies, the probability for a relative velocity v = v · sˆ between an emitting/absorbing molecule and an observer is !   m 1/2 mv2 p (v) = exp − , (11.42) 2πkT 2kT where m is the mass of the radiating molecule. For small v this leads to a Doppler shift in observed wavenumber of v η − η0 = η0 . (11.43) c Substituting equation (11.43) into (11.42) one can calculate the line profile as [20]  √ !2   η − η0  ln 2 (11.44) κη = S φDη (γD , η − η0 ) = S √ exp −(ln 2)  , γD γD π where γD is the Doppler line half-width, given by r η0 2kT ln 2. γD = c0 m

(11.45)

Note that, unlike during collision and natural line broadening, the Doppler line width depends on its spectral position. The different line shapes are compared in Fig. 11-8. For equal overall strength, the Doppler line is much more concentrated near the line center.

Combined Effects

√ In most engineering applications collision broadening, which is proportional to p/ T, is by far the most important broadening mechanism. Only at very high temperatures (when, owing to the distribution of the Planck function, transitions at large η are most important; and/or through the opposing temperature dependencies of γL and γD ) and/or low pressures may Doppler broad√ ening, with its proportionality to η T, become dominant. Figure 11-9 shows typical line halfwidths for CO2 and water vapor in their 2.7 µm bands as a function of temperature. It is seen that at low pressures (p = 0.1 bar) Doppler broadening always dominates. At higher pressures (p ≥ 1 bar) collision broadening dominates, unless extremely high temperatures (T > 2000 K)

11.4 LINE RADIATION

-1

CO2 H 2O

Line half-widths, cm

-1

10

319

gL (1bar)

10

gD

-2

gL (0.1bar) 10

-3

0

500

1000

1500 2000 Temperature T, K

2500

3000

FIGURE 11-9 Lorentz and Doppler line half-widths for the 2.7 µm bands of CO2 and H2 O.

are encountered. Even then the lines retain their Lorentz shape in the all-important line wings (since in gas columns line centers tend to be opaque, regardless of line shape, radiative behavior is usually governed by the strengths of the line wings). A study by Wang and Modest [42] quantifies the conditions under which combined pressure–Doppler broadening must be considered. Combined broadening behavior is also encountered in low-pressure plasmas, where both Doppler and Stark broadening can be substantial, especially for monatomic gases. If combined effects need to be considered, it is customary to assume collision and Doppler broadening to be independent of one another (which is not strictly correct). In that case a collision-broadened line would be displaced by the Doppler shift, equation (11.43), and averaged over its probability, equation (11.42). This leads to the Voigt profile [2], SγL κη = 3/2 π

Z

+∞ −∞

2

e−x dx xγD η − η0 − √ ln 2

!2

, x=v

r

2

+ γL

m . 2kT

(11.46)

No closed-form solution exists for the Voigt profile. It has been tabulated in the meteorological literature in terms of the parameter 2γL /γD . How the shape of the Voigt profile changes from pure Doppler broadening (γL /γD = 0) to pure collision broadening (γL /γD → ∞) is also shown in Fig. 11-8 (for constant line half-widths). Several fast algorithms for the calculation of the Voigt profile have also been reported [43–46]. A Fortran subroutine voigt is given in Appendix F, that calculates the Voigt κη as a function of S, γL , γD , and |η−η0 | based on the Huml´ıc˘ ek algorithm [46]. Example 11.1. The half-width of a certain spectral line of a certain gas has been measured to be 0.05 cm−1 at room temperature (300 K) and 1 atm. When the line half-width is measured at 1 atm and 3000 K, it turns out that the width has remained unchanged. Estimate the contributions of Doppler and collision broadening in both cases. Solution As a first approximation we assume that the widths of both contributions may be added to give the total line half-width (this is a fairly good approximation if one makes a substantially larger contribution than the other). Therefore, we may estimate γC1 + γD1 ≈ γ1 = γ2 ≈ γC2 + γD2

320

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

and, from equations (11.38) and (11.45), r γC2 T1 1 = = √ , γC1 T2 10

γD2 = γD1

r

T2 √ = 10. T1

Eliminating the Doppler widths from these equations we obtain √ √ γC1 γC1 γ2 = √ + 10γD1 = √ + 10(γ1 − γC1 ), 10 10 √ ! √ γC1 γ2 10 = 10 − = 0.76, γ1 9 γ1 and ! γC2 1 √ γ1 = 10 − 1 = 0.24. γ2 9 γ2 We see that at room temperature, collision broadening is about three times stronger than Doppler broadening, while exactly the reverse is true at 3000 K.

Radiation from Isolated Lines Combining equations (11.16) and (11.22) gives the complete equation of transfer for an absorbing– emitting (but not scattering) medium, dIη ds

= κη (Ibη − Iη ),

(11.47)

where the first term of the right-hand side represents augmentation due to emission and the second term is attenuation due to absorption. Let us assume we have a layer of an isothermal and homogeneous gas of thickness L. Then neither Ibη nor κη is a function of location and the solution to the equation of transfer is   (11.48) Iη (X) = Iη (0) e−κη X + Ibη 1 − e−κη X ,

where the optical path length X is equal to L if a linear absorption coefficient is used (geometric path length), or equal to L multiplied by partial density (density path length) or pressure (pressure path length) of the radiating gas if either mass or pressure absorption coefficient is used. Thus, the difference between entering and exiting intensity, integrated over the entire spectral line, is Z Z   I(X) − I(0) = [Iη (X) − Iη (0)] dη ≈ [Ibη − Iη (0)] 1 − e−κη X dη, (11.49) ∆η

∆η

where the assumption has been used that neither incoming nor blackbody intensity can vary appreciably over the width of a single spectral line. The integrand of the factor Z   1 − e−κη X dη (11.50) W= ∆η

is the fraction of incoming radiation absorbed by the gas layer at any given wavenumber, and it is also the fraction of the total emitted radiation that escapes from the layer (not undergoing self-absorption). W is commonly called the equivalent line width since a line of width W with infinite absorption coefficient would have the identical effect on absorption and emission; the dependence of the increase of W with increasing optical path X is sometimes called the curve of growth. The equivalent line width for a Lorentz line may be evaluated by substituting equation (11.36) into equation (11.50) to yield W = 2πγL x e−x [I0 (x) + I1 (x)] = 2πγL L (x),

(11.51)

11.5 NONEQUILIBRIUM RADIATION

where γL ≡ γC + γN ,

x ≡ SX/2πγL ,

321

(11.52)

the I0 and I1 are modified Bessel functions, and L(x) is called the Ladenburg–Reiche function, after the authors who originally developed it [47]. For simpler evaluation, equation (11.51) may be approximated as reported by [2] as "

πx L(x) ≃ x 1 + 2 

5/4 #−2/5

,

(11.53)

with a maximum error of approximately 1% near x = 1. Asymptotic values for W are easily obtained as W = SX, p W = 2 SXγL ,

x ≪ 1,

(11.54a)

x ≫ 1.

(11.54b)

Comparing equation (11.52) with equation (11.36), evaluated at half-height (|η − η0 | = γL ), shows that x is the nondimensional optical thickness of the gas layer, κη X, at that location. Therefore, the parameter x gives an indication of the strength of the line. For a weak line (x ≪ 1) little absorption takes place so that every position in the gas layer receives the full irradiation, resulting in a linear absorption rate (with distance). In the case of a strong line (x ≫ 1) the radiation intensity has been appreciably weakened before exiting the gas layer, resulting in locally lesser absorption and causing the square-root dependence of equation (11.54b).

11.5

NONEQUILIBRIUM RADIATION

There are many radiation applications, in which local thermal equilibrium cannot be assumed, such as in the plasma generated during atmospheric entry of spacecraft, ballistic ranges, highspeed shock tubes, arc jets, etc. When a gas is not in thermal equilibrium, its state cannot be described by a single temperature [48], and the populations of internal energy states do not follow Boltzmann distributions, equation (11.5). The thermodynamic state may then be described using a multitemperature approach (i.e., a Boltzmann distribution is assumed for each internal mode with a specific temperature) [49]. Alternatively, level population distributions may be calculated directly, taking into account collisional and radiative processes. This is known as the Collisional–Radiative model (CR) [50, 51] or, if infinitely fast reaction rates are assumed, the Quasi-Steady State (QSS) approximation [49]. Most often the more closely spaced energy levels for translation, rotation, vibration, and free electrons are assumed to have individual equilibrium distributions with up to four different temperatures (Tt , Tr , Tv , Te ), while the widely spaced electronic energy levels are modeled using the QSS/CR approach. Once all energy state distributions have been determined, the emission is given by equation (11.20). Relating it to the absorption coefficient one may define a nonequilibrium Planck function, from equation (11.21), as (in terms of wavenumbers) jη nu ne Ibη = = 2hc20 η3 . (11.55) κη nl − nu An example is given in Fig. 11-10, showing the nonequilibrium Planck function for diatomic CN (a strongly radiating ablation product from thermal protection systems) [52]. In this graph a two-temperature model was adopted with Tt = Tr = 15,000 K and Tv = Tel = Te = 10,000 K (with electronic energy levels in equilibrium at Tel ), and only Doppler broadening was considered. The ultraviolet CN band (1 ↔ 3 electronic transition) is shown, including many vibration–rotation subbands. For example, the lines labeled ∆v = vu − vl = −2 imply that the vibrational energy of the upper (electronic) level is two levels lower than that of the lower (electronic) energy state, and so on. The nonequilibrium Planck function displays line structure similar to that of the

322

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

0.75

∆v=-1

∆v=0 ∆v=+1

0.5

∆v=+2

2

-1

IBη W/cm -cm -sr

∆v=-2

∆v=+3 ∆v=+4 ∆v=+5

0.25 Nonequilibrium (Tv = 10,000 K, TR = 15,000 K) Equilibrium (T = 10,000 K)

0 22000

25000

28000 31000 Wavenumber η (cm -1)

34000

FIGURE 11-10 Nonequlibrium Planck function for CN for a two-temperature model (electronic, vibrational, and electron states at equilibrium with Tv , rotational and translational states with Tr ).

absorption coefficient. This can be better understood by looking at the special case of negligible stimulated emission and no line overlap (both good approximations for the present case). Then [cf. equation (11.31)] ne Ibη (Tv , Tr )

Ibη (Tv )

=

   nu hc 0 η/ kTv [Qvr,l /Qvr,u ]ne (Tv , Tr ) Eru − Erl 1 1 , exp e = − nl [Qvr,l /Qvr,u ](Tv ) k Tv Tr

(11.56)

where Qvr is the rovibrational partition function (depending on temperature only), and Er is the rotational energy level. Note that u and l refer to the upper and lower states of the total transition, always determined by the electronic level, i.e., Eru −Erl is the rotational energy change for a given transition (spectral line), which can be negative (lines below the equilibrium Planck function in Fig. 11-10). As can be appreciated from the discussion in this section, and on electronic transitions in Section 11.3, radiation in high-temperature nonequilibrium plasmas is considerably more complicated than usually encountered in engineering, and is beyond the scope of the present text. The reader is referred to the literature dedicated to such problems [49, 53].

11.6 HIGH-RESOLUTION SPECTROSCOPIC DATABASES During the past 40 years or so, due to the advent of high-resolution spectroscopy (mostly FTIR spectrometers), it has become possible to measure strengths and positions of individual spectral lines. A first collection of spectral data was assembled in the late 1960s by the Air Force Cambridge Research Laboratories for atmospheric scientists, including low-temperature data for the major constituents of the Earth’s atmosphere, and was published in 1973 as an Air Force report [54]. With contributions from many researchers across the world this grew into the HITRAN database (an acronym for HIgh resolution TRANsmission molecular absorption), first published in 1987 [55]. The database is maintained by the Harvard–Smithsonian Center for Astrophysics, with periodic updates [32, 56–59]. The latest version at present is HITRAN

11.6 HIGH-RESOLUTION SPECTROSCOPIC DATABASES

323

2008 [32], which includes detailed information on 39 species with a total of about 2.7 million lines. As the popularity of HITRAN grew, the need for a database valid at elevated temperatures became obvious. A first attempt was made by the group around Taine in France, who augmented HITRAN 1986 data for water vapor and carbon dioxide through theoretical calculations [60,61]. A development by the HITRAN group resulted in a first version of HITEMP (1995) [62] for H2 O, CO2 , CO, and OH, using theoretical models. Comparison with experiment [63–66] indicated that HITEMP 1995 greatly overpredicted CO2 emissivities above 1000 K, while agreement for H2 O was acceptable. More accurate and extensive calculations for CO2 were carried out in Russia, resulting in several versions of the CDSD-1000 database [67, 68] (with the 2008 version containing 4 million lines), which were shown to agree well with experiment. The latest version of CDSD, called CDSD-4000 [69], aims to be accurate up to 4000 K, and has 628 million lines, requiring 23 GB of storage. Several extensive high-temperature collections were developed for H2 O: the Ames database [70] includes 300 million lines, SCAN [71] contains 3 billion, and the BT2 collection [72] has 500 million lines; building up on the Ames database, Perez et al. [73] rejected lines from the Ames collection that remain weak below 3000 K, and combined it with wellestablished lines from HITRAN 2001 and HITEMP 1995, culminating in a manageable collection with 1.3 million lines. Finally, in 2010 a new version of HITEMP was released [74], designed for temperatures up to 3000 K. Citing best agreement against experimental data, HITEMP 2010 incorporates and extends CDSD 2008 for CO2 (11 million lines) and a slimmed-down version of BT2 for H2 O (111 million lines). HITEMP 2010 also includes data for three diatomic gases (CN, CO, and OH) with their relatively few lines. Approximate high-temperature data for methane (up to 2000 K) are available from [75]. An example calculation is given in Fig. 11-11, showing a small part of the artificial spectrum of the 4.3 µm CO2 band, generated from the HITRAN database [32], assuming Lorentz broadening, and containing more than 1,500 spectral lines. The top frame of Fig. 11-11 shows the pressure-based absorption coefficient of CO2 at low partial pressure in air at a total pressure of 10 mbar. Because of the relatively low total pressure, the lines are fairly narrow, resulting in little overlap. If the total pressure is raised to 1 bar, shown in the center frame, lines become strongly broadened, leading to substantial line overlap, and a smoother variation in the absorption coefficient (with considerably lower maxima and higher minima). At the high temperatures usually encountered during combustion the spectral lines narrow considerably [see equation (11.38)], decreasing line overlap; at the same time the strengths of the lines that were most important at low temperature decrease according to equation (11.35) and finally, at high temperatures “hot lines,” that were negligible at room temperature, become more and more important. To be valid up to 3000 K HITEMP 2010 [74] lists more than 22,000 spectral lines for this small wavenumber range. The result is a fairly erratic looking absorption coefficient as depicted in the bottom frame of Fig. 11-11. If high temperatures are combined with low total pressures (not shown), the spectral behavior of the absorption coefficient resembles high-frequency electronic noise. Fortunately, heat transfer calculations in media at low total pressure are rare (they are important, though, in meteorological applications dealing with the low-pressure upper atmosphere). Similar efforts have been made by the plasma radiation community. RAD/EQUIL is perhaps the earliest attempt, including contributions from atomic lines and continua, and approximate models for molecules, but only for thermodynamic equilibrium conditions [76]. The NonEQuilibrium AIr Radiation (NEQAIR) model [77] was originally developed for the study of radiative properties of nonequilibrium, low density air plasmas. The updated NEQAIR96 model [78] includes spectral line data for spontaneous emission, stimulated emission, and absorption for 14 monatomic and diatomic species, as well as bound–free and free–free transition data for atoms. Nonequilibrium electronic level populations are determined using the QSS approximation (cf. Section 11.5). Since the creation of NEQAIR various improvements have been made by Laux [79] and others, leading to the SPECAIR database [80]. In Japan the SPRADIAN database was assembled [81], which was recently updated in cooperation with KAIST [82]. A new High-

324

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

105 T = 300 K, p = 10 mbar, pCO2 = 0 bar

κpη, cm–1 bar–1

104

103

102

101

100

10-1

κpη, cm–1 bar–1

›2 103 10

10

2

10

1

2

κpη, cm–1 bar–1

0 10

10

T = 300 K, p = 1 bar, pCO2 = 0 bar

T = 1000 K, p = 1 bar, pCO2 = 0 bar

1

0

10 2320

2322

2324

η, cm–1

2326

2328

2330

FIGURE 11-11 Spectral absorption coefficient for small amounts of CO2 in nitrogen, across a small portion of the CO2 4.3 µm band; top frame: p = 10 mbar, T = 300 K; center frame: p = 1 bar, T = 300 K; bottom frame: p = 1 bar, T = 1000 K.

11.7 SPECTRAL MODELS FOR RADIATIVE TRANSFER CALCULATIONS

325

temperature Aerothermodynamic RAdiation model (HARA) developed by Johnston [50, 83] utilizes comprehensive and updated atomic line data obtained from the National Institute of Standards and Technology (NIST) online database [84] and the Opacity Project [85], as well as atomic bound-free cross-sections from the TOPbase [86]. Since the above databases are generally stand-alone programs, incorporating several other tools, such as primitive RTE solvers, Sohn et al. [87] extracted the relevant data from NEQAIR96 to form an efficient radiative property module. This database has very recently been updated for high-speed retrieval rates and to incorporate the state-of-the-art data in HARA [88].

11.7 SPECTRAL MODELS FOR RADIATIVE TRANSFER CALCULATIONS A single spectral line at a certain spectral position is fully characterized by its strength (the intensity, or integrated absorption coefficient) and its line half-width (plus knowledge of the broadening mechanism, i.e., collision and/or Doppler broadening). However, a vibration– rotation band has many closely spaced spectral lines that may overlap considerably. While the absorption coefficients for individual lines may simply be added to give the absorption coefficient of an entire band at any spectral position, X κη = κηj , (11.57) j

the resulting function tends to gyrate violently across the band (as seen in Figs. 11-6 and 11-11), unless the lines overlap very strongly. This tendency, plus the fact that there may be literally millions of spectral lines, makes radiative transfer calculations a truly formidable task, if the exact relationship is to be used in the spectral integration for total intensity [equation (10.28)], total radiative heat flux [equation (10.52)], or the divergence of the heat flux [equation (10.59)]. This has prompted the development of a number of approximate spectral models. Exact and approximate methods may be loosely put into four groups (in order of decreasing complexity and accuracy): (1) line-by-line calculations, (2) narrow band calculations, (3) wide band calculations, and (4) global models. Line-By-Line Calculations With the advent of powerful computers and the necessary highresolution spectroscopic databases, a number of spectrally resolved or “line-by-line calculations” have been performed, a few for actual heat transfer calculations, e.g., [89–91], some to prepare narrow band model correlations, e.g., [92, 93], and others to validate global spectral models, e.g., [94–96]. Such calculations rely on very detailed knowledge of every single spectral line, taken from one of the high-resolution spectroscopic databases described in Section 11.6. Because of strongly varying values of the absorption coefficient (see Fig. 11-11), the spectral radiative transfer problem must be solved for up to one million wavenumbers, followed by integration over the spectrum. While such calculations may be the most accurate to date, they require vast amounts of computer resources. This is and will remain undesirable, even with the availability of powerful computers, since radiative calculations are usually only a small part of a sophisticated, overall fire/combustion code. In addition, high-resolution gas property data (resolution of better than 0.01 cm−1 ), which are required for accurate line-by-line calculations, are generally found from theoretical calculations and mostly still remain to be validated against experimental data. In particular, temperature and pressure dependence of spectral line broadening is very complicated and simply not well enough understood to extrapolate room temperature data to the high temperatures important in combustion environments. For these reasons it is fair to assume that, for the foreseeable future, line-by-line calculations will only be used as benchmarks for the validation of more approximate spectral models. Narrow Band Models When calculating spectral radiative fluxes from a molecular gas one finds that the gas absorption coefficient (and with it, the radiative intensity) varies much more rapidly across the spectrum than other quantities, such as blackbody intensity, etc. It is, therefore,

326

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

in principle possible to replace the actual absorption coefficient (and intensity) by smoothened values appropriately averaged over a narrow spectral range. A number of such “narrow band models” were developed some 40–50 years ago, and will be examined in the following section. In principle, narrow band calculations can be as accurate as line-by-line calculations, provided an “exact” narrow band average can be found. The primary disadvantages of such narrow band models are that they are difficult to apply to nonhomogeneous gases and the fact that heat transfer calculations, based on narrow band data and using general solution methods, are limited to nonscattering media within a black-walled enclosure. An alternative to the “traditional” narrow band models is the so-called “correlated k-distribution.” In this method it is observed that, over a narrow spectral range, the rapidly oscillating absorption coefficient κη attains the same value many times (at slightly different wavenumbers η), each time resulting in identical intensity Iη and radiative flux (provided the medium is homogeneous, i.e., has an absorption coefficient independent of position). Since the actual wavenumbers are irrelevant (across the small spectral range), in the correlated k-distribution method the absorption coefficient is reordered, resulting in a smooth dependence of absorption coefficient vs. artificial wavenumber (varying across the given narrow range). This, in turn, makes spectral integration very straightforward. k-distributions are relatively new, and are still undergoing development. While attractive, they also are difficult to apply to nonhomogeneous media. Wide Band Models Wide band models make use of the fact that, even across an entire vibration–rotation band, blackbody intensity does not vary substantially. In principle, wide band correlations are found by integrating narrow band results across an entire band, resulting in only slightly lesser accuracy. Wide band model calculations have been very popular in the past, due to the facts that the necessary calculations are relatively simple and that much better spectral data were not available. However, it is well recognized that wide band correlations have a typical correlational accuracy of ±30%, and in some cases may be in error by as much as 70%; substantial additional but unquantified errors may be expected due to experimental inaccuracies. One of the attractions of the correlated k-distributions is that they can be readily adapted to wide band calculations. Global Models In heat transfer calculations it is generally only the (spectrally integrated) total radiative heat flux or its divergence that are of interest. Global models attempt to calculate these total fluxes directly, using spectrally integrated radiative properties. Most early global methods employ the total emissivities and absorptivities of gas columns, but more recently full-spectrum correlated k-distributions have also been developed. During the remainder of this chapter we will discuss the smoothing of spectral radiative properties of molecular gases over narrow bands and wide bands, as well as the evaluation of total properties. Actual heat transfer calculations using these data will be deferred until Chapter 20 (i.e., until after the discussion of particulate properties and of solution methods for the radiative transfer equation). Global models require manipulation of the RTE and, thus, will also be deferred to Chapter 20.

11.8

NARROW BAND MODELS

Examination of the formal solution to the equation of radiative transfer, equation (10.28), shows that all spectral integrations may be reduced to four cases, namely, Z ∞ Z ∞   R  X I(b)η 1 − exp − 0 κη dX dη, (11.58) κη I(b)η dη and 0

0

where I(b)η denotes that either Ibη or Iη can occur, and X is the optical path length introduced in equation (11.48). It is clear from inspection of Fig. 1-5 that the Planck function will never vary appreciably over the spectral range of a few lines, considering that adjacent lines are very closely spaced (measured in fractions of cm−1 ). Local radiation intensity Iη , on the other hand,

327

Spectral absorption coefficient κη

Spectral absorption coefficient κη

11.8 NARROW BAND MODELS

d

η 0 – 2d η 0 – d η0 η0 + d η0 + 2 d Wavenumber η (a) FIGURE 11-12 Typical spectral line arrangement for (a) Elsasser and (b) statistical model.

Wavenumber η (b)

may vary just as strongly as the absorption coefficient, since emission within the gas takes place at those wavenumbers where κη is large [see equation (10.10)]. However, if we limit our consideration to nonscattering media bounded by black (or no) walls, the formal solution of the radiative equation of transfer, equation (10.29), shows that all spectral integrations involve only the Planck function, and not the local intensity. For such a restricted scenario4 we may simplify expressions (11.58), with extremely good accuracy, to Z

∞ 0

Ibη

(

1 ∆η

Z

η+∆η/2 ′

κη dη η−∆η/2

)



(11.59a)

and Z

∞ 0

Ibη

(

1 ∆η

Z

η+∆η/2 η−∆η/2

)   R  X 1 − exp − 0 κη dX dη′ dη.

(11.59b)

The expressions within the large braces are local averages of the spectral absorption coefficient and of the spectral emissivity, respectively, indicated by an overbar:5 Z η+∆η/2 1 κη dη′ , κη (η) = ∆η η−∆η/2 Z η+∆η/2   R  1 X 1 − exp − 0 κη dX dη′ . ǫη (η) = ∆η η−∆η/2

(11.60) (11.61)

One can expect the spectral variation of κ and ǫ to be relatively smooth over the band, making spectral integration of radiative heat fluxes feasible. To find spectrally averaged or “narrow band” values of the absorption coefficient and the emissivity, some information must be available on the spacing of individual lines within the group and on their relative strengths. A number of models have been proposed to this purpose, of which the two extreme ones are the Elsasser model, in which equally spaced lines of equal intensity are considered, and the statistical models, in which the spectral lines are assumed to have random spacing and/or intensity. A typical spectral line arrangement for these two extreme models is shown in Fig. 11-12. The main distinction between the two models is the difference in line overlap. Both models will predict the same narrow band parameters for optically thin situations or nonoverlap conditions (since overlap has no effect), as well as for optically very strong situations (since no beam can penetrate through the gas, regardless of the overlapping 4 If the Monte Carlo method is employed as the solution method, this restriction is not necessary, since integration over local intensity is avoided even for reflecting walls/scattering media; see Section 21.3. 5 It should be understood that the definition of κ in equation (11.60) is not sufficient since ǫ , 1 − exp(−κs). This fact will be demonstrated in Example 11.2.

328

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

characteristics). Under intermediate conditions the Elsasser model will always predict a higher emissivity/absorptivity than the statistical models, since regular spacing always results in less overlap (for the same average absorption coefficient) [3]. The deviation between the models is never more than 20%. In the following we will limit our discussion to lines of Lorentz shape, since collision broadening generally dominates at the relatively high pressures encountered in heat transfer applications. Discussion on models for Doppler and Voigt line shapes can be found in the meteorological literature, e.g., [2].

The Elsasser Model We saw earlier in this chapter that diatomic molecules and linear polyatomic molecules have only two, identical rotational modes, resulting in a single set of lines (consisting of two or three branches, as shown in Fig. 11-2 and Fig. 11-5). For these gases one may expect spectral lines with nearly constant spacing and slowly varying intensity, in particular if the Q-branch is unimportant (or “forbidden”) and if the folding back of the R-branch gives also only a small contribution. Summing up the contributions from infinitely many Lorentz lines on both sides of an arbitrary line with center at η0 , we get κη =

∞ X γL S , π (η − η0 − id)2 + γ2L

(11.62)

i=−∞

where d is the (constant) spacing between spectral lines.6 This series may be evaluated in closed form, as was first done by Elsasser, resulting in [97] κη =

sinh 2β S , d cosh 2β − cos(z − z0 )

where β ≡ πγL /d,

z ≡ 2πη/d.

(11.63) (11.64)

From equation (11.60), the average absorption coefficient is simply κη =

S . d

(11.65)

This also follows without integration from the fact that S is each line’s contribution to the integrated absorption coefficient [see equation (11.36)], and that the lines are spaced d wavenumbers apart, i.e., for every d wavenumbers S is added to the integrated absorption coefficient. The spectrally averaged emissivity may be evaluated from equation (11.61) as ! Z π 2βx sinh 2β 1 ǫη = 1 − exp − dz, (11.66) 2π −π cosh 2β − cos z where, since the absorption coefficient is a periodic function, one full period was chosen for the averaging wavenumber range and, thus, the arbitrary location z0 could be eliminated. As one may see from its definition, equation (11.64), β is the line overlap parameter: β gives an indication of how much the individual lines overlap each other, and x, already defined in equation (11.52), is the line strength parameter. At this point we may also define another nondimensional parameter, the narrow band optical thickness τ = κX, so that we now have three characterizing parameters, namely, γL SX S x= (11.67) , β = π , τ = X = 2βx. 2πγL d d 6 Since we are using wavenumber here, the value for d is measured in units of wavenumbers, cm−1 . If we were to use frequency or wavelength, the definition and units of d would correspondingly change.

11.8 NARROW BAND MODELS

329

Equation (11.66) cannot be solved in closed form, but an accurate approximate expression, known as the Godson approximation, has been given [2]: ǫη ≈ erf

! ! √ √ √  πW π S −x = erf X e [I0 (x) + I1 (x)] = erf πβL(x) 2 d 2 d

(11.68)

where erf is the error function and is tabulated in standard mathematical texts [98]. The Godson approximation is reasonably accurate for small-to-moderate line overlap (β < 1). For larger values of β, and for hand calculations it is desirable to have simpler expressions. We can distinguish among three different limiting regimes: weak lines (x ≪ 1) : strong overlap (β > 1) :

  S ǫη = 1 − exp − X = 1 − e−τ , d r  p   S γL   ǫη = erf  π X = erf τβ , d d

strong lines (x ≫ 1) : no overlap (β ≪ 1) :

ǫη =

W = 2βL(x), d

(11.69a) (11.69b) (11.69c)

where the W/d in equation (11.69c) can possibly be further simplified using equations (11.54a) and (11.54b). These relations are summarized in Table 11.1.

The Statistical Models In the statistical models it is assumed that the spectral lines are not equally spaced and of equal strength but, rather, are of random strength and are randomly distributed across the narrow band. This assumption can be expected to be an accurate representation for complex molecules for which lines from different rotational modes overlap in an irregular fashion. In several early studies Goody [99] and Godson [100] showed that any narrow band model with randomly placed spectral lines, with arbitrary strengths and line shape (i.e., Lorentzian or other) leads to the same expression for the spectrally averaged emissivity, ! W , ǫη = 1 − exp − d

(11.70)

where W is an average over the N lines contained in the spectral interval, W=

N 1 X Wi , N

(11.71)

i=1

and d is the average line spacing, defined as d=

∆η . N

(11.72)

A number of statistical models have been developed, in which lines are placed at random across ∆η with random strengths picked from different probability distributions. We will limit our brief discussion to three different models, which excel due to their simplicity and/or their success to model actual spectral distributions. The simplest statistical model is the uniform statistical model, in which all lines have equal strengths, or Uniform statistical model:

S = S = const.

(11.73)

330

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

A more realistic representation must allow for varying lines strengths, given by a probability density function p(S). The properties of the narrow band are then found by averaging line properties with the probability density function. A frequently used such probability distribution is the exponential form proposed by Goody [99], ! 1 S Goody model: p(S) = exp − , 0 ≤ S < ∞, (11.74) S S which is popular due to its simplicity. However, Malkmus [101] recognized that in many cases this exponential intensity distribution severely underpredicts the number of low-strength lines. He modified the physically plausible 1/S distribution proposed by Godson [100] to obtain an exponential-tailed 1/S distribution, now known as the Malkmus model: ! 1 S Malkmus model: p(S) = exp − , 0 ≤ S < ∞. (11.75) S S All three distribution functions, equations (11.73), (11.74), and (11.75), have identical average line strengths S. Finding the average equivalent line width W for the uniform statistical model is trivial, since every equivalent line width from equation (11.73) is identical, and W = W (single line). For the Goody and Malkmus model the sum in equation (11.71) can, for a large statistical sample, be replaced by an integral: Z ∞ Z ∞ Z +∞   1 − e−κη (S)X dη dS. (11.76) W −→ p(S)W(S) dS = p(S) N→∞

0

0

−∞

Substituting equations (11.74) and (11.75) and carrying out the integrations leads to, for Lorentz lines, ! γ W SX Uniform statistical model: = 2π L L = 2βL(x) = 2βL(τ/2β), (11.77) d d 2πγL , !1/2 . W S SX = X 1+ = τ (1 + τ/β)1/2 , (11.78) Goody model: d d πγL  !1/2  i  β h πγL  4SX W 1/2  =  1 + (1 + 4τ/β) = − 1 , (11.79) − 1 Malkmus model:   2 d 2d  πγL

where L(x) is the Ladenburg–Reiche function given by equation (11.51). In these models the narrow band parameters γL /d and S/d are either found by fitting experimental data, or from highresolution spectral data, such as the HITRAN database [32]. In the latter case, it is desirable to have the models yield exact results in the limits of weak lines (x ≪ 1) as well as strong lines (x ≫ 1). In the weak line limit we have, for all three models, weak lines (x ≪ 1) :

W S → X = 2βx = τ, d d

(11.80)

while the models lead to slightly different strong line limits, i.e., strong lines (x ≫ 1) : Uniform statistical: Goody/Malkmus:

W → d W → d

q 2 γL SX

= 2β(2x/π)1/2 = 2(τβ/π)1/2 ,

(11.81a)

q

= β(2x)1/2 = (τβ)1/2 .

(11.81b)

d

πγL SX d

11.8 NARROW BAND MODELS

331

TABLE 11.1

Summary of effective line widths and narrow band emissivities for Lorentz lines.

Single line, W

Weak line

Strong line

No overlap

x≪1

x≫1 p 2 SXγL

β≪1

SX

W d Elsasser model W d

Statistical models W (S = const) d

τ

2βL(τ/2β)

τ

p 2 τβ/π p  erf τβ

2βL(τ/2β)  √π W  erf 2 d

τ

W (Goody) d

p 2 τβ/π

τ

W (Malkmus) d

p τβ

τ

p τβ

W d

2βL(τ/2β) τ β 2

W d

  1 − exp −W/d

1 − e−τ

ǫη

2πγL L(τ/2β)

p 2 τβ/π

1 − e−τ

ǫη

All regimes

.p 1 + τ/β

hp i 1 + 4τ/β − 1

  1 − exp −W/d

Definitions: x=

SX ; 2πγL

β=π

γL ; d

τ=

S X = 2βx; d

"  5/4 #−2/5 πx L(x) ≃ x 1 + 2

Satisfying these two conditions requires [2] N

1 X S = Si , d ∆η i=1

Cγ γL = d ∆η

hP

i2 N 1/2 i=1 (Si γLi ) PN

i=1

Si

,

(11.82)

with Cγ = 1 for the uniform statistical model, and Cγ = 4/π for the Goody and Malkmus models; the latter two models will always have some weak lines, resulting in a smaller value for W/d, even in the strong line limit (based on average line strength). The results from the statistical models have also been summarized in Table 11.1. The narrow band emissivities from all four models are compared in Fig. 11-13 as a function of the optical path of an average spectral line (i.e., average absorption coefficient S/2πγL multiplied by distance X). Note that all predictions are relatively close to each other, although the statistical models may predict up to 20% lower emissivities for optically thick situations. The Goody and Malkmus models more or less coincide for small values of β, giving somewhat lower emissivities than the uniform statistical model because of their different strong line behavior. For optically thin situations (x < 1) the uniform statistical and Goody’s model move toward the Elsasser model, with lower emissivities predicted by the Malkmus model. Note that the Elsasser lines were drawn from numerical evaluations of equation (11.66), not from equation (11.68), which would show serious error for the β = 1 line. Example 11.2. The following data are known at a certain spectral location for a pure gas at 300 K and 0.75 atm: The mean line spacing is 0.6 cm−1 , the mean line half-width is 0.03 cm−1 , and the mean line strength (or integrated absorption coefficient) is 0.08 cm−2 atm−1 . What is the mean spectral emissivity for geometric path lengths of 1 cm and 1 m, if the gas is diatomic (such as CO), or if the gas is polyatomic (such as water vapor)?

332

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

1

β → ∞ (all models) Elsasser model Uniform statistical model Goody model Malkmus model

Emissivity



η

0.8

β = 1.0 β = 0.1

0.6

β = 0.01 0.4

0.2

0 10

2

10

1

0

10 10 _ τ = 2β x = (S/d )X

1

10

2

10

3

FIGURE 11-13 Mean spectral emissivities for Lorentz lines as a function of average optical depth (S/d)X.

Solution Since the units of the given line strength tell us that a pressure absorption coefficient has been used, we need to employ a pressure path length X = ps. For a path length of 1 cm we get X = 0.75 atm × 1 cm = 0.75 cm atm and x = SX/2πγ = 0.08 cm−2 atm−1 × 0.75 cm atm/(2π 0.03 cm−1 ) = 1/π, while the overlapping parameter turns out to be β = πγ/d = π × 0.03 cm−1 /0.6 cm−1 = π/20, and τ = 2βx = 2(π/20)(1/π) = 0.1. For a diatomic gas for which the Elsasser model should be more accurate, we can use either equation (11.68) or (since β ≪ 1) equation (11.69c). Evaluating the Ladenburg–Reiche function from (11.53) gives   i−2/5 1h 1 = 1 + 0.55/4 = 0.2766, L π π and  √ π π ǫη = erf π 0.2766 = erf (0.0770) = 0.0867 ≃ 2 0.2766 = 0.0869 = 8.7%. 20 20 If the gas is polyatomic we may want to use one of the statistical models. Choosing the Malkmus model, equation (11.79), we obtain    !1/2     4 × 0.1   1 π  = 0.0670. − 1 ǫη = 1 − exp  −  1 +     2 20 (π/20)

If the path length is a full meter, we have X = 75 cm atm and x = 100/π while β is still β = π/20 and now τ = 10. Thus we are in the strong-line region. For the diatomic gas, from equation (11.69b) p ǫη = erf[ 10(π/20)] = erf(1.2533) = 0.924. For the polyatomic gas, again using equation (11.79), we get ǫη = 0.692. In the first two cases, using the simple relation ǫ = 1 − exp(−κs) actually would have given fairly good results (0.095) because the gas is optically thin resulting in essentially linear absorption at every wavenumber. For the larger path we would have gotten 1 − e−10 ≈ 1. Thus, using an average value for the absorption coefficient makes the gas opaque at all wavenumbers rather than only near the line centers. Example 11.3. For a certain polyatomic gas the line-width-to-spacing ratio and the average absorption coefficient for a vibration–rotation band in the infrared are known as       S S −2|η−η0 |/ω S (11.83) ≈ e , = 10 cm−1 , d η d 0 d 0 γ ≈ 0.1 ≈ const. ω = 50 cm−1 , d

11.8 NARROW BAND MODELS

333

Find an expression for the averaged spectral emissivity and for the total band absorptance, defined by Z ∞ Z  ǫη dη = 1 − e−κη X dη, A≡ band

0

for a path length of 20 cm. Solution Calculating the optical thickness τ0 = (S/d) 0 X = 10 × 20 = 200, the overlap parameter β = π/10, and the line strength x0 = τ0 /2β = 1000/π, we find that this band falls into the “strong-line” regime everywhere except in the (unimportant) far band wings. Since we have a polyatomic molecule with exponential decay of intensity, one of the statistical models should provide the best answer. As seen from Fig. 11-13, all three statistical models give very similar results, and the (more appropriate) Goody and Malkmus models go to the same strong line limit, equation (11.81b), or  p  ǫη = 1 − e −W/d ≈ 1 − exp − τβ , since τ/β ≫ 1. Substituting yields the spectral emissivity,   p ǫη = 1 − exp − τ0 β e−|η−η0 |/ω .

Integrating this equation over the entire band gives the total band absorptance, Z ∞h  p i A= 1 − exp − τ0 β e−|η−η0 |/ω dη. 0

Realizing that this integral has two symmetric parts and setting ln z = −(η − η0 )/ω, we have A = 2ω

Z

1 0

h  p i dz 1 − exp − τ0 βz . z

This integral may be solved in terms of exponential integrals7 as given, for example, in Abramowitz and Stegun [98]. This leads to   p p A = 2ω E1 ( τ0 β) + ln( τ0 β) + γE = 264.7 cm−1 ,

where γE = 0.57721 . . . is Euler’s constant.

Most available narrow band property data, such as the RADCAL database [102, 103], have been correlated with the Goody model. The correlation by Malkmus is a relative latecomer, but is today recognized as the best model for polyatomic molecules. While commonly used in the atmospheric sciences this correlation was widely ignored by the heat transfer community for many years. Taine and coworkers [92, 93, 104] have generated artificial narrow band properties from HITRAN 1992 line-by-line data. Employing the Malkmus model with a resolution of 25 cm−1 they observed a maximum 10% error between line-by-line and narrow band absorptivities. Using two narrow spectral ranges of H2 O and CO2 Lacis and Oinas [105] showed that (for a resolution of 10 cm−1 , and for total gas pressures above 0.1 atm) the correlational accuracy of the Malkmus model can be improved to better than 1% if the model parameters are found through least square fits of the HITRAN 1992 line-by-line data. Soufiani and Taine [106] have assembled the Malkmus-correlated EM2C narrow band database (25 cm−1 resolution) for various gases, using the HITRAN 1992 database together with some proprietary French high-temperature extensions. However, to date very few experimental narrow band data have been correlated with the Malkmus model: Phillips has measured and correlated the 2.7 µm H2 O band [107] and the 4.3 µm CO2 band [108], both between room temperature and 1000 K. Both the RADCAL and the EM2C databases are included in Appendix F. More recently, two generalizations of the Malkmus model have been developed, a multiscale model for nonhomogeneous gases [109] (see also below) and a generalized model more appropriate for Doppler-dominated regimes [110]. 7

Exponential integrals are discussed in some detail in Appendix E.

334

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

Gas Mixtures Experimental data for narrow band properties, such as line overlap (γ/d) and average absorption coefficient (S/d), are usually given from correlations of measurements performed on a homogeneous column involving a single absorbing gas species. In practical applications, on the other hand, radiative properties of mixtures that contain several absorbing gas species, such as CO2 , H2 O, CO, etc., are generally required. Over large portions of the spectrum spectral lines from different species do not overlap each other, and the expressions given in Table 11.1 remain valid. However, there are regions of the spectrum where spectral line overlap is substantial and must be accounted for. For example, the two most important combustion gases, water vapor and CO2 , both have strong bands in the vicinity of 2.7 µm. Mixture values for (γ/d) and (S/d) are found from their definitions, equation (11.82), by setting X XXp XX Xp Si γi = Sni γni , (11.84) Si = Sni ; i

n

i

n

i

i

where the subscript n identifies the gas species. Comparing equation (11.82) for the mixture and its individual components readily leads to   X S S = ; d mix d n n

r 2    X  γ   S   S  .  = d mix d mix  n d n d n

γ

(11.85)

Expressions in Table 11.1 together with equation (11.85) can then be used to evaluate the transmissivity of a gas mixture. Other expressions for mixture values of (γ/d) and (S/d) have been discussed by Liu and coworkers [111]. Taine and Soufiani [4] pointed out that there is no physical reason why there should be any significant correlation between the spectral variation of absorption coefficients of different gas species. If one treats the absorption coefficients of the M species as statistically independent random variables, the transmissivity of a mixture can be evaluated as the product of the individual species’ transmissivities, M Y τη,mix = 1 − ǫη,mix = τη,m . (11.86) m=1

For example, comparing the mixture transmissivity of a room temperature water vapor–carbon dioxide mixture for the overlapping 2.7 µm region, calculated directly from the HITRAN database and from equation (11.86), they found them to be virtually indistinguishable.

Nonhomogeneous Gases Up to this point in calculating narrow band emissivities we have tacitly assumed that the gas is isothermal, and has constant total and partial pressure of the absorbing gas everywhere, i.e., RX we replaced the integral 0 κ dX in equation (11.61) by κX. We now want to expand our results to include nonhomogeneous gases. For the Elsasser model the solution to equation (11.66) is possible, but too cumbersome to allow a straightforward solution if properties are pathdependent. For the more important statistical models the same is true, especially if not only line strength, S, but also the line overlap parameter, β, varies along the path. Instead, one resorts to approximations. The best known and most widely used approximation is known as the Curtis–Godson two parameter scaling approximation [2, 112], which has been fairly successful. Other scaling approximations have been developed, e.g., the one by Lindquist and Simmons [113]. In the Curtis–Godson approximation the values of τ and β used in equation (11.66), or (11.68) (Elsasser model) and equations (11.70) plus (11.77) through (11.79) (statistical models) are replaced by path-averaged values e τ and e β. The proper values (scaling) for e τ and

11.8 NARROW BAND MODELS

335

e β are found by satisfying both the optically thin and optically thick limits. Thus, we find from equations (11.54a) and (11.54b), for a single line “i”, x≪1:

Wi =

Z

X

Si (X) dX,

0

s Z

x≫1:

Wi = 2

(11.87)

X

Si (X) γLi (X) dX.

0

(11.88)

For many lines, from equation (11.71), Z X N Z X 1 X Si (X) dX = W= S(X) dX. N 0 0

x≪1:

(11.89)

i=1

Now, from equation (11.69a) or (11.80), W e = τ= d

For strong lines we obtain

Z

X 0

N 2 X W= N

x≫1:

! S dX. d

s

i=1

Z

(11.90)

X

Si (X) γLi (X) dX.

0

(11.91)

If one assumes Si and γLi to be separable, i.e., they can be written as, e.g., Si (X) = Si0 fs (X), where Si0 is a different constant for each line, and fs (X) is a function of the path (but the same for each line), one can—after some manipulation—rewrite equation (11.91) as [4] 2 W = N 2

x≫1:



2 Z

X 0

 N 2 X p   Si (X) γLi (X)  dX. 

(11.92)

i=1

Comparing with equation (11.54b) [or (11.81)], and utilizing equation (11.82) we obtain W d or

!2

4/π e 4 e = τβ = 2 Cγ d 1 e β= e τ

Z

X 0

Z

X 0

S(X) γL (X) dX

S β dX. d

(11.93)

(11.94)

Equations (11.68) and (11.77) through (11.79) may now be used with e τ and e β to calculate narrow band emissivities for nonhomogeneous paths. The accuracy of various scaling approximations was tested by Hartmann and coworkers [93, 104] for various nonhomogeneous conditions in CO2 –N2 and H2 O–N2 mixtures. It was found that the Malkmus model together with the Curtis–Godson scaling approximation generally gave the most accurate results, except in the presence of strong (total) pressure gradients. More recently a multiscale Malkmus model was developed by Bharadwaj and Modest [109] to improve its accuracy for nonhomogeneous paths. In this scheme it is assumed that high-temperature spectral lines (coming from elevated vibrational energy levels, i.e., with larger lower level energy El ) are uncorrelated from lower temperature lines. This implies that transmissivities of the individual “scales” are multiplicative [equation (11.86)]. Separating the gas accordingly into

336

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

scales and applying equation (11.75) to each scale m as well as the Curtis–Godson approximation leads to   !1/2  4˜τm W X β˜m   , 1 + = − 1 (11.95)   d 2  β˜m m

with τ˜ m and β˜m from equations (11.90) and (11.94). Bharadwaj and Modest also outlined how scales are to be defined, whether using experimental data or data from spectroscopic databases. Testing the method with various nonhomogeneous CO2 –H2 O–N2 mixtures, they found the 2scale Malkmus model to be a factor of 2 to 5 more accurate than the standard Curtis–Godson approach.

11.9

NARROW BAND k-DISTRIBUTIONS

As in the case of “traditional” narrow band models (i.e., Elsasser and statistical models), we will start by looking at a homogeneous medium (constant temperature, pressure, and concentrations), i.e., a medium whose absorption coefficient is a function of wavenumber alone. In such a medium the spectral intensity depends on geometry, the Planck function, Ibη , emittance of bounding surfaces, ǫη , the absorption and scattering coefficients of suspended particles, κpη and σsη , and finally the absorption coefficient of any absorbing gas. Over a small spectral interval, such as a few tens of wavenumbers, the Planck function and nongaseous radiation properties remain essentially constant. Thus, across such a small spectral interval the intensity varies with gas absorption coefficient alone. On the other hand, Fig. 11-11 shows that the gas absorption coefficient varies wildly even across a very narrow spectrum, attaining the same value for κη many times, each time producing the identical intensity field within the medium. Thus, carrying out line-by-line calculations across such a spectrum would be rather wasteful, repeating the same calculation again and again. It would, therefore, be advantageous to reorder the absorption coefficient field into a smooth, monotonically increasing function, assuring that each intensity field calculation is carried out only once. This reordering idea was first reported in the Western literature by Arking and Grossman [114], but they give credit to Kondratyev [115], who in turn credits a 1939 Russian paper. Other early publications on k-distributions are by Goody and coworkers [116], Lacis and Oinas [105], and Fu and Liou [117], all in the field of meteorology (atmospheric radiation). In the heat transfer area most of the work on k-distributions again is due to the group around Taine and Soufiani in France [106, 118–120]. The narrow band average of any spectral quantity that depends only on the gaseous absorption coefficient, such as intensity Iη , transmissivity τη , etc., can be rewritten in terms of a k-distribution f (k) as follows (here expressed for transmissivity τη ): Z Z ∞ 1 −κη X τη (X) = (11.96) e dη = e−kX f (k) dk. ∆η ∆η 0 The nature of k-distributions and how to evaluate them is best illustrated by looking at a very small part of the spectrum with very few lines. Figure 11-14a shows a fraction of the CO2 15 µm band at 1 bar and 296 K and, to minimize irregularity, with only the strongest 10 lines considered (two of them having their centers slightly outside the depicted spectral range). It is seen that the absorption coefficient goes through a number of minima and maxima; between any two of these the integral may be rewritten as Z Z κη,max −κη X −κη X dη e dη = e dκη . dκη κη,min

The absolute value sign comes from the fact that, where dκη /dη < 0, we have changed the direction of integration (always from κη,min to κη,max ). Therefore, integration over the entire

337

11.9 NARROW BAND K-DISTRIBUTIONS (A

..000

(B

.001

.002

.003

F,

bar

101

K,

κ pη , cm−1bar−1

cm−1bar−1

101

100

δKJ

100

10−1 660

10−1

661

||

662

δη I(KJ)

663

664

η , cm−1

665 0

0.2

0.4

0.6

δ g(KJ)

0.8

g

1

FIGURE 11-14 Extraction of k-distributions from spectral absorption coefficient data: (a) simplified absorption coefficient across a small portion of the CO2 15 µm band (p = 1.0 bar, T = 296 K); (b) corresponding k-distribution f (k) and cumulative k-distribution k(1).

range ∆η gives f (k) as a weighted sum of the number of points where κη = k, 1 X dη f (k) = . ∆η dκη

(11.97)

i

i

Mathematically, this can be put into a more elegant form as Z 1 f (k) = δ(k − κη ) dη, ∆η ∆η

where δ(k − κη ) is the Dirac-delta function defined by   0, |x| > δǫ,   1 δ(x) = lim   , |x| < δǫ, δǫ→0  2δǫ or Z ∞ δ(x) dx = 1.

(11.98)

(11.99a)

(11.99b)

−∞

The k-distribution of the absorption coefficient in Fig. 11-14a is shown as the thin solid line in Fig. 11-14b. Even for this minuscule fraction of the spectrum with only three dominant lines, f (k) shows very erratic behavior: wherever the absorption coefficient has a maximum or minimum f (k) → ∞ since |dκη /dη| = 0 at those points (6 in the present case); and wherever a semistrong line produces a wiggle in the absorption coefficient f (k) has a strong maximum. Thankfully, the k-distribution itself is not needed during actual calculations. Introducing the cumulative k-distribution function 1(k) as Z k 1(k) = f (k) dk, (11.100) 0

we may rewrite the transmissivity (or any other narrow band-averaged quantity) as τη (X) =

Z

∞ 0

e−kX f (k) dk =

Z

1 0

e−k(1)X d1,

(11.101)

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

f (k), bar

338

10

6

10

10

10

4

10

8

10

2

10

6

10

0

10

4

10

2

10

0

T = 300K, p = 0.1bar

T = 300K, p = 1bar

10 10

T = 1000K, p = 1bar 10

10

10 10

10

10

0

1

2

10 10 –1 –1 k, cm bar

10

3

10

4

10 5 10

FIGURE 11-15 CO2 k-distributions for the three cases depicted in Fig. 11-11.

with k(1) being the inverse function of 1(k), which is shown in Fig. 11-14b as the thick solid line. Sticking equation (11.100) into (11.98) leads to 1(k) =

Z

k 0

f (k) dk =

1 ∆η

Z

∆η

Z

k 0

δ(k − κη ) dk dη =

1 ∆η

Z

H(k − κη ) dη,

(11.102)

∆η

where H(k) is Heaviside’s unit step function, H(x) =



0, x < 0, 1, x > 0.

(11.103)

Thus, 1(k) represents the fraction of the spectrum whose absorption coefficient lies below the value of k and, therefore, 0 ≤ 1 ≤ 1 [this can also be seen by setting X = 0 in equations (11.96) or (11.101), leading to τη = 1]. 1 acts as a nondimensional wavenumber (normalized by ∆η), and the reordered absorption coefficient k(1) is a smooth, monotonically increasing function, with minimum and maximum values identical to those of κη (η). In actual reordering schemes values of k are grouped over small ranges k j ≤ k < k j +δk j = k j+1 , as depicted in Fig. 11-14, so that 1 X 1 X δη d1(k j ) = f (k j )δk j ≃ δηi (k j ), (11.104) δk j = ∆η δκη i ∆η i

i

where the summation over i collects all the occurrences where k j < κη < k j+1 , as also indicated in the figure. If the absorption coefficient is known from line-by-line data, the k-distribution is readily calculated from equation (11.104). The k-distributions for the three cases in Fig. 11-11 are shown in Fig. 11-15. Because of the many maxima and minima in the absorption coefficient these functions show very erratic behavior, as expected. Numerically, one can never obtain the singularities f (k) → ∞, and they appear as sharp peaks [strongly dependent on the spacing used for η and δk in equation (11.104)]. Inaccurate evaluation of f (k) (such as its peaks) has little influence on k(1), which is much easier to determine accurately. This, and the fact that 1(k) represents the fraction of wavenumbers with kη ≤ k, suggests a very simple method to evaluate f (k)δk and 1(k): the wavenumber range ∆η is broken up into N intervals δη of equal width. The absorption coefficient at the center of each

11.9 NARROW BAND K-DISTRIBUTIONS

339

105 10

4

102

−1

cm bar

−1

103

K,

101

= 1bar T = 300K, P = 1bar T = 1000K, P

100 0K, P

T = 30

−1

10

10−2 0

0.2

0.4

= 0.1b

0.6

ar

g

0.8

1

FIGURE 11-16 k-values as a function of cumulative k-distribution 1 for the three CO2 cases depicted in Fig. 11-11.

interval is evaluated and, if k j ≤ κη < k j+1 , the value of f (k j )δk j is incremented by 1/N. After all intervals have been tallied f (k j )δk j contains the fraction of wavenumbers with k j ≤ κη < k j+1 , and j X 1(k j+1 ) = f (k j′ )δk j′ = 1(k j ) + f (k j )δk j . (11.105) j′ =1

The k(1) for the three cases in Fig. 11-11 are shown in Fig. 11-16. Program nbkdistdb in Appendix F is a Fortran code that calculates such a 1(k) distribution directly from a spectroscopic database, while nbkdistsg determines a single k-distribution from a given array of wavenumber–absorption coefficient pairs. As an example for the determination of k-distributions, the instructions to nbkdistdb show how to obtain the distributions of Figs. 11-15 and 11-16. The k-distribution can be found more easily if accurate narrow band transmissivity data are available: inspection of equation (11.96) shows that τη is the Laplace transform of f (k), i.e., f (k) = L −1 {τη (X)},

(11.106)

where L −1 indicates inverse Laplace transform. This was first recognized by Domoto [121], who also found an analytical expression for the k-distribution based on the Malkmus model, equation (11.79): r " !# β 1 κβ S κ k f (k) = exp 2 − − (11.107) , κ= . 2 πk3 4 k κ d The cumulative k-distribution can also be determined analytically as r  r   p r  p r  1 β  β  κ  β  κ 1 k k   + e erfc    − + 1(k) = erfc   ,      2 2 k 2 2 k κ κ 

(11.108)

where erfc is the complementary error function [98] and, by convention, erfc(−∞) = 2. Example 11.4. A certain diatomic gas is found to have an absorption coefficient that obeys Elsasser’s model across a narrow band of width ∆η = 10 cm−1 . The gas conditions are such that mean absorption coefficient (S/d) and overlap parameter β are known for the N = ∆η/d lines across the narrow band. Determine the narrow band k-distribution of the gas.

340

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

Solution From equation (11.64) the absorption coefficient may be written as κη =

S d

sinh 2β cosh 2β − cos 2β

η − ηc γ

!,

ηl < η < ηl + ∆η,

(11.109)

where ηl is the minimum wavenumber of the narrow band and ηc is the line center position of any one line in the band. Because of the periodic nature of an Elsasser band (see Fig. 11-12a), there will be exactly 2N wavelengths where kmin =

S sinh 2β S sinh 2β < k = κη < kmax = d cosh 2β + 1 d cosh 2β − 1

with identical |dκη /dη| each time. Therefore, from equation (11.97) or (11.98) Z dη 2N dη 1 δ(k−κη ) dκη = . f (k) = dκη ∆η ∆η dκη k=κη

k=κη

But

dκη S = dη d "

! ! κ2η 2β 2β η − ηc η − ηc sin 2β = sin 2β , !#2 γ S γ γ γ η − ηc sinh 2β cosh 2β − cos 2β d γ sinh 2β

and ! " #! η − ηc S sinh 2β −1 sin 2β = sin cos cosh 2β − = γ d κη

s

#2 S sinh 2β 1 − cosh 2β − . d κη "

Therefore, s S #2 " , 2 γ d sinh 2β S sinh 2β f (k) = 1 − cosh 2β − d 2β d k k2 S sinh 2β 1 d = . r 2  π S k k2 − k cosh 2β − sinh 2β d Integrating f (k) according to equation (11.101) we obtain (using integration tables), # " 1 S sinh 2β 1(k) = 1 − cos−1 cosh 2β − π d k or, after inversion, k=

sinh 2β S . d cosh 2β − cos π(1 − 1)

(11.110)

This is, of course, just equation (11.109) with 2β(η − ηc )/γ replaced by π(1 − 1): the k-distribution recognizes that, in the Elsasser scheme, the same structure is repeated 2N times (of that N times as a mirror image), and a single half-period is stretched across the entire reordered range 0 ≤ 1 ≤ 1. The present k-distribution can also be obtained by precalculating an array of absorption coefficients across ∆η from equation (11.109) and using subroutine nbkdistsg in Appendix F.

Comparing equation (11.101) with the first expression in equation (11.96), we note that the integration in equation (11.101) is equivalent in difficulty to the integration over half of a single line. Given that a narrow spectral range can contain thousands of little overlapping lines, we conclude that the CPU time savings over line-by-line calculations can be enormous! However, the generation of the necessary k-distributions from the large number of spectral lines contained in the various spectroscopic databases is tedious and time consuming. A first database of

11.9 NARROW BAND K-DISTRIBUTIONS

341

narrow band k-distributions for CO2 and H2 O was offered by Soufiani and Taine [106] as part of their EM2C narrow band database. It contains k-distribution data for fairly wide spectral intervals (larger than 100 cm−1 ; 17 bands for CO2 and 44 for H2 O), and are valid for atmospheric pressure and temperatures up to 2500 K. Each k-distribution is defined by 7 k-values, to be used with a 7-point Gaussian quadrature for spectral integration. Like their Malkmus parameter counterparts they are generated from the HITRAN 1992 database plus proprietary extensions (cf. p. 333). A more accurate, highly compact database, also for CO2 and H2 O, was generated by Wang and Modest [122], valid for total pressures between 0.1 bar and 30 bar, and temperatures between 300 K and 2500 K. The spectrum is divided into 248 narrow bands for all gases (allowing the determination of mixture k-distributions from those of individual species). Nested Gauss– Chebychev quadrature with up to 128 quadrature points is used to guarantee 0.5% accuracy for all absorption coefficient and emissivity calculations, and to allow for variable order spectral quadrature. The original Wang and Modest database employed the CDSD-1000 database [67] (for CO2 ) and HITEMP 1995 [62] (for H2 O). This Narrow Band K-Distribution for InfraRed (NBKDIR) database has since then been augmented to include additional species (CO, CH4 , and C2 H4 ), and is continuously updated to incorporate the newest spectroscopic data; at the time of print all k-distributions have been obtained from HITEMP 2010 [74] (H2 O, CO2 , and CO) and HITRAN 2008 [32], (CH4 and C2 H4 ). Both EM2C and NBKDIR are included in Appendix F.

Gas Mixtures k-distributions for mixtures can, in principle, be calculated directly, simply by adding the linear, spectral absorption coefficients of all components in the mixture before applying the reordering process, equation (11.104). Since assembling k-distributions is a tedious, time-consuming affair, it is desirable to obtain them from databases. However, determining an exact k-distribution for a mixture from those of individual species is in general impossible, because k-distributions never retain any information pertaining to the spectral location of individual absorption lines. Only in two simple situations is exact manipulation of k-distributions feasible: (1) a gas “mixing” with itself, i.e., changing the concentration of the absorbing gas species, and (2) adding a gray (across the given narrow band) material to the nongray absorbing gas. Variable Mole Fraction of a Single Absorbing Gas Consider a gas whose absorption coefficient is linearly dependent on its partial pressure, i.e., a gas whose line broadening is unaffected by its own partial pressure. This is always true for molecules that have the same size as the surrounding broadening gas (such as CO2 in air), and for all gases whenever Doppler broadening dominates. Then κxη (T, p, x; η) = xκη (T, p; η), (11.111) where κη is the absorption coefficient of the pure gas and x is its mole fraction in a mixture. Comparing the two k-distributions Z 1 δ(k − κη ) dη, (11.112) f (T, p; k) = ∆η ∆η Z 1 δ(kx − κxη ) dη, (11.113) fx (T, p, x; kx = xk) = ∆η ∆η we see that they both are populated by exactly the same spectral locations (i.e., kx = κxη wherever k = κη ), so that fx (T, p, x; kx ) d(xk) = f (T, p; k) dk or fx (T, p, x; kx ) =

1 f (T, p; kx /x). x

(11.114)

342

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

15

β = 10, κ-- = 5 cm−1 κ p = 2, X = 0.5

K,

cm

−1

10

5

KP =Kκ P K KX=XK

0 0

0.2

0.4

0.6

g

0.8

FIGURE 11-17 Scalability of narrow band k-distributions: k: pure gas; kx : gas with mole fraction x = 0.5; kp : gas mixed with gray medium of κp = 2 cm−1 .

1

Integrating equation (11.114) leads to 1(T, p; k) =

Z

k 0

f (T, p; k) dk =

Z

kx 0

fx (T, p, x; kx ) dkx = 1x (T, p, x; kx ),

(11.115)

i.e., the k vs. 1 behavior is independent of mole fraction. In a k vs. 1 plot the lines are simply vertically displaced by a multiplicative factor of x, or kx (1) = xk(1),

(11.116)

as demonstrated in Fig. 11-17 for a k-distribution based on the Malkmus model, equation (11.108) (using an unrealistically large overlap parameter of β = 10 for better visibility). Single Absorbing Gas Mixed with Gray Medium Consider a gas that is mixed with a gray medium (say, particles), with constant absorption coefficient κp . Then κpη (T, p, κp ; η) = κη (T, p, η) + κp .

(11.117)

Proceeding as in the previous paragraph we obtain Z 1 fp (T, p, κp ; kp ) = δ(kp − [κxη + κp ]) dη ∆η ∆η Z 1 δ([kp − κp ] − κxη ) dη = ∆η ∆η = f (T, p; k = kp − κp )

(11.118)

and 1(T, p; k) = 1p (T, p, κp ; kp = k + κp ),

(11.119)

i.e., the k vs. 1 behavior is also independent of any gray additions. In a k vs. 1 plot the lines are simply vertically displaced by a constant amount of κp , kp (1) = κp + k(1),

(11.120)

as also shown in Fig. 11-17. Multispecies Mixtures Several approximate mixing models for k-distributions have been proposed that rely on assumptions about the statistical relationships between the absorption lines of the individual species, mostly by Solovjov and Webb [123] (full spectrum models only),

11.9 NARROW BAND K-DISTRIBUTIONS

343

such as their convolution, superposition, multiplication, and hybrid approaches, and by Modest and Riazzi [124], exploiting the uncorrelatedness between species. All of these approaches produce a single mixture k-distribution, but rely on different assumptions and methodologies to achieve their goal. It was found that the approach of Modest and Riazzi results in negligible errors for all conditions tested (low to moderate pressures). Very recently, Pal and Modest [125] found that their methodology works equally well at very high pressures (up to 30 bar), even though broadened spectral lines overlap much more strongly. Consequently, we will present here only the Modest and Riazzi mixing scheme. Earlier it was shown how the idea of uncorrelated absorption coefficients can be used to obtain the transmissivity of a mixture, as given by equation (11.86). Through simple mathematical manipulation, it is possible to extend this logic to the mixing of cumulative k-distributions. We begin by recalling that the definition of the transmissivity, in terms of the k-distribution for a single absorbing species, is also the definition of the Laplace transform of f (k) [121], equation (11.106). Using this and the product of transmissivities model, the transmissivity of a mixture of M species may be expressed as the product of the Laplace transforms of the component k-distributions, or τ¯ η,mix = L [ fmix (k)] =

M Y

τη,m =

m=1

M Y

L [ fm (k)].

(11.121)

i=m

In terms of the cumulative k-distributions, the transmissivity of an individual component is given by Z 1 τ¯ m = e−km L d1m , (11.122) 0

and for a binary mixture this becomes τ¯ mix = L [ fmix (k)] =

Z

1 0

e−k1 L d11

Z

1 0

e−k2 L d12 =

Z

1 11 =0

Z

1

e−[k1 (11 )+k2 (12 )]L d12 d11 .

(11.123)

12 =0

Using the integral property of the Laplace transform we obtain

L

"Z

k 0

# Z fmix (k) = L [1mix (k)] = =

Z

1 11 =0 1

11 =0

Z

Z

1

−[k1 (11 )+k2 (12 )]L

e 12 =0 1

12 =0

d12 d11

!

1 L

e−[k1 (11 )+k2 (12 )]L d12 d11 , L

(11.124)

or, when the inverse transform is taken, with H being the Heaviside step function, 1mix (kmix ) =

Z

1 11 =0

Z

1

H[kmix − (k1 + k2 )]d12 d11 = 12 =0

Z

1

12 (kmix − k1 ) d11 .

(11.125)

11 =0

In the second, once integrated expression, it is assumed that 1m (k < km,min ) = 0 (i.e., all absorption coefficients are above km,min ) and 1m (k > km,max ) = 1 (i.e., all absorption coefficients are below km,max ). This relation may also be readily extended to a mixture of M species, 1mix (kmix ) =

Z

1

.... 11 =0

Z

1

H[kmix − (k1 + .... + kM )]d1M ....d11 .

(11.126)

1M =0

This integral may be evaluated by multiple Gaussian quadrature, leading to a single mixture kdistribution at specific k-values while using the component k-distributions stored at quadrature points with their associated weights. The k-values for this new mixture distribution must be

344

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

1

0.02

0.9

0

0.8 50% CO2 - 50% H2O T=1200K, P =1bar, L=100cm

-0.02 Absolute Error (τ )

0.7

τ

0.6 0.5

τ mix (exact)

0.4

τ mix Eq. (11.122) Error

0.3 0.2 0.1 0

3400

3600 η , cm-1

3800

4000

FIGURE 11-18 Narrow band transmissivity of a CO2 – H2 O mixture from individual species kdistributions, equation (11.126).

predetermined and chosen such that they cover the entire range of values of all component species. Carrying out mixing with this model consistently outperforms the models of Solovjov and Webb [123] (by a factor of 10 or more). Its accuracy is demonstrated in Fig. 11-18 for a mixture of water vapor and carbon dioxide in the 2.7 µm region (where both gases heavily overlap), with absolute errors mostly below 0.005 (roughly the same as obtained by direct multiplication of transmissivities). The mixing scheme described here is incorporated into the NBKDIR database in Appendix F, i.e., NBKDIR allows for the retrieval of mixture k-distributions. Example 11.5. Consider a mixture of two diatomic gases, both having absorption coefficients, κ1η and κ2η , that obey Elsasser’s model across a narrow band of width ∆η = 10 cm−1 . The following is known for the two gases: γ1 π S1 = 1 cm−1 ; β1 = π = = 0.314; d1 d1 10 γ2 S2 7π = 2 cm−1 ; β2 = π = = 0.110. d2 = 0.1429 cm−1 , γ2 = 0.0050 cm−1 , d2 d2 200

Gas 1 : d1 = 0.2500 cm−1 , γ1 = 0.0250 cm−1 , Gas 2 :

Determine the narrow band k-distribution for this mixture. Solution The individual k-distributions for the two component gases are given from the previous example as   sinh 2βi S ki = ; i = 1, 2. (11.127) d i cosh 2βi − cos π(1 − 1i ) The k-distribution of the mixture is immediately found from the rightmost expression in equation (11.125) as Z 1 1mix (kmix ) = 12 (kmix − k1 ) d11 , kmin = k1 min + k2 min ≤ kmix ≤ kmax = k1 max + k2 max , (11.128) 11 =0

where k1 is obtained from equation (11.127), while 12 is found from its inverse, or  0, k < k2 min ,  "    sinh 2β #   1 S 2  −1 1 − cos cosh 2β2 − , k2 min < k < k2 max , 12 (k) =    π d 2 k   1, k > k2 max .

The integration in equation (11.128) is best carried out numerically. Here care must be taken that the argument of cos−1 does not fall outside its allowable range (between −1 and +1). The same holds true

11.9 NARROW BAND K-DISTRIBUTIONS

A 

1

Absorption coefficient, κη, cm -1

100

Gas 1 Gas 2 Mixture

0.2

0.25 Relative wavenumber, η/∆η

0.3

Reordered absorption coefficient, K, cm −1

B 

10

10-1

345

101

Gas 1 Gas 2 Mixture (exact) Mixture, Eq. (11.129)

0

10

10−1 0

0.2 0.4 0.6 0.8 Reordered wavenumber, g

1

FIGURE 11-19 Narrow band k-distribution for a two-component mixture (Example 11.5): (a) absorption coefficients, (b) k-distributions.

for the mixing of any two k-distributions, i.e., 12 ≡ 0 for kmix − k1 ≤ k2 min , and 12 ≡ 1 for kmix − k1 ≥ k2 max . The result of a simple trapezoidal rule integration is shown in Fig. 11-19. Frame (a) shows the absorption coefficients for the mixture and the two component gases, and Frame (b) the corresponding k-distributions. The mixture k-distribution is calculated in two ways: “exactly,” using the absorption coefficient in Fig. 11-19a, or equation (11.109) (with random and different η1 for each gas), and with equation (11.128). Semilog plots are employed to better separate the various absorption coefficients and k-distributions. It is apparent that both mixture k-distributions virtually coincide (in fact, transmissivities calculated with both k-distributions coincide to within 5 digits).

Nonhomogeneous Gases Correlated-k Like the statistical models the k-distribution is not straightforward to apply to nonhomogeneous paths. However, it was found that for many important situations the kdistributions are essentially “correlated,” i.e., if k-distributions k(1) are known at two locations in a nonhomogeneous medium, then the absorption coefficient can essentially be mapped from one location to the other (documented to some extent by Lacis and Oinas [105]). This implies that all the values of η that correspond to one value of κ and 1 at one location, more or less map to the same value of 1 (but a different κ) at another location [105, 117]: pressure changes affect all lines equally (causing more or less broadening by higher/lower total pressure p, increasing line strengths uniformly by changes in partial pressure of the absorbing gas, pa ). We may then write, with good accuracy, τη (0 → X) =

1 ∆η

Z

∆η

Z  R  X exp − 0 κη dX dη ≃

1 0

 R  X exp − 0 k(X, 1) dX d1.

(11.129)

This assumption of a correlated k-distribution has proven very successful in the atmospheric sciences, where temperatures change only from about 200 K to 320 K, but pressure changes can be very substantial [105, 116, 117]. Scaled-k A more restrictive, but mathematically precise condition for correlation of k-distributions is to assume the dependence on wavenumber and location in the absorption coefficient to be separable, i.e., κη (η, T, p, pa ) = kη (η)u(T, p, pa ), (11.130)

346

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

where kη (η) is the absorption coefficient at some reference condition, and u(T, p, pa ) is a nondimensional function depending on local conditions of the gas, but not on wavenumber. This is commonly known as the scaling approximation. Substituting this into equation (11.129) gives Z Z     RX 1 1 τη (0 → X) = (11.131) exp −kη (η) 0 u dX dη = exp −kη X dη , ∆η ∆η ∆η ∆η where X is now a path-integrated value for X. Comparing with equation (11.96), we find that in this case there is only a single k-distribution, based on the reference absorption coefficient kη , and Z Z 1

τη (0 → X) =

0

X

e−k(1)X d1;

X=

0

u dX.

(11.132)

As for homogeneous media equations (11.129) and (11.132) provide reordered absorption coefficients, which can be used in arbitrary radiation solvers without restrictions. At first glance, equation (11.129) looks superior to equation (11.132), since the assumption of a scaled absorption coefficient is more restrictive. However, in practice one needs to approximate an actual absorption coefficient, which is neither scaled nor correlated: if the scaling method is employed, the scaling function u(T, p, pa ) and its reference state for kη can be freely chosen and, thus, optimized for a problem at hand. On the other hand, if the correlated-k method is used, the absorption coefficient is simply assumed to be correlated (even though it is not), and the inherent error cannot be minimized. Following Modest and Zhang [126] and assuming constant total pressure, reference state temperature T0 and partial pressure pa0 may be chosen from Z 1 pa dV, (11.133) pa0 = V V Z 1 κη (T0 , x0 )Ibη (T0 ) = κη (T, x)Ibη (T) dV, (11.134) V V R where κη = ∆η κη dη/∆η is the average absorption coefficient, i.e., volume-averaged partial pressure and a mean temperature based on average emission from the volume. For the scaling function Modest and Zhang suggest equating exact and approximate radiation leaving from a homogeneous slab of the length under consideration, or Z

1 0

  exp −k(T, pa , 1)L d1 =

Z

1 0

  exp −k(T0 , pa0 , 1)u(T, p, pa )L d1.

(11.135)

Correlated-k and scaled-k are about equally efficient numerically: both require evaluation of the local k-distribution k(T, pa , 1) everywhere along the path. As an illustration a simple (yet severe) example is shown in Fig. 11-20, showing transmissivity through, and emissivity from, a slab of hot gas at 1000 K adjacent to a cold slab at 300 K. Both layers are at the same total and partial pressures, and are of equal width [127]. The transmissivity for a blackbody beam Ibη (Th = 1000 K), through such a double layer is, from Chapter 10, τη =

Iη (L) tr Ibη (Th )

=

1 ∆η

Z

exp[−κη (Th , x)Lh − κη (Tc , x)Lc ] dη,

(11.136)

∆η

while the emissivity is defined here as the intensity of emitted radiation exiting the cold layer, as compared to the Planck function of the hot layer. Employing equation (10.29) this is readily evaluated as # Z "  Ibη (Tc )  Iη (L) em 1 −κη (Tc ,x)Lc −κη (Tc ,x)Lc −κη (Th ,x)Lh −κη (Tc ,x)Lc ǫη = = −e + 1−e e dη. (11.137) Ibη (Th ) ∆η ∆η Ibη (Th )

11.9 NARROW BAND K-DISTRIBUTIONS

A 

1

1 0.9

0.8 0.7 0.6 0.5 0.4 0.3

Transmissivity, τη ; emissivity, ∋η

Transmissivity, τη ; emissivity, ∋η

0.9

∆η = 5cm-1 LBL scaled-K correlated-K ∆η = 25cm-1 LBL scaled-K correlated-K

0.2

B 

∆η = 25cm-1 LBL scaled-K correlated-K

0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.1 0 3300

0.8

347

3400 3500 3600 -13700 Wavenumber, η , cm

3800

0 1000

1500 2000 -1 Wavenumber, η , cm

2500

FIGURE 11-20 Narrow band transmissivities and emissivities for two-temperature slab, as calculated by the LBL, scaled-k, and correlated-k methods: (a) 2.7 µm band of CO2 with pCO2 = 0.1 bar, (b) 6.3 µm band of H2 O with pH2 O = 0.2 bar.

Note that, while transmissivities are more regularly shown in the narrow band literature, the emissivity is generally more descriptive of heat transfer problems. Figure 11-20a shows these narrow band transmissivities and emissivities for the 2.7 µm band of CO2 for a partial pressure of pCO2 = 0.1 bar, as calculated by the LBL, scaled-k, and correlated-k methods, using the original HITEMP 1995 database [62], and all for a resolution of ∆η = 5 cm−1 (lines) and 25 cm−1 (symbols). Both correlated and scaled k-distributions predict transmissivity very accurately with the exception of small discrepancies near the minima at 3600 cm−1 and 3700 cm−1 . Similar errors also show up in the emissivity, but are somewhat amplified. This amplification was observed for all bands studied (i.e., the effect is not limited to regions of small emissivities, as in this figure). For both, transmissivity and emissivity, results from the two k-distributions are virtually identical, although correlated-k performs slightly better for the 2.7 µm band (in the case of the 4.3 µm band, not shown, roles are reversed and scaled-k slightly outperforms correlated-k). Figure 11-20b shows transmissivities and emissivities for the wide 6.3 µm water vapor band. Conditions are the same as for Fig. 11-20a, except that pH2 O = 0.2 bar and only a ∆η = 25 cm−1 resolution is shown (a resolution of 5 cm−1 results in a very irregular shape which, while the k-distributions follow this behavior accurately, makes them difficult to compare). Again, both k-distributions predict transmissivities rather accurately, and the slight errors are somewhat amplified in the emissivities. And, again, both k-distributions give virtually the same results, with scaled-k being a little more accurate for this band. In summary, one may say that both models perform about equally well; this implies that—for narrow bands and for temperatures not exceeding 1000 K—the absorption coefficients for water vapor and carbon dioxide are relatively well correlated. Note also that the present case, with a sharp step in temperature, is rather extreme; accuracy can be expected to be significantly better in more realistic combustion systems. Unfortunately, for nonhomogeneous media with even more extreme temperature gradients the correlation between k-distributions at different temperatures breaks down. The reason for this is that different lines can have vastly different temperature dependence through the exponential term in equations (11.32): at low temperatures lines near the band center are strongest (with largest κη ), while at high temperatures lines away from the band center exhibit the largest κη . Since the correlated k-distribution pairs values of equal absorption coefficients, this results in pairing wrong spectral values in hot and cold regions. This is not only true for wide spectral ranges, but also on a narrow band level, since a vibration–rotation band consists of

348

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

many slightly displaced subbands, generated by different levels of vibrational energies (different Bv ), some of which undergo transitions only at elevated temperatures [large values for El in equation (11.32)], known as “hot lines.” For more detail the reader may want to consult the monograph by Taine and Soufiani [4]. The lack of correlation in nonisothermal media was first recognized by Rivi`ere and coworkers [118–120], who devised the so-called “fictitious gas technique”: starting with a high-resolution database, they grouped lines according to the values of their lower energy levels, E j = hcBv j(j + 1) (i.e., according to their temperature dependence), found the k-distribution for each of the fictitious gases and, in a further approximation, estimated the gas transmissivity as the product of the transmissivities of the fictitious gases, τη =

n1 Z Y i=1

1 0

 R  X exp − 0 ki (1, X) dX d1,

(11.138)

where n1 is the number of fictitious gases. A very similar approach was taken by Bharadwaj and Modest [109], employing the fictitious gas approach applied to k-distributions obtained from the Malkmus model. Unfortunately, these methods can only supply the mean transmissivity for a gas layer, i.e., they lose all the advantages of the k-distributions, and are limited in their application in the same way as the statistical narrow band models.

Comparison of k-Distributions and Statistical Models The k-distribution method has a number of important advantages over the statistical narrow band models, although the statistical models, in particular the Malkmus model combined with the Curtis–Godson scaling approximation, outperform k-distributions in a couple of respects: 1. Perhaps the greatest advantage that k-distributions have is that they formulate radiative properties in terms of a (reordered) absorption coefficient. This implies that radiative heat transfer rates may be calculated using any desired solution method for the radiative transfer equation. If based on exact line-by-line property data, the method is essentially exact (for a homogeneous medium). Statistical narrow band models, on the other hand, calculate gas column transmissivity, and heat transfer rates can only be determined in terms of these transmissivities. 2. Statistical narrow band models are, due to the transmissivity approach, limited to application in black enclosures without scattering. No such restriction is necessary for kdistributions (as long as wall reflectance and scattering properties remain constant across the narrow band). 3. The k-distribution method is valid for spectral lines of any shape; statistical narrow band models, on the other hand, are generally limited to Lorentz lines (although some formulations for Doppler and Voigt profiles exist). This is not unimportant, since in combustion applications the lines often have Voigt profiles as seen from Fig. 11-9. 4. Statistical narrow band models return an explicit expression for averaged transmissivity, while the k-distribution requires integration (quadrature) over the (reordered) narrow spectrum. On the other hand, the narrow band is limited to several tens of wavenumbers for statistical models (to avoid significant changes in statistical parameters, such as S and d), but can span several hundreds of wavenumbers for k-distributions (only limited by changes in Planck function and, if present, spectral variations of wall emittances and scattering properties). 5. Neither method treats nonhomogeneous paths to complete satisfaction. In fields with moderate temperature gradients and moderate-to-strong pressure variations the correlated-

11.10 WIDE BAND MODELS

349

(S/d )

κ

∆ηe

FIGURE 11-21 The box model for the approximation of total band absorptance.

ηc

k approach performs extremely well, while the Curtis–Godson approximation loses accuracy in the presence of strong pressure variations. On the other hand, in fields with extreme temperature fields all methods have some problems; under such conditions only the correlated-k, fictitious-gas approach performs well. However, the fictitious-gas approach calculates gas layer transmissivities only, i.e., it is under the same limitations as the statistical methods.

11.10

WIDE BAND MODELS

The heat transfer engineer is usually only interested in obtaining heat fluxes or divergences of heat fluxes integrated over the entire spectrum. Therefore, it is desirable to have models that can more readily predict the total absorption or emission from an entire band as was done in Example 11.3. These models are known as wide band models since they treat the spectral range of the entire band. It is theoretically possible to use quantum mechanical relations, such as equations (11.33), to accurately predict the radiative behavior of entire bands. This has been attempted by Greif and coworkers [128, 129] in a series of papers. While such calculations are more accurate, they tend to be too involved, so simpler methods are sought for practical applications.

The Box Model In this very simple model the band is approximated by a rectangular box of width ∆ηe (the effective band width) and height κ as shown in Fig. 11-21. With these assumptions we can calculate the total band absorptance for a homogeneous gas layer as Z Z ∞    A≡ ǫη dη = (11.139) 1 − e−κη X dη = ∆ηe 1 − e−κX , band

0

where both ∆ηe and κ may be functions of temperature and pressure. The box model was developed by Penner [20] and successfully applied to diatomic gases. However, the determination of the effective band width is something of a “black art.” Once ∆ηe has been found (by using the somewhat arbitrary criterion given by Penner [20] or some other means), κ may be related to the band intensity α, defined as Z ∞ Z ∞  S α≡ κη dη = dη, (11.140) d η 0 0 leading to κ = α/∆ηe . (11.141) If the molecular gas layer forms a radiation barrier between two surfaces of unequal temperature, then a suitable choice for the effective band width can give quite reasonable results. However,

350

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

S/d

(S/d)0

(1/e)×(S/d)0

ω

ω

ηu

ηc

ω

ηl

Wavenumber η FIGURE 11-22 Band shapes for exponential wide band model.

if emission from a hot gas is considered, then the results become very sensitive to the correct choice of ∆ηe . Nevertheless, the box model—because of its great simplicity—enjoys considerable popularity for use in heat transfer models (see Chapter 20). Example 11.6. Calculate the effective band width ∆ηe for which the box model predicts the correct total band absorptance for Example 11.3. Solution Integrating equation (11.83) over the entire band gives α = (S/d) 0 × ω = 500 cm−2 and κX = αX/∆ηe = 10,000 cm−1 /∆ηe . Equation (11.139) then, with A = 264.7 cm−1 , results in ∆ηe = 264.7 cm−1 by trial and error. ∆ηe is seen to be substantially larger than ω and essentially equal to A, because the band in this example is optically very thick. Even in the band wings far away from the band center the band is optically opaque (τ ≫ 1). This result must be accounted for in the choice of ∆ηe . For optically thick gases finding the correct ∆ηe is equivalent to finding A itself. Drawing a box seemingly best approximating the actual band shape can lead to large errors!

The Exponential Wide Band Model The exponential wide band model, first developed by Edwards and Menard [130], is by far the most successful of the wide band models. The original model has been further developed in a series of papers by Edwards and coworkers [131–134]. The word “successful” here implies that the model is able to correlate experimental data for band absorptances with an average error of approximately ±20% (but with maximum errors as high as 50% to 80%). We present here the latest version of Edwards, together with its terminology (based on Goody’s narrow band model), followed by a short discussion of newer models by Felske and Tien [135] (Goody’s model) and Wang [136] (Malkmus’ model). For a more exhaustive discussion on Edwards’ model the reader may want to consult Edwards’ monograph on gas radiation [1]. Since it is known from quantum mechanics that the line strength decreases exponentially in the band wings far away from the band center,8 Edwards assumed that the smoothed absorption coefficient S/d has one of the following three shapes, as shown in Fig. 11-22: with upper limit head symmetric band with lower limit head

α S = e−(ηu −η)/ω , d ω

(11.142a)

S α = e−2|ηc −η|/ω , d ω

(11.142b)

8 This fact is easily seen by letting j ≫ 1 in equations (11.28a) and (11.33a) for the P-branch, and in equations (11.28c) and (11.33b) for the R-branch.

11.10 WIDE BAND MODELS

351

TABLE 11.2

Exponential wide band correlation for an isothermal gas. β≤1

0 ≤ τ0 ≤ β

Square root regime

A∗ = ln(τ0 β) + 2 − β

Logarithmic regime

0 ≤ τ0 ≤ 1

A∗ = τ0

Linear regime

1 ≤ τ0 < ∞

A∗ = ln τ0 + 1

Logarithmic regime

β ≤ τ0 ≤ 1/β 1/β ≤ τ0 < ∞ β≥1

Linear regime

A∗ = τ0 p A∗ = 2 τ0 β − β

α, β, and ω from Table 11.3 and equations (11.144) through (11.147), τ0 = αX/ω.

α S = e−(η−ηl )/ω , d ω

(11.142c)

where α is the integrated absorption coefficient or the band strength parameter (or area under the curves in Fig. 11-22), which was defined in equation (11.140), and ω is the band width parameter,9 giving the width of the band at 1/e of maximum intensity. The band can be expected to be fairly symmetric if, during rotational energy changes, the B does not change too much [recall equations (11.28a) through (11.28c)]. ηc is then the wavenumber connected with the vibrational transition. On the other hand, if the change in B is substantial, then either the R- or the P-branch may fold back, leading to bands with upper or lower head. Thus, the wavenumbers ηu and ηl are the wavenumbers where this folding back occurs, and not the band center. The sharp exponential apex is, of course, not very realistic. The rationale is that, if the band center is optically thick, then it is opaque no matter what the shape, while if it is thin, then only the total α is of importance. Edwards and Menard [130] proceeded to evaluate the band absorptance using the general statistical model by substituting expressions (11.142) into equation (11.78) and carrying out the integration in an approximate fashion. Since equation (11.78) contains the line overlap parameter β and the optical thickness τ, the authors were able to describe the total band absorptance as a function of three parameters, namely, A∗ = A/ω = A∗ (α, β, τ0 ),

(11.143)

where τ0 is the optical thickness at the band center (symmetric band) or the band head. Their results are summarized in Table 11.2.10 Example 11.7. Determine the total band absorptance of the previous two examples by the exponential wide band model. Solution From Example 11.3 we have τ0 = 200 and β = π/10. Thus, since τ0 > 1/β, we find from Table 11.2 A∗ = ln(τ0 β) + 2 − β = ln(200 × π/10) + 2 − π/10 = 5.826 and A = A∗ ω = 5.826 × 50 = 291.3 cm−1 . The difference between the two results is primarily due to the fact that in Example 11.3 we treated the optically thin band wings as optically thick.

The parameters α, β, and ω are functions of temperature and must be determined experimentally. Values for the most important combustion gases—H2 O, CO2 , CO, CH4 , NO, and 9 The band width parameter ω, as used here, applies only to the wide band correlation. If equations (11.142) are used for spectral (i.e., narrow band) calculations, Edwards [1] suggests increasing the value of ω by 20% for better agreement between wide band model and band-integrated narrow band model calculations. p 10 In the original version the parameters C1 = α, C3 = ω, and C2 = 4C1 C3 β∗ were used, where β∗ is the value of β for a gas mixture at a total pressure of 1 atm with zero partial pressure of the absorbing gas. Also, limits between regimes were slightly different, using A itself rather than τ0 .

352

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

SO2 —for a reference temperature of T0 = 100 K are given in Table 11.3. Most of these correlation data are based on work by Edwards and coworkers and are summarized in [1]. Data for the purely rotational band of H2 O have been taken from the more modern work of Modak [137]. Values for other bands and other gases may be found in the literature, e.g., for H2 O, CO2 , and CH4 [1, 131, 134, 138–142], for CO [1, 131, 134, 143–145], for SO2 [1, 134, 146], for NH3 [147], for NO [148], for N2 O [149], and for C2 H2 [150] (in the older of these references the parameters for the slightly different original model are given; in a number of papers a pressure path length has been used instead of a density path length). The temperature dependence of the band correlation parameters for vibration–rotation bands is given by Edwards [1] as Ψ(T) α(T) = α0 , Ψ(T0 ) r T0 Φ(T) β(T) = β∗ Pe = β∗0 Pe , T Φ(T0 ) r T ω(T) = ω0 , T0 and

"

p pa Pe = 1 + (b − 1) p0 p

!#n

,

(11.144) (11.145) (11.146)

(p0 = 1 atm, T0 = 100 K),

(11.147)

where m ∞ Y X (vk + 1k + |δk | − 1)! −u (T)v e k k    m (1k − 1)! vk !    X   vk =v0,k k=1    , uk (T)δk  Ψ(T) =  1 − exp −   m X ∞ Y   (vk + 1k − 1)! −u (T)v k=1 k k e (1k − 1)! vk ! v =0 k=1

Φ(T) =

k

s  2 m ∞    (vk + 1k + |δk | − 1)! −u (T)v  Y X  e k k      (1 − 1)! v ! k k v =v k=1

0,k

k

m ∞ Y X (vk + 1k + |δk | − 1)! −u (T)v e k k (1k − 1)! vk ! v =v k=1

k

(11.148)

,

(11.149)

0,k

and uk (T) = hcηk /kT,

v0,k =

(

0 |δk |

for for

δk ≥ 0, δk ≤ 0.

(11.150)

In these rather complicated expressions the vk are vibrational quantum numbers, δk is the change in vibrational quantum number during transition (±1 for a fundamental band, etc.), and the 1k are statistical weights for the transition (degeneracy = number of ways the transition can take place). Values for the ηk , δk , and 1k are given in Table 11.3. The effective pressure Pe gives the pressure dependence of line broadening due to collisions of absorbing molecules with other absorbing molecules and with nonabsorbing molecules that may be present (for example, nitrogen and other inert gases contained in a mixture). Note that the definition for Pe is slightly different here from equation (11.39) (this was done for empirical reasons, to achieve better agreement with experimental data). For the case of nonnegative δk or v0,k = 0 (the majority of gas bands listed in Table 11.3), the series in the expression for Ψ and the denominator of Φ may be simplified [98] to ∞ X (vk + 1k + δk − 1)! −uk vk (1k + δk − 1)! −1 −δ e 1 − e−uk k k . = (1k − 1)!vk ! (1k − 1)! v =0 k

(11.151)

353

11.10 WIDE BAND MODELS

TABLE 11.3

Wide band model correlation parameters for various gases. Band Location λ ηc [µm] [cm−1 ]

Vibr. Quantum Step (δk )

Pressure Parameters n b

m = 3, η1 = 3652 cm−1 , η2 = 1595 cm−1 , η3 = 3756 cm−1 , 1k = (1, 1, 1) q 71 µma ηc = 140 cm−1 (0, 0, 0) 1 8.6 TT0 + 0.5 q 6.3 µm ηc = 1600 cm−1 (0, 1, 0) 1 8.6 TT0 + 0.5 q (0, 2, 0) (1, 0, 0) 1 8.6 TT0 + 0.5 2.7 µm ηc = 3760 cm−1 (0, 0, 1) q −1 (0, 1, 1) 1 8.6 TT0 + 0.5 1.87 µm ηc = 5350 cm q 1.38 µm ηc = 7250 cm−1 (1, 0, 1) 1 8.6 TT0 + 0.5

Correlation Parameters α0 β∗0 ω0 [cm−1 /(g/m2 )] [cm−1 ]

H2 O

CO2

5.455

0.143

69.3

41.2

0.094

56.4

0.2 2.3 23.4

0.132b,c

60.0b

3.0

0.082

43.1

2.5

0.116

32.0

m = 3, η1 = 1351 cm−1 , η2 = 666 cm−1 , η3 = 2396 cm−1 , 1k = (1, 2, 1) 15 µm

ηc = 667 cm−1

(0, 1, 0)

0.7

1.3

19.0

0.062

12.7

10.4 µmd

ηc = 960 cm−1

(−1, 0, 1)

0.8

1.3

2.47×10−9

0.040

13.4

9.4 µmd

ηc = 1060 cm−1

10.1

(0, −2, 1)

0.8

1.3

2.48×10−9

0.119

4.3 µm

−1

ηu = 2410 cm

(0, 0, 1)

0.8

1.3

110.0

0.247

11.2

2.7 µm

ηc = 3660 cm−1

(1, 0, 1)

0.65

1.3

4.0

0.133

23.5

2.0 µm

ηc = 5200 cm−1

(2, 0, 1)

0.65

1.3

0.060

0.393

34.5

m = 1, η1 = 2143 cm−1 , 11 = 1

CO

4.7 µm

ηc = 2143 cm−1

(1)

0.8

1.1

20.9

0.075

25.5

2.35 µm

ηc = 4260 cm−1

(2)

0.8

1.0

0.14

0.168

20.0

CH4

NO

m = 4, η1 = 2914 cm−1 , η2 = 1526 cm−1 , η3 = 3020 cm−1 , 1k = (1, 2, 3, 3)

7.7 µm

ηc = 1310 cm−1

(0, 0, 0, 1)

0.8

1.3

28.0

0.087

21.0

3.3 µm

ηc = 3020 cm−1

(0, 0, 1, 0)

0.8

1.3

46.0

0.070

56.0

2.4 µm

ηc = 4220 cm−1

(1, 0, 0, 1)

0.8

1.3

2.9

0.354

60.0

1.7 µm

ηc = 5861 cm

(1, 1, 0, 1)

0.8

1.3

0.42

0.686

45.0

(1)

0.65

1.0

9.0

0.181

20.0

m = 1, η1 = 1876 cm−1 , 11 = 1 5.3 µm

SO2

−1

ηc = 1876 cm−1

m = 3, η1 = 1151 cm−1 , η2 = 519 cm−1 , η3 = 1361 cm−1 , 1k = (1, 1, 1)

19.3 µm

ηc = 519 cm−1

(0, 1, 0)

0.7

1.28

4.22

0.053

33.1

8.7 µm

ηc = 1151 cm−1

(1, 0, 0)

0.7

1.28

3.67

0.060

24.8

7.3 µm

ηc = 1361 cm−1

(0, 0, 1)

0.65

1.28

29.97

0.493

8.8

4.3 µm

ηc = 2350 cm−1

(2, 0, 0)

0.6

1.28

0.423

0.475

16.5

4.0 µm

ηc = 2512 cm−1

(1, 0, 1)

0.6

1.28

0.346

0.589

10.9

  √ √ For the rotational band α = α0 exp −9( T0 /T − 1) , β∗ = β∗0 T0 /T. b Combination of three bands, all but weak (0, 2, 0) band are fundamental bands, α0 = 25.9 cm−1 /(g/m2 ). c Line overlap for overlapping bands from equation (11.154). d “Hot bands,” very weak at room temperature, exponential growth in strength at high temperatures. q q in hp  p α = α0 ΨΨ0 , ω = ω0 TT0 , β = β∗ Pe = β∗0 TT0 ΦΦ0 Pe , Pe = p0 1 + (b − 1) pa .

a

Ψ from equations (11.144) and (11.148), Φ from equation (11.149), T0 = 100 K, p0 = 1 atm.

354

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

If v0,k , 0, then v0,k terms need to be subtracted from the above result. Because of the low reference temperature of T0 = 100 K, the values for u0,k are relatively large, so both Φ0 and Ψ0 are very simple to evaluate and, for v0,k = 0, Ψ0 ≈

m Y (1k + δk − 1)! k=1

(1k − 1)!

,

Φ0 ≈ 1.

(11.152)

If only one of the vibrational modes undergoes a transition (only one δk , 0), then all other modes cancel out of the expression for Ψ; and if the transition results in a fundamental band (single transition with δk = 1), then Ψ ≡ 1. This implies that, for a fundamental band, α(T) = α0 = const. Unfortunately, the temperature dependence of the broadening mechanism is always more complicated, and Φ must generally be evaluated from equation (11.149). If several bands overlap each other (e.g., the three H2 O bands situated around 2.7 µm), then also the individual lines overlap lines from other bands, resulting in an effective overlap parameter β that is larger than for any of the individual bands. The band strength and overlap parameter for overlapping bands are calculated [1] from α=

J X

α j,

(11.153)

j=1

 2 J q  1 X  α j β j  , β =   α

(11.154)

j=1

where J is the number of overlapping bands. When the exponential wide band model was first presented by Edwards and Menard, the temperature dependence for the broadening parameter was not calculated by quantum statistics but was rather correlated from experimental data that, because of their scatter, generally resulted in fairly simple formulae; but extrapolation to higher temperatures tended to be very inaccurate. Most of the bands listed in Table 11.3 are fundamental bands, not because calculations for these bands are simpler, but because fundamental bands tend to be much stronger than overtones or combined-mode bands, often making them the only important ones for heat transfer calculations. To facilitate hand calculations, the temperature dependence of band strength parameters α (for nonfundamental bands) and overlap parameters β∗ are shown in graphical form in Fig. 1123 for water vapor. A similar plot is given in Fig. 11-24 for the important bands of carbon dioxide, and Fig. 11-25 shows the temperature dependence of the line overlap parameter for the fundamental bands of methane and carbon monoxide (with α = α0 = const). For more accurate computer calculations the subroutines wbmh2o, wbmco2, wbmch4, wbmco, wbmno, and wbmso2 are given in Appendix F. Alternatively, very accurate polynomial fits for these functions have been given by Lallemant and Weber [151]. Example 11.8. Consider a water vapor–air mixture at 3 atm and 600 K, with 5% water vapor by volume. What is the most important H2 O band and what is its total band absorptance for a path of 10 cm? Solution At 600 K the Planck function has its maximum around 5 µm. Since total emission will depend on the blackbody intensity [see equation (11.58)], we seek a band with large α in the vicinity of 5 µm. Inspection of Table 11.3 shows that the strongest vibration–rotation band for water vapor lies at 6.3 µm 2 −1 and is,√therefore, the band we are interested in. √From the table √ we find α = α0 = −141.2 cm /(g/m ), ∗ ∗ β = β0 T0 /T(Φ/Φ0 )Pe with β0 = 0.094, and ω = ω0 T/T0 = 56.4 600/100 = 138.15 cm . To evaluate the √ effective broadening pressure we find n = 1 and b = 8.6 100/600 + 0.5 = 4.01 and with a volume fraction x = pa /p the effective pressure becomes Pe = {(p/1 atm)[1+(b−1)x]}n = 3[1+3.01×0.05] = 3.452. Estimating the temperature dependence of the line overlap parameter from Fig. 11-23 leads to β∗ /β∗0 ≃ 0.65 and

11.10 WIDE BAND MODELS

355

2.0

Normalized band strength parameter α /α0

Normalized overlap parameter b */b *0

5 H 2O

α /α0 b */b *0

4

nd

µ 2.7

3

m

ba

1

µ µm .38 1 .87

m

1.5

6.3 µm

2

1

m 1.38 µ 2.7 µm

µm 1.87

0

5

10 15 Normalized temperature T/T 0

1.0 20

FIGURE 11-23 Temperature dependence of the line overlap parameter, β∗ , and band strength parameter, α, for water vapor.

Normalized overlap parameter b */b *0

Normalized band strength parameter α/α0

3.0

70 CO2

60

α /α0 b */b *0

50

2.5

40

nd

30

µ 2.0

20

m

ba

2.0

µm 2.7

4.3

µm 2. 7

µm

µm µm 15

2.0

1.5

10 0

5

10 15 Normalized temperature T/T 0

1.0 20

FIGURE 11-24 Temperature dependence of line overlap parameter, β∗ , and band strength parameter, α, for carbon dioxide.

β = 0.094 × 0.65 × 3.452 = 0.211. Since all values for α in Table 11.3 are based on a mass absorption coefficient, we must calculate X as X = ρa s, where ρa is the partial density of the absorbing gas (not the density of the gas mixture). For our water vapor with a partial pressure of pa = 0.05 × 3 = 0.15 atm and a molecular weight of M = 18 g/mol, we get from the ideal gas law ρa =

Mpa Ru T

=

18 g/mol × 0.15 atm 1.0132 × 105 J/m3 = 54.84 g/m3 8.3145 J/mol K × 600 K 1 atm

and X = 54.84 × 0.1 = 5.48 g/m2 . Finally, from τ0 = αX/ω we get τ0 = 41.2 × 5.48/138.15 = 1.634. Since the value of τ0 lies between the values of β and 1/β we are in the square-root regime and √ p A∗ = 2 τ0 β − β = 2 1.634 × 0.211 − 0.211 = 0.964

or A = 0.964 × 138.15 = 133 cm−1 .

356

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

Normalized overlap parameter b */b 0*

10

10

2

7.7

1

µ 3.3

10

µ

H4 mC

ban

d

H4

mC

0

4.7 µm CO

5

10 15 Normalized temperature T/T 0

20

FIGURE 11-25 Temperature dependence of the line overlap parameter, β∗ , for the fundamental bands of methane and carbon monoxide.

The calculation of exact values for Φ and Ψ for nonfundamental bands is rather tedious and is best left to computer calculations with the subroutines given in Appendix F. While the correlation in Table 11.2 is simple and straightforward (aside from the temperature dependence of α and β), it is often preferable to have a single continuous correlation formula. A simple analytical expression can be obtained for the high-pressure limit, i.e., when the lines become very wide from broadening resulting in very strong overlap, or β → ∞, leading to κη = (S/d)η and A∗ = E1 (τ0 ) + ln τ0 + γE = Ein(τ0 ), β → ∞, (11.155) where E1 (τ) is known as an exponential integral function, which is discussed in some detail in Appendix E. Felske and Tien [135] have given a formula for all ranges of β, based on results from the numerical quadrature of equation (11.78): s     s  s  1  1 + 2β  τ0 β  τ0 /β  τ0 /β      + E1     A = 2E1   2 1 + β/τ  − E1  2  1 + β/τ0  1 + β/τ0  0 ∗

+ ln

! τ0 β + 2γE , (1 + β/τ0 )(1 + 2β)

(11.156)

!

(11.157)

or, more compactly, ! ! h1 i w A = 2 Ein(w) + Ein − Ein +1 w , 2β 2β ∗

W w= d

0,Goody

= p

τ0 1 + τ0 /β

.

A previous, somewhat simpler expression by Tien and Lowder [152] is known today to be seriously in error for small values of β [135, 153], and is not recommended. Edwards’ wide band model, given in Table 11.2, as well as the continuous correlation by Felske and Tien, are based on equation (11.70) together with Goody’s statistical model, equation (11.78), the best narrow band model available at the time of Edwards and Menard’s [130] original paper. Since then it has been found that the Malkmus model introduced in 1967, equation (11.79), describes the radiative behavior of most gases better than Goody’s model [104]. It was shown by Wang [136] that an exact closed-form solution for the band absorptance can be

11.10 WIDE BAND MODELS

357

101

β→∞

Nondimensional band absorptance A *

Edwards & Menard

β=1

Felske & Tien Wang 10

0

β=

0.1

β= 10

-1

10

-2

1 0.0

β=

0. 0

01

001 0.0 = β

10

-2

-1

10

0

1

10 10 Gas band optical thickness τ0

10

2

10

3

FIGURE 11-26 Comparison of various band absorptance correlations.

found, if equation (11.142) is combined with Malkmus’ narrow band model, leading to ∗

  A = e E1 (β + w) − E1 (β) + ln(1 + w/β) + Ein(w), β

W w= d

!

0,Malkmus

=

i β hp 1 + 4τ0 /β − 1 . 2 (11.158)

Results from Wang’s model, equation (11.158), are compared in Fig. 11-26 with those of Edwards and Menard, Table 11.2, and Felske and Tien, equation (11.156). The agreement between all three models is good. However, the band absorptance based on Malkmus’ model, equation (11.158), is always slightly below that predicted by Goody’s model, equation (11.156). Both the Felske and Tien and the Wang models go to the correct strong-overlap limit (β → ∞), equation (11.155), while the older Edwards and Menard model shows its more approximate character, substantially overpredicting band absorptances for large β, particularly for intermediate values of τ0 . A considerable number of other band correlations are available in the literature, based on numerous variations of the Elsasser and statistical models. An exhaustive discussion of the older (up to 1978) correlations and their accuracies (as compared with numerical quadrature results based on the plain Elsasser and the general statistical models) has been given by Tiwari [154]. Example 11.9. Repeat Example 11.8, using the Felske and Tien and the Wang models. Solution All relations developed for Edwards and Menard’s model, equations (11.144) through (11.147), are equally valid for these two models, as are the data in Table 11.3. Thus, we have again τ0 = 1.634 and β = 0.211. Sticking these numbers into equations (11.156) and (11.158) (or, rather, using the Fortran functions ftwbm and wangwbm, or the stand-alone program wbmodels, all supplied in Appendix F) gives A∗FT = 0.6916,

A∗Wang = 0.6427.

As expected, the results are fairly close to each other, with the Malkmus-based Wang correlation predicting an about 7% lower band absorptance. Both values are significantly lower than those predicted by Edwards and Menard’s model, which—as inspection of Fig. 11-26 shows—considerably overpredicts band absorptances for strong line overlap (large β) at intermediate optical thicknesses τ0 .

358

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

Wide Band Model for Nonhomogeneous Gases As indicated in the previous section on narrow band models, the spectral emissivity for a nonhomogeneous path (with varying temperature and/or gas pressures) [cf. equation (11.61)] is  R  X ǫη = 1 − exp − 0 κη dX , (11.159) from which we may calculate the total band absorptance as Z ∞ Z ∞  R  X 1 − exp − 0 κη dX dη. A= ǫη dη = 0

0

(11.160)

Here we have replaced the geometric path s by X in case a linear absorption coefficient is not used, but rather one based on density (as was done for the correlation parameters in Table 11.3) or pressure. Since we would still like to use the simple wide band model, appropriate path-averaged values for the correlation parameters α, β, and ω must be found. Attempts at such scaling were made by Chan and Tien [155], Cess and Wang [156], and Edwards and Morizumi [157], and are summarized by Edwards [1]. The average value for α follows readily from the weak line limit (linear regime in Table 11.2) as Z XZ ∞ Z XZ ∞  Z X 1 S 1 1 e α≡ dη dX = κη dη dX = α dX. (11.161) X 0 X 0 d η X 0 0 0

The definition of an average value for ω is

1 e≡ ω e αX

Z

X 0

ωα dX,

(11.162)

while the averaged value for β is found by comparison with the square root regime in Table 11.2 as Z X 1 e βωα dX. (11.163) β≡ ee ω αX 0

e and e β,11 but comparison with spectral There is little theoretical justification for the choice of ω calculations using equations (11.90), (11.94), and (11.78) showed that they give excellent results [157]. Example 11.10. Reconsider Example 11.8, but assume that the water vapor–air mixture temperature varies linearly between 400 K and 800 K over its path of 10 cm. How does this affect the total band absorptance for the 6.3 µm band? Solution We may express the temperature variation as T = 400 K(1 + s′ /s), where s′ is distance along path s, and the density variation as 3 ρ600 600 K T0 = 6ρ600 = 2 ′ . ρa = ρ600 T T 1 + s /s Thus, Z s  Z 1 Z s 3 dξ T0 ′ ′ ds = ρ600 s ρa ds = 6ρ600 = 32 X600 ln 2 = 1.040 X600 = 5.702 g/m2 . X= T 2 1 +ξ 0 0 0 The path-averaged band strength becomes Z s 1 1 e α= αρa ds = α0 X = α0 = 41.2 cm−1 /(g/m2 ), X 0 X

11 Note that there are two different definitions for e β, one for narrow band calculations and the present one for the wide band model.

11.10 WIDE BAND MODELS

359

e since the 6.3 µm band is a fundamental band and α is independent of temperature. For the averaged ω √ √ √ we get, from ω = ω0 T/T0 = ω0 4 1 + s′ /s, Z sr Z 1 Z s 6ω0 ρ600 T T0 ′ 3ω0 ρ600 s dξ 1 ′ e ωαρa ds = ds = ω = √ X T0 T X e αX 0 1+ξ 0 0 √  1  X600 √ 6 2−1 p 3ω0 X600 = × 56.4 cm−1 × 2 1 + ξ = 6 2 − 1 ω0 = 3 X X ln 2 0 2 = 134.8 cm−1 .

And, finally, the overlap parameter is obtained from ! r   Z s β∗  6ρ600 T  T0 ∗ ds′ βωαρa ds = β0 Pe ∗ ω0  β T T e ω X 0 0 0 0 Z 1 ∗r β T0 ′ ω0 X600 ∗ dξ . = 6 β0 Pe ∗ β T e X ω 0 0

e β =

1 ee ω αX

Z

s



√ √ Inspection of Fig. 11-23 reveals that the integrand varies between 0.59/ 4 ≃ 0.30 (at 400 K), to 0.66/ 6 ≃ √ 0.27 (at 600 K), back to 0.80/ 8 ≃ 0.29 (at 800 K); i.e., the integrand is relatively constant. Keeping in mind the inherent inaccuracies of the wide band model, the integral may be approximated by using an average value of 0.28. Then 0.28×6β∗0 Pe ω0 X600 0.28×0.09427×3.4515 e  = = √ = 0.220. β ≃ 0.28 × 6β∗0 Pe √ e X ω 2−1 6 2−1

The effective optical thickness at the band center is now

αX/e ω = 41.2 × 5.702/134.8 = 1.743. τ0 = e

Again we are in the square root regime and q √ β−e β = 2 1.743×0.220 − 0.220 = 1.018 and A = 137 cm−1 . A∗ = 2 τ0 e

Thus, although the temperature varied considerably over the path (by a factor of two) values for α, β, and ω changed only slightly, and the final value for the band absorptance changed by less than 3%. In view of the accuracy of the wide band correlation, the assumption of an isothermal gas can often lead to satisfactory results. This has been corroborated by Felske and Tien [158], who suggested a linear average for temperature, and a second independent linear average for density (as opposed to density evaluated at average temperature). They found negligible discrepancy for a large number of nonisothermal examples.

Wide Band k-Distributions Wide band models allow us to determine the radiative emission (or the absorption of incoming radiation) from a volume of gas over an entire vibration–rotation band with a single calculation; but they are inherently less accurate than narrow band models, and they have the same limitations, i.e., they are difficult to apply to nonhomogeneous gases, and they cannot be used at all in enclosures that have nonblack walls and/or in the presence of scattering particles. The k-distribution method, on the other hand, smoothes the spectrum by simply reordering it, rather than supplying an effective transmissivity, and, therefore, it can readily be applied to nonblack walls as well as to scattering media. For a homogeneous medium the method is essentially exact, even for an entire vibration–rotation band, except for the assumption that the Planck function, Ibη , is invariable across the band. This has prompted a number of researchers to generate wide band k-distributions based on exponential wide band correlation data. The first such k-distribution was generated by Wang and Shi [159], using the Malkmus narrow band

360

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

model together with exponentially decaying average line strength. In order to obtain a finiterange reordered wavenumber, 0 ≤ 1 ≤ 1, as was done for narrow band k-distributions, they truncated the exponentially decaying band wings [see Fig. 11-22 and equation (11.142b)]. This resulted in an analytical expression for theR wide band k-distribution, F(k). However, evaluation of the reordered wavenumber, 1(k) = F dk, and its inversion to k(1) required numerical integration. Marin and Buckius [160] took a very similar approach but used the exponential wide band model together with both the Malkmus model and also the Goody model; they also provided approximate, explicit expressions for water vapor and carbon dioxide [161–163]. Lee et al. [164, 165] were able to find the k-distribution directly from wide band correlations, using a rather obscure version of Edwards’ model. This approach was further refined by Parthasarathy et al. [166], using Wang’s wide band model [136]. Denison and Fiveland [167] also provided closed-form approximations for the cumulative k-distribution, based on Edwards’ original wide band model given in Table 11.2. Comparison with narrow band calculations has shown that results from this model have very respectable accuracy [168]. The band absorptance for a vibration–rotation band is given by equation (11.139). Assuming a symmetric band, such as given by equation (11.142b), and reordering according to Section 11.9 leads to Z ∞ Z ∞ Z ∞    A=2 1 − e−κη X d|η − ηc | = 2 1 − e−κX F(κ) dκ = 2 1 − e−κ(1)X d1, (11.164) 0

0

0

where the k-distribution F(κ) =

Z



δ(κ − κη ) dη,

0

(11.165)

is defined over an unbounded (wide band) spectral range ∆η → ∞ and, thus, 1 is also unbounded [cf. equations (11.98) and (11.101)] and equivalent to |η − ηc |. The reordered band can also be regarded as symmetrical, if desired (with 1 going into both directions away from ηc ). Nondimensionalizing equation (11.164) gives A A = =2 ω ∗

Z

∞ 0



1−e

−κ∗ τ0









F (κ ) dκ =

Z

∞ 0

τ0 =

 ∗ ∗ 1 − e−κ (1 )τ0 d1∗ ,



α X, ω

κ∗ =

κω , α

F∗ =

αF , ω2

1 1∗ = 2 . ω

(11.166)

Differentiating equation (11.166) with respect to τ0 , and using Wang’s expression for band absorptance, equation (11.158), yields s     Z ∞   β    dA∗ 1  4τ ∗   0         = 1 − 1 + e−κ τ0 κ∗ F∗ (κ∗ ) dκ∗ . (11.167) 1 − exp = 2       dτ0 τ0  2 β  0

Comparing both sides of this equation it is apparent that F∗ (κ∗ ) is related to the inverse Laplace transform of dA∗ /dτ0 , ! ∗ −1 dA ∗ ∗ ∗ 2κ F (κ ) = L . (11.168) dτ0 Using Wang’s model an analytical expression can be obtained for the inverse [166]:  p p ! !     β √  β √ 1  1  1    ∗ ∗ β ∗ ∗   erfc  κ − √  − e erfc  κ + √  F (κ ) = ∗  . ∗ ∗  4κ  2 2 κ κ 

(11.169)

The cumulative k-distribution 1∗ , or reordered wavenumber, must be found and inverted numerically from Z ∞ 1 1 F∗ (κ∗ ) dκ∗ = 1∗ = . (11.170) 2 ω ∗ κ

11.10 WIDE BAND MODELS

10

β = 0.0001 β = 0.001 β = 0.01 β = 0.1 β=1 β =10 β =100

0

10

1

10

0

κ*

10

1

361

10

–1

10

–2

0

0.25

0.5

0.75

1

g*

2

3

4

5

10

–1

10

–2

FIGURE 11-27 Nondimensional reordered absorption coefficient κ∗ for an exponential wide band vs. nondimensional cumulative k-distribution 1∗ .

Figure 11-27 shows the resulting reordered, nondimensional absorption coefficient κ∗ vs. artificial, normalized wavenumber 1∗ . For large values of β there is strong line overlap and κη ≃ (S/d) η , and essentially no reordering is necessary. For that case F∗ approaches F∗ → 1/2κ∗ ∗ for κ∗ < 1 and F∗ → 0 for κ∗ > 1, leading to κ∗ → e−1 , 1∗ & 0.1.12 For smaller values of β, or less line overlap, but with identical average absorption coefficient the maximum value of the spectral absorption coefficient increases, and fewer spectral positions will have intermediate values, making the distribution more and more compressed toward small 1∗ , with larger values near 1∗ = 0. Example 11.11. The water vapor–air mixture of Example 11.8 is contained in a nonblack furnace of varied dimensions mixed with soot and scattering particles. In order to make accurate predictions of the radiative heat flux possible across the 6.3 µm water vapor band, determine a reordered correlated k-distribution for this mixture. Solution For the water vapor–air mixture of Example 11.8 we have for the 6.3 µm band α = 41.2 cm−1 /(g/m2 ), ω = 138.15 cm−1 , β = 0.211, and ρa = 54.84 g/m3 . Obtaining a reordered, nondimensional absorption coefficient κ∗ = κ∗ (1∗ ) from equation (11.170) [by utilizing the Fortran subroutine wbmkvsg given in Appendix F], we get from equation (11.166) ! ! ρa α ∗ 2|η − ηc | ρa α ∗ 21 κ = κ , κ(1) = κ(|η − ηc |) = ω ω ω ω where we have replaced the α in equation (11.166) by ρa α in order to obtain a linear, rather than density-based, absorption coefficient [see equation (11.18)], which is generally preferred for spectral calculations. This equivalent spectral absorption coefficient for the 6.3 µm water vapor band, centered at ηc = 1600 cm−1 , is shown in Fig. 11-28, and is compared with the spectral narrow band average absorption coefficient, (S/d) η for the same conditions. Since, for β = 0.211, there is relatively little line overlap, average values (S/d) η must come from strongly varying κη with values much larger and much smaller than the average; thus the abundance of large κ (near η = ηc ) with a quick drop-off away from the band center.

Figure 11-28 makes the band appear less wide than indicated by the band width parameter ω. This was done for mathematical convenience: as Fig. 11-11 shows, a band with small β 12

By convention erfc(x) = 0 for x → +∞, and erfc(x) = 2 for x → −∞ [98].

362

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

κg or S/d, cm–1

0.4 0.3 0.2 0.1 0 1500

1550

1600 1650 wavenumber η, cm–1

1700

FIGURE 11-28 Reordered absorption coefficient for Example 11.11.

contains many strong lines separated by small κ; we have simply chosen to collect all the large values of κ near the band center. The wide band k-distribution presented here requires numerical integration of equation (11.170) and its inversion to obtain the reordered absorption coefficient k(1); the reordered absorption coefficient recovers the total band absorptance as defined by exponential wide band model parameters. On the other hand, in the work of Marin and Buckius [161–163] explicit (albeit cumbersome) expressions are given for k(1), which approximate the wide band k-distributions obtained from the HITEMP 1995 database [62]. While probably more accurate below 1000 K (the limit of applicability of HITEMP), the Marin and Buckius formulation depends strongly on the arbitrary and nonphysical choice for the cutoff wavenumber (chosen to find a best fit with HITEMP-generated k-distributions).

11.11 TOTAL EMISSIVITY AND MEAN ABSORPTION COEFFICIENT Total Emissivity In less sophisticated, more practical engineering treatment it is usually sufficient to evaluate the emission from a hot gas (usually considered isothermal) that reaches a wall. The total emissivity is defined as the portion of total emitted radiation over a path X that is not attenuated by self-absorption, divided by the maximum possible emission or, from equation (11.48) and considering only emission within the gas,  R ∞  R ∞ ! Z ! −κη X N N X   I dη X 1 − e I ǫ dη πIbη0 πIbη0 bη η bη 0 0 −κη X R∞ ǫ≡ R ∞ 1−e dη = = = Ai , σT 4 i ∆ηband σT 4 i Ibη dη Ibη dη i=1 i=1 0

0

(11.171)

where two simplifying assumptions have been made: (i) The spectral width of each of the N bands is so narrow that the Planck function varies only negligibly over this range, and (ii) the bands do not overlap. While the first assumption is generally very good (with the exception of pure rotational bands such as the one for water vapor listed in Table 11.3), bands do sometimes overlap (for example, the 2.7 µm bands in a water vapor–CO2 mixture). If two or more bands of the species contained in a gas mixture overlap, the emission from the mixture will be smaller than the sum of the individual contributions (because of increased self-absorption). This problem has been dealt with, in an approximate fashion, by Hottel and Sarofim [11]. They argued that the transmissivities of species a and b over the overlapping region ∆η are independent from one another, that is, Z Z Z 1 1 1 −κηa X −κηb X −κηa X τa+b = e e dη ≈ e dη e−κηb X dη = τa τb . (11.172) ∆η ∆η ∆η ∆η ∆η ∆η

363

11.11 TOTAL EMISSIVITY AND MEAN ABSORPTION COEFFICIENT

If we define the total emissivity for a single band as ! πIbη0 ǫi ≡ Ai , σT 4 i

(11.173)

then this expression leads to the total emissivity of two overlapping bands, or (11.174)

ǫa+b = ǫa + ǫb − ǫa ǫb .

This equation is only accurate if both bands fully overlap. If the overlap is only partial, then the correction term, ǫa ǫb , should be calculated based on the fractions of band emissivity that pertain to the overlap region (i.e., a quantity that is not available from wide band correlations). An approximate way of dealing with this problem has been suggested by Felske and Tien [158]. A total absorptivity for the gas may be defined in the same way as equation (11.171). However, as for surfaces, in the absorptivity the absorption coefficient must be evaluated at the temperature of the gas, while the Planck function is based on the blackbody temperature of the radiation source. It is clear from equation (11.171) that the total emissivity is equal to the sum of band absorptances multiplied by the weight factor (πIbη0 /σT 4 ). Since the band absorptance is roughly proportional to the band strength parameter α (exactly proportional for small values of optical path X), comparison of the factors [α(πIbη0 /σT 4 )]i gives an idea of which bands need to be considered for the calculation of the total emissivity. Example 11.12. What is the total emissivity of a 20 cm thick layer of pure CO at 800 K and 1 atm? Solution For these conditions CO has a single important absorption band in the infrared. Comparing αIbη0 for the 4.7 µm and 2.35 µm bands (see Table 11.3) we find with (η0 /T) 4.3 = 2143 cm−1 /800 K = 2.679 cm−1 /K and (η0 /T) 2.35 = 4260 cm−1 /800 K = 5.325 cm−1 /K, ! , ! αEbη0 αEbη0 20.9 × 1.5563 = 874. = T3 4.7 T3 2.35 0.14 × 0.2659 Therefore, since the  4.7 µm band is much stronger (α4.7 /α2.35 ≃ 150) and located in a more important part of the spectrum Ebη4.7 /Ebη2.35 ≃ 6 , the influence of the 2.35 µm band can be neglected. We first need to calculate the band absorptance for the 4.7 µm band. Since values in Table 11.3 are based on the mass absorption coefficient, we need to calculate the density of the CO from the ideal gas law, as we did in Example 11.8: ρa =

Mpa Ru T

=

28 g/mol × 1 atm 1.0132 × 105 J/m3 = 426.6 g/m3 8.3145 J/mol K × 800 K 1 atm

0.8 and X = ρa s = 85.32 g/m2 . We also find from Table 11.3 that n = 0.8 and b = 1.1, so √ that Pe = 1.1 = 1.079 and β∗0 Pe = 0.075×1.079 = 0.081. Further we find α = 20.9 cm−1 /(g/m2 ), ω = 25.5 800/100 = 72.125 cm−1 , and τ0 = αX/ω = 20.9×85.32/72.125 = 24.72. From Fig. 11-25 or subroutine wbmco we obtain β∗ /β∗0 = 0.529 and β = (β∗ /β∗0 )β∗0 Pe = 0.529 × 0.081 = 0.043. Thus, τ0 > 1/β and we are in the logarithmic regime, and

A∗ = ln(τ0 β) + 2 − β = 2.018 and A = 145.6 cm−1 . Sticking this into equation (11.171), ǫCO (800 K, 1 atm) =

πIbη0 σT 4

!

×A= η0

=2143 cm−1

= 1.5563×10−8 = 0.0500.

Ebη0 T3

!

η0

× =2143 cm−1

A σT

W 145.6 cm−1 × 2 −1 3 m cm K 5.670×10−8 ×800 W/m2 K3

364

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

0.70 0.50

Water vapor Total pressure 1 bar Partial pressure 0 bar

400

150

Emissivity ∋( pa L, p = 1 bar, T )

80

40 b

ar c

m

20

0.10

10 6

0.05

3

1.5

2

0. m

rc

ba

0.01

0.

5

200 400

600 800 1000 1200 1400 1600 1800 2000 2200

Temperature T, °C

FIGURE 11-29 Total emissivity of water vapor at a total gas pressure of 1 bar and zero partial pressure, from Hottel [18] (solid lines) and Leckner [169] (dashed lines).

If only total emissivities are desired, it would be very convenient to have correlations, tables, or charts from which the total emissivity can be read directly, rather than having to go through the algebra of the wide band correlations plus equation (11.171). A number of investigators have included total emissivity charts with their wide band correlation data; for example, Brosmer and Tien [141,150] compiled data on CH4 and C2 H2 , and Tien and coworkers [149] did the same for N2 O. However, by far the most monumental work has been collected by Hottel [18] and Hottel and Sarofim [11]. They considered primarily combustion gases, but they also presented charts for a number of other gases. Their data for total emissivity and absorptivity are presented in the form ǫ = ǫ(pa L, p, T1 ), α = α(pa L, p, T1 , Ts ) ≈

(11.175) T1 Ts

!1/2

ǫ pa L

!

Ts , p, Ts , T1

(11.176)

where T1 is the gas temperature and Ts is the temperature of an external blackbody (or gray) source such as a hot surface. Originally, the power for T1 /Ts recommended by Hottel was 0.65 for CO2 and 0.45 for water vapor, but with greater theoretical understanding the single value of 0.5 has become accepted [11]. In equation (11.176) pa is the partial pressure of the absorbing gas and p is the total pressure. (Hottel and Sarofim preferred a pressure path length over the density path length used by Edwards.) The emissivities were given in chart form vs. temperature, with pressure path length as parameter, and for an overall pressure of 1 atm. Later work by Leckner [169], Ludwig and coworkers [170,171], Sarofim and coworkers [172] and others has shown that the original charts by Hottel [11, 18], while accurate for many conditions (in particular, over the ranges covered by experimental data of the times), are seriously in error for some conditions (primarily those based on extrapolation of experimental data). New charts, based on the integration of spectral data, have been prepared by Leckner [169] and Ludwig and coworkers [170, 171], and show good agreement among each other. Emissivity charts, comparing the newly calculated data by Leckner [169] with Hottel’s [18], are shown in Fig. 1129 for water vapor and in Fig. 11-30 for carbon dioxide. These charts give the emissivities for the limiting case of vanishing partial pressure of the absorbing gas (pa → 0).

11.11 TOTAL EMISSIVITY AND MEAN ABSORPTION COEFFICIENT

365

0.30 100 b ar cm 40

Emissivity ∋ ( paL, p = 1 bar, T )

0.20

15 8

0.10

4b

ar c

2 0.05

m

1 0.6 0.3

0.1 5b ar 0.01

0.0 0.005

cm

5

Carbon dioxide Total pressure 1 bar Partial pressure 0 bar 200

400 600 800 1000 1200 1400 1600 1800 2000 2200

Temperature T, °C

FIGURE 11-30 Total emissivity of carbon dioxide at a total gas pressure of 1 bar and zero partial pressure, from Hottel [18] (solid lines) and Leckner [169] (dashed lines).

The original charts by Hottel also included pressure correction charts for the evaluation of cases with pa , 0 and p , 1 bar, as well as charts for the overlap parameter ∆ǫ. Again, these factors were found to be somewhat inaccurate under extreme conditions and have been improved upon in later work. Particularly useful for calculations are the correlations given by Leckner [169], which (for temperatures above 400 K) have a maximum error of 5% for water vapor and 10% for CO2 , respectively, compared to his spectrally integrated emissivities (i.e., the dashed lines in Figs. 11-29 and 11-30). In his correlation the zero-partial-pressure emissivity is given by  !j !i  N M X X T  p L 1 a   , T0 = 1000 K, (pa L) 0 = 1 bar cm, ǫ0 (pa L, p=1 bar, T1 ) = exp  log10 c ji  T0 (pa L) 0  i=0 j=0

(11.177)

and the c ji are correlation constants given in Table 11.4 for water vapor and carbon dioxide. The emissivity for different pressure conditions is then found from  " #2  ǫ(pa L, p, T1 )  (pa L) m  (a−1)(1−PE ) (11.178) =1− exp −c log10  , ǫ0 (pa L, 1 bar, T1 ) a+b−1+PE pa L

where PE is an effective pressure, and a, b, c, and (pa L) m are correlation parameters, also given in Table 11.4. As noted before, in a mixture that contains both carbon dioxide and water vapor, the bands partially overlap and another correction factor must be introduced, which is found from ζ ∆ǫ = − 0.0089ζ10.4 10.7 + 101ζ with



ζ=



log10

pH2 O . pH2 O + pCO2

(pH2 O + pCO2 )L (pa L) 0

!2.76

,

(11.179) (11.180)

366

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

TABLE 11.4

Correlation constants for the determination of the total emissivity for water vapor and carbon dioxide [169]. Gas

Water Vapor

Carbon Dioxide

M, N

2, 2

2, 3

c00

...

cN0

. . .

..

.

. . .

c0M

...

cNM

PE (pa L) m /(pa L) 0 a

−2.2118 0.85667 −0.10838

−1.1987 0.93048 −0.17156

0.035596 −0.14391 0.045915

√ (p + 2.56pa / t)/p0

2.7669 −1.1090 0.19731

−2.1081 1.0195 −0.19544

t < 0.75 t > 0.75

0.39163 −0.21897 0.044644

(p + 0.28pa )/p0 0.054/t2 , 0.225t2 ,

13.2t2 2.144, 1.888 − 2.053 log10 t,

−3.9893 1.2710 −0.23678

t < 0.7 t > 0.7

1 + 0.1/t1.45

b

1.10/t1.4

0.23

c

0.5

1.47

T0 = 1000 K, p0 = 1 bar, t = T/T0 , (pa L) 0 = 1 bar cm

This factor is directly applicable to emissivity and absorptivity. To summarize, the total emissivity and absorptivity of gases containing CO2 , water vapor, or both, may be calculated from:   ǫ ǫi (pi L, p, T1 ) = ǫ0i (pi L, 1 bar, T1 ) (pi L, p, T1 ), i = CO2 or H2 O, (11.181a) ǫ0 i ! ! T1 1/2 Ts ǫi pi L , p, Ts , (11.181b) i = CO2 or H2 O, αi (pi L, p, T1 , Ts ) = Ts T1   ǫCO2 +H2 O = ǫCO2 + ǫH2 O − ∆ǫ pH2 O L, pCO2 L , (11.181c) ! Ts Ts . (11.181d) αCO2 +H2 O = αCO2 + αH2 O − ∆ǫ pH2 O L , pCO2 L T1 T1 For the convenience of the reader Appendix F contains the Fortran routines totemiss and totabsor, which calculate the total emissivity or absorptivity of a CO2 –water vapor mixture from Leckner’s correlation, and which can also be called from the stand-alone program Leckner through user prompts. Example 11.13. Consider a 1 m thick layer of a gas mixture at 1000 K and 5 bar that consists of 10% carbon dioxide, 20% water vapor, and 70% nitrogen. What is the total normal intensity escaping from this layer? Solution From equations (11.48) and (11.171) we see that the exiting total intensity is Z ∞ Z ∞   ǫσT 4 , Ibη 1 − e−κη X dη = Ibη ǫη dη = I= π 0 0 where ǫ is the total emissivity of the water vapor–carbon dioxide mixture. First we calculate the emissivity of CO2 at a total pressure of 1 bar from Table 11.4: With pCO2 L = 0.1 × 5 m bar = 50 bar cm and T1 = 1000 K we find ǫCO2 ,0 (1 bar) = 0.157 (which may also be estimated from Fig. 11-30); for a total pressure of 5 bar we find from Table 11.4 the effective pressure is PE = 5.14, a = 1.1, b = 0.23, c = 1.47,

11.11 TOTAL EMISSIVITY AND MEAN ABSORPTION COEFFICIENT

367

and (pa L) m = 0.225 bar cm. Thus, from equation (11.178) "  #    0.1 × (−4.14) 0.225 2 ǫ = 1− exp −1.47 × log10 ≈ 1.00, ǫ0 CO2 0.33 + 5.14 50 and

ǫCO2 ≈ 0.157. Similarly, for water vapor with pH2 O L = 0.2 × 5 m bar = 100 bar cm we find ǫH2 O,0 (1 bar) ≈ 0.359 and the pressure correction factor becomes, with PE = 7.56, a = 1.88, b = 1.1, c = 0.5, and (pa L) m = 13.2 bar cm, "     # 0.888 × (−6.56) ǫ 13.2 2 exp −0.5 × log10 = 1− = 1.414, ǫ0 H 2 O 1.988 + 7.56 100 and

ǫH2 O ≈ 0.359 × 1.414 = 0.508. Finally, since we have a mixture of carbon dioxide and water vapor, we need to deduct for the band overlaps: From equation (11.179), with ζ = 32 , ∆ǫ = 0.072. Thus, the total emissivity is ǫ = 0.157 + 0.508 − 0.072 = 0.593. Alternatively, and more easily, using subroutine totemiss with ph2o = 1., pco2 = .5, ptot = 5, L = 100, and Tg = 1000 returns the same numbers. The total normal intensity is then I = 0.593 × 5.670×10−8 W/(m2 K4 ) × (1000 K)4 /π sr = 10.70 kW/m2 sr.

It is apparent from this example that the calculation of total emissivities is far from an exact science and carries a good deal of uncertainty. Carrying along three digits in the above calculations is optimistic at best. The reader should understand that accurate emissivity values are difficult to measure, and that too many parameters are involved to make simple and accurate correlations possible.

Mean Absorption Coefficients We noted in the previous chapter that the emission term in the equation of transfer, equation (10.21), and in the divergence of the radiative heat flux, equation (10.59), is proportional to κη Ibη . Thus, for the evaluation of total intensity or heat flux divergence it is convenient to define the following total absorption coefficient, known as the Planck-mean absorption coefficient: R ∞ Z ∞ Ibη κη dη π 0 κP ≡ R ∞ = Ibη κη dη. (11.182) σT 4 0 Ibη dη 0

Using narrow band averaged values for the absorption coefficient, and making again the assumption that the Planck function varies little across each vibration–rotation band, equation (11.182) may be restated as κP =

! Z N X πIbη0 i=1

σT 4

i

! N   X πIbη0 S dη = αi , σT 4 i ∆ηband d i=1

(11.183)

where the sum is over all N bands, and the Ibη0 are evaluated at the center of each band. It is interesting to note that the Planck-mean absorption coefficient depends only on the band strength parameter α and, therefore, on temperature (but not on pressure). Values for α have been measured and tabulated by a number of investigators for various gases and, using them, Planck-mean absorption coefficients have been presented by Tien [3], but these values are today known to be seriously in error. Alternatively, the Planck-mean absorption coefficient can be

368

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

Planck-mean absorption coefficient, (cm⋅bar)-1

0.5 HITRAN 1996 HITRAN 2008 HITEMP 2010 Tien

0.4

0.3 CO2 0.2 H 2O 0.1

0

500

1000

1500 2000 Temperature, K

2500

FIGURE 11-31 Planck-mean absorption coefficients for carbon dioxide, and water vapor.

calculated directly from high-resolution databases such as HITRAN [32] and HITEMP [74] as [173] Z ∞ X X πIbη0 ! Z ∞ X πIbη0 ! π κP = Ibη κηj dη = κηj dη = S j, (11.184) σT 4 0 σT 4 j 0 σT 4 j j j j where the summation is now over all the spectral lines of the gas, and the Ibη0 are evaluated at the center of each line. Figures 11-31 through 11-33 show Planck-mean absorption coefficients calculated from the HITEMP 2010 (CO2 , H2 O, and CO) and HITRAN 2008 databases (all gases). For some gases, which saw major updates in the most recent HITRAN 2008 version, the values obtained from HITRAN 1996 [57] are also shown for comparison. At higher temperatures the Planck-mean absorption coefficients from HITRAN 2008 are generally larger than those from HITRAN 1996, due to the inclusion of many more lines from higher vibrational energy levels. Accordingly, today’s HITRAN 2008 can be used with confidence up to about 1000 K. The latest version of HITEMP [74] includes many more “hot lines,” and strives to be accurate for temperatures up to 3000 K. Sometimes the Planck-mean absorption coefficient is required for absorption (rather than emission), for example, when gas and radiation source are at different temperatures. This expression is known as the modified Planck-mean absorption coefficient, and is defined as R∞ Ibη (Ts )κη (T) dη κm (T, Ts ) ≡ 0 R ∞ . (11.185) I (T ) dη s bη 0

An approximate expression relating κm to κP has been given by Cess and Mighdoll [174] as   Ts . (11.186) κm (T, Ts ) = κP (Ts ) T In later chapters we shall see that in optically thick situations the radiative heat flux becomes proportional to 1 1 dIbη ∇Ibη = ∇T. (11.187) κη κη dT

This has led to the definition of an optically thick or Rosseland-mean absorption coefficient as ,Z ∞ Z ∞ Z ∞ dIbη 1 dIbη π 1 dIbη 1 ≡ dη dη = dη. (11.188) 3 κR κη dT dT κη dT 4σT 0 0 0

11.12 EXPERIMENTAL METHODS

369

Planck-mean absorption coefficient, (cm⋅bar)-1

0.5

0.4

HITRAN 2008 Tien

NH3 0.3 N2O 0.2

NH3

0.1 SO2 0

500

1000

1500 2000 Temperature, K

2500

FIGURE 11-32 Planck-mean absorption coefficients for ammonia, nitrous oxide, and sulfur dioxide.

Planck-mean absorption coefficient, (cm⋅bar)-1

0.06 HITRAN 1996 HITRAN 2008 HITEMP 2010 Tien

CH4

0.05 0.04 0.03

CO

0.02 NO

0.01 0

500

1000

1500 2000 Temperature, K

2500

FIGURE 11-33 Planck-mean absorption coefficients for carbon monoxide, nitric oxide, and methane.

Even though they noted the difficulty of integrating equation (11.188) over the entire spectrum (with zero absorption coefficient between bands), Abu-Romia and Tien [145] and Tien [3] attempted to evaluate the Rosseland-mean absorption coefficient for pure gases. Since the results are, at least by this author, regarded as very dubious they will not be reproduced here. We shall return to the Rosseland absorption coefficient when its use is warranted, i.e., when a medium is optically thick over the entire spectrum (for example, an optically thick particle background with or without molecular gases).

11.12

EXPERIMENTAL METHODS

Before going on to employ the above concepts of radiation properties of molecular gases in the solution of the radiative equation of transfer and the calculation of radiative heat fluxes, we want to briefly look at some of the more common experimental methods of determining these properties. While light sources, monochromators, detectors, and optical components are similar to the ones used for surface property measurements, as discussed in Section 3.10, gas property measurements result in transmission studies (as opposed to reflection measurements for surfaces). All transmission measurements resemble one another to a certain extent: They consist of a

370

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

Vacuum

Water-cooled window holder

M1

Absorption cell M3

M4 Furnace

Globar source

N2 or argon inlet

Preamplifier

M2 Chopper

M5

Vacuum

Source unit N2 or argon inlet

Amplifier

Optical path Electrical signal Pipe line M1 Spherical mirror M2 , M4 Parabolic mirrors M3 , M5 Flat mirrors Valve

Monochromator

Recorder (potentiometer)

Motor control Wavelength drive motor

FIGURE 11-34 General setup of gas radiation measurement apparatus [175].

light source, a monochromator or FTIR spectrometer (unless, for measurements over a narrow spectral range, a tunable laser is used as source), a chopper, a test cell with the (approximately isothermal) gas whose properties are to be measured, a detector, associated optics, and an amplifier–recorder device. The chopper often serves two purposes: (i) A pyroelectric detector cannot measure radiative intensity, rather, it measures changes in intensity; and (ii) if the beam is chopped before going through the sample gas then, by measuring the difference in intensity between chopper open and closed conditions, indeed only transmission of the incident light beam is measured. That is, any emission from the (possibly very hot) test gas and/or stray radiation will not be part of the signal. A typical setup is shown in Fig. 11-34, depicting an apparatus used by Tien and Giedt [175]. A chopper is not required if an FTIR spectrometer is used, since the light is modulated inside the unit. However, for high test gas temperatures care must be taken to eliminate sample emission from the signal [176,177]. Usually, gas temperatures are measured independently, and knowledge of gas absorption coefficients is acquired. But it is also possible to radiatively determine the gas temperature, if accurate knowledge of the absorption coefficient is given, such as detailed line structure of diatomic molecules together with FTIR spectrometry [178–180]. Measurements of radiative properties of gases may be characterized by the nature of the test gas containment and by the spectral width of the measurements. As indicated by Edwards [1], we distinguish among (1) hot window cell, (2) cold window cell, (3) nozzle seal cell, and (4) free jet devices; these may be used to make (a) narrow band measurements, (b) total band absorptance measurements, or (c) total emissivity/absorptivity measurements. The hot window cell uses an isothermal gas within a container that is closed off at both ends by windows that are kept at the same temperature as the gas. While this setup is the most nearly ideal situation for measurements, it is generally very difficult to find window material that (i) can withstand the high temperatures at which gas properties are often measured, (ii) are transparent in the spectral regions where measurements are desired (usually near-infrared to infrared) and do not experience “thermal runaway” (strong increase in absorptivity at a certain temperature level), and (iii) do not succumb to chemical attack from the test gas and other gases. Such cells have been used, for example, by Penner [20], Goldstein [181], and Oppenheim and Goldman [182]. The cold window cell, as the name implies, lets the probing beam enter and exit the test

11.12 EXPERIMENTAL METHODS

Water in

Test gas supply

Water out

Radiation shield Graphite heater Zirconia tube ( ZrO2 )

Carbon powder fill

371

Inert gas exhaust Power in Water in Window holder Water out

Window (Irtran 4) Moly radiation shield

Water in

9´´ Diameter

Test gas exhaust Graphite heater inert gas supply

Pyrometer viewing tube

Water out

Window 18 12´´ FIGURE 11-35 Schematic of the high-temperature gas furnace used by Tien and Giedt [175].

cell through water-cooled windows. This method has the advantage that the problems in a hot window cell are nearly nonexistent. However, if the geometric path of the gas is relatively short, this method introduces serious temperature and density variations along the path. Tien and Giedt [175] designed a high-temperature furnace, consisting of a zirconia tube surrounded by a graphite heater, that allowed temperatures up to 2000 K. The furnace was fitted with watercooled, movable zinc selenide windows, which are transmissive between 0.5 µm and 20 µm and stay inert to reactions with water vapor and carbon oxides for temperatures below 550 K. A schematic of their furnace is shown in Fig. 11-35. While allowing high temperatures, it is impossible to obtain truly isothermal gas columns with such a device. For example, for a nominal cell at 1750 K of 30 cm length, they found that the temperature gradually varied by a rather substantial 350 K over the central 2/3 of the cell, and then rapidly dropped to 330 K over the outer 1/3. This apparatus was used by Tien and coworkers to measure the properties of various gases [144, 146–149, 183]. Nozzle seal cells are open flow cells in which the absorbing gas is contained within the cell by layers on each end of inert gases such as argon or nitrogen. This system eliminates some of the problems with windows, but may also cause density and temperature gradients near the seal; in addition, some scattering may be introduced by the turbulent eddies of the mixing flows [184]. This type of apparatus has been used by Hottel and Mangelsdorf [13] and Eckert [185] for total emissivity measurements of water vapor and carbon dioxide. Most of the measurements made by Edwards and coworkers also used nozzle seal cells [131, 132, 134, 143, 184, 186, 187]. A schematic of the apparatus used by Bevans and coworkers [186] is shown in Fig. 11-36. Using a burner and jet for gas radiation measurements eliminates the window problems, and is in many ways similar to the nozzle seal cell. Free jet devices can be used for extremely high temperatures, but they also introduce considerable uncertainty with respect to gas temperature and density distribution and to path length. Ferriso and Ludwig [188] used such a device for spectral measurements of the 2.7 µm water vapor band. More recently, Modest has constructed a high-temperature gas transmissometer, shown schematically in Fig. 11-37 and used by Bharadwaj et al. [63, 65, 66] to measure transmissivities of carbon dioxide and water vapor. The device is based on the infrared emissometer [189–191] shown in Fig. 3-44 and combines the advantages of hot-window and cold-window absorption

372

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

cells. In essence, the apparatus consists of a hermetically sealed high-temperature furnace, a motorized tube fitted with an optical window, a sealed optical path, and an FTIR spectrometer with internal infrared light source and an external detector, which can only detect the modulated light from the FTIR. Light from the FTIR is imaged onto a platinum mirror inside the furnace; the reflected light, in turn, is imaged onto the external detector. The cold drop-tube with an optical window is placed into position and retracted by a high-speed motor. The gas column between platinum mirror and optical window forms an isothermal absorption cell and, since the optical window resides within the furnace’s hot zone for only a few seconds at a time, this device is able to measure transmissivities of truly isothermal high-temperature gas columns. All multispectral diagnostic techniques discussed so far have employed single-detector monochromator or FTIR spectroscopy. Such devices can provide spectral scans in a wide range of resolutions and of great accuracy, but to obtain a spectrally resolved measurement with good signal-to-noise ratio takes tens of seconds for low-resolution narrow-band scans to hours for high-resolution full-spectrum measurements. Very few attempts have been made to date to obtain time-resolved multispectral signals from turbulent systems, because—to obtain snapshots of a turbulent flowfield—exposure times must be of order of 0.1 ms or less. Richardson et al. [192, 193] were perhaps the first to attempt such measurements, using a 32-element InSb linear array detector fitted with a grating monochromator. The apparatus was quite similar to the one shown in Fig. 11-34, except that there is no need to rotate the monochromator’s prism or grating, with the spectrally separated light hitting different elements of the array detector simultaneously. Their device was able to collect a 32-spectrum signal over 160 µs, storing 250 samples for each detector element. This resulted in an equivalent FTIR resolution of 32 cm−1 when collecting a spectrum of 250 cm−1 , with a signal-to-noise ratio of about 50. Their improved second device was able to hold 2048 full spectra collected every 16 µs. A similar apparatus was built by Keltner et al. [194], using a 256 × 256 MCT array detector. They argued that the use of (dual) prisms is preferable to grating monochromators in connection with array detectors. This dual prism arrangement was also used by Ji et al. [195], together with a 160-element PbSe linear array detector. The resulting high-speed spectrometer, is capable of taking near-instantaneous snapshots at a rate of 390 Hz. The device was calibrated against a blackbody, and spectra from a laminar premixed flame were compared with measurements using a grating spectrometer– InSb detector combination. Later measurements have been carried out with this high-speed infrared array spectrometer, to provide radiation data for the otherwise well-documented Sandia Workshop flames [196–198], and for a sooty ethylene air diffusion flame [199].

KBr window

Inactive gas line

Active gas lines Power leads Preheat coils Preheat helix

Exhaust chamber Vacuum ports Exhaust lines

Radiation shielding End chamber Test cell

Cooling coils

FIGURE 11-36 Schematic of nozzle seal gas containment system by Bevans and coworkers [186].

11.12 EXPERIMENTAL METHODS

373

Optical path Detector

FM FM

M

Drop tube

M

WW

FTIR spectrometer

Furnace

F W Gas column

F

M

SiC tube

Gas supply

Temperature controller

M Mirror FM Focusing Mirror W Window F Flow meter FIGURE 11-37 Schematic of a drop-tube transmissometer [63].

Data Correlation The half-width of a typical spectral line in the infrared is on the order of 0.1 cm−1 . To get a strong enough signal with a monochromator, any spectral measurement is by experimental necessity an average over several wavenumbers and, therefore, dozens or even hundreds of lines, unless an extremely monochromatic laser beam is employed. Thus, the measured transmissivity or (after subtracting from unity) absorptivity/emissivity is of the narrow band average type. Most FTIR measurements also fall into this category, although they generally have much better resolution than monochromators; resolutions better than 0.1 cm−1 are possible with high-end spectrometers. A correlation for the average absorption coefficient may be found by inverting equation (11.68) or equation (11.70), depending on whether the Elsasser or one of the statistical models is to be used, in either case yielding S S = (ǫη , X, γ/d), d d

(11.189)

where the ǫη and X (density or pressure path length) are measured quantities, and the widthto-spacing ratio must be determined independently. Most early measurements have assumed a constant γ/d for the entire band, in which case the width-to-spacing ratio can be obtained in a number of ways: (i) direct prediction of γ and d, (ii) using an independently determined band intensity, α, as the closing parameter, or (iii) finding a best fit for β (which is directly related to γ/d) in the exponential wide band model. With the recent advent of high-resolution databases it has been recognized that line spacing can vary dramatically across a band. The first narrow band correlation with variable β was done by Brosmer and Tien [200] for propylene, using Goody’s model and least-mean-square-error fits. In medium-resolution measurements of CO2 Modest and Bharadwaj [63] correlated their experimental transmissivities to the Malkmus model through a least-mean-square-error fit. As an example the 2.7 µm bands of CO2 at 300 and 1000 K are shown in Fig. 11-38 and compared with

11 RADIATIVE PROPERTIES OF MOLECULAR GASES

Spectral line intensity to spacing ratio S/d, cm–2 bar–1

374

1

∆η = 4 cm–1 HITRAN eq. (10.72) HITRAN avg. κ η (300K) HITRAN/FTIR avg. Experiment 300K Experiment 1000K ∆η = 25 cm–1 EM2C database 300K EM2C database 1000K

0.35 0.3 0.25 0.2

0.5

0.15 0.1 0.05

0 3500

3600 3700 Wavenumber η , cm–1

0 3800

FIGURE 11-38 Narrow band correlation for the 2.7 µm band of carbon dioxide; experimental data from [63].

data obtained from the most accurate databases of the time, HITRAN 1996 [57] and EM2C [106]. CO2 is seen to have two bands around 2.7 µm, one centered at 3615 cm−1 , the other at 3715 cm−1 . Agreement between experiment-based correlation and HITRAN 1996 is seen to be excellent except near the four S/d peaks, where the absorption coefficient is dominated by a few widelyspaced strong lines (about 1.8 cm−1 apart). This leads to a jagged appearance if the statistical definition for S/d is used, equation (11.82), and even if straight averaging over 4 cm−1 (equal to the experimental resolution) is carried out. The line labeled “HITRAN/FTIR avg.” was obtained by averaging the absorption coefficient with the FTIR’s instrument response function [201] as weight factor, which comes close to simulating the actual experiment. Results from the EM2C database are also shown for comparison. Because of its relatively low resolution of 25 cm−1 this database cannot capture the dual peaks, but agreement with experiment is excellent if the lower resolution is accounted for. Measured spectral absorptivities may be integrated to determine total band absorptances. Plotting those band absorptances that fall into the logarithmic regime vs. XPe on semilog paper gives a straight√line whose slope is the band width parameter (cf. Table 11.2). Preparing a linear plot of A/Pe vs. X/Pe for data in the square root regime gives again a straight line, this time with p αωβ∗ as the slope (where β∗ = β/Pe = πγ/d is the width-to-spacing ratio for a dilute mixture, cf. Tables 11.2 and 11.3). Finally, total emissivity values may be calculated by substituting the measured total band absorptances into equation (11.171).

Experimental Errors Most of the earlier gas property measurements were subject to considerable experimental errors, as listed by Edwards [184]: (1) inhomogeneity and uncertainty in the values of temperature, pressure, and composition, (2) scattering by mixing zones in nozzle seals and free jets, (3) reflection and scattering by optical windows, and/or (4) deterioration of the window material due to adsorption or “thermal runaway.” In addition, essentially all data until the 1980s were poorly correlated, using fixed values for γ/d (across an entire vibration–rotation band), with a resulting correlational accuracy of ±20% at best. Only the more modern measurements by Phillips [107,108] and Bharadwaj et al. [63,65,66] apparently have experimental accuracies better than 5% and have been accurately correlated.

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Shi: “Total band absorptance and k-distribution function for atmospheric gases,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 39, pp. 387–398, 1988. 160. Marin, O., and R. O. Buckius: “Wide band correlated-k method applied to absorbing, emitting and scattering media,” Journal of Thermophysics and Heat Transfer, vol. 10, pp. 364–371, 1996. 161. Marin, O., and R. O. Buckius: “A model of the cumulative distribution function for wide band radiative properties,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 59, pp. 671–685, 1998. 162. Marin, O., and R. O. Buckius: “A simplified wide band model of the cumulative distribution function for water vapor,” International Journal of Heat and Mass Transfer, vol. 41, pp. 2877–2892, 1998. 163. Marin, O., and R. O. Buckius: “A simplified wide band model of the cumulative distribution function for carbon dioxide,” International Journal of Heat and Mass Transfer, vol. 41, pp. 3881–3897, 1998. 164. Lee, P. Y. C., G. D. Raithby, and K. G. T. Hollands: “The “reordering” concept of the absorption coefficient for modelling nongray gases,” in Radiative Heat Transfer: Current Research, vol. HTD-276, ASME, pp. 21–30, 1994. 165. Lee, P. Y. C., K. G. T. Hollands, and G. D. Raithby: “Reordering the absorption coefficient within the wide band for predicting gaseous radiant exchange,” ASME Journal of Heat Transfer, vol. 118, no. 2, pp. 394–400, 1996. 166. Parthasarathy, G., J. C. Chai, and S. V. Patankar: “A simple approach to nongray gas modeling,” Numerical Heat Transfer, vol. 29, pp. 394–400, 1996. 167. Denison, M. K., and W. A. Fiveland: “A correlation for the reordered wave number of the wideband absorptance of radiating gases,” ASME Journal of Heat Transfer, vol. 119, pp. 853–856, 1997. 168. Strohle, J., and P. J. Coelho: “On the application of the exponential wide band model to the calculation of ¨ radiative heat transfer in one- and two-dimensional enclosures,” International Journal of Heat and Mass Transfer, vol. 45, pp. 2129–2139, 2002. 169. Leckner, B.: “Spectral and total emissivity of water vapor and carbon dioxide,” Combustion and Flame, vol. 19, pp. 33–48, 1972. 170. Boynton, F. P., and C. B. Ludwig: “Total emissivity of hot water vapor – II, semi-empirical charts deduced from long-path spectral data,” International Journal of Heat and Mass Transfer, vol. 14, pp. 963–973, 1971. 171. Ludwig, C. B., W. Malkmus, J. E. Reardon, and J. A. L. Thomson: “Handbook of infrared radiation from combustion gases,” Technical Report SP-3080, NASA, 1973. 172. Sarofim, A. F., I. H. Farag, and H. C. Hottel: “Radiative heat transmission from nonluminous gases. Computational study of the emissivities of carbon dioxide,” ASME paper no. 78-HT-55, 1978. 173. Zhang, H., and M. F. Modest: “Evaluation of the Planck-mean absorption coefficients from HITRAN and HITEMP databases,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 73, no. 6, pp. 649–653, 2002. 174. Cess, R. D., and P. Mighdoll: “Modified Planck mean coefficients for optically thin gaseous radiation,” International Journal of Heat and Mass Transfer, vol. 10, pp. 1291–1292, 1967.

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381

175. Tien, C. L., and W. H. Giedt: “Experimental determination of infrared absorption of high-temperature gases,” in Advances in Thermophysical Properties at Extreme Temperatures and Pressures, ASME, pp. 167–173, 1965. 176. Tanner, D. B., and R. P. McCall: “Source of a problem with Fourier transform spectroscopy,” Applied Optics, vol. 23, no. 14, pp. 2363–2368, 1994. 177. Tripp, C. P., and R. A. McFarlane: “Discussion of the stray light rejection efficiency of FT-IR spectrometers: The effects of sample emission on FT-IR spectra,” Applied Spectroscopy, vol. 48, no. 9, pp. 1138–1142, 1994. 178. Anderson, R. J., and P. R. Griffiths: “Determination of rotational temperatures of diatomic molecules from absorption spectra measured at moderate resolution,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 17, pp. 393–401, 1977. 179. Gross, L. A., and P. R. Griffiths: “Temperature estimation of carbon dioxide by infrared absorption spectrometry at medium resolution,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 39, no. 2, pp. 131–138, 1988. 180. Medvecz, P. J., K. M. Nichols, D. T. Clay, and R. Atalla: “Determination of gas temperatures at 295–1273 K using CO vibrational–rotational absorption spectra recorded with an FT-IR spectrometer,” Applied Spectroscopy, vol. 45, no. 8, pp. 1350–1359, 1991. 181. Goldstein, R. J.: “Measurements of infrared absorption by water vapor at temperatures to 1000 K,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 4, pp. 343–352, 1964. 182. Oppenheim, U. P., and A. Goldman: “Spectral emissivity of water vapor at 1200 K,” in Tenth Symposium (International) on Combustion, The Combustion Institute, pp. 185–188, 1965. 183. Abu-Romia, M. M., and C. L. Tien: “Spectral and integrated intensity of CO fundamental band at elevated temperatures,” International Journal of Heat and Mass Transfer, vol. 10, pp. 1779–1784, 1967. 184. Edwards, D. K.: “Thermal radiation measurements,” in Measurements in Heat Transfer, eds. E. R. G. Eckert and R. J. Goldstein, ch. 10, Hemisphere, Washington, DC, 1976. 185. Eckert, E. R. G.: “Messung der Gesamtstrahlung von Wasserdampf und Kohlens¨aure in Mischung mit nichtstrahlenden Gasen bei Temperaturen bis 1300◦ C,” VDI Forschungshefte, vol. 387, pp. 1–20, 1937. 186. Bevans, J. T., R. V. Dunkle, D. K. Edwards, J. T. Gier, L. L. Levenson, and A. K. Oppenheim: “Apparatus for the determination of the band absorption of gases at elevated pressures and temperatures,” Journal of the Optical Society of America, vol. 50, pp. 130–136, 1960. 187. Edwards, D. K.: “Absorption by infrared bands of carbon dioxide gas at elevated pressures and temperatures,” Journal of the Optical Society of America, vol. 50, pp. 617–626, 1960. 188. Ferriso, C. C., and C. B. Ludwig: “Spectral emissivities and integrated intensities of the 2.7 µm H2 O band between 530 and 2200 K,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 4, pp. 215–227, 1964. 189. Postlethwait, M. A., K. K. Sikka, M. F. Modest, and J. R. Hellmann: “High temperature normal spectral emittance of silicon carbide based materials,” Journal of Thermophysics and Heat Transfer, vol. 8, no. 3, pp. 412–418, 1994. 190. Postlethwait, M. A., M. F. Modest, M. A. Botch, and J. R. Hellmann: “Normal spectral emittance of alumina based materials,” in Radiative Heat Transfer: Current Research; 6th AIAA/ASME Thermophysics and Heat Transfer Conference, vol. HTD-276, ASME, pp. 73–77, 1994. 191. Challingsworth, M. J., J. R. Hellmann, and M. F. Modest: “Tailoring the spectral emittance of rare earth oxides via doping,” Proceedings of the 97th Annual Meeting and Exposition of the American Ceramic Society, 1995. 192. Richardson, H. H., V. W. Pabst, and J. A. Butcher: “A novel infrared spectrometer using a linear array detector,” Applied Spectroscopy, vol. 44, no. 5, pp. 822–825, 1990. 193. Alawi, S. M., T. Krug, and H. H. Richardson: “Characterization and application of an infrared linear array spectrometer for time-resolved infrared spectroscopy,” Applied Spectroscopy, vol. 47, no. 10, pp. 1626–1630, 1993. 194. Keltner, Z., K. Kayima, A. Lanzarotta, L. Lavalle, M. Canepa, A. E. Dowrey, G. M. Story, C. Marcott, and A. J. Sommer: “Prism-based infrared spectrographs using modern-day detectors,” Applied Spectroscopy, vol. 61, no. 9, pp. 909–915, 2007. 195. Ji, J., J. P. Gore, Y. R. Sivathanu, and J. Lim: “Fast infrared array spectrometer with a thermoelectrically cooled 160-element pbse detector,” Review of Scientific Instruments, vol. 75, no. 2, pp. 333–339, 2004. 196. Barlow, R. S.: International Workshop on Measurement and Computation of Turbulent Nonpremixed Flames (TNF) website: http://www.sandia.gov/TNF/abstract.html. 197. Zheng, Y., R. S. Barlow, and J. P. Gore: “Measurements and calculations of spectral radiation intensities for turbulent non-premixed and partially premixed flames,” ASME Journal of Heat Transfer, vol. 125, pp. 678–686, 2003. 198. Zheng, Y., R. S. Barlow, and J. P. Gore: “Spectral radiation properties of partially premixed turbulent flames,” ASME Journal of Heat Transfer, vol. 125, pp. 1065–1073, 2003. 199. Zheng, Y., and J. P. Gore: “Measurements and inverse calculations of spectral radiation intensities of a turbulent ethylene/air jet flame,” in Thirtieth Symposium (International) on Combustion, The Combustion Institute, pp. 727–734, 2005. 200. Brosmer, M. A., and C. L. Tien: “Thermal radiation properties of propylene,” Combustion Science and Technology, vol. 48, pp. 163–175, 1986. 201. Griffiths, P. R., and J. A. de Haseth: Fourier Transform Infrared Spectrometry, vol. 83 of Chemical Analysis, John Wiley & Sons, New York, 1986.

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Problems 11.1 Estimate the eigenfrequency for vibration, νe , for a CO molecule. 11.2 A certain gas at 1 bar pressure has a molecular mass of m = 10−22 g and a diameter of D = 5 × 10−8 cm. At what temperature would Doppler and collision broadening result in identical broadening widths for a line at a wavenumber of 4000 cm−1 ? 11.3 Water vapor is known to have spectral lines in the vicinity of λ = 1.38 µm. Consider a single, broadened spectral line centered at λ0 = 1.33 µm. If the water vapor is at a pressure of 0.1 atm and a temperature of 1000 K, what would you expect to be the main cause for broadening? Over what range of wavenumbers would you expect the line to be appreciable, i.e., over what range is the absorption coefficient at least 1% of its value at the line center? 11.4 Compute the half-width for a spectral line of CO2 at 2.8 µm for both Doppler and collision broadening as a function of pressure and temperature. Find the temperature as a function of pressure for which both broadening phenomena result in the same half-width. (Note: The effective diameter of the CO2 molecule is 4.0×10−8 cm.) 11.5 Methane is known to have a vibration-rotation band around 1.7 µm. It is desired to measure the Doppler half-width of a spectral line in that band at room temperature (T = 300 K). In order to make sure that collision broadening is negligible, the pressure of the CH4 is adjusted so that the expected collision half-width is only 1/10 of the Doppler half-width. What is this pressure? (For methane: D = 0.381 nm.) 11.6 Repeat Problem 11.4 for CO at a spectral location of 4.8 µm (Note: The effective diameter of the CO molecule is 3.4×10−8 cm.) 11.7 A certain gas has two important vibration–rotation bands centered at 4 µm and 10 µm. Measurements of spectral lines in the 4 µm band (taken at 300 K and 1 bar = 105 N/m2 ) indicate a half-width of γη = 0.5 cm−1 . Predict the half-width in the 10 µm band for the gas at 500 K, 3 bar. (The diameter of the gas molecules is known to be between 5 Å < D < 40 Å.) 11.8 It is desired to measure the volume fraction of CO in a hot gas by measuring the transmissivity of a 10 cm long column, using a blackbody source and a detector responsive around 4.7 µm. The conditions in the column are 1000 K, 1 atm, and properties for CO around 4.7 µm are known to −1 be S = 0.8 cm−2 atm , γ = 0.02 cm−1 , and d = 0.05 cm−1 . Give an expression relating measured transmissivity to CO volume fraction. 11.9 A polyatomic gas has an absorption band in the infrared. For a certain small wavelength range the following is known: Average line half-width: 0.04 cm−1 , Average integrated absorption coefficient: 2.0 × 10−4 cm−1 /(g/m2 ), Average line spacing: 0.25 cm−1 , The density of the gas at STP is 3 × 10−3 g/cm3 . For a 50 cm thick gas layer at 500 K and 1 atm calculate the mean spectral emissivity for this wavelength range using (a) the Elsasser model, (b) the statistical model. Which result can be expected to be more accurate? 11.10 Consider a gas for which the semistatistical model is applicable, i.e., ǫη = 1−exp(−Wη /d). To predict ǫη for arbitrary situations, a band-averaged (or constant) value for γη /d must be known. Experimentally R available are values for α = ∆η (Sη /d) dη and ǫη = ǫη (η) (for optically thick situations) for given pe and T. It is also known that  1/2 γη  γη  T0 ≃ pe . d d 0 T Outline how an average value for (γη /d) 0 can be found. 11.11 The following is known for a gas mixture at 600 K and 2 atm total pressure and in the vicinity of a certain spectral position: The gas consists of 80% (by volume) N2 and 20% of a diatomic absorbing

383

PROBLEMS

gas with a molecular weight of 20 g/mol, a mean line half-width γ = 0.01 cm−1 , a mean line spacing of d = 0.1 cm−1 , and a mean line strength of S = 8 × 10−5 cm−2 /(g/m3 ). (a) For a gas column 10 cm thick determine the mean spectral emissivity of the gas. (b) What happens if the pressure is increased to 20 atm? (Since no broadening parameters are known you may assume the effective broadening pressure to be equal to the total pressure.) 11.12 Repeat Problem 11.11 for a four-atomic gas. 11.13 1 kg of a gas mixture at 2000 K and 1 atm occupies a container of 1 m height. The gas consists of 70% nitrogen (by volume) and 30% of an absorbing species. It is known that, at a certain spectral location, the line half-width is γ = 300 MHz, the mean line spacing is d = 2000 MHz, and the line strength is S = 100 cm−1 MHz. (a) Calculate the mean spectral emissivity under these conditions. (b) What will happen to the emissivity if the sealed container is cooled to 300 K? 11.14 A 50 cm thick layer of a pure gas is maintained at 1000 K and 1 atm. It is known that, at a certain spectral location, the mean line half-width is γ = 0.1 nm, the mean line spacing is d = 2 nm, and the mean line strength is S = 0.002 cm−1 nm atm−1 = 2 × 10−10 atm−1 . What is the mean spectral emissivity under these conditions? (1 nm = 10−9 m) 11.15 The following data for a diatomic gas at 300 K and 1 atm are known: The mean line spacing is 0.6 cm−1 and the mean line half-width is 0.03 cm−1 ; the mean line strength (= integrated absorption coefficient) is 0.8 cm−2 atm−1 (based on a pressure absorption coefficient). Calculate the mean spectral emissivity for a path length of 1 cm. In what band approximation is the optical condition? 11.16 The average narrow band transmissivity of a homogeneous gas mixture has, at a certain wavenumber η, been measured as 0.70 for a length of 10 cm, and as 0.58 for a length of 20 cm. What is the expected transmissivity for a gas column of 30 cm length, assuming the Malkmus model to hold? 11.17 1 kg of a gas mixture at 2000 K and 1 atm occupies a container of 1 m height. The gas consists of 70% nitrogen (by volume) and 30% of an absorbing species. It is known that, at a certain spectral location, the nitrogen-broadening line half-width at STP (1 atm and 300 K) is γn0 = 0.05 cm−1 , the self-broadening line half-width is γa0 = 0.02 cm−1 , the mean line spacing is d = 0.4 cm−1 , and the density and mean line strength (for the given mixture conditions) are ρ = 0.800 kg/m3 and S¯ = 4 × 10−3 cm−1 /(g/m2 ), respectively. Under these conditions collision broadening is expected to dominate. (a) Calculate the mean spectral emissivity based on the height of the container. (b) What will happen to the emissivity if the sealed container is cooled to 300 K at constant pressure (with fixed container cross-section and sinking top end)? Note: The mean line intensity is directly proportional to the number of molecules of the absorbing gas and otherwise constant. The line half-width is given by γ = [γn0 pn + γa0 pa ]

r

T0 T

(p in atm, T0 = 300 K),

where pn and pa are partial pressures of nitrogen and absorbing species. 11.18 A certain gas is known to behave almost according to the rigid-rotor/harmonic-oscillator model, resulting in gradually changing line strengths (with wavenumber) and somewhat irregular line spacing. Calculate the mean emissivity for a 1 m thick layer of the gas at 0.1 atm pressure. In the wavelength range of interest, it is known that the integrated absorption coefficient is equal to 0.80 cm−2 atm−1 , the line half-width is 0.04 cm−1 and the average line spacing is 0.40 cm−1 . 11.19 A narrow band of a certain absorbing gas contains a single spectral line of Lorentz shape at its center. For a narrow band width of ∆η = 10γ, determine the corresponding reordered k vs. 1 distribution. Hint: This can be achieved without a lot of math. 11.20 Consider the spectral absorption coefficient for a narrow κη band range of ∆η as given by the sketch. Carefully sketch 3κ1 the corresponding k-distribution. Determine the mean 2κ1 narrow band emissivity of a layer of thickness L from κ1 this k-distribution. ∆η 5

∆η 5

∆η 5

∆η 5

∆η 5

η

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11 RADIATIVE PROPERTIES OF MOLECULAR GASES

11.21 Consider the spectral absorption coefficient for a narκ row band range of ∆η as given by the sketch. Care- 3κ1 η fully sketch the corresponding k-distribution. Verify your 2κ1 sketch through calculations. κ1

11.22 Consider the (highly artificial) absorption coefficient shown. Mathematically, this may be expressed as κη = A[η + 3h(η)]



∆η 2

η

κη

0 3 µm. For hand calculations you may approximate the index of refraction by a single average value (say, at 3 µm), and the emissive power by Wien’s law. 12.20 Redo Problem 12.19 for the case that the soot has agglomerated into mass fractal aggregates of 1000 soot particles each (D f = 1.77 and k f = 8.1). 12.21 Consider a particle cloud with a distribution function of n(a) = Ca2 e−ba , where a is particle radius and b and C are constants. The particles are soot (m ≃ 1.5 − 0.5i), and measurements show the soot occupies a volume fraction of 10−5 , while the number density has been measured as NT = 1012 /cm3 . Calculate the extinction, absorption, and scattering coefficients of the cloud for the wavelength range 1 µm < λ < 4 µm.

CHAPTER

13 RADIATIVE PROPERTIES OF SEMITRANSPARENT MEDIA

13.1

INTRODUCTION

Any solid or liquid that allows electromagnetic waves to penetrate an appreciable distance into it is known as a semitransparent medium. What constitutes an “appreciable distance” depends, of course, on the physical system at hand. If a thick film on top of a substrate allows a substantial amount of photons to propagate, say, 100 µm into it, the film material would be considered semitransparent. On the other hand, if heat transfer within a large vat of liquid glass is of interest, the glass cannot be considered semitransparent for those wavelengths that cannot penetrate several centimeters through the glass. Pure solids with perfect crystalline or very regular amorphous structures, as well as pure liquids, gradually absorb radiation as it travels through the medium, but they do not scatter it appreciably within that part of the spectrum that is of interest to the heat transfer engineer. If a solid crystal has defects, or if a solid or liquid contains inclusions (foreign molecules or particles, bubbles, etc.), the material may scatter as well as absorb. In some instances semitransparent media are inhomogeneous and tend to scatter radiation as a result of their inhomogeneities. An example of such material is aerogel [1], a highly transparent, low heat-loss window material made of tiny hollow glass spheres pressed together. A number of theoretical models exist to predict the absorption and scattering characteristics of semitransparent media. As for opaque surfaces, the applicability of theories is limited, and they must be used in conjunction with experimental data. In this chapter we shall limit ourselves to absorption within semitransparent media. The models describing scattering behavior are the same as the ones presented in the previous chapter and will not be further discussed here. In particular, scattering from turbid media, insulation, foams, etc., has been summarized near the end of Section 12.12.

13.2 ABSORPTION BY SEMITRANSPARENT SOLIDS The absorption behavior of ionic crystals can be rather successfully modeled by the Lorentz model, which was discussed in some detail in Chapters 2 and 3. The Lorentz theory predicts 440

13.2 ABSORPTION BY SEMITRANSPARENT SOLIDS

441

Al2O3

Absorption coefficient κλ , cm–1

NaF

LiF

15

KBr

KRS-6 MgO – Al2O3 NaCl TlCl 10

SiO2

MgO

KCl

KRS-5 KI TlBr

CaF2 5

0

2

4

6

8 10

20

40

60

80 100

Wavelength λ , µ m

FIGURE 13-1 Spectral absorption coefficients of several ionic crystals at room temperature [2].

Absorption coefficient κλ , cm–1

CsI KI

Tl(Br,I) TlCl

2.0

TlBr

CsBr AgCl

AgBr

KBr

1.0

NaCl

0

LiF NaF KCl

0.2

0.3

0.4 Wavelength λ , µ m

0.5

0.6

FIGURE 13-2 Spectral absorption of several halides temperature [2].

coefficient at room

that an ionic crystal has one or more Reststrahlen bands in the midinfrared (λ > ≃ 5 µm) (photon excitation of lattice vibrations). The wavelength at which strong absorption commences because of Reststrahlen bands is often called the long-wavelength absorption edge. The spectral absorption coefficients and their long-wavelength absorption edges are shown for a number of ionic crystals in Fig. 13-1. Note that these crystals are essentially transparent over much of the near infrared, and become very rapidly opaque at the onset of Reststrahlen bands. The Lorentz model also predicts that the excitation of valence band electrons, across the band gap into the conduction band, results in several absorption bands at short wavelengths (usually around the ultraviolet). Figure 13-2 shows the absorption coefficient and short-wavelength absorption edge for several halides: Materials that are essentially opaque in the ultraviolet become highly transparent in the visible and beyond. Pure solids are generally highly transparent between the two absorption edges. If large amounts of localized lattice defects and/or dopants (foreign-material molecules called color centers) are present, electronic excitations may occur at other wavelengths in between. A number of models predict the absorption characteristics of such defects, some sophisticated, some simple

442

13 RADIATIVE PROPERTIES OF SEMITRANSPARENT MEDIA

Absorption coefficient κλ , cm–1

103

102

1017 cm–3

10 1016 cm–3

1015 cm–3

1 300

500 700 900 Temperature T, K

1100

FIGURE 13-3 Spectral absorption coefficient of phosphorus-doped Si at 10.6 µm; solid lines: model of Blomberg and coworkers [4]; square symbols ( ): data of Boyd and coworkers [6] (dopant concentration of 1.1×1015 cm−3 ); circular symbols (•): data of Siregar and coworkers [5] (dopant concentration unknown).

and semiempirical. For example, Bhattacharyya and Streetman [3] and Blomberg and coworkers [4] developed models predicting the effect of dopants on the absorption coefficient of silicon. Figure 13-3 shows a comparison of the model by Blomberg and coworkers with experimental data of Siregar and colleagues [5] and Boyd and coworkers [6] for phosphorus-doped silicon at 10.6 µm (a wavelength of great importance for materials processing with CO2 lasers). The absorption coefficient increases strongly with dopant concentration and with temperature. According to both models, the rise with temperature is due to increases in the number of free electrons and to their individual contributions. The same trends were observed by Timans [7] for the wavelength range between 1.1 and 1.6 µm. The absorption behavior of amorphous, i.e., noncrystalline solids is much more difficult to predict, although the general trends are quite similar. By far the most important semitransparent amorphous solid is soda–lime glass (ordinary window glass, as opposed to the quartz or silicon dioxide crystals depicted in Fig. 13-1). A number of investigators measured the absorption behavior of window glass, notably Genzel [8], Neuroth [9, 10], Grove and Jellyman [11], and Bagley and coworkers [12]. Figure 1-17 shows the behavior of the spectral absorption coefficient of window glass for a number of different temperatures. As expected from the data for the transmissivity of window panes (Figs. 3-28 and 3-29), glass is fairly transparent for wavelengths λ < 2.5 µm; beyond that it tends to become rather opaque. The temperature dependence for quartz has been observed to be similar to that of silicon by Beder and coworkers [13], who reported a fourfold increase of the absorption coefficient between room temperature and 1500◦ C.

13.3 ABSORPTION BY SEMITRANSPARENT LIQUIDS The absorption properties of semitransparent liquids are quite similar to those of solids, while they also display some behavior similar to molecular gases. Remnants of intermolecular vibrations (Reststrahlen bands) are observed in many liquids, as are remnants of electronic band gap transitions in the ultraviolet. In the wavelengths in between, molecular vibration bands

13.3 ABSORPTION BY SEMITRANSPARENT LIQUIDS

0

10

4

-1

10

3

-2

10

2

10

1

Absorption coefficient κ, cm-1

10

10

10

-3

10

10-4 0.2

Water Ice

0.5

1.0 2 Wavelength λ, µm

5

10

443

100 20

FIGURE 13-4 Spectral absorption coefficient of clear water (at room temperature) and clear ice (at −10◦ C [14] and −25◦ C [17]); from [14] (thick lines), [18] (medium line), and [17] (thin lines).

are observed for molecules with permanent dipole moments, similar to the vibration–rotation bands of gases. Because of its abundance in the world around us (and, indeed, inside our own bodies) the absorption properties of water (and its solid form as ice) are by far the most important and, therefore, have been studied extensively, indeed for centuries. The data of many investigators for clear water and clear ice have been collected and interpreted by Irvine and Pollack [14] and by Ray [15]. Another review, limited to pure water, has been given by Hale and Querry [16]. More recent measurements have been reported by Kou and colleagues [17] (water and ice for wavelengths below 2.5 µm) and by Marley and coworkers [18] (water between 3.3 µm and 11 µm). The spectral absorption coefficient of clear water (at room temperature) and of clear ice (at −10◦ C) is shown in Fig. 13-4, based on the tabulations of Irvine and Pollack [14], Kou and colleagues [17] and Marley and coworkers [18]. Note the similarity between solid ice and liquid water. The lowest points of the absorption spectra of water and ice lie in the visible, making them virtually transparent over short distances. The minimum point lies in the blue part of the visible (λ ≃ 0.45 µm): Large bodies of water (or clear ice) transmit blue light the most, giving them a bluish hue. In the near- to midinfrared water and ice display several absorption bands (at 1.45, 1.94, 2.95, 4.7, and 6.05 µm in water, somewhat shifted for ice). These bands are very similar to the water vapor bands at 1.38, 1.87, 2.7, and 6.3 µm (see Table 11.3). Agreement between the data of Irvine and Pollack, and that of Kou and colleagues is excellent, while the data of Marley and coworkers in the longer wavelength region are considerably lower than those of Irvine and Pollack: measurement of such large absorption coefficients is extremely difficult, and the modern measurements of Marley and coworkers list an average estimated error of better than 3%. The temperature dependence of the absorption coefficient of water has been investigated by Goldstein and Penner [19] (up to 209◦ C) and by Hale and coworkers [20] (up to 70◦ C) and was found to be fairly weak. As temperature increases, water becomes somewhat more transparent in relatively transparent regions and somewhat more opaque in absorbing regions. A rather detailed discussion of the absorption behavior of clean water and ice has been given by Bohren and Huffman [21]. Natural waters and ice generally contain significant amounts of particulates (small organisms, detritus) and gas bubbles, which tend to increase the absorption rate as well

444

13 RADIATIVE PROPERTIES OF SEMITRANSPARENT MEDIA

Wavelength λ , µ m 10

6 7 B C A D E 8

5

4.5

Absorption coefficient κ η , cm–1

8

6

4

2

0

1200

1800 Wavenumber η, cm–1

2400

FIGURE 13-5 Spectral absorption coefficient of LiF for various temperatures; A: 300 K; B: 705 K; C: 835 K; D: 975 K; E: 1160 K. The melting point of LiF is 1115 K [22].

as to scatter radiation. While a number of measurements have been made on varieties of natural waters and ice, the results are difficult to correlate since the composition of natural waters varies greatly. The similarity of absorption behavior between the solid and liquid states of a substance is not limited to water. Barker [22] has measured the absorption coefficient of three alkali halides (KBr, NaCl, and LiF) for several temperatures between 300 K and temperatures above the melting point. Since Reststrahlen bands tend to widen with increasing temperature (see Section 3.5), the long-wavelength absorption edge moves toward shorter wavelengths. No distinct discontinuity in absorption coefficient was observed as the material changed phase from solid to liquid. As an example, the behavior of lithium fluoride (LiF) is depicted in Fig. 13-5. Semiempirical models for the absorption coefficient of alkali halide crystals, resulting in simple formulae, have been given by Skettrup [23] and Woodruff [24], while a similar formula for alkali halide melts has been developed by Senatore and coworkers [25].

13.4 RADIATIVE PROPERTIES OF POROUS SOLIDS The applicability of the RTE to heterogeneous media was studied by several investigators, e.g., [26–38]. In this section we will assume that heterogeneous media can be modeled as homogeneous with radiative intensity described by a local average value based on appropriate continuum properties. The radiative properties of open cell carbon foam were studied using experimental techniques and a predictive model by Baillis et al. [39]. The model combined elements of geometric optics and diffraction theory applied to the foam geometry determined by microscopic techniques. Extinction, scattering, and absorption coefficients were determined by assuming open cells to consist of struts with varying thickness and strut junctions, as schematically shown in Fig. 13-6, leading to

13.4 RADIATIVE PROPERTIES OF POROUS SOLIDS

b

Struts

Fc

445

dw

a

Walls (if closed cell)

Fs

FIGURE 13-6 Schematic of ideal foam cell, consisting of struts (with lengths a and curved triangular crosssection diameter b), strut junctions, and, in the case of closed-cell foams, thin walls of thickness dw [40].

! G¯ 2 ¯ , βλ =N G1 + 2

(13.1)

σsλ =ρλ βλ ,

(13.2)

κλ = 1 − ρλ βλ , 

(13.3)

where N is the number of struts per unit volume, G¯ 1 and G¯ 2 are the average geometric cross sections of struts and strut junctions, respectively, and ρλ is the spectral hemispherical reflectance of the solid.1 Hemispherical reflectances of foam slabs obtained by solving the RTE with the predicted properties agreed well with measured ones, as shown in Fig. 13-7. Larger discrepancies were observed for the very small, and thus difficult to measure, hemispherical transmittance of a 4.3 mm thick sample. The radiative properties of highly-porous open-cell metallic foams with inhomogeneities in the size range of geometric optics were studied using simple predictive models by Loretz et al. [42]. The foam structure was determined using microscopic and tomographic techniques. The cells (Fig. 13-6) were assumed to consist of struts and strut junctions. The extinction coefficient of the cells modeled as pentagon dodecahedrons or tetracaedecahedrons was obtained using the Glicksman and Torpey model [43]: r 1−ε , (13.4) β = 4.09 D2

where ε and D are the porosity and average cell diameter, respectively. For pentagon dodecahedrons with neglected strut junctions equation (13.4) becomes β=

3 b 1.305 2 , 4 a

(13.5)

1 The factor of 21 in equation (13.1) is not present in the original paper [39], but was added in more recent work, e.g., [41], perhaps to account for the fact that foam contains fewer strut junctions than struts.

Hemispherical reflectance, %

0.08 0.07 0.06 RT RE

0.05 0.04 0.03

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Wavelength,

mm

FIGURE 13-7 Hemispherical reflectance for carbon foam sample 4.3 mm thick for normal incidence; experimental (Re ) and theoretical (Rt ) results [39].

446

13 RADIATIVE PROPERTIES OF SEMITRANSPARENT MEDIA

6

C

F F F

C

= 46 m m

C

= 76 m m

C

= 106 m m

KR,

KR,

mW/(m K)

8

mW/(m K)

6

4

2 0%

5

F

C

= 78 m m

F

C

= 108 m m

C

= 138 m m

4

With correlation Without correlation Experimental 10%

F

With correlation Without correlation Experimental

20%

30%

40%

3

0%

10%

20%

(A

30%

40%

FS

FS

(B )

FIGURE 13-8 Radiative conductivity for two different extruded polystyrene foams [40]. Φc is the diameter of the foam cell measured as (a) Φc = 76 ± 30 µm and (b) Φc = 108 ± 30 µm, respectively. Both predicted and “measured” conductivities depend on the unknown solid fraction contained in struts, fs .

where a and b are the strut length and average thickness, respectively, as indicated in Fig. 13-6. The radiative properties of closed-cell foams were studied for expanded polystyrene foam by Coquard et al. [44,45], and those for extruded polystyrene foams were predicted and verified experimentally by Kaemmerlen and coworkers [40]. The properties were determined using the integration method of [46] applied to the curved-triangular foam cell wall and strut geometries of Fig. 13-6. Radiative conductivity kr was calculated employing the Rosseland-mean extinction coefficient, which in turn was calculated by independently determining the extinction coefficients of struts and of thin films of polystyrene. Due to the low density of the foam, independent scattering was assumed to hold, and the bulk extinction coefficient was determined by adding contributions from struts and walls, similar to equation (13.1). Figure 13-8 shows kr with and without a correction factor to the scattering efficiency to account for the concave shape of circular struts, which leads to a noticeable decrease in the variation of kr with the strut fraction as compared to the uncorrected results. However, as the authors noticed, the trends between predicted and measured radiative conductivities are different. A more extensive discussion of the radiative properties of open-cell and closed-cell foams may be found in the book by Dombrovsky and Baillis [41]. Monte Carlo ray tracing methods have been employed in a number of studies for the determination of effective radiative properties of heterogeneous media based on the geometry and properties of individual medium components. Tancrez and Taine [29] presented methodology for porous media with opaque solid phase, which was extended to media with semitransparent solids [33]. Coquard and Baillis applied ray tracing to determine the radiative properties of beds of opaque, diffusely or specularly reflecting particles [47]. The latter study was extended to beds of spheres containing an absorbing and scattering medium [48], and also applied to the actual geometry of polymeric foams obtained by tomography [49]. Also using tomography, the geometry of reticulated porous ceramics (RPC) with an opaque solid phase was obtained by Petrasch et al. [50] and by Haussener et al. [51]; the latter also used this technique for reacting packed beds with an opaque solid phase [52]. Finally, mullite foam with a semitransparent solid phase was studied by Zeghondy and coworkers [33]. In the tomography-based Monte Carlo methods used to study radiative properties of reticulated porous ceramics (Fig. 13-9) [29, 33, 50, 51] the media were assumed to be statistically homogeneous and isotropic, and the solid phase was assumed to be opaque. Diffraction effects were neglected and geometric optics was assumed to be valid. A large number Nr of stochastic rays were launched in the void phase of a subvolume V0 of a representative elementary volume V. Rays were traced until they interacted with the solid–void interface or were lost at the faces of V. For each ray colliding with the solid phase the

13.5 EXPERIMENTAL METHODS

447

FIGURE 13-9 3D rendering of Rh-coated reticulated porous ceramics with nominal pore diameter dnom = 2.54 mm obtained using computed tomography techniques [50].

distance to collision was recorded, and rays were either absorbed or reflected, either specularly or diffusely. The distribution function for attenuation path length was then computed as Z s 1 dN(s) = 1 − exp(−βs), (13.6) Fs = Nr s∗ =0

where dN(s) is the number of rays attenuated within ds around s; Fs quantifies the probability of a ray hitting the solid–void interface at a location between 0 and s. The scattering and absorption coefficients were then obtained from equations (13.2) and (13.3). Figure 13-10 shows the radiative intensity obtained numerically and experimentally as a function of normalized path length. The relative difference of 10% between experimental (βex ) and Monte Carlo-determined (βMC ) extinction coefficient was attributed to the effect of local material anisotropy for finite and relatively small RPC samples. Monte Carlo results were integrated over all solid angles, while the experimental measurements were carried out only along a single direction.

13.5

EXPERIMENTAL METHODS

The spectral absorption coefficient of a semitransparent solid or liquid can be measured in several ways. The simplest and most common method is to measure the transmissivity of a sample of known thickness, as described in Section 12.12 for particulate clouds. Since solids and liquids reflect energy at the air interfaces, the transmissivity is often determined by forming a ratio between the transmitted signals from two samples of different thickness. However, the transmission method is not capable of measuring very small or very large absorption coefficients: For samples with large transmissivity small errors in the determination of transmissivity, τ, lead to very large errors for the absorption coefficient, κ (since κ is proportional to ln τ). On the other hand, for a material with large κ sufficient energy for transmission measurements can be passed only through extremely thin samples. Such samples are usually prepared as vacuum-deposited thin films, which do not have the same properties as the parent material [54]. The absorption coefficient may also be determined through a number of different reflection techniques. The reflectivity of an optically smooth interface of a semitransparent medium depends, through the complex index of refraction, on the refractive index n as well as the absorptive index k. In turn, k is related to the absorption coefficient through equation (3.79) as κ = 4πηk/n, where η = 1/λ is the wavenumber of the radiation inside the medium. Thus,

448

13 RADIATIVE PROPERTIES OF SEMITRANSPARENT MEDIA

1.0

0.1

1−FS , Monte Carlo I /(1 − Fv )I0, Experiment exp(−βMC S ) exp(−βex S ) 0.01 0

5

S,

10 mm

15

FIGURE 13-10 Variation of radiative intensity in Rh-coated reticulated porous ceramics obtained numerically (squares) and experimentally (circles) as a function of the normalized path length, along with exponential fits with βMC = 210 m−1 and βex = 230 m−1 [53].

two data points are necessary to determine n and k. Noting the directional dependence of reflectivity on m = n − ik, some researchers have measured the specular reflectivity at two different angles. Leupacher and Penzkofer [55] showed that this can lead to very substantial errors. Other researchers have measured the reflectivity at a single angle, using parallel- and perpendicular-polarized light (known as an ellipsometric technique). However, this may also lead to large errors [55]. A new method overcoming these problems has been proposed by Lu and Penzkofer [56]. Using parallel-polarized light they vary the incidence angle until the point of minimum reflectivity at Brewster’s angle is found (cf. Figs. 2-8 and 2-11). Another reflection technique exploits the fact that a causal relationship exists between n and k, i.e., they are not independent of one another. This causal relationship is known as the Kramers–Kronig relation, which may be expressed as Z ∞ ln ρn (η′ ) ′ η dη , (13.7) δ(η) = π 0 η2 − η′2

where ρn (η) is the spectral, normal reflectivity of the sample surface [cf. equation (2.114)], and δ(η) is the phase angle of the complex reflection coefficient, equation (2.111), e rn =



ρn eiδ =

n − ik − 1 . n − ik + 1

(13.8)

Thus, if ρn is measured for a large part of the spectrum, the phase angle δ may be determined from equation (13.7) for wavenumbers well inside the measured spectrum; n and k are then readily found from equation (13.8). The method is particularly well suited to experiments employing an FTIR (Fourier transform infrared) spectrometer, which can take broad spectrum measurements over very short times, and which often have a built-in Kramers–Kronig analysis capability. More detailed discussions on the various Kramers–Kronig relations may be found, for example, in the books by Wooten [57] and Bohren and Huffman [21]. A description of the numerical evaluation of equation (13.7) has been given by Wooten [57]. Measurement of physical properties at high temperatures is always difficult, but particularly so for semitransparent media since two properties need to be measured (absorption coefficient as well as interface reflectivity, or equivalently, n and k). Myers and coworkers [58] have given a good review of such methods for solid samples. They also developed a new method

13.5 EXPERIMENTAL METHODS

449

10 mm

20 mm

Heating tube

Equalizing block Semitransparent slab sample

Blackbody reference Cavity-hole Pressure plate Load screw Through-hole Blackbody reference

Support disk Pins

FIGURE 13-11 Sample and holder, mounted within heating tube, for device to determine the optical properties of small, semitransparent solid samples [58].

to determine the optical properties of small, semitransparent, solid samples. Their device is essentially a compact arrangement of that employed by Stierwalt [59], which takes three different radiance measurements in rapid succession. A front and cross-sectional view of their sample heating arrangement is shown in Fig. 13-11. The slab-shaped sample is mounted within an equalizing nickel block, which is coupled radiatively to the electrically heated tube. The nickel block has four cavities and holes serving as radiance targets. A water-cooled graphite block (not shown) is positioned behind the heating tube to provide a room-temperature background for the through-hole as well as a reference for the detector. Three radiance measurements are made and compared with the reference: (i) the slab sample positioned in front of the blackbody (cavity-hole), (ii) the freely radiating sample (through-hole), and (iii) the blackbody reference. With the relations given in Section 3.8 one can use these measurements to deduce the optical properties (n, k, and κ). The method has the advantages that measurements at high temperatures (≃ 1000◦ C) can be taken, that only a single sample is necessary, and that no optically smooth surfaces are required. On the other hand, the method suffers from the standard weaknesses of transmission methods (see discussion at the beginning of this section), and is restricted to high temperatures (to produce a strong enough emission signal). Measurements of the optical properties of a high-temperature liquid are even more challenging. It is more difficult to confine a liquid in a sample holder (which must be horizontal), and more difficult to measure the thickness of the liquid layer. In addition, the layer thickness may be nonuniform because of (often unknown) surface tension effects. Furthermore, high-temperature liquids are often highly reactive, making a sealed chamber necessary. If the vapor pressure becomes substantial at high temperatures, the windows of the sealed chamber will be attacked. Shvarev and coworkers [60] have measured the optical properties of liquid silicon in the wavelength range of 0.4–1.0 µm with such a sealed-chamber furnace apparatus, using an ellipsometric technique. Barker [22, 61] designed an apparatus to measure the optical properties of semitransparent solid slabs and corrosive melts. To isolate the specimen he relied on a windowless chamber with continuous inert-gas purging. His data evaluation required independent measurements of the interface reflectivity, the reflectivity of a platinum mirror, the sample overall reflectivity, and the thickness of the sample. In addition, the reflectivity of the platinum–liquid interface must be estimated. As such, Barker’s method appears to be very vulnerable to experimental error. A more accurate device, limited to absorption coefficients of liquids, has been reported by Ebert and Self [62]. A schematic of their apparatus is shown in Fig. 13-12a. The aperture of a blackbody source at 1700◦ C is imaged (by the spherical mirror M3) onto the platinum mirror located in an alumina crucible inside the furnace. The reflected signal is focused onto

450

13 RADIATIVE PROPERTIES OF SEMITRANSPARENT MEDIA

Incident Collected beam beam

M1 Monochromator (Ebert type) Aperture M3 M5

Mirror support rods (3 platinum)

Source Nernst glower Chopper

Molten slag

M2 M4

Platinum mirror

Filter

Electric furnace Platinum mirror Crucible support tube Mirror support tubes (3)

Adjustable

M6

Adjustable

HgCdTe detector

First surface reflection not collected Multiple reflections not collected

Alumina crucible Adjustable crucible support tube (1–alumina) Adjustable mirror support tubes (3–alumina)

(a) (b) FIGURE 13-12 Measurement of absorption coefficients of high-temperature liquids. (a) Schematic of apparatus of Ebert and Self [62], (b) schematic of their submerged reflector arrangement.

the monochromator and detector via another spherical mirror (M5). The beam is chopped to eliminate emission as well as background radiation from the signal. The transmissivity of the liquid is measured by what they called a “submerged reflector method,” illustrated in Fig. 1312b: A platinum mirror, which may be adjusted via three support rods, is submerged below the surface of the liquid filling the crucible. The platinum mirror is tilted slightly from the horizontal to allow the first surface reflection and multiple internal reflections to be rejected from the collection optics. The thickness of the liquid layer is adjusted by raising and lowering the crucible (leaving the platinum mirror in place). As in the transmission technique, signals for two different layer thicknesses (d1 and d2 ) are ratioed, giving the transmissivity for a layer of thickness 2 (d2 − d1 ). By rejecting the first reflection, and by being able to produce and measure very thin liquid layers, they were able to measure absorption coefficients an order of magnitude higher than Barker, reporting values as high as 70 cm−1 for synthetic molten slags [62]. Similar measurements have been carried out by Gupta and Modest [63] (lithium salts), by Makino and coworkers [64] (alkali metal carbonates), and by Zhang and colleagues [65] (liquid glasses). Foams and Packed Fibers. Measurements on foams were done by Kuhn and coworkers [66] (polystyrene and polyurethane foam insulation), Sacadura et al. [67–69] (fiberglass and carbon foam), Mital and colleagues [70], and Hendricks and Howell [71] (reticulated porous ceramics). The bidirectional reflectance of mullite foam has been measured by Zeghondy and coworkers [72], which agreed well with model results based on the Monte Carlo tool of Tancrez and Taine [29, 33]. Cunnington and coworkers [73] measured the scattering from individual, coated silica fibers, and found qualitative agreement with a theoretical model. Cunnington and Lee measured direct transmissivity and hemispherical reflectivity of randomly packed, high-porosity fibrous material (tiles from the Space Shuttle) [74], and for aerogel-reinforced fibrous material [75]; comparison with Lee’s models [76–79] showed excellent agreement for both materials.

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PROBLEMS

453

59. Stierwalt, D. L.: “Infrared spectral emittance of optical materials,” Applied Optics, vol. 5, no. 12, pp. 1911–1915, 1966. 60. Shvarev, K. M., B. A. Baum, and P. V. Gel’d: “Optical properties of liquid silicon,” Sov. Phys. Solid State, vol. 16, no. 11, pp. 2111–2112, May 1975. 61. Barker, A. J.: “A compact, windowless reflectance furnace for infrared studies of corrosive melts,” Journal of Physics E: Scientific Instruments, vol. 6, pp. 241–244, 1973. 62. Ebert, J. L., and S. A. Self: “The optical properties of molten coal slag,” in Heat Transfer Phenomena in Radiation, Combustion and Fires, vol. HTD-106, ASME, pp. 123–126, 1989. 63. Gupta, S. B., and M. F. Modest: “Measurement of infrared absorption coefficient of molten LiF and Li2 S,” 28th AIAA Thermophysics Conference, Orlando, Florida, AIAA paper no. 93-2760, 1993. 64. Makino, T., M. Edamura, A. Kato, and A. Yoshida: “Thermal radiation properties of molten salt (properties of alkali metal carbonates),” Heat Transfer – Japanese Research, vol. 21, no. 4, pp. 331–339, 1992. 65. Zhang, Z., M. F. Modest, and S. P. Bharadwaj: “Measurement of infrared absorption coefficients of molten glasses,” Experimental Heat Transfer, vol. 14, no. 3, pp. 145–156, 2001. 66. Kuhn, J., H. P. Ebert, M. C. Arduini-Schuster, D. Buettner, and J. Fricke: “Thermal transport in polystyrene and polyurethane foam insulations,” International Journal of Heat and Mass Transfer, vol. 35, no. 7, pp. 1795–1801, 1992. 67. Nicolau, V. P., M. Raynaud, and J.-F. Sacadura: “Spectral radiative properties identification of fiber insulating materials,” International Journal of Heat and Mass Transfer, vol. 37, pp. 311–324, 1994. 68. Doermann, D., and J.-F. Sacadura: “Heat transfer in open cell foam insulation,” ASME Journal of Heat Transfer, vol. 118, no. 1, pp. 88–93, 1996. 69. Baillis, D., M. Raynaud, and J.-F. Sacadura: “Spectral radiative properties of open-cell foam insulation,” Journal of Thermophysics and Heat Transfer, vol. 13, no. 3, pp. 292–298, 1999. 70. Mital, R., J. P. Gore, and R. Viskanta: “Measurements of radiative properties of cellular ceramics at high temperatures,” Journal of Thermophysics and Heat Transfer, vol. 10, no. 1, pp. 33–38, January-March 1996. 71. Hendricks, T. J., and J. R. Howell: “Absorption/scattering coefficients and scattering phase functions in reticulated porous ceramics,” ASME Journal of Heat Transfer, vol. 118, no. 1, pp. 79–87, 1996. 72. Zeghondy, B., E. Iacona, and J. Taine: “Experimental and RDFI calculated radiative properties of a mullite foam,” International Journal of Heat and Mass Transfer, vol. 49, pp. 3702–3707, 2006. 73. Cunnington, G. R., T. W. Tong, and P. S. Swathi: “Angular scattering of radiation from coated cylindrical fibers,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 48, no. 4, pp. 353–362, 1992. 74. Cunnington, G. R., and S. C. Lee: “Radiative properties of fibrous insulations: Theory versus experiments,” Journal of Thermophysics and Heat Transfer, vol. 10, no. 3, pp. 460–466, 1996. 75. Cunnington, G. R., S. C. Lee, and S. M. White: “Radiative properties of fiber-reinforced aerogel: Theory versus experiment,” Journal of Thermophysics and Heat Transfer, vol. 12, no. 1, pp. 17–22, 1998. 76. Lee, S. C.: “Radiative transfer through a fibrous medium: Allowance for fiber orientation,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 36, no. 3, pp. 253–263, 1986. 77. Lee, S. C.: “Radiation heat-transfer model for fibers oriented parallel to diffuse boundaries,” Journal of Thermophysics and Heat Transfer, vol. 2, no. 4, pp. 303–308, Oct 1988. 78. Lee, S. C.: “Effect of fiber orientation on thermal radiation in fibrous media,” International Journal of Heat and Mass Transfer, vol. 32, no. 2, pp. 311–320, 1989. 79. Lee, S. C.: “Scattering phase function for fibrous media,” International Journal of Heat and Mass Transfer, vol. 33, no. 10, pp. 2183–2190, 1990.

Problems 13.1 The absorption coefficient of a liquid, confined between two parallel and transparent windows, is to be measured by the transmission method. The detector signals from transmission measurements with varying liquid thickness are to be used. (a) Using transmission measurements for two thicknesses, show how the absorption coefficient κ may be deduced. Determine how errors in the transmissivity value and the liquid layer thickness affect the accuracy of κ. (b) If transmission measurements are made for many thicknesses, can you devise a method that measures small absorption coefficients more accurately? 13.2 Show how the optical properties (n, k, and κ) of a semitransparent solid may be deduced from the three measurements taken with the apparatus of Myers and coworkers [58], as depicted in Fig. 13-11.

CHAPTER

14 EXACT SOLUTIONS FOR ONE-DIMENSIONAL GRAY MEDIA 14.1

INTRODUCTION

The governing equation for radiative transfer of absorbing, emitting, and scattering media was developed in Chapter 10, resulting in an integro-differential equation for radiative intensity in five independent variables (three space coordinates and two direction coordinates). The problem becomes even more complicated if the medium is nongray (which introduces an additional variable, such as wavelength or frequency) and/or if other modes of heat transfer are present (which make it necessary to solve simultaneously for overall conservation of energy, to which intensity is related in a nonlinear way). Consequently, exact analytical solutions exist for only a few extremely simple situations. The simplest case arises when one considers thermal radiation in a one-dimensional plane-parallel gray medium that is either at radiative equilibrium (i.e., radiation is the only mode of heat transfer) or whose temperature field is known. Analytical solutions for such simple problems have been studied extensively, partly because of the great importance of one-dimensional plane-parallel media, partly because the simplicity of such solutions allows testing of more general solution methods, and partly because such a solution can give qualitative indications for more difficult situations. In the present chapter we develop some analytical solutions for one-dimensional planeparallel media and also include a few solutions for one-dimensional cylindrical and spherical media (without development). In general, we shall assume the medium to be gray, and all radiative R ∞ intensity-related quantities are total, i.e., frequency-integrated quantities, for example, Ib = 0 Ibν dν = n2 σT 4 /π. Most relations also hold, on a spectral basis, for nongray media, except for those that utilize the statement of radiative equilibrium, ∇ · q = 0 (since this relation does not hold on a spectral basis).

14.2 GENERAL FORMULATION FOR A PLANE-PARALLEL MEDIUM The governing equation for the intensity field in an absorbing, emitting, and scattering medium is, from equation (10.24), Z σs sˆ · ∇I = κIb − βI + (14.1) I(ˆs i ) Φ(ˆs i , sˆ ) dΩ i , 4π 4π 454

14.2 GENERAL FORMULATION FOR A PLANE-PARALLEL MEDIUM

455

θ I+(τ,

A2

θ

Qs

Q

τ τ´

´

τs

A2

τ

θ

s

s z

θ

s

τ

I1(θ )

θ)

τ ´s

P

τs

τL τ´

I2 (θ )

A1

A1

z

I –(τ ,θ )

(a) (b) FIGURE 14-1 Coordinates for radiative intensities in a one-dimensional plane-parallel medium: (a) upward directions, (b) downward directions.

which describes the change of radiative intensity along a path in the direction of sˆ . The formal solution to equation (14.1) is given by equation (10.28) as Z τs ′ −τs I(r, sˆ ) = Iw (ˆs) e + S(τ′s , sˆ ) e−(τs −τs ) dτ′s , (14.2) 0

where S is the radiative source term, equation (10.25), Z ω ′ ′ S(τs , sˆ ) = (1 − ω)Ib (τs ) + I(τ′ , sˆ i ) Φ(ˆs, sˆ i ) dΩ i , (14.3) 4π 4π s Rs and τs = 0 β(s) ds is optical thickness or optical depth based on extinction coefficient1 measured from a point on the wall (τ′s = 0) toward the point under consideration (τ′s = τs ), in the direction of sˆ . For a plane-parallel medium the change of intensity is illustrated in Fig. 14-1a, measuring polar angle θ from the direction perpendicular to the plates (z-direction), and azimuthal angle ψ in a plane parallel to the plates (x-y-plane): Radiative intensity of strength Iw (ˆs) = Iw (θ, ψ) leaves the point on the bottom surface into the direction of θ, ψ, toward the point under consideration, P. This intensity is augmented by the radiative source (by emission and by in-scattering, i.e., scattering of intensity from other directions into the direction of P). The amount of energy S(τ′s , θ, ψ) dτ′s is released over the infinitesimal optical depth dτ′s and travels toward P. Since this energy also undergoes absorption and out-scattering along its path from τ′s to τs , only the ′ fraction e−(τs −τs ) actually arrives at P. In general, the intensity leaving the bottom wall may vary across the bottom surface, and radiative source and medium properties may vary throughout the medium, i.e., in the directions parallel to the plates as well as normal to them. We shall now assume that both plates are isothermal and isotropic, i.e., neither temperature nor radiative properties vary across each plate and properties may show a directional dependence on polar angle θ, but not on azimuthal angle ψ. Thus, the intensity leaving the bottom plate at a certain location is the same for all azimuthal angles and, indeed, for all positions on that plate; it is a function of polar angle θ alone. We also assume that the temperature field and radiative properties of the medium vary only in the direction perpendicular to the plates. This assumption implies that the radiative source at position Q, S(τ′ , θ), is identical to the one at R z′ position Qs , S(τ′s , θ), or any horizontal position with identical z-coordinate τ′ = 0 β dz (based on extinction coefficient). Therefore, radiative source, S(τ, θ), and radiative intensity, I(τ, θ), both depend only on a single space coordinate plus a single direction coordinate. The radiative 1

We use here the notation τs to describeRoptical depth along s so that we will be able to use the simpler τ for optical z depth perpendicular to the plates, i.e., τ = 0 β dz.

456

14 EXACT SOLUTIONS FOR ONE-DIMENSIONAL GRAY MEDIA

source term may be simplified for the one-dimensional case to S(τ′ , θ) = (1 − ω)Ib (τ′ ) +

ω 4π

Z

2π ψi =0

Z

π

I(τ′ , θi ) Φ (θ, ψ, θi , ψi ) sin θi dθi dψi .

(14.4)

θi =0

For isotropic scattering, Φ ≡ 1, and we find immediately from the definition for incident radiation, G [equation (10.32)], that ω G(τ′ ). S(τ′ ) = (1 − ω)Ib (τ′ ) + (14.5) 4π In other words, the source term does not depend on direction, that is, the radiative source due to isotropic emission and isotropic in-scattering is also isotropic. If the scattering is anisotropic, we may write, from equation (12.99),2 Φ(ˆs · sˆ i ) = 1 +

M X

Am Pm (ˆs · sˆ i ),

(14.6)

m=1

where it is assumed that the series may be truncated after M terms. Measuring the polar angle from the z-axis and the azimuthal angle from the x-axis (in the x-y-plane) for both sˆ and sˆ i , we get the direction vectors ˆ sˆ = sin θ(cos ψˆı + sin ψˆ) + cos θk, ˆ sˆ i = sin θi (cos ψi ˆı + sin ψi ˆ) + cos θi k,

(14.7) (14.8)

and Φ(θ, ψ, θi , ψi ) = 1 +

M X

Am Pm [cos θ cos θi + sin θ sin θi cos(ψ − ψi )].

(14.9)

m=1

Using a relationship between Legendre polynomials [1], one may separate the directional dependence in the last relationship by Pm [cos θ cos θi + sin θ sin θi cos(ψ − ψi )] = Pm (cos θ)Pm (cos θi ) m X (m − n)! m P (cos θ)Pnm (cos θi ) cos m(ψ − ψi ), +2 (m + n)! n

(14.10)

n=1

where the Pnm are associated Legendre polynomials. Thus, the scattering phase function may be rewritten as Φ(θ, ψ, θi , ψi ) = 1 +

M X

Am Pm (cos θ)Pm (cos θi )

m=1

+2

M X m X m=1 n=1

Am

(m − n)! m P (cos θ)Pnm (cos θi ) cos m(ψ − ψi ). (m + n)! n

(14.11)

For a one-dimensional plane-parallel geometry, the intensity does not depend on azimuthal angle, and we may carry out the ψi -integration in equation (14.4). This integration leads to a one-dimensional scattering phase function of Φ(θ, θi ) = 2

1 2π

Z



Φ(ˆs · sˆ i ) dψi = 1 + 0

M X

Am Pm (cos θ)Pm (cos θi ),

(14.12)

m=1

In Chapter 12 we used Θ to denote the angle between the incoming and scattered ray and, therefore, cos Θ = sˆ · sˆ i .

14.2 GENERAL FORMULATION FOR A PLANE-PARALLEL MEDIUM

since

R

2π 0

457

cos m(ψ − ψi ) dψi = 0. The radiative source then becomes ω S(τ , θ) = (1 − ω)Ib (τ ) + 2 ′



Z

π

I(τ′ , θi ) Φ(θ, θi ) sin θi dθi .

(14.13)

0

For linear-anisotropic scattering, with Φ(ˆs · sˆ i ) = 1 + A1 P1 (ˆs · sˆ i ) = 1 + A1 sˆ · sˆ i ,

M = 1,

(14.14)

and, using the definitions for incident radiation and radiative heat flux, equations (10.32) and (10.52), respectively, equation (14.13) reduces to S(τ′ , θ) = (1 − ω)Ib (τ′ ) +

 ω  G(τ′ ) + A1 q(τ′ ) cos θ . 4π

(14.15)

We may now simplify the equation of radiative transfer, equation (14.1), using the geometric relations τs = τ/ cos θ and τ′s = τ′ / cos θ (see Fig. 14-1a), Z π dI ω dI 1 dI = cos θ I(τ, θi ) Φ(θ, θi ) sin θi dθi . (14.16) = = (1 − ω)Ib − I + β ds dτs dτ 2 0 Similarly, the expression for intensity, equation (14.2), may be simplified to Z τ π dτ′ ′ I+ (τ, θ) = I1 (θ) e−τ/ cos θ + , 0 π/2) we obtain (see Fig. 14-1b) Z τ dτ′ ′ − (τL −τ)/ cos θ I (τ, θ) = I2 (θ) e + S(τ′ , θ) e(τ −τ)/ cos θ cos θ τ Z LτL π dτ′ ′ = I2 (θ) e(τL −τ)/ cos θ − S(τ′ , θ) e(τ −τ)/ cos θ , < θ < π, (14.18) cos θ 2 τ where I2 (θ) is the intensity leaving the wall at τ = τL (Wall 2). It is customary (and somewhat more compact) to rewrite equations (14.16) through (14.18) in terms of the direction cosine µ = cos θ, or Z 1 dI ω µ + I = (1 − ω)Ib + I(τ, µi ) Φ(µ, µi ) dµi = S(τ, µ), (14.19) dτ 2 −1 +

−τ/µ

I (τ, µ) = I1 (µ) e

+

Z

τ



S(τ′ , µ) e−(τ−τ )/µ 0

I− (τ, µ) = I2 (µ) e(τL −τ)/µ −

Z

τL τ



dτ′ , µ

S(τ′ , µ) e(τ −τ)/µ

dτ′ , µ

0 < µ < 1,

(14.20a)

−1 < µ < 0.

(14.20b)

458

14 EXACT SOLUTIONS FOR ONE-DIMENSIONAL GRAY MEDIA

For heat transfer purposes the incident radiation, G, and radiative heat flux, q, are of interest. From the definition of incident radiation, equation (10.32), it follows that Z 2π Z π Z +1 G(τ) = I(τ, θ) sin θ dθ dψ = 2π I(τ, µ) dµ 0

= 2π

"Z

= 2π

"Z

= 2π +

I− (τ, µ) dµ + −1 1 −

Z

I (τ, −µ) dµ + 0

I+ (τ, µ) dµ 0

Z

I1 (µ) e−τ/µ dµ + 0

1 0

+1

1

(Z

Z

−1

0 0

"Z

τ ′

S(τ , µ) e

#

1 +

I (τ, µ) dµ 0

Z

#

1

I2 (−µ) e−(τL −τ)/µ dµ 0

−(τ−τ′ )/µ



dτ +

0

Z

τL

−(τ′ −τ)/µ



S(τ , −µ) e τ

) dµ dτ . µ ′

#

(14.21)

Similarly, for the radiative heat flux for a plane-parallel medium, equation (10.52), Z 2π Z π Z +1 q(τ) = I(τ, θ) cos θ sin θ dθ dψ = 2π I(τ, µ)µ dµ 0

= 2π +

(Z

Z

1 0

−1

0 1

I1 (µ) e−τ/µ µ dµ − 0

"Z

τ ′

S(τ , µ) e 0



Z

−(τ−τ )/µ

1

I2 (−µ) e−(τL −τ)/µ µ dµ 0 ′

dτ −

Z

τL



−(τ′ −τ)/µ

S(τ , −µ) e τ

) dτ dµ . ′

#

(14.22)

During a large part of this chapter we shall study the solution to equations (14.21) and (14.22) for a number of different situations. We shall assume either that the temperature across the medium and, therefore, Ib (τ) is known or that radiative equilibrium prevails, dq/dτ = 0. In either case we are interested in the direction-integrated form of the equation of transfer, equation (14.1), which has been given by equation (10.59) as ∇ · q = κ(4πIb − G),

(14.23)

or, for the present one-dimensional case after division by extinction coefficient β (and remembering that κ/β = 1 − σs /β), dq = (1 − ω)(4πIb − G). (14.24) dτ We note in passing that, up to this point, all relations, and in particular equations (14.21), (14.22), and (14.24), hold on a total basis for a gray medium and on a spectral basis for any medium. If radiative equilibrium prevails, then dq/dτ = 0 or, in the presence of a heat source,3 ′′′ dq Q˙ = (τ), dτ β ′′′

(14.25)

where Q˙ is local heat generation per unit time and volume. Equation (14.25) is valid only for total radiative heat flux and may, therefore, in this form be applied only to gray media. For such a case we see that the incident radiation is closely related to the blackbody intensity (and, therefore, temperature) by ′′′ Q˙ 4πIb (τ) = G(τ) + (τ). (14.26) κ 3

Such heat sources are often used to couple the radiation problem with overall energy conservation.

14.3 PLANE LAYER OF A NONSCATTERING MEDIUM

459

14.3 PLANE LAYER OF A NONSCATTERING MEDIUM Enclosure with Black Bounding Surfaces Since this is the most basic of cases, we shall rederive the relationships for this simple problem. From equation (14.3), with ω = 0, it follows that S(τ′ , sˆ ) = Ib (τ′ ); for black bounding surfaces, the intensity leaving the lower plate is I1 (θ) = Ib1 and the intensity leaving the top plate is I2 (θ) = Ib2 . Thus, for this simple case, neither radiative source nor boundary intensities are direction-dependent. Equations (14.17) and (14.18) may then be rewritten as Z τ π dτ′ ′ + −τ/cos θ I (τ, θ) = Ib1 e , 0 0

Remember that equation (16.5) is truncated beyond l = N, so that IN+1 (τ) = 0.

(16.15)

16.4 BOUNDARY CONDITIONS FOR THE PN -METHOD

499

k

ψ s

θ n

r

ψ´ θ´

rw FIGURE 16-2 Prescribed boundary intensities for PN -method.

0

everywhere on the surface, that is, the intensity leaving a surface (described by the vector rw ) must be prescribed in some fashion for all outgoing directions n· ˆ sˆ > 0 (with nˆ being the outward surface normal), as shown in Fig. 16-2. When the PN -approximation is applied [truncating equation (16.1) after l = N] this boundary condition can no longer be satisfied and must be replaced by one that either satisfies equation (16.15) at selected directions sˆ i or satisfies it in an integral sense. Mark [15, 16] and Marshak [17] proposed two different sets of boundary conditions for the spherical harmonics method as applied to neutron transport within a one-dimensional plane-parallel medium.

Mark’s Boundary Condition For a one-dimensional slab of optical thickness τL , equation (16.15) may be rewritten as I(0, µ) = Iw1 (µ),

0 < µ < 1,

(16.16a)

I(τL , µ) = Iw2 (µ),

−1 < µ < 0,

(16.16b)

where Iw1 and Iw2 are the prescribed intensities at Surfaces 1 (τ = 0) and 2 (τ = τL ).6 The PN -method for such a medium, equation (16.14), requires (N + 1) boundary conditions, say 21 (N + 1) each, at τ = 0 and τ = τL (assuming that N is odd). Noting that the equation PN+1 (µ) = 0

(16.17)

has precisely 21 (N + 1) roots µi with values between 0 and 1, Mark suggested replacing the boundary conditions of equation (16.16) by I(0, µ = µi ) = Iw1 (µi ), I(τL , µ = −µi ) = Iw2 (−µi ),

i = 1, 2, . . . , 21 (N + 1), i = 1, 2, . . . ,

1 2 (N

+ 1),

(16.18a) (16.18b)

where the µi are the positive roots of equation (16.17). A detailed explanation for this choice has been given by Mark [15, 16] and by Davison [3]. For example, for the P1 -approximation √ for a medium bounded by black walls we get with P2 (µ) = 21 (3µ2 − 1), µ1 = 1/ 3 and, from equation (16.5), ! I1 (0) 1 I 0, µ = √ = I0 (0) + √ = Ib1 , (16.19a) 3 3 ! I1 (τL ) 1 (16.19b) I τL , µ = − √ = I0 (τL ) − √ = Ib2 . 3 3 6

We include the subscript w here to distinguish the Iwi from the intensity moments Ii defined by equation (16.5).

500

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

One serious drawback of Mark’s boundary conditions is the fact that they are difficult, if not impossible, to apply to more complicated geometries.

Marshak’s Boundary Conditions An alternative set of boundary conditions for the one-dimensional plane-parallel PN -approximation was proposed by Marshak, who suggested that equation (16.16) be satisfied in an integral sense by setting Z Z

1

I(0, µ)P2i−1 (µ) dµ = 0 0

I(τL , µ)P2i−1 (µ) dµ = −1

Z Z

1

Iw1 (µ)P2i−1 (µ) dµ,

i = 1, 2, . . . , 21 (N + 1);

(16.20a)

Iw2 (µ)P2i−1 (µ) dµ,

i = 1, 2, . . . , 21 (N + 1).

(16.20b)

0 0 −1

Again, the reason for choosing all the Legendre polynomials of odd order has been explained in detail by Marshak [17] and Davison [3]. Substituting equation (16.5) and assuming diffuse surfaces, i.e., Iw = Jw /π, leads to N X

Z

Jw2 Il (τL ) Pl (µ)P2i−1 (µ) dµ = π −1

Z

l=0

N X l=0

1

Jw1 π

Il (0)

Z Z

Pl (µ)P2i−1 (µ) dµ = 0 0

1

P2i−1 (µ) dµ,

i = 1, 2, . . . , 21 (N + 1);

(16.21a)

P2i−1 (µ) dµ,

i = 1, 2, . . . , 21 (N + 1).

(16.21b)

0 0 −1

As an example we again consider the P1 -approximation for a medium bounded by black walls. Then, with P1 (µ) = µ, Z

1

I(0, µ)µ dµ = 0

Z

1 0

  I0 (0) + I1 (0)µ µ dµ =

Z

1

Ib1 µ dµ, 0

or I0 (0) + 32 I1 (0) = Ib1 ,

(16.22a)

2 3 I1 (τL )

(16.22b)

I0 (τL ) −

= Ib2 .

√ We note that replacing the factor 2 in Marshak’s boundary condition by a 3 converts it to Mark’s boundary condition. One advantage of Marshak’s boundary condition is that it may be extended to more general problems, although not painlessly. Note that the integration in equation (16.20) is carried out over all directions above the surface (i.e., a hemisphere) with the Legendre polynomials of equation (16.5) as weight factors. Thus, it appears natural to generalize the boundary condition to (see Fig. 16-2) Z Z m m I(rw , sˆ )Y2i−1 (ˆs) dΩ = Iw (ˆs)Y2i−1 (ˆs) dΩ, n·ˆ ˆ s>0

n·ˆ ˆ s>0

i = 1, 2, . . . , 12 (N + 1), m

all relevant m,

(16.23)

where the Y2i−1 (ˆs) are expressed in terms of a local coordinate system, in which polar angle θ′ is measured from the surface normal (i.e., cos θ′ = nˆ · sˆ ), and azimuthal angle ψ′ is measured on the surface, as indicated in Fig. 16-2. The statement “all relevant m′′ rather than −i ≤ m ≤ +i appears in equation (16.23) since it may provide more boundary conditions than are required. For example, for a one-dimensional plane-parallel medium there is no azimuthal dependence,

16.4 BOUNDARY CONDITIONS FOR THE PN -METHOD

501

z, τz

s T2

k s

θ´ n2

θ

τy n1

s

θ τx

T1

x,τx

FIGURE 16-3 Geometry for Example 16.1.

so that all Inm with m , 0 vanish. and the only “relevant” value for m is m = 0. This leads to a single boundary condition on each surface for the P1 -approximation (as already seen to be correct), two for the P3 -approximation, and so on. Generally, equation (16.23) leads to too many boundary conditions in multidimensional situations. For example, for the P1 -approximation for a general three-dimensional medium without symmetry, equation (16.23) leads to three boundary conditions everywhere (i = 1, m = 0, ±1), while only one is needed (as explained in the following section). Davison [3] has shown that the number of superfluous conditions is always at least one less than the possible m at i = 21 (N + 1). Thus, on intuitive grounds it was accepted practice to satisfy equation (16.23) for all m for i = 1, 2, . . . , 21 (N − 1), and for as many relevant m as possible for i = 12 (N + 1). Recently, Modest [18] has shown that a self-consistent set of boundary conditions is obtained if, for i = 12 (N + 1), only the even values for m are chosen, discarding all odd m. Example 16.1. Consider the infinite quarter-space τx > 0, τz > 0 bounded by isothermal black surfaces at T1 and T2 as shown in Fig. 16-3. Develop the boundary conditions for the P1 -approximation at both surfaces (i.e., τx = 0 and τz = 0). Solution For the P1 -approximation equation (16.1) reduces to I(τx ,τz , θ, ψ) = I00 (τx ,τz ) − I1−1 (τx ,τz ) sin ψP1−1 (cos θ) + I10 (τx ,τz )P10 (cos θ) + I11 (τx ,τz ) cos ψP11 (cos θ). For this two-dimensional problem it is convenient to measure polar angle θ from the τz -axis, and azimuthal angle ψ in the τx -τ y -plane from the τx -axis. Then I(ψ) = I(−ψ) and, with P10 (cos θ) = cos θ, and P11 = P1−1 (cos θ) = − sin θ, I(τx , τ y , θ, ψ) = I00 + I10 cos θ − I11 cos ψ sin θ, since the term involving sin ψ must vanish owing to symmetry. Therefore, equation (16.23) is able to provide two boundary conditions everywhere on the surface (i = 1 and m = 0, 1), while we need only one (as to be developed in the next section). Thus, following the discussion of equation (16.23), we introduce local direction coordinate systems on the surfaces and satisfy equation (16.23) only for m = 0. For the bottom surface, τz = 0, the problem is simple since the surface normal is parallel to the τz -axis, from which the polar angle is measured. Thus, Z or

2π ψ=0

Z

π/2 θ=0

Z  I00 + I10 cos θ − I11 cos ψ sin θ cos θ sin θ dθ dψ =



2π 0

Z

π/2

Ib1 cos θ sin θ dθ dψ, 0

502

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

I00 (τx , 0) + 32 I10 (τx , 0) = Ib1 . At the vertical surface (τx = 0) P10 = cos θ′ , where θ′ is the angle between a direction vector and the ˆ it follows that surface normal nˆ = ˆı. Thus, with cos θ′ = sˆ · ˆı and sˆ = sin θ(cos ψˆı + sin ψˆ) + cos θk, ′ cos θ = sin θ cos ψ and Z

π/2 ψ=−π/2

Z

or

π θ=0



   I00 + I10 cos θ − I11 cos ψ sin θ sin θ cos ψ sin θ dθ dψ = π I00 − 23 I11 = πIb2 , I00 (0, τz ) − 23 I11 (0, τz ) = Ib2 .

We shall see in the next section that I00 is directly proportional to incident radiation, while I10 and I11 are proportional to radiative heat flux into the τ y - and τx -directions, respectively.

Davison [3] stated that for low-order approximations Marshak’s boundary conditions would give superior results, but that for high-order approximations Mark’s boundary conditions should be more accurate. However, subsequent numerical work by Pellaud [19] and Schmidt and Gelbard [20] showed Marshak’s boundary condition leads to more accurate results, even in high-order approximations.

16.5

THE P1 -APPROXIMATION

If the series in equation (16.1) is truncated beyond l = 1 (i.e., Ilm ≡ 0 for l ≥ 2), we get the lowest-order, or P1 , approximation, or I(r, sˆ ) = I00 Y00 + I1−1 Y1−1 + I10 Y10 + I11 Y11 .

(16.24)

From standard mathematical texts, such as MacRobert [21], or directly from equation (16.3) we find the associated Legendre polynomials as P00 = 1, P10 = cos θ, P11 = P1−1 = − sin θ, and, using equation (16.2), I(r, θ, ψ) = I00 + I10 cos θ + I1−1 sin θ sin ψ − I11 sin θ cos ψ. (16.25) We notice that equation (16.25) has four terms: The first term is independent of direction, the second is proportional to the z-component of the direction vector sˆ = sin θ cos ψˆı + sin θ sin ψˆ + ˆ the third is proportional to s y and the last to sx .7 Each term is preceded by an unknown cos θk, function of the space coordinates, which are to be determined. Equation (16.25) may be written more compactly by introducing two new functions, a (a scalar) and b (a vector having three components) as I(r, sˆ ) = a(r) + b(r) · sˆ .

(16.26)

The four unknowns—a and the three components of b, or the four components of Inm —can be related to physical quantities. Substituting equation (16.26) into the definition for incident radiation yields Z Z Z G(r) = I(r, sˆ ) dΩ = a(r) dΩ + b(r) · sˆ dΩ = 4πa(r), (16.27) 4π





since Z 7

sˆ dΩ = 4π

Z

2π 0

Z

π 0

    0 sin θ cos ψ    sin θ sin ψ   sin θ dθ dψ = 0 = 0.      0 cos θ

Provided the polar angle is measured from the z-axis, and the azimuthal angle from the x-axis.

(16.28)

16.5 THE P1 -APPROXIMATION

503

Similarly, substituting equation (16.26) into the definition for the radiative heat flux gives Z Z Z 4π q(r) = I(r, sˆ ) sˆ dΩ = a(r) b(r), (16.29) sˆ dΩ + b(r) · sˆ sˆ dΩ = 3 4π 4π 4π since Z

sˆ sˆ dΩ = 4π

=

Z Z

2π 0

π 0

Z

π 0

  sin2 θ cos2 ψ  2 sin θ sin ψ cos ψ  sin θ cos θ cos ψ

 π sin2 θ   0  0

 1 4π  = 0 3  0

0 1 0

0 π sin2 θ 0

 0 4π 0 = δ,  3 1

sin2 θ sin ψ cos ψ sin2 θ sin2 ψ sin θ cos θ sin ψ

 0    sin θ dθ 0  2  2π cos θ

 sin θ cos θ cos ψ  sin θ cos θ sin ψ  × sin θ dθ dψ  cos2 θ

(16.30)

where δ is the unit tensor, and b · δ = b. Therefore, we may rewrite equation (16.26) in terms of incident radiation and radiative heat flux as I(r, sˆ ) =

1 [G(r) + 3q(r) · sˆ ]. 4π

(16.31)

We find that, except for a constant factor, I00 is the incident radiation, while I11 , I1−1 , and I10 are the x-, y-, and z-components of the radiative heat flux, respectively. The preceding development is useful to show that equation (16.31) indeed corresponds to the lowest order of the PN approximation, equation (16.1). Of course, equation (16.31) should have physical significance and it should be possible to derive it from physical principles. This was done by Modest [22], who treated radiation as a “photon gas” with momentum and energy, and derived the intensity field through quantum statistics. He showed that the average photon velocity (which is proportional to heat flux) is inversely proportional to optical thickness, and that equation (16.31) holds for a location a large optical distance away from any points not at thermodynamic equilibrium (sharp temperature gradients, steps in temperature, etc.). Now, substituting equation (16.31) into equation (16.4) and assuming linear-anisotropic scattering,8 Φ(ˆs · sˆ ′ ) = 1 + A1 sˆ · sˆ ′ ,

(16.32)

leads to Z Z 1 ′ ′ ′ I(ˆs ) Φ(ˆs · sˆ ) dΩ = (G + 3q · sˆ ′ )(1 + A1 sˆ · sˆ ′ ) dΩ′ 4π 4π 4π "Z "Z # ! # Z Z 3q G ′ ′ ′ ′ ′ ′ ′ ′ = dΩ + A1 sˆ · sˆ dΩ + · sˆ dΩ + A1 sˆ sˆ dΩ · sˆ 4π 4π 4π 4π 4π 4π = G + A1 q · δ · sˆ = G + A1 q · sˆ ,

(16.33)

where equations (16.28) and (16.30) have been employed (and the last step is easily verified by, say, using Cartesian coordinates and carrying out the dot product). Thus, equation (16.4) 8 Because of the orthogonality of spherical harmonics the P1 -approximation remains unchanged for nonlinear anisotropic scattering. The choice of the functional form for intensity, equation (16.31), does not allow such scattering behavior, i.e., the medium must be so optically thick that any nonlinear anisotropically scattered intensity is smoothed out in the immediate vicinity of the scattering point. In reality, this smoothing implies that a “best” linear-anisotropic scattering factor A∗1 must be determined.

504

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

becomes   1 ω 1 ∇τ · sˆ (G + 3q · sˆ ) + (G + 3q · sˆ ) ≃ (1−ω)Ib + (G + A1 q · sˆ ), 4π 4π 4π

(16.34)

∇τ · q = (1 − ω)(4πIb − G),

(16.35)

where we were able to pull the direction vector sˆ inside the gradient, since direction is independent of position. Multiplying equation (16.34) by Y00 = 1 and integrating over all solid angles gives

where again equations (16.28) and (16.30) have been invoked. Equation (16.35) is, of course, identical to equation (10.59) since it does not depend on the functional form for intensity. To obtain additional equations we may multiply equation (16.34) by Y1m (m = −1, 0, +1) or equivalently, by the components of the direction vector sˆ . Choosing the latter and integrating over all directions leads to " Z # " Z # Z Z 1 1 ∇τ · G sˆ sˆ dΩ + 3q · sˆ sˆ sˆ dΩ + G sˆ dΩ + 3q · sˆ sˆ dΩ 4π 4π 4π 4π 4π 4π # " Z Z Z ω = (1 − ω)Ib sˆ dΩ + sˆ dΩ + A1 q · sˆ sˆ dΩ . (16.36) G 4π 4π 4π 4π R It is easy to show that 4π sˆ sˆ sˆ dΩ = 0 (and, indeed, the integral over any odd multiple of sˆ ) and, therefore, this equation reduces to

or

1 ωA1 ∇τ · (Gδ) + q · δ = q · δ, 3 3 ∇τ G = − (3 − A1 ω) q.

(16.37)

Equations (16.35) and (16.37) are a complete set of one scalar and one vector equation in the unknowns G and q, and are the governing equations for the P1 or differential approximation. The heat flux may be eliminated from these equations by taking the divergence of equation (16.37) after dividing by (1 − A1 ω/3):   1 ∇τ · (16.38) ∇τ G = −3∇τ · q = −3(1 − ω)(4πIb − G). 1 − A1 ω/3 If A1 ω is constant (does not vary across the volume) this equation reduces to ∇τ2 G − (1 − ω) (3 − A1 ω) G = −(1 − ω) (3 − A1 ω) 4πIb .

(16.39)

Equation (16.39) is a Helmholtz equation, closely related to Laplace’s equation, and is elliptic in nature (see, for example, a standard mathematics text such as Pipes and Harvill [23]). As such, it requires a single boundary condition specified everywhere on the enclosure surface. If radiative equilibrium prevails, then ∇ · q = 0, and

or

∇τ2 G = 0,

(16.40)

∇τ2 Ib = 0.

(16.41)

In either case we get the elliptic Laplace’s equation with the same boundary condition requirements. Once the incident radiation and/or blackbody intensity has been determined, the radiative heat flux is found from equation (16.37) as q=−

1 ∇τ G. 3 − A1 ω

(16.42)

16.5 THE P1 -APPROXIMATION

505

Equation (16.23) can supply three boundary conditions for the P1 -approximation, while equations (16.39) or (16.40) only require a single one. Thus, following the discussion of Marshak’s boundary condition, equation (16.23), we choose only the case of m = 0 for the weight function in equation (16.23), with polar angle measured from the surface normal. Thus, 0

Y1 (ˆs) = P10 (cos θ′ ) = cos θ′ = sˆ · n, ˆ

(16.43)

where θ′ is the polar angle of sˆ in the local coordinate system as shown in Fig. 16-2. Physically, that is, without reference to the general PN -approximation, this choice of boundary condition implies that the directional distribution of the outgoing intensity along the enclosure wall is satisfied in an integral sense, by requiring the normal heat flux to be continuous (from enclosure surface into the participating medium). Then the boundary condition becomes Z Z 1 (G + 3q · sˆ ) sˆ · nˆ dΩ Iw (ˆs) sˆ · nˆ dΩ = 4π n·ˆ ˆ s>0 n·ˆ ˆ s>0 Z 2π Z π/2   1 G + 3qt1 sin θ′ cos ψ′ + 3qt2 sin θ′ sin ψ′ + 3qn cos θ′ cos θ′ sin θ′ dθ dψ′ = 4π ψ′ =0 θ′ =0 Z π/2 1 1 (G + 3qn cos θ′ ) cos θ′ sin θ′ dθ′ = (G + 2qn ) = 2 0 4 or Z Iw (ˆs) sˆ · nˆ dΩ.

G + 2q · nˆ = 4

(16.44)

n·ˆ ˆ s>0

Here qt1 and qt2 are the two components of the heat flux vector tangential to the surface and qn = q · nˆ is the normal component. For an opaque surface which emits and reflects radiation diffusely, Iw (ˆs) = Jw /π, where Jw is the surface’s radiosity. Substituting this into equation (16.44) leads to Z 2π Z π/2 4 G + 2q · nˆ = Jw cos θ′ sin θ′ dθ′ dψ′ = 4Jw . (16.45) π 0 0 Recalling equation (5.26), q · nˆ =

ǫ (πIbw − Jw ) , 1−ǫ

(16.46)

equation (16.44) finally becomes 2q · nˆ = 4Jw − G =

ǫ (4πIbw − G), 2−ǫ

(16.47)

where ǫ is the local surface emittance. Modest [22] has shown that equation (16.47) also holds if the surface reflectance consists of purely diffuse and purely specular components, i.e., if ǫ = 1 − ρd − ρs .

(16.48)

Thus, within the accuracy of the P1 , or differential, approximation, the results for enclosures with diffusely and/or specularly reflecting surfaces are identical. Since equation (16.39) is a second-order equation in G, it is of advantage to eliminate q · nˆ from the boundary condition using equation (16.42). Thus, −

2−ǫ 2 nˆ · ∇τ G + G = 4πIbw ǫ 3 − A1 ω

(16.49)

is the correct boundary condition to go with equation (16.38) or (16.39). Equation (16.49) is known as a boundary condition of the third kind (since it incorporates both the dependent variable and its normal gradient). Appendix F provides subroutine P1sor for the solution to this system for a two-dimensional (rectangular or axisymmetric-cylindrical) enclosure.

506

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

Summary of the P1 -Approximation For convenience we will summarize here the pertinent equations and boundary conditions that constitute the P1 -approximation for a medium bounded by diffuse, gray walls. This can be done in two ways: (i) simultaneous first-order PDEs in incident radiation and radiative heat flux, or (ii) a single elliptic second-order PDE in incident radiation. The former is the preferred formulation for the case of radiative equilibrium in a gray medium; the latter is more useful if the temperature field is known (or must be found through iteration). Simultaneous Equations:

r = rw :

∇ · q = κ(4πIb − G),  ∇G = − 3β − A1 σs q, ǫ 2q · nˆ = 4Jw − G = (4πIbw − G). 2−ǫ

(16.50a) (16.50b) (16.50c)

Incident Radiation Formulation: ! 1 1 ∇· ∇G − G = −4πIb , 3κ β − A1 σs /3 2−ǫ 2 r = rw : − nˆ · ∇G + G = 4πIbw , ǫ 3β − A1 σs and q=−

1 ∇G. 3β − A1 σs

(16.51a) (16.51b)

(16.52)

Example 16.2. Consider an isothermal, gray slab at temperature T and of optical thickness τL , bounded by two isothermal black surfaces at temperature Tw . The medium scatters linear-anisotropically. Determine an expression of the nondimensional heat flux as a function of the optical parameters. Solution Since the temperature field is given we use the incident radiation formulation, and we may write equation (16.39) or equation (16.51a) as

or

d2 G − (1 − ω) (3 − A1 ω) G = −(1 − ω) (3 − A1 ω) 4n2 σT 4 , dτ2 G(τ) = C1 cosh γτ + C2 sinh γτ + 4n2 σT 4 ,

where γ=

p (1 − ω) (3 − A1 ω).

Because of the symmetry of the problem it is advantageous to place the origin at the center of the slab, i.e., −τL /2 ≤ τ ≤ +τL /2. Then dG (τ = 0) = 0 = γC1 sinh(γ × 0) + γC2 cosh(γ × 0) + 0, dτ or C2 = 0. Applying equation (16.49) [or (16.51b)] at τ = τL /2, with ǫ = 1, we get 2 dG (τL /2) + G(τL /2) = 4n2 σTw4 , 3 − A1 ω dτ or

and

2γ C1 sinh 21 γτL + C1 cosh 12 γτL + 4n2 σT 4 = 4n2 σTw4 , 3 − A1 ω 4n2 σ(T 4 − Tw4 ) , C1 = − q 1−ω cosh 12 γτL + 2 3−A sinh 12 γτL 1ω

Nondimensional heat flux Ψ = q/n2σ (Tw4 – Tw4)

16.5 THE P1 -APPROXIMATION

1.0

507

ω=0

0.8 Exact P1-approximation

ω = 0.5 A1 = 0

0.6

A1 = +1 A1 = –1

0.4

0.2

0

1

0

2

3

4 6 5 Optical thickness τ L

7

8

9

10

FIGURE 16-4 Nondimensional wall heat fluxes for a constant-temperature slab with linear-anisotropic scattering.

G(τ) = 4n2 σT 4 − 4n2 σ(T 4 − Tw4 )

cosh γτ . q 1−ω 1 cosh 12 γτL + 2 3−A sinh γτ L ω 2 1

The heat flux is determined from equation (16.42) as Ψ=

q n2 σ(T 4 − Tw4 )

=−

1 1 dG = n2 σ(T 4 − Tw4 ) 3 − A1 ω dτ

sinh

1 2 γτL

2 sinh γτ . q 1 1ω + 12 3−A cosh γτ L 1−ω 2

Some sample results for the heat flux at the wall (τ = τL /2) are given in Fig. 16-4. We note that in this case the P1 -approximation goes to the correct optically thin limit Ψ → 4τ/τL (emission, but no self-absorption of emission), but not to the correct optically thick limit (since, as a result of the temperature step at the wall, there will always be an intensity discontinuity at the wall). In fact, for this problem the results of the P1 -approximation are worst (in absolute magnitude) close to that location. Example 16.3. Let us look at a gray medium at radiative equilibrium placed between two black concentric cylinders of radius R1 and R2 that are isothermal at temperatures T1 and T2 . For simplicity, we shall assume that the medium does not scatter (σs = 0), and that its absorption coefficient, κ, is constant. We desire to find the heat flux from inner to outer cylinder as a function of the ratio R1 /R2 and the optical thickness of the medium, τ12 = τ2 − τ1 = κ(R2 − R1 ). Solution For one-dimensional radiative equilibrium problems such as this, it is advantageous to use the simultaneous equation formulation, equations (16.50a) and (16.50b). Then, from equation (16.50a) we have, in cylindrical coordinates (with ω = 0 and τ = κr), 1 d (τq) = 4n2 σT 4 − G = 0. τ dτ If we multiply by τ and integrate, we find τq = C1

or

q=

C1 . τ

Substituting this expression into equation (16.37) gives

or

dG 3C1 = −3q = − , dτ τ

508

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

Nondimensional heat flux at inner cylinder Ψ(R1) = q(R1)/σ (T14 − T24)

1.25

Diffusion approximation (Deissler) P1-approximation P3-approximation Exact (Table 14.4)

1.00

0.75

0.50

R2/R1 = 2

0.25

0.00

0

1

2 3 Optical thickness τ2 − τ1 = κ (R 2 − R 1)

4

5

FIGURE 16-5 Nondimensional heat fluxes between concentric black cylinders at radiative equilibrium.

G = −3C1 ln τ + C2 . The boundary conditions are, from equation (16.47) with ǫ = 1, τ = τ1 :

2q · nˆ = 2q = 4n2 σT14 − G,

τ = τ2 :

2q · nˆ = −2q = 4n2 σT24 − G,

from which C1 and C2 may be determined as C1 =

4n2 σ(T14 − T24 ) , 2 τ2 2 + + 3 ln τ1 τ2 τ1

C2 = 4n2 σT24 + C1



 2 + 3 ln τ2 . τ2

Heat flux and temperature then follow as q

2 τ2 3 τ2 n2 σ T14 − T24 1+ + τ2 ln τ1 2 τ1 3 τ2 1 + τ2 ln T 4 − T24 2 τ Φ = 4 = . 4 τ2 3 τ2 T1 − T2 1+ + τ2 ln τ1 2 τ1

Ψ =



 =



 τ2 , τ

The resulting nondimensional heat flux, Ψ, evaluated at the inner cylinder, is shown in Fig. 16-5 for the case of R2 /R1 = 2 together with exact results (Table 14.4), results from the diffusion approximation with jump boundary condition (Example 15.3) and results from the P3 -approximation given by Bayazitoglu ˘ and Higenyi [24]. As expected, the P1 -approximation does well for optically thick media. For the optically thin case, however, as κ → 0 the heat flux goes to   R2 2 R1 , =2 1+ Ψ1 → 1 + R2 /R1 R1 R2 while the correct answer should be Ψ1 → 1, as we know from Chapter 5, equation (5.35). Therefore, for R1 /R2 → 1 the correct optically thin limit is obtained (and the gap between such cylinders becomes a plane-parallel slab), while for small inner cylinders, R1 /R2 ≪ 1, the error becomes larger and may be as large as 100%!

The P1 -approximation is a very popular method since it reduces the (spectral or gray) equation of transfer from a very complicated integral equation to a relatively simple partial differential equation, e.g., [25–37]. The method is powerful (allowing nonblack surfaces, nonconstant

16.6 P3 - AND HIGHER-ORDER APPROXIMATIONS

509

properties, anisotropic scattering, etc.), and the average heat transfer engineer is much better trained in solving differential equations than integral equations. Furthermore, if overall energy conservation (also a partial differential equation) is computed, compatibility of the solution methods is virtually assured. However, it is important to remember that the P1 -approximation may be substantially in error in optically thin media with strongly anisotropic intensity distributions, in particular in multidimensional geometries with large aspect ratios (i.e., long and narrow configurations) and/or when surface emission dominates over medium emission. Attempts to improve the method’s accuracy, by modifying Marshak’s boundary condition, were made by Liu and coworkers [38] and by Su [39]. In one-dimensional geometries accuracy can also be improved by applying the P1 -approximation separately to different solid angle ranges, as done by Menguc ¨ ¸ and Subramaniam [40]. Most of the shortcomings of the P1 -approximation are overcome by the modified differential approximation discussed in Section 16.8 below.

16.6 P3 - AND HIGHER-ORDER APPROXIMATIONS The general PN -approximation for one-dimensional absorbing/emitting, and anisotropically scattering cylindrical media has been given by Kofink [41], and the P3 -approximation for onedimensional slabs, concentric cylinders, and concentric spheres has been developed in terms of moments by Bayazitoglu ˘ and Higenyi [24]. Higher-order solutions, up to P11 , for a gray, anisotropically scattering medium between concentric spheres have been considered by Tong and Swathi [42] (uniform heat generation) and by Li and Tong [43] (isothermal medium). Onedimensional fibrous material was considered by Tong and Li [44] and a packed bed by Wu and Chu [45]. For multidimensional geometries, the process described in equations (16.11) through (16.14) can also be carried out in three dimensions, as outlined by Davison [3], resulting in a set of (N + 1)2 simultaneous, first-order partial differential equations in the unknown Inm . The general PN -formulation for three-dimensional Cartesian coordinate systems has been derived by Cheng [8, 9], including Marshak’s boundary conditions for surfaces normal to one of the coordinates. A three-dimensional problem was solved by Park and coworkers, analyzing radiative equilibrium in a rectangular box filled with a gray, nonscattering medium [26]. Menguc ¨¸ and Viskanta [46, 47] limited their development to the P3 -approximation in terms of moments (rather than spherical harmonics), but considered three-dimensional Cartesian coordinates [46] as well as axisymmetric cylindrical geometries [47]. The three-dimensional PN -approximation for arbitrary coordinate systems has been derived by Ou and Liou [10]. With the exception of Cheng [8], no boundary conditions beyond a reference to equation (16.23) have been given in these publications. Recently, Modest and coworkers [13,18,48] outlined a methodology that reduces the (N + 1)2 simultaneous equations of the standard PN -formulation to N(N + 1)/2 simultaneous, secondorder elliptic partial differential equations for a given odd order N, allowing for variable properties, anisotropic scattering, and arbitrary three-dimensional geometries. They further showed how to extract a completely defined, self-consistent set of boundary conditions from equation (16.23). The analysis is very tedious, to say the least, and we will present here only the final result for the (somewhat simpler) case of isotropic scattering. Defining a second-order operator ! 1 ∂ 1 ∂ Lxy = , (16.53) β ∂x β ∂y etc., and eliminating spherical harmonics coefficients Inm of odd order n, leads to the following set of second-order PDEs:

510

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

TABLE 16.1

Elliptic PN -approximation coefficients for isotropic scattering k=1 k=2 1 1 − 4(2n + 5)(2n + 3) 2(2n + 3)(2n − 1) 2m − 1 n+m+1 nm (a) bk − 2(2n + 5)(2n + 3) 2(2n + 3)(2n − 1) π2 (n + m + 1) n2 + n − 1 + m2 cknm − 2(2n + 5)(2n + 3) (2n + 3)(2n − 1) π3 (n + m + 1) (2m + 1)(n + m + 1)(n − m) nm dk − − 2(2n + 5)(2n + 3) 2(2n + 3)(2n − 1) π (n + m + 1) (n + m + 1)π2 (n − m − 1) π 4 2 eknm − 4(2n + 5)(2n + 3) 2(2n + 3)(2n − 1) (a) nm ak = 0 for m ≤ 1, bknm = 0 for m = 0; k−1 Y πk (n) = (n + j)

aknm (a)

k=3 1 4(2n − 1)(2n − 3) n−m − 2(2n − 1)(2n − 3) π2 (n − m − 1) − 2(2n − 1)(2n − 3) π3 (n − m − 2) 2(2n − 1)(2n − 3) π4 (n − m − 3) 4(2n − 1)(2n − 3)

j=0

Ynm : n = 0, 2, . . . , N − 1, 0 ≤ m ≤ n : 3 (   X δm1 nm m m+2 m−2 ck In+4−2k + eknm In+4−2k + (Lxx − L yy ) (1 + δm2 )aknm In+4−2k 2 k=1 i h m−1 m+1 +(Lxz + Lzx ) (1 + δm1 )bknm In+4−2k + dknm In+4−2k   δm1 nm −m −(m−2) −(m+2) +(Lxy + L yx ) −(1 − δm2 )aknm In+4−2k + ck In+4−2k + eknm In+4−2k 2 i h nm −(m−1) nm −(m+1) +(L yz + Lzy ) −(1 − δm1 )bk In+4−2k + dk In+4−2k ) m + [Lzz − (1 − ωδ0n )] Inm = −(1 − ω)Ib δ0n +(Lxx + L yy − 2Lzz )cknm In+4−2k

(16.54a)

and Yn−m : n = 0, 2, . . . , N − 1, 1 ≤ m ≤ n : 3 (   X δm1 nm m m+2 m−2 ck In+4−2k − eknm In+4−2k + (Lxy + L yx ) (1 + δm2 )aknm In+4−2k 2 k=1 i h m−1 m+1 +(L yz + Lzy ) (1 + δm1 )bknm In+4−2k − dknm In+4−2k   δm1 nm −m −(m+2) −(m−2) +(Lxx − L yy ) (1 − δm2 )aknm In+4−2k − ck In+4−2k + eknm In+4−2k 2 h i nm −(m−1) nm −(m+1) +(Lxz + Lzx ) (1 − δm1 )bk In+4−2k + dk In+4−2k ) −m +(Lxx + L yy − 2Lzz )cknm In+4−2k + [Lzz − 1] In−m = 0.

(16.54b)

The necessary constants9 are listed in Table 16.1. For anisotropic scattering, not presented here, 9 There is a slight error in the original paper [18], introducing a constant fn , which after correction is fn ≡ 1 and, thus, has been eliminated from equations (16.54).

511

16.6 P3 - AND HIGHER-ORDER APPROXIMATIONS

ˆ Zxn

q

s

d

Y,tY

Xx tˆ

tW s FIGURE 16-6 Local and global coordinates for a two-dimensional enclosure.

f XxtX

O

the constants for k = 1 and 3 undergo only minor changes, but for k = 2 [involving two different anisotropy constants Am from equation (16.6)] the operators become nonsymmetric. Since the orientation of the Cartesian coordinate system is arbitrary, one would expect to see equation (16.54) to show similar operators in x, y, and z. The reason that this is not the case is that the global direction angles (θ, ψ) and, thus, the results for Inm are tied to the choice of the coordinate system, i.e., we may write I(r, sˆ ) =

n N X X

n m=−n

Inm (r)Ynm (ˆs) =

n N X X

m

m

In (r)Yn (ˆs),

(16.55)

n m=−n

where the barred values refer to a rotated coordinate system (x, y, z). Example 16.4. Consider an isothermal medium at temperature T, confined inside a two-dimensional enclosure as shown in Fig. 16-6. The medium is gray and absorbs and emits, but does not scatter. Determine the set of governing equations for the P3 -approximation. Solution For a two-dimensional problem with polar angle θ measured from the z-axis we must have I(θ, ψ) = I(π − θ, ψ), i.e., all Inm , for which the accompanying associated Legendre polynomials Pnm (cos θ) have an odd-power dependence on cos θ, must vanish. This is the case whenever n + m is odd. Therefore, Inm = 0 for n + m = odd and, since the governing equations are cast in terms of even n, terms with odd m in equations (16.54) vanish. Using this, and eliminating all terms with z-derivatives, we get from equations (16.54) Y00 :

(Lxx − L yy )e100 I22 + (Lxy + L yx )e100 I2−2 + (Lxx + L yy )c100 I20 + (Lxx + L yy )c200 I00 − I00 = −Ib ,

Y20 :

(Lxx − L yy )e220 I22 + (Lxy + L yx )e220 I2−2 + (Lxx + L yy )c220 I20 + (Lxx + L yy )c320 I00 − I20 = 0, (Lxx − L yy )2a222 I20 + (Lxx + L yy )c222 I22 + (Lxx − L yy )2a322 I00

− I22 = 0,

Y2−2 : (Lxy + L yx )2a222 I20 + (Lxx + L yy )c222 I2−2 + (Lxy + L yx )2a322 I00

− I0−2 = 0.

Y22 :

m For n = 0 the case of k = 3 is not needed, since this leads to nonexistent I−2 , and, similarly, for n = 2 the m case of k = 1, producing I4 , i.e., terms omitted in the P3 -approximation. In addition, all Inm with odd m and with m > n are dropped. Equations (16.54) are also valid for n = 2, m = ±1, but every term in these equations vanishes. Thus the above set constitutes the needed four equations for the four unknowns. The coefficients are evaluated from Table 16.1 as

1 1 1·2·3·4 3·4·1·2 1 1 2 4 = − ; a322 = = ; e 00 = = ; e220 = − =− ; 2·7·3 42 4·3·1 12 1 4·5·3 5 2·7·3 7 1 −1 1 20 5 5 9 3 20 1·2 1 1·2 00 22 = − ; c2 = = ; c2 = = ; c = = ; c3 = − =− . =− 2·5·3 15 3 · (−1) 3 7·3 21 2 7·3 7 2·3·1 3 a222 = −

c100

512

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION) Z

Z Z

b

Z

Y

g a

Y

Y

Y

Y

X

a X

X

b

g X

X

X

FIGURE 16-7 Definition of Euler angles for an arbitrary rotation

Substituting these values into the four governing equations, we find  2 2 1 0 Y00 : (Lxx − L yy )I22 + (Lxy + L yx )I2−2 − (Lxx + L yy ) I − 5 5 15 2  4 5 0 4 I − Y20 : − (Lxx − L yy )I22 − (Lxy + L yx )I2−2 + (Lxx + L yy ) 7 7 21 2  3 1 0 (Lxx + L yy )I22 − (Lxx − L yy ) I − Y22 : 7 21 2  3 1 0 (Lxx + L yy )I2−2 − (Lxy + L yx ) I − Y2−2 : 7 21 2

 1 0 I0 − I00 = −Ib , 3  1 0 I0 − I20 = 0, 3  1 0 I0 − I22 = 0, 6  1 0 I0 − I2−2 = 0. 6

(16.56a) (16.56b) (16.56c) (16.56d)

Boundary Conditions Equation set (16.54) consists of N(N + 1)/2 simultaneous, elliptic PDEs, requiring N(N + 1)/2 boundary conditions everywhere along the domain boundary, which must be determined from the general Marshak condition, equation (16.23). Unfortunately, equation (16.23) is cast in terms of a local coordinate system. Thus, in order to obtain a generic boundary condition for arbitrary geometries, the global spherical harmonics must be rotated into the local coordinate system. Such rotation, according to Euler’s rotation theorem, may be described using three angles, which are called Euler angles. In the literature, there are several notation and rotation conventions for Euler angles. Here, the notation (α, β, γ) is used for three Euler angles following Varshalovich et al.’s definition [49]. In Varshalovich’s convention, as shown in Fig. 16-7, an arbitrary rotation is defined by Euler angles (α, β, γ), where the first rotation is by an angle α about the z-axis, the second is by an angle β about the y′ -axis, and the third is by an angle γ about the z′ -axis. As indicated in Fig. 16-7 all three rotations are, following the right-hand rule, in counterclockwise direction about the center axis. The three rotations can, in general, be carried out by (1) rotating x-y so that y′ is perpendicular to nˆ (nˆ · ˆ′ = 0), ıˆ′ = cos α ˆı + sin α ˆ,

ˆ′ = − sin α ˆı + cos α ˆ

and tan α =

ny nx

,

(16.57) (16.58)

ˆ or (2) rotating x′ -z such that z′ becomes parallel to n, ′ kˆ = sin β ˆı′ + cos β kˆ

(16.59)

(nx cos α + n y sin α) sin β + nz cos β = 1.

(16.60)



and nˆ · kˆ = 1 gives

513

16.6 P3 - AND HIGHER-ORDER APPROXIMATIONS

(3) The third rotation is arbitrary and serves to place the local x-y-coordinates into convenient locations. Example 16.5. Determine the Euler angles for the local coordinate system for the boundary location indicated in Fig. 16-6. Solution To perform the transformation indicated in Fig. 16-6 (with the global z-axis pointing toward the reader), the local surface normal is determined as ˆ nˆ = − sin δ ˆı + cos δ ˆ + 0 k,

(16.61)

and the first rotation angle α follows from tan α = − tan δ, or α = δ ±

π . 2

(16.62)

If we choose α = δ − π/2 (y′ points into the indicated x-direction), the second rotation angle becomes      π 3π π + cos δ sin δ − sin β = 1, or β = − sin δ cos δ − . 2 2 2

(16.63)

This has x′′ pointing out of the paper, and a final (optional) rotation of γ = π/2 rotates x′′′ into the desired local x-direction.

It can be shown that, for a given rotation, the spherical harmonics of order n are transformed into a linear combination of spherical harmonics of the same order n. Such an operation can be represented in the form of a rotation matrix, where each element of this matrix is a function of Euler angles, n X m ′ Ynm (θ, φ) = (16.64) ∆nmm′ (α, β, γ)Yn (θ, φ), m=−n

where ∆nmm′ (α, β, γ) is the representation matrix of the rotation operation for the real spherical harmonics Ynm of order n. Blanco10 et al. [50] developed a closed-form expression to specify all the elements based on so-called Wigner-D functions, from which the ∆n matrices can be obtained in terms of the Euler angles as ′

n m n ∆nmm′ = sign(m′ )Ψm (α)Ψm′ (γ)[d|m|,|m d|m|,−|m′ | (β)] ′ | (β) + (−1) ′

n m n − sign(m)Ψ−m (α)Ψ−m′ (γ)[d|m|,|m d|m|,−|m′ | (β)] ′ | (β) − (−1)

where sign(0) = 1 and the function Ψm is defined as ( cos mξ, for Ψm (ξ) = sin |m|ξ, for

m ≥ 0, m < 0.

(16.65)

(16.66)

To determine the ∆n matrices by equation (16.65) the d n matrices are needed, which are modified versions of the real parts of the Wigner-Dnmm′ functions, and may be calculated from n dmm ′ (β) =

(−1)

m+m′



(n − |m|)!(n + |m |)! 1 + δm,0

′ min(n−m,n+m ) X

k=max(0,m′ −m)

′  ′    β 2k+m−m β 2n−2k−m+m sin 2 (−1)k cos 2

k!(n − m − k)!(n + m′ − k)!(m − m′ + k)!

.

(16.67) With the rotation of spherical harmonics between local and global coordinates as indicated m by equation (16.64), relationships between Inm and In can be revealed accordingly by expressing 10 In Blanco’s derivation, a normalization factor is employed. In order to be consistent with the real spherical harmonics used in the current study, a modification coefficient was included in the transformation.

514

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

intensity in terms of, both, local and global coordinates, as given by equation (16.55). This leads to n n X X m′ m ¯ n ′ (−γ, −β, −α)Inm′ , Inm = ∆nmm′ (α, β, γ)In , and In = ∆ (16.68) mm m′ =−n

m′ =−n

¯ n ′ implies backward rotation from local to global coordinates, as where the bar on the ∆ mm indicated by the arguments. Substitution of equation (16.55) into (16.23), and assuming the surface intensity Iw to be diffuse, reduces the boundary conditions to N "Z X n=0

1 0

m ¯ 2i−1 ¯ µ¯ Pnm (µ)P (µ)d

#

m In (τw )

"Z

=

1

#

¯ µ¯ δm,0 Iw , P2i−1 (µ)d 0

i = 1, 2, ..., 12 (N + 1), all relevant m.

(16.69)

m

Before these boundary conditions can be applied to equations (16.54) the In with odd n must be eliminated. Boundary conditions are usually formulated in terms of local normal and tangential gradients, and this leads to   N−1 N−1  X 2 2 X 0 0 1 ∂    Y2i−1 : p02l,2i−1 I2l + v0li I2l   ∂τx   l=0 l=1  N−1   N−1   2   2  ∂ X 0 −1  ∂ X 0 0      = Iw p00,2i−1 , v w I I m = 0, (16.70a) + −   li 2l  li 2l   ∂τ y   ∂τz  l=1 l=0   N−1 N−1 N−1  X 2 2 2 X X m m m−1 m+1 ∂   m  (1+δ )um I  − − Y2i−1 : I I pm v m,1  2l 2l 2l 2l,2i−1 li li   ∂τx  l=1 l=0 l=1  N−1   N−1  N−1  2  X  2 2 X −(m−1) −(m+1) m ∂ X ∂    m m  −   (1−δm,1 )um  = 0, m > 0, (16.70b) + I + I I v w   2l 2l 2l li li li  ∂τz   ∂τ y  l=1 l=1 l=1   N−1 N−1 N−1  X 2 2 2 X X −m −m −(m−1) −(m+1) ∂   m  (1−δ )um I   Y2i−1 : I − − I pm v m,1  2l 2l 2l 2l,2i−1 li li  ∂τx  l=1 l=1 l=1  N−1   N−1  N−1  2  X  2 2 X m−1 m+1 −m ∂ X ∂    m m −   (1+δm,1 )um    = 0, m > 0, (16.70c) − I + v I w I   2l 2l 2l li li li  ∂τz   ∂τ y  l=0

l=1

l=1

where the pm are defined as n,j

pm n,j

=

pmj,n

=

Z

1 0

¯ m ¯ µ, ¯ Pnm (µ)P j (µ)d

(16.71)

, vm , wm are related to them by and the coefficients um li li li um li

=

vm li = wm li =

− pm pm 2l−1,2i−1 2l+1,2i−1

, 2(4l+1) − π2 (2l−m)pm π2 (2l+m)pm 2l−1,2i−1 2l+1,2i−1 2(4l+1) + (2l−m+1)pm (2l+m)pm 2l−1,2i−1 2l+1,2i−1 (4l+1)

.

(16.72a) ,

(16.72b) (16.72c)

515

16.6 P3 - AND HIGHER-ORDER APPROXIMATIONS

TABLE 16.2

Half-moments of associated Legendre polynomials, 10−m × pm . n,j m 0

1

2

3

4 5

/j

0

1

2

3

4

5

0 1 2 3 4 5 1 2 3 4 5 2 3 4 5 3 4 5 4 5 5

1.00000 0.50000 0.00000 –0.12500 0.00000 0.06250 . . . . . . . . . . . . . . .

. 0.33333 0.12500 0.00000 –0.02083 0.00000 0.06667 0.07500 0.00000 –0.04167 0.00000 . . . . . . . . . .

. . 0.20000 0.12500 0.00000 –0.03906 . 0.12000 0.07500 0.00000 –0.02344 0.04800 0.07500 0.00000 –0.06563 . . . . . .

. . . 0.14286 0.07031 0.00000 . . 0.17143 0.14062 0.00000 . 0.17143 0.14062 0.00000 0.10286 0.19687 0.00000 . . .

. . . . 0.11111 0.07031 . . . 0.22222 0.14062 . . 0.40000 0.39375 . 0.56000 0.55125 0.44800 0.99225 .

. . . . . 0.09091 . . . . 0.27273 . . . 0.76364 . . 1.83273 . 3.29891 3.29891

n

m

In equations (16.70) and (16.72) it is implied that coefficients in front of nonsensical In (i.e., with nonsensical subscripts (n < m) are zero. The pm may be determined |m| > n) and pm nj n,j through recursion relationships [18] and are listed in Table 16.2 (scaled by a factor of 10−m ) for up to the P5 -approximation. m It remains to rotate the In in equations (16.70) to global values Inm , which results in  N−1  N−1   2l 2l 2 2   X X X X  0 ∂ ′ ′  0 0 ¯ 2l 2l m m  ¯ Y2i−1 : p2l,2i−1 ∆0,m′ I2l + vli ∆1,m′ I2l      ∂τx   l=1 m′ =−2l  l=0 m′ =−2l  N−1  N−1   X    2l 2l 2 2    X X X   ∂  ∂ ′ ′    0 ¯ 2l 0 ¯ 2l m m  vli ∆−1,m′ I2l  wli ∆0,m′ I2l  + − = Iw p00,2i−1 , m = 0, (16.73a)         ∂τ y   l=1 m′ =−2l  l=0 m′ =−2l  ∂τz     N−1 N−1  2l 2l h 2 2   X X X X i ′   m ∂ ′  m ¯ 2l m  ¯ 2l ¯ 2l m ∆ Y2i−1 : pm (1+δm,1 )um ′ − vli ∆m+1,m′ I2l  m−1,m li 2l,2i−1 ∆m,m′ I2l − ∂τ      x  l=0 m′ =−2l  l=1 m′ =−2l  N−1  N−1   X   2l h 2l 2 2    X X X i ′    ∂ ∂  ′    m ¯ 2l m ¯ 2l m ¯ 2l m  m (1−δm,1 )uli ∆−(m−1),m′ + vli ∆−(m+1),m′ I2l  − wli ∆m,m′ I2l  + = 0,         ∂τ y   l=1 m′ =−2l  l=1 m′ =−2l   ∂τz  −m

Y2i−1

m > 0, (16.73b)   N−1  2l 2l h 2   X X X X i ′   ∂ ′  m ¯ 2l m  m ¯ 2l ¯ 2l (1−δm,1 )um ∆ : pm ′ − vli ∆−(m+1),m′ I2l  −(m−1),m li 2l,2i−1 ∆−m,m′ I2l − ∂τ     x    l=1 m′ =−2l l=1 m′ =−2l  N−1   N−1  X   2l h 2l 2 2    X X X i ′    ∂  ∂ ′    m m ¯ 2l m  m ¯ 2l m ¯ 2l wli ∆−m,m′ I2l  − (1+δm,1 )uli ∆m−1,m′ + vli ∆m+1,m′ I2l  − = 0,         ∂τ y   l=0 m′ =−2l  ∂τz   l=1 m′ =−2l  N−1 2

m > 0.

(16.73c)

516

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

Equations (16.73) are a set of (N +2)(N +1)/2 boundary conditions for N(N +1)/2 variables I2lm (l = 0, 1, ..., (N−1)/2; m = −2l, ..., +2l), containing normal as well as tangential derivatives, or N + 1 too many. Commercial PDE solvers generally allow for boundary conditions containing normal derivatives. In principle, i.e., if the coefficients in front of the I2lm inside the normal derivatives form a nonsingular matrix, linear combination of the boundary conditions leads to a set of “natural” boundary conditions for each variable, or   ′ ′   m′ ∂I2lm′ ∂I2lm′ ′ ′ ′ ′ 1 = f I2l′ , , ; l = 0, ... 2 (N−1); m = −2l , ..., +2l  , ∂τz ∂τx ∂τ y

∂I2lm

l = 0, ..., 21 (N−1), m = −2l, ..., +2l,

(16.74)

which can be used with FlexPDE [51] and other commercial programs. Modest [18] has shown that such a nonsingular matrix can be found only if, for the largest value of i = 12 (N +1), only the even values of m are employed (omitting the N+1 odd values). Therefore, the qualifier “all relevant m” in equations (16.69), (16.70), and (16.73) may be restated precisely as ( i = 1, 2, ..., 12 (N − 1), all m, All relevant m = (16.75) i = 12 (N + 1), all even m, which supplies a consistent set of N(N + 1)/2 boundary conditions for an equal number of variables. Other codes, such as PDE2D [52] or FDEM [53], use derivatives in global coordinates in the boundary conditions. In that case, the transformation to global Inm using equation (16.68) is carried out first, followed by elimination of odd orders. The resulting boundary conditions are given in [13]. Example 16.6. Determine the necessary boundary conditions for the problem of Example 16.4 for the surface location indicated in Fig. 16-6. The surface is black and at temperature Tw . Solution The boundary conditions are usually expressed in terms of local coordinates (i.e., in terms of gram dients into the surface normal and tangential directions), either using local spherical harmonics In , m equation (16.70), followed by rotation to global spherical harmonics In , or by directly applying equation (16.73). We will follow the first track here. With local azimuthal angle ψ defined from the x-axis in the x–y–plane, for this two-dimensional problem independent of y we must have I(θ, ψ) = I(θ, −ψ) m and, therefore, all In with negative m vanish. Thus, from equation (16.70), eliminating all terms with negative m and y-gradients, we obtain 0

Y1 : 1

Y1 : 0

Y3 : 2

Y3 :

∂ ∂τx 1 ∂ p121 I2 − ∂τx 0 0 ∂ p003 I0 + p023 I2 + ∂τx 2 ∂ p223 I2 − ∂τx 0

0

p001 I0 + p021 I2 +

−1

−2

∂ ∂τz   0 0 2 ∂ 2u101 I0 + 2u111 I2 − v111 I2 − ∂τz   1 ∂ − v012 I2 ∂τz   1 ∂ u212 I2 − ∂τz   1 v011 I2



m

  0 0 w001 I0 + w011 I2 = Ibw p001 ,   1 w111 I2

= 0,

  0 0 w002 I0 + w012 I2 = Ibw p003 ,   2 w212 I2

= 0.

The equations for Y1 and Y3 contain only In with negative m and, thus, vanish identically, leaving us m with the proper four boundary conditions for the four unknown In . The coefficients pm , um , vm , and wm nj li li li

517

16.6 P3 - AND HIGHER-ORDER APPROXIMATIONS

are found from Table 16.2 [or, more easily from program pnbcs.f90 in Appendix F] as 1 1 1 1 15 3 , p021 = , p121 = , p003 = − , p023 = , p223 = ; 2 8 4 8 8 2     1 1 1 p −p31 p2 −p233 −p11 1 1 2 1 1 120 12 = − , u111 = 11 = −0 = , u212 = 13 = 0− =− ; u101 = 2·1 3 2·5 10 3 15 2·5 10 7 7     2 · 3p011 − 2 · 3p031 3 · 4p111 − 1 · 2p131 2 3 1 1 1 4 0 1 12 × − 0 = , v11 = = − 0 = , v11 = = 2·5 5 3 5 2·5 10 3 5   0 0 − 2 · 3p 2 · 3p 3 1 3 33 13 = 0− =− ; v012 = 2·5 5 7 35     0 0 0 + 3p 2p 3p1 + 2p131 1 · p 1 1 2 2 1 2 2 31 11 w001 = = , w011 = 11 = −0 = , w111 = 11 = 3× +0 = , 1 3 5 5 3 15 5 5 3 5     1 · p013 2p0 + 3p033 4p2 + 1 · p233 3 3 120 24 1 1 0+ = 0+ = = 0, w012 = 13 = , w212 = 13 = . w002 = 1 5 5 7 35 5 5 7 7 p001 =

Therefore, after normalization with the leading term, 1

0

Y1 :

0

I0 +

1

Y1 :

1 0 2 ∂I2 I + − 4 2 5 ∂τx   1 ∂ 8 0 8 0 16 2 I2 + I0 − I2 + I2 − 45 15 ∂τx 9 1

0

Y3 :

0

0

24 ∂I2 35 ∂τx

2

8 ∂I2 35 ∂τx

I0 − I2 + I2 +

0

(16.76a)

1

8 ∂I2 15 ∂τz

= 0,

(16.76b)

= Ibw ,

(16.76c)

= 0.

(16.76d)

0

+

24 ∂I2 35 ∂τz



16 ∂I2 35 ∂τz

1

2

Y3 :

0

2 ∂I0 4 ∂I2 − = Ibw , 3 ∂τz 15 ∂τz

2

m

Next, the local In must be converted to global Inm with equation (16.68). For n = 0 this simply gives

0 I0

= I00 , i.e., I00 is nondirectional and does not vary with rotation, and we will drop the unnecessary superscript from I0 . Remembering that, in global coordinates, Inm with odd m vanish (as opposed to negative m in local coordinates), for n = 2 this leads to 0 ¯ 2 I −2 + ∆ ¯ 2 I0 + ∆ ¯ 2 I 2, I2 = ∆ 0,−2 2 0,0 2 0,2 2 1

¯ 2 I −2 + ∆ ¯ 2 I0 + ∆ ¯ 2 I 2, I2 = ∆ 1,−2 2 1,0 2 1,2 2 2 ¯ 2 I −2 + ∆ ¯ 2 I0 + ∆ ¯ 2 I 2. I2 = ∆ 2,−2 2 2,0 2 2,2 2

¯ 2 ′ (−γ = − π , −β = − 3π , −α = The necessary ∆ m,m 2 2 tion (16.65) with

β

 −1,     0,      π  1, = Ψm −   2   −1,    0,

m=2 1 0 , −1 −2

β

π 2

− δ) are determined via backward rotation from equa-

 − cos 2δ,     sin δ,      π  1, Ψm′ −δ =   2   cos δ,    sin 2δ,

m′ = 2 1 0 , −1 −2

2 √1 and cos( 2 ) = sin( 2 ) = cos(− 3π 4 ) = − 2 . The dmm′ follow from equation (16.67) after some painful algebra (or, more easily, by manipulating program Delta.f90 in Appendix F). Finally, 0

I2 = −3 sin 2δ I2−2 −

1 0 I − 3 cos 2δ I22 , 2 2

1

I2 = −2 cos 2δ I2−2 + 2 sin 2δ I22 , 2

I2 =

1 1 1 sin 2δ I2−2 − I20 + cos 2δ I22 . 2 4 2

518

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

Sticking this into equation (16.76) delivers the desired local boundary conditions as 0

Y1 :

1

Y1 :

0

Y3 :

2

Y3 :

i 3 1 3 4 ∂ h cos 2δ I2−2 − sin 2δ I22 sin 2δ I2−2 − I20 − cos 2δ I22 − 4 8 4 5 ∂τx i 2 ∂ h 5I0 − 6 sin 2δ I2−2 − I20 − 6 cos 2δ I22 = Iw , − 15 ∂τz i 8 ∂ h 5I0 − I2 + 6 sin 2δ I2−2 + 6 cos 2δ I22 −2 cos 2δ I2−2 + 2 sin 2δ I22 + 45 ∂τx i 48 ∂ h cos 2δ I2−2 − sin 2δ I22 = 0, + 45 ∂τz i 48 ∂ h 1 cos 2δ I2−2 − sin 2δ I22 I0 + 3 sin 2δ I2−2 + I20 + 3 cos 2δ I22 − 2 35 ∂τx   1 24 ∂ 3 sin 2δ I2−2 + I20 + 3 cos 2δ I22 = Iw , − 35 ∂τz 2 h i ∂ 1 16 1 1 cos 2δ I2−2 − sin 2δ I22 sin 2δ I2−2 − I20 + cos 2δ I22 − 2 4 2 35 ∂τx i 4 ∂ h 2 sin 2δ I2−2 − I20 + 2 cos 2δ I22 = 0. − 35 ∂τz I0 −

Once all Inm for even n have been determined, the remaining Inm (odd n) may be determined from relations given in Modest and Yang [13]. Normally, only incident radiation G = 4πI0 and radiative flux are of interest, the latter being related to the I1m : comparing equations (16.24), (16.25), and (16.31) and noting that higher-order terms drop out because of the orthogonality of spherical harmonics [14], leads to

q(r) =

Z

where the I1m are given by [13]

 1  −I1   4π  −1 I(r, sˆ ) sˆ dΩ = −I1  ,  3  0  4π I1

(16.77)

I10 = −

0 1 −1 ∂I0 2 ∂I2 3 ∂I2 3 ∂I2 − + + , ∂τz 5 ∂τz 5 ∂τx 5 ∂τ y

(16.78a)

I11 = +

0 1 2 −2 ∂I0 1 ∂I2 3 ∂I2 6 ∂I2 6 ∂I2 − − + + , ∂τx 5 ∂τx 5 ∂τz 5 ∂τx 5 ∂τ y

(16.78b)

0 −1 2 −2 1 ∂I2 3 ∂I2 6 ∂I2 6 ∂I2 ∂I0 − − − + . 5 ∂τ y 5 ∂τx ∂τ y 5 ∂τ y 5 ∂τz

(16.78c)

I1−1 = +

Since equation (16.1) is valid for any coordinate system orientation, equations (16.77) and (16.78) are valid for both the global coordinate system (x-y-z, Inm ) as well as a local coordinate system at m a boundary (x-y-z, In ). Finally, for nonblack surfaces the boundary radiosity Jw = πIw must be related to the wall’s emissive power and/or net radiative flux. From equations (16.1) and (16.77) we have qn =

4π 0 ǫπ [Ibw − Iw ] = I , 1−ǫ 3 1

(16.79)

0

where ǫ is the surface’s emittance, and with I1 transformed to global I1m through equation (16.68). If the temperature of the surface, Tw , is specified, Iw is determined from Iw = Ibw −

  0 4 1 − 1 I1 . 3 ǫ

(16.80)

16.6 P3 - AND HIGHER-ORDER APPROXIMATIONS

519

For three-dimensional geometries, it is obvious that anything but low-order approximations quickly become extremely cumbersome to deal with. Already the P3 -approximation may result in as many as six simultaneous partial differential equations (depending on the symmetry), and it includes cross-derivatives, which do not ordinarily occur in engineering problems (and which complicate numerical solutions). In addition, complicated boundary conditions need to be developed from equation (16.73). As a result of this complexity, very few multidimensional problems have been solved by the P3 -approximation, and apparently none by higher orders. First results using the new elliptic formulation of equations (16.54) and (16.73) have been reported by Modest and coworkers [13, 18, 48]. We shall limit ourselves here to a simple example for a one-dimensional plane-parallel slab. Example 16.7. Consider an isothermal medium at temperature T, confined between two large, parallel black plates that are isothermal at the (same) temperature Tw . The medium is gray and absorbs and emits, but does not scatter. Determine an expression for the heat transfer rates within the medium using the P3 -approximation. Employ the results from the previous three examples. Solution For such a one-dimensional problem it is, generally, advantageous to choose τz as the (nondimensional) space coordinate between the plates, as was done in Example 16.2, since this will make all Inm vanish with m , 0. However, for demonstrative purposes, and to utilize results from the previous three examples, we will choose the global coordinate system of Fig. 16-6, i.e., the problem becomes one-dimensional in the y-direction, with the bottom surface corresponding to δ = 0, and the top to δ = π. Since now we have no x-dependence we must have I(θ, ψ) = I(θ, π − ψ), which implies that we will not have any odd positive or even negative m terms in equation (16.56a). Together with n + m = even (no z-dependence) that reduces the set of equations developed in Example 16.4 to Y00 : Y20 : Y22 :

 d2 2 2 I + dτ2y 5 2  d2 4 2 I + dτ2y 7 2  d2 3 2 I + dτ2y 7 2

 1 0 1 I2 − I0 +I0 = Ib , 15 3  5 0 1 I2 − I0 −I20 = 0, 21 3  1 0 1 I2 − I0 −I22 = 0, 21 6

and all terms vanish for the Y2−2 -equation, i.e., we now have three equations in three unknowns (since I2−2 = 0). To exploit the symmetry of the problem, we choose the origin for τ y to be at the midpoint between the two plates. Then the first derivatives of all three unknowns will be zero at the midpoint: τy = 0 :

dI 0 dI 2 dI0 = 2 = 2 = 0. dτ y dτ y dτ y

The necessary second set of boundary conditions follows from Example 16.6 with δ = 0 at τ y = −τL /2 (and τL is the total optical thickness of the medium) as 0

Y1 : 0

Y3 : 2

Y3 : 1

i 1 0 3 2 2 d h 5I0 − I20 − 6I22 = Ibw , I − I − 8 2 4 2 15 dτ y i 1 12 d h 0 I2 + 6I22 = Ibw , I0 + I20 + 3I22 − 2 35 dτ y h i 1 1 4 d I 0 − 2I22 = 0, − I20 + I22 + 4 2 35 dτ y 2

I0 −

with all terms in the Y1 boundary condition vanishing. While the given set of three simultaneous ordinary differential equations in I0 , I20 , and I22 , together with their boundary conditions, can be solved as they are, we do know from Section 16.3 that, for a one-dimensional problem, there should be only

520

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

a single Inm for every n (i.e., In0 ). Inspecting the governing equations and boundary conditions, we find that I20 and I22 always occur in one of two combinations, viz.  1 0 I + 6I22 , 2 2 K2 = I20 − 2I22 , I2 = −

where the factor − 12 was included for convenience (i.e., I2 just so happens to be I20 for the case that the z-axis points from plate to plate). Then Y00 :

2 ′′ I − 15 2 11 − I2′′ − 21 −

Y20 + 6Y22 : Y20 − 2Y22 :

1 ′′ I + I0 = Ib , 3 0 2 ′′ I + I2 = 0, 3 0 1 ′′ K − K2 = 0, 7 2

where the primes have been introduced as shorthand for d/dτ y . The boundary conditions at τ y = −τL /2 follow as 0 2 1 4 ′ I = Ibw , Y1 : I0 + I2 − I0′ − 4 3 15 2 0 24 ′ I = Ibw , Y3 : I0 − I2 + 35 2 2 1 4 ′ K2 K = 0. Y3 : + 2 35 2 It follows that K2 ≡ 0, since both its governing equation and its boundary conditions are homogeneous. I2 can be eliminated from the remaining equations: first we eliminate I2′′ from the first two equations, leading to 9 14 I2 = Ib , − I0′′ + I0 − 55 55 or 55 9 (I0 − Ib ) . I2 = − I0′′ + 14 14 Differentiating twice and eliminating I2′′ from the Y00 equation, we obtain 3 (iv) 6 ′′ I − I0 + I0 = Ib . 35 0 7 The general solution to the above equation (keeping in mind that Ib = const) is I0 (τ y ) = Ib + (Ibw − Ib )[C1 cosh λ1 τ y + C2 cosh λ2 τ y + C3 sinh λ1 τ y + C4 sinh λ2 τ y ], where the constant factor (Ibw − Ib ) was included to make the Ci dimensionless. The λ1 and λ2 are the positive roots of the equation 3 4 6 2 λ − λ + 1 = 0, 35 7 or λ1 = 1.1613 and λ2 = 2.9413. With τ y = 0 placed at the midpoint between the two plates I0′ (0) = I0′′′ (0) = 0 and C3 = C4 = 0. The two needed boundary conditions at one of the plates, say at τ = −τL /2, are found by again eliminating I2 , or 0

Y1 : 0

Y3 : leading to

I0 + I0

 9 1 − I′′ + 4 14 0  9 − − I0′′ + 14

 2 55 4 (I0 − Ib ) − I0′ − 14 3 15  24 55 (I0 − Ib ) + 14 35

9 ′′′ I + 14 0  9 − I0′′′ + 14 



 55 ′ I0 = Ibw , 14  55 ′ I0 = Ibw , 14

16.6 P3 - AND HIGHER-ORDER APPROXIMATIONS

Ibw − Ib =

521

111 9 ′′ 6 ′′′ 12 ′ (I0 − Ib ) − I − I + I , 56 7 0 56 0 35 0

Ibw − Ib = −

132 ′ 9 ′′ 108 ′′′ 41 (I0 − Ib ) + I + I − I . 14 49 0 14 0 245 0

Now, substituting the solution for I0 into these boundary conditions leads to 1 = a1 C1 + a2 C2 = b1 C1 + b2 C2 , where

   τL 9 2 6 3 τL 111 12 − λi cosh λi + λi − λi sinh λi , i = 1, 2, 56 56 2 7 35 2     9 2 τL 108 3 τL 132 41 − λ cosh λi − λi − λ sinh λi , i = 1, 2. bi = − 74 14 i 2 49 245 i 2 ai =



Finally, we get C1 =

b2 − a2 , a1 b2 − a2 b1

C2 =

a1 − b1 . a1 b2 − a2 b1

The heat flux through the medium is determined from equations (16.77) and (16.78) as   ! 0 2 4π ∂I0 4π 4π  ∂I0 1 ∂I2 6 ∂I2  2 ∂I2 − − + . q(τ y ) = − I1−1 = −   = − 3 3 ∂τ y 5 ∂τ y 5 ∂τ y 3 ∂τ y 5 ∂τ y Substituting for I2 we obtain

  9 ′′′ 11 ′ 4π ′ I0 − I0 + I0 , 3 35 7 and the heat flux may be expressed in nondimensional form as q(τ y ) = −

Ψ=

q(τ y ) n2 σ(Tw4 − T 4 )

= −

2 ′ ′′′ 12 10I0 − I0 12 X =− (10λi − λ3i )Ci sinh λi τ y , 35 Ibw − Ib 35 i=1

where, for simplicity, it was assumed that the medium is gray, or Ib = n2 σT 4 /π. The nondimensional heat flux at the top surface (τ y = τL /2) is shown in Fig. 16-8, as a function of optical depth of the slab. The results are compared with those of the P1 - or differential approximation (Example 16.2), and with the exact result, Ψ = 1 − 2E3 (τL ), which is readily found from equation (14.35). For this particular example the P1 -approximation is very accurate (maximum error ∼15%) and, as to be expected, the P3 -approximation performs even better (maximum error ∼7%).

It should be clear from the above example that P3 - and higher-order PN -approximations quickly become very tedious, even for simple geometries. However, P3 results can be substantially more accurate than P1 results, particularly in optically thin media and/or geometries with large aspect ratios. Another example, shown in Fig. 16-5, depicts nondimensional heat flux through a gray, nonscattering medium at radiative equilibrium, confined between infinitely long, concentric, black and isothermal cylinders, in which the P3 -solution of Bayazitoglu ˘ and Higenyi [24] is compared with the P1 -solution (Example 16.3). Observe that the P3 -approximation introduces roughly half the error of the P1 -method, which appears to be approximately true for all problems. One outstanding advantage of the P3 -method is that, once the problem has been formulated (setting up the governing equations suitable for a numerical solution), the increase in computer time required (compared with the P1 -method) is relatively minor. In addition, P3 -calculations are also usually very grid-compatible with conduction/convection calculations, if one must account for combined modes of heat transfer. Three additional twodimensional examples will be presented in the final section of this chapter, comparing results from different orders and different schemes of the spherical harmonics method.

522

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

Nondimensional heat flux at wall Ψ(τL ) = Q(τL )/N2σ (Tw4 − T 4)

1.00

0.75

0.50

Exact P1-approximation P3-approximation

0.25

0.00

0

1

2 3 Optical thickness τL

4

5

FIGURE 16-8 Nondimensional wall heat fluxes for an isothermal slab; comparison of P1 - and P3 -approximations with the exact solution.

16.7

SIMPLIFIED PN -APPROXIMATION

As noted in the previous section, higher-order PN -formulations for anything but one-dimensional slabs become extremely cumbersome mathematically, and they also introduce cross-derivatives, which make a numerical solution considerable more involved. Facing these mathematical difficulties Gelbard [5] introduced the Simplified PN -Approximation some 50 years ago, as an intuitive three-dimensional extension to the one-dimensional slab PN -formulation, equation (16.14), and its Marshak boundary conditions, equations (16.21). Gelbard formulated his set of simplified-PN or SPN equations, such that they reduced to the standard PN -approximation for a one-dimensional slab and some other narrow circumstances, but the method lacked any theoretical foundation, which impeded its acceptance. Theoretical justifications were found many years later by Larsen et al. [54] (showing SPN to be an asymptotic correction to the diffusion approximation of Section 15.2) and by Pomraning [55] (showing the SPN to be asymptotically related to the PN -equations for the slab geometry). A fine review of the SPN -method has recently been given by McClarren [56]. While the developments of Larsen and Pomraning provide theoretical credentials to the method, they are rather tedious, and we will here only provide the intuitive development of Gelbard, further developed for radiative heat transfer applications by Modest [57]. Depending on whether k is odd or even, Gelbard made the following substitutions in equations (16.14) and (16.21): dIk k odd : Ik (τ) → Ik (τx , τ y , τz ), → ∇τ · Ik , (16.81a) Ik′ = dτ dIk Ik′ = → ∇τ Ik , (16.81b) k even : Ik (τ) → Ik (τx , τ y , τz ), dτ i.e., for every odd k the Ik becomes a vector and differentiation is replaced by the divergence operator, while even Ik remain scalars and their differentiation is replaced by the gradient operator. Substituting equations (16.81) into equation (16.14) leads to k = 0, 2, . . . , N − 1 k = 1, 3, . . . , N

(even) :

(odd) :

k+1 k ∇τ · Ik+1 + ∇τ · Ik−1 + αk Ik = αk Ib δ0k , 2k + 3 2k − 1

(16.82a)

k k+1 ∇τ Ik+1 + ∇τ Ik−1 + αk Ik = 0, 2k + 3 2k − 1

(16.82b)

16.7 SIMPLIFIED PN -APPROXIMATION

523

where αk = 1 −

ωAk . 2k + 1

(16.82c)

Solving equation (16.82b) for Ik and substituting the result into (16.82a) produces a set of simultaneous elliptic partial differential equations in the unknown scalars Ik (k even): k = 0,2, . . . , N − 1

(even) :     (k + 1)2 (k + 1)(k + 2) 1 1 ∇τ · ∇τ Ik+2 + ∇τ · ∇τ Ik (2k + 3)(2k + 5) αk+1 (2k + 3)(2k + 1) αk+1     k(k − 1) k2 1 1 ∇τ · ∇τ Ik + ∇τ · ∇τ Ik−2 = αk (Ik − Ib δ0k ). + (2k − 1)(2k + 1) αk−1 (2k − 1)(2k − 3) αk−1

(16.83)

Similarly, sticking equations (16.81) into the PN boundary conditions, equations (16.21), gives us a consistent set of conditions for the SPN -equations: N−1 X

k even

Ik

Z1

Pk (µ)P2i−1 (µ)dµ +

N X

nˆ · Ik

k odd

0

Z1

Jw Pk (µ)P2i−1 (µ)dµ = π

Z1

P2i−1 (µ)dµ,

0

0

i = 1, 2, . . . , 12 (N + 1),

(16.84)

or, with the definition of the Legendre polynomial half-moments pm given by equation (16.71), n,j N−1 X

p0k,2i−1 Ik +

k even

N X

p0k,2i−1 nˆ · Ik =

k odd

p00,2i−1 π

Jw ,

i = 1, 2, . . . , 12 (N + 1).

(16.85)

Again, eliminating the odd Ik with equation (16.82b), this set of boundary conditions reduces to N−1 X

k even

p0k,2i−1 Ik



" N p0 X k,2i−1

k odd

αk

# p00,2i−1 k+1 k nˆ · ∇τ Ik−1 + nˆ · ∇τ Ik+1 = Jw 2k − 1 2k + 3 π i = 1, 2, . . . , 12 (N + 1).

(16.86)

No direct formula for intensity is derived, but one may assume a series of the form I(r, sˆ ) = I0 (r) + I1 (r) · sˆ + I2 (r)P02 (ˆs) + . . . ,

(16.87)

which is no longer a complete series of orthogonal functions and, therefore, is not guaranteed to approach the exact answer in the limit. However, assuming this to be an orthogonal set, we can obtain incident radiation G and radiative flux q from their definitions as Z G(r) = I(r, sˆ ) dΩ = 4πI0 (r), (16.88) Z 4π   2 4π 4π ∇τ I0 + ∇τ I2 . (16.89) q(r) = I1 (r) = − I(r, sˆ ) sˆ dΩ = 3 3α1 5 4π While equations (16.83) and (16.86) form a self-consistent set of (N + 1)/2 simultaneous elliptic partial differential equations and their boundary conditions, the problem can be further simplified by recognizing that the combination of variables Jk =

k+1 k+2 Ik + Ik+2 2k + 1 2k + 5

(16.90)

524

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

appears repeatedly in both the governing equations and boundary conditions. In addition, inspection of Table 16.2 shows that p0n,j = 0 if n + j = even, with the exception of n = j. Thus we may rewrite equations (16.83) as k = 0, 2, . . . ,N − 1

(even) :     k k+1 1 1 ∇τ · ∇τ Jk + ∇τ · ∇τ Jk−2 = αk (Ik − Ib δ0k ), 2k + 3 αk+1 2k − 1 αk−1

(16.91)

and boundary conditions (16.86) as p02i−1,2i−1 α2i−1

N−1

nˆ · ∇τ J2i−2 =

2 X

p02k,2i−1 I2k −

k=0

p00,2i−1 π

Jw ,

i = 1, 2, . . . , 12 (N + 1).

(16.92)

The Ik on the right-hand sides may be eliminated by inverting equation (16.90), starting with k = N − 1 (and noting that IN+1 ≡ 0). This results in individual partial differential equations for each Jk , in which Jl (l , k) occur only as source terms without derivatives. Once the Jk have been determined, incident radiation and radiative flux are obtained from equations (16.88) and (16.89) as   24 2 G(r) = 4π J0 (r) − J2 (r) + J4 (r) − + . . . , (16.93) 3 55 4π ∇τ J0 (r). (16.94) q(r) = − 3α1 We will demonstrate this by looking in more detail at the SP1 - and SP3 -approximations (even orders, such as SP2 , have also been formulated [58], but—based on the development shown here—appear to be as inappropriate as for the standard PN -method).

SP1 -Approximation With N = 1 we obtain a single equation and a single boundary condition from equations (16.91) and (16.92), i.e.: Governing equation:   1 1 k=0: ∇τ · ∇τ J0 = α0 (I0 − Ib ); (16.95) 3 α1 Boundary condition: i=1: With p00,1 =

1 2

and p01,1 =

1 3

with boundary condition

p01,1 α1

nˆ · ∇τ J0 = p00,1 (I0 − Jw /π).

(16.96)

from Table 16.2, and I0 = J0 from equation (16.90), we obtain   1 1 ∇τ · ∇τ J0 = α0 (J0 − Ib ), 3 α1 1 1 nˆ · ∇τ J0 = (J0 − Jw /π). 3α1 2

(16.97)

(16.98)

Not surprisingly, comparison with equations (16.38) and (16.49) and using G = 4πI0 = 4πJ0 shows that the SP1 -approximation is identical to the P1 -method.

16.7 SIMPLIFIED PN -APPROXIMATION

525

SP3 -Approximation Setting N = 3 we get two simultaneous equations and two boundary conditions: Governing equations:     1 2 1 k=0: ∇τ · ∇τ J0 = α0 (I0 − Ib ) = α0 J0 − J2 − Ib , 3 α1 3     2 5 3 1 1 k=2: ∇τ · ∇τ J2 + ∇τ · ∇τ J0 = α2 I2 = α2 J2 , 7 α3 3 α1 3 or, subtracting 2 × equation (16.99a),     5 4 3 1 ∇τ · k=2: ∇τ J2 = α2 + α0 J2 − 2α0 (J0 − Ib ). 7 α3 3 3

(16.99a) (16.99b)

(16.99c)

Boundary conditions: i=1: i=2:

p01,1 α1 p03,3 α3

nˆ · ∇τ J0 = p00,1 (I0 − Jw /π) + p02,1 I2 ,

(16.100a)

nˆ · ∇τ J2 = p00,3 (I0 − Jw /π) + p02,3 I2 .

(16.100b)

With p02,1 = p02,3 = 81 , p03,3 = 17 , p00,3 = − 81 , and eliminating the Ik , the boundary conditions become i=1: i=2:

1 nˆ · ∇τ J0 = 12 (J0 − 23 J2 − Jw /π) + 18 53 J2 = 12 (J0 − Jw /π) − 81 J2 , 3α1 1 7 nˆ · ∇τ J2 = − 18 (J0 − 32 J2 − Jw /π) + 18 53 J2 = − 18 (J0 − Jw /π) + 24 J2 . 7α3

(16.100c) (16.100d)

Unlike the regular P3 -approximation, SP3 has only two, and nearly separated, elliptic partial differential equations: equations (16.99a) and (16.100c) for J0 and equations (16.99c) and (16.100d) for J2 , the only connection being the other Jk appearing in source terms. Example 16.8. Repeat Examples 16.4, 16.6, and 16.7 using the SP3 -approximation. Solution For a nonscattering medium without z-dependence equations (16.99) reduce to 2 1 (Lxx + L yy )J0 − J0 = − J2 − Ib , 3 3 1 2 (Lxx + L yy )J2 − J2 = − (J0 − Ib ), 7 3 where we have used the operators defined in equation (16.53) for better comparison with the equivalent P3 set of Example 16.4. The boundary conditions for a general location simplify to 1 1 1 ∂J0 = (J0 − Ib ) − J2 , 3 ∂τz 2 8 1 ∂J2 1 7 J2 . = − (J0 − Ib ) + 7 ∂τz 8 24 Finally, for the one-dimensional case with only y-dependence, and again taking advantage of the symmetry by placing τ y = 0 at the midplane, the equations and boundary conditions further reduce to 2 1 ′′ J − J0 = − J 2 − I b , 3 0 3 1 ′′ 2 J − J2 = − (J0 − Ib ), 7 2 3

526

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

J0′ = J2′ = 0,

τy = 0 :

1 ′ 1 1 J = (J0 − Ibw ) − J2 , 3 0 2 8 1 ′ 1 7 J = − (J0 − Ibw ) + J2 . 7 2 8 24

τ y = −τL /2 :

The set of two simultaneous equations is readily reduced to one, by solving the first for J2 : 3 1 (J0 − Ib ) − J0′′ , 2 2

J2 =

then substituting for J2 and J2′′ in the second, or     2 1 1 3 ′′ 1 (iv) 3 J0 − J0 − (J0 − Ib ) − J0′′ = − (J0 − Ib ), 7 2 2 2 2 3 or 3 (iv) 6 ′′ J − J0 + J0 = Ib . 35 0 7 Similarly, we eliminate J2 from the boundary conditions: J0′ = J0′′′ = 0,

τy = 0 :

  1 ′ 1 1 3 1 J0 = (J0 − Ibw ) − (J0 − Ib ) − J0′′ , 3 2 8 2 2     1 1 3 ′ 1 ′′′ 7 3 1 J0 − J0 = − (J0 − Ibw ) + (J0 − Ib ) − J0′′ , 7 2 2 8 24 2 2

τ y = −τL /2 :

leading to

5 2 1 (J0 − Ib ) − J0′ + J0′′ 8 3 8 5 12 ′ 7 ′′ 4 ′′′ Ibw − Ib = − (J0 − Ib ) + J + J − J . 2 7 0 6 0 7 0

τ y = −τL /2 :

Ibw − Ib =

Since the governing fourth-order equation is exactly the same as the one for I0 in Example 16.7, the solution is also the same, J0 (τ y ) = Ib + (Ibw − Ib )[C1 cosh λ1 τ y + C2 cosh λ2 τ y ], (here given right away without the C3 and C4 , which are eliminated through the τ y = 0 boundary condition). Again, C1 =

b2 − a2 , a1 b2 − a2 b1

C2 =

a1 − b1 , a1 b2 − a2 b1

but with the ai and bi replaced by   5 1 2 τL 2 τL ai = + λi cosh λi + λi sinh λi , 8 8 2 3 2     τL 4 3 τL 5 7 2 12 λi − λi sinh λi , bi = − + λi cosh λi − 2 6 2 7 7 2

i = 1, 2, i = 1, 2.

The heat flux through the medium is determined from equation (16.89) as   4π 4π ′ 2 ′ I0 + I2 = − J0′ . q(τ y ) = − 3 5 3 Substituting for J0 we may express the heat flux for a gray medium again in nondimensional form as Ψ=

q(τ y ) n2 σ(Tw4 − T 4 )

= −

2 4X Ci λi sinh λi τ y . 3 i=1

As mentioned in the beginning of this section, for a one-dimensional slab the SPN -method reduces to the regular PN solution. Therefore, the solution here must be identical to that of Example 16.4, which can be shown to be true after considerable algebra.

16.8 THE MODIFIED DIFFERENTIAL APPROXIMATION

527

A

s

θ n

s, τ s n´

θ´

s ´,τ

´s r

rw

FIGURE 16-9 Radiative intensity within an arbitrary enclosure.

0

16.8 THE MODIFIED DIFFERENTIAL APPROXIMATION As indicated earlier, the P1 - or differential approximation enjoys great popularity because of its relative simplicity and because of its compatibility with standard methods for the solution of the (overall) energy equation. The fact that the P1 -approximation may become very inaccurate in optically thin media—and thus of limited use—has prompted a number of investigators to seek enhancements or modifications to the differential approximation to make it reasonably accurate for all conditions [59–70]. We shall briefly describe here the so-called modified differential approximation. The directional intensity at any given point inside the medium is due to two sources: radiation originating from a surface (due to emission and reflection), and radiation originating from within the medium (due to emission and in-scattering). The contribution due to radiation emanating from walls may display very irregular directional behavior, especially in optically thin situations (due to surface radiosities varying across the enclosure surface, causing irradiation to change rapidly over incoming directions). Intensity emanating from inside the medium generally varies very slowly with direction because emission and isotropic scattering result in an isotropic radiation source. Only for highly anisotropic scattering may the radiation source—and, therefore, at least locally also the intensity—display irregular directional behavior. In what they termed the modified differential approximation (MDA) Olfe [59–62] and Glatt and Olfe [71] separated wall emission from medium emission in simple black and gray-walled enclosures with gray, nonscattering media, evaluating radiation due to wall emission with exact methods, and radiation from medium emission with the differential (or P1 ) approximation. While very accurate, their model was limited to nonscattering media in simple, mostly one-dimensional enclosures. Wu and coworkers [63] demonstrated, for one-dimensional planeparallel media, that the MDA may be extended to scattering media with reflecting boundaries. Finally, Modest [64] showed that the method can be applied to three-dimensional linearanisotropically scattering media with reflecting boundaries. While until recently only used in conjunction with the P1 -approximation, higher order PN - and SPN -methods can also benefit from this approach, as recently shown by Modest and Yang [13], who demonstrated the accuracy of a modified P3 -approach. Consider an arbitrary enclosure as shown in Fig. 16-9. The equation of transfer is, from equation (16.4), dI (r, sˆ ) = sˆ · ∇τ I = S(r, sˆ ) − I(r, sˆ ), dτs

(16.101)

528

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

where, for linear-anisotropic scattering with a phase function given by equation (16.32), the radiative source term is, from equation (16.33), S(r, sˆ ) = (1 − ω)Ib (r) +

ω [G(r) + A1 q(r) · sˆ ]. 4π

(16.102)

For diffusely reflecting walls, equations (16.101) and (16.102) are subject to the boundary condition I(rw , sˆ ) =

Jw 1−ǫ (rw ) = Ibw (rw ) − q · n(r ˆ w ), π πǫ

(16.103)

where Jw is the surface radiosity related to Ibw and qw = q · nˆ through equation (16.46). We now break up the intensity at any point into two components: one, Iw , which may be traced back to emission from the enclosure wall (but may have been attenuated by absorption and scattering in the medium, and by reflections from the enclosure walls), and the remainder, Im , which may be traced back to the radiative source term (i.e., radiative intensity released within the medium into a given direction by emission and scattering). Thus, we write I(r, sˆ ) = Iw (r, sˆ ) + Im (r, sˆ )

(16.104)

dIw (r, sˆ ) = −Iw (r, sˆ ), dτs

(16.105)

Jw (rw ) e −τs , π

(16.106)

and let Iw satisfy the equation

leading to Iw (r, sˆ ) =

as indicated in Fig. 16-9. Since for Iw no radiative source within the medium is considered, the radiosity in equation (16.106) is the one caused by wall emission only (with attenuation within the medium). The radiosity variation along the enclosure wall may be determined by invoking the definition of the radiosity as the sum of emission plus reflected irradiation, or Z Jw (r) = ǫπIbw (r) + (1 − ǫ) Iw (r, sˆ ) |ˆs · n| ˆ dΩ Z sˆ ·nˆ ri ). (d) What is the surface temperature of the sun? 16.15 Repeat Problem 16.14 but replace assumption (iv) by the following: The fusion process may be approximated by assuming that the sun releases heat uniformly throughout its volume corresponding to the total heat loss of the sun. 16.16 Consider a sphere of very hot dissociated gas of radius 5 cm. The gas may be approximated as a gray, linear-anisotropically scattering medium with κ = 0.1 cm−1 , σs = 0.2 cm−1 , A1 = 1. The gas is suspended magnetically in vacuum within a large cold container and is initially at a uniform temperature T1 = 10,000 K. Using the P1 -approximation and neglecting conduction and convection, specify the total heat loss per unit time from the entire sphere at time t = 0. Outline the solution procedure for times t > 0. Hint: Solve the governing equation by introducing a new dependent variable 1(τ) = τ(4πIb − G). 16.17 A spherical test bomb of 1 m radius is coated with a nonreflective material and cooled. Inside the sphere is nitrogen mixed with spherical particles at a rate of 108 particles/m3 . The particles have a radius of 300 µm, are diffuse-gray with ǫ = 0.5, and generate heat at a rate of 150 W/cm3 of particle volume. Using absorption and scattering coefficients found in Problem 12.12, determine the temperature distribution inside the bomb, using the P1 -approximation and two simplified phase functions: (i) isotropic scattering, and (ii) linear-anisotropic back scattering with A1 = −1. In particular, what is the gas temperature at the center and at the wall? How much do the two scattering treatments differ from one another? 16.18 A revolutionary new fuel is ground up into small particles, magnetically confined to remain within a spherical cloud of radius R. This cloud of particles has a constant, gray absorption coefficient, ′′′ does not scatter, and releases heat uniformly at Q˙ (W/m3 ). The cloud is suspended in a vacuum chamber, isothermal chamber (at Tw ). Heat transfer is solely by radiation, i.e.,    by a large,  enclosed ′′′ ∇ · q = 1/r2 d r2 q /dr = Q˙ . (a) Assuming the P1 -approximation to be valid, set up the necessary equations and boundary conditions to determine the heat transfer rates, and temperature distribution within the spherical cloud. (b) Determine the maximum temperature in the cloud.

16.19 Repeat Problem 16.5 using subroutine P1sor and/or program P1-2D. How do the answers change for a quadratic enclosure (side walls also cold and black)? 16.20 Repeat Problem 16.6 using subroutine P1sor and/or program P1-2D. How do the answers change for a quadratic enclosure (side walls also black, with a linear surface temperature variation from T(x = 0) = T1 to T(x = L) = T2 )?

540

16 THE METHOD OF SPHERICAL HARMONICS (PN -APPROXIMATION)

16.21 Consider a gray, isotropically scattering medium at radiative equilibrium contained between large, isothermal, gray plates at temperatures T1 and T2 , and emittances ǫ1 and ǫ2 , respectively. Determine the radiative heat flux between the plates using the P3 -approximation. Compare the results with the answer from Problem 16.2. 16.22 Do Problem 16.3 using the P3 -approximation with Marshak’s boundary condition. 16.23 A hot gray medium is contained between two concentric black spheres of radius R1 = 10 cm and R2 = 20 cm. The surfaces of the spheres are isothermal at T1 = 2000 K and T2 = 500 K, respectively. The medium absorbs and emits with n = 1, κ = 0.05 cm−1 , but does not scatter radiation. Determine the heat flux between the spheres using the modified differential approximation (MDA). Note: This problem requires the numerical solution of a simple ordinary differential equation. 16.24 Repeat Problem 16.23 for concentric cylinders of the same radii. Compare your result with those of Fig. 16-5. Note: This problem requires the numerical solution of a simple ordinary differential equation.

CHAPTER

17 THE METHOD OF DISCRETE ORDINATES (SN-APPROXIMATION)

17.1

INTRODUCTION

Like the spherical harmonics method, the discrete ordinate method is a tool to transform the equation of transfer (for a gray medium, or on a spectral basis) into a set of simultaneous partial differential equations. Like the PN -method, the discrete ordinates or SN -method may be carried out to any arbitrary order and accuracy, although the mathematical formulation of high-order SN -schemes is considerably less involved. First proposed by Chandrasekhar [1] in his work on stellar and atmospheric radiation, the SN -method originally received little attention in the heat transfer community. Again like the PN -method, the discrete ordinates method was first systematically applied to problems in neutron transport theory, notably by Lee [2] and Lathrop [3, 4]. There were some early, unoptimized attempts to apply the method to onedimensional, planar thermal radiation problems (Love et al. [5, 6], Hottel et al. [7], Roux and Smith [8, 9]). But only during the past thirty years has the discrete ordinates method been applied to, and optimized for, general radiative heat transfer problems, primarily through the pioneering works of Fiveland [10–13] and Truelove [14–16]. The discrete ordinates method is based on a discrete representation of the directional variation of the radiative intensity. A solution to the transport problem is found by solving the equation of transfer for a set of discrete directions spanning the total solid angle range of 4π. As such, the discrete ordinates method is simply a finite differencing of the directional dependence of the equation of transfer. Integrals over solid angle are approximated by numerical quadrature (e.g., for the evaluation of the radiative source term, the radiative heat flux, etc.). Today, many numerical heat transfer models use finite volumes rather than finite differences. Similarly, one may also use finite “solid angle volumes” for directional discretization. This variation of the discrete ordinates method is commonly known as the finite volume method (for radiative transfer), and enjoys increasing popularity. As a result of the relatively straightforward formulation of high-order implementations, the discrete ordinates method (DOM) and its finite volume cousin (FVM) have received great attention and are today probably the most popular RTE solvers (together with the P1 -approximation), and some version of them is incorporated in most commercial CFD codes. Detailed reviews of the capabilities and shortcomings of the 541

542

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

DOM and FVM have been given by Charest et al. [17] and by Coelho [18]. The latter provides the most complete description of the method for general geometries, far exceeding the details we can provide in this book. In this chapter we shall first develop the set of partial differential equations for the standard SN -method and their boundary conditions. This is followed by a section describing how the method may be applied to one-dimensional plane-parallel media, and another dealing with spherical and cylindrical geometries, and then its application to two- and three-dimensional problems will be outlined. This is followed by the development of the finite volume method and, finally, the chapter will close with a brief look at other, related methods.

17.2

GENERAL RELATIONS

The general equation of transfer for an absorbing, emitting, and anisotropically scattering medium is, according to equation (10.21), Z σs (r) dI = sˆ · ∇I(r, sˆ ) = κ(r)Ib (r) − β(r)I(r, sˆ ) + I(r, sˆ ′ ) Φ(r, sˆ ′ , sˆ ) dΩ′ . (17.1) ds 4π 4π Equation (17.1) is valid for a gray medium or, on a spectral basis, for a nongray medium, and is subject to the boundary condition Z ρ(rw ) I(rw , sˆ ) = ǫ(rw )Ib (rw ) + I(rw , sˆ ′ ) |nˆ · sˆ ′ | dΩ′ , (17.2) ′ π ˆ s 0.

(17.5)

n·ˆ ˆ sj 0), and once where it strikes the wall, to be absorbed or reflected (n·ˆ ˆ s i < 0). The governing equation is first order, requiring only one boundary condition (for the emanating intensity, nˆ · sˆ i > 0). Equations (17.4) together with their boundary conditions (17.5) constitute a set of n simultaneous, first-order, linear partial differential equations for the

17.2 GENERAL RELATIONS

543

unknown Ii (r) = I(r, sˆ i ). The solution for the Ii may be found using any standard technique (analytical or numerical). If scattering is present (σs , 0), and/or if the bounding walls are reflecting, the equations are coupled in such a way that generally an iterative procedure is necessary. Even in the absence of scattering and surface reflections, the temperature field may not be known, but must be calculated from the intensity field if radiative equilibrium persists, again making iterations necessary. Only in the absence of scattering and wall reflections, and if the temperature field is given, then the solution to the intensities Ii is straightforward (as is the exact solution). Once the intensities have been determined the desired direction-integrated quantities are readily calculated. The radiative heat flux, inside the medium or at a surface, may be found from its definition, equation (10.52), q(r) =

Z

n X

I(r, sˆ ) sˆ dΩ ≃ 4π

wi Ii (r) sˆ i .

(17.6)

i=1

The incident radiation G [and, through equation (10.59), the divergence of the radiative heat flux] is similarly determined as G(r) =

Z

I(r, sˆ ) dΩ ≃ 4π

n X

wi Ii (r).

(17.7)

i=1

At a surface the heat flux may also be determined from surface energy balances [equations (4.1) and (3.16)] as q · n(r ˆ w ) = ǫ(rw ) [πIb (rw ) − H(rw )] X   ≃ ǫ(rw ) πIb (rw ) − wi Ii (rw ) |nˆ · sˆ i | .

(17.8)

n·ˆ ˆ si 0

(17.11)

at the enclosure surface. Of course, radiative heat flux and incident radiation are unknowns to be determined from directional intensities from the series in equations (17.6) and (17.7). Equations (17.10) and (17.11) are convenient forms for the iterative solution procedure: For each iteration values of G and q are estimated, and the n intensities Ii are evaluated. The values for G and q are then updated, and so on.

Selection of Discrete Ordinate Directions The choice of quadrature scheme is arbitrary, although restrictions on the directions sˆ i and quadrature weights wi may arise from the desire to preserve symmetry and to satisfy certain

544

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

conditions. It is customary to choose sets of directions and weights that are completely symmetric (i.e., sets that are invariant after any rotation of 90◦ ), and that satisfy the zeroth, first, and second moments, or Z Z Z

dΩ = 4π = 4π

sˆ dΩ = 0 = 4π

sˆ sˆ dΩ = 4π

4π δ= 3

n X i=1 n X

i=1 n X

wi ,

(17.12a)

wi sˆ i ,

(17.12b)

wi sˆ i sˆ i ,

(17.12c)

i=1

where δ is the unit tensor [cf. equation (16.30)]. Different sets of directions and weights satisfying all these criteria have been tabulated, for example, by Lee [2] and Lathrop and Carlson [19]. Fiveland [12] and Truelove [15] have observed that different sets of ordinates may result in considerably different accuracy. They noted that (i) the intensity may have directional discontinuity at a wall, and (ii) the important radiative heat fluxes at the walls are evaluated through a first moment of intensity over a half range of 2π [equation (17.8)]. They concluded that the set of ordinates and weights should also satisfy the first moment over a half range, that is, Z Z X |nˆ · sˆ | dΩ = nˆ · sˆ dΩ = π = wi nˆ · sˆ i . (17.13) n·ˆ ˆ s0

n·ˆ ˆ si >0

While it is impossible to satisfy equation (17.13) for arbitrary orientations of the surface normal, ˆ Sets of ordinates and weights it can be satisfied for the principal orientations, if nˆ = ˆı, ˆ, or k. that satisfy (i) the symmetry requirement, (ii) the moment equations (17.12), and (iii) the halfmoment equation (17.13) (for the three principal directions of n) ˆ 1 have been given by Lathrop and Carlson [19]. The first four sets labeled S2 -, S4 -, S6 -, and S8 -approximation are reproduced in Table 17.1. In the table the ξi , ηi , and µi are the direction cosines of sˆ i , or ˆ kˆ = ξi ˆı + ηi ˆ + µi k. ˆ sˆ i = (ˆs i · ˆı) ˆı + (ˆs i · ˆ) ˆ + (ˆs i · k)

(17.14)

Only positive direction cosines are given in Table 17.1, covering one eighth of the total range of solid angles 4π. To cover the entire 4π any or all of the values of ξi , ηi , and µi may be positive or negative. Therefore, each row of ordinates contains eight different directions. For ˆ sˆ 2 = example, for the S2 -approximation the different directions are sˆ 1 = 0.577350(ˆı + ˆ + k), ˆ ˆ 0.577350(ˆı + ˆ − k), . . . , sˆ 8 = −0.577350(ˆı + ˆ + k). Since the symmetric S2 -approximation does not satisfy the half-moment condition, a nonsymmetric S2 -approximation is also included in Table 17.1, as proposed by Truelove [15]. This approximation satisfies equation (17.13) for two principal directions and should be applied to one- and two-dimensional problems, from which the nonsymmetric term drops out (as seen in Example 17.1 in the following section). The name “SN -approximation” indicates that N different direction cosines are used for each principal direction. For example, for the S4 -approximation ξi = ±0.295876 and ±0.908248 (or ηi or µi ). Altogether there are always n = N(N+2) different directions to be considered (because of symmetry, many of these may be unnecessary for one- and two-dimensional problems). Several other quadrature schemes can be found in the literature. Carlson [20] proposed a set with equal weights wi (such as the S2 and S4 sets in Table 17.1). Two more quadratures and a good review of the applicability of all discrete ordinate sets have been given by Fiveland [21]. Other publications documenting procedures for the generation of quadrature sets are those of S´anchez and Smith [22] and El-Wakil and Sacadura [23]. A new family of quadrature sets, like the Sn 1

With the exception of the symmetric S2 -approximation.

17.3 THE ONE-DIMENSIONAL SLAB

545

TABLE 17.1

Discrete ordinates for the SN -approximation (N = 2, 4, 6, 8), from [19]. Order of

Ordinates

Weights

ξ

η

µ

w

S2 (symmetric)

0.5773503

0.5773503

0.5773503

1.5707963

S2 (nonsymmetric)

0.5000000

0.7071068

0.5000000

1.5707963

S4

0.2958759 0.2958759 0.9082483 0.1838670 0.1838670 0.1838670 0.6950514 0.6950514 0.9656013 0.1422555 0.1422555 0.1422555 0.1422555 0.5773503 0.5773503 0.5773503 0.8040087 0.8040087 0.9795543

0.2958759 0.9082483 0.2958759 0.1838670 0.6950514 0.9656013 0.1838670 0.6950514 0.1838670 0.1422555 0.5773503 0.8040087 0.9795543 0.1422555 0.5773503 0.8040087 0.1422555 0.5773503 0.1422555

0.9082483 0.2958759 0.2958759 0.9656013 0.6950514 0.1838670 0.6950514 0.1838670 0.1838670 0.9795543 0.8040087 0.5773503 0.1422555 0.8040087 0.5773503 0.1422555 0.5773503 0.1422555 0.1422555

0.5235987 0.5235987 0.5235987 0.1609517 0.3626469 0.1609517 0.3626469 0.3626469 0.1609517 0.1712359 0.0992284 0.0992284 0.1712359 0.0992284 0.4617179 0.0992284 0.0992284 0.0992284 0.1712359

Approximation

S6

S8

sets symmetric in 90◦ rotations, but with different arrangement of directions, have been given by Thurgood and coworkers [24], and have been dubbed Tn sets by the authors. These always generate positive weights and are claimed to reduce the so-called “ray effect” (which will be discussed a little later on p. 560). These sets have been further refined by Li and coworkers [25]. A comprehensive review of directional quadrature schemes, including an evaluation of their accuracies, has recently been given by Koch and Becker [26]. None of the above ordinate sets can treat collimated (i.e., unidirectional) irradiation accurately. To address this problem Li and coworkers [27] developed the ISW scheme adding a single ordinate of “infinitely small weight” to the regular quadrature set.

17.3

THE ONE-DIMENSIONAL SLAB

We will first demonstrate how the SN discrete ordinates method is applied to the simple case of a one-dimensional plane-parallel slab bounded by two diffusely emitting and reflecting isothermal plates. As in previous chapters we shall limit ourselves to linear-anisotropic scattering, although extension to arbitrarily anisotropic scattering is straightforward. We avoid it here to make the steps in the development a little easier to follow. If we choose z as the spatial coordinate between the two plates (0 ≤ z ≤ L), and introduce the optical coordinate τ with dτ = β dz (0 ≤ τ ≤ τL ), equation (17.4) is transformed to µi

n h i ω X dIi = (1 − ω) Ib − Ii + w j I j 1+A1 (µi µ j +ξi ξ j +ηi η j ) , dτ 4π j=1

i = 1, 2, . . . , n.

(17.15)

546

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

TABLE 17.2

Discrete ordinates for the one-dimensional SN -approximation (N = 2, 4, 6, 8). Order of Approximation

Ordinates µ

Weights w′

S2 (symmetric)

0.5773503

6.2831853

S2 (nonsymmetric)

0.5000000

6.2831853

S4

0.2958759 0.9082483 0.1838670 0.6950514 0.9656013 0.1422555 0.5773503 0.8040087 0.9795543

4.1887902 2.0943951 2.7382012 2.9011752 0.6438068 2.1637144 2.6406988 0.7938272 0.6849436

S6

S8

For a one-dimensional slab intensity is independent of azimuthal angle. Since for every ordinate j (with a given µj ) with a positive value for ξ j there is another with the same, but negative, value, and since the intensity is the same for both ordinates, the terms involving ξ j in equation (17.15) add to zero. The same is true for the terms involving η j , but not for those with µ j (since the intensity does depend on polar angle θ, and µ = cos θ). However, the terms involving µ j are repeated several times: Each value of µ (counting positive and negative µ-values separately) shown in one row of Table 17.1 corresponds to four different ordinates (combinations of positive and negative values for ξ and η). In addition, a particular value of µ may occur on more than one line of Table 17.1. If all the quadrature weights corresponding to a single µ-value are added together, equation (17.15) reduces to N

µi

ω X ′ dIi = (1 − ω) Ib − Ii + w j I j (1 + A1 µi µ j ), dτ 4π

i = 1, 2, . . . , N,

(17.16)

j=1

where the w′j are the summed quadrature weights. For example, for µ = 0.2958759 in the S4 -approximation the summed quadrature weight is w′ = 4 × (0.5235987 + 0.5235987) = 4π/3, and so forth. The ordinates and quadrature weights for the one-dimensional slab are listed in Table 17.2. Equation (17.16) could have been found less painfully by using equation (17.10) instead of (17.4), leading directly to µi

ω dIi + Ii = (1 − ω) Ib + (G + A1 qµi ), dτ 4π

i = 1, 2, . . . , N.

(17.17)

Before proceeding to the boundary conditions of equation (17.17) we should recognize that, of the N different intensities, half emanate from the wall at τ = 0 (with µi > 0), and the other half from the wall at τ = τL (with µi < 0). Following the notation of Chapter 14, we replace the N different Ii by + I1+ , I2+ , . . . , IN/2

− and I1− , I2− , . . . , IN/2 .

17.3 THE ONE-DIMENSIONAL SLAB

547

Then equation (17.17) may be rewritten as µi −µi

dIi+ dτ dIi− dτ

ω (G + A1 qµi ), 4π ω = (1 − ω) Ib + (G − A1 qµi ), 4π i = 1, 2, . . . , N/2;

+ Ii+ = (1 − ω) Ib +

(17.18a)

+ Ii−

(17.18b) µi > 0.

With this notation the boundary conditions for equation (17.18) follow from equations (17.5) or (17.11) as τ=0: τ = τL :

1 − ǫ1 q1 , ǫ1 π 1 − ǫ2 Ii− = J2 /π = Ib2 + q2 , ǫ2 π i = 1, 2, . . . , N/2, Ii+ = J1 /π = Ib1 −

(17.19a) (17.19b) µi > 0.

(For the boundary condition at τL the sign switches since nˆ points in the direction opposite to z.) Radiative heat flux q and incident radiation G are related to the directional intensities through equations (17.6) and (17.7), or q =

N/2 X

w′i µi (Ii+ − Ii− ),

(17.20a)

N/2 X

w′i (Ii+ + Ii− ).

(17.20b)

i=1

G =

i=1

At the two surfaces the radiative heat flux is more conveniently evaluated from equation (17.8) as τ=0:

q1 =

q(0) =

N/2 X   w′i µi Ii− , ǫ1 Eb1 −

(17.21a)

N/2 X

(17.21b)

i=1

τ = τL :



q2 = −q(τL ) = −ǫ2 Eb2 −

i=1

 w′i µi Ii+ .

Example 17.1. Consider two large, parallel, gray-diffuse and isothermal plates, separated by a distance L. One plate is at temperature T1 with emittance ǫ1 , the other is at T2 with ǫ2 . The medium between the two plates is a gray, absorbing/emitting and linear-anisotropically scattering gas (n = 1) with constant extinction coefficient β and single scattering albedo ω. Assuming that radiative equilibrium prevails, determine the radiative heat flux between the two plates using the S2 -approximation. Solution For radiative equilibrium we have, from equation (10.59), Ib = G/4π and q = const; equations (17.18) and (17.19) become µ1 −µ1 τ=0:

dI1+ dτ dI1− dτ

1 (G + A1 ωµ1 q), 4π 1 = (G − A1 ωµ1 q), 4π

+ I1+ = + I1−

I1+ = J1 /π,

τ = βL = τL :

I1− = J2 /π.

For the S2 -approximation we have only a single ordinate direction µ1 (pointing toward τL for I1+ , and toward 0 for I1− ), where µ1 = 0.57735 for the symmetric S2 -approximation, and µ1 = 0.5 for the nonsymmetric S2 -approximation [which satisfies the half-range moment, equation (17.13)]. For the simple

548

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

S2 -approximation the simultaneous equations (only two in this case) may be separated. We do this here by eliminating I1+ and I1− in favor of G and q. From equation (17.20), with w′i = 2π, G = 2π(I1+ + I1− ), q = 2π µ1 (I1+ − I1− ). Therefore, adding and subtracting the two differential equations and multiplying by 2π leads to dq dq + G = G, or = 0, dτ dτ   1 1 dG dG q = A1 ωµ1 q, or + = − 2 − A1 ω q. µ1 dτ µ1 dτ µ1 The first equation is simply a restatement of radiative equilibrium, while the second may be integrated (since q = const), or   1 G = C − 2 − A1 ω qτ. µ1 This relation contains two unknown constants (C and q), which must be determined from the boundary conditions, that is, ! q 1 G+ τ=0: I1+ = = J1 /π, 4π µ1 ! q 1 G− I1− = = J2 /π, τ = τL : 4π µ1 or q q =C+ , 4J1 = G + τ=0: µ1 µ1   q q 1 τ = τL : 4J2 = G − = C − 2 − A1 ω q τL − . µ1 µ1 µ1 Subtracting, we obtain, Ψ=

2µ1 q   = , J1 − J2 1 + 1/µ21 − A1 ω µ1 τL /2

from which the radiosities may be eliminated through equation (14.48). For the symmetric S2 -approxi√ mation, µ1 = 0.57735 = 1/ 3, and with isotropic scattering, A1 = 0, this expression becomes 1 Ψsymmetric = √ . 3/2 + 3τL /4 On the other hand, for the nonsymmetric S2 -approximation (µ1 = 0.5), also with isotropic scattering, Ψnonsymmetric =

1 . 1 + τL

The S2 -approximation is the same as the two-flux method discussed in Section 15.3, and the nonsymmetric S2 -method is nothing but the Schuster–Schwarzschild approximation. Results from the two S2 -approximations are compared in Table 17.3 with those from the P1 -approximation and the exact solution. It is seen that the accuracy of the S2 -method is roughly equivalent to that of the P1 -approximation. The nonsymmetric S2 -approximation is superior to the symmetric one, since the symmetric S2 does not satisfy the half-moment condition, equation (17.13), and causes substantial errors in the optically thin limit.

As a second example for the one-dimensional discrete ordinates method we shall repeat Example 16.4, which was originally designed to demonstrate the use of the P3 -approximation. Example 17.2. Consider an isothermal medium at temperature T, confined between two large, parallel black plates that are isothermal at the (same) temperature Tw . The medium is gray and absorbs and

17.3 THE ONE-DIMENSIONAL SLAB

549

TABLE 17.3

Radiative heat flux through a one-dimensional plane-parallel medium at radiative equilibrium; comparison of S2 - and P1 -approximations. Ψ = q/(J1 − J2 ) τL

Exact

S2 (sym)

S2 (nonsym)

P1

0.0 0.1 0.5 1.0 5.0

1.0000 0.9157 0.7040 0.5532 0.2077

1.1547 1.0627 0.8058 0.6188 0.2166

1.0000 0.9091 0.6667 0.5000 0.1667

1.0000 0.9302 0.7273 0.5714 0.2105

emits, but does not scatter. Determine an expression for the heat transfer rates within the medium using the S2 and S4 discrete ordinates approximations. Solution For this particularly simple case equations (17.18) reduce to µi −µi

dIi+ dτ dIi− dτ

+ Ii = Ib , + Ii = Ib .

Since Ib = const, these equations may be integrated right away, leading to Ii+ = Ib + C+ e−τ/µi , Ii− = Ib + C− eτ/µi . The integration constants C+ and C− may be found from boundary conditions (17.19) as τ=0:

Ii+ = Ibw = Ib + C+ ,

τ = τL :

Ii−

or

− τL /µi

= Ibw = Ib + C e

,

C+ = Ibw − Ib ; or

C− = (Ibw − Ib ) e−τL /µi .

Thus, Ii+ = Ib + (Ibw − Ib ) e−τ/µi , Ii− = Ib + (Ibw − Ib ) e−(τL −τ)/µi . The radiative heat flux follows then from equation (17.20) as q=

N/2 X i=1

  w′i µi (Ibw −Ib ) e−τ/µi − e−(τL −τ)/µi ,

or, in nondimensional form, Ψ=

q n2 σ(Tw4 −T 4 )

=

N/2  1 X ′  −τ/µi wi µi e − e−(τL −τ)/µi . π i=1

For the nonsymmetric S2 -approximation we have w′1 = 2π and µ1 = 0.5, or ΨS2 = e−2τ − e−2(τL −τ) . P For the S4 -approximation, w′1 = 4π/3, w′2 = 2π/3, µ1 = 0.2958759, µ2 = 0.9082483, and w′i µi = π, so that     ΨS4 = 0.3945012 e−τ/0.2958759 − e−(τL −τ)/0.2958759 + 0.6054088 e−τ/0.9082483 − e−(τL −τ)/0.9082483 .

550

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

The results should be compared with those of Examples 16.2 and 16.4 for the P1 - and P3 -approximations. Note that the SN -method goes to the correct optically thick limit (τL → ∞) at the wall, i.e., Ψ → 1 [if the half moment of equation (17.13) is satisfied]. The PN -approximations, on the other hand, overpredict the optically thick limit for this particular example.

It should be emphasized that this last example—dealing with a nonscattering, isothermal medium—is particularly well suited for the discrete ordinates method. One should not expect that, for a general problem, the S4 -method is easier to apply than the P3 -approximation. A number of researchers have solved more complicated one-dimensional problems by the discrete ordinates method. Fiveland [12] considered the identical case as presented in this section, but allowed for arbitrarily anisotropic scattering. Solving the system of equations by a finite difference method, he noted that higher-order SN -methods demand a smaller numerical step ∆τ, in order to obtain a stable solution. Kumar and coworkers [28] not only allowed arbitrarily anisotropic scattering, but also considered boundaries with specular reflectances as well as boundaries with collimated irradiation (as discussed in Chapter 19). To solve the set of simultaneous first-order differential equations they employed a subroutine from the IMSL software library [29], which is available on many computers. Stamnes and colleagues [30, 31] investigated the same problem as Kumar and coworkers but also allowed for variable radiative properties and a general bidirectional reflection function at the surfaces. They decoupled the set of simultaneous equations using methods of linear algebra and found exact analytical solutions in terms of eigenvalues and eigenvectors that, in turn, were determined using the EISPACK software library [32]. Other examples of the use of the one-dimensional discrete ordinates model as a tool to solve more complex problems may be found in [33–42].

17.4 ONE-DIMENSIONAL CONCENTRIC SPHERES AND CYLINDERS Applying the discrete ordinates method and taking advantage of the symmetries in a one-dimensional problem is considerably more difficult for concentric spheres and cylinders than for a plane-parallel slab. The reason is that the local direction cosines change while traveling along a straight line of sight through such enclosures.

Concentric Spheres Consider two concentric spheres of radius R1 and R2 , respectively. The inner sphere surface has an emittance ǫ1 and is kept isothermal at temperature T1 , while the outer sphere is at temperature T2 with emittance ǫ2 . If the temperature within the medium is a function of radius only, then the equation of transfer is given by equation (14.69), µ

∂I 1 − µ2 ∂I + + βI = βS, r ∂µ ∂r

(17.22a)

or, alternatively, i µ ∂ 2 1 ∂ h (1 − µ2 ) I + βI = βS, (r I) + 2 r ∂µ r ∂r

(17.22b)

where µ = cos θ is the cosine of the polar angle, measured from the radial direction (see Fig. 14-5). S is the radiative source function, S(r, µ) = (1 − ω) Ib + or S(r, µ) = (1 − ω) Ib +

ω 2

Z

1

I (r, µ′ ) Φ(µ, µ′ ) dµ′ ,

(17.23a)

−1

ω (G + A1 qµ), 4π

(17.23b)

17.4 ONE-DIMENSIONAL CONCENTRIC SPHERES AND CYLINDERS

N + 1/2

µN

551

N – 1/2

µN –1

θN

N – 3/2

θ1

5/2

µ2 3/2

µ1 1/2

FIGURE 17-1 Directional discretization and discrete ordinate values for one-dimensional problems.

if the scattering is limited to the linear-anisotropic case. The additional difficulty lies in the fact that equation (17.22) contains a derivative over direction cosine, µ, that is to be discretized in the discrete ordinates method. Applying the SN -method to equation (17.22), we obtain ( ) i µi d 2 1 ∂ h 2 (1−µ I ) + )I (17.24) (r + β Ii = β Si , i = 1, 2, . . . , N, i r ∂µ r2 dr µ=µi where Si is readily determined from equation (17.23) (and is independent of ordinate direction unless the medium scatters anisotropically). Equation (17.24) is only applied to the N principal ordinates since, similar to the slab, there is no azimuthal dependence. Since the direction vector µ is discretized, its derivative must be approximated by finite differences. We may write ( ) i αi+1/2 Ii+1/2 − αi−1/2 Ii−1/2 ∂ h 2 (1−µ )I ≃ , (17.25) w′i ∂µ µ=µi which is a central difference with the Ii±1/2 evaluated at the boundaries between two ordinates, as shown in Fig. 17-1. Since the differences between any two sequential µi are nonuniform, the geometrical coefficients α are nonconstant and need to be determined. The values of α depend only on the differencing scheme and, therefore, are independent of intensity and may be determined by examining a particularly simple intensity field. For example, if both spheres are at the same temperature, then Ib1 = Ib2 = Ib = const, and also I = Ib = const. This then leads to # " ∂ αi+1/2 − αi−1/2 = w′i = −2 w′i µi , i = 1, 2, . . . , N. (17.26) (1 − µ2 ) ∂µ µ=µi This expression may be used as a recursion formula for αi+1/2 , if a value for α1/2 can be determined. That value is found by noting that I1/2 is evaluated at µ = −1 (Fig. 17-1), where (1 − µ2 )I = 0 and,

552

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

therefore, α1/2 = 0. Similarly, IN+1/2 is evaluated at µ = +1 and also αN+1/2 = 0. The finite-difference scheme of equations (17.25) and (17.26) satisfies the relation [4] Z

+1 −1

+1 i ∂ h (1 − µ2 )I dµ = (1 − µ2 ) I = 0 −1 ∂µ ( ) N N  X X i  ∂ h = w′i = (1 − µ2 )I αi+1/2 Ii+1/2 − αi−1/2 Ii−1/2 ∂µ µ=µi i=1

i=1

= α3/2 I3/2 −α1/2 I1/2 +α5/2 I5/2 −α3/2 I3/2 +− · · · αN+1/2 IN+1/2 −αN−1/2 IN−1/2 = 0. Finally, the intensities at the node boundaries, Ii±1/2 , need to be expressed in terms of node center values, Ii . We shall use here simple, linear averaging, i.e., Ii+1/2 ≃ 12 (Ii + Ii+1 ). Equation (17.24) may now be rewritten as αi+1/2 Ii+1 + (αi+1/2 − αi−1/2 )Ii − αi−1/2 Ii−1 µi d 2 + βIi = βSi , (r Ii ) + 2 2rw′i r dr or, carrying out the differentiation and using equation (17.26), µi

αi+1/2 Ii+1 − αi−1/2 Ii−1 dIi µi + Ii + + βIi = βSi , dr r 2rw′i αi+1/2 = αi−1/2 − 2w′i µi ,

α1/2 = αN+1/2 = 0,

(17.27a) i = 1, 2, . . . , N.

(17.27b)

Equations (17.27) constitute a set of N simultaneous differential equations in the N unknown intensities Ii , subject to the boundary conditions [cf. equation (17.19)] N N +1, +2, . . . , N (µi > 0), 2 2 N i = 1, 2, . . . , (µi < 0). 2

1−ǫ1 q1 , ǫ1 π 1−ǫ2 q2 , r = R2 : Ii = J2 /π = Ib2 + ǫ2 π r = R1 : Ii = J1 /π = Ib1 −

i=

(17.28a) (17.28b)

As for the one-dimensional slab the radiative heat flux and incident radiation are evaluated [cf. equations (17.20) and (17.21)] from N X

w′i Ii (r),

(17.29a)

N X

w′i µi Ii (r),

(17.29b)

N/2 X   q(R1 ) = q1 = ǫ1 Eb1 + w′i µi Ii ,

(17.29c)

N X   w′i µi Ii . −q(R2 ) = q2 = ǫ2 Eb2 −

(17.29d)

G(r) =

i=1

q(r) =

i=1

and

i=1 (µi 0)

Example 17.3. Consider a nonscattering medium at radiative equilibrium that is contained between two isothermal, gray spheres. The absorption coefficient of the medium may be assumed to be gray

17.4 ONE-DIMENSIONAL CONCENTRIC SPHERES AND CYLINDERS

553

and constant. Using the S2 -approximation determine the radiative heat flux between the two concentric spheres. Solution From equation (17.27) we find, with N = 2, that α1/2 = α5/2 = 0, α3/2 = −2w′1 µ1 = 2w′2 µ2 = 4πµ (since µ2 = −µ1 > 0; we keep µ = µ2 as a nonnumerical value to allow comparison between the symmetric and nonsymmetric S2 -approximations). For a gray, nonscattering medium at radiative equilibrium we have β = κ and ∇ · q = 0, and the source function is, from equations (10.61) and (17.39), S = Ib = G/4π. µ dI1 µ G 1 − I1 + I2 + I1 = = (I1 + I2 ), dτ τ τ 4π 2 µ  1 dI1 − − (I1 − I2 ) = 0, −µ dτ τ 2 −µ

i=1:

µ 1 dI2 µ + I2 − I1 + I2 = (I1 + I2 ), dτ τ τ 2  µ 1 dI2 (I1 − I2 ) = 0. − + µ dτ τ 2

i=2:

µ

While addition of the two equations simply leads to a restatement of radiative equilibrium (as in Example 17.1), subtracting them (and multiplying by w′i = 2π) leads to −µ or

d [2π(I1 + I2 )] + 2π(I1 − I2 ) = 0, dτ q τ2 q 1 dG = − 2 = − 2 2. dτ µ µ τ

Since for a medium at radiative equilibrium between concentric spheres Q = 4πr2 q = const and, therefore, τ2 q = const, the incident radiation may be found by integration, G(τ) =

τ2 q 1 + C, µ2 τ

where the two constants (τ2 q) and C are still unknown and must be determined from the boundary conditions, equations (17.28): I2 (τ1 ) = J1 /π,

I1 (τ2 ) = J2 /π.

Using the definitions for q and G, equations (17.29), q = 2πµ (I2 − I1 ) or

and

G = 2π(I2 + I1 ),

!

! q q 1 1 G− , I2 = G+ , I1 = 4π µ 4π µ

the boundary conditions may be restated in terms of q and G as τ = τ1 : τ = τ2 :

µ τ 1 q1 q1 τ2 q  1 q1 = 2 +C+ = 2 + 2 + C, µ µ µ µ τ1 τ1 µ q2 τ 2 q2 q2 τ2 q  1 4J2 = G − = 2 +C− = 2 − 2 + C. µ µ µ µ τ2 τ2 4J1 = G +

Subtracting the second boundary condition from the first we obtain Ψ=

1 τ2 q =  .  2 τ21 J1 − J2 τ τ1 τ1 1 1 + 21 + 2 1 − 4µ τ2 4µ τ2

554

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

√ For the symmetric S2 -approximation, with µ = 1/ 3, this equation becomes 1 Ψsymmetric = √  ,  2 τ 3τ1 τ1 3 1 + 12 + 1− 4 4 τ2 τ2 and for the nonsymmetric approximation with µ = 0.5, Ψnonsymmetric =

1

 . τ1 1 1 + 2 + τ1 1 − 2 τ2 τ2 

τ21 

The accuracy of the S2 -approximation is very similar to that of the P1 -approximation, for which ΨP1 =

1 .  τ21  3τ1  1 τ1 1+ 2 + 1− 2 4 τ2 τ2

Note that the method is very accurate for large τ1 (large optical thickness) but breaks down for optically thin conditions (κ → 0), in particular for small ratios of radii, R1 /R2 . In the limit (κ → 0, R1 /R2 → 0) we find ΨP1 = ΨS2 ,nonsym → 2, while the correct limit should go to Ψexact → 1.

Numerical solutions to equations (17.27), allowing for anisotropic scattering, variable properties, and external irradiation, have been reported by Tsai and colleagues [43] using the S8 discrete ordinates method with the equal-weight ordinates of Fiveland [12]. The same method was used by Jones and Bayazitoglu ˘ [44,45] to determine the combined effects of conduction and radiation through a spherical shell.

Concentric Cylinders The analysis for two concentric cylinders follows along similar lines. Again we consider an absorbing, emitting, and scattering medium contained between two isothermal cylinders with radii R1 (temperature T1 , diffuse emittance ǫ1 ) and R2 (temperature T2 , emittance ǫ2 ), respectively. For this case the equation of transfer is given by equation (14.88), sin θ cos ψ

∂I sin θ sin ψ ∂I − + βI = βS, r ∂r ∂ψ

(17.30)

where polar angle θ is measured from the z-axis, and azimuthal angle ψ is measured from the local radial direction (cf. Fig. 14-6). S is the radiative source function and has been given by equation (17.23). Introducing the direction cosines ξ = sˆ · eˆ z = cos θ, µ = sˆ · eˆ r = sin θ cos ψ, and η = sˆ · eˆ ψc = sin θ sin ψ, we may rewrite equation (17.30) as µ ∂ 1 ∂ (rI) − (η I) + βI = βS. r ∂r r ∂ψ

(17.31)

For a one-dimensional cylindrical medium the symmetry conditions are not as straightforward as for slabs and spheres. Here we have I(r, θ, ψ) = I(r, π − θ, ψ) = I(r, θ, −ψ).

(17.32)

Therefore, the intensity is the same for positive and negative values of ξ, as well as for positive and negative values of η. Thus, we only need to consider positive values for ξi and ηi from Table 17.1, leading to Nc = N(N + 2)/4 different ordinates for the SN -approximation, with

17.4 ONE-DIMENSIONAL CONCENTRIC SPHERES AND CYLINDERS

555

quadrature weights w′′ = 4wi . Equation (17.31) may then be written in discrete ordinates form i as ( ) µi d 1 ∂ (rIi ) − + βIi = βSi , i = 1, 2, . . . , Nc . (17.33) (ηI) r dr r ∂ψ ψ=ψi As for the concentric spheres case the term in braces is approximated as ( ) αi+1/2 Ii+1/2 − αi−1/2 Ii−1/2 ∂ ≃ , i = 1, 2, . . . , Ni , ξi fixed. (ηI) w′′ ∂ψ ψ=ψi i

(17.34)

In this relation the subscript i + 1/2 implies “toward the next higher value of ψi , keeping ξi constant.” The value of Ni depends on the value of ξi . For example, for the S4 -approximation we have from Table 17.1 Ni = 4 for ξi = 0.2958759 (four different values for µi , two positive and two negative) and Ni = 2 for ξi = 0.9082483. In the case of concentric cylinders the recursion formula for α, by letting I = S = const in equation (17.31), is obtained as ∂η αi+1/2 − αi−1/2 = w′′ (17.35) i = 1, 2, . . . , Ni , ξi fixed. = w′′ i i µi , ∂ψ ψ=ψi Again, α1/2 = 0 since at that location ψ1/2 = 0 and, therefore, η = 0. Similarly, αNi +1/2 = 0 since ψNi +1/2 = π and η = 0. Finally, using linear averaging for the half-node intensities leads to µi

αi+1/2 Ii+1 − αi−1/2 Ii−1 dIi µi + Ii − + βIi = βSi , dr 2r 2rw′′ i

αi+1/2 = αi−1/2 + w′′ i µi ,

α1/2 = αN+1/2 = 0,

i = 1, 2, . . . , Nc ,

(17.36a)

i = 1, 2, . . . Ni , ξi fixed.

(17.36b)

Equation (17.36) is the set of equations for concentric cylinders, for the Nc = N(N+2)/4 unknown directional intensities Ii , and is equivalent to the set for concentric spheres, equation (17.27). The boundary conditions for cylinders and spheres are basically identical [equations (17.28)], except for some renumbering, as are the expressions for incident intensity and radiative heat flux [equations (17.29)], that is, r = R1 : r = R2 :

J1 = Ib1 − π J2 = Ib2 + Ii = π Ii =

Nc Nc 1−ǫ1 q1 , i = +1, +2, . . . , Nc (µi > 0), ǫ1 π 2 2 Nc 1−ǫ2 q2 , i = 1, 2, . . . , (µi < 0), ǫ2 π 2 Nc X w′′ G(r) = i Ii (r),

(17.37a) (17.37b) (17.37c)

i=1

q(r) = and

Nc X

w′′ i µi Ii (r),

(17.37d)

i=1

N c /2 X   w′′ q(R1 ) = q1 = ǫ1 Eb1 + i µi Ii ,

(17.37e)

Nc X   − q(R2 ) = q2 = ǫ2 Eb2 − w′′ i µi Ii .

(17.37f )

i=1 (µi 0)

An example of the use of the discrete ordinates method in a one-dimensional medium is the work of Krishnaprakas [46], who considered combined conduction and radiation in a gray, constant property medium with various scattering behaviors.

556

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

y si

AN N

ηi W AW

ξi

E

P

x AE

S FIGURE 17-2 A general two-dimensional control volume.

AS

17.5

MULTIDIMENSIONAL PROBLEMS

While the discrete ordinates method is readily extended to multidimensional configurations, the method results in a set of simultaneous first-order partial differential equations that generally must be solved numerically. As for one-dimensional geometries, the equation of transfer is slightly different whether a Cartesian, cylindrical, or spherical coordinate system is employed. We shall first describe the method for Cartesian coordinate systems, followed by a brief description of the differences for cylindrical and spherical geometries.

Enclosures Described by Cartesian Coordinates For Cartesian coordinates equation (17.4) becomes, using equation (17.14), ξi

∂Ii ∂Ii ∂Ii + ηi + µi + β Ii = β Si , ∂x ∂y ∂z

i = 1, 2, . . . , n,

(17.38)

where Si is again shorthand for the radiative source function n

Si = (1 − ω) Ib +

ω X w j Φij I j , 4π

i = 1, 2, . . . , n.

(17.39)

j=1

Equation (17.38) is subject to the boundary conditions in equation (17.5) along each surface. For example, for a surface parallel to the y-z-plane, with nˆ = ˆı and nˆ · sˆ j = sˆ j · ˆı = ξ j , we have for all i with ξi > 0 (n/2 boundary conditions) Ii = Jw /π = ǫw Ibw +

1 − ǫw X w j I j |ξ j |. π

(17.40)

ξ j 0, and = AE for ξi < 0), Axe is the x-direction face area through which the beam exits (= AE for ξi > 0, and = AW for ξi < 0), I yi i and I ye i are the corresponding y-direction face intensities, and so on. Then equation (17.44) may be generalized to Ipi =

βVSpi + |ξi | Ax Ixi i /γx + |ηi | A y I yi i /γ y

where

βV + |ξi | Axe /γx + |ηi | A ye /γ y

,

(17.48)

Ax = (1 − γx )Axe + γx Axi ,

(17.49a)

A y = (1 − γ y )A ye + γ y A yi .

(17.49b)

If all walls are black and in the absence of scattering, all unknown quantities can be calculated with a single pass, since all wall radiosities, Jw , and all internal sources Spi = (Ib ) pi are known a priori (if the temperature field is given or assumed). If the walls are reflecting and/or the medium is scattering, iterations are necessary. After a pass over all directions and over all finite volumes has been completed, the values for the wall radiosities and the radiative source terms are updated, and the procedure is repeated until convergence criteria are met. And finally, internal values of incident radiation and radiative heat flux are determined from equations (17.6) and (17.7), while heat fluxes at the walls may be calculated from equations (17.8). For highly reflecting walls (ǫw ≪ 1) and strongly scattering media (1 − ω ≪ 1), the discrete ordinates method will become extremely inefficient. As pointed out by Chai and coworkers [47], the number of iterations caused by scattering can be reduced by removing forward scattering from the phase function, and treating it as transmission. This can be done in equations (17.38) and (17.39) by defining a modified extinction coefficient and a modified source as σs wi Φii , 4π n ω X = (1 − ω)Ib + w j Φij I j , 4π

βmi = β − Smi

(17.50) i = 1, 2, ..., n.

(17.51)

j=1 j,i

This leads to faster convergence, particularly if the phase function has a strong forward peak (as is often the case for large particles; see also the discussion in Section 12.9). Spatial Differencing Schemes Expressing unknown intensities in terms of upstream values, such as defining INi and IEi in terms of Ipi , ISi , and IWi in equation (17.43) for ξi , ηi > 0, is known as spatial differencing (of intensity). Many different schemes have been proposed over the years. We give here only a brief description of the most basic and popular ones.

17.5 MULTIDIMENSIONAL PROBLEMS

559

Step Scheme The step scheme is the simplest differencing scheme, setting γx = γ y = 1, which leads to INi = Ipi and IEi = Ipi for ξi , ηi > 0, etc. Akin to a fully implicit finite difference of a first derivative, it has the largest truncation error of all methods, but is the only one that never produces unphysical results. Diamond Scheme This is the most popular differencing scheme, in which the interpolation factors are set to γx = γ y = 21 . However, already Carlson and Lathrop [4] noticed that this may lead to physically impossible negative intensities at the control volume faces (i.e., INi and IEi for ξi , ηi > 0, etc.). While they simply suggest setting negative intensities to zero and continuing computations, this may lead to oscillations and instability. Fiveland [13] showed that such negative intensities may be minimized (but not totally avoided) if finite volume dimensions are kept within ∆x
0 (i.e., a single direction for the S2 -approximation). For this direction xi = West and yi = South. To distinguish among the different nodes we attach the node number after the W, etc. For example, IW2,1 is the intensity at the West face of volume element 2, pointing into the direction of sˆ 1 .   i = 1 sˆ 1 = 0.5(ˆı + ˆ) : For all nodes   Ip j,1 = 15 Spj + 2IW j,1 + 2ISj,1 , IE j,1 = 2Ipj,1 − IW j,1 , IN j,1 = 2Ipj,1 − ISj,1 ,

j = 1, 2, 3, 4.

Starting at Element 1 we have IW1,1 = 0, IS1,1 = Ibw , and   Ip1,1 = 51 Sp1 + 2Ibw , Ip2,1

Ip3,1

Ip4,1

IE1,1 = 2Ip1,1 = IW2,1 , IN1,1 = 2Ip1,1 − Ibw = IS3,1 ;     = 15 Sp2 + 2IW2,1 + 2IS2,1 = 15 Sp2 + 4Ip1,1 + 2Ibw ,

IN2,1 = 2Ip2,1 − Ibw = IS4,1 ;     = 15 Sp3 + 2IS3,1 = 15 Sp3 + 4Ip1,1 − 2Ibw , IE3,1 = 2Ip3,1 = IW4,1 ;   = 51 Sp4 + 2IW4,1 + 2IS4,1   = 51 Sp4 + 4Ip3,1 + 4Ip2,1 − 2Ibw .

  i = 2 sˆ 2 = 0.5(−ˆı + ˆ) : In a problem without symmetry we would start in the lower right corner, scanning again over all elements. However, in this problem we can determine the intensities right away through symmetry, as Ip1,2 = Ip2,1 , Ip2,2 = Ip1,1 , Ip3,2 = Ip4,1 , Ip4,2 = Ip3,1 .

562

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

  i = 3 sˆ 3 = −0.5(ˆı + ˆ) : Starting in the upper right corner, we have, for all nodes,   Ip j,3 = 51 Spj + 2IEj,3 + 2IN j,3 , IW j,3 = 2Ipj,3 − IWE,3 , IS j,3 = 2Ipj,3 − IN j,3 . Starting at Element 4 with IE4,3 = IN4,3 = 0, we find Ip4,3 = 15 Sp4 ,

Ip3,3

Ip2,3

Ip1,3 Also

IS4,3 = 2Ip4,3 = IN2,3 , IW4,3 = 2Ip4,3 = IE3,3 ;     = 51 Sp3 + 2IE3,3 = 15 Sp3 + 4Ip4,3 ,

IS3,3 = 2Ip3,3 = IN1,3 ;     = 15 Sp2 + 2IN2,3 = 15 Sp2 + 4Ip4,3 ,

IW2,3 = 2Ip2,3 = IE1,3 ;   = 15 Sp1 + 2IE1,3 + 2IN1,3 =

1 5

 Sp1 + 4Ip2,3 + 4Ip3,3 .



IS1,3 = 2Ip1,3 − IN1,3 = 2(Ip1,3 − Ip3,3 ), IS2,3 = 2Ip2,3 − IN2,3 = 2(Ip2,3 − Ip4,3 ), which will be needed later for the calculation of wall heat fluxes from equation (17.8).   i = 4 sˆ 4 = 0.5(ˆı − ˆ) : Again, by symmetry it follows immediately that Ip1,4 = Ip2,3 , Ip2,4 = Ip1,3 , Ip3,4 = Ip4,3 , Ip4,4 = Ip3,3 ,

and also IS1,4 = IS2,3 , IS2,4 = IS1,3 . Summarizing, we have Ip1,1 = Ip2,2 =

1 5

Ip2,1 = Ip1,2 =

1 5

Ip3,1 = Ip4,2 =

1 5

Ip4,1 = Ip3,2 =

1 5

Ip1,3 = Ip2,4 =

1 5

Ip2,3 = Ip1,4 =

1 5

Ip3,3 = Ip4,4 =

1 5



 Sp1 + 2Ibw ,

 Sp2 + 4Ip1,1 + 2Ibw ,   Sp3 + 4Ip1,1 − 2Ibw ,   Sp4 + 4Ip3,1 + 4Ip2,1 − 2Ibw ,   Sp1 + 4Ip2,3 + 4Ip3,3 ,   Sp2 + 4Ip4,3 ,   Sp3 + 4Ip4,3 , 

Ip4,3 = Ip3,4 = 51 Sp4 ,

IS1,3 = IS2,4 = 2(Ip1,3 − Ip3,3 ), IS2,3 = IS1,4 = 2(Ip2,3 − Ip4,3 ). The source functions are readily evaluated from equation (17.7) and symmetry as Sp1 = Sp2 = 14 (Ip1,1 + Ip1,2 + Ip1,3 + Ip1,4 ), Sp3 = Sp4 = 41 (Ip3,1 + Ip3,2 + Ip3,3 + Ip3,4 ). Since the equations are linear, one could substitute the relations for the Spj into the above equations and solve for the unknown Ip j,i by matrix inversion. However, in general one would have many more, and much more complicated, equations, which are best solved by iteration. We start by setting all Spj = 0,

17.5 MULTIDIMENSIONAL PROBLEMS

563

TABLE 17.4

Nodal intensities of Example 17.4 as a function of iteration, normalized by I bw . Ip3,1

Ip4,1

Ip1,3

Ip2,3

Ip3,3

Ip4,3

Sp1

Sp3

Diamond scheme 1 0.4000 0.7200 2 0.4560 0.8208 3 0.4729 0.8513 ≥9 0.4815 0.8667

0.0000* 0.0000* 0.0000* 0.0037

0.1760 0.2654 0.2955 0.3148

0.0000 0.1191 0.1615 0.1852

0.0000 0.0630 0.0846 0.0963

0.0000 0.0158 0.0261 0.0333

0.0000 0.0088 0.0145 0.0185

0.2800 0.3647 0.3926 0.4074

0.0440 0.0725 0.0840 0.0926

Step scheme 1 0.3333 2 0.3981 3 0.4252 ≥ 10 0.4459

0.1111 0.1574 0.1808 0.2027

0.1852 0.2541 0.2883 0.3198

0.0000 0.1001 0.1442 0.1802

0.0000 0.0730 0.1049 0.1306

0.0000 0.0329 0.0521 0.0721

0.0000 0.0247 0.0391 0.0541

0.1944 0.2755 0.3103 0.3378

0.0741 0.1173 0.1401 0.1622

Iter.



Ip1,1

Ip2,1

0.4444 0.5309 0.5669 0.5946

negative values set to zero finding values for the Ip j,i , updating the Sp j , reevaluating the Ipj,i , and so on, until convergence has been reached. The changing values of the intensity (normalized with Ibw ) as a function of iteration are given in Table 17.4. Values accurate to ≃ 5% are reached after three iterations, and fully converged values (to four significant digits) are obtained after nine iterations. The converged intensities are used to determine the net radiative heat flux from the bottom wall at x = L/4 and x = 3L/4. From equation (17.8) we have q(x = 0.25 L) = q(x = 0.75 L) = πIbw −

4 X

wi IS1,i |ηi | = πIbw −

i=3

or Ψ=

π (IS1,3 + IS1,4 ), 2

Ip1,3 − Ip3,3 + Ip2,3 − Ip4,3 q0.25L q0.75L = =1− = 0.7704. Ebw Ebw Ibw

For comparison we will work this example also with the simpler, but more stable step differencing scheme, i.e., γx = γ y = 1. Then we obtain from equations (17.47) and (17.48) Ipi =

1 (Sp + Ixi i + I yi i ), 3

Ixe i = I ye i = Ipi .

Then, following the same procedure we obtain (a little more easily) Ip1,1 Ip2,1 Ip3,1 Ip4,1

= = = =

1 3 (Sp1 1 3 (Sp2 1 3 (Sp3 1 3 (Sp4

+ 0 + Ibw ) = Ip2,2 + Ip1,1 + Ibw ) = Ip1,2 + 0 + Ip1,1 ) = Ip4,2 + Ip3,1 + Ip2,1 ) = Ip3,2

Ip4,3 Ip3,3 Ip2,3 Ip1,3

= = = =

1 3 (Sp4 1 3 (Sp3 1 3 (Sp2 1 3 (Sp1

+ 0 + 0) = Ip3,4 + Ip4,3 + 0) = Ip4,4 + 0 + Ip4,3 ) = Ip1,4 + Ip2,3 + Ip3,3 ) = Ip2,4

The formulas for the source functions and heat fluxes remain the same, but the nondimensional flux becomes, after substituting for IS = Ip , Ψ=1−

Ip1,3 + Ip1,4 2Ibw

= 0.844.

The iteration results for the step scheme are also included in Table 17.4. Results for the nondimensional heat flux from both schemes are shown in Fig. 17-5 along with exact results reported by Razzaque and coworkers [59], and with S2 - and S4 -calculations of Truelove [15] (for a much finer mesh). Truelove’s results demonstrate the importance of good ordinate sets, at least for low-order approximations: The S′2 and S′4 results were obtained with sets that do not obey the half-moment condition of equation (17.13) (as used by Fiveland [11] in a first investigation of rectangular enclosures), while the S2 and S4 results were

564

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

Nondimensional radiative heat flux Ψ = qy /σ T w4

1.0 S´2 0.9 S´4 0.8 S2 S4

0.7

Example 16.4 Step scheme Diamond scheme Exact [48] S2 , S4 Truelove [15] S´2 , S´4 Fiveland [11]

0.6

0.5

0.4

0

0.5 Nondimensional distance, x/L

1

FIGURE 17-5 Nondimensional heat flux along the bottom wall of the square enclosure of Example 17.4.

obtained with the sets given in Table 17.1 (using the nonsymmetric ordinates for S2 ). Not surprisingly, the diamond scheme (similar to a Crank–Nicolson finite differencing scheme) is more accurate than the step scheme (similar to fully implicit finite differencing). The step scheme shows a smoother distribution for the Ip and Sp and is always stable. The diamond scheme, on the other hand, gives nonphysical negative intensities for Ip3,1 during the first few iterations, which were set to zero. Ray effects, while present, are not apparent in this example because of the large cells used to allow for hand calculations. They become very noticeable when repeating this example with a fine mesh, as will be done in the context of the finite volume method, the subject of the next section (see Example 17.7).

In his early calculations Fiveland [11] applied the S2 -, S4 -, and S6 -approximations to purely scattering rectangular media (ω = 1), and to isothermal, nonscattering media bounded by cold black walls. Truelove [15] repeated some of those results to demonstrate the importance of good ordinate sets, and gave some new results for radiative equilibrium in a square enclosure. Jamaluddin and Smith [50] applied the S4 -approximation to a rectangular, nonscattering enclosure with known temperature profile. Kim and Lee investigated the effects of strongly anisotropic scattering, using high-order approximations (up to S16 ) [60], and the effects of collimated irradiation [61]. Finally, combined conduction and radiation in a linear-anisotropically scattering rectangular enclosure has been studied by Baek and Kim [62]. They also investigated the influence of radiation in compressible, turbulent flow over a backward facing step, using the same method (gray constant properties, here without scattering) [63]. Finally, radiation in two-dimensional packed beds, together with conduction and convection, was studied by Lu and coworkers [36]. While they also assumed gray properties, they allowed them to vary locally; for scattering, they used the large diffuse sphere phase function, equation (12.85). Other applications of the two-dimensional Cartesian form of the discrete ordinates method can be found in [64–67], all dealing with combined-mode heat transfer. Particularly noteworthy here is the study of Selc¸uk and Kayakol [68], who compared the performance of the S4 method with that of the related discrete transfer method [69] (see p. 575), finding the methods to have comparable accuracy, while the S4 solution required three orders of magnitude less computer time. Three-Dimensional Problems The method can be extended immediately to three-dimensional geometries by giving the control volume Front and Back surfaces, AF and AB , and rewriting equation (17.48) as Ipi =

βVSpi + |ξi | Ax Ixi i /γx + |ηi | A y I yi i /γ y + |µi | Az Izi i /γz βV + |ξi | Axe /γx + |ηi | A ye /γ y + |µi | Aze /γz

,

(17.57)

17.5 MULTIDIMENSIONAL PROBLEMS

where

565

Ax = (1 − γx ) Axe + γx Axi ,

(17.58a)

A y = (1 − γ y ) A ye + γ y A yi ,

(17.58b)

Az = (1 − γz ) Aze + γz Azi ,

(17.58c)

and the sub-subscript i again denotes the face where the beam enters, and e where it exits, as explained in the context of equation (17.48). A three-dimensional Cartesian enclosure has eight corners, from each of which 81 N(N + 2) directions must be traced (covering one octant of directions), for a total of N(N + 2) ordinates. Some such calculations have been performed by Jamaluddin and Smith [70] (nonscattering medium with prescribed temperature), and by Fiveland [13] and Truelove [16] (both studying the idealized furnace of Menguc ¨ ¸ and Viskanta [71], considering a linear-anisotropically scattering medium with internal heat generation at radiative equilibrium), by Park and Yoon [72] (combined conduction and radiation, using inverse analysis to determine constant, gray values for κ and σs , for given temperature profiles), and Lacroix and colleagues (radiation in a plasma formed by the laser welding process) [73], and others. Also, Gonc¸alves and Coelho [74] have shown how the discrete ordinates method can be implemented on parallel computers. Fiveland and Jessee [75] discussed several acceleration schemes for optically thick geometries, for which the discrete ordinate method is known to converge very slowly (or not at all). An extensive review up to the year 2000 of the discrete ordinate method from a computer science point of view, emphasizing convergence rates and multigrid and parallel implementations, has been given by Balsara [76].

Multidimensional Non-Cartesian Geometries A few investigations have dealt with the application of the discrete ordinates method to two- and three-dimensional cylindrical enclosures, and more recently the method has also been applied to irregular geometries. A two-dimensional axisymmetric enclosure was first considered by Fiveland [10], who calculated radiative heat flux rates for a cylindrical furnace with known temperature profile. A very similar problem was treated by Jamaluddin and Smith [70] who, a little later, also addressed the case of a three-dimensional cylindrical furnace [77, 78]. Kim and Baek [79] investigated fully developed nonaxisymmetric pipe flow with a gray, constant property, absorbing/emitting and isotropically scattering medium. Kaplan and coworkers [80] modeled an unsteady ethylene diffusion flame, treating soot and combustion gasses as gray and nonscattering, but with spatial variation. Ramamurthy and colleagues [81] investigated reacting, radiating flow in radiant tubes, using a more sophisticated model for the spectral behavior of the combustion gases, and a molten glass jet was studied by Song and coworkers [82]. All of these used the S4 -method in two-dimensional, cylindrical geometries, although the S14 scheme was used by Jendoubi et al. [83] to evaluate different scattering behaviors. Complex three-dimensional geometries are difficult to treat with the standard discrete ordinates method. This was attempted by Howell and Beckner [84], who used “embedded boundaries” to simulate irregular surfaces, and by Adams and Smith [85], who modeled a complex furnace. Their results clearly demonstrate the ray effect: using a coarse ordinate mesh (up to S8 ) together with a very fine spatial mesh, their calculated radiative fluxes undergo very strong unphysical oscillations. Sakami and colleagues [86, 87] showed how spatial differentiation can be done across unstructured, triangular, two-dimensional meshes. They trace back each ray through each cell, integrating over the entire cell using a finite-element Galerkin scheme. A somewhat similar approach was suggested by Cheong and Song [88–91]. Through careful spatial differencing, they showed how the standard discrete ordinates method can be applied to unstructured grids and irregular geometries. This method was also further refined by Seo and Kim [92]. Discretization of equation (17.4) for, both, rectangular Cartesian and irregular structured or unstructured grids may also be carried out using the finite volume approach of Patankar [93],

566

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

z si n2 n1

2

P n3

1

3 4

n4

R nq Q

y x

(a) (b) FIGURE 17-6 Spatial and directional discretization in a two-dimensional domain: (a) finite volume elements with nodes at the centers of the elements, (b) typical subdivision of all directions into solid angle elements.

as is done in the “finite volume method” (for radiation) described in the following section. This can then be combined with any spatial differencing scheme [18]. Alternatively, equation (17.4) can be solved using the finite element approach (e.g., [94,95]), meshless methods (e.g., [96]), etc.

17.6

THE FINITE VOLUME METHOD

The discrete ordinates method, in its standard form, suffers from a number of serious drawbacks, such as false scattering and ray effects. The fact that half-range moments, equation (17.13), must be satisfied for the accurate evaluation of surface fluxes makes it very difficult to apply the method to irregular geometries. Perhaps the most serious drawback of the method is that it does not ensure conservation of radiative energy. This is a result of the fact that the standard discrete ordinates method uses simple quadrature for angular discretization, even though generally a finite volume approach is used for spatial discretization, as outlined in the previous sections. Thus, it was a logical step in the evolution of the method to move to a fully finite volume approach, in space as well as in direction. This was first proposed by Briggs and colleagues [97] in the field of neutron transport. The first formulations for radiative heat transfer were given by Raithby and coworkers [98–101]. Slightly different schemes have been proposed by Chai and colleagues [102–104]. A good review has been given by Raithby [105]. The finite volume method uses exact integration to evaluate solid angle integrals, which is analogous to the evaluation of areas and volumes in the finite volume approach. The method is fully conservative: exact satisfaction of all full- and half-moments can be achieved for arbitrary geometries, and there is no loss of radiative energy. The angular grid can be adapted to each special situation, such as collimated irradiation [102].

Two-Dimensional Formulation As in the development of the standard discrete ordinates method, for clarity we will limit our development to two-dimensional geometries; extension to three dimensions is straightforward. However, in view of the finite volume method’s ability to easily accommodate irregular geometries, we will consider a general two-dimensional domain with irregularly-shaped finite volumes as depicted in Fig. 17-6a. The quadrilateral volumes follow “practice B” of Patankar [93], the ones used by Chai and coworkers [104] (i.e., nodes are placed at the center of each finite volume). However, other finite volume schemes may be used, as well. Similar to the spatial domain, the

17.6 THE FINITE VOLUME METHOD

567

directional domain of 4π steradians is broken up into n solid angles Ω i (i = 1, 2, . . . , n), which exactly fill the directional domain without overlap. This can be done in many ways, and without restrictions, but it is usually easiest to define the Ω i as the areas on a unit sphere defined by lines of longitude and latitude, as shown in Fig. 17-6b. The starting point for the analysis is again equation (17.1) together with its boundary condition, equation (17.2). For each volume element, such as the one surrounding point P in Fig. 17-6a, equation (17.1) is integrated over the volume element and over each of the solid angle elements Ω i . The volume integration over ∂I/∂s is the same as in equation (17.41), but is now for an element of arbitrary shape, for which we obtain Z Z Z Z ∂I sˆ · ∇I dV = ∇ · (ˆsI) dV = Iˆs · nˆ dΓ, (17.59) dV = V V Γ V ∂s where Γ is the surface of the volume element consisting of four (two-dimensional) or six (threedimensional) faces and nˆ is the outward surface normal as indicated in the figure. In equation (17.59) the unit direction vector sˆ can be moved inside the spatial ∇-operator since directional coordinates are independent from spatial coordinates. Conversion to a surface integral in the last step follows from the divergence theorem [106]. Thus, integrating equation (17.1) over the volume element V and solid angle Ω i leads to Z Z Z Z Z Z Z σs Φ(ˆs′ , sˆ )I(ˆs′ ) dΩ′ dV dΩ. (17.60) Iˆs · nˆ dΓ dΩ = (κIb − βI) dV dΩ + 4π Ωi Γ Ωi V Ωi V 4π In the simplest implementation of the finite volume method it is assumed, for the term on the left-hand side, that the intensity is constant across each face of the element as well as over the solid angle Ω i . Similarly, it is assumed for the volume integrals that values are constant throughout and equal to the value at point P. Equation (17.60) then becomes X Iki (si · nˆ k )Ak = βp (Spi − Ipi )VΩ i , (17.61a) k

Spi = (1 − ωp )Ibp +

n ωp X



¯ ij , Ip j Φ

(17.61b)

j=1

Z Z 1 ¯ ¯ s′ , sˆ ) dΩ′ dΩ, Φi j = Φ(ˆ Ω i Ωi Ωj Z si = sˆ dΩ,

(17.61c) (17.61d)

Ωi

where subscripts k and p imply evaluation at the center of the volume’s faces Ak (as indicated by an × in Fig. 17-6a) and element center P, respectively; subscript i denotes a value associated with solid angle Ω i . The radiative source Spi is similar to the one in equation (17.39), but now ¯ ij . Finally, the si is a vector (of varying length has an analytically averaged phase function Φ indicative of the size of Ω i ) pointing into an average direction within solid angle element Ω i . Of course, the forward-scattering term in Spi can, and should, be removed as was done in the standard discrete ordinate formulation [cf. equation (17.50)]. What remains to be done is to relate the intensities at the face centers, Iki , to those at volume centers, Ipi . There are many different ways to do this. Raithby and coworkers [98], in particular, have developed schemes of high accuracy. However, such sophisticated schemes require substantial analytical and computational overhead. In light of the stability considerations discussed by Chai and colleagues [48], the simple step scheme has generally been preferred. Therefore, similar to equation (17.43) with γ = 1, we assume that for intensities leaving control volume P (i.e., for si · nˆ k > 0) Iki = Ipi . All incoming intensities (si · nˆ k < 0) are assigned the value of the

568

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

element center from which they came. Substituting Iki = Ipi for si · nˆ k > 0 into equation (17.61) then leads to the final expression P βp Spi VΩ i + Iki |si · nˆ k | Ak k,in

Ipi =

P

βp VΩi +

(si · nˆ k )Ak

(17.62)

,

k,out

where the “in” and “out” on the summation signs denote summation over volume faces with incoming (si · nˆ k < 0) or outgoing (si · nˆ k > 0) intensities, only. Lately, the CLAM scheme has become popular (e.g., [58, 107]), requiring an iterative approach as discussed in Section 17.5. The boundary conditions are developed in a similar manner, except that—for diffusely emitting and reflecting surfaces—it is advantageous to make an energy balance to ensure conservation of radiative energy for surfaces not lined up with the solid angles Ω i . Multiplying equation (17.2) by nˆ · sˆ and integrating over all outgoing directions gives an expression for surface radiosity as Z Z Z Jw = I nˆ · sˆ dΩ = ǫw Ibw nˆ · sˆ dΩ + (1 − ǫw ) I |nˆ · sˆ | dΩ. (17.63) n·ˆ ˆ s>0

n·ˆ ˆ s>0

n·ˆ ˆ s 0), nˆ q is the unit surface normal at Q pointing out of the boundary (but into the adjacent volume element R). The Iqi are intensities leaving the adjacent volume element R going into boundary element Q. Using the step scheme we can set Iqi = IRi for si · nˆ q < 0. Once all internal intensities Ipi and boundary intensities Iqi have been determined, internal values for incident radiation and radiative flux are found from X X Gp = Ipi Ω i , qp = Ipi si , (17.66) i

i

while wall fluxes are given by   X     X  qq = ǫq Ebq − Hq = ǫq Ibq si · nˆ q − Iqi si · nˆ q  . i,out

(17.67)

i,in

Note that, for arbitrarily oriented surfaces, the sums of |si · nˆ q | may not add up to π (for either incoming or outgoing directions); therefore, for consistency, the finite volume rendition for Ebq given in the right-most part of equation (17.67) is preferred. Example 17.5. Repeat Example 17.1 for the finite volume method, using the upper and lower hemispheres as solid angle ranges. Solution The governing equation is, as before, µ

ω dI + I = (1 − ω) Ib + (G + A1 qµ). dτ 4π

If we want to apply the finite volume method in a similar fashion as in Example 17.1, i.e., to obtain a differential equation for each solid angle range, then we need only integrate the governing equation

17.6 THE FINITE VOLUME METHOD

569

over these solid angles, not over volume. Assuming a constant intensity I+ over the upper hemisphere, and I− over the lower one, we obtain with Ib = G/4π # Z 2π Z 1 " 1 dI+ + I+ = (G + A1 ωµq) dµ dψ, µ upper hemisphere: dτ 4π 0 0 # Z 2π Z 0 " − 1 dI + I− = (G + A1 ωµq) dµ dψ, µ lower hemisphere: dτ 4π 0 −1 or 1 1 dI+ + 2πI+ = G + A1 ωq, π dτ 2 4 1 1 dI− − + 2πI = G − A1 ωq. −π dτ 2 4 From the definitions for heat flux and incident radiation we have again ( ) Z 2π Z +1 ( ) ) ( G 1 2π (I+ + I− ) = , I dµ dψ = q π (I+ − I− ) µ 0 −1 as in Example 17.1. Thus adding and subtracting the equations for the upper and lower hemispheres we obtain dq +G=G dτ 1 1 dG + 2q = A1 ωq 2 dτ 2

or or

dq = 0, dτ dG = −(4 − A1 ω) q. dτ

For the boundary conditions, equation (17.65), we need to first calculate the si : s1 = s2 =

Z Z

2π 0 2π 0

Z Z

π/2

ˆ sin θ dθ dψ = πk, ˆ (sin θ cos ψˆı + sin θ sin ψˆ + cos θk)

0 π

ˆ sin θ dθ dψ = −πk. ˆ (sin θ cos ψˆı + sin θ sin ψˆ + cos θk)

π/2

For the bottom boundary we have nˆ = kˆ and s1 · nˆ = −s2 · nˆ = π, so that at τ=0:

I+ = ǫ1 Ib1 + (1 − ǫ1 )I− .

Subtracting (1 − ǫ1 )I+ , using the definition for q, and dividing by ǫ1 leads to τ=0:

I+ = J1 /π = Ib1 −

1 − ǫ1 q, ǫ1 π

which is, of course, the same as for Example 17.1 (and for any diffuse surface). Similarly, at the top wall τ = τL :

I− = J2 /π.

The solution is then found immediately from Example (17.1) (setting 1/µ21 = 4) as Ψ=

q 1  ,  = J1 − J2 1 + 1 + A41 ω τL

which is the same as the answer from the nonsymmetric S2 -approximation. More importantly, the analysis in this example shows that the Schuster–Schwarzschild (or two-flux) approximation is simply the lowest-level finite volume method. Example 17.6. Repeat Example 17.4 using the finite volume method, by splitting the total solid angle into four equal ranges. Solution As in Example 17.4 we will put the z-axis perpendicular to the paper in Fig. 17-4, from which the polar angle θ is measured. Because of the two-dimensionality, it is best to assign each Ω i the entire range of

570

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

polar angles, and a quarter of the azimuthal range. Thus, breaking up by quadrant we choose 0≤ψ
0 (pointing out of volume element) may actually overlap into the element. Similarly, it is unlikely to have solid angle boundaries lined up perfectly with the solid boundaries everywhere. They improved the accuracy of the method through pixelation, i.e., by breaking up Ω i into smaller pieces, to determine overlap fractions. Also, noting that the standard line iteration method leads to unacceptably slow convergence in optically thick situations, they introduced a new scheme, which updates all directional intensities within a cell simultaneously, leading to convergence rates essentially independent of optical thickness [111,112]. Hassanzadeh and coworkers [113] also developed a method to accelerate convergence for optically thick media by carrying out iterations in terms of mean intensity, G/4π, as opposed to all directional intensities. Several other improvements to the method have been suggested. Kim and Huh [114] noted that most researchers broke up the total solid angle of 4π into N × N

572

17 THE METHOD OF DISCRETE ORDINATES (SN -APPROXIMATION)

segments of equal polar angles θ and azimuthal angles ψ. This makes the Ωi very small near the poles (θ = 0, π), and large near the equator (θ = π/2). They suggest that, for n different polar angles θi , one should pick fewer azimuthal angles near the poles, namely a distribution of 4, 8, ..., 2n − 4, 2n, 2n, 2n − 4, ..., 8, 4 with growing θi . This results in n(n + 2) different solid angles (equal to the number of ordinates in the standard Sn scheme), with all Ωi being roughly equally large. Finally, Liu and coworkers [115] have shown how the finite volume method with unstructured grids can be parallelized using domain decomposition. The method has also been employed in a number of combined heat transfer problems [116, 117] and is included in several important commercial CFD codes, such as FLUENT [118].

Comparison with Standard Discrete Ordinates Method The radiative transfer equation (RTE), equation (17.1), is a five-dimensional integro-differential equation, with three spatial and two directional coordinates. For a numerical solution both, spatial and directional dependencies must be discretized. Various methods of discretization are available, such as finite differences, finite volumes, finite elements, etc., and one or the other may be applied for the spatial and for the directional discretization. Originally, for the standard discrete ordinates method finite differences were used for both. As the method has evolved to more general geometries, different spatial discretization schemes have been employed, but directional discretization has remained in finite difference form. In contrast, in the original form of the finite volume method finite volumes were used for both spatial and directional discretization. However, recently other spatial discretization schemes have also been used, e.g., Cui and Li [119] and Grissa et al. [120] employed the finite element method. Therefore, the one defining difference between the standard discrete ordinates method and the finite volume method is the fact that the standard discrete ordinates method uses finite differences for directional discretization, while the finite volume method employs finite volumes. Liu and coworkers [121] have expressed the RTE in general boundary-fitted coordinates [122], and applied both the standard discrete ordinates method and the finite volume method to a number of two- and three-dimensional problems. They found both methods to require similar amounts of CPU time, while the finite volume method was always slightly more accurate. Similar conclusions were drawn by Fiveland and Jessee [123] and by Kim and Huh [124], noting that the finite volume (FV) method outperforms standard discrete ordinates particularly in optically thin media, since it is less sensitive to ray effects. Coelho and coworkers [125] compared the performance of the FV method with that of the discrete transfer method [69] and, like Selc¸uk and Kayakol [68], found the FV method to be much more economical. Major advantages of the finite volume method are greater freedom to select ordinates, and the fact that the FV method conserves radiative energy. In addition, treatment of complex enclosures comes more natural to the FV method. For example, Baek and colleagues [126–128] used boundaryfitted coordinates to investigate radiation in several three-dimensional enclosures with gray, constant-property media.

17.7 THE MODIFIED DISCRETE ORDINATES METHOD It was noted in Section 17.5 that the discrete ordinates method (in its standard or finite volume form) can suffer from ray effects, if directional discretization is coarse compared to spatial discretization, and if the medium contains small sources of strong emission (from walls or from within the medium). This prompted Ramankutty and Crosbie [129, 130] to separate boundary emission from medium emission, as is done in the modified differential approximation of Section 16.8, i.e., letting I(r, sˆ ) = Iw (r, sˆ ) + Im (r, sˆ ). (17.68)

17.8 EVEN-PARITY FORMULATION

573

The wall-related intensity field can be solved by any standard method as outlined in Section 17.5, while the RTE and boundary conditions for Im become Z Z dIm σs σs ′ ′ ′ = κIb − βIm (ˆs) + (17.69) Im (ˆs ) Φ(ˆs , sˆ ) dΩ + Iw (ˆs′ ) Φ(ˆs′ , sˆ ) dΩ′ , ds 4π 4π 4π 4π Z 1−ǫ Im (rw , sˆ ) = Im sˆ ′ ) |nˆ · sˆ ′ | dΩ′ . (17.70) ′ π n·ˆ ˆ s 0. (17.76) ′ π ˆ s J). From the above example it can be seen that it is generally necessary to apply stabilizing methods even to the solution of linear inverse problems, such as the techniques discussed below. The various techniques to solve ill-conditioned problems may be loosely collected under the titles regularization, gradient-based optimization, and metaheuristics, and some of the most common methods will be briefly discussed below. More detail can be found in books on the subject, e.g., Hansen [30], as well as several review articles [33, 34]

23.3

REGULARIZATION

We saw in the previous section that an ill-conditioned matrix has a large condition number, i.e., some of the singular values are very small, causing the solution to become unstable. Decreasing the condition number of a matrix A by modifying it (or its inverse) is known as regularization. We will briefly describe here the simple truncated singular value decomposition and the perhaps most popular Tikhonov regularization methods.

Truncated Singular Value Decomposition (TSVD) The simplest form of regularization consists of simply omitting parts of the inverse of A corresponding to the (offending) smallest singular values. This is justified by the fact that the higher terms in the series correspond to “high frequency” components, and often have less physical significance. Our prior knowledge (or desire) of a smooth solution is used as justification for

786

23 INVERSE RADIATIVE HEAT TRANSFER 4

3

w/h = 5, ∋ = 0.5; N = 20

3.5

2.5

7

3

2

3

1.5 4 T top

4 Tact4 , Tdes

2.5 2

1

5

1.5

K=1 7

1

K=1

0.5

3

0

5

0.5 0

0

0.2

0.4

0.6

x/w

0.8

1

FIGURE 23-4 Predicted top surface temperatures and recovery of desired bottom surface temperatures for Example 23.2.

truncation [30]. The matrix A, as given in the normal equation set (23.10), is first singular value decomposed as given by equation (23.16b). The full solution to equation (23.10) would then be obtained from equation (23.18a). Eliminating the largest values of 1/Skk is achieved by keeping only the first K terms in equation (23.18a) (i.e., dropping terms with k > K, thereby decreasing the condition number) N K X Vik X p∗i = U jk b j , i = 1, . . . , N, (23.20) Skk j=1

k=1

where p is the regularized solution. The proper value for K must be determined through external, often subjective criteria. Large values of K force the result vector i (e.g., the achieved nondimensional temperature of the bottom plate in Example 23.1) to more closely follow the prescribed data vector y (desired temperature), but may result in strongly oscillatory and/or unphysical parameter vectors p (power setting on heater plate). Small values of K, on the other hand, lead to a smooth variation for p, but the result vector i may depart substantially from the desired value y. ∗

Example 23.2. Repeat the control problem Example 23.1 using truncated singular value decomposition (TSVD). Solution The solution proceeds exactly as in Example 23.1, but the series in equation (23.18a) is truncated to give nondimensional heater temperatures as Θ2i =

K N X Vik X U jk b j , Skk j=1

k=1

and the resulting design surface temperatures are found from equation (23.9) Θ1i =

N X

Xij Θ2j .

j=1

Figure 23-4 shows the results, again for w/h = 5, ǫ = 0.5, and N = 20 strips on each plate, for several odd values of K (even values produce essentially identical results as the next lower K because of symmetry). It is observed that retaining a single singular value (K = 1) results in a very smooth heater setting, and also a smooth design surface temperature (but departing substantially from the desired value of “1”). Larger values of K bring the design plate temperatures closer to the desired value (albeit with slight oscillations), but at a cost of oscillatory heater settings. Values of K > 7 result in some strips having negative emissive power (cooling), which would be undesirable at best.

23.3 REGULARIZATION

787

Tikhonov Regularization Most regularization methods transform an ill-posed inverse problem into a well-behaved one by adding auxiliary information based on desired or assumed solution characteristics [34, 38]: F = (i − y) · W·(i − y) + λ2 Ω(p),

(23.21)

where Ω(p) is an arbitrary (positive) function and λ is the (positive) regularization parameter. One of the earliest and most popular examples is Tikhonov regularization [2], employing Ω = p · L · p, where L is an operator. In the simplest 0th order discrete Tikhonov regularization we have L = δ and Ω = p · p. Thus, equation (23.10) is changed to (A + λδ) · p = b,

with A = XT · W · X, b = XT · W · y,

(23.22)

where δ is again an Nth order unity tensor. Many different and higher order versions of Tikhonov’s regularization exist, and the reader is referred to [32, 38]. The regularization parameter determines the smoothness of the solution: a small value of λ implies little regularization, while a large λ prioritizes some presumed information, which in the case of standard Tikhonov forces the solution vector toward zero. Several schemes exist to find an optimal value of λ. Numerical Recipes [37] suggests a starting value for λ of λ ≃ Tr(A)/N,

(23.23)

where Tr is the trace of the matrix (sum of the N diagonal elements), giving both parts in the minimization equal weights. An optimum value for λ is then found by trial and error. More sophisticated schemes include construction of a so-called L-curve, which leads to a semiquantitative determination of λ [30, 39]. Example 23.3. Repeat Example 23.1 using 0th order discrete Tikhonov regularization. Solution As in the previous example we calculate A = XT · X and b = X · 1. Before inverting A we modify the matrix to A∗ = A + λδ, or A∗ij = Aij + λδij , i.e., all diagonal elements are incremented by λ, which is evaluated as λ=C

N 1 X Aii , N i−1

where C is a constant whose optimal value is to be found by trial and error. Heater emissive powers Θ2 and design surface emissive powers Θ1 are then determined from Θ2i =

N  X

A∗−1

j=1

Θ1i =

N X



ij

bj,

bj =

N X

Xk j ,

k=1

Xi j Θ2j .

j=1

Results for Tikhonov regularization are shown in Fig. 23-5, again for w/h = 5, ǫ = 0.5, and N = 20 strips on each plate, for five fractional values of C = 2−(5−k) , with larger C implying more regularization. It is seen from the figure that the Numerical Recipes’ suggested value (C = 1) gives a reasonable (perhaps slightly over-regularized) solution with smoothly varying heater values, but with design surface temperatures dropping near the edges of the plate. Smaller values of the regularization parameter lead to somewhat better design surface temperatures, at the cost of stronger heater surface variations. In general, it appears that Tikhonov regularization gives better results than TSVD, at least for the present problem.

788

23 INVERSE RADIATIVE HEAT TRANSFER 1.3

3

1

1: λ=Tr(A)/16N 2: λ=Tr(A)/8N 3: λ=Tr(A)/4N 4: λ=Tr(A)/2N 5: λ=Tr(A)/N

2 3 1.2

4

W /H

= 5, ∋ = 0.5; N = 20 2

4 T top

4 Tact4 , Tdes

5 1.1

1

5 1

1

0.9

0

0

0.2

0.4

XxW

0.6

0.8

1

FIGURE 23-5 Predicted top surface temperatures and recovery of desired bottom surface temperatures for Example 23.3.

23.4

GRADIENT-BASED OPTIMIZATION

In optimization the objective function F, most often using least square norms as given by equations (23.2) or (23.3), is minimized in an iterative process. Iteration is always necessary for nonlinear problems, but may also be employed for linear ones to overcome ill-conditioning, which in optimization manifests itself in the form of a difficult objective function topography having a minimum (or several minima in nonlinear problems) surrounded by a long, shallow valley, as shown in Fig. 23-1b. Many different optimization schemes have been developed to minimize F. When F is continuously differentiable over the feasible region of p, it is generally best to use analytically defined search directions, with gradient-based methods being used most often [40]. In all schemes, during each iteration a step of appropriate size is taken along a direction of descent, which is based on the local curvature of the objective function at the previous iteration. Thus, after the kth iteration a new solution vector is found from pk+1 = pk + βk dk ,

(23.24)

where β is the search step size, and d is the direction of descent. The main difference between gradient minimization techniques is how the search direction is chosen, which usually is how they got their name. As indicated by Daun and coworkers [40], whose development we will follow here, nearly all of the methods require first-order curvature information as contained in the gradient vector, !T ∂F ∂F ∂F g(p) = ∇p F(p) = , ,··· , = 2(i − y) · W · X, (23.25) ∂p1 ∂p2 ∂p J k

k

where equation (23.7) has been invoked. Some methods also use second-order curvature information contained in the Hessian matrix   2 ∂2 F ∂2 F   ∂ F · · ·    ∂p2 ∂p1 ∂p2 ∂p1 ∂p J  1    ∂2 F ∂2 F   ∂2 F   · · ·  ∂p2 ∂p J  . ∂p22 (23.26) H(p) = ∇p ∇p F(p) =  ∂p2 ∂p1    .. .. .. ..   .   . . .   2 2   ∂2 F ∂ ∂ F F   ···   ∂p ∂p 2 ∂p ∂p ∂p J 1 J 2 J

23.4 GRADIENT-BASED OPTIMIZATION

789

Some of the more common gradient minimization techniques are steepest descent, Newton and quasi-Newton methods, the Levenberg–Marquardt method, and conjugate gradient methods. Steepest descent is the simplest, but has a linear or even slower convergence rate and is, therefore, not recommended. The other four are briefly described below.

Newton’s Method In Newton’s method the direction of descent is calculated using both first- and second-order curvature information, by expanding the objective function into a second-order Taylor series. Assuming the desired parameter vector p∗ is a distance sk away from the latest approximation for pk , i.e., p∗ = pk + sk , the gradient vector of the objective function can be written as a two-term Taylor expansion g(p∗ ) = g(pk + sk ) ≃ g(pk ) + skT · H(pk ), (23.27) which is exact with constant Hessian if the objective function is quadratic (which tends to be approximately true, if p∗ is reasonably close to pk ). Since F has a global minimum at p∗ all elements of the gradient vector g(p∗ ) are equal to zero, and sk is determined from sk ≃ −H(pk )−1 · g(pk ).

(23.28)

In Newton’s method, dk is set equal to sk , which is called Newton’s direction (with an implied step size βk = 1). While the Hessian matrix is generally not constant near the minimum, using Newton’s direction results in much better convergence (typically quadratic), compared with the steepest descent method. However, calculating the Hessian matrix at each iteration tends to require significant extra CPU time, which can make Newton’s method actually less efficient than the steepest descent method. Thus, Newton’s method should only be used when the second derivatives can be calculated easily.

The Quasi-Newton Method The quasi-Newton method avoids calculating the Hessian matrix by approximating it using only first-order curvature data collected at previous iterations. At each iteration, the search direction dk = sk is calculated from equation (23.28) with an approximate Hessian B as dk = −(Bk )−1 · g(pk ).

(23.29)

Initially, (Bk )−1 is set equal to the identity matrix δ (which makes it the search direction for the steepest descent method) times an appropriate step size β0 [usually found from a single-value minimization of F(p0 − β0 g0 )]. At each subsequent iteration, the approximation of the Hessian matrix is improved upon by adding an update matrix, Uk , Bk = Bk−1 + Uk ,

(23.30)

and Uk is determined using only values of the objective function and gradient vectors from previous iterations. The most common quasi-Newton scheme is the Broyden-Fletcher-GoldfarbShanno (BFGS) scheme [31]; in this method, the update matrix is calculated from Uk =

zk · z k Bk−1 · dk−1 · dk−1 · Bk−1 − , where dk−1 = pk − pk−1 , zk = g(pk ) − g(pk−1 ), (23.31a) k k−1 z ·d dk−1 · Bk−1 · dk−1

or, in expanded notation Uikj = P

zki zkj

k k−1 m zm dm



P

P k−1 k−1 Bk−1 dk−1 p dp Bp j il l P P k−1 k−1 k−1 q p dq Bqp dp

l

(23.31b)

790

23 INVERSE RADIATIVE HEAT TRANSFER

Since it takes a few iterations for B to accurately approximate the Hessian matrix, the convergence rate of the quasi-Newton scheme is less than the Newton’s method, requiring a few more iterations to find the global minimum for F. However, since no second derivatives are needed, the quasi-Newton scheme is usually computationally more efficient. We will here illustrate the method by presenting a very simple example, this time a problem to infer radiative properties of a participating medium through intensity measurements. Extension to more complicated geometries and/or radiative property fields affects only the direct-solution part of the problem, which has been discussed extensively in previous chapters. Example 23.4. Consider a one-dimensional, absorbing–emitting (but not scattering) slab of width L, bounded by two cold, black walls. The temperature distribution within the slab is unknown, and is to be estimated with the quasi-Newton method, by measuring exit intensities on both bounding walls for various angles. The absorption coefficient of the medium at the detector wavelength, κ, is known and constant. Solution Direct Problem. The direct solution for this simple problem is immediately found from equation (14.20) as Z L dx′ ′ , µ < 0, Ib (x′ ) eκ(x −x)/µ κ I(x, µ) = − µ x Z x dx′ ′ Ib (x′ ) e−κ(x−x )/µ κ , µ > 0, = µ 0 with I1 = I2 = 0 (cold walls) and S = Ib (no scattering). Letting τL = κL, ξ = x/L, and evaluating only the necessary intensities exiting from the faces at ξ = 0, 1, leads to Z 1 τL Ib (ξ) eτL ξ/µ dξ, µ 0 Z 1 τL I(1, µ) = Ib (ξ) e−τL (1−ξ)/µ dξ, µ 0 I(0, µ) = −

µ < 0, µ > 0.

Inverse Problem. We will assume that the unknown Planck function field Ib (ξ) can be approximated by a simple Nth order polynomial, or N X pn ξ n . Ib (ξ) = n=0

(Power series, while simple and adequate for the present example, are generally not a good practice because the coefficients will vary over a wide range of magnitudes [24]). Substituting this into the direct solution for exiting intensity gives I(0, µ) = −

N X

! τL , µ

pn fn

n=0

I(1, µ) = e−τL /µ

N X

pn fn

n=0

fn (τ) = τ

Z

1 0

! τL , µ

µ < 0,

(23.32a)

µ > 0,

(23.32b)

n ξn eτξ dξ = eτ − fn−1 (τ). τ

Since the temperature (or Planck function) is to be found by measuring I(0, µ) and I(1, µ) for a set of I exit angles −1 < µi < +1, and assuming constant weights, the objective function becomes F=

I X (Ii − Yi )2 , i=1

where the Ii are evaluated from equation (23.32a) or (23.32b), depending on whether µi is negative or positive, and the Yi are the corresponding experimental data.

23.4 GRADIENT-BASED OPTIMIZATION

791

The sensitivity matrix is readily found by differentiating equations (23.32a) and (23.32b) with respect to pn , leading to !  τ    − fn L , µi < 0,   µi  ! Xin =   τL   −τ /µ  , µi > 0,  e L i fn µi and

Ii =

N X

(23.33)

pn Xin ,

n=0

since the problem is linear. In order to use the quasi-Newton method, we first need to calculate the gradient vector from equation (23.25), or, assuming unity weights W = δ, 1kn = 2

I  X

Iik − Yi

i=1

 ∂Iik

∂pn

=2

I  X

 Iik − Yi Xin .

i=1

(23.34)

In the first iteration we set B−1 = δ, and p1 = p0 − β0 g0 , using a first guess for p of pn = δn0 (constant temperature slab). The proper step size β0 is found by minimizing F with respect to β0 , i.e., by setting I

X ∂Ii ∂F (Ii − Yi ) 0 = 0, =2 ∂β0 ∂β i=1

or

 N  N  I X X     X   0 0 0 0     2 pn − β 1n Xin − Yi  − 1n Xin  = 0  i=1

i=1

and, finally

n=0

n=0

 N  N   N 2 I X I X X X  X    0 0 0 0      pn Xin − Yi   1n Xin  − β 1n Xin  = 0   n=0

n=0

β0 =

I P

i=1

(Xi0 − Yi )

i=1 N I P P

i=1 n=0

N P

n=0

10n Xin

10n Xin !2

n=0

!

.

For all following iterations we need to update Bk according to equations (23.30) and (23.31a). Since we are only interested in the inverse of Bk , it is usually more efficient to calculate it directly from the Sherman–Morrison formula [31]:  −1    −1 −1  dk−1 ·zk + zk · Bk−1 ·zk dk−1 dk−1 −1  −1  Bk−1 ·zk dk−1 + dk−1 zk · Bk−1 k k−1 = B + − , (23.35a) B 2 dk−1 ·zk dk−1 ·zk

or, in expanded notation, P P P k k−1 −1 k  k−1 k−1 P k−1 −1 k k−1 k−1 P k k−1 −1 k−1 k )il zl d j +di )l j )lm zm di d j −1  −1  l zl (B l (B l dl zl + l m zl (B + = Bk−1 − . (23.35b) Bk P   P k−1 k 2 k−1 k ij ij l dl zl l dl zl

After each iteration the objective function is recalculated, and the procedure is stopped when F no longer decreases (substantially). Figure 23-6 shows the simulation results for a Planck function field of Ib (ξ) = 1 + 3ξ2 − 4ξ4

for various optical thicknesses, and using 20 equally spaced measurement directions. For errorless measurements Yi , the exact result is recovered for all optical thicknesses. Figure 23-6 shows the estimated

792

23 INVERSE RADIATIVE HEAT TRANSFER

1.6 1.4

IBη ,ex, IBη ,ap

1.2 1 exact τ L = 0.1 τ L = 0.5 τL = 1 τL = 2 τL = 4

0.8 0.6 0.4 0.2 0

0

0.2

0.4 X

L

0.6

0.8

1

FIGURE 23-6 Planck function distribution for Example 23.4 predicted by the quasi-Newton method.

Planck function field for measurements that have been given a random Gaussian error, with a relative variance of 3%. It is seen that the Planck function field recovery is rather poor for optically thin slabs, getting more and more accurate as the optical thickness increases (up to a point: at very large τL the exiting intensities become independent of the internal temperature field and, thus, the temperature field cannot be recovered).

The Levenberg–Marquardt Method The Levenberg–Marquardt method was originally devised for nonlinear parameter estimation problems, but has also proved useful for the solution of ill-conditioned linear problems [3, 25, 41, 42]. In this method the problem of inverting a near-singular matrix is avoided by increasing the value of each diagonal term in the matrix, i.e., by regularizing the Gauss-Newton method of equation (23.13) to pk+1 = pk + (XTk · W · Xk + µk Ω k )−1 · XTk · W · (y − ik ),

(23.36)

where µk is a positive scalar called the damping parameter, and Ω k is a diagonal matrix. In this equation the inverse is an approximation of the Hessian matrix, and the remainder is the negative of the gradient vector, as given by equation (23.7). Levenberg suggested several choices for the diagonal matrix Ω k , among them Ω k = δ (each diagonal term is increased by a fixed amount µk ) and Ωkii = (XTk · W · Xk )ii (each diagonal term is increased by a fixed percentage). As with regularization, large values for µk dampen out oscillations in the ill-conditioned system, but also change the solution. Thus, after starting the iteration with a relatively large value of µk , its value is gradually decreased as the iteration approaches convergence. Comparison with equation (23.21) shows that the method is related to Tikhonov regularization, but using a gradually decreasing regularization parameter. Different versions of the Levenberg–Marquardt method have been incorporated into various numerical libraries, such as the Numerical Recipes [37] and IMSL routines [43].

The Conjugate Gradient Method The conjugate gradient method is another simple and powerful iterative technique to solve linear and nonlinear minimization problems. The method is explained in detail in a number of books, such as [21, 24, 44–46]. In this method the direction of descent is found as a conjugate of the gradient direction and the previous direction of descent, or dk = −g(pk−1 ) + γk dk−1 ,

(23.37)

23.4 GRADIENT-BASED OPTIMIZATION

793

with γk being the conjugation coefficient and g(pk−1 ) = ∇p F(pk−1 ) evaluated from equation (23.25). The search step size βk is taken as the value that minimizes the objective function at the next iteration, F(pk+1 ): using equations (23.3) and (23.24) together with the Taylor expansion, equation (23.12), leads to     F(pk+1 ) = i(pk + βk dk ) − y · W · i(pk + βk dk ) − y     (23.38) ≃ i(pk ) − y + βk Xk · dk · W · i(pk ) − y + βk Xk · dk . Differentiating with respect to βk , setting ∂Fk+1 /∂βk = 0, and solving for βk results in βk =

(Xk · dk ) · W · (y − ik ) , (Xk · dk ) · W · (Xk · dk )

(23.39a)

or, in expanded notation, J I Yi − I k P P i

Xijk djk 2 σ j=1 i βk = 2 .  J I  P 1  P  Xijk djk  2  i=1 σi j=1 i=1

(23.39b)

Several different expressions are in use for the conjugation coefficient γk . We mention here only the simple Fletcher–Reeves expression



2

g k γk = (23.40a) , k = 1, 2, . . . ,

k−1

2 g = 0,

k = 0.

(23.40b)



2 In expanded notation gk becomes, from equation (23.25),

2  I J X



2 X Iik − Yi  k k  

g = 4 Xij  . 2  σ i j=1 i=1

(23.41)

Example 23.5. Repeat Example 23.4 using the conjugate gradient method. Solution The solution proceeds exactly as in the previous example up to and including the evaluation of the gradient vector. But, in order to use the conjugate gradient method the γk and βk coefficients need to be calculated from equations (23.39) through (23.41), i.e.,  I 2 N   N X X X  



2  2  k k k k 

g = 1n = 4 Ii − Yi Xin  ,  n=0

n=0

 N 2 ,X I X N I  X X   k k k k k  Yi − I i Xin dn Xin dn  . β =  k

i=1

n=0

(23.42)

i=1

i=1

(23.43)

n=0

The calculation proceeds as follows:

1. Since the problem is linear, the sensitivity matrix is precalculated once and for all. 2. An initial guess is made for the parameter vector (such as pn = 0, all n), and the iteration counter is set to k = 0. 3. The direct solution Iik is found from equation (23.33), and the objective function F k is calculated; if it meets certain stopping criteria, the iteration is terminated.

794

23 INVERSE RADIATIVE HEAT TRANSFER

TABLE 23.1

Recovery of slab temperature distribution using various inversion techniques.

τL 0.1 0.5 1.0 2.0 4.0

Quasi-Newton with BFGS without line search with line search iterations time (ms) iterations time (ms) 17 18 22 10 11

0.88 0.91 1.00 0.75 0.77

17 20 20 19 19

1.02 1.13 1.12 1.14 1.11

Conjugate Gradient

Steepest Descent

Tikhonov

iterations

time (ms)

iterations

time (ms)

time (ms)

6 5 5 5 5

0.45 0.47 0.47 0.46 0.48

4,446 51,914 28,286 40,779 30,282

280 4,330 1,800 2,750 1,990

0.34 0.44 0.47 0.48 0.48

4. The gradient of F k is found from equation (23.34); γk is calculated by division with the previous value



2 of gk (for the first iteration, the “old” value is set to a very large number to force γ0 = 0). A new search direction dk is set from equation (23.37).

5. The search step size is determined from equation (23.39), and the parameter vector is updated with equation (23.24). The calculation returns then to step 3 above (alternatively, the step size βk , or the change in the parameter vector can also be used as stopping criteria). The simulation results for the same field as in Example 23.4, again using 20 equally spaced measurement directions, give essentially identical results when using the conjugate gradient approach, i.e., for errorless measurements the exact result is recovered for all optical thicknesses, and for random Gaussian error are similar to those of Fig. 23-6. The problem was also solved using various other inversion techniques, viz., quasi-Newton BFGS with line search (i.e., BFGS with βk , 1 found from the relation for β0 in Example 23.4, with 10n replaced by −dkn ), Tikhonov regularization, and the method of steepest descent.. All methods return very similar temperature profiles. The number of iterations and CPU times required for the different methods is compared in Table 23.1. Tikhonov regularization does not require any iteration (for this linear problem) and is, together with the conjugate gradient method, the fastest. Of the iterative methods conjugate gradients requires the fewest iterations and is thus the fastest, while BFGS with line search does not appreciably increase convergence, thus taking a little longer than BFGS without it. Not surprisingly, the method of steepest descent requires many more iterations.

23.5

METAHEURISTICS

Metaheuristics also belong to the family of optimization. They received their name because they are not based on a mathematically rigorous minimization formulation—in contrast to gradientbased methods, which usually approximate the objective function as locally quadratic, and then find the minimum via a Taylor series expansion. The algorithms of many metaheuristics are inspired by physics or biology (genetic algorithms and swarm algorithms are important examples of biomimicry). One popular algorithm is simulated annealing, which is based on the changing arrangement of atoms in metals. The simulated annealing algorithm is analogous to nature, where the objective function is the lattice energy, and the design parameters specify the lattice arrangement. The Second Law of Thermodynamics drives a system toward a lower energy state, so the atoms in a metal will preferentially move into lower energy configurations, but can spontaneously move into a high energy configuration. The same idea applies in metaheuristics, and the nomenclature “annealing schedule,” “temperature,” etc. carries over. At each iteration a candidate step is proposed, analogous to atoms randomly moving. A new candidate objective function is generated and compared to the present one. If the new objective function is lower, the candidate step is always accepted (probability of unity). If the new objective function is larger, the candidate step is accepted with a probability proportional to exp(−∆F/T), where the “annealing ”temperature” is defined in terms of the iteration number k. Thus, higher T make uphill steps more likely (smaller k) and, as temperature decreases (cooling the metal), accepting

23.6 SUMMARY OF INVERSE RADIATION RESEARCH

795

an uphill step becomes increasingly improbable (large k). As in actual metal annealing, the underlying idea behind simulated annealing is that the method allows the design parameters to transition through a temporary higher energy state (a crest in the objective function topography) in their quest for the lowest energy level (global minimum). By their nature metaheuristics are inevitably less efficient than gradient-based methods at finding local minima. Therefore, they should only be used when gradient-based methods are unreliable or impractical, or if the objective function topography is suspected to have multiple local minima.

23.6 SUMMARY OF INVERSE RADIATION RESEARCH Inverse Surface Radiation While inverse radiation problems involving a participating medium received the earliest attention, more recently a number of researchers have concerned themselves with inverse surface radiation problems. Harutunian et al. [47], Fedorov et al. [48], Jones [17], Erturk ¨ et al. [49] and Franc¸a et al. [50], were the first to recognize the potential of inverse radiation analysis for control: they investigated the needed energy input into a heating element, in order to achieve a prespecified result at a design surface. This was followed with considerable more work by the group around Howell [35, 40, 51–54] and a few others [55]. That inverse analysis can also be used to deduce surface reflectances was demonstrated by Wu and Wu [18]. Various solution techniques were employed. For example, TSVD was used by Franc¸a et al. [50,53] to predict heater performance in the presence of convection, and by Daun and coworkers [35] for 3D surface heating; the latter also used Tikhonov regularization, quasi-Newton and conjugate gradient techniques (optimization), and simulated annealing (metaheuristics). The conjugate gradient method was also used by Erturk ¨ et al. [51], who optimized transient heating control of a furnace, while Porter and Howell [52] used metaheuristic methods (simulated annealing and tabu search) to control a surface heater. Daun and coworkers [40, 56] and Leduc et al. [55] performed geometric optimization of radiant enclosures using Tikhonov regularization [55], the quasi-Newton method [40], and Kiefer-Wolfowitz stochastic programming (a variation on the steepest descent scheme) [56]. The only work reporting experimental verification seems to be the one by Erturk ¨ et al. [54], who investigated radiative heating control of silicon wafers. They found that accurate knowledge of radiative properties is crucial, and obtained wafer temperatures to within 3% of the target value.

Inverse Radiation in Participating Media Most research to date on inverse radiation within a participating medium has centered around the retrieval of temperature distributions, with some also deducing various radiative properties, such as surface reflectances, scattering albedos, and phase functions. Much of the work dealt with pure radiation in mostly gray [9–16, 57–68], and a few nongray [69, 70], constant-property, one-dimensional media. Others have dealt with multidimensional geometries [71–85], and interactions between conduction and radiation have also received growing attention [76, 77, 86– 89], along with, to a lesser extent, inverse radiation combined with convection [90]. Most of these investigations have concentrated on developing an inverse method using artificial data. Only a few experiments have been combined with inverse analysis to measure particle distributions and scattering properties of pulverized coal [91,92], and to infer temperature and concentration distributions in axisymmetric flames [93–99]. Most of these determined spatial averages [93] or used Abel’s transformation [94–98] (reconstruction from spatial scans). However, it has been shown that these profiles can also be determined from a single transmission measurement through spectrometry (reconstruction from spectral scan) [70, 98–100].

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23 INVERSE RADIATIVE HEAT TRANSFER

More recently, there has been growing interest in optical tomography, the reconstruction of property fields based on radiative field measurements. Two areas of interest have been identified. One is the detection of internal tumors in biomedical applications, generally using ultrafast lasers with transient radiation effects (see also Chapter 19) [101–106]. A recent review with many references has been given by Charette and colleagues [36]. Secondly, optical tomography is increasingly applied to the diagnosis of combustion systems [39, 107–109]. As for surface radiation problems, several different inverse methodologies have been employed, such as TSVD [69], Tikhonov regularization [39, 108], Tikhonov regularization plus Kalman filtering (to connect information from transient signals) [109], conjugate gradient methods [68, 81, 101–106, 110, 111], and metaheuristics [67, 70, 80, 81, 84].

Comparison of Inverse Solution Methods A few studies used several inversion techniques to allow for comparison. Daun and coworkers [35], in order to investigate surface heater control in a 3D furnace, used five different inversion techniques, viz., TSVD and Tikhonov regularization, two optimizations (the quasi-Newton and conjugate gradient methods), and one metaheuristic scheme (simulated annealing). They found that all techniques predicted solutions within acceptable accuracy, but the methods in some cases provided widely different distributions that achieve the same final result. The regularization, conjugate gradient, and simulated annealing methods provided smooth distributions of heater inputs across the heater surface, whereas the quasi-Newton technique tended to give uneven distributions. In another study Deiveegan et al. [67] retrieved surface emittances and gas properties in gray participating media, using the Levenberg–Marquardt method, and several metaheuristics schemes, i.e., genetic algorithms, artificial neural networks, and Bayesian statistics. They also found that all methods gave acceptable results, with Bayesian statistics being least susceptible to random noise, and genetic algorithms being considerably more computationally expensive. We conclude our discussion of solution methods with one simple, nonlinear example. Example 23.6. Repeat Example 23.5 for the case that the absorption coefficient is also unknown and, thus, must be estimated, as well. Compare performance and effort of the quasi-Newton, Levenberg– Marquardt, and conjugate gradient methods. Solution The solution is identical to the previous example, only now the parameter vector p has one additional member, κ, or equivalently, τL . The sensitivity matrix is identical to the one of Example 23.4, except that it has one additional row, namely Xi,N+1

! !# " N X 1 τL 1 τL ∂Ii =− an fn fn+1 + , = τL µi µi µi ∂τL n=0 " ! ! !# N X 1 1 τL 1 τL an − fn+1 fn + , = e−τL /µi τL µi µi µi µi n=0

µi < 0, µi > 0.

The problem is now nonlinear, since all Xin contain the unknown parameter τL , and Xi,N+1 also contains the an . This causes no problem in the conjugate gradient method, except that the sensitivity matrix now has to be evaluated anew after each iteration (i.e., in the calculation procedure of Example 23.5 steps 1 and 2 are interchanged, and the iteration always repeats from step 2). Results for the conjugate gradient method are shown in Fig. 23-7. Again, the exact relations are recovered for undisturbed measurements, and the cases shown are for measurements with a random Gaussian error with 3% relative variance. Results are very similar to Example 23.3, perhaps just a little worse, and recovery of the absorption coefficient is well within the variance of the data, except for Levenberg–Marquardt, which incurs errors up to 5% for small and large τL . On the other hand, Levenberg–Marquardt also is the fastest of the different methods for this problem, as seen in Table 23.2, which shows the time requirements for the different methods.

797

REFERENCES

1.6 1.4

Ibη ,ex, Ibη ,ap

1.2 1 0.8

exact τ L = 0.1 (0.100) τ L = 0.5 (0.501) τ L = 1 (1.003) τ L = 2 (2.006) τ L = 4 (4.022)

0.6 0.4 0.2 0

0

0.2

0.4

x/L

0.6

0.8

1

FIGURE 23-7 Absorption coefficient and Planck function distribution for Example 23.6 as predicted by the conjugate gradient method.

TABLE 23.2

Recovery of slab temperature distribution and absorption coefficient using various inversion techniques.

τL 0.1 0.5 1.0 2.0 4.0

Quasi-Newton with BFGS Conjugate Gradient Steepest Descent Levenberg–Marquardt without line search with line search iterations time (ms) iterations time (ms) iterations time (ms) iterations time (ms) iterations time (ms) – 24 20 21 23

—2.03 1.80 1.88 1.98

9 22 19 20 24

5.69 4.36 3.13 2.94 3.57

2334 716 235 350 752

6.94 2.30 0.96 1.31 2.61

57,118 130,001 34,101 13,512 19,435

33,200 61,080 16,350 5,760 8,770

13 4 3 3 6

0.97 0.52 0.45 0.59 0.69

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F., and S. T. Thynell: “Line-of-sight temperature and species profiles determined from spectral transmittances,” Journal of Thermophysics and Heat Transfer, vol. 11, no. 3, pp. 367–374, 1997. 101. Klose, A. D., and A. H. Hielscher: “Optical tomography using the time-independent equation of radiative transfer — part 2: Inverse model,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 72, pp. 715–732, 2002. 102. Kim, H. K., and A. H. Hielscher: “A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer,” Inverse Problems, vol. 25, no. 1, pp. 1–20, 2009. 103. Boulanger, J., and A. Charette: “Numerical developments for short-pulsed near infra-red laser spectroscopy. Part II: Inverse treatment,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 91, pp. 297–318, 2005. 104. Boulanger, J., and A. Charette: “Reconstruction optical spectroscopy using transient radiative transfer equation and pulsed laser: a numerical study,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 93, pp. 325–336, 2005.

801

PROBLEMS

105. Boulanger, J., A. El Akel, A. Charette, and F. Liu: “Direct imaging of turbid media using long-time back-scattered photons, a numerical study,” International Journal of Thermal Sciences, vol. 45, pp. 537–552, 2006. 106. Kim, H. K., and A. Charette: “A sensitivity function-based conjugate gradient method for optical tomography with the frequency-domain equation of radiative transfer,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 104(1), pp. 24–39, 2007. 107. Ayrancı, I., R. Vaillon, N. Selc¸uk, F. Andr´e, and D. Escudi´e: “Determination of soot temperature, volume fraction and refractive index from flame emission spectrometry,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 104, no. 2, pp. 266–276, 2007. 108. Daun, K. J.: “Infrared species limited data tomography through Tikhonov reconstruction,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 111, no. 1, pp. 105–115, 2010. 109. Daun, K. J., S. L. Waslander, and B. B. Tulloch: “Infrared species tomography of a transient flow field using Kalman filtering,” Applied Optics, vol. 50, no. 6, pp. 891–900, 2011. 110. Li, H. Y.: “Estimation of thermal properties in combined conduction and radiation,” International Journal of Heat and Mass Transfer, vol. 42, no. 3, pp. 565–572, 1999. 111. Park, H. M., and T. Y. Yoon: “Solution of the inverse radiation problem using a conjugate gradient method,” International Journal of Heat and Mass Transfer, vol. 43, no. 10, pp. 1767–1776, 2000.

Problems 23.1 Repeat Example 23.2, but determine the necessary heat flux distribution, q2 (x2 ), along the plate. 23.2 A black plate of width w is irradiated by two line sources as shown. The plate is insulated at the bottom, while the top loses heat by radiation to the (cold) environment. Ideally, the plate should be at a uniform temperature of 500 K. Breaking up the plate into four equally wide segments, determine the optimal heater powers (without exploiting the symmetry):

w/2



S´ w/2 w

A2

a) using TSVD on the direct equations (23.19), b) using TSVD and the normal equations (23.10), c) using Tikhonov regularization and the normal equations.

w/2 w = 20 cm

23.3 Soot volume fraction and temperature are to be determined by measuring the transmissivity of a gas–soot layer for several wavelengths. Consider a homogeneous layer of thickness L = 0.2 m, whose absorption coefficient obeys equation (12.123), where C0 is a known function of wavelength and temperature, such that C0 (λ, T) = 5[1 + aλ(T − T0 )], T0 = 300 K, a = 0.01 (µmK)−1 . Transmissivity measurements are conducted at four wavelengths as shown in the table: one set of data has been taken with high precision (i.e., zero error), and the other has some noise in the data. Wavelength λ High-fidelity data Noisy data

1 µm 0.6065 0.617

2 µm 0.7788 0.763

3 µm 0.8465 0.826

4 µm 0.8825 0.891

Determine soot volume fraction and temperature using Tikhonov regularization. 23.4 Repeat Problem 23.3 using the quasi-Newton method. 23.5 Consider a one-dimensional, absorbing–emitting (but not scattering) slab of width L, bounded by two cold, black walls. The temperature distribution within the slab is unknown, and is to be estimated by measuring spectral exit heat fluxes on both bounding walls for various wavenumbers in a range over which the absorption coefficient of the medium, κ, is known, is linearly proportional to wavenumber, and is spatially constant. Use the P1 -approximation and Tikhonov regularization. Hint: Set up a 1D finite difference solution for the P1 -approximation by breaking up the slab into N isothermal layers; then determine M > N wall fluxes in terms of the Ibη (Ti ).

802

23 INVERSE RADIATIVE HEAT TRANSFER

23.6 Repeat Problem 23.5 for a medium that also scatters radiation isentropically, with a gray scattering coefficient. 23.7 Repeat Problem 23.5 for the case of an unknown absorption coefficient (except for the fact that it is linearly proportional to wavenumber). Use the P1 -approximation together with the quasi-Newton algorithm. 23.8 In laser absorption tomography, the concentration of a target species (e.g., gas or soot) is inferred from the transmittance of multiple lasers passing through the flow field. If the domain is split into n regions in each of which the concentration is assumed uniform, the Beer-Lambert law along the ith beam becomes ln(I0i /Ii ) =

n X

Ai j κ j ,

j=1

where Aij is the chord length of the ith beam subtended by the jth element. Writing this equation for m beams results in an m × n matrix equation, A · p = b, which relates the beam transmittance data, b, to the unknown species concentration (through the absorption coefficient, p = κ), equivalent to equation (23.19). However, even if n = m the matrix is ill-conditioned, and its inversion must be regularized to suppress measurement noise amplification. Consider the axisymmetric problem shown to the right. Laser transmittance measurements made along the center of each annular element are summarized in the table below. It is known that each data point is contaminated by normally-distributed error having a standard deviation of 0.025. y (cm) 0 0.3158 0.6316 0.9474 1.2632 1.5789 1.8947 2.2105 2.5263 2.8421 ln(I0i /Ii ) 0.6258 0.5494 0.4652 0.2883 0.1183 0.0831 0.0171 0.0259 –0.0179 –0.0056 (a) Derive the A matrix and perform a singular value decomposition. What do the singular values imply about this problem? (b) Attempt to recover p using no regularization, and plot the values as a function of y. Comment on the solution. (c) Use first-order Tikhonov regularization to recover the solution. The truncated equation (23.19) for first-order Tikhonov becomes    1 −1    1 −1     ′ ′ (A + λL ) · p = b, where L =  . .  .  .. ..     1 −1 Attempt to recover the solution using different values of λ. What is the optimal level of regularization?

CHAPTER

24 NANOSCALE RADIATIVE TRANSFER

24.1

INTRODUCTION

In the last chapter of this book we will provide a brief introduction to radiative heat transfer in geometries where the pertinent dimensions are measured in nanometers (nm). Research in the field of nanoscale energy transfer has exploded during the past few years, leading to fascinating new problems and devices in microelectronics and microfabrication technology, such as quantum structures, optoelectronics, molecular- and atomic-level imaging techniques, etc. Most radiation is incoherent (multispectral, as well as random in polarization and direction) in the “far field” (a distance of a wavelength or so away from the source), and the radiative transfer equation (RTE) and its solution methods described over the previous chapters are only valid for such incoherent radiation. We noticed in Chapters 2 and 3 (optically smooth surfaces) and Chapter 12 (small particles) that, when distances of the order of the wavelength λ are relevant, radiative transfer must be calculated from the full Maxwell’s equations presented in Chapter 2. However, Maxwell’s equations do not include any radiative emission sources, which must be modeled via what is known as fluctuational electrodynamics, pioneered by Rytov [1, 2]. In the following we will give very brief accounts of some interesting radiative phenomena that are observed at the nanoscale, culminating in the prediction of radiative flux between two plates, spaced a tiny distance apart. The reader interested in detailed knowledge of the subject area should consult the books by Chen [3], Novotny and Hecht [4], and Zhang [5], review articles by Basu and coworkers [6] and Zhang and Park [7], as well as the large number of recent research papers in the field.

24.2

COHERENCE OF LIGHT

No radiation source is perfectly coherent, i.e., perfectly monochromatic and unidirectional, not even lasers or emission from single atoms. On the other hand, no source is truly incoherent: even the most chaotic blackbody radiation has a small coherence length, which is related to the distance the wave travels within a coherence time [5]. If the wave nature of light is completely preserved, we speak of coherent light. If light travels longer than the coherence time, or a distance larger than the coherence length, fluctuations in the waves will diminish wave interference effects (see Fig. 2-13 and the discussion of reflection from a thin layer). The coherence of light in space and time (or, equivalently, frequency) is measured by the mutual coherence function of any two waves, defined as hE(r1 , t)E∗ (r2 , t)i, where the angular brackets denote time-averaging, and the r1 and r2 are two different locations; the electric field can be expressed in either the frequency 803

804

24 NANOSCALE RADIATIVE TRANSFER

n z

z

w´i, si

sr

Medium 1

ef

ro nt

θ1

W

av

x

θc

Medium 1, n1> n2

θ2

Medium 2

x

w´t , st Medium 2, n 2

(b)

(a)

FIGURE 24-1 Total internal reflection and evanescent waves: (a) propagation of waves at critical angle of incidence, (b) evanescent wave propagating along x-direction and exponentially decaying in −z-direction.

domain, or time domain [5]. For our purposes we simply note that the coherence length of random blackbody radiation is about λ/2 [4, 8], and longer for more coherent sources.

24.3

EVANESCENT WAVES

We observed in Section 2.5, equation (2.100), that at an interface between two dielectrics total reflection takes place if light attempts to enter a less dense material (n2 < n1 ) at an incidence angle θ1 larger than the critical angle sin θ1 > sin θc =

n2 , n1

(24.1)

with no energy penetrating into Medium 2 (see Fig. 24-1a). This is true as far as far-field radiation is concerned, and also for net (time- and space-averaged) energy. However, if one carefully inspects the electromagnetic wave theory relationships, one observes that a wave traveling parallel to the interface enters Medium 2, with its strength decreasing exponentially away from the interface, known as an evanescent wave (from Latin for “vanishing”). To simplify the analysis we will, without loss of generality, consider here only the case of a parallel polarized (TM) wave (E⊥ = 0), and only concern ourselves with the electric field. Then, from equations (2.73) and (2.75), we have E c1 = Eki eˆ ki e−2πi(wi ·r−νt) + Ekr eˆ kr e−2πi(wr ·r−νt) , −2πi(wt ·r−νt)

E c2 = Ekt eˆ kt e

,

(24.2a) (24.2b)

and the wave vector w, as defined1 by equation (2.31) has x- and z-components ˆ w = η0 nˆs = wx ˆı + wz k.

(24.3)

Since the tangential components of the electrical field must be conserved, equation (2.67), we have wxi = wxr = wxt = wx , and wx = η0 n1 sin θ1 = η0 n2 sin θ2 , 1

(24.4)

Recall that this book’s definition of the wave vector differs by a factor of 2π and in name from the definition k = 2πw in most optics texts in order to conform with our definition of wavenumber.

24.4 RADIATION TUNNELING

Z

propagating waves

805

Medium 1, N 1> N 2

θ1 X

evanescent waves Medium 2, N 2

D

propagating waves

θ3

Medium 3, N 3 = N 1

FIGURE 24-2 Photon tunneling through a layer of lesser refractive index, adjacent to two optically denser materials.

which is Snell’s law. If θ1 exceeds the critical angle, then n1 sin θ1 > n1 sin θc > n2 , and the z-component of the transmitted wave becomes q q 2 η0 n2 − w2x = iη0 (n1 sin θ1 )2 − n22 = i|wzt | = iη0 n2 | cos θ2 |, wzt =

(24.5)

(24.6)

i.e., wzt and cos θ2 are purely imaginary (and |wzt | and | cos θ2 | are their magnitudes). Substituting this into equation (24.2b) we have E c2 = Ekt kˆ e−2π|wzt |+2πiνt ,

(24.7)

with the magnitude of |wzt | = O{η0 = 1/λ0 }, i.e., we have a wave inside Medium 2 traveling along the interface, exponentially decaying in strength over the distance of one wavelength or so (depending on θ1 ). This is depicted in Fig. 24-1b. Performing the same analysis for the magnetic field (with Hk = 0), it is easy to show that the z-component of the time-averaged Poynting vector, see equation (2.42), is zero, i.e., no net energy crosses the interface [3]. However, if the instantaneous Poynting vector is examined, one finds that there is periodic in- and outflow of energy carried by the evanescent field.

24.4

RADIATION TUNNELING

We have seen in the previous section that, if a radiative wave train is reflected at the interface to an optically less dense medium, an evanescent wave exists within the optically rarer medium with exponentially decaying strength away from the interface. Furthermore, the evanescent wave does not carry any net (time-averaged) energy into the direction normal to the surface. However, if a second denser medium is brought into close proximity to the first, net energy can be transported across the gap or intermediate layer. This phenomenon is known as radiation tunneling (or sometimes as photon tunneling, or frustrated total internal reflection [5]), and is very important for heat transfer between two media a distance of a wavelength or less apart, as schematically shown in Fig. 24-2. While this phenomenon has been known since Newton’s time, in the heat transfer area it was probably first discovered by Cravalho and coworkers [9], who investigated closely spaced cryogenic insulation. Today’s important applications range from thermophotovoltaic devices to nanothermal processing and nanoelectronics thermal management [5].

806

24 NANOSCALE RADIATIVE TRANSFER

If a second optically dense material is close to the first, the evanescent wave in the layer in between is reflected back toward the first interface. Interference between the two waves cause the Poynting vector to have a nonzero net component in the z-direction. However, if the gap is too wide (i.e., well more than one wavelength away), the evanescent wave reaching the second interface is too weak and net energy transfer becomes negligible. To calculate the transmissivity of the gap or intermediate film for above-critical angles of incidence we may use the thin film relations developed in Chapter 2, keeping in mind that cos θ2 may become imaginary for large incidence angles. Limiting ourselves here to three dielectrics with n1 = n3 > n2 , equations (2.131b) and (2.133) may be rewritten as t=

d t12 t21 eiβ , T = tt∗ , β = 2πw2z d = 2πn2 cos θ2 , λ 1 − r221 e2iβ

(24.8)

with r21 , t12 , and t21 determined from equations (2.89) through (2.92). For θ1 < θc the interface reflection and transmission coefficients are real, and Tλ =

(1 − r221 )2 (t12 t21 )2 = , 1 − 2r221 cos 2β + r421 1 − 2r221 cos 2β + r421

θ1 < θc = sin



 n2 . n1

(24.9)

If θ1 exceeds the critical angle an evanescent wave enters Medium 2 and w2z and cos θ2 become purely imaginary. From equation (24.6) we find that the phase shift β now becomes imaginary (the exponential decay of the evanescent wave), β = i (2πn2 | cos θ2 |)

d = i|β|, λ

(24.10)

and the r21 , t12 , and t21 become complex [i.e., replacing cos θ2 by i| cos θ2 | in equations (24.8) and (24.9)]. Therefore, t= and Tλ = tt∗ =

t12 t21 e−|β| , 1 + r221 e−2|β|

(t12 t21 )(t∗12 t∗21 )e−2|β|  , e−2|β| + r221 r∗2 1 + r221 + r∗2 e−4|β| 21 21 

(24.11)

(24.12)

which, after some algebra (left as an exercise), may be reduced to  n1 | cos θ2 |    , parallel (TM) polarization,  2  sin 2α  n2 cos θ1 Tλ = , where tan α =    n2 | cos θ2 | sin2 2α + sinh2 |β|   , perpendicular (TE) polarization.  n1 cos θ1 (24.13) Again, equation (24.13) is valid for, both, parallel- and perpendicular-polarized light, except for the different definition of tan α (due to the different structure of rk and r⊥ ). Example 24.1. Consider a vacuum gap surrounded by a dielectric medium with refractive index √ n1 = n3 = 2 = 1.4142. Determine the transmissivity for parallel-polarized light for all angles of incidence and as a function of gap width. Solution With n2 = 1 we have sin θc = 1/√2, or θc = 45◦ . Writing a small computer code, using equation (24.9) for θ1 < 45◦ , and equation (24.13) for θ1 > 45◦ , and with tan α =

n1 | cos θ2 | , n2 cos θ1

24.5 SURFACE WAVES (POLARITONS)

807

30°

1

θ1 = 0°

40°

10°

20 °

Transmissivity, Tl

0.8 N1 = N3 =

1.4142; θ C = 45 °

0.6 44°

0.4 45° 46°

0.2 80 °

0

0

70 °

50°

60 °

0.5

Dd λ

1

1.5

FIGURE 24-3 Transmissivity of a vacuum gap surrounded by identical dielectrics (n1 = n3 = 1.4142), for parallel-polarized light.

we obtain the gap transmissivity shown in Fig. 24-3. It is observed that for small θ1 we have noticeable interference effects, but the transmissivity remains high for all gap widths (Tλ > 0.9). Wavelength of interference and magnitude increase with θ1 until, reaching Brewster’s angle (≈ 35◦ ), we have total transmission of a parallel-polarized wave (see also Fig. 2-9). Beyond Brewster’s angle ρk increases rapidly, with decreasing transmissivity (but still increasing wavelength of interference). At θ1 = 45◦ we have rk = −1, and an evanescent wave forms, and the larger the incident angle, the faster the strength of the evanescent wave decays across the gap. It is straightforward to verify that, at 45◦ , both equations (24.9) and (24.13), go to the same limit, i.e., Tλ (θ1 = 45◦ ) =

1 !. πd 1+ 2λ

24.5 SURFACE WAVES (POLARITONS) The interaction between electromagnetic waves and the oscillatory movement of free charges (electrons) near the surface of metallic materials is known as surface plasmons or surface plasmon polaritons. Surface plasmons are usually found in the visible to near-infrared part of the spectrum in highly conductive metals, such as gold, silver, and aluminum. They are of importance in near-field microscopy and nanophotonics [5, 10–12]. In some polar dielectrics lattice vibrations (phonons) and/or oscillations of bound charges can also interact with electromagnetic waves in the mid-infrared; these are known as surface phonon polaritons, and are of interest in the tuning of emission properties [13] and nanoscale imaging [14]. In either case they result in the generation of an electromagnetic wave traveling along, and only in the immediate vicinity of both sides of an interface, i.e., a surface wave. In our brief discussion here we will mostly follow the presentation of Zhang [5]. One requirement of a surface wave, i.e., a wave decaying in both directions normal to the surface, is that there are evanescent waves on both sides of the interface. Consider the arrangement shown in Fig. 24-4, consisting of a thin layer and a thick substrate, with the thin layer bound at the top by a third medium. The thin layer may be air with a metallic substrate (Otto configuration), or a metal layer bounded by air at the bottom (Kretschmann configuration) [15]. If light is incident from the top medium, it is possible for evanescent waves to occur simultaneously in both the underlying air and metal layers, as also indicated in Fig. 244. A second requirement for polaritons is that the polariton dispersion relations must be satisfied,

808

24 NANOSCALE RADIATIVE TRANSFER

dielectric

θ X

air or metal, e1 evanescent waves metal or air, e2

FIGURE 24-4 Typical configuration for the generation of surface polaritons, consisting of a dielectric for incident light, and an air/metal thin layer/substrate combination.

Z

which are the poles of the Fresnel reflection coefficients, since infinite reflection coefficients are an indication of resonance. If one writes the reflection coefficients in terms of wave-vector components [4, 5] as     w1z w2z w1z w2z rk = , (24.14a) − + ε1 ε2 ε ε2 !, 1 ! w1z w2z w1z w2z r⊥ = − + , (24.14b) µ1 µ2 µ1 µ2 the polariton dispersion relations are defined by w1z w2z + = 0, ε1 ε2 w1z w2z + = 0, µ1 µ2

for parallel-polarized light,

(24.15a)

for perpendicular-polarized light.

(24.15b)

The nature of the dispersion relations is more easily understood by first looking at the case of two dielectric media: in order to have evanescent waves we must have both w1z and w2z purely imaginary, with w1z = −i|w1z | and w2z = −i|w2z |, i.e., both with a negative sign in order to have e−2πiwr ·r = e−2πi(w1x x−w1z z) = e−2πiw1x x+2π|w1z |z (reflected wave) decay toward negative z, and e−2πiwt ·r = e−2πi(w2x x+w2z z) = e−2πiw2x x−2π|w2z |z (transmitted wave) toward positive z (see Fig. 24-4). This implies that in order to produce a surface wave with parallel-polarized incident light, the electrical permittivities of the two materials must have opposite signs. Since metals display negative permittivities over large parts of the spectrum, this condition is easily fulfilled. To produce a surface polariton with perpendicular-polarized light, on the other hand, requires a medium with negative magnetic permeability. While so-called negative index materials (NIM) exhibit both negative permittivity and permeabilty [16], most materials are nonmagnetic, for which surface polaritons cannot be generated with perpendicular-polarized light. Employing equation (2.31) together with m2 = ε, we may write for a general nonmagnetic medium w21 = w2x + w21z = η20 ε1 ,

(24.16a)

η20 ε2 ,

(24.16b)

w22

=

w2x

+

w22z

=

where we have made use of the fact that the tangential component of the wave vector must be continuous across the interface, w1x = w2x = wx . Using these relations the z-components may be eliminated from equation (24.15a), leading to r ε1 ε2 . (24.17) wx = η0 ε1 + ε2

24.6 FLUCTUATIONAL ELECTRODYNAMICS

2.0.10

809

+05

propagating wave η

0

=

+05

1.0.10

X

’=

0

C

ν/ W

−1

Frequency, ν /C0 = η 0 , cm

1.5.10

ηp

+05

ηp /1.4142

5.0.10+04

evanescent wave

+00

0.0.10 +00 0.0.10

5.0.10

+04

+05 1.0.10

1.5.10

+05

+05 2.0.10

Real part of tangential wave vector, WX, cm-1 w

FIGURE 24-5 Dispersion relation for aluminum and air; top left solid line: propagating waves; dashed line: light line; bottom right solid line: evanescent waves.

This equation relates the tangential component of the wave vector to wavenumber (or frequency), and is a popular alternative statement of the polariton dispersion relation. If one of the media is vacuum or air (ε = 1), an evanescent wave exists if wx > η0 (i.e., wz has an imaginary component). Note that equation (24.17) also gives the roots to the numerator of equation (24.14a): for wx < η0 equation (24.17) describes propagating waves. Example 24.2. Determine the dispersion relation between aluminum and air, assuming that the dielectric function of Al obeys the Drude theory. Solution The Drude equation has been given by equation (3.64), when written in complex form, as εAl = 1 −

ν2p ν(ν + iγ)

;

νp = 3.07 × 1015 Hz,

γ = 3.12 × 1013 Hz,

with plasma frequency νp and damping factor γ from Fig. 3-7. With εair = 1 the tangential wave vector component may be calculated from equation (24.17). Since εAl is complex, so is wx = w′x + iw′′ x . It is common to show a dispersion relationship by plotting the real part of wx vs. frequency or wavenumber, which has been done in Fig. 24-5. The dashed line w′x = η0 is called the light line. On its left wz is real in air, and a propagating wave exists. On its right, w′x > η0 and the wz in air becomes imaginary, and only evanescent waves are found. It is seen that, for the evanescent waves, w′x increases rapidly, reaching an √ asymptote at ν = νp / 2, when the real part of the dielectric function of Al approaches −1. For ν > νp metal becomes transparent and the real part of the dielectric function becomes positive. The solution to equation (24.17) for ν > νp corresponds to rk = 0 in equation (24.14a) and shows, therefore, propagating waves.

24.6 FLUCTUATIONAL ELECTRODYNAMICS As indicated earlier, Maxwell’s equations do not contain a thermal radiation emission term. Such a source must be added by considering radiative transitions by elementary energy carriers (such as electrons, lattice vibrations called phonons, etc.) from a higher energy state to a lower one, accompanied by the release of a photon. Such a quantum-mechanical process, similar to emission from gas molecules covered in Chapter 11, must be linked to the equations describing the electromagnetic waves. This is achieved through the concept of fluctuational electrodynamics, originally developed by Rytov [1, 2]. At any finite temperature above absolute zero, chaotic thermal motions takes place inside any material. Charged particles of opposite sign pair up (known as dipoles), and the random motion of the dipoles induce a fluctuating electromagnetic

810

24 NANOSCALE RADIATIVE TRANSFER

field. Thus, in this fluctuational electrodynamics model the random thermal fluctuations generate a space- and time-dependent (but random) electric current density j′ (r, t) inside the medium, whose time average is zero [5]. To include the stochastic current density in the electromagnetic wave equations, several approaches are possible. The most common technique is to employ a dyadic Green’s function Ge (r, r′ , ν) (a 3 × 3 matrix). The induced electric and magnetic fields in the frequency domain can then be determined from Z E(r, ν) = 2πiµ0 Ge (r, r′ , ν) · j(r′ , ν)dr′ , (24.18a) ZV Gh (r, r′ , ν) · j(r′ , ν)dr′ , (24.18b) H(r, ν) = V

where the integral is over the volume, which contains the fluctuating dipoles, j(r′ , ν) is the Fourier transform of the electric current density source j′ (r, t) into frequency space, and µ0 is the magnetic permeability of vacuum. The dyadic Green’s function for the magnetic field is, by equation (2.13), directly related to Ge through Gh = −∇ × Ge . Physically, Ge may be interpreted as a transfer function relating the electric field at location r and frequency ν to a vector source located at r’. Mathematically, the dyadic Green’s function is found as the solution to a vector Helmholtz equation, which may be reduced to a scalar one as [4] ! 1 ′ Ge (r, r , ν) = δ + ∇∇ G0 (r, r′ , ν), (24.19) (2πw)2 with G0 the solution to





 (2πw)2 + ∇2 G0 (r, r′ , ν) = −δ(r − r′ ),

(24.20)

hS(r, ν)i = 21 hℜ{E c × H ∗c }i,

(24.21)

where δ(r − r ) is a 3D Dirac-delta function as defined on p. 610, and w is the magnitude of the wave vector w. The time-averaged emitted energy flux may be calculated from the average Poynting vector, equation (2.41),

where the angle brackets denote the ensemble average over the random fluctuations. Sticking equations (24.18) into equation (24.21) requires a two-point ensemble average of the random current density, which must be a function of local temperature. This is achieved through the fluctuation–dissipation theorem pioneered by Rytov [1], leading to

jm (r, ν)jn (r′ , ν) = 8νǫ0 ε′′ Θ(ν, T)δmn δ(r − r′ ),

(24.22)

where ǫ0 is the electrical permittivity of vacuum, ε′′ is the imaginary part of the medium’s dielectric function, and subscripts m and n denote the x-, y- and z-components of j. The function Θ(ν, T) is the mean energy of a Planck oscillator given by [4] Θ(ν, T) =

hν . ehν/kT − 1

(24.23)

A multiplicative factor of 4 is included on the right-hand side of equation (24.22), since only positive frequencies are considered in the Fourier transform for the electric current density [17]. Sticking equations (24.18) and (24.22) into equation (24.21) yields for the individual terms arising in the Poynting vector, after some manipulation,   Z  2   X E D   ν  ′ ∗ ′ ′ 1 Θ(ν, T)ℜ  Ge,im (r, r , ν)Ge,jm (r, r , ν)dr  , (24.24) iε 2 Ei (r, ν)H j (r, ν) = 8π c     0 V m

24.7 HEAT TRANSFER BETWEEN PARALLEL PLATES

811

Z

Medium 2: T2, e2 X

D

Medium 0: vacuum, N 0 = 1

Medium 1: T1, e1

R

FIGURE 24-6 Closely spaced parallel plates separated by a vacuum gap.

where the subscripts again denote the various x-, y- and z-components. For example, the z-component of the Poynting vector becomes D E hSz (r, ν)i = 12 Ex H∗y − E y Hx∗   Z  2   X    ν   Ge,xm G∗h,ym − Ge,ym G∗h,xm (r, r′ , ν)dr′  Θ(ν, T)ℜ  = 8π . (24.25) iε     c0 V m

As given, the time-averaged Poynting vector constitutes the local radiative flux caused by the surrounding electromagnetic field.

24.7 HEAT TRANSFER BETWEEN PARALLEL PLATES Consider Medium 1 separated from Medium 2 by a small, perfectly parallel vacuum gap of width d, as shown in Fig. 24-6. To calculate the radiative flux between them we need to determine the normal component of the Poynting vector for the energy transmitted from Medium 1 across the gap, as well as the counter-flow from Medium 2 to 1. Since the problem is one-dimensional in the z-direction, it has no azimuthal (or x- and y-) dependence, making the analysis a little simpler if cylindrical coordinates are employed, i.e., we define position and wave vectors as ˆ r = rˆer + zk,

ˆ w = wr eˆ r + wz k.

(24.26)

The dyadic Green’s function for two semi-infinite media separated by a parallel gap may be determined from [4, 5, 18, 19] Z ∞  wx ′ ′ ′ ′ sˆ t⊥ sˆ + pˆ 1 tk pˆ 2 e−2πi(w2z z−w1z z ) e−2πiwx (r−r ) dwx , (24.27a) Ge (r, r , z, z ; ν) = i 2w 1z 0 where

ˆ sˆ = eˆ r × k,

pˆ i = (wx kˆ − wiz eˆ r )/wi ,

i = 1, 2.

(24.27b)

Here t⊥ and tk are the transmission coefficients from Medium 1 to Medium 2, as evaluated from Airy’s formula, equation (2.131b), and the interrelationship between wx , wiz , and wi is given by equation (24.16). Sticking this into equation (24.25) one obtains, after considerable algebra, for the spectral radiative flux from Medium 1 to Medium 2 Z ∞ qν,1→2 = 8πΘ(ν, T1 ) Z12 (ν, wx )wx dwx (24.28a) 0

812

24 NANOSCALE RADIATIVE TRANSFER

where 4ℜ {w1z } ℜ {w2z } w20 e−2iw0 d

Z12 (ν, wx ) =  2 (w0z + w1z )(w0z + w1z ) 1 − r⊥01 r⊥02 e−2iw0 d o n o n 4ℜ ε1 w∗1z ℜ ε2 w∗2z w20 e−2iw0 d +  2 . (ε1 w0z + w1z )(ε2 w0z + w1z ) 1 − rk01 rk02 e−2iw0 d

(24.28b)

The Z12 (ν, wx ) may be interpreted as an exchange function, identifying the contribution of a given tangential wave vector component, wx (related to incidence angle), to the spectral flux. Observing that Z12 = Z21 , the net heat exchange between the two surfaces is readily found, after integration over all frequencies, as Z ∞ Z ∞ Z ∞  (Θ(ν, qν,1→2 − qν,2→1 dν = qnet = Z12 (ν, wx )wx dwx dν. (24.29) T1 ) − Θ(ν, T2 )) 0

0

0

Equations (24.28) and (24.29) include contributions from both propagating and evanescent waves. We observed in Section 24.3 that we have propagating waves for wx < w0 = η0 = ν/c0 (real w0z in the vacuum layer), and evanescent waves for wx > ν/c0 (imaginary w0z ). Using the expressions for transmission coefficients developed in Section 24.4, we find Zprop (ν, wx ) =

(1 − r2k01 )(1 − r2k02 ) (1 − r2⊥01 )(1 − r2⊥02 ) 2 + 2 , 4 1 − r⊥01 r⊥02 e−2iw0 d 4 1 − rk01 rk02 e−2iw0 d

wx < η0 .

(24.30a)

For the evanescent waves the exchange function reduces to

  ℑ {r⊥01 } ℑ {r⊥02 } e−2|w0 |d ℑ rk01 ℑ rk02 e−2|w0 |d Zevan (ν, wx ) = + 2 2 , 1 − rk01 rk02 e−2|w0 |d 1 − r⊥01 r⊥02 e−2|w0 |d

wx > η0 .

(24.30b)

Clearly, similar to the evanescent transmissivity of Section 24.3, the contribution from Zevan to the flux decreases exponentially with distance between the plates. Far Field Heat Flux. As discussed in Section 2.5, as d becomes large, d ≫ λ0 , the radiation will lose coherence, and the gap transmissivity will obey equation (2.133) (with κ = 0 for the vacuum gap). Then the exchange function reduces to, with |r|2 = ρ, Zprop,ff (ν, wx ) =

(1 − ρ⊥01 )(1 − ρ⊥02 ) (1 − ρk01 )(1 − ρk02 )  .  + 4 1 − ρ⊥01 ρ⊥02 4 1 − ρk01 ρk02

(24.31)

Integration over wx may be replaced by wx = (ν/c0 ) sin θ, where θ is the polar angle in vacuum, and equation (24.29) becomes, with Zevan = 0 and 1 − ρ = ǫ,    Z ∞ Z π/2   1 2π 1    cos θ sin θ dθ ν2 dν.  [Θ(ν, T1 ) − Θ(ν, T2 )] qnet,far = 2 +   1 1 1 1  c0 0 0  + −1 + − 1  ǫ⊥01 ǫ⊥02 ǫk01 ǫk02 (24.32) Comparison with equation (5.35) shows that these results are identical if the emissivities are assumed to be gray and diffuse. Example 24.3. Determine the total radiative flux between two plates of aluminum, separated by a vacuum gap, assuming that the dielectric function of Al obeys the Drude theory as in the previous example. The plates are isothermal and maintained at 400 K and 300 K, respectively. Determine the

Radiative flux Qnet, W/m2

24.7 HEAT TRANSFER BETWEEN PARALLEL PLATES

10

6

10

5

10

4

10

3

10

2

813

evanescent, TM evanescent, TE evanescent, TM+TE propagating, TM propagating, TE propagating, TM+TE total, TM+TE far field, TM far field, TE far field, TM+TE

101 10

0

T1 = 400K, T2 = 300K 10−1 −9 10

10−8

10−7

Gap width D, m

10−6

10−5

FIGURE 24-7 Total radiative heat fluxes between aluminum plates separated by a vacuum microgap of varying width (dashed = evanescent waves; dash-dot = propagating waves; triangles = parallel/TM polarization; squares = perpendicular/TE polarization; thick lines = both polarizations).

total radiative flux as a function of gap thickness. Distinguish contributions from propagating and evanescent waves, and compare the influence of parallel and perpendicular polarizations. Solution With the dielectric function of Al given in the previous example, and with the wiz related to wx and εi by equation (24.16), the reflection coefficients in equations (24.30) may be calculated from equations (24.14). Integrating over all frequencies ν and all tangential wave vectors wx , separately 0 ≤ wx < η0 for propagating waves, and η0 < wx < ∞ for evanescent waves, yields the desired total radiative flux between the two aluminum plates, as shown in Fig. 24-7 for gap widths ranging from 1 nm to 10 µm. For the far-field solution equation (24.30a) is replaced by equation (24.31) and Zevan = 0. Integration may again be over tangential wave vectors 0 ≤ wx < η0 or, alternatively, over polar angle θ. It is seen that, for gap sizes of less than about 2 µm, the heat flux is dominated by the evanescent waves, in particular its TE component. For small gap widths the propagating component approaches an asymptotic limit, which is about an order of magnitude larger than the far-field solution, but still considerably smaller than the blackbody limit of σ(T14 − T24 ) ≃ 992 W/m2 (due to the small emissivity of aluminum, see Fig. 3-7).

The plasma frequency of aluminum corresponds to a wavelength slightly less than 0.1 µm, while heat transfer at the example’s temperatures occurs at wavelengths between roughly 2.5 and 60 µm. Therefore, the spectral variations in heat flux essentially follow a Planck function pattern. Silicon carbide, on the other hand, has a band around 12 µm (see Fig. 3-13), giving rise to interesting spectral variations. Example 24.4. Determine the spectral radiative flux between two plates of silicon carbide, separated by a 10 nm vacuum gap, assuming that the dielectric function of SiC obeys the Lorentz model with parameters given by Fig. 3-13. The plates are again isothermal and maintained at 400 K and 300 K, respectively. Distinguish contributions from propagating and evanescent waves, as well as the influence of parallel and perpendicular polarizations, and compare with the far-field solution. Solution As noted in Chapter 3, the dielectric function of SiC is well-described by the single oscillator Lorentz model of equation (3.63), with ε0 = 6.7, νpi = 4.327 × 1013 Hz, νi = 2.380 × 1013 Hz (corresponding to a wavenumber of 793 cm−1 ), and γi = 1.428 × 1011 Hz. Aside from the different dielectric function

814

24 NANOSCALE RADIATIVE TRANSFER

10−7

evanescent, TM evanescent, TE evanescent, TM+TE propagating, TM propagating, TE propagating, TM+TE total, TM+TE far field, TM+TE

−8

2

Spectral radiative flux Qν ,net, W/m Hz

10

−9

10

−10

10

−11

10

−12

10

−13

10

−14

10

D

−15

10

10−16 600

800

1000

= 10nm, T1 = 400K, T2 = 300K

1200 1400 1600 Frequency, ν /C0 = η 0, cm−1

1800

2000

FIGURE 24-8 Spectral radiative heat fluxes between silicon carbide plates separated by a 10 nm vacuum microgap.

and the fixed gap width, the solution proceeds as in the previous example, but without carrying out the actual integration over frequency. Results are shown in Fig. 24-8 for the spectral region between 600 cm−1 and 2,000 cm−1 surrounding the resonance band of SiC. It is seen that the TE evanescent wave has a maximum at the resonance frequency of 793 cm−1 , before dropping by several orders of magnitude similar to the propagating waves. On the other hand, the TM evanescent wave has a maximum at 969 cm−1 (corresponding to the wavelength with near-zero reflectivity in Fig. 3-13). The far-field flux follows the behavior given in Fig. 3-13, i.e., flux decreases over wavelengths with large reflectivities.

A number of researchers have investigated near-field radiative transfer theoretically, primarily looking at different aspects of the heat flow across plane-parallel gaps [17, 20–26]. Other geometries that have also received attention are spheres in close contact with flat plates [18,19,27], and with another sphere [28–31].

24.8 EXPERIMENTS ON NANOSCALE RADIATION It has been recognized for some time that radiative heat transfer can exceed blackbody limits at the nanoscale, and thus plays an important role in a number of applications, such as near-field microscopy, nanoelectronics thermal management, photovoltaics, etc. Correspondingly, the problem of heat transfer between closely spaced objects has been studied theoretically in some detail, as outlined in the previous sections. On the other hand, experimental verification has been limited, mostly because of the difficulties of maintaining a precise nanoscale gap between the emitter and receiver. The earliest experiments were carried out in the field of cryogenic insulation by Domoto and coworkers [32] (accompanied by some theoretical attempts [20, 21]), and by Hargreaves [33, 34]. At cryogenic temperatures, say below 10 K, according to Wien’s displacement law, equation (1.16), heat transfer is maximized around a wavelength of 300 µm, i.e., even plates tens of µm apart should display tunneling effects. Domoto and coworkers measured heat flow between two copper plates as close as 10 µm together, and at temperatures between 5 K and 15 K. While the measured heat transfer was only about 3% of that between blackbodies (because of copper’s small emittance), and agreement with their model was only fair, they were able to show that—contrary to far-field analysis—the heat transfer increased

24.8 EXPERIMENTS ON NANOSCALE RADIATION

815

12

DT = 19.0 K

Heat transfer coefficient

HR ,W/m 2K

10

DT = 15.0 K

8

6

DT = 11.2 K

4

DT = 6.8 K DT = 19.0 K DT = 15.0 K DT = 11.2 K DT = 6.8 K

2

0 1

10 Separation D, m m

100

FIGURE 24-9 Heat transfer coefficients between sapphire plates separated by a vacuum microgap; curves are vertically offset by 2 W/m2 K, with respective zeros indicated by the horizontal lines extending from the left axis. Solid lines = predictions from equation (24.29); dashed lines = predictions for slightly convex plates.

by a factor of 2.3 between the far field and their closest spacing of 10 µm. Hargreaves carried out similar experiments, using chromium plates with vacuum gaps down to 1.5 µm. He was able to demonstrate a factor of five heat transfer increase from far field to near field (but still considerably less than the blackbody limit). Small gaps are more easily achieved by moving a small tip close to a surface. For example, Xu et al. [35] tried to measure near-field radiative transfer by moving a 100 µm diameter indium probe of a scanning thermal microscope as close as 12 nm to a thermocouple probe, but could not detect any substantial increase in heat transfer. Kittel and colleagues [36] used a scanning tunneling microscope (STM) to measure near-field radiation between the thermocouple tip and a plate, observing the expected 1/d3 increase in heat transfer down to a gap width of 10 nm. Below that distance, there was disagreement between theory and experiment. Narayanaswamy et al. [27] measured near-field radiation with a bimetallic atomic force microscope (AFM) cantilever with a silica microsphere at its tip. The plate was heated to maintain a temperature difference with the sphere, leading to near-field radiative transfer rates in the order of nW, which was measured by monitoring the deflection of the bimetallic cantilever. Their measurements confirmed that the near-field radiation between the flat surface and the microsphere was more than two orders of magnitude larger than between blackbodies, with a 1/d-dependence. Successful measurements between parallel plates have been carried out by Hu and coworkers [27]. They employed two precise optical glass flats spaced a fixed 1.6 µm apart by using polystyrene spacer beads. Applying various temperature differences they measured heat transfer rates approximately 35% higher than the blackbody limit, and observed good agreement with theoretical predictions. Very recently, Ottens et al. [37] carried out high-precision heat transfer measurements between two sapphire plates spaced a variable distance as little as 2 µm apart. They also used cryogenic temperatures to emphasize near-field effects. Figure 24-9 shows the pertinent results of their experiments, compared with theoretical results from equation (24.29), displayed in the form of a heat transfer coefficient, i.e., hr = qnet /(T1 − T2 ). Agreement between theory and experiment is good, except for a slight systematic error, which may be due to imperfect flatness of the plates, as demonstrated by the dashed lines, which correspond to near-field radiative heat transfer between two convex plates, each having a radius of curvature of ≃ 1 km. Note that the highest heat transfer coefficient measured, 8.5 W/m2 K for the ∆T = 6.8 K case, exceeds the blackbody limit of σ(T14 − T24 )/(T1 − T2 ) ≃ 6.7 W/m2 K.

816

24 NANOSCALE RADIATIVE TRANSFER

References 1. Rytov, S. M.: “Correlation theory of thermal fluctuations in an isotropic medium,” Soviet Physics JETP, vol. 6, no. 1, pp. 130–140, 1958. 2. Rytov, S. M., Y. A. Kravtsov, and V. I. Tatarskii: Principles of Statistical Radiophysics III: Elements of Random Fields, Springer Verlag, Berlin, 1987. 3. Chen, G.: Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons, Oxford University Press, New York, 2005. 4. Novotny, L., and B. Hecht: Principles of Nano-Optics, Cambridge University Press, New York, 2006. 5. Zhang, Z. M.: Nano/Microscale Heat Transfer, McGraw-Hill, New York, 2007. 6. Basu, S., Z. H. Zhang, and C. J. Fu: “Review of near-field thermal radiation and its application to energy conversion,” International Journal of Energy Research, vol. 33, pp. 1203–1232, 2009. 7. Zhang, Z. H., and K. Park: “Fundamentals and applications of near-field radiative energy transfer,” in ThermalFluidsPedia, Thermal-Fluids Central, 2012, https://www.thermalfluidscentral.org/encyclopedia/index.php/Nearfield thermal radiation. 8. Carminati, R., and J.-J. Greffet: “Near-field effect in spatial coherence of thermal sources,” Physics Review Letters, vol. 82, no. 8, pp. 1660–1663, 1999. 9. Cravalho, E. G., C. L. Tien, and R. P. Caren: “Effect of small spacings on radiative transfer between two dielectrics,” ASME Journal of Heat Transfer, vol. 89, pp. 351–358, 1967. 10. Kawata, S. (ed.): Near-Field Optics and Surface Plasmon Polaritons, Springer, Berlin, 2001. 11. Tominaga, J., and D. P. Tsai (eds.): Optical Nanotechnologies — The Manipulation of Surface and Local Plasmons, Springer, Berlin, 2003. 12. Homola, J., S. S. Yee, and G. Gauglitz: “Surface plasmon resonance sensors: Review,” Sensors and Actuators B, vol. 54, pp. 3–15, 1999. 13. Greffet, J.-J., R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen: “Coherent emission of light by thermal sources,” Nature, vol. 416, pp. 61–64, 2002. 14. Hillenbrand, R., T. Taubner, and F. Kellmann: “Phonon-enhanced light–matter interaction at the nanometer scale,” Nature, vol. 418, pp. 159–162, 2002. 15. Raether, H. (ed.): Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer, Berlin, 1988. 16. Rupin, R.: “Surface polaritons of a left-handed medium,” Physics Letters A, vol. 277, pp. 61–64, 2000. 17. Fu, C. J., and Z. H. Zhang: “Nanoscale radiation heat transfer for silicon at different doping levels,” International Journal of Heat and Mass Transfer, vol. 49, pp. 1703–1718, 2006. 18. Mulet, J. P., K. Joulain, R. Carminati, and J.-J. Greffet: “Nanoscale radiative heat transfer between a small particle and a plane surface,” Applied Physics Letters, vol. 78, pp. 2931–2933, 2001. 19. Mulet, J. P., K. Joulain, R. Carminati, and J.-J. Greffet: “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophysical Engineering, vol. 6, pp. 209–222, 2002. 20. Cravalho, E. G., C. L. Tien, and R. P. Caren: “Effect of small spacings on radiative transfer between two dielectrics,” ASME Journal of Heat Transfer, vol. 89, pp. 351–358, 1967. 21. Boehm, R. F., and C. L. Tien: “Small spacing analysis of radiative transfer between parallel metallic surfaces,” ASME Journal of Heat Transfer, vol. 92, pp. 412–417, 1970. 22. Narayanaswamy, A., and G. Chen: “Thermal radiation in 1D photonic crystals,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 93(1-3), pp. 175–183, 2005. 23. Hu, L., A. Narayanaswamy, X. Chen, and G. Chen: “Near-field thermal radiation between two closely spaced glass plates exceeding Planck’s blackbody radiation law,” Applied Physics Letters, vol. 92, p. 133106, 2008. 24. Narayanaswamy, A., S. Shen, L. Hu, X. Chen, and G. Chen: “Breakdown of the Planck blackbody radiation law at nanoscale gaps,” Applied Physics A, vol. 96, pp. 357–362, 2009. 25. Basu, S., B. J. Lee, and Z. M. Zhang: “Near-field radiation calculated with an improved dielectric function model for doped silicon,” ASME Journal of Heat Transfer, vol. 132, no. 2, p. 023302, 2010. 26. Rousseau, E., M. Laroche, and J.-J. Greffet: “Radiative heat transfer at nanoscale: Closed-form expression for silicon at different doping levels,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 111, no. 7-8, pp. 1005–1014, 2010. 27. Narayanaswamy, A., S. Shen, and G. Chen: “Near-field radiative heat transfer between a sphere and a substrate,” Physical Review B, vol. 72, p. 115303, 2008. 28. Volokitin, A. I., and B. N. J. Persson: “Radiative heat transfer between nanostructures,” Physical Review B, vol. 63, p. 205404, 2001. 29. Volokitin, A. I., and B. N. J. Persson: “Resonant photon tunneling enhancement of the radiative heat transfer,” Physical Review B, vol. 69, p. 045417, 2004. 30. Domingues, G., S. Volz, K. Joulain, and J.-J. Greffet: “Heat transfer between two nanoparticles through near field interaction,” Physical Review Letters, vol. 94, p. 085901, 2005. 31. Narayanaswamy, A., and G. Chen: “Thermal near-field radiative transfer between two spheres,” Physical Review B, vol. 77, p. 075125, 2005. 32. Domoto, G. A., R. F. Boehm, and C. L. Tien: “Experimental investigation of radiative transfer between metallic surfaces at cryogenic temperatures,” ASME Journal of Heat Transfer, vol. 92, pp. 405–411, 1970. 33. Hargreaves, C. M.: “Anomalous radiative transfer between closely-spaced bodies,” Physics Letters A, vol. 30, pp. 491–492, 1969.

PROBLEMS

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34. Hargreaves, C. M.: “Radiative transfer between closely spaced bodies,” Technical Report 5, Philips Research Report, 1973. 35. Xu, J. B., K. Lauger, R. Moller, K. Dransfeld, and I. H. Wilson: “Heat transfer between two metallic surfaces at small distances,” Journal of Applied Physics, vol. 76, pp. 7209–7216, 1994. 36. Kittel, A., W. Muller-Hirsch, J. Parisi, S. Biehs, D. Reddig, and M. Holthaus: “Near-field heat transfer in a scanning ¨ thermal microscope,” Physical Review Letters, vol. 95, p. 224301, 2005. 37. Ottens, R. S., V. Quetschke, S. Wise, A. A. Alemi, R. Lundock, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting: “Near-field radiative heat transfer between macroscopic planar surfaces,” Physical Review Letters, vol. 107, no. 1, p. 014301, 2011.

Problems 24.1 Show that the transmissivity of a thin dielectric film, surrounded by two identical, but different dielectrics, is described by equation (24.13) for incidence angles θ1 > θc . Solve the problem separately for both TM and TE waves. 24.2 Consider an interface in the x-y-plane at z = 0 between two dielectrics (n1 , z < 0 and n2 < n1 , z > 0), and determine the z-component of the Poynting vector in Medium 2 for incidence in Medium 1 at angles exceeding the critical angle. Show that the time average of the Poynting vector is zero.

APPENDIX

A

CONSTANTS AND CONVERSION FACTORS

TABLE A.1

Physical constants. Speed of light in vacuum First Planck function constant Second Planck function constant Wien’s constant Electron charge Planck’s constant Modified Planck’s constant Boltzmann’s constant Electron rest mass Neutron rest mass Proton rest mass Avogadro’s number Solar constant (at mean RSE ) Radius of Earth (mean) Radius of solar disk Earth–sun distance (mean) Universal gas constant Effective surface T of sun Molar volume of ideal gas (at 273.15 K, 101.325 kPa) Electrical permittivity of vacuum Magnetic permeability of vacuum Stefan–Boltzmann constant

c0 C1 C2 C3 e h ~ k me mn mp NA qsol REarth Rsun RSE Ru Tsun Vmol ǫ0 µ0 σ

= 2.9979×108 m/s = 3.7418×10−16 W m2 = 2πhc20 = 14,388 µm K = hc 0 /k = 2897.8 µm K = 1.6022×10−19 C = 6.6261×10−34 J s = 1.0546×10−34 J s = h/2π = 1.3807×10−23 J/K = 9.1094×10−31 kg = 1.6749×10−27 kg = 1.6726×10−27 kg = 6.0221×1023 molecules/mol = 1367 W/m2 = 6.371×106 m = 6.955×108 m = 1.4960×1011 m = 8.3145 J/mol K = 5777 K = 22.4140 ℓ/mol = 22.4140 m3 /kmol = 8.8542×10−12 C2 /N m2 = 4π×10−7 N s2 /C2 = 5.6704×10−8 W/m2 K4

818

A CONSTANTS AND CONVERSION FACTORS

TABLE A.2

Conversion factors. Acceleration Area Diffusivity Energy

Specific heat Temperature

1 m/s2 1 m2 1 m2 /s 1J 1 eV = 1.6022×10−19 J 1N 1W 1 W/m2 1 W/m3 1 W/m2 K 1 W/m2 sr 1 m2 /s 1 J/kg 1m 1 km 1 kg 1 kg/m3 1 kg/s 1W 1 Pa = 1 N/m2 1.0133×105 N/m2 1 J/kg K T(K)

Temperature difference Thermal conductivity Thermal resistance Velocity and speed

1K 1 W/m K 1 K/W 1 m/s

Viscosity (dynamic) Volume

1 N s/m2 = 1 kg/s m 1 m3

Volume flow rate

1 m3 /s

Force Heat transfer rate Heat flux Heat generation rate Heat transfer coefficient Intensity Kinematic viscosity Latent heat Length Mass Mass density Mass flow rate Power Pressure and stress

= 4.2520×107 ft/h2 = 1550.0 in2 = 10.764 ft2 = 3.875×104 ft2 /h = 9.4787×10−4 Btu = 1.5187×10−22 Btu = 0.22481 lb f = 3.4123 Btu/h = 0.3171 Btu/h ft2 = 0.09665 Btu/h ft3 = 0.17612 Btu/h ft2 ◦ F = 0.3171 Btu/h ft2 sr = 3.875×104 ft2 /h = 4.2995×10−4 Btu/lbm = 39.370 in = 3.2808 ft = 0.62137 mi = 2.2046 lbm = 0.062428 lbm /ft3 = 7936.6 lbm /h = 3.4123 Btu/h = 1.4504×10−4 lb f /in2 = 1 standard atmosphere = 2.3886×10−4 Btu/lbm ◦ F = (5/9)T(◦ R) = (5/9)(T(◦ F) + 459.67) = T(◦ C) + 273.15 = 1◦ C = (9/5)◦ R = (9/5)◦ F = 0.57782 Btu/h ft ◦ F = 0.52750 ◦ F h/Btu = 3.2808 ft/s = 2.2364 mph = 2419.1 lbm /ft h = 6.1023×104 in3 = 35.314 ft3 = 1.2713×105 ft3 /h = 2.1189×103 ft3 /min

TABLE A.3

Conversion factors for spectral variables. Wavelength to energy to frequency to wavenumber Energy to frequency to wavelength to wavenumber Wavenumber to energy to frequency to wavelength Frequency to energy to wavelength to wavenumber

a µm = a × 103 nm a µm = a × 104 Å a µm a eV a eV a eV a cm−1 a cm−1 a cm−1 a Hz a Hz a Hz

= ˆ 1.240/a eV = ˆ 2.9979×1014 /a Hz = ˆ 104 /a cm−1 = ˆ 2.418×1014 a Hz = ˆ 1.240/a µm = ˆ 8.066×103 a cm−1 = ˆ 1.240×10−4 a eV = ˆ 2.9979×1010 a Hz = ˆ 10+4 /a µm = ˆ 4.136×10−15 a eV = ˆ 2.9979×1014 /a µm = ˆ 3.336×10−11 a cm−1

819

APPENDIX

B TABLES FOR RADIATIVE PROPERTIES OF OPAQUE SURFACES In this appendix, tables of total normal emittances, as well as a number of total normal solar absorptances, are given. The data have been collected from several surveys [1–8] that, in turn, have assembled their data from a multitude of references dating back all the way into the 1920s. As seen from the tables, there can sometimes be considerable differences in total emittance for ostensibly the same material, as reported by different researchers. While these discrepancies are partially due to varying accuracy, the primary reason is, as outlined in Chapter 3, the fact that surface layers, surface roughness, oxidation, etc., strongly affect the emittance of materials. Therefore, it should be realized that the total normal emittance or absorptance of a given surface may, in actuality, differ considerably from these reported values. In estimating the total hemispherical emittance from total normal data, one should keep in mind that: 1. Materials with high emittance tend to behave like dielectrics, resulting in a hemispherical emittance that is 3% to 5% smaller than the normal one (cf. Fig. 3-19). 2. Materials with low emittance tend to behave like metals, resulting in hemispherical emittances that may be up to 25% larger than normal ones (cf. Fig. 3-9).

References 1. Edwards, D. K., A. F. Mills, and V. E. Denny: Transfer Processes, 2nd ed., Hemisphere/McGraw-Hill, New York, 1979. 2. Hottel, H. C.: “Radiant heat transmission,” in Heat Transmission, ed. W. H. McAdams, 3rd ed., ch. 4, McGraw-Hill, New York, 1954. 3. Hottel, H. C., and A. F. Sarofim: Radiative Transfer, McGraw-Hill, New York, 1967. 4. Gubareff, G. G., J. E. Janssen, and R. H. Torborg: “Thermal radiation properties survey,” Honeywell Research Center, Minneapolis, MI, 1960. 5. Wood, W. D., H. W. Deem, and C. F. Lucks: Thermal Radiative Properties, Plenum Publishing Company, New York, 1964. 6. Touloukian, Y. S., and D. P. DeWitt (eds.): Thermal Radiative Properties: Metallic Elements and Alloys, vol. 7 of Thermophysical Properties of Matter, Plenum Press, New York, 1970. 7. Touloukian, Y. S., and D. P. DeWitt (eds.): Thermal Radiative Properties: Nonmetallic Solids, vol. 8 of Thermophysical Properties of Matter, Plenum Press, New York, 1972. 8. Svet, D. I.: Thermal Radiation: Metals, Semiconductors, Ceramics, Partly Transparent Bodies, and Films, Plenum Publishing Company, New York, 1965. 9. Gale, W. F., and T. C. Totemeier (eds.): Smithells Metals Reference Book, 8th ed., Butterworth-Heinemann, Oxford, 2002.

820

B TABLES FOR RADIATIVE PROPERTIES OF OPAQUE SURFACES

821

TABLE B.1

Total emittance and solar absorptance of selected surfaces (compiled by Edwards et al. [1]). Temperature [◦ C]

Total normal emittance

Extraterrestrial solar absorptance

Alumina, flame-sprayed Aluminum foil, as received Bright dipped Aluminum, vacuum-deposited on mylar Aluminum alloy 6061, as received Aluminum alloy 75S-T6, weathered 20,000 h on a DC6 aircraft Aluminum, hard-anodized, 6061-T6 Aluminum, soft-anodized, Reflectal alloy Aluminum, 7075-T6, sandblasted with 60 mesh silicon carbide grit Aluminized silicone resin paint Dow Corning XP-310

−25 20 20 20 20

0.80 0.04 0.025 0.025 0.03

0.28

65 −25 −25

0.16 0.84 0.79

0.54 0.92 0.23

20 95 425

0.30 0.20 0.22

0.55 0.27

Beryllium

150 370 600 150 370 600

0.18 0.21 0.30 0.90 0.88 0.82

0.77

−25

0.95

0.975

0.93

0.94

−25 95 425

0.89 0.81 0.80

0.95

95 400 35

0.12 0.15 0.15

0.78

20 35

0.03 0.16

0.47 0.91

Glass, second surface mirror Aluminized Silvered

−25 −25

0.83 0.83

0.13 0.13

Gold, coated on stainless steel Heated in air at 540◦ C Coated on 3M tape Y9814

95 400 20

0.09 0.14 0.025

0.21

Graphite, crushed on sodium silicate

−25

0.91

0.96

Inconel X, oxidized 4 h at 1000 C Oxidized 10 h at 700◦ C

−25 95 425

0.71 0.81 0.79

0.90

Magnesium–thorium alloy

95 260 370

0.07 0.06 0.36

Beryllium, anodized

Black paint, Parson’s optical black Black silicone, high-heat National Lead Co. 46H47 Black epoxy paint, Cat-a-lac Finch Paint and Chem. Co. 463-1-8 Black enamel paint, Rinshed-Mason Heated 1000 h at 375◦ C in air Chromium plate Heated 50 h at 600◦ C Copper, electroplated Black-oxidized in Ebonol C



Magnesium, Dow 7 coating

−25 to 750

0.10 0.10 0.37

822

B TABLES FOR RADIATIVE PROPERTIES OF OPAQUE SURFACES

TABLE B.1

Total emittance and solar absorptance of selected surfaces (cont’d). Total normal emittance

Extraterrestrial solar absorptance

20 20 20

0.37 0.63 0.81

0.17 0.17 0.24

20

0.03

0.22

35 35

0.05 0.11

0.85 0.85

95 400 95 425

0.13 0.15 0.11 0.13

Silica, Corning Glass 7940M Sintered, powdered, fused silica Silica, second surface mirror, aluminized Silvered

35 20 20

0.84 0.83 0.83

0.08 0.14 0.07

Silicon solar cell, boron-doped, no coverglass

35

0.32

0.94

95 400 95 425

0.06 0.08 0.11 0.13

20

0.24

0.20

95 425 −25 35

0.27 0.32 0.75 0.13

0.89 0.76

95

0.42

0.68

95 425 35 425 35 35 35 −25

0.10 0.19 0.21 0.25 0.16 0.20 0.14 0.73

White acrylic resin paint Sherwin-Williams M49WC8-CA-10144

95 200

0.92 0.87

White epoxy paint, Cat-a-lac Finch Paint and Chemical Co. 483-1-8

Temperature [◦ C] Mylar film, aluminized on second surface 0.0625 mm thick 0.025 mm thick 0.075 mm thick Nickel, electroplated Nickel, electro-oxidized on copper 110-30 125-30 Platinum-coated stainless steel Annealed in air 300 h at 375◦ C

Silver, plated on nickel on stainless steel Heated 300 h at 375◦ C Silver Chromatone paint Stainless steel Type 312, heated 300 h at 260◦ C Type 301 with Armco black oxide Type 410, heated to 700◦ C in air Type 303, sandblasted heavily with 80 mesh aluminum oxide grit Titanium, 75A 75A, oxidized 300 h at 450◦ C C-110M, oxidized 100 h at 425◦ C in air C-110M, oxidized 300 h at 450◦ C in air Evaporated 80–100 µm, oxidized 3 h at 400◦ C Anodized

0.80 0.52 0.77 0.75 0.51

−25

0.88

0.25

White potassium zirconium silicate coating

20

0.89

0.13

Zinc, blackened by electrochemical treatment

35

0.12

0.89

B TABLES FOR RADIATIVE PROPERTIES OF OPAQUE SURFACES

823

TABLE B.2

Total normal emittance of various surfaces. Temperaturea [◦ C]

Total normal emittancea

225–575 100 100 40 200–600 95–500 275–500 500–825 40

0.039–0.057 0.09 0.18 0.055–0.07 0.11–0.19 0.20–0.31 0.63–0.42 0.42–0.26 0.216

25 230–480 230–425 230–500 25 230–485 230–505 230–460

0.11, 0.10, 0.08 0.22–0.16 0.20–0.18 0.22–0.15 0.09 0.17–0.15 0.20–0.16 0.16–0.13

200–600 200–600

0.18–1.19 0.52–0.57

Antimony, polished

35–260

0.28–0.31

Beryllium, polished

1000–1200

0.37

75

0.34

245–355 255–375 275 100 40–315 22 22 50–350 200–600

0.028–0.031 0.033–0.037 0.030 0.06 0.10 0.06 0.20 0.22 0.61–0.59

40–1100

0.08–0.36

80 115 100 19 22 25 200–600 800–1100 1075–1275

0.018 0.023 0.052 0.030 0.072 0.78 0.57 0.66–0.54 0.16–0.13

A. Metals and their oxides Aluminum Highly polished plate, 98.3% pure Commercial sheet Rough polish Rough plate Oxidized at 600◦ C Heavily oxidized Aluminum oxide Al-surfaced roofing Aluminum alloysb Alloy 75 ST: A, B1 , C Alloy 75 ST: Ac Alloy 75 ST: B1 c Alloy 75 ST: Cc Alloy 24 ST: A, B1 , C Alloy 24 ST: Ac Alloy 24 ST: B1 c Alloy 24 ST: Cc Calorized surfaces, heated at 600◦ C Copper Steel

Bismuth, bright Brass Highly polished 73.2% Cu, 26.7% Zn 62.4% Cu, 36.8% Zn, 0.4% Pb, 0.3% Al 82.9% Cu, 17.0% Zn Polished Rolled plate, natural surface Rolled plate, rubbed with coarse emery Dull plate Oxidized by heating at 600◦ C Chromium, polished Copper Carefully polished electrolytic copper Polished Commercial emeried, polished, pits remaining Commercial, scraped shiny, not mirror-like Plate heated long time, with thick oxide layer Plate heated at 600◦ C Cuprous oxide Molten copper

824

B TABLES FOR RADIATIVE PROPERTIES OF OPAQUE SURFACES

TABLE B.2

Total normal emittance of various surfaces (cont’d). Temperaturea [◦ C]

Total normal emittancea

25 230–400 230–425 230–405

0.15, 0.15, 0.12 0.24–0.20 0.16 0.21–0.18

800

0.55

225–625

0.018–0.035

1400

0.45

Inconel: Types X and B: surface A, B2 , C Type X: surface Ac Type X: surface B2 c Type X: surface Cc Type B: surface Ac Type B: surface B2 c Type B: surface Cc

25 230–880 230–855 230–900 230–880 230–950 230–1000

0.19–0.21 0.55–0.78 0.60–0.75 0.62–0.73 0.35–0.55 0.32–0.51 0.35–0.40

Iron and steel (not including stainless) Metallic surfaces (or very thin oxide layer) Electrolytic iron, highly polished Steel, polished Iron, polished Iron, roughly polished Iron, freshly emeried Cast iron, polished Cast iron, newly turned Cast iron, turned and heated Wrought iron, highly polished Polished steel casting Ground sheet steel Smooth sheet iron Mild steelb : A, B2 , C Mild steelb : Ac Mild steelb : B2 c Mild steelb : Cc

175–225 100 425–1025 100 20 200 22 880–990 40–250 770–1035 935–1100 900–1040 25 230–1065 230–1050 230–1065

0.052–0.064 0.066 0.14–0.38 0.17 0.24 0.21 0.44 0.60–0.70 0.28 0.52–0.56 0.55–0.61 0.55–0.60 0.12, 0.15, 0.10 0.20–0.32 0.34–0.35 0.27–0.31

Oxidized surfaces Iron plate, pickled, then rusted red Iron plate, completely rusted Iron, dark gray surface Rolled sheet steel Oxidized iron Cast iron, oxidized at 600◦ C Steel, oxidized at 600◦ C Smooth, oxidized electrolytic iron Iron oxide Rough ingot iron Sheet steel, strong, rough oxide layer Dense, shiny oxide layer

20 20 100 21 100 200–600 200–600 125–525 500–1200 925–1115 25 25

0.61 0.69 0.31 0.66 0.74 0.64–0.78 0.79 0.78–0.82 0.85–0.89 0.87–0.95 0.80 0.82

Dow metal:b A; B1 ; C Ac B1 c Cc Germanium, polished Gold, pure, highly polished Hafnium, polished b

B TABLES FOR RADIATIVE PROPERTIES OF OPAQUE SURFACES

825

TABLE B.2

Total normal emittance of various surfaces (cont’d). Temperaturea [◦ C] 23 23 40–250 20–360 40–370

Total normal emittancea 0.80 0.82 0.95 0.94 0.94–0.97

1300–1400 1600–1800

0.29 0.28

1560–1710 1500–1650 1520–1650 1515–1770 1520–1690

0.27–0.39 0.42–0.53 0.43–0.40 0.42–0.45 0.40–0.41

125–225 25 200

0.057–0.075 0.28 0.63

275–825 900–1705 35–260

0.55–0.20 0.20 0.07–0.13

0–100

0.09–0.12

725–2595 100 35–260 540–1370 2750

0.096–0.202 0.071 0.05–0.08 0.10–0.18 0.29

Monel metalb Oxidized at 600◦ C K Monel 5700: A, B2 , C K Monel 5700: Ac K Monel 5700: B2 c K Monel 5700: Cc

200–600 25 230–875 230–955 230–975

0.41–0.46 0.23, 0.17, 0.14 0.46–0.65 0.54–0.77 0.35–0.53

Nickel Electroplated, polished Technically pure (98.9% Ni, + Mn), polished Polished Electroplated, not polished Wire Plate, oxidized by heating at 600◦ C Nickel oxide

23 225–375 100 20 185–1005 200–600 650–1255

0.045 0.07–0.087 0.072 0.11 0.096–0.186 0.37–0.48 0.59–0.86

Nickel alloys Chromnickel Copper–nickel, polished Nichrome wire, bright

50–1035 100 50–1000

0.64–0.76 0.059 0.65–0.79

Cast plate, smooth Cast plate, rough Cast iron, rough, strongly oxidized Wrought iron, dull oxidized Steel plate, rough Molten surfaces Cast iron Mild steel Steel, several different kinds with 0.25– 1.2% C (slightly oxidized surface) Steel Pure iron Armco iron Lead Pure (99.96%), unoxidized Gray oxidized Oxidized at 150◦ C Magnesium Magnesium oxide Magnesium, polished Mercury Molybdenum Filament Massive, polished Polished

826

B TABLES FOR RADIATIVE PROPERTIES OF OPAQUE SURFACES

TABLE B.2

Total normal emittance of various surfaces (cont’d).

Nichrome wire, oxidized Nickel–silver, polished Nickelin (18–32% Ni; 55–68% Cu; 20% Zn), gray oxidized Type ACI-HW (60% Ni; 12% Cr), smooth, black, firm adhesive oxide coat from service Platinum Pure, polished plate Strip Filament Wire Silver Polished, pure Polished Stainless steelb Polished Type 301: A, B2 , C Type 301: Ac Type 301: B2 c Type 301: Cc Type 316: A, B2 , C Type 316: Ac Type 316: B2 c Type 316: Cc Type 347: A, B2 , C Type 347: Ac Type 347: B2 c Type 347: Cc Type 304: (8% Cr; 18% Ni) Light silvery, rough, brown after heating After 42 h heating at 525◦ C Type 310 (25% Cr; 20% Ni), brown, splotched, oxidized from furnace service Allegheny metal no. 4, polished Allegheny alloy no. 66, polished Tantalum filament Thorium oxide Tin Bright tinned iron Bright Commercial tin-plated sheet iron Tungsten Filament, aged Filament Polished coat Yttrium

Temperaturea [◦ C] 50–500 100

Total normal emittancea 0.95–0.98 0.135

20

0.262

270–560

0.89–0.82

225–625 925–1625 27–1225 225–1375

0.054–0.104 0.12–0.17 0.036–0.192 0.073–0.182

225–625 40–370 100

0.020–0.032 0.022–0.031 0.052

100 25 230–950 230–940 230–900 25 230–870 230–1050 230–1050 25 230–900 230–875 230–900

0.074 0.21, 0.27, 0.16 0.57–0.55 0.54–0.63 0.51–0.70 0.28, 0.28, 0.17 0.57–0.66 0.52–0.50 0.26–0.31 0.39, 0.35, 0.17 0.52–0.65 0.51–0.65 0.49–0.64

215–490 215–525

0.44–0.36 0.62–0.73

215–525 100 100

0.90–0.97 0.13 0.11

1340–3000

0.19–0.31

275–500 500–825

0.58–0.36 0.36–0.21

25 50 100

0.043, 0.064 0.06 0.07, 0.08

27–3300 3300 100

0.032–0.35 0.39 0.066

1400

0.35

B TABLES FOR RADIATIVE PROPERTIES OF OPAQUE SURFACES

827

TABLE B.2

Total normal emittance of various surfaces (cont’d). Temperaturea [◦ C] Zinc Commercial 99.1% pure, polished 225–325 400 Oxidized by heating at 400◦ C Galvanized sheet iron, fairly bright 27 Galvanized sheet iron, gray oxidized 25 Zinc, galvanized sheet 100 B. Refractories, building materials, paints, and miscellaneous Alumina (99.5–85% Al2 O3 ; 0–12% SiO2 ; 0–1% Fe2 O3 ) Effect of mean grain size 1010–1565 10 µm 50 µm 100 µm Alumina on Inconel

540–1100

Alumina–silica (showing effect of Fe) 80–58% Al2 O3 ; 16–38% SiO2 ; 0.4% Fe2 O3 36–26% Al2 O3 ; 50–60% SiO2 ; 1.7% Fe2 O3 61% Al2 O3 ; 35% SiO2 ; 2.9% Fe2 O3

1010–1565

Asbestos Board Paper Brick Red, rough, but no gross irregularities Grog brick, glazed Building Fireclay White refractory Carbon Filament Rough plate Graphitized Candle soot Lampblack–waterglass coating Thin layer on iron plate Thick coat Lampblack, 0.075 mm or thicker Lampblack, rough deposit Lampblack, other blacks Graphite, pressed, filed surface Carborundum (87% SiC; density 2.3 g/cm3 ) Concrete tiles Concrete, rough Enamel, white fused, on iron Glass Smooth Pyrex, lead, and soda

Total normal emittancea 0.045–0.053 0.11 0.23 0.28 0.21

0.30–0.18 0.39–0.28 0.50–0.40 0.65–0.45 0.61–0.43 0.73–0.62 0.78–0.68

23 35–370

0.96 0.93–0.94

20 1100 1000 1000 1100

0.93 0.75 0.45 0.75 0.29

1040–1405 100–320 320–500 100–320 320–500 95–270 100–275 20 20 40–370 100–500 50–1000 250–510

0.526 0.77 0.77–0.72 0.76–0.75 0.75–0.71 0.952 0.96–0.95 0.927 0.967 0.945 0.84–0.78 0.96 0.98

1010–1400

0.92–0.81

1000 38

0.63 0.94

20

0.90

20 260–540

0.94 0.95–0.85

828

B TABLES FOR RADIATIVE PROPERTIES OF OPAQUE SURFACES

TABLE B.2

Total normal emittance of various surfaces (cont’d). Temperaturea [◦ C]

Total normal emittancea

Gypsum, 5 mm thick on smooth or blackened plate

20

0.903

Ice Smooth Rough crystals

0 0

0.966 0.985

Magnesite refractory brick

1000

0.38

Marble, light gray, polished

20

0.93

72 25 20 75–145 35–95 35–95 100

0.906 0.875 0.821 0.91 0.80–0.95 0.96–0.98 0.92–0.96

100 100 20 150–315

0.52 0.27–0.67 0.39 0.35

35–150

0.87–0.97

260 260 260 260

0.66 0.68, 0.75, 0.75 0.74 0.77, 0.82

260

0.29

35 20 20

0.95 0.92, 0.94 0.91

10–88

0.91

20

0.92

20 280–840 280–840 280–840

0.93 0.90–0.41 0.93–0.47 0.92–0.68

23 25

0.94 0.86

35–260

0.83–0.90

1010–1565 1010–1565

0.42–0.33 0.62–0.46

Paints, lacquers, varnishes White enamel varnish on rough iron plate Black shiny lacquer, sprayed on iron Black shiny shellac on tinned iron sheet Black matte shellac Black or white lacquer Flat black lacquer Oil paints, 16 different, all colors Aluminum paints and lacquers 10% Al, 22% lacquer body, on rough or smooth surface Other Al paints, varying age and Al content Al lacquer, varnish binder, on rough plate Al paint, after heating at 325◦ C Lacquer coatings, 0.025–0.37 mm thick on aluminum alloys Clear silicone vehicle coatings, 0.025–0.375 mm On mild steel On stainless steels, 316, 301, 347 On Dow metal On Al alloys 24 ST, 75 ST Aluminum paint with silicone vehicle, two coats on Inconel Paper White Thin, pasted on tinned or blackened plate Roofing Plaster, rough lime Porcelain, glazed Quartz Rough, fused Glass, 1.98 mm thick Glass, 6.88 mm thick Opaque Rubber Hard, glossy plate Soft, gray, rough (reclaimed) Sandstone Silica (98% SiO2 ; Fe-free), grain size 10 µm 70–600 µm

B TABLES FOR RADIATIVE PROPERTIES OF OPAQUE SURFACES

829

TABLE B.2

Total normal emittance of various surfaces (cont’d).

Silicon carbide

Temperaturea [◦ C] 150–650

Total normal emittancea 0.83–0.96

Slate

35

0.67–0.80

Soot, candle

90–260

0.95

Water

0–100

0.95–0.963

35 20 70

0.75 0.90 0.94

240–500 500–830

0.92–0.80 0.80–0.52

Wood, sawdust Oak, planed Beech Zirconium silicate a

Temperatures and emittances in pairs separated by dashes correspond; use linear interpolation.

b

Surface treatment: A, cleaned with toluene, then methanol; B1 , cleaned with soap and water, toluene, then methanol; B2 , cleaned with abrasive soap and water, toluene, and methanol; C, polished, then cleaned with soap and water.

c

Results after repeated heating and cooling.

TABLE B.3

Spectral, normal emittance of metals at room temperature [9]. Metal Aluminum Antimony Bismuth Cadmium Chromium Cobalt Copper Gold Iridium Iron Lead Magnesium Molybdenum Nickel Niobium Palladium Platinum Rhodium Silver Tantalum Tellurium Tin Titanium Tungsten Vanadium Zinc

0.5 0.75 0.45 0.36 0.45 0.49 0.28 0.42 0.40 0.24 0.03 0.62 0.43–0.59 -

0.6 0.47 0.76 0.44 0.080 0.080 0.48 0.27 0.55 0.37 0.36 0.21 0.03 0.55 0.51 0.44–0.49 0.42–0.57 0.42–0.58

Wavelength, µm 1.0 3.0 0.08–0.27 0.03–0.12 0.45 0.35 0.72 0.26 0.30 0.07 0.43 0.30 0.32 0.23 0.030 0.026 0.020 0.015 0.22 0.09 0.41 0.26 0.20 0.42 0.19 0.27 0.12 0.29 0.14 0.28 0.12 0.24 0.11 0.16 0.08 0.03 0.02 0.22 0.08 0.50 0.47 0.46 0.32 0.37–0.49 0.25–0.33 0.40 0.07 0.36–0.50 0.10–0.17 0.50–0.61 0.08

5.0 0.03–0.08 0.31 0.12 0.04 0.19 0.15 0.024 0.015 0.06 0.08 0.14 0.16 0.06 0.06 0.10 0.06 0.07 0.02 0.07 0.43 0.24 0.10–0.18 0.05 0.07–0.11 0.05

10.0 0.02–0.04 0.28 0.08 0.02 0.08 0.04 0.021 0.015 0.04 0.06 0.07 0.15 0.04 0.04 0.03 0.05 0.05 0.02 0.06 0.22 0.14 0.05–0.12 0.03 0.06–0.09 0.03

830

B TABLES FOR RADIATIVE PROPERTIES OF OPAQUE SURFACES

TABLE B.4

Total, normal emittance of metals for elevated temperatures [9]. Metal Aluminum Beryllium Bismuth Chromium Cobalt Copper Germanium Gold Hafnium Iron Lead Magnesium Mercury Molybdenum Nickel Niobium Palladium Platinum Rhenium Rhodium Silver Tantalum Tin Titanium Tungsten α-Uranium γ-Uranium Zinc Zirconium

100 0.038 0.06 0.08 0.15–0.24 0.02 0.07 0.63 0.12h 0.12 0.08 0.02–0.03 0.04 0.07 0.11 0.07 -

500 0.064 0.11–0.14 0.34–0.46 0.02 0.54 0.02 0.14 0.13 0.09–0.15 0.06 0.086 0.035 0.02–0.03 0.06 0.05 0.33h -

Temperature, [◦ C] 1000 1200 0.55 0.87 0.12m 0.30 0.24 0.19 0.22 0.14–0.22 0.12 0.14 0.12 0.15 0.14 0.16 0.22 0.25 0.07 0.08 0.11 0.13 0.11 0.14 0.29–0.40h 0.22 0.25

Alloys Brass Cast iron, cleaned Nichrome Steel, polished cleaned

0.059 0.21 0.13–0.21 0.21–0.38

0.95 0.18–0.26 0.25–0.42

0.98 0.55–0.80 0.50–0.77

h

Total, hemispherical emittance

m

Value for molten state

-

1400 0.31 0.24 0.16 0.27 0.09 0.15 0.17 0.27

1600 0.32 0.27 0.18 0.29 0.18 0.19 -

2000 0.21 0.23 0.23 -

-

0.29m -

-

831

TABLE B.5

B TABLES FOR RADIATIVE PROPERTIES OF OPAQUE SURFACES

Spectral, normal emittance of metals at a wavelength of 0.65 µm [9]. Temperature, [◦ C] Metal 600 Chromium Cobalt Copper Erbium Gold 0.16–0.18 Iridium Iron Manganese Molybdenum Niobium Osmium Palladium Platinum Rhenium Rhodium Ruthenium Silicon Silver Tantalum 0.47 Thorium Titanium Tungsten Uranium Zirconium -

800 0.11 0.16–0.19 0.37 0.37–0.43 0.40 0.29–0.31 0.25 0.63 0.055 0.46 0.48 0.19–0.36 -

1000 0.33–0.38 0.10 0.55 0.16–0.21 0.36 0.36 0.36–0.42 0.37 0.52 0.37 0.29–0.31 0.22 0.42 0.57 0.055 0.45 0.38 0.48 0.46–0.48 0.19–0.36 0.48

1200 0.34–0.37 0.10m 0.55 0.13m 0.34 0.35 0.59 0.35–0.42 0.37 0.44 0.34 0.29–0.31 0.19 0.35 0.52 0.44 0.38 0.48 0.43–0.48 0.34m 0.45

1400 0.35–0.37 0.11m 0.55 0.32 0.35 0.59m 0.34–0.41 0.37 0.40 0.30 0.29–0.31 0.42 0.18 0.32 0.46 0.42 0.38 0.47 0.42–0.47 0.34m 0.42

1600 0.39 0.37m 0.12m 0.38m 0.37m 0.34–0.41 0.37 0.38 0.37m 0.29–0.31 0.42 0.16 0.31 0.48m 0.41 0.42–0.47 0.39

Alloys Cast iron Nichrome Steel

0.37 0.35 0.35–0.40

0.37 0.35 0.32–0.40

0.37 0.35 0.30–0.40

0.37 0.35 -

0.40m 0.37m

m

-

Value for molten state

1800 0.14m 0.33–0.40 0.37 0.38 0.41 0.31 0.40 0.41–0.47 0.36 -

2000 0.30 0.32–0.39 0.37 0.38 0.41 0.31 0.39 0.40–0.47 -

2500 0.39 0.31–0.37 0.40 0.40 0.38 0.38–0.46 -

3000 0.36 0.36–0.45 -

TABLE B.6

Spectral, normal emittance of metals at high temperatures [9]. TemperaMetal Cobalt

Copper

832

B TABLES FOR RADIATIVE PROPERTIES OF OPAQUE SURFACES

Iron

Molybdenum

Nickel

Platinum Rhenium

Titanium Tungsten

Zirconium

ture [◦ C] 800 1000 1200 762 901 985 800 1000 1200 1245 1327 1727 2527 800 1000 1200 1110 1127 1537 2118 2772 750 1327 2127 2527 1127 1327 1727

Wavelength, µm 1.0 0.049 0.340 0.335 0.300 0.260 0.36 0.36 0.36 0.490 0.385 0.37 0.36 0.46 0.444 0.442

1.2 0.26 0.26 0.26 0.294 0.294 0.291 0.316 0.295 0.293 0.290 0.292 0.257 -

1.4 0.298 0.267 0.269 0.271 0.270 0.510 -

1.5 0.031 0.079 0.037 0.290 0.185 0.195 0.210 0.250 0.227 0.29 0.30 0.32 0.500 0.28 0.292 0.30 0.422 0.375

1.6 0.264 0.267 0.300 0.282 0.250 0.252 0.253 -

1.8 0.034 0.268 0.230 0.232 0.235 -

2.0 0.21 0.21 0.22 0.029 0.065 0.237 0.245 0.252 0.260 0.140 0.170 0.193 0.215 0.219 0.223 0.290 0.193 0.25 0.27 0.29 0.455 0.21 0.245 0.26 0.386 0.368 0.357

2.5 0.052 0.032 0.217 0.227 0.235 0.248 0.205 0.23 0.24 0.26 0.360 0.351

3.0 0.18 0.19 0.043 0.031 0.240 0.115 0.155 0.185 0.187 0.151 0.525 0.13 0.18 0.348 0.343 0.342

3.5 0.038 0.235 0.174 0.575 0.330

4.0 0.025 0.032 0.030 0.225 0.114 0.145 0.185 0.162 0.130 0.600 0.095 0.15 0.325 -

4.5 0.218 -

APPENDIX

C BLACKBODY EMISSIVE POWER TABLE

nλT [µm K] 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000

η/nT [cm−1 /K] 10.0000 9.0909 8.3333 7.6923 7.1429 6.6667 6.2500 5.8824 5.5556 5.2632 5.0000 4.7619 4.5455 4.3478 4.1667 4.0000 3.8462 3.7037 3.5714 3.4483 3.3333 3.2258 3.1250 3.0303 2.9412 2.8571 2.7778 2.7027 2.6316 2.5641 2.5000

Ebλ /n3 T5 [W/m2 µm K5 ] 0.02110 ×10−11 0.04846 0.09329 0.15724 0.23932 0.33631 0.44359 0.55603 0.66872 0.77736 0.87858 0.96994 1.04990 1.11768 1.17314 1.21659 1.24868 1.27029 1.28242 1.28612 1.28245 1.27242 1.25702 1.23711 1.21352 1.18695 1.15806 1.12739 1.09544 1.06261 1.02927 833

Ebη /nT3 [W/m2 cm−1 K3 ] 0.00211 ×10−8 0.00586 0.01343 0.02657 0.04691 0.07567 0.11356 0.16069 0.21666 0.28063 0.35143 0.42774 0.50815 0.59125 0.67573 0.76037 0.84411 0.92604 1.00542 1.08162 1.15420 1.22280 1.28719 1.34722 1.40283 1.45402 1.50084 1.54340 1.58181 1.61623 1.64683

f (nλT) 0.00032 0.00091 0.00213 0.00432 0.00779 0.01285 0.01972 0.02853 0.03934 0.05210 0.06672 0.08305 0.10088 0.12002 0.14025 0.16135 0.18311 0.20535 0.22788 0.25055 0.27322 0.29576 0.31809 0.34009 0.36172 0.38290 0.40359 0.42375 0.44336 0.46240 0.48085

834 nλT [µm K] 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 6000 6200 6400 6600 6800 7000 7200 7400 7600 7800 8000 8200 8400 8600 8800 9000 9200 9400 9600 9800 10,000 10,200 10,400 10,600 10,800 11,000 11,200 11,400 11,600 11,800 12,000

C BLACKBODY EMISSIVE POWER TABLE

η/nT [cm−1 /K] 2.4390 2.3810 2.3256 2.2727 2.2222 2.1739 2.1277 2.0833 2.0408 2.0000 1.9608 1.9231 1.8868 1.8519 1.8182 1.7857 1.7544 1.7241 1.6949 1.6667 1.6129 1.5625 1.5152 1.4706 1.4286 1.3889 1.3514 1.3158 1.2821 1.2500 1.2195 1.1905 1.1628 1.1364 1.1111 1.0870 1.0638 1.0417 1.0204 1.0000 0.9804 0.9615 0.9434 0.9259 0.9091 0.8929 0.8772 0.8621 0.8475 0.8333

Ebλ /n3 T5 [W/m2 µm K5 ] 0.99571 ×10−11 0.96220 0.92892 0.89607 0.86376 0.83212 0.80124 0.77117 0.74197 0.71366 0.68628 0.65983 0.63432 0.60974 0.58608 0.56332 0.54146 0.52046 0.50030 0.48096 0.44464 0.41128 0.38066 0.35256 0.32679 0.30315 0.28146 0.26155 0.24326 0.22646 0.21101 0.19679 0.18370 0.17164 0.16051 0.15024 0.14075 0.13197 0.12384 0.11632 0.10934 0.10287 0.09685 0.09126 0.08606 0.08121 0.07670 0.07249 0.06856 0.06488

Ebη /nT3 [W/m2 cm−1 K3 ] 1.67380 ×10−8 1.69731 1.71758 1.73478 1.74912 1.76078 1.76994 1.77678 1.78146 1.78416 1.78502 1.78419 1.78181 1.77800 1.77288 1.76658 1.75919 1.75081 1.74154 1.73147 1.70921 1.68460 1.65814 1.63024 1.60127 1.57152 1.54126 1.51069 1.48000 1.44933 1.41882 1.38857 1.35866 1.32916 1.30013 1.27161 1.24363 1.21622 1.18941 1.16319 1.13759 1.11260 1.08822 1.06446 1.04130 1.01874 0.99677 0.97538 0.95456 0.93430

f (nλT) 0.49872 0.51599 0.53267 0.54877 0.56429 0.57925 0.59366 0.60753 0.62088 0.63372 0.64606 0.65794 0.66935 0.68033 0.69087 0.70101 0.71076 0.72012 0.72913 0.73778 0.75410 0.76920 0.78316 0.79609 0.80807 0.81918 0.82949 0.83906 0.84796 0.85625 0.86396 0.87115 0.87786 0.88413 0.88999 0.89547 0.90060 0.90541 0.90992 0.91415 0.91813 0.92188 0.92540 0.92872 0.93184 0.93479 0.93758 0.94021 0.94270 0.94505

C BLACKBODY EMISSIVE POWER TABLE

nλT [µm K] 12,200 12,400 12,600 12,800 13,000 13,200 13,400 13,600 13,800 14,000 14,200 14,400 14,600 14,800 15,000 16,000 17,000 18,000 19,000 20,000 21,000 22,000 23,000 24,000 25,000 26,000 27,000 28,000 29,000 30,000 31,000 32,000 33,000 34,000 35,000 36,000 37,000 38,000 39,000 40,000 41,000 42,000 43,000 44,000 45,000 46,000 47,000 48,000 49,000 50,000

η/nT [cm−1 /K] 0.8197 0.8065 0.7937 0.7813 0.7692 0.7576 0.7463 0.7353 0.7246 0.7143 0.7042 0.6944 0.6849 0.6757 0.6667 0.6250 0.5882 0.5556 0.5263 0.5000 0.4762 0.4545 0.4348 0.4167 0.4000 0.3846 0.3704 0.3571 0.3448 0.3333 0.3226 0.3125 0.3030 0.2941 0.2857 0.2778 0.2703 0.2632 0.2564 0.2500 0.2439 0.2381 0.2326 0.2273 0.2222 0.2174 0.2128 0.2083 0.2041 0.2000

Ebλ /n3 T5 [W/m2 µm K5 ] 0.06145 ×10−11 0.05823 0.05522 0.05240 0.04976 0.04728 0.04494 0.04275 0.04069 0.03875 0.03693 0.03520 0.03358 0.03205 0.03060 0.02447 0.01979 0.01617 0.01334 0.01110 0.00931 0.00786 0.00669 0.00572 0.00492 0.00426 0.00370 0.00324 0.00284 0.00250 0.00221 0.00196 0.00175 0.00156 0.00140 0.00126 0.00113 0.00103 0.00093 0.00084 0.00077 0.00070 0.00064 0.00059 0.00054 0.00049 0.00046 0.00042 0.00039 0.00036

Ebη /nT3 [W/m2 cm−1 K3 ] 0.91458 ×10−8 0.89540 0.87674 0.85858 0.84092 0.82374 0.80702 0.79076 0.77493 0.75954 0.74456 0.72998 0.71579 0.70198 0.68853 0.62643 0.57194 0.52396 0.48155 0.44393 0.41043 0.38049 0.35364 0.32948 0.30767 0.28792 0.26999 0.25366 0.23875 0.22510 0.21258 0.20106 0.19045 0.18065 0.17158 0.16317 0.15536 0.14810 0.14132 0.13501 0.12910 0.12357 0.11839 0.11352 0.10895 0.10464 0.10059 0.09677 0.09315 0.08974

835

f (nλT) 0.94728 0.94939 0.95139 0.95329 0.95509 0.95680 0.95843 0.95998 0.96145 0.96285 0.96418 0.96546 0.96667 0.96783 0.96893 0.97377 0.97765 0.98081 0.98340 0.98555 0.98735 0.98886 0.99014 0.99123 0.99217 0.99297 0.99367 0.99429 0.99482 0.99529 0.99571 0.99607 0.99640 0.99669 0.99695 0.99719 0.99740 0.99759 0.99776 0.99792 0.99806 0.99819 0.99831 0.99842 0.99851 0.99861 0.99869 0.99877 0.99884 0.99890

APPENDIX

D VIEW FACTOR CATALOGUE

In this appendix a small number of view factor relations and figures are presented. A much larger collection from a variety of references has been compiled by Howell [1, 2], from which the present list has been extracted. The latest edition of this collection can be accessed on the Internet via http://www.engr.uky.edu/rtl/Catalog/. View factors for all configurations given in this appendix, as well as those between two arbitrarily orientated rectangular plates lying in perpendicular planes, as given by equations (4.41) and (4.42), can be calculated with the stand-alone program viewfactors (prompting for user input) or from within another program through calls to Fortran function view, both given in Appendix F. A number of commercial and noncommercial computer programs are available for the evaluation of more complicated view factors [3–13]. A list of papers and monographs that either deal with evaluation methods for view factors, or present results for specified configurations (ordered by date of publication) is also given. No attempt at completeness has been made. Note: In all expressions in which inverse trigonometric functions appear, the principal value is to be taken; i.e., for any argument ξ, −

π π ≤ sin−1 ξ ≤ + ; 2 2

0 ≤ cos−1 ξ ≤ π;

1



π π ≤ tan−1 ξ ≤ + . 2 2

Differential strip element of any length z to infinitely long strip of differential width on parallel line; plane containing element does not intercept strip dA2

dFd1−d2 = dφ

φ

dA1

836

cos φ dφ 2

D VIEW FACTOR CATALOGUE

2

837

Differential planar element to differential coaxial ring parallel to the element r

R = r/l dA2

l

dFd1−d2 =

2R dR (1 + R2 )2

dA1

3

Differential planar element on and normal to ring axis to inside of differential ring

dA2 dA1

X = x/r r

dFd1−d2 =

2X dX (X 2 + 1)2

x

4

Element on surface of right-circular cylinder to coaxial differential ring on cylinder base, r2 < r1

dA2

Z = z/r1 , r1

R = r2 /r1

X = 1 + Z 2 + R2

r2

dFd1−d2 = z

2Z(X − 2R2 )R dR (X 2 − 4R2 )3/2

dA1

5

Parallel differential strip elements in intersecting planes Y = y/x

dy

dA2

dFd1−d2 =

Y sin2 φ dY 2(1 + Y 2 − 2Y cos φ)3/2

y

φ

x dA1

6

dA2 dφ

φ

r b

dA1

Strip of finite length b and of differential width, to differential strip of same length on parallel generating line B = b/r cos φ dFd1−d2 = tan−1 B dφ π

838

D VIEW FACTOR CATALOGUE

7

Differential ring element to ring element on coaxial disk R = r2 /r1 , L = l/r1

dA1

r1

dFd1−d2 = l

2RL2 [L2 + R2 + 1] dR [(L2 + R2 + 1)2 − 4R2 ]3/2

r2 dA2

8

Ring element on base to circumferential ring element on interior of right-circular cylinder

x dr r1

X = x/r2 ,

r2

dFd1−d2 = 

dA1 dA2

9

R = r1 /r2

2X(X 2 − R2 + 1) dX  (X 2 + R2 + 1)2 − 4R2 3/2

Two ring elements on the interior of right-circular cylinder X = x/2r " # X(2X 2 + 3) dFd1−d2 = 1 − dX2 2(X 2 + 1)3/2

dx 2 x dx 1

dA2 r

dA1

10 b

Differential planar element to finite parallel rectangle; normal to element passes through corner of rectangle A2

c

A = a/c,

B = b/c

a dA1

Fd1−2 =

1 2π

(



B B A tan−1 √ + √ tan−1 √ 1+A2 1+A2 1+B2 1+B2 A

)

Differential planar element to rectangle in plane 90◦ to plane of element

11 b

X = a/b, A2

Fd1−2

Y = c/b

1 1 1 Y = tan−1 − √ tan−1 √ 2 2 2 2π Y X +Y X + Y2

a

!

dA1 c

12 dA1 A2

h

r

Differential planar element to circular disk in plane parallel to element; normal to element passes through center of disk H = h/r 1 Fd1−2 = 2 H +1

D VIEW FACTOR CATALOGUE

13

839

Differential planar element to circular disk in plane parallel to element

a

H = h/a,

h

dA1

A2

Fd1−2

r

14

R = r/a

Z = 1 + H 2 + R2 " # Z − 2R2 1 1− √ = 2 Z 2 − 4R2

Differential planar element to circular disk; planes containing element and disk intersect at 90◦ ; l ≥ r

dA1

H = h/l, h A2

Fd1−2 r

R = r/l

Z = 1 + H 2 + R2 " # Z H −1 = √ 2 Z 2 − 4R2

l

15

Differential planar element to right-circular cylinder of finite length and radius; normal to element passes through one end of cylinder and is perpendicular to cylinder axis L = l/r, H = h/r

l

r

A2

X = (1 + H)2 + L2

dA1 h

Fd1−2 A2

16 r

Y = (1 − H)2 + L2   s r X −2H L X(H −1) H −1  L  1 −1 −1 −1 + √ tan − tan =  tan √  πH  L Y(H +1) H +1  XY H 2 −1

Differential planar element to sphere; normal to center of element passes through center of sphere Fd1−2 =

r 2 h

h dA1

A2

17 r

Differential planar element to sphere; tangent to element passes through center of sphere

Fd1−2

h dA1

H = h/r   √ 1  H 2 −1  1 = tan−1 √ −  π H2  H 2 −1

840

D VIEW FACTOR CATALOGUE

18

Differential planar element to sphere; element plane does not intersect sphere

A2 r

θ ≤ cos−1

h

θ

Fd1−2 = n

r 2 h

r h cos θ

dA1

19

Differential planar element to sphere A2

r

L = l/r, H≥1:

h

H = h/r

Fd1−2 =

H L2 + H 2

dA1

−1 < H < 1 :

l

Fd1−2 = −

20

1 π p

(

H L2

(L2

+

+

 H 2 3/2

H2

cos−1 L

− 1)(1 − L2 + H 2

H2)



−H

L2 +H 2

− sin−1

−1 √ ) H 2 + L2 − 1 π + L2 2

Differential element on longitudinal strip inside cylinder to inside cylinder surface

A2

Z = z/2r,

dA1

3/2

r h z r

21

H = h/2r

Z 2 + 12 (H − Z)2 + 21 Fd1−2 = 1 + H − √ − p Z2 + 1 (H − Z)2 + 1

Differential element on longitudinal strip on inside of right-circular cylinder to base of cylinder Z = z/r

dA1 r

Fd1−2 = z

Z Z2 + 2 − √ 2 2 Z +4 2

r A2

22

Differential element on surface of right-circular cylinder to disk on base of cylinder, r2 < r1 (see Configuration 13) Z = z/r1 , R = r2 /r1

r1 dA1

r2

A2

z

Fd1−2

X = 1 + Z 2 + R2 ( ) Z X = −1 √ 2 X 2 − 4R2

D VIEW FACTOR CATALOGUE

23

841

Infinite differential strip to parallel infinite plane of finite width; plane and plane containing strip intersect at arbitrary angle φ

A2

X = x/l φ

Fd1−2 =

x l

dA1

24

1 cos φ − X + p 2 2 1 + X 2 − 2X cos φ

Differential strip element of any length to an infinitely long strip of finite width; cross-section of A2 is arbitrary (but does not vary perpendicular to the paper); plane of dA1 does not intersect A2

A2

φ1 φ2

Fd1−2 = 21 (sin φ2 − sin φ1 ) dA1

25

Differential strip element of any length to infinitely long parallel cylinder; r < a

r

A = a/r,

B = b/r

A2 a

Fd1−2 =

A A2 + B2

dA1

b

26

Differential strip element to rectangle in plane parallel to strip; strip is opposite one edge of rectangle b

c

X = a/c,

A2

Y = b/c

a dA1

Fd1−2 =

! 1 √ X XY Y 1+Y 2 tan−1 √ − tan−1 X + √ tan−1 √ πY 1+Y 2 1+X 2 1+X 2 Differential strip element to rectangle in plane 90◦ to plane of strip

27 b

X = a/b,

Y = c/b

A2 a

dA1 c

Fd1−2 =

# " 1 1 Y Y 2 (X 2 +Y 2 +1) Y 1 −1 tan−1 + ln 2 − tan √ √ π Y 2 (Y +1)(X 2 +Y 2 ) X 2 +Y 2 X 2 +Y 2

842

D VIEW FACTOR CATALOGUE

28 s

x

r h

Differential strip element to exterior of right-circular cylinder of finite length; strip and cylinder are parallel and of equal length; plane containing strip does not intersect cylinder

A2

S = s/r,

dA1

X = x/r,

H = h/r

A = H2 + S 2 + X2 − 1 B = H2 − S 2 − X2 + 1

Fd1−2 =

    √ 2 2     1 B 1 − 1 cos−1 B − A + 4H cos−1 √ B  − A  − sin−1 √ π A 2H 4H A S 2 + X 2 2H S 2 + X2

S S 2 + X2

29

Differential strip element of any length on exterior of cylinder to plane of infinite length and width Fd1−2 = 12 (1 + cos φ)

φ dA1 A2

30

Differential ring element on surface of disk to coaxial sphere R1 = r1 /a, R2 = r2 /a

r2 A2

Fd1−2 = 

a dA1

R22 1 + R21

3/2

r1

31 dA1

Differential ring element on interior of right-circular cylinder to circular disk at end of cylinder

r

X = x/2r X 2 + 12 Fd1−2 = √ −X X2 + 1

x

r A2

32

Two infinitely long, directly opposed parallel plates of the same finite width

w A2 h A1 w

F1−2

H = h/w √ = F2−1 = 1 + H 2 − H

D VIEW FACTOR CATALOGUE

33

Two infinitely long plates of unequal widths h and w, having one common edge, and at an angle of 90◦ to each other H = h/w  √ 1 F1−2 = 1 + H − 1 + H2 2

A2 h

843

A1 w

34

Two infinitely long plates of equal finite width w, having one common edge, forming a wedge-like groove with opening angle α w

A2

F1−2 = F2−1 = 1 − sin

A1

α 2

α w

35

Infinitely long parallel cylinders of the same diameter X =1+

r

r

F1−2 s

A2

A1

36

s 2r

! √ 1 −1 1 2 = sin + X −1−X π X

Two infinite parallel cylinders of different radius r1

R = r2 /r1 ,

r2

S = s/r1 ,

C =1+R+S A1

A2

s

F1−2 =

( p p 1 π + C 2 − (R + 1)2 − C 2 − (R − 1)2 2π +(R − 1) cos−1

37

R−1 R+1 − (R + 1) cos−1 C C

)

Exterior of infinitely long cylinder to unsymmetrically placed, infinitely long parallel rectangle; r ≤ a

r

B1 = b1 /a, A1

a

A2

F1−2 =

B2 = b2 /a

 1  −1 tan B1 − tan−1 B2 2π

b2 b1

38

Identical, parallel, directly opposed rectangles

a b

X = a/c, A2

c

F1−2 A1

Y = b/c

2 = πXY

 " #1/2  √  X  (1+X 2 )(1+Y 2 ) ln + X 1+Y 2 tan−1 √   2 2  1+X +Y 1+Y 2

  √  Y −1 −1 −1  2 +Y 1+X tan √ − X tan X − Y tan Y    2 1+X

844

D VIEW FACTOR CATALOGUE

39

Two finite rectangles of same length, having one common edge, and at an angle of 90◦ to each other

l

H = h/l,

W = w/l

A2 h

A1

90°

 √ 1 1  1 1 W tan−1 F1−2 = + H tan−1 − H 2 + W 2 tan−1 √ πW W H H2 + W 2   " #W 2 " 2 #H 2     1  H (1+H 2 +W 2 )   (1+W 2 )(1+H 2 ) W 2 (1+W 2 +H 2 )  + ln     1+W 2 +H 2 4  (1+W 2 )(W 2 +H 2 ) (1+H 2 )(H 2 +W 2 ) 

w

40

Disk to parallel coaxial disk of unequal radius

r1

R1 = r1 /a,

A1 A2

a

R2 = r2 /a 1 + R22 R21 s

X =1+

r2

F1−2 41

1 = 2

     X−    

X2

R2 −4 R1

!2         

Outer surface of cylinder to annular disk at end of cylinder R = r1 /r2 , L = l/r2

A1

A = L 2 + R2 − 1

A2 l

B = L 2 − R2 + 1

r2 r1

F1−2 42

r    (A + 2)2 1  −1 A 1 A B −1  −1 AR −4 cos + − − sin R = cos 8RL 2π  B 2L R2 B 2RL Inside surface of right-circular cylinder to itself

r

F1−1 h

A1

43 r

h

A2

A1

H = h/2r √ = 1 + H − 1 + H2

Base of right-circular cylinder to inside surface of cylinder H = h/2r h√ i F1−2 = 2H 1 + H 2 − H

D VIEW FACTOR CATALOGUE

44

Interior of finite-length, right-circular coaxial cylinder to itself R = r2 /r1 , H = h/r1

A2 r1

845

r2

h

 √ √ 1 H 2 + 4R2 − H 1  2 2 R2 − 1 − +  tan−1 R 4R π R H  √   2 2 2 2 2 2 2  H   4R + H −1 H +4(R −1)−2H /R −1 R − 2   − sin − sin    2 2 2  2R  H H + 4(R − 1) R 

F2−2 = 1 −

45 A1

Interior of outer right-circular cylinder of finite length to exterior of inner right-circular coaxial cylinder

A2 r2

r1

R = r2 /r1 ,

H = h/r1

h

F2−1 =

46

Interior of outer right-circular cylinder of finite length to annular end enclosing space between coaxial cylinders

A1 r2

( H 2 − R2 + 1 1 H 2 + R2 − 1 1 1− cos−1 2 − R 4H π H + R2 − 1 p )! (H 2 +R2 +1)2 −4R2 H 2 −R2 +1 H 2 −R2 +1 −1 1 −1 cos − sin − 2H R(H 2 +R2 −1) 2H R

r1

H = h/r2 , R = r1 /r2 √ X = 1 − R2 R(1 − R2 − H 2 ) Y= 1 − R2 + H 2

h

A2

F1−2

47

 !   i X2  π  H h −1 2  −1 X −1 2X R tan − tan + sin (2R − 1) − sin−1 R + + sin−1 R    H H 4 4H 2 p √ " !#    (1 + R2 + H 2 )2 − 4R2  π 4 + H2 π 2R2 H 2  −1 −1 − + sin Y + + sin 1 −  2 2  4H 2 4 2 4X + H 

1 = π

r

Sphere to rectangle, r < d

A1 d A2

F1−2 l2

l1

D1 = d/l1 , D2 = d/l2 s 1 1 −1 = tan 4π D21 + D22 + D21 D22

846

D VIEW FACTOR CATALOGUE

48

Sphere to coaxial disk A1

F1−2

a

R = r/a " # 1 1 1− √ = 2 1 + R2

r A2

49 A2

Sphere to interior surface of coaxial right-circular cylinder; sphere within ends of cylinder

r

R = r/a a

A1

1 F1−2 = √ 1 + R2

a

50

Sphere to coaxial cone r1

r2

ω

S = s/r1 ,

1 : S +1    1  1+S +R cot ω  = 1 − p 2 (1+S +R cot ω)2 +R2

for ω ≥ sin−1

s A1

R = r2 /r1

A2

F1−2 51 A2

s D A1

Infinite plane to row of cylinders s  D 2 D D F1−2 = cos−1 + 1 − 1 − s s s

References 1. Howell, J. R.: A Catalog of Radiation Configuration Factors, McGraw-Hill, New York, 1982. 2. Howell, J. R., and M. P. Meng¨uc¸: “Radiative transfer configuration factor catalog: A listing of relations for common geometries,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 112, pp. 910–912, 2011. 3. Wong, R. L.: “User’s manual for CNVUFAC–the General Dynamics heat transfer radiation view factor program,” Technical report, University of California, Lawrence Livermore National Laboratory, 1976. 4. Shapiro, A. B.: “FACET–a computer view factor computer code for axisymmetric, 2D planar, and 3D geometries with shadowing,” Technical report, University of California, Lawrence Livermore National Laboratory, August 1983 (maintained by Nuclear Energy Agency under http://www.oecd-nea.org/tools/abstract/detail/nesc9578/). 5. Burns, P. J.: “MONTE–a two-dimensional radiative exchange factor code,” Technical report, Colorado State University, Fort Collins, 1983. 6. Emery, A. F.: “VIEW–a radiation view factor program with interactive graphics for geometry definition (version 5.5.3),” Technical report, NASA computer software management and information center, Atlanta, 1986, (available from http://www.openchannelfoundation.org/projects/VIEW). 7. Ikushima, T.: “MCVIEW: A radiation view factor computer program or three-dimensional geometries using Monte Carlo method,” Technical report, Japan Atomic Energy Research Institute (JAERI), 1986, (maintained by Nuclear Energy Agency under http://www.oecd-nea.org/tools/abstract/detail/nea-1166). 8. Jensen, C. L.: “TRASYS-II user’s manual–thermal radiation analysis system,” Technical report, Martin Marietta Aerospace Corp., Denver, 1987.

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848

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42. Tripp, W., C. Hwang, and R. E. Crank: “Radiation shape factors for plane surfaces and spheres, circles, or cylinders,” Spec. Rept. 16, Kansas State Univ. Bull., 1962. 43. Dunkle, R. V.: “Configuration factors for radiant heat-transfer calculations involving people,” ASME Journal of Heat Transfer, vol. 85, no. 1, pp. 71–76, 1963. 44. Haller, H. C., and N. O. Stockman: “A note on fin-tube view factors,” ASME Journal of Heat Transfer, vol. 85, no. 4, pp. 380–381, 1963. 45. Sparrow, E. M.: “A new and simpler formulation for radiative angle factors,” ASME Journal of Heat Transfer, vol. 85, pp. 73–81, 1963. 46. Sparrow, E. M., and V. K. Jonsson: “Radiant emission characteristics of diffuse conical cavities,” Journal of the Optical Society of America, vol. 53, pp. 816–821, 1963. 47. Sparrow, E. M., and V. K. Jonsson: “Thermal radiation absorption in rectangular-groove cavities,” ASME Journal of Applied Mechanics, vol. E30, pp. 237–234, 1963. 48. Sparrow, E. M., and V. K. 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Stasiek: “Application of generalized Pythagoras theorem to calculation of configuration factors between surfaces of channels of revolution,” International Journal of Heat and Fluid Flow, vol. 4, no. 3, pp. 157–160, 1983. 120. Lipps, F. W.: “Geometric configuration factors for polygonal zones using Nusselt’s unit sphere,” Solar Energy, vol. 30, no. 5, pp. 413–419, 1983. 121. Chung, B. T. F., M. M. Kermani, and M. H. N. Naraghi: “A formulation of radiation view factors from conical surfaces,” AIAA Journal, vol. 22, no. 3, pp. 429–436, 1984. 122. Mahbod, B., and R. L. Adams: “Radiation view factors between axisymmetric subsurfaces within a cylinder with spherical centerbody,” ASME Journal of Heat Transfer, vol. 106, no. 1, p. 244, 1984. 123. Yarbrough, D. W., and C. L. Lee: “Monte Carlo calculation of radiation view factors,” in Integral Methods in Sciences and Engineering, eds. F. R. Payne et al., Harper and Rowe/Hemisphere, 1984. 124. Eichberger, J. I.: “Calculation of geometric configuration factors in an enclosure whose boundary is given by an arbitrary polygon in the plane,” W¨arme- und Stoff¨ubertragung, vol. 19, no. 4, p. 269, 1985. 125. Mathiak, F. U.: “Berechnung von konfigurationsfaktoren polygonal berandeter ebener gebiete (calculation of formfactors for plane areas with polygonal boundaries),” W¨arme- und Stoff¨ubertragung, vol. 19, no. 4, p. 273, 1985. 126. Shapiro, A. B.: “Computer implementation, accuracy and timing of radiation view factor algorithms,” ASME Journal of Heat Transfer, vol. 107, no. 3, pp. 730–732, 1985. 127. Shukla, K. N., and D. Ghosh: “Radiation configuration factors for concentric cylinder bodies in enclosure,” Indian Journal of Technology, vol. 23, pp. 244–246, 1985. 128. Lin, S., P. M. Lee, J. C. Y. Wang, Y. L. Dai, and Y. S. Lou: “Radiant-interchange configuration factors between disk and segment of parallel concentric disk,” ASME Journal of Heat Transfer, vol. 29, no. 3, pp. 501–503, 1986. 129. Maxwell, G. M., M. J. Bailey, and V. W. Goldschmidt: “Calculations of the radiation configuration factor using ray casting,” Computer Aided Design, vol. 18, no. 7, p. 371, 1986. 130. Stefanizzi, P.: “Reliability of the Monte Carlo method in black body view factor determination,” Termotechnica, vol. 40, no. 6, p. 29, 1986. 131. Wang, J. C. Y., S. Lin, P. M. Lee, W. L. Dai, and Y. S. Lou: “Radiant-interchange configuration factors inside segments of frustum enclosures of right circular cones,” International Communications in Heat and Mass Transfer, vol. 13, pp. 423–432, 1986. 132. Eddy, T. L., and G. E. Nielsson: “Radiation shape factors for channels with varying cross-section,” ASME Journal of Heat Transfer, vol. 110, no. 1, pp. 264–266, 1988. 133. Frankel, J. I., and T. P. Wang: “Radiative exchange between gray fins using a coupled integral equation formulation,” Journal of Thermophysics and Heat Transfer, vol. 2, no. 4, pp. 296–302, Oct 1988. 134. Modest, M. F.: “Radiative shape factors between differential ring elements on concentric axisymmetric bodies,” Journal of Thermophysics and Heat Transfer, vol. 2, no. 1, pp. 86–88, 1988. 135. Mel’man, M. M., and G. G. Trayanov: “View factors in a system of parallel contacting cylinders,” Journal of Engineering Physics, vol. 54, no. 4, p. 401, 1988. 136. Naraghi, M. H. N.: “Radiation view factors from differential plane sources to disks—a general formulation,” Journal of Thermophysics and Heat Transfer, vol. 2, no. 3, pp. 271–274, 1988. 137. Naraghi, M. H. N.: “Radiative view factors from spherical segments to planar surfaces,” Journal of Thermophysics and Heat Transfer, vol. 2, no. 4, pp. 373–375, Oct 1988. 138. Naraghi, M. H. N., and J. P. Warna: “Radiation configuration factors from axisymmetric bodies to plane surfaces,” International Journal of Heat and Mass Transfer, vol. 31, no. 7, pp. 1537–1539, 1988. 139. Sabet, M., and B. T. F. Chung: “Radiation view factors from a sphere to nonintersecting planar surfaces,” Journal of Thermophysics and Heat Transfer, vol. 2, no. 3, pp. 286–288, 1988.

D VIEW FACTOR CATALOGUE

851

140. Chung, B. T. F., and M. M. Kermani: “Radiation view factors from a finite rectangular plate,” ASME Journal of Heat Transfer, vol. 111, no. 4, p. 1115, 1989. 141. van Leersum, J.: “A method for determining a consistent set of radiation view factors from a set generated by a nonexact method,” International Journal of Heat and Fluid Flow, vol. 10, no. 1, p. 83, 1989. 142. Bornside, D. E., and R. A. Brown: “View factor between differing-diameter, coaxial disks blocked by a coaxial cylinder,” Journal of Thermophysics and Heat Transfer, vol. 4, no. 3, pp. 414–416, 1990. 143. Saltiel, C., and M. H. N. Naraghi: “Radiative configuration factors from cylinders to coaxial axisymmetric bodies,” International Journal of Heat and Mass Transfer, vol. 33, no. 1, pp. 215–218, 1990. 144. Tseng, J. W. C., and W. Strieder: “View factors for wall to random dispersed solid bed transport,” ASME Journal of Heat Transfer, vol. 112, pp. 816–819, 1990. 145. Emery, A. F., O. Johansson, M. Lobo, and A. Abrous: “A comparative study of methods for computing the diffuse radiation viewfactors for complex structures,” ASME Journal of Heat Transfer, vol. 113, no. 2, pp. 413–422, 1991. 146. Rushmeier, H. E., D. R. Baum, and D. E. Hall: “Accelerating the hemi-cube algorithm for calculating radiation form factors,” ASME Journal of Heat Transfer, vol. 113, no. 4, pp. 1044–1047, 1991. 147. Sika, J.: “Evaluation of direct-exchange areas for a cylindrical enclosure,” ASME Journal of Heat Transfer, vol. 113, no. 4, pp. 1040–1043, 1991. 148. Ambirajan, A., and S. P. Venkateshan: “Accurate determination of diffuse view factors between planar surfaces,” International Journal of Heat and Mass Transfer, vol. 36, no. 8, pp. 2203–2208, 1993. 149. Beard, A., D. Drysdale, and P. Holborn: “Configuration factor for radiation in a tunnel or partial cylinder,” Fire Technology, vol. 29, no. 3, pp. 281–288, 1993. 150. Byrd, L. W.: “View factor algebra for two arbitrary sized non-opposing parallel rectangles,” ASME Journal of Heat Transfer, vol. 115, pp. 517–518, 1993. 151. Ehlert, J. R., and T. F. Smith: “View factors for perpendicular and parallel, rectangular plates,” Journal of Thermophysics and Heat Transfer, vol. 7, no. 1, pp. 173–174, 1993. 152. Guelzim, A., J. M. Souil, and J. P. Vantelon: “Suitable configuration factors for radiation calculation concerning tilted flames,” ASME Journal of Heat Transfer, vol. 115, no. 2, pp. 489–491, May 1993. 153. Murty, C. V. S.: “Evaluation of radiation reception factors in a rotary kiln using a modified Monte Carlo scheme,” International Journal of Heat and Mass Transfer, vol. 36, no. 1, pp. 119–132, 1993. 154. Noboa, H. L., D. O’Neal, and W. D. Turner: “Calculation of the shape factor from a small rectangular plane to a triangular surface perpendicular to the rectangular plane without a common edge,” ASME Journal of Solar Energy Engineering, vol. 115, pp. 117–119, 1993. 155. Brockmann, H.: “Analytic angle factors for the radiant interchange among the surface elements of two concentric cylinders,” International Journal of Heat and Mass Transfer, vol. 37, no. 7, pp. 1095–1100, 1994. 156. Flouros, M., S. Bungart, and W. Leiner: “Calculation of the view factors for radiant heat exchange in a new volumetric receiver with tapered ducts,” ASME Journal of Solar Energy Engineering, vol. 117, pp. 58–60, 1995. 157. Hollands, K. G. T.: “On the superposition rule for configuration factors,” ASME Journal of Heat Transfer, vol. 117, no. 1, pp. 241–244, 1995. 158. Lawson, D. A.: “An improved method for smoothing approximate exchange areas,” International Journal of Heat and Mass Transfer, vol. 38, no. 16, pp. 3109–3110, 1995. 159. Loehrke, R. I., J. S. Dolaghan, and P. J. Burns: “Smoothing Monte Carlo exchange factors,” ASME Journal of Heat Transfer, vol. 117, no. 2, pp. 524–526, 1995. 160. Rao, V. R., and V. M. K. Sastri: “Efficient evaluation of diffuse view factors for radiation,” International Journal of Heat and Mass Transfer, vol. 39, pp. 1281–1286, 1996. 161. Krishnaprakas, C. K.: “View factor between inclined rectangles,” Journal of Thermophysics and Heat Transfer, vol. 11, no. 3, pp. 480–482, 1997. 162. Li, B. W., W. Q. Tao, and R. X. Liu: “Ray effect in ray tracing method for radiative heat transfer,” International Journal of Heat and Mass Transfer, vol. 40, no. 14, pp. 3419–3426, 1997. 163. Mavroulakis, A., and A. Trombe: “A new semianalytical algorithm for calculating diffuse plane view factors,” ASME Journal of Heat Transfer, vol. 120, no. 1, pp. 279–282, 1998. 164. Tso, C. P., and S. P. Mahulikar: “View factors between finite length rings on an interior cylindrical shell,” Journal of Thermophysics and Heat Transfer, vol. 13, no. 3, pp. 375–379, 1999. 165. Katte, S. S., and S. P. Venkateshan: “Accurate determination of view factors in axisymmetric enclosures with shadowing bodies inside,” Journal of Thermophysics and Heat Transfer, vol. 14, no. 1, pp. 68–76, 2000. 166. Howell, J. R., R. Siegel, and M. P. Meng¨uc¸: Thermal Radiation Heat Transfer, 4th ed., Taylor and Francis-Hemisphere, Washington, 2011.

APPENDIX

E EXPONENTIAL INTEGRAL FUNCTIONS

The exponential integral functions En (x) and their derivatives occur frequently in radiative heat transfer calculations; therefore, a summary of their properties as well as a brief tabulation are given here. More detailed discussions of their properties may be found in the books by Chandrasekhar [1] and Kourganoff [2], or in mathematical handbooks such as [3]. Detailed tabulations are given in [3], and formulae for their numerical evaluation are listed in [3, 4]. The exponential integral of order n is defined as Z ∞ dt e−xt n , n = 0, 1, 2, . . . , En (x) = t 1

(E.1)

or, setting µ = 1/t, En (x) =

Z

1

e−x/µ µn−2 dµ,

n = 0, 1, 2, . . . .

(E.2)

0

Differentiating equation (E.1), a first recurrence relationship is found as dEn (x) = −En−1 (x), dx

n = 1, 2, . . . ,

(E.3)

e−x . x

(E.4)

where E0 (x) =

Z



e−xt dt =

1

A second recurrence is found by integrating equation (E.3), or Z ∞ En (x) dx = En+1 (x), n = 0, 1, 2, . . . .

(E.5)

x

An algebraic recurrence between consecutive orders may be obtained by integrating equation (E.1) by parts, or En+1 (x) =

 1  −x e − xEn (x) , n

852

n = 1, 2, 3, . . . .

(E.6)

E EXPONENTIAL INTEGRAL FUNCTIONS

853

The integral of equation (E.1) may be solved in a general series expansion as [3] En (x) =

∞ X (−x)m (−x)n−1 (ψn − ln x) + , (n−1)! m!(n−1−m) m=0

n = 1, 2, 3, . . . ,

(E.7a)

m,n−1

where

 n = 1, −γE ,     n−1  X 1 ψn =   , n ≥ 2, −γ +    E m

(E.7b)

m=1

and

γE =

Z



1



1 − e−t

 dt t

= 0.577216 . . .

(E.7c)

is known as Euler’s constant. Substituting values for n, one obtains x2 x3 x4 + − + −..., 2!2 3!3 4!4 x3 x4 x2 + − + −..., E2 (x) = 1 + x(γE − 1 + ln x) − 2!1 3!2 4!3 ! 2 3 4 3 1 x x x −γE + − ln x + − + −.... E3 (x) = − x + 2 2 2 3!1 4!2 E1 (x) = −(γE + ln x) + x −

A function related to E1 that often occurs in radiation calculations is Z 1  dt Z ∞   −ξ Ein(x) = 1 − e−xt = 1 − e−xe dξ t 0 0 x2 x3 = E1 (x) + ln x + γE = x − + − +.... 2!2 3!3

(E.8) (E.9) (E.10)

(E.11)

For vanishing values of x it follows from equation (E.7), or directly integrating equation (E.1), that  n = 1,    +∞, 1 En (0) =  (E.12a)  , n ≥ 2,  n−1 Ein(0) = 0. (E.12b) For large values of x, the asymptotic expansion for the exponential integrals is given by [3] # " e−x n n(n+1) n(n+1)(n+2) − + − . . . , n = 0, 1, 2, . . . . 1− + En (x) = x x x2 x3

(E.13)

To estimate the relative magnitude of different orders of exponential integrals, the following inequalities are sometimes handy [3]:

References

n−1 En (x) < En+1 (x) < En (x), n = 1, 2, 3, . . . , n 1 1 < e x En (x) ≤ , n = 1, 2, 3, . . . . x+n x+n−1

(E.14) (E.15)

1. Chandrasekhar, S.: Radiative Transfer, Dover Publications, New York, 1960, (originally published by Oxford University Press, London, 1950). 2. Kourganoff, V.: Basic Methods in Transfer Problems, Dover Publications, New York, 1963. 3. Abramowitz, M., and I. A. Stegun (eds.): Handbook of Mathematical Functions, Dover Publications, New York, 1965. 4. Breig, W. F., and A. L. Crosbie: “Numerical computation of a generalized exponential integral function,” Math Comp., vol. 28, no. 126, pp. 575–579, 1974.

854

E EXPONENTIAL INTEGRAL FUNCTIONS

TABLE E.1

Values of exponential integral functions. x 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.50 3.00 4.00 5.00

Ein 0.000000 0.009975 0.019900 0.029776 0.039603 0.049382 0.059112 0.068794 0.078428 0.088015 0.097554 0.144557 0.190428 0.235204 0.278920 0.321609 0.363305 0.404039 0.443842 0.520769 0.594310 0.664669 0.732039 0.796600 0.858517 0.917946 0.975031 1.029907 1.082700 1.133528 1.182499 1.229716 1.275274 1.319263 1.518421 1.688876 1.967289 2.187802

E1 ∞ 4.037929 3.354707 2.959118 2.681263 2.467898 2.295307 2.150838 2.026941 1.918744 1.822924 1.464461 1.222650 1.044283 0.905677 0.794215 0.702380 0.625331 0.559773 0.454379 0.373769 0.310597 0.260184 0.219384 0.185991 0.158408 0.135451 0.116219 0.100020 0.086308 0.074655 0.064713 0.056204 0.048901 0.024915 0.013048 0.003779 0.001148

E2 1.000000 0.949671 0.913105 0.881672 0.853539 0.827835 0.804046 0.781835 0.760961 0.741244 0.722545 0.641039 0.574201 0.517730 0.469115 0.426713 0.389368 0.356229 0.326644 0.276184 0.234947 0.200852 0.172404 0.148496 0.128281 0.111104 0.096446 0.083890 0.073101 0.063803 0.055771 0.048815 0.042780 0.037534 0.019798 0.010642 0.003198 0.000996

E3 0.500000 0.490277 0.480968 0.471998 0.463324 0.454919 0.446761 0.438833 0.431120 0.423610 0.416291 0.382276 0.351945 0.324684 0.300042 0.277669 0.257286 0.238663 0.221604 0.191551 0.166061 0.144324 0.125703 0.109692 0.095881 0.083935 0.073576 0.064576 0.056739 0.049906 0.043937 0.038716 0.034143 0.030133 0.016295 0.008931 0.002761 0.000878

E4 0.333333 0.328382 0.323526 0.318762 0.314085 0.309494 0.304986 0.300559 0.296209 0.291935 0.287736 0.267789 0.249447 0.232543 0.216935 0.202501 0.189135 0.176743 0.165243 0.144627 0.126781 0.111290 0.097812 0.086062 0.075801 0.066824 0.058961 0.052064 0.046007 0.040682 0.035997 0.031870 0.028232 0.025023 0.013782 0.007665 0.002423 0.000783

APPENDIX

F COMPUTER CODES

This appendix contains a listing and brief description of a number of computer programs that may be helpful to the reader of this book, and that can be downloaded from its dedicated website located at http://booksite.elsevier.com/9780123869449. Some of the codes are very basic and are entirely intended to aid the reader with the solution to the problems given at the end of the more basic chapters. Some of the codes were born out of research, but are basic enough to aid a graduate student with more complicated assignments or a semester project. And a few programs are so sophisticated in nature that they will be useful only to the practicing engineer conducting his or her own research. Finally, it is anticipated that the website will be kept up-to-date and augmented once in a while. Thus, there may be a few additional programs not described in this appendix. It is a fact that most engineers have done, and still do, their programming in Fortran, and the author of this book is no exception. It is also true that computer scientists and most commercial programmers do their work in C++; more importantly, the younger generation of engineers at many universities across the U.S. are now also learning C++. Both compiled languages have in recent years been trumped by Matlabr [1], which—while an interpreted rather than compiled language— has many convenient mathematical and graphical tools. Since all the programs in this listing were written by the author, either for research purposes or for the creation of this book, they all started their life in Fortran (older programs as Fortran77, and the later ones as Fortran90). However, as a gesture toward the C++ and Matlabr communities, the most basic codes have all been converted to C++ as well as Matlabr , as indicated below by the program suffixes .cpp and .m. If desired, all other programs are easily converted with freeware translators such as f2c (resulting in somewhat clumsy, but functional codes). Finally, self-contained programs that have been precompiled for Microsoft Windows have the suffix .exe. The programs are listed in order by chapter in which they first appear. More detailed descriptions, sometimes with an example, can be found on the website. Third-party codes that are also provided at the website are listed at the end. Chapter 1 bbfn.f, bbfn.cpp, bbfn.m:

Function bbfn(x) calculates the fractional blackbody emissive power, as defined by equation (1.23), where the argument is x = nλT with units of µm K.

planck.f, planck.cpp, planck.m, planck.exe:

planck is a small stand-alone program that prompts the user for input (temperature and wavelength or wavenumber), then calculates the spectral blackbody emissive powers Ebλ /T 5 , Ebη /T 3 and the fractional blackbody emissive power f (λT ). 855

856

F COMPUTER CODES

Chapters 2 and 3 fresnel.f, fresnel.cpp, fresnel.m:

Subroutine fresnel calculates Fresnel reflectivities from equation (2.113) for a given complex index of refraction and incidence angle.

Chapter 3 emdiel.f90, emdiel.cpp, emdiel.m: emmet.f90, emmet.cpp, emmet.m: callemdiel.f90, callemdiel.cpp, callemdiel.m, callemdiel.exe: callemmet.f90, callemmet.cpp, callemmet.m, callemmet.exe: dirreflec.f, dirreflec.cpp, dirreflec.m, dirreflec.exe: totem.f90, totem.cpp, totem.m:

Function emdiel calculates the unpolarized, spectral, hemispherical emissivity of an optical surface of a dielectric material from equation (3.82). Function emmet calculates the unpolarized, spectral, hemispherical emissivity of an optical surface of a metallic material from equation (3.77). Program callemdiel is a stand-alone front end for function emdiel, prompting for input (refractive index n) and returning the unpolarized, spectral, hemispherical as well as normal emissivities. Program callemmet is a stand-alone front end for function emmet, prompting for input (complex index of refraction n, k) and returning the unpolarized, spectral, hemispherical as well as normal emissivities. Program dirreflec is a stand-alone front end for subroutine fresnel, returning perpendicular polarized, parallel polarized, and unpolarized reflectances. Program totem is a routine to evaluate the total, directional or hemispherical emittance or absorptance of an opaque material, based on an array of spectral data.

Chapter 4 and Appendix D view.f90, view.cpp, view.m: parlplates.f90, parlplates.cpp parlplates.m:

A function to evaluate any of the 51 view factors given in Appendix D.

perpplates.f90, perpplates.cpp, perpplates.m:

A function to evaluate the view factor between two displaced perpendicular plates, as given by equation (4.41).

A function to evaluate the view factor between two displaced parallel plates, as given by equation (4.42).

viewfactors.f90, A stand-alone front end to functions view, parlplates, and perpplates. viewfactors.cpp, The user is prompted to input configuration number and arguments; the proviewfactors.m, gram then returns the requested view factor. viewfactors.exe:

F COMPUTER CODES

857

Chapter 5 Subroutine graydiff provides the solution to equation (5.38) for an enclosure consisting of N gray-diffuse surfaces. For each surface the area, emittance, external irradiation, and either heat flux or temperature must be specified. In addition, the upper triangle of the view factor matrix must be provided (Fi− j ; i = 1, N; j = i, N). For closed configurations, the diagonal view factors Fi−i are not required, since they can be calculated from the summation rule. The remaining view factors are calculated from reciprocity. On output, the program provides all view factors, and temperatures and radiative heat fluxes for all surfaces. graydiffxch.f90, Program graydiffxch is a front end for subroutine graydiff, generating graydiffxch.cpp, the necessary input parameters for a three-dimensional variation to Example graydiffxch.m: 5.4, primarily view factors calculated by calls to function view. This program may be used as a starting point for more involved radiative exchange problems. graydiff.f90, graydiff.cpp, graydiff.m:

Chapter 6 graydifspec.f90, Subroutine graydifspec provides the solution to equation (6.23) for an engraydifspec.cpp, closure consisting of N diffusely emitting surfaces with diffuse and specular graydifspec.m: reflectance components. For each surface the area, emittance, specular reflectance, external irradiation, and either heat flux or temperature must be specified. In addition, the upper triangle of the specular view factor matrix s must be provided (Fi− j ; i = 1, N; j = i, N). Otherwise same as graydiff. grspecxch.f90, grspecxch.cpp, grspecxch.m:

Program grspecxch is a front end for subroutine graydifspec, similar to graydiffxch.

Chapter 7 semigray.f90, semigray.cpp, semigray.m:

Subroutine semigray provides the solution to equations (7.5) for an enclosure consisting of N diffusely emitting surfaces with diffuse and specular reflectance components, considering two spectral ranges (one for external irradiation, one for emission). For each surface the area, emittance and specular reflectance (two values each), external irradiation, and either heat flux or temperature must be specified. Otherwise same as graydiff.

semigrxch.f90, semigrxch.cpp, semigrxch.m:

Program semigrxch is a front end for subroutine semigray providing the necessary input for Example 7.1. This program may be used as a starting point for more involved radiative exchange problems.

bandapp.f90, bandapp.cpp, bandapp.m:

Subroutine bandapp provides the solution to equations (7.6) for an enclosure consisting of N diffusely emitting surfaces with diffuse and specular reflectance components, considering M spectral bands. For each surface the area, emittance, specular reflectance and external irradiation (one value for each spectral band), and either heat flux or temperature must be specified. Otherwise same as graydiff.

bandmxch.f90, bandmxch.cpp, bandmxch.m:

Program bandmxch is a front end for subroutine bandapp providing the necessary input for Example 7.2. This program may be used as a starting point for more involved radiative exchange problems.

858

F COMPUTER CODES

Chapter 8 MCintegral.f90:

MCintegral is a little program that evaluates the integral specified function by the Monte Carlo method.

R

b a

f (x) dx for any

Chapter 11 voigt.f:

Subroutine voigt calculates the spectral absorption coefficient for a Voigtshaped line based on the fast algorithm by Huml´ıc˘ ek [2], as a function of line intensity, and Lorentz and Doppler line widths.

nbkdistdb.f90:

Program nbkdistdb is a Fortran90 code to calculate narrow band kdistributions for a number of temperatures and a number of wavenumber ranges, for a gas mixture containing CO2 , H2 O, CH4 , and soot. The spectral absorption coefficient is calculated directly from the HITRAN or HITEMP databases. Program nbkdistsg is a Fortran90 code to calculate a single narrow band kdistribution from a given array of wavenumber–absorption coefficient pairs.

nbkdistsg.f90: wbmxxx.f, wbmxxxcl.f, wbmxxxcl.exe:

emwbm.f, ftwbm.f, wangwbm.f:

Subroutines wbmxxx, where xxx stands for the different gases h20, co2, ch4, co, no, and so2, calculate for a given temperature the ratios Ψ∗ (T )/Ψ∗ (T 0 ) [from equations (11.144) and (11.148)] and Φ(T )/Φ(T 0 ) [from equation (11.149)], i.e., the functions shown in Figs. 11-23 through 11-25. The stand-alone programs wbmxxxcl.f are front ends for the wbmxxx.f, prompting the user for input, and printing the ratios Ψ∗ (T )/Ψ∗ (T 0 ) and Φ(T )/Φ(T 0 ) to the screen for all bands listed in Table 11.3. Fortran functions to calculate the nondimensional total band absorptance A∗ from the Edwards and Menard model, Table 11.2 (emwbm), the Felske and Tien model, equation (11.156) (ftwbm), and the Wang model, equation (11.158) (wangwbm).

wbmodels.f, wbmodels.exe:

Stand-alone front end for the functions emwbm, ftwbm, and wangwbm; the nondimensional total band absorptance A∗ is printed to the screen, as calculated from three band models (Edwards and Menard, Felske and Tien, and Wang models).

wbmkvsg.f:

Fortran subroutine wbmkvsg calculates the κ∗ vs. g∗ distribution of equation (11.170).

totemiss.f:

Fortran subroutine totemiss calculates the total emissivity of an isothermal gas mixture, using Leckner’s model, equations (11.177) through (11.181).

totabsor.f:

Fortran subroutine totabsor calculates the total absorptivity of an isothermal gas mixture, using Leckner’s model, equations (11.177) through (11.181).

Leckner.f, Leckner.exe:

Stand-alone front end for totemiss and totabsor, with total emissivities and absorptivities printed to the screen.

Chapter 12 coalash.f90:

This file contains subroutine coalash (plus a front end for screen input and output) to determine nondimensionalized spectral absorption and extinction coefficients κ∗ and β∗ , as listed in Table 12.3, from the Buckius and Hwang [3] and the Meng¨uc¸ and Viskanta [4] models, as functions of complex index of refraction m = n − ik and size parameter x.

F COMPUTER CODES

mmmie.f:

859

Program mmmie calculates Mie coefficients (scattering coefficients an and bn , efficiencies Qsca , Qext , and Qabs , and asymmetry factor g; see Section 12.2 for definitions), and relates them to particle cloud properties (extinction coefficient β, absorption coefficient κ, scattering coefficient σ s , cloud asymmetry factor g, scattering phase function Φ for specified scattering angles, and phase function expansion coefficients An , as defined in Section 12.3).

Chapter 16 P1sor.f90, P1sor.cpp:

Subroutine P1sor provides the solution to equation (16.38) with its boundary condition (16.49) for a two-dimensional (rectangular or axisymmetric cylinder) enclosure with reflecting walls and an absorbing, emitting, linearanisotropically scattering medium. For each surface the emittance and blackbody intensities must be specified; for the medium spatial distributions of radiation properties and blackbody intensities must be input. Internal incident radiation (G) and wall flux (q) fields are calculated. Can be used for gray problems or on a spectral basis.

P1-2D.f90, P1-2D.cpp:

Program P1-2D is a front end for subroutine P1sor, setting up the problem for a gray medium with spatially constant radiative properties; it may be used as a starting point for more involved applications.

Delta.f90:

Program Delta is a stand-alone program to calculate the rotation matrix ∆nmm′ (α, β, γ) required for the boundary conditions of higher-order PN approximations, as given by equations (16.64) through (16.67).

pnbcs.f90:

Program pnbcs is a stand-alone program to calculate the Legendre halfm m m moments pm n, j and coefficients uli , vli , wli , which are required for the boundary conditions of higher-order PN -approximations, as given by equations (16.71) through (16.72).

Chapter 19 transPN.f90:

Program transPN calculates energy from a pulsed collimated laser source transmitted through an absorbing, isotropically scattering slab as a function of time, using the P1 and P1/3 methods.

Chapter 20 fskdist.f90:

Program fskdist is a Fortran90 code to calculate full-spectrum kdistributions for a number of Planck function temperatures and a single gas property state (temperature, partial and total pressures), for a gas mixture containing CO2 , H2 O, CH4 , and soot; weight functions a(T, T 0 , g) are calculated, as well. The spectral absorption coefficient is either calculated directly from the HITRAN or HITEMP databases, or is supplied by the user.

fskdco2.f90, fskdh2o.f90:

These subroutines determine full spectrum cumulative k-distributions for CO2 and H2 O, respectively, employing the correlations of Modest and Mehta [5] and of Modest and Singh [6].

fskdco2dw.f90, fskdh2odw.f90:

Equivalent to fskdco2.f90 and fskdh2o.f90, but employing the older correlations of Denison and Webb [7, 8].

860

F COMPUTER CODES

kdistmix.f90:

Subroutine kdistmix finds the cumulative k-distribution for an n-component mixture from a given set of individual species cumulative k-distributions (narrow band, wide band, or full spectrum), employing the mixing scheme of Modest and Riazzi [9].

fskdistmix.f90:

This Fortran90 routine finds the full spectrum cumulative k-distribution for a CO2 –H2 O mixture, employing the correlations of Modest and Mehta [5] and Modest and Singh [6], using one of three mixing schemes described by equations (20.162) (superposition), (20.163) (multiplication), or (20.167) (uncorrelated mixture).

Chapter 21 mocacyl.f, rnarray.f:

Program mocacyl is a Monte Carlo routine for a nongray, nonisothermal, isotropically scattering medium confined inside a two-dimensional, axisymmetric cylindrical enclosure bounded by nongray, diffusely emitting and reflecting walls. Temperature and radiative properties are assumed known everywhere inside the enclosure and along the walls. Requires use of program rnarray to set up random number relationships (locations and wavenumbers of emission vs. random numbers). Calculates internal radiative heat sources ∇ · qR as well as local radiative fluxes to the walls qRw .

FwdMCcs.f90, FwdMCck1.f90, FwdMCck2.f90:

Program FwdMCcs is a standard forward Monte Carlo code for a narrow collimated beam penetrating through a nonabsorbing, isotropically scattering slab, calculating the flux onto a small, directionally selective detector, as given in Example 21.3. FwdMCck1 and FwdMCck2 are forward Monte Carlo codes for the same problem, but also allow for absorption in the medium; FwdMCck1 uses standard ray tracing, while FwdMCck2 uses energy partitioning; see Example 21.4.

FwdMCps.f90:

Program FwdMCps is a standard forward Monte Carlo code for a radiative energy emitted by a point source penetrating through a nonabsorbing, isotropically scattering slab, calculating the flux onto a small, directionally selective detector. These programs are backward Monte Carlo implementations of the equivalent FwdMCcs, FwdMCcka1, and FwdMCcka2, as also discussed in Examples 21.3 and 21.4. The backward Monte Carlo equivalent of FwdMCps.

RevMCcs.f90, RevMCck1.f90, RevMCck2.f90: RevMCps.f90: Software Packages MONT3D

This code, developed at Colorado State University by Burns et al. [10–14], calculates radiative exchange factors for complicated, three-dimensional geometries by the Monte Carlo method, as given by equations (8.15) and (8.21). Diffuse and specular view factors may be calculated as special cases.

VIEW3D

This code, developed at National Institute of Standards and Technology (NIST) by Walton [15], calculates radiative view factors with obstructions by adaptive integration.

RADCAL

This code, developed at NIST by Grosshandler [16, 17], is a narrow band database for combustion gas properties, using tabulated values and theoretical approximations.

REFERENCES

861

EM2C

This package contains a number of Fortran codes, developed at the Ecole Centrale de Paris by Soufiani and Taine [18], calculating statistical narrow band properties as well as narrow band k-distributions for CO2 and H2 O, using the HITRAN92 database together with some proprietary French hightemperature extensions.

NBKDIR

This package contains a number of Fortran codes, developed at the Pennsylvania State University and the University of California at Merced by the author and his students/postdocs A. Wang, G. Pal, and J. Cai, for the assembly of full spectrum k-distributions from a narrow band k-distributions database. At the time of printing NBKDIR contained data for five species (CO2 , H2 O, CO, CH4 , C2 H4 ), as well as nongray soot, for temperatures up to 3000 K and pressures up to 80 bar. Spectroscopic data are taken from the HITEMP 2010 (CO2 , H2 O, CO) [19] and HITRAN 2008 (CH4 , C2 H4 ) [20].

FVM2D

This Fortran77 code, developed at the University of Minnesota and Nanyang Technological University by Chai and colleagues [21–23], calculates radiative transfer in participating media using the finite-volume method of Chapter 17 for a two-dimensional, rectangular enclosure with reflecting walls and an absorbing, emitting, anisotropically scattering medium. For each surface the emittance and blackbody intensities must be specified; for the medium spatial distributions of radiation properties and blackbody intensities must be input. Internal incident radiation (G) and wall flux (q) fields are calculated. Can be used for gray problems or on a spectral basis.

References 1. MathWorks MATLAB website, http://www.mathworks.com/products/matlab/. 2. Huml´ıc˘ ek, J.: “Optimized computation of the Voigt and complex probability functions,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 27, p. 437, 1982. 3. Buckius, R. O., and D. C. Hwang: “Radiation properties for polydispersions: Application to coal,” ASME Journal of Heat Transfer, vol. 102, pp. 99–103, 1980. 4. Meng¨uc¸, M. P., and R. Viskanta: “On the radiative properties of polydispersions: A simplified approach,” Combustion Science and Technology, vol. 44, pp. 143–159, 1985. 5. Modest, M. F., and R. S. Mehta: “Full spectrum k-distribution correlations for CO2 from the CDSD-1000 spectroscopic databank,” International Journal of Heat and Mass Transfer, vol. 47, pp. 2487–2491, 2004. 6. Modest, M. F., and V. Singh: “Engineering correlations for full spectrum k-distribution of H2 O from the HITEMP spectroscopic databank,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 93, pp. 263–271, 2005. 7. Denison, M. K., and B. W. Webb: “Development and application of an absorption line blackbody distribution function for CO2 ,” International Journal of Heat and Mass Transfer, vol. 38, pp. 1813–1821, 1995. 8. Denison, M. K., and B. W. Webb: “An absorption-line blackbody distribution function for efficient calculation of total gas radiative transfer,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 50, pp. 499–510, 1993. 9. Modest, M. F., and R. J. Riazzi: “Assembly of full-spectrum k-distributions from a narrow-band database; effects of mixing gases, gases and nongray absorbing particles, and mixtures with nongray scatterers in nongray enclosures,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 90, no. 2, pp. 169–189, 2005. 10. Burns, P. J.: “MONTE–a two-dimensional radiative exchange factor code,” Technical report, Colorado State University, Fort Collins, 1983. 11. Maltby, J. D.: “Three-dimensional simulation of radiative heat transfer by the Monte Carlo method,” M.S. thesis, Colorado State University, Fort Collins, CO, 1987. 12. Burns, P. J., and J. D. Maltby: “Large-scale surface to surface transport for photons and electrons via Monte Carlo,” Computing Systems in Engineering, vol. 1, no. 1, pp. 75–99, 1990. 13. Maltby, J. D., and P. J. Burns: “Performance, accuracy and convergence in a three-dimensional Monte Carlo radiative heat transfer simulation,” Numerical Heat Transfer – Part B: Fundamentals, vol. 16, pp. 191–209, 1991. 14. Zeeb, C. N., P. J. Burns, K. Branner, and J. S. Dolaghan: “User’s manual for Mont3d – Version 2.4,” Colorado State University, Fort Collins, CO, 1999. 15. Walton, G. N.: “Calculation of obstructed view factors by adaptive integration,” Technical Report NISTIR–6925, National Institute of Standards and Technology (NIST), Gaithersburg, MD, 2002. 16. Grosshandler, W. L.: “Radiative transfer in nonhomogeneous gases: A simplified approach,” International Journal of Heat and Mass Transfer, vol. 23, pp. 1447–1457, 1980.

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F COMPUTER CODES

17. Grosshandler, W. L.: “RADCAL: a narrow-band model for radiation calculations in a combustion environment,” Technical Report NIST Technical Note 1402, National Institute of Standards and Technology, 1993. 18. Soufiani, A., and J. Taine: “High temperature gas radiative property parameters of statistical narrow-band model for H2 O, CO2 and CO, and correlated-k model for H2 O and CO2 ,” International Journal of Heat and Mass Transfer, vol. 40, no. 4, pp. 987–991, 1997. 19. Rothman, L. S., I. E. Gordon, R. J. Barber, H. Dothe, R. R. Gamache, A. Goldman, V. I. Perevalov, S. A. Tashkun, and J. Tennyson: “HITEMP, the high-temperature molecular spectroscopic database,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 111, no. 15, pp. 2139–2150, 2010. 20. Rothman, L. S., I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Simeckova, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. V. Auwera: “The HITRAN 2008 molecular spectroscopic database,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 110, pp. 533–572, 2009. 21. Chai, J. C., H. S. Lee, and S. V. Patankar: “Finite volume method for radiation heat transfer,” Journal of Thermophysics and Heat Transfer, vol. 8, no. 3, pp. 419–425, 1994. 22. Chai, J. C., H. S. Lee, and S. V. Patankar: “Treatment of irregular geometries using a Cartesian coordinates finite-volume radiation heat transfer procedure,” Numerical Heat Transfer – Part B: Fundamentals, vol. 26, pp. 225–235, 1994. 23. Chai, J. C., G. Parthasarathy, H. S. Lee, and S. V. Patankar: “Finite volume method radiative heat transfer procedure for irregular geometries,” Journal of Thermophysics and Heat Transfer, vol. 9, no. 3, pp. 410–415, 1995.

ACKNOWLEDGMENTS

c 1990 John Figs. 1-1, 1-8: Incropera, F. P., and DeWitt, D. P. Fundamentals of Heat and Mass Transfer, 3rd ed. Copyright Wiley & Sons, Inc., New York. Reprinted with permission of John Wiley & Sons, Inc. Fig. 1-3: Courtesy of NASA. c 1984 by Addison-Wesley Publishing Co. Reprinted by permission of AddisonFig. 1-14: White, F. M. Heat Transfer, Wesley Publishing Co., Inc., Reading, MA. Fig. 1-16: Edwards, D. K. “Radiation Interchange in a Nongray Enclosure Containing an Isothermal CO2 –N2 Gas Mixture,” ASME Journal of Heat Transfer, vol. 84C, 1962, pp. 1–11. Fig. 1-17: Neuroth, N. “Der Einfluss der Temperatur auf die Spektrale Absorption von Gl¨asern im Ultraroten, I (Effect of Temperature on Spectral Absorption of Glasses in the Infrared),” Glastechnische Berichte, 25, 242–249, 1952. ¨ ¸ ik, M. N. Radiative Transfer and Interactions with Conduction and Convection, John Figs. 2-2, 9-1a, b, 9-3, 22-8, 22-11: Ozis Wiley & Sons, Inc., New York, 1973. Reprinted by permission of the author. Figs. 2-12, 2-14, 12-8, 12-22: Bohren, C. F., and Hoffman, D. R. Absorption and Scattering of Light by Small Particles. c 1983 John Wiley & Sons, Inc., New York. Reprinted with permission of John Wiley & Sons, Inc. Copyright ¨ Fig. 3-1a, b: Schmidt, E., and Eckert, E. R. G. “Uber die Richtungsverteilung der W¨armestrahlung von Oberfl¨achen,” Forschung auf dem Gebiete des Ingenieurwesens, vol. 17, 1935, p. 175. Fig. 3-5: Torrance, K. E., and Sparrow, E. M. “Biangular Reflectance of an Electric Nonconductor as a Function of Wavelength and Surface Roughness,” ASME Journal of Heat Transfer, vol. 87, 1965, pp. 283–292. Figs. 3-8, 3-11: Parker, W. J., and Abbott, G. L. “Theoretical and Experimental Studies of the Total Emittance of Metals.” In Symposium on Thermal Radiation of Solids, NASA SP-55, 1965, pp. 11–28. Fig. 3-10: Dunkle, R. V. “Emissivity and Inter-Reflection Relationships for Infinite Parallel Specular Surfaces.” In Symposium on Thermal Radiation of Solids, NASA SP-55, 1965, pp. 39–44. Fig. 3-13: From: Spitzer, W. G., Kleinman, D. A., Frosch, C. J., and Walsh, D. J. “Optical Properties of Silicon Carbide.” In O’Connor, J. R., and Smittens, J., eds., Silicon Carbide—A High Temperature Semi-Conductor, Proceedings of the 1959 Conference on Silicon Carbide, Boston, MA. Pergamon Press, 1960, pp. 347–365. Figs. 3-14, 3-20: Jasperse, J. R., Kahan, A., Plendl, J. N., and Mitra, S. S. “Temperature Dependence of Infrared Dispersion in Ionic Crystals LiF and MgO,” Physical Review, vol. 140, no. 2, 1966, pp. 526–542. Fig. 3-15: Touloukian, Y. S., and DeWitt, D. P., eds. “Thermal Radiative Properties: Nonmetallic Solids.” In Thermophysical Properties of Matter, vol. 8, 1972. Plenum Press Publishing Corp., New York. c 1972. Fig. 3-16: American Institute of Physics Handbook, 3rd ed. Ch. 6, McGraw-Hill Publishing Co., New York, Reprinted with permission of McGraw-Hill Publishing Co., Inc. Fig. 3-17: Brandenberg, W. M. “The Reflectivity of Solids at Grazing Angles.” In Measurement of Thermal Radiation of Solids, 1963, pp. 75–82, NASA. Fig. 3-21: Riethof, T. R., and DeSantis, V. J. “Techniques of Measuring Normal Spectral Emissivity of Conductive Refractory Compounds at High Temperatures.” In Measurement of Thermal Radiation Properties of Solids, NASA SP-31, 1963, pp. 565–584. Fig. 3-23: Torrance, K. E., and Sparrow, E. M. “Theory for Off-Specular Reflection from Roughened Surfaces,” Journal of the Optical Society of America, vol. 57, no. 9, 1967, pp. 1105–1114. Fig. 3-24: Tang, K., and R. O. Buckius, “A statistical model of wave scattering from random rough surfaces,” International Journal of Heat and Mass Transfer, vol. 44, no. 21, 2001, pp. 4059–4073, Figure 6. Reprinted with the permission of Elsevier Science. Fig. 3-25: Sparrow, E. M., and Cess, R. D. Radiation Heat Transfer, Hemisphere Publishing Corp., New York, 1978. Reprinted by permission. Fig. 3-26: Pezdirtz, G. F., and Jewell, R. A. “A Study of the Photodegradation of Selected Thermal Control Surfaces.” In Symposium on Thermal Radiation of Solids, NASA SP-55, 1965, pp. 433–441. Fig. 3-30: Fan, J. C. C., and Bachner, F. J. “Transparent Heat Mirrors for Solar-Energy Applications,” Applied Optics, vol. 15, no. 4, pp. 1012–1017, 1976.

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ACKNOWLEDGMENTS

c 1974 John Wiley & Sons, Inc., Fig. 3-32: Duffie, J. A., and Beckman, W. A. Solar Energy Thermal Processes. Copyright New York. Reprinted with permission of John Wiley & Sons, Inc. Fig. 3-33, 3-47b: Edwards, D. K., Nelson, K. E., Roddick, R. D., and Gier, J. T. “Basic Studies on the Use and Control of Solar Energy.” Technical Report, The University of California, 1960, Report No. 60-93. Fig. 3-35: Trombe, F., Foex, M., and Lephat, V. M. “Research on Selective Surfaces for Air Conditioning Dwellings,” Proceedings of the UN Conference on New Sources of Energy, vol. 4, 1964, pp. 625–638. Fig. 3-36: Brandenberg, W. M., and Clausen, O. W. “The Directional Spectral Emittance of Surfaces Between 200 and 600 C,” in Symposium on Thermal Radiation of Solids, ed. S. Katzoff, NASA SP-55, 1965, pp. 313–319. Figs. 3-37, 3-38, 3-39: Courtesy of the Oriel Product Catalog. Fig. 3-41: Funai, A. I. “A Multichamber Calorimeter for High-Temperature Emittance Studies.” In Measurement of Thermal Radiation Properties of Solids, NASA SP-31, 1963, pp. 317–327. Figs. 3-42, 3-43: Fussell, W. B., and Stair, F. “Preliminary Studies Toward the Determination of Spectral Absorption Coefficients of Homogeneous Dielectric Material in the Infrared at Elevated Temperatures.” In Symposium on Thermal Radiation of Solids, NASA SP-55, 1965, pp. 287–292. Fig. 3-45: Birkebak, R. C., and Eckert, E. R. G. “Effect of Roughness of Metal on Angular Distribution of Monochromatic Related Radiation,” ASME Journal of Heat Transfer, vol. 87, pp. 85–94. Reprinted with permission of ASME. Fig. 3-46: Dunkle, R. V. “Spectral Reflection Measurements,” in First Symposium—Surface Effects on Spacecraft Materials, John Wiley & Sons, Inc., New York, pp. 117–137, 1960. Fig. 3-48: Touloukian, Y. S., and DeWitt, D. P., eds. Thermal Radiative Properties: Metallic Elements and Alloys, vol. 7 of Thermophysical Properties of Matter, Plenum Press Publishing Corp., New York, 1970. Fig. 6-1: Sarofim, A. F., and Hottel, H. C. “Radiation Exchange Among Non-Lambert Surfaces,” ASME Journal of Heat Transfer, vol. 88C, 1966, pp. 37–44. Fig. 6-11: Reprinted with permission from Solar Energy, vol. 7, no. 3, Hollands, K. G. T. “Directional Selectivity Emittance and Absorptance Properties of Vee Corrugated Specular Surfaces,” 1963, pp. 108–116. Fig. 6-17: Toor, J. S. “Radiant Heat Transfer Analysis Among Surfaces Having Direction Dependent Properties by the Monte Carlo Method,” M.S. Thesis, Purdue University, 1967. Fig. 8-2: Modest, M. F., and Poon, S. C. “Determination of Three-Dimensional Radiative Exchange Factors for the Space Shuttle by Monte Carlo,” ASME paper no. 77-HT-49, 1977. Reprinted with permission of ASME. Fig. 9-2: Sparrow, E. M., Eckert, E. R. G., and Irvine, T. P. “The Effectiveness of Radiating Fins with Mutual Irradiation,” Journal of the Aerospace Sciences, no. 28, 1961, pp. 763–772. Figs. 11-5: Tien, C. L. Thermal Radiation Properties of Gases, vol. 5 of Advances in Heat Transfer, Academic Press, 1968, pp. 253–324. Figs. 11-29, 11-30: Reprinted by permission of Elsevier Science Publishing Co., Inc. from “Spectral and Total Emissivity c 1972 by the Combustion of Water Vapor and Carbon Dioxide” by B. Leckner, Combustion and Flame, vol. 19, pp. 33–48. Institute. Figs. 11-34, 11-35: Tien, C. L., and Giedt, W. H. “Experimental Determination of Infrared Absorption of HighTemperature Gases,” Advances in Thermophysical Properties at Extreme Temperatures and Pressures, ASME, 1965, pp. 167–173. Reprinted by permission of ASME. Fig. 11-36: Bevans, J. T., Dunkle, R. V., Edwards, D. K., Gier, J. T., Levenson, L. L., and Oppenheim, A. K. “Apparatus for the Determination of the Band Absorption of Gases at Elevated Pressures and Temperatures,” Journal of the Optical Society of America, vol. 50, 1960, pp. 130–136. Fig. 12-2: Tien, C. L., and Drolen, B. L. “Thermal Radiation in Particulate Media with Dependent and Independent Scattering.” In Annual Review of Numerical Fluid Mechanics and Heat Transfer, vol. 1, Hemisphere Publishing Corp., New York, 1987, pp. 1–32. Figs. 12-3: Van de Hulst, H. C. Light Scattering by Small Particles, Dover Publications, New York, 1981. Figs. 12-4a, b: Kattawar, G. W., and Plass, G. N. “Electromagnetic Scattering from Absorbing Spheres,” Applied Optics, vol. 6, no. 8, 1967, pp. 1377–1383. Figs. 12-6, 12-7, 14-4: Modest, M. F., and Azad, F. H. “The Influence and Treatment of Mie-Anisotropic Scattering in Radiative Heat Transfer,” ASME Journal of Heat Transfer, vol. 102, 1980, pp. 92–98. Reprinted by permission of ASME. Fig. 12-16: Reprinted with permission from Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 33, no. 4, Crosbie, A. L., and Davidson, G. W. “Dirac-Delta Function Approximations to the Scattering Phase Function,” 1985, Pergamon Press. Fig. 12-19: Millikan, R. C. “Optical Properties of Soot,” Journal of the Optical Society of America, vol. 51, 1961, pp. 698–699. ¨ O., ¨ and G. M. Faeth, “Radiative properties of flame-generated soot,” ASME Journal of Heat Figs. 12-21a, b: Koyl ¨ u, ¨ U. Transfer, vol. 115, no. 2, 1993, pp. 409–417, Figures 3 and 5. Reprinted by permission of ASME.

ACKNOWLEDGMENTS

865

Figs. 12-23, 12-26: Hottel, H. C., Sarofim, A. F., Vasalos, I. A., and Dalzell, W. H. “Multiple Scatter: Comparison of Theory with Experiment,” ASME Journal of Heat Transfer, vol. 92, 1970, pp. 285–291. Reprinted by permission of ASME. Figs. 12-24, 12-25a, b: Smart, C., Jacobsen, R., Kerker, M., Kratohvil, P., and Matijevic, E. “Experimental Study of Multiple Light Scattering,” Journal of the Optical Society of America, vol. 55, no. 8, 1965, pp. 947–955. Figs. 12-27: Mulholland, G. W., and M. Y. Choi: “Measurement of the mass specific extinction coefficient for acetylene and ethene smoke using the large agglomerate optics facility,” Proceedings of the Combustion Institute, vol. 27, pp. 1515–1522, 1998. Reprinted by permission of the Combustion Institute. Figs. 13-1, 13-2: Smakula, A. “Synthetic Crystals and Polarizing Materials,” Optica Acta, vol. 9, 1962, pp. 205–222. Fig. 13-3: Boyd, I. W., Binnie, J. I., Wilson, B., and Colles, M. J., “Absorption of Infrared Radiation in Silicon,” Journal of Applied Physics, vol. 55, no. 8, 1984, pp. 3061–3063. Fig. 13-5: Barker, A. J. “The Effect of Melting on the Multiphonon Infrared Absorption Spectra of KBr, NaCL, and LiF,” Journal of Physics C: Solid State Physics, vol. 5, 1972, pp. 2276–2282. Reprinted with permission of The Institute of Physics. Fig. 13-11: Myers, V. H., Ono, A., and DeWitt, D. P. “A Method for Measuring Optical Properties of Semitransparent Materials at High Temperatures,” AIAA Journal, vol. 24, no. 2, 1986, pp. 321–326. Figs. 13-12a, b: Ebert, J. L., and Self, S. A., “The Optical Properties of Molten Coal Slag.” In Heat Transfer Phenomena in Radiation, Combustion and Fires, vol. HTD-106, ASME, 1989, pp. 123–126. Reprinted by permission of ASME. Fig. 16-4: Reprinted with permission from Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 23, Modest, M. F., and Azad, F. H. “The Differential Approximation for Radiative Transfer in an Emitting, Absorbing and Anisotropically Scattering Medium,” 1980, Pergamon Press. Figs. 16-6, 16-7: Reprinted with permission from Journal of Quantitative Spectroscopy and Radiative Transfer, Modest, M. F., and Yang, J. “Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries,” vol. 109, 2008, pp. 1641–1666, Elsevier Science. Figs. 16-11, 16-12: Reprinted with permission from Proceedings of Eurotherm Seminar 95, Modest, M. F., and Lei, S. “Simplified Spherical Harmonics Method For Radiative Heat Transfer,” Elsevier Science. Fig. 17-5: Abstracted from: Truelove, J. S. “Discrete-ordinate Solutions of the Radiation Transport Equation,” ASME Journal of Heat Transfer, vol. 109, no. 4, 1987, pp. 1048–1051. Reprinted with permission of ASME. Figs. 22-13, 22-14: Reprinted with permission from International Journal of Heat and Mass Transfer, vol. 24, Azad, F. H. and Modest, M. F. “Combined Radiation and Convection in Absorbing, Emitting and Anisotropically Scattering Gas-Particulate Tube Flow,” 1981, Pergamon Press. Figs. 21-2: Reprinted with permission from International Journal of Heat and Mass Transfer, Wang, A. and Modest, M. F., “Spectral Monte Carlo Models for Nongray Radiation Analyses in Inhomogeneous Participating Media,”vol. 26, pp. 3877–3889, 2007, Elsevier Science. Figs. 21-3, 21-4: Reprinted with permission from ASME Journal of Heat Transfer, Wang, A. and Modest, M. F., “Photon Monte Carlo Simulation for Radiative Transfer in Gaseous Media Represented by Discrete Particle Fields,”vol. 128, pp. 1041–1049, 2006. Figs. 22-9a, b: Reprinted with permission from International Journal of Heat and Mass Transfer, vol. 5, Viskanta, R., and Grosh, R. J. “Effect of Surface Emissivity on Heat Transfer by Simultaneous Conduction and Radiation,” 1962, Pergamon Press. Fig. 22-10: Viskanta, R. “Radiation Transfer and Interaction of Convection with Radiation Heat Transfer.” In Advances in Heat Transfer, vol. 3, Academic Press, Inc., New York, 1966, pp. 175–251. Fig. 22-12: Kurosaki, Y. “Heat Transfer by Simultaneous Radiation and Convection in an Absorbing and Emitting Medium in a Flow Between Parallel Plates.” In Proceedings of the Fourth International Heat Transfer Conference, vol. 3, no. R2.5, 1970. Figs. 22-13, 22-14: Reprinted with permission from International Journal of Heat and Mass Transfer, vol. 24, Azad, F. H. and Modest, M. F. “Combined Radiation and Convection in Absorbing, Emitting and Anisotropically Scattering Gas-Particulate Tube Flow,” 1981, Pergamon Press. Fig. 22-15: Negrelli, D. E., J. R. Lloyd, and J. L. Novotny, “A Theoretical and Experimental Study of Radiation– Convection Interaction in a Diffusion Flame,” ASME Journal of Heat Transfer, vol. 99, 1977, pp. 212–220, Figure 3. Reprinted by permission of ASME. Fig. 22-16: Kaplan, C. R., S. W. Baek, E.S. Oran, and J.L. Ellzey, “Dynamics of a Strongly Radiating Unsteady Ethylene Jet Diffusion Flame,” Combustion and Flame, vol. 96, 1994, pp. 1–21, Figure 10. Reprinted with the permission of Elsevier Science Limited. Figs. 22-17, 22-18: Badinand, T., and T. Fransson, “Improvement of the Finite Volume Method for Coupled Flow and Radiation Calculations by the Use of Two Grids and Rotational Periodic Interface,” from Radiative Transfer 2001—The Third International Symposium on Radiative Transfer, edited by M. P. Menguc ¨ ¸ and N. Selc¸uk, 2001, Begell House, Figures 4 and 6. Reprinted by permission.

866

ACKNOWLEDGMENTS

Figs. 22-19: Reprinted with permission from Journal of Quantitative Spectroscopy and Radiative Transfer, Wang, A., Modest, M. F., Haworth, D. C. and Wang, L. “Monte Carlo Simulation of Radiative Heat Transfer and Turbulence Interactions in Methane/Air Jet Flames,”vol. 109, pp. 269–279, 2008, Elsevier Science. Figs. 22-20: Proceedings of 2011 ASME/JSME Thermal Engineering Joint conference, Pal, G., Gupta, A., Modest, M. F., and Haworth, D. C. “Comparison of accuracy and computational expense of radiation models in simulation of nonpremixed turbulent jet flames,”, 2011. Reprinted with permission of ASME. Fig. 22-21a: Reprinted with permission from International Journal of Heat and Mass Transfer, Maag, G., Lipinski, W. and ´ Steinfeld, A. “Particle–gas reacting flow under concentrated solar irradiation,” vol. 52, pp. 4997–5004, 2009. Elsevier Science. Fig. 22-21b: ASME Journal of Thermal Science and Engineering Applications, Hischier, I. Hess, D., Lipinski, W., Modest, ´ M. F. and Steinfeld, A. “Heat transfer analysis of a novel pressurized air receiver for concentrated solar power via combined cycles,” vol. 1, pp. 041002, 2009. Reprinted with permission of ASME. Unnumbered figures in Appendix D: Howell, J. R. A Catalog of Radiation Configuration Factors, McGraw-Hill, New York, 1982. Reprinted with permission.

INDEX

Index Terms

Links

A Abel’s transformation

795

Absorbing medium

50

Absorptance

21

65–68

103

821

spectral, directional

65

66

spectral, hemispherical

66

71

total, directional

67

68

total, hemispherical

67

solar

Absorption

2

gray, diffuse in a participating medium

304

129 280–281

multiphoton

612

negative

304

saturable

612

Absorption band

24

58

307

60

84

24

37

86 Absorption coefficient

xxi 85

band-integrated

xx

correlated

345

database

669

effective

307

634

for a particle cloud

394

395

for coal particles

416

for Rayleigh scattering

399

line-integrated

xix

316

linear

280

307

mass

280

307

mean

367–369 This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

660

635

Index Terms

Links

Absorption coefficient (Cont.) modified Planck-mean

368

narrow band average

327

328

Planck-mean

367

395

416

417

424

635

709 pressure

280

307

Rosseland-mean

368

395

416

417

424

635

scaled

345

660

spectral

327

of carbon dioxide

313

of clear ice

443

of clear water

443

of halides

441

of ionic crystals

441

of lithium fluoride

444

of nitrogen

314

of silicon

442

of window glass true

25

324

442

307

Absorption cross-section

389

Absorption Distribution Function model

654

748

Absorption edge

441

444

Absorption efficiency factor

389

for absorbing spheres

393

for specularly reflecting spheres

404

Absorption suppression Absorptive index

Absorptivity

713 xviii

7

35

36

85

86

281

290

xx

see also Absorptance of a gas layer

24

of a thick slab

96

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

Absorptivity (Cont.) of an isothermal medium of carbon dioxide

650 24

spectral of a participating medium

281

644

of a gas

363

646

of a gas-particulate suspension

646

of a participating medium

644

of an isothermal medium

647

total

Acetylene

430

Acrylic paint

94

ADF method

654

750

751

679

743

26

27

748 Aerogel

729

Aggregate fractal

414

soot

414

Air mass

6

Air plasma

313

Airy’s formulae

57

Albedo, scattering

xxi 285

Alumina

93

112

Aluminum

76

93

112

393

809

812

Ammonia

369

Amorphous solid

89

442

Amplitude function

390

398

for diffraction

402

Angle azimuthal

xxi

Brewster’s

48

critical

48

divergence

427

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

12

290

Index Terms

Links

Angle (Cont.) of incidence

45

of refraction

45

opening

48

71

104

phase

33

55

448

polar

xxi

12

15

polarizing

48

scattering

xxi

391

solid

xxi

11–13

zenith

6

Angle factor, see View factor

131

Angular frequency

xxi

3

Anomalous diffraction

401–402

Anomalous skin effect

79

82

172

179

213

412

Apparent emittance

217 Ash particle

415

Asymmetric top

308

Asymmetry factor

391

410

for a particle cloud

394

396

for coal particles

416

Atomic force microscope

815

Attenuation by absorption

280–281

by scattering

281

Attenuation vector

33

Azimuth

38

Azimuthal angle

xxi

12

B Babinet’s principle

402

Band absorption

84

electron energy

58

electronic-vibration-rotation

313

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

86

Index Terms

Links

Band (Cont.) fundamental

311

molecular vibration

442

overtone

311

Reststrahlen

84

86

441

442

88

symmetric

350

vibration-rotation

304

308

with a head

312

351

xvii

333

349

351

638

647

234

242

Band absorptance

for nonisothermal gas

358

slab

648

Band absorptance correlation

357

Band approximation

233

311

761 Band center

311

Band gap

58

Band intensity

441

349

Band model narrow band

326–336

wide band

349–362

Band origin

311

Band overlap

362

Band strength parameter

xx

312

351

355

363 Band width

637

effective

xvii

parameter

351

349

Band wing

643

bandapp

234

246

857

bandmxch

234

246

857

Bandpass filter

84

Basis function

780

Bayesian statistics

796

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms bbfn

Links 11

Beam channeling

222

Beam splitter

110

Beer’s law

855

24

BFGS scheme

789

Bidirectional reflection function

198

235

for magnesium oxide

70

91

spectral

68

90

total

71

Black chrome

103

Black nickel

103

Black surface

5

Black-walled enclosure

5

238

160

161

7

15

10

833–835

14

15

Blackbody manufacture of

179

reference

449

Blackbody cavity

172

Blackbody cavity source

107

Blackbody emissive power

xvii 833–835

fraction of total

xviii 10

Blackbody intensity

xviii 305

Bleaching

612

Boltzmann number

xvii

741

Boltzmann’s constant

xviii

7

Boltzmann’s distribution law

305

Boltzmann, Ludwig Erhard

10

Bound electron transition

76

Boundary layer

738–743

Box model

349

637–643

Brass, oxidized

197

198

Bremsstrahlung

304

Brewster’s angle

48

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

448

807

Index Terms Brewster, Sir David

Links 48

Broadening collision

316

Doppler

316–318

line

24

Lorentz

316

natural line

316

Stark

316

Voigt

319

Bundle, energy

249

304

316

317

C Calcium carbonate

763

callemdiel

856

callemmet

856

Candela

18

Carbon foam

450

particle

760

Carbon capture

751

Carbon dioxide

312

315

323

351

353–355

360

364

365

374

651

652

654

655

662

669

671 Planck-mean absorption coefficient

368

total emissivity

365

366

Carbon foam

445

Carbon monoxide

351

353

354

356

363

369

Carbon particle

418

763

Case’s normal-mode expansion technique

475

Causal relationship

448

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

Cavity conical

213

cylindrical

171

hemispherical

184

spherical

213

CDSD database

179

213

323

341

669

670

678

684

Cell cold-window

370

hot-window

370

nozzle seal

371

Cement

763

Central limit theorem

251

372

Ceramics reticulated porous

Cerium dioxide Cesium

446

447

760

761

762 77

Chamotte

760

Char

415

Charge density

xxi

32

Chemical reaction

752

754

CHEMKIN

750

Chopper

111

Chrome-oxide coating

103

Chromium

815

CLAM scheme

559

Cluster T-matrix method

414

Coal gasification

763

Coal particle properties

370

415–418

coalash

858

Coating

53

antireflective

53

chrome-oxide

103

for glass

103

98–99

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

450

Index Terms

Links

Coating (Cont.) nickel-oxide

103

reflectivity

98

surface

95

Cobalt

77

Coherence

803

Coherence length

803

Coherence time

803

Cold medium approximation

287

Collimated irradiation

66

216

545

610–625

Collision broadening

316

Collisional interference

316

Color center

441

Colors of the sky

399

Combustion

222

748–751

Complex index of refraction

xix

35

74

388

of ash

415

of coal

415

of metals

77

of semiconductors

77

of various soots

421

Composition PDF method

756

Composition variable

754

Computational fluid dynamics

751

Computer codes

73

756

855–862

Computing, parallel

261

Concentrator, compound parabolic

760

Condition number

783

785

1

2

460

xix

270

725

741

745

Conduction

706 Conduction-to-radiation parameter

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

Conductivity dc-

76

79

electrical

xxi

32

36

57

76

77

79 radiative

483

thermal

xviii

Configuration factor, see View factor Conjugate gradient method

131 792–796

Conjugation coefficient

793

Conservation of energy

2

overall

2

297–298

radiative

295

Contour integration

134

Convection

458

1

2

460

706 free

743

in boundary layers

738–743

in internal flow

744–753

interfacing with radiation Convection-to-radiation parameter Copper

751 xvii

741

76

77

93

814 Correlated k-distribution

326

global

326

narrow band

345

wide band

326

359–362

Correlation length

xxi

91

Cosine law

15

Couette flow

746

Critical angle

48

661

806 Cross-section for absorption

389

for extinction

389

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

290

804

Index Terms

Links

Cross-section (Cont.) for scattering

389

Crossed-strings method

135

Crossover wavelength

82

Crystal lattice

57

83

337

360

658

484

490

201

205

Cumulative k-distribution

147

162

663 Cumulative wavenumber

665

Current density

810

Curtis-Godson approximation

334

Cutoff wavelength

102

Cylinders, concentric at radiative equilibrium

474 508

discrete ordinates method without participating medium

554–555 176 589

Cylindrical fiber absorption and scattering by Cylindrical medium discrete ordinates method

408 471–474 554–555

D Damping parameter

792

Darkening

612

Database absorption coefficient

669

CDSD

323

341

669

670

678

684

333

341

374

652

655

671

323

341

347

362

368

627

652

655

657

EM2C

672 HITEMP

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

Database (Cont.) 662

663

667–672

676

678

684

312

315

323

330

333

334

341

368

374

627

655

669

671

861

861 HITRAN

k-distribution

671

NBKDIR

341

NEQAIR

323

NIST

325

RADCAL

333

SPECAIR

323

SPRADIAN

323

Dc-conductivity Degeneracy

76

79

xviii

305

352 Degrees of freedom, of a molecule

308

Deissler’s jump boundary conditions

484

Delta-Eddington approximation

411

Density

xxi

charge

xxi

optical

48

partial, of absorbing gas

32

320

Density path length

320

Detectivity

xvii

110

108–110

450

Detector fiber-optic

428

photon or quantum

109

pyroelectric

370

thermal

109

Diamond differencing

557

Diamond scheme

559

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

308

Index Terms

Links

Dielectric film

56

Dielectric function

xx

57–59

75

810

Dielectric layer

93

Dielectric medium

36

74

45

55

497

502–509

74 Differential approximation, see P1-approximation

488 730

modified

527–531

Diffraction

25

by a particle

387

from large spheres

402

Diffraction peak

397

Diffraction theory

90

Diffuse emission

15

Diffuse emitter

62

Diffuse irradiation

71

Diffuse reflectance

69

Diffuse reflector

70

Diffuse view factor, see View factor

131

Diffusion approximation

299

129

482–486

741

761–763 Diffusion flame laminar

750

751

Dipole

809

Dipole element

314

Dipole moment

309

443

xx

337

411

610

751

759

Dirac-delta function

Direct numerical simulation

409

Direction of incidence of propagation specular

71 281 69

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

236

289

Index Terms Direction cosine

Direction vector

Links xviii

xxi

457

490

xix

11

36

135

456

697

dirreflec

856

Discrete dipole approximation

414

Discrete ordinate method

761

Discrete ordinates method

299

300

532

541–584

even-parity modified

135

422

488

573 572–573

Discrete transfer method

300

564

572

575 Dispersion

57

anomalous

59

normal

59

Dispersion exponent

419

Dispersion relation

807

Dispersion theory Dissipation function

420

809

73–75 xxi

297

Distribution function cumulative

249

Gaussian

398

particle

xix

395

416

417 Divergence angle

427

Dopant

441

Doppler broadening

316–318

Doppler effect

318

Doppler shift

318

Drude theory

75–78

809

812

xviii

810

811

Dyadic Green’s function

E Eddington approximation

299

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

426

Index Terms Eddy dissipation model Efficacy, luminous Efficiency factor absorption

Links 753 18 389 389

for absorbing spheres

393

for specularly reflecting spheres

404

extinction

389

for dielectric spheres

392

for long cylinders

409

for water droplets

398

Rayleigh scattering

398

scattering

389

for absorbing spheres

393

for long cylinders

409

for specularly reflecting spheres

404

Efficiency, luminous

19

18

19

Eigenfrequency

310

Einstein coefficients

xvii

305–308

Electric field

xvii

32

Electrical conductivity

xxi

32

36

57

76

77

58

60

173–177

214

215

xx

32

57

808

810

313

79 Electrical conductor

36 75

Electrical network analogy Electrical permittivity Electrical resistivity

76

Electromagnetic energy

36

Electromagnetic wave

1

3

Electromagnetic wave spectrum

3

4

Electromagnetic wave theory

3

31–60

Electron

32 73–75

809

bound

58

60

free

32

58

75

442

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

60

Index Terms Electron energy Electron volt

Links 58

59

4

Electronic transition

86

442

Electronic-vibration-rotation band

313

Electrostatic approximation

423

Ellipsometric parameter

38

40

Ellipsometric technique

448

449

Ellipticity

38

Elsasser model EM2C database

327–329 333

341

374

652

655

671

672

860

emdiel

88

856

emdielr

88

Emission

1

blackbody diffuse

7 15

62

63

129 from a gas volume

362

from a volume element

282

from any isothermal volume

293

gray, diffuse

129

luminous

415

spontaneous

304

305

307

stimulated

304

305

307

315 Emission coefficient Emission measurement Emissive power

282

307

111–113 xvii

apparent

217

blackbody

xvii

5–11

7

833–835 blackbody spectrum directional

8 15

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

9

15

Index Terms

Links

Emissive power (Cont.) effective

166

maximum

8

spectral

5

spectral, directional total

62

62 5

weighted

10

604

Emissivity

xx see also Emittance

narrow band

327

of a nonhomogeneous layer

345

of an isothermal medium

282

spectral

327

of a participating medium

644

of an isothermal layer

282

650

629

spectral, directional of nonconductors spectral, hemispherical

87 80

of nickel

80

of nonconductors

87

spectrally averaged

328

total of a gas

362–367

of an isothermal layer

633

of carbon dioxide

365

366

of water vapor

364

366

total, directional Emittance apparent

81 22

62–64

172

179

217 hemispherical infrared

22 103

of selected materials

22

spectral, directional

62

spectral, hemispherical

62

of tungsten

83

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

68

213

Index Terms

Links

Emittance (Cont.) spectral, normal of aluminum

93

of zirconium carbide

90

total, directional

62

of several metals

63

of several nonmetals

63

total, hemispherical of a metal total, normal of polished metals

64

64

68

68

82 78

79

79

tables

821

Emitted energy

13

emmet

80

emwbm

858

Enclosure

130

black-walled

94

856

5

160

161

531

532

643

closed

130

164

idealized

129

130

203

214

isothermal

203

long

147

open

164

160

206

Energy electromagnetic

36

internal

xx 304

of a photon

3

solar

2

Energy bundle

249

path

253

Energy conservation equation

297

Energy density, radiation

292

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

23

297

Index Terms

Links

Energy level electronic

23

molecular

303

310

rotational

23

304

308

vibrational

23

83

304

308

310

312

Energy partitioning

262

713

Enthalpy

752

754

Environment, large, isothermal

170

Epoxy coating

94

103

Equation of transfer, see Radiative transfer equation (RTE) Equilibrium radiation

305

Error, statistical

248

Ethene

430

Ethylene

750

758

Euler angles

xx

512

513

Euler’s constant

xx

333

853

xviii

xix

585–586

590–592

596–597

Evanescent wave Even-parity formulation

804–806 573

Exchange area direct

determination of total

Exchange factor

606 xviii

xix

593–596

598–600

xviii

252

254

585

586

695

333

459

Exchange function

812

Exponential integral

xvii 852–854

Exponential kernel approximation, see Kernel approximation Exponential scheme

559

Extinction

281

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

389

586–590

Index Terms Extinction coefficient

Links xx

25

281

426 for a particle cloud

394

395

for coal particles

415

416

modified

558

Planck-mean

395

416

417

395

416

417

424

446

483

424 Rosseland-mean

634 Extinction efficiency factor

389

for dielectric spheres

392

for long cylinders

409

for water droplets

398

Extinction paradox

402

F False scattering

560

Favre averaging

753

FDF method

759

Fiberglass

450

Fibers, scattering by

409

450

Fictitious gas technique

348

654

Figure of merit

712

Film dielectric

56

metallic

176

nonmetallic

103

porous

99

slightly absorbing

56

thick

56

thin

53

Filter, bandpass

84

Filter, optical

107

Filtered density function

759

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

447

679

Index Terms

Links

Fin efficiency

270

Fin radiator

268

Finite volume method

300

566–572

Flame axisymmetric

533

laminar diffusion

749

luminous

758

nonluminous

758

Flame D, Sandia

533

759

668

757

758 Fluctuation-dissipation theorem

810

Fluctuational electrodynamics

803

Fluidized bed

408

Fluidized bed, solar

760

809

Flux heat

xix

2

luminous

xix

19

momentum

17

Flux method

574

Fly ash properties FN -method

415–418 475

Foam closed cell

446

open cell

444

Foam insulation

450

Forced collisions

713

Fourier’s law

2

28

Fractal aggregate

414

422

Fractal prefactor

xviii

422

424

178

179

462

32

75

442

Fractal surface

91

Fredholm integral equation Free electron Freezing Frequency angular

297

733–738 xxi

3

xxi

3

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

Frequency (Cont.) of radiation

2

plasma

59

resonance

59

fresnel

47

76

52

87

47

52

73

74

79

85

86

290

856 Fresnel’s relation

Fresnel, Augustin-Jean FSCK method

47 534

662

684

672

678

704 FSCK Monte Carlo FSK method

704 656–686

fskdco2

670

fskdco2

859

fskdco2dw

670

fskdco2dw

859

fskdh2o

670

fskdh2o

859

fskdh2odw

670

fskdh2odw

859

fskdist

669

fskdistmix

859

FSSK method

664

684

FTIR spectrometer

108

370

ftwbm

357

858

Fuel sprays

706

Full spectrum k-distribution

656–686

Function estimation

779

Fundamental band

311

Furnace

449

high-temperature

371

sealed-chamber

449

FVM2D

861 This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

678

672

859

Index Terms

Links

FwdMCcs

719

FwdMCxx

860

G Galerkin method

475

Gamma distribution

395

Gamma rays

94

Gas emission from

362

mixture

334

341

sum of gray gases

603

649–654

total absorptivity

363

total emissivity

362

673

Gas layer isothermal

282

nonisothermal

334

Gas-particulate mixture

635

Gauss’ theorem

782

Gaussian distribution function

398

Genetic algorithms

796

Geometric path length Glass

646

43

Gauss-Newton method

Geometric optics

321

26

53

92

95

389

402

320 86

multiple panes

99

single pane

95

soda-lime

97

Global model

326

655

2

303

Globar light source

107

427

Godson approximation

329

Global warming

Gold Goody model

77 330

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

701

651

Index Terms

Links

Graphite pyrolytic

653

Gray medium

295

Gray source

68

Gray surface

64

Gray, diffuse surface

64

299

160

165

288 graydiff

172

857

graydiffxch

172

194

195

204

228

857

2

96

857 graydifspec Greenhouse effect Grid system

751

Groove right-angled

164

170

V-corrugated

104

105

211

212

222

236

204

228

857

77

78

Half-width, line

316

318

Halide

444

grspecxch

H Hagen-Rubens relation

Harmonic oscillator

58

74

311

314

Heat conduction

297

Heat flux

xix

at a surface

202

average

162

directional

2

15

outgoing

202

prescribed

272

radiative

13

15–17

292–293 reflected

310

69 This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

66

Index Terms

Links

Heat flux vector

297

Heat of fusion

735

Heat rate

xix

2

Heat rejector, radiative

101

103

Heat source

297

298

xviii

2

convective

174

271

radiative

270

Heat transfer coefficient

Heaviside’s unit step function

Helmholtz equation Hemisphere

815

xviii

338

620

637

665

718

504

810

11

Hemispherical cavity

184

Hemispherical volume

629

Henyey-Greenstein phase function

410

411

Hessian matrix

788

789

HITEMP database

323

341

347

362

368

627

652

655

657

662

663

667–672

676

678

684

312

315

323

330

333

334

341

368

374

627

655

669

671

861

Hohlraum

107

112

Hole, cylindrical

171

179

Hot band

315

Hot line

315

861 HITRAN database

368 Hottel, Hoyte Clark

147

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

323

348

Index Terms

Links

I Ice

443

Ill-conditioned problem

782

Ill-posed problem

779

780

19

217

199

207

In-scattering

27

282

Incidence angle

45

71

Incidence direction

71

Illumination Image

Incident radiation for a plane-parallel medium

xviii

287

292

304

441

458

Index of refraction, complex, see Complex index of refraction Induced emission, see Emission, stimulated

304

Infrared emittance

103

Infrared radiation

4

Inside sphere method

135

Insulation, foam

450

Insulator

58

Integral equation for outgoing intensity

235

for radiosity

166

for specular reflections

220

Fredholm

178

179

426

430

xviii

13–15

xviii

14

Integrating sphere Intensity blackbody

203

305 in vacuum

20–21

outgoing

235

reflected

69

weighted

651

Interaction radiation and combustion

748–751

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

234

235

15

Index Terms

Links

Interaction (Cont.) radiation and conduction

268–271

724–733

radiation and convection

271–275

738–748

radiation and melting/freezing

733–738

radiation and turbulence

753–759

Interface moving

736

optically smooth

45

73

plane

43

44

43

44

50

49

51

56

98

23

297

Interface condition

736 Interface reflectivity

48 52

Interfacing, convection with radiation

751

Interference structure

398

Interference, wave

53 99

Internal energy

xx 304

Invariance, principle of

475

Inverse Bremsstrahlung

304

Inverse heat transfer

779

Inverse radiation Ionic crystal

779–796 440

Iron

441

77

Irradiation

xviii

130

166

532

66

216

545

610–625

diffuse

66

71

diffuse and gray

67

directional

65

collimated

external

161

gray

67

laser

229 This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

160

222

162

202

610

611

Index Terms

Links

Irradiation (Cont.) polarized solar

86

197

6

611

spectral, directional

65

spectral, hemispherical

70

total

67

Isotropic medium

32

Isotropic scattering

283

287

391

411

Isotropic surface

299

63

J Jacobian

781

Jeans, Sir James Hopwood

6

Jet diffusion flame nonluminous

756

Jet engine

752

Jet flame

533

Jump boundary condition

485

K k-distribution

xviii

326

359–362

xviii

337

360

658

663

748 cumulative

databases

671

for mixtures

673

global

326

narrow band

656–686

336–349

Planck function weighted

656

wide band

326

Kalman filtering

796

kdistmix

859

Kernel

178

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

661

491

Index Terms Kernel approximation

Links 179

182

300

14

74

164

204

491–493 Kirchhoff approximation Kirchhoff’s law

92 5 236

for absorptance

66

for bidirectional reflection

69

Kirchhoff, Gustav Robert Kramers-Kronig relation Kronecker’s delta

5 448 xx 781

L Ladenburg-Reiche function

321

Lambert

18

62

Lambert’s law

15

62

Lambert, Johann Heinrich

15

Laplace transform

xix

Laplace’s equation

504

Large eddy simulation

751

Laser

729

759

pulsed

610

619–622

Laser irradiation

229

610

Laser light source

106

Latex particles

427

430

Lattice defect

79

82

83

85

103

441

Lattice vibration

58

88

809

Lattice, crystal

57

83

Law of reciprocity for bidirectional reflection function

69

for diagonally opposed pairs

145

for direct exchange areas

586

for exchange factors

253

for specular view factors

200

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

592

611

Index Terms

Links

Law of reciprocity (Cont.) for total exchange areas for view factors

587

593

131–133

Layer dielectric

93

of alumina

93

of silica

94

opaque

85

oxide

93

94

surface

79

84

thick dielectric

98

thin

93

93

LBL, see Line-by-line calculations Least squares norm

780

Leckner

366

858

Legendre polynomials

xix

391

456

associated

456

496

502

orthogonality of

498

polyadic

497

Leibniz’s rule

103

Lens

110

Levenberg-Marquardt method

792

796

3

18

Light polarized

448

Light guide

223

Light line

809

Light source

106

blackbody cavity

107

globar

107

laser

106

Nernst glower

107

Lighting Lime

183

18 763

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

370

427

218

Index Terms

Links

Line Absorption-emission

23

collision-broadened

316

Doppler-broadened

316

hot

348

isolated

316

Lorentz

316

no overlap

329

rotational

304

spectral

304

308

strong

329

330

Voigt-broadened

316

weak

329

330

Line broadening

24

304

316

Line half-width

xx

316

318

Line intensity

306

313

Line mixing

316

Line overlap parameter

xx

315

328

355

356 Line shape

316

Line shape function

xxi

306

316

Line spacing

xvii

327

328

Line strength

xix

306

316

xx

313

314

321

328

317 Line strength parameter

Line structure effects

702

Line width, equivalent

xx

320

329

325

627

652–654

659

667

668

673

678

702

330 nonhomogeneous path Line-by-line calculations

335

Line-by-line Monte Carlo

702

Linear-anisotropic phase function

411

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms Linear-anisotropic scattering

Links 299

411

Liquid high-temperature

449

semitransparent

442

Lithium

77

Lithium fluoride

444

Lorentz broadening

316

Lorentz model

75

440

441

813 single oscillator Lorentz, Hendrik Anton Lorenz, Ludvig

83

813

58 388

Lorenz-Mie scattering, see Mie scattering Lumen

18

217

Luminance

xviii

18–19

Luminous Efficacy

xviii

18

19

Luminous efficiency

xxi

18

19

Luminous emission

415

Luminous flame

758

759

Luminous flux

xix

19

Lux

18

M Magnesium oxide

bidirectional reflection function Magnetic field Magnetic permeability

Malkmus model

84

88

90

91

116

69

70

xviii

32

xxi

32

47

808

330

332

339

512–519

522

37

655 Manganese sulfide

55

Mark’s boundary condition

499

Marshak’s boundary condition

500 529

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms Mass fractal dimension

Links xvii

Mass fraction

xx

Maxwell’s equations

32

Maxwell, James Clerk

32

MCintegral

422

424

389

858

Mean beam length definition for an isothermal gas layer for optically thin media geometric spectrally averaged

xviii

628–633

639

628–630 629 630–631 630 631–633

Mean free path for a photon for absorption for collision for scattering Mean square deviation of the mean

2

281

281 2 281 251

Measurement absolute

427

emission

111–113

gas properties

369–374

multiple-scattering reflection

428 113–118

relative

427

scattering

427

semitransparent media transmission

447–450 369

Medium absorbing cold

50 287

conducting

73

cylindrical, see Cylindrical medium dielectric gray isotropic

45

55

295

299

32 This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

74

Index Terms

Links

Medium (Cont.) nonabsorbing

38

nongray

299

603–606

nonhomogeneous

334

345

nonmagnetic

36

37

nonparticipating

20

129

nonscattering opaque participating

57

286 61

optically thick

626–686

74

462 21

279

plane-parallel, see Plane-parallel medium scattering semitransparent

287 85

95

440–450

733–738

280

spherical, see Spherical medium transparent Melting

462 733–738

Mesh, numerical

752

Metaheuristics

785

Metallic foam

445

Methane

323

351

353

354

356

369

533

749

750

760

762

Microgravity

750

Mie scattering

26

794–796

389

467

14

69

110

117

197

platinum

449

450

spherical

449

450

equivalent-sphere

423

Mie scattering coefficient

391

Mie theory

762

Mie, Gustav

388

Milne-Eddington approximation Minimization Mirror

488–491 792

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

Mixture gas

334

673

gas-particulate

635

646

mmmie

393

396

859

mocacyl

860

488–491

495

450

Modified differential approximation Mole fraction

527–531 xx

Moment method Momentum, of photons Monochromatic radiation

299 17 6

Monochromator

108

370

MONT3D

254

860

Monte Carlo method

134

247–266

300

532

694–723

761

762 results for a V-groove

222

results for a gas slab

637

MSFSK method

679

Mueller matrix

42

Mullite

446

Multigrid algorithm

752

Multiphoton absorption

612

Multisphere Mie solution, generalized

414

Mushy zone

734

236

684

686

450

735

N Nanoscale radiation

803–815

Narrow band model

325–336

700

Narrow band model Monte Carlo

700

Narrow band parameter

327

Natural line broadening

316

NBKDIR database

341

670

671

678

684

861

nbkdistdb

339

858

nbkdistsg

339

858

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

Negative index materials

808

Nephelometer

427

NEQAIR database

323

Nernst glower

107

Net radiation method

166

Neural networks

796

Neutron transport theory

495

Newton’s direction

789

Newton’s method

789

Nickel

430

238–242

763

541

77

Nickel-oxide coating

103

NIST database

325

Nitric oxide

353

369

Nitrous oxide

351

369

Nonconductor, of electricity

58

Nonequilibrium radiation

306

321

Nonequlibrium radiation

285

Nongray medium

299

Nongray surface

230

Nonluminous flame

758

Number density

396

molecular

307

317

Numerical quadrature

179

181

Nusselt number

xix

745

xviii

780

781

784

790

793

Obstruction, visual

147

208

Off-specular peak

91

603–606

626–686

542

O Objective function

Opacity Project Opaque

325 4

Opaque medium

61

74

Opaque surface

5

11

61 This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

21–22

Index Terms Opening angle

Links 104

Optical constants

57–60

Optical coordinate

xxi

Optical density

48

Optical depth

455

Optical filter

107

285

Optical path length

xx

320

Optical thickness

xxi

455

for absorption

281

for extinction

281

for scattering

281

narrow band

328

of a spectral line

321

Optically thick approximation Optically thin approximation

482–486 299

480–481

Optics collection

450

geometric

26

53

389

402

thin film

95

53

Optimization

795

gradient-based

785

788–794

Oscillator double

84

harmonic

58

74

311

314

isolated

75

single

83

813

755

756

27

281

OTFA Out-scattering Overlap band

362

line

327

Overlap parameter

xxi

(for MSFSK)

681

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

636

310

Index Terms Overtone band

Links 311

Oxide film

93–94

Oxy-fuel

651

750

P P1-2D

859

P1-approximation

497

for box model

634

transient

620

with conduction

761

642–643

semigray

with collimated irradiation

502–509

616–618 730

P1/3-approximation

620

P1sor

505

859

P3-approximation

497

509

Packed bed

763

Palladium

77

Parallel computing

261

Parallel plates

173

176

179

200

204

220

811–812

815

xix

780

793

796

parlplates

146

159

parlplates.cpp

856

parlplates.m

856

Parameter vector

Particle ash

415

carbon

760

coal

415

large

402

model

707

soot

398

spherical

387

stochastic

706

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

781

Index Terms

Links

Particle beds

408

Particle distribution function

395

416

417

427

428

646

706

746

Particle size parameter, see Size parameter Particle suspension

Partition function

314

Path length density

320

geometric

xix

320

optical

xx

320

pressure

320

Pathlength method

713

PDF method

706

754

756

407

410

397

401

758 Peak backward-scattering

397 412

diffraction

397

forward-scattering

393

off-specular

91

specular

69

Pellet-reflection technique

420

421

Pencil of rays

15

16

Permeability, magnetic

xxi

32

47

808

complex

xx

33

43

electrical

xx

32

57

808

810

37

Permittivity

relative

57

perpplates

146

perpplates.cpp

856

perpplates.m

856

Phase angle

33

55

xx

40

of polarization

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

159

448

Index Terms Phase function

approximate

Links xxi

25

390

391

283

410–413

for a particle cloud

394

for absorbing particles

396

for dielectric particles

397

for diffraction

402

for diffusely reflecting spheres

407

for large spheres

405

for Rayleigh scattering

399

for Rayleigh-Gans scattering

401

for single sphere

394

for specularly reflecting spheres

404

Henyey-Greenstein

410

isotropic

411

linear-anisotropic

411

396

400

411

Phase velocity

34

36

401

Phenomenological coefficient

32

36

57

Phonon

58

807

809

Photoacoustic

426

Photolysis

612

Photometer, scattering

426

Photon

1–3

Photon detector

109

Photon energy

4

Photon gas

503

Photon momentum

17

Photon pressure

17

Photon-phonon interaction

83

Photovoltaics

759

planck

11

855

Planck function

xviii

15

Planck number

270

Planck oscillator

xxi

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

810

Index Terms Planck’s constant modified

Links xviii

3

309

Planck’s law

7

Planck, Max

7

Planck-mean absorption coefficient

367–369

9

306

635

705

709 for coal particles

416

for particles

395

for soot

424

modified

368

417

Planck-mean extinction coefficient for coal particles

416

for particles

395

for soot

424

Planck-mean temperature

417

666

Plane of equal amplitude

34

44

of equal phase

33

44

of incidence

45

91

33

38

Plane wave Plane-parallel medium approximate methods

480–494

at radiative equilibrium

461–467

discrete ordinates method

545–550

exact formulation

454–458

639–643

isothermal, nongray gas

633

648–649

isothermal, nonscattering

522

595

nonscattering

459–465

optically thick

482–486

optically thin

480–481

scattering

465–467

specified temperature field

460–461

Plasma

463–466

317

Plasma frequency

59

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

76

44

Index Terms Platinum

Links 77

79

80

230 PN -approximation Simplified

495–534 495

Point collocation method

475

Poiseuille flow

744

Polar angle

xxi

Polaritons

807

Polarization

522–526

12

15

37–42

612

circular

38

41

degree of

41

elliptical

38

linear

38

41

parallel

40

73

101

perpendicular

40

73

101

plane

38

state of

38

197

Polarization ellipse

xvii

xx

xx

40

Polarization phase angle Polarized light

448

Polarizer

427

Polarizing angle

48

Pollutants

750

Polystyrene

446

Porous film

99

Position vector

38

135

138

Potassium

77

Poynting vector

xix

36–38

390

805

810

811

Poynting, John Henry

36

Prevost’s law



Prandtl number

xix

741

Prefactor, fractal

422

424

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms Pressure

Links xix

correction chart

365

correction factor

365

effective

352

partial

317

partial, of absorbing gas

320

photon

17

radiation

xix

solar

18

297

xviii

252

706

754

756

758

181

542

320

Principle of invariance

475

Probability

756

Probability distribution Profilometer

364

17–18

Pressure path length

Probability density function

365

248 89

Property, radiative, see Radiative properties Pseudorandom number

250

Pulverized coal

706

Pyroelectric detector

109

Pyrometer

30

Q Quadrature, numerical

179

Quantum detector

109

Quantum mechanics

3

Quantum number rotational vibrational Quartz

xviii

309

xx

310

352

7

112

442

761 Quasi-Newton method

789–792

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

794

Index Terms

Links

R RADCAL database

333

860

Radiation background

450

external

216

from isolated lines

320

midinfrared

4

monochromatic

6

sky

216

transient

619–622

ultraviolet

4

76

94

3

4

18–19

Radiation energy density

xx

292

Radiation pressure

xix

17–18

176–178

215–216

98 visible

Radiation shield Radiation tunneling Radiation-turbulence interaction

805 753–759

Radiative combination

304

Radiative conductivity

483

Radiative equilibrium

298

571

474

508

between concentric cylinders between concentric spheres

297

469–471

in a gray medium

705

in a nongray gas slab

636

639–643

in a nongray medium

647

706

in a nonscattering slab

461–465

in a scattering slab

466–467

547–548

568–569

13

15–17

66

Radiative heat flux

292–293 divergence of

293–295

for a cylindrical medium

473

for a plane-parallel medium

458

for a spherical medium

469

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms Radiative heat transfer Radiative intensity, see Intensity Radiative properties definitions for surfaces

Links 1 13–15 3 62–72

directional

22

hemispherical

22

of coal particles

415–418

of fly ash

415–418

of gases

23–24

of materials

75–83

of nonconductors

83–89

of particles

25–26

387–439

103

of semitransparent media

24–25

of semitransparent sheets

95–101

of soot

303–386

2

of metals

of selective absorbers

28

440–450

418–425

of window glass

95

spectral

22

summary for surfaces

72

73

78–79

82–83

88–89

173

174

27

285

temperature dependence

600 total

22

Radiative resistance

xix 214

Radiative source

xix 455

for anisotropic scattering

457

for isotropic scattering

456

for linear-anisotropic scattering

457

modified

558

time-averaged

754

Radiative transfer equation Radiative transfer equation (RTE) boundary conditions

320

454

26–27

279–292

288

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

Radiative transfer equation (RTE) (Cont.) integral formulation solution methods Radiative transport theory Radiosity

295–297 299 27–28 xviii

165

200

168

178

247

249

202 artificial

587

for a semitransparent wall

217

spectral

166

volume zone

597

Radiosity equation

166 203

Raman effect

387

Random number

xix

Random number generators

250

Random number relation for absorption

697

700

for absorption and reflection

257

for direction of emission

256

696

for point of emission

254

696

for scattering

698

for wavelength of emission

256

inversion

262

696

700

Ray effect

560

570–572

Ray model

707

708

Ray tracing

95

99

200

26

393

398–401

423

611

259–261 Rayleigh scattering

Rayleigh, John William Strutt, Lord

6

Rayleigh-Debye Gans-scattering

423

Rayleigh-Gans scattering

393

Rayleigh-Jeans distribution

9

Reaction mechanism

749

Reactive flow

756 This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

401

Index Terms

Links

Reciprocity, see Law of reciprocity Reflectance

21

bidirectional components of

68–72

68 197

diffuse

69

of silicon carbide

88

spectral, directional of platinum

80

spectral, directional-hemispherical

69

spectral, hemispherical

71

spectral, hemispherical-directional

71

spectral, normal of magnesium oxide

89

of silicon

85

total, directional-hemispherical

72

total, hemispherical

72

total, hemispherical-directional

72

total, normal

78

Reflection

25

by a slab

53–57

by a thin film

53–56

42–57

387

from large spheres

402

gray, diffuse

129

irregular

222

specular

45

69

197

xix

47–49

51

448

808

Reflection coefficient

for a thin film

54

Reflection function bidirectional

198

spectral

68

total

71

Reflection measurement Reflection technique

113–118 447

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

235

238

Index Terms Reflectivity

Links xxi

see also Reflectance

47

coating

98

for polarized light

79

interface

48

49

52 of a dielectric thin film of a slab

55 450

of a thick slab

56

96

of a thin film

54

56

of aluminum

52

spectral, directional

79

spectral, directional, polarized of glass

87

spectral, normal

83

of aluminum

76

of an In2O3 film on glass

99

of copper

76

of magnesium oxide

84

of metals

75

of silicon carbide

84

of silver

76

85

78

Reflectometer heated cavity

115

integrating mirror

117

integrating sphere

116

Reflector diffuse

70

perfect

71

specular

69

73

25

387

Refraction in large spheres Refraction angle

402 45

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

51

Index Terms Refractive index

Links xix

3

35

290 for semitransparent materials

85

of air

3

of vacuum

3

varying

86

612

Refractive index function

xvii

419

Regularization

779

785–787

parameter

xxi

787

Relaxation parameter

274

792

Relaxation time

76

77

Resistance, radiative

xix

173

174

88

214 Resistivity, electrical

76

Resonance frequency

59

Reststrahlen band

84

86

441

442

446

447

760

761

Reticulated porous ceramics

RevMCcs

719

RevMCxx

860

Reynolds number

xix

741

Rigid rotator

309

311

Ripple

398

rnarray

860

Rosseland approximation

483

761–763

Rosseland-mean absorption coefficient

368

635

for coal particles

416

417

for particles

395

for soot

424

Rosseland-mean extinction coefficient

446

483

for coal particles

416

417

for particles

395

for soot

424

Rotation matrix

xx

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

513

450

314

634

Index Terms Rotational energy level

Links 304

308

xviii

309

309

311

314

root-mean-square

xxi

89

91

surface

69

89–93

533

668

757

761

815

Rotational quantum number Rotator, rigid Roughness

RPC, see Reticulated porous ceramics RTE (Radiative transfer equation), see Radiative transfer equation (RTE)

S Sandia Flame D

758 Sapphire

112

Saturable absorption

612

Scaled k-distribution global

661

narrow band

345

Scaling approximation

346

654

xx

345

660

26

299

Scaling function Scanning tunneling microscope Scattering

815 25

attenuation by augmentation by

281 282–283

by fibers

409

by nonspherical particles

422

dependent

388

elastic

387

false

560

independent

388

inelastic or Raman

387

isotropic

283

287

391

411

linear-anisotropic

299

411

multiple

427 This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

450

299

Index Terms

Links

Scattering (Cont.) Rayleigh

393

398–401

Rayleigh-Gans

393

401

single

427

Scattering albedo

xxi

26

611

27

285 Scattering angle

xxi

391

Scattering coefficient

xxi

25

for a particle cloud

394

395

Scattering cross-section

389

Scattering efficiency factor

389

for absorbing spheres

393

for long cylinders

409

for specularly reflecting spheres

404

281

Scattering measurement

427

Scattering peak

393

397

401

407

410

412

Scattering phase function, see Phase function Scattering photometer

426

Scattering regimes

389

Schrödinger’s wave equation

308

310

Schuster-Schwarzschild approximation

299

486–488

548

574 Search direction

788

792

794

Search step size

788

793

794

Selection rule

309

310

Selective surface

101

102

207

229 Self-broadening coefficient

xvii

Self-correlation Planck function

757

temperature

755

756

58

83

84

232

246

857

Semiconductor semigray

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms Semigray approximation

Links 230

233

242

246

857

733–738

634–637 semigrxch

232

Semitransparent Semitransparent liquid

4 442

Semitransparent medium

85

440–450

Semitransparent sheet

74

216–219

Semitransparent surface Semitransparent wall Semitransparent window Sensitivity matrix

Shading, partial Shadowing

216–219 202

290

216–219

761

xx

781

782

784

793

796

208 91

Shape factor, see View factor Sheet, semitransparent

131 74

216–219

176–178

215–216

Signal velocity

36

619

Silica

94

760

Silicon

85

112

795

83

88

89

94

760

813

76

77

Shield, radiation

absorption coefficient

442

phosphorus-doped

442

Silicon carbide

Silver Simplified PN -approximation Simulated annealing

522–526 795

Single scattering albedo, see Scattering albedo Singular value

783

Singular value decomposition

783

truncated

785

785

Six-flux method

299

488

575

Size parameter

xx

26

388

Sky radiation

216

Skylight

217 This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms

Links

Slab, see Plane-parallel medium reflection by

53–57

transmission through

53–57

Slab absorptivity

xvii

Slab band absorptance

648

Slab reflectivity

xix

96

96

spectral, normal for several glass panes

97

of soda-lime glass

97

Slab transmissivity

xx

96

spectral, normal for several glass panes

97

of soda-lime glass

97

Slag

450

SLW method

654

659

664

672

673

743

45

290

805

51

73

SN -approximation Snell’s law generalized

541–576

Soda-lime glass

97

Sodium

77

Solar absorptance

103

Solar cell

109

Solar collector

101

Solar concentration ratio

759

Solar constant

17

Solar energy

2

Solar furnace

762

Solar irradiation concentrated

6 759–763

Solar pressure

18

Solar reactor

760

Solar receiver

760

Solar sail

102

18

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611

759

Index Terms Solar temperature

Links 8

11

17

19 Solar transmittance

103

Solid amorphous

89

high-temperature

448

semitransparent

440

Solid angle

xxi

infinitesimal overhang

442

11–13

12 571

total

12

Solidification

733–738

Soot

398

415

652

685

aggregate

414

422

cylindrical

422

size distribution

424

Soot model

651

758

Soot properties Soot radiation

418–425

428

750

Source, radiative, see Radiative source Space radiator

268

SPECAIR database

323

Special surfaces

101–104

Species concentration

752

Spectral line

304

strength

308

315

7

279

604

640

426

428

313

Spectral models

325

Spectral range

231

Spectral variable

xx 309

Spectral window

24 650

Spectrometer

108 448 This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms Spectroscopic database

Links 322

Spectrum electromagnetic wave

3

of the sun

6

vibration-rotation band

312

Specular direction

69

Specular peak

69

Specular reflection

45

paths

198

peak

197

Specular reflector Specular view factor

4

236

289

69

197

69

73

xviii

131

198–202

20

230 Specularity index

421

Speed of light

xvii

3

3

35

integrating

426

430

large, diffusely reflecting

406

large, opaque

402

large, specularly reflecting

403

near-dielectric

401

in vacuum Sphere

762

Spheres, concentric at radiative equilibrium discrete ordinates method without participating medium

Spherical harmonics Spherical harmonics method Spherical medium discrete ordinates method

470 550–554 163

170

201

205

xx

491

300

495–534

176

496

467–471 550–554

isothermal

286

isothermal, nongray gas

633

Spherical top

308

Spline

475 This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

292

295

Index Terms

Links

SPRADIAN database

323

Stabilization

779

Stanton number

xix

273

Stark broadening

316

317

Stark effect

318

Stark number

xix

Statistical error

248

Statistical model

327

general

351

329–331

rough surface

92

Statistical sampling

247

Statistical uncertainty

780

Steepest descent

789

794

Stefan number

xix

737

Stefan, Josef

10

Stefan-Boltzmann constant

xxi

Step scheme

559

Step size

789

Stepwise-gray model Steradian

10

637–643 12

Stimulated emission

315

Stochastic particle

706

Stokes’ parameter

xviii

40–42

xix

xx

43

138

139

353

369

for polarization Stokes’ theorem Successive approximation, method of

179

Sulfur dioxide

352

Summation relation for exchange factors

253

for specular view factors

200

for view factors

134

203

Sun, see Solar Surface artificial black

130 5 This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

164

Index Terms

Links

Surface (Cont.) concave

134

convex

134

147

170

curved

199

213

221

cylindrical

187

160

165

directionally nonideal flat

234–242 134

fractal

91

gray

64

gray, diffuse

64 288

grooved

105

ideal

102

129

isotropic

63

nongray

230

nonideal

229–246

289

5

11

21–22

69

78

102

207

opaque

61 optically smooth

45 197

polished

78

rough

222

selective

101 229

semitransparent

216–219

solar collector

101

specularly reflecting

197

289

V-grooved

104

105

211

212

222

236

84

93

vector description

258

Surface coating

95

Surface damage

93–94

Surface integration

134

Surface layer

79

Surface modification

94

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Index Terms

Links

Surface normal

xix

12

135

138

Surface phonon polaritons

807

Surface plasmons

807

Surface polaritons

807

Surface preparation

76

Surface radiosity

165

Surface roughness

69

808

89–93

Surface waves

807

Suspension, particle

427

428

706

746

Symmetric top

308

Symmetry number

315

T T-matrix method, cluster

414

Tables: apparent emittance for cylindrical cavities associate Legendre polynomial half-moments p

180 m n,j

515

blackbody emissive powers

833

coefficients for full spectrum k-distributions

670

comparison of different Monte Carlo implementations

720

conversion factors

819

discrete ordinates (one-dimensional)

546

discrete ordinates (three-dimensional)

545

Drude parameters for metals

77

exponential integrals

854

mean beam lengths

632

narrow band correlations

331

optical properties of coal and ash

415

physical constants

818

radiative equilibrium between concentric cylinders

474

between concentric spheres

470

in a plane-parallel medium

463

131

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

671

646

Index Terms

Links

Tables: (Cont.) radiative heat flux from an isothermal cylinder

473

radiative properties of coal particles

417

spectral, normal emittances of metals

831

Stokes’ parameters for polarization

41

total emissivity correlation for CO2

366

total emissivity correlation for H2O

366

total, normal emittances

821

total, normal emittances of metals

830

total, normal solar absorptances

821

view factor catalogue

836

weighted-sum-of-gray-gases coefficients

652

wide band model correlation

351

wide band model parameters

353

TE wave

832

823

40

Temperature bulk

272

Planck-mean

666

solar

8

11

17

19 Temperature dependence of radiative properties Temperature discontinuity

82

88

462

Temperature measurement of gases

370

Thermal conductivity Thermal detector Thermal radiation

xviii

2

109 1

2

28 Thermal runaway

370

Thermopile

109

Thick film

56

Thin eddy approximation

755

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

756

4

Index Terms Thin film

Links 53

reflectivity transmission through

56 53–56

Thin film optics

53

Thin layer

93

Tikhonov regularization

447

780

785

792

794–796

Titanium dioxide

94

430

TM wave

40

Tomography

445

796

totabsor

366

858

totem

856

totemiss

366

367

787

633

858 Transient radiation

619–622

Transition bound electron

76

bound-bound

23

303

bound-free

23

304

electronic

86

442

forbidden

312

free-bound

23

free free

23

interband

58

vibrational

83

84

4

154

Translucent Transmission

304

42–57

through a slab

53–57

Transmission coefficient

xix

for a thin film

54

47

Transmission measurement

369

428

Transmission method

447

449

Transmissivity

xxi

see also Transmittance full spectrum

48 675

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

85

312

312

Index Terms

Links

Transmissivity (Cont.) narrow band of a dielectric thin film of a fictitious gas

336 55 348

of a gas layer

24

of a material layer

25

of a nonhomogeneous layer

334

of a slab

450

of a thick slab

56

of a thin film

54

of a thin gap

806

of an In2O3 film on glass

99

of multiple glass sheets

101

of window glass vacuum gap

96

96 807

Transmittance

21

solar

103

Transparent

345

22

4

Transparent medium

462

transPN

621

Transverse electric

40

Transverse magnetic

40

622

859

TRI, see Turbulence-radiation interaction Truncated singular value decomposition

780

Tunneling, of radiation

805

796

Turbulence interaction with radiation

753–759

Turbulence model

753

Turbulence moment

754

Turbulence-radiation interaction

754

753–759

Turbulent diffusivity

754

Two-flux approximation

299 760

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

488

574

Index Terms

Links

U Ultraviolet radiation

4

76

94

98

304

442

503

589

xix

11

36

260

456

surface normal

xix

12

258

surface tangent

258

104

105

211

212

222

236

Uncertainty, statistical

780

Unit sphere method

135

Unit tensor

140

Unit vector

135

for direction

Unity tensor

781

V V-groove

Vacuum

129

Van Royen, Willebrord van Snel

45

Variance

251

Variational calculus

179

Velocity

xx

780

297

mean

272

phase

34

36

signal

36

619

Vibration ellipse

38

Vibration, lattice

58

88

Vibration-rotation band

24

304

308

83

304

308

310

312

310

352

83

84

401

311 spectrum

312

Vibrational energy level

Vibrational quantum number Vibrational transition

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

Index Terms view

Links 134

146

159

836 View factor

xviii

by area integration

135

by contour integration

138

by crossed-strings method

147

by inside sphere method

151

by unit sphere method

153

catalogue

836–846

definition of

131

diffuse

131

evaluation methods

134

specular

129

xviii

131

198–202

230 View factor algebra

134

view.cpp

856

view.m

856

VIEW3D

860

viewfactors

134

143

159

836

4

18–19

856 Visible radiation

3 443

voigt

319

858

Voigt broadening

319

Voigt profile

319

Volume fraction

395

of particles

388

Wall, semitransparent

202

290

wangwbm

357

858

Water

443

399

W

Water droplets

410–412

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Index Terms Water vapor

Links 351

353–355

360

364

365

651

652

654

655

662

669

670

677 Planck-mean absorption coefficient

368

total emissivity

364

366

Wave homogeneous

34

inhomogeneous

34

plane

33

Wave equation, Schrödinger’s Wave interference

Wave vector for transmission

38

44

53

56

98

99

103

xx

33

308

50

Wavefront

44

Wavelength

xxi

crossover Cutoff

804

3

82 102

Wavenumber

xx

cumulative

665

reordered

658

3

33

wbmkvsg

361

858

wbmodels

357

858

wbmxxx

354

858

Weight factor (for WSGG)

650

652

654

Weight function

708

Weight function (for FSK)

659

661

664

666–668

681

603

649–654

Weighted sum of gray gases Weighting matrix

xx

Wide band model

326

exponential

350

for isothermal media

349–362

647–649

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

743

Index Terms

Links

Wien’s displacement law

8

814

Wien’s distribution

9

10

Wien’s law

9

Wien, Wilhelm

7

Wigner-D functions Window semitransparent spectral

xvii

513

95

449

216–219

761

604

640

Window glass absorption coefficient

442

WSGG, see Weighted sum of gray gases

Y YIX method

575

Z Zenith angle

6

Zinc oxide

762

Zinc selenide

371

Zirconia

760

Zirconium carbide Zonal method

89 300

585–609

This p a g e ha s b e e n re fo rm a tte d b y Kno ve l to p ro vid e e a sie r na vig a tio n.

307

650