Radiative Heat Transfer in Participating Media. With MATLAB Codes 9783030990442, 9783030990459, 9789385462061

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Rahul Yadav · C. Balaji · S. P. Venkateshan

Radiative Heat Transfer in Participating Media With MATLAB Codes

Radiative Heat Transfer in Participating Media

Rahul Yadav · C. Balaji · S. P. Venkateshan

Radiative Heat Transfer in Participating Media With MATLAB Codes

Rahul Yadav Application Engineer—Thermo Fluids Systems Altair Engineering Inc. Bengaluru, Karnataka, India

C. Balaji Department of Mechanical Engineering Indian Institute of Technology Madras Chennai, Tamil Nadu, India

S. P. Venkateshan Department of Mechanical Engineering Indian Institute of Information Technology, Design and Manufacturing Kancheepuram, Tamil Nadu, India

ISBN 978-3-030-99044-2 ISBN 978-3-030-99045-9 (eBook) https://doi.org/10.1007/978-3-030-99045-9 Jointly published with ANE Books Pvt. Ltd. In addition to this printed edition, there is a local printed edition of this work available via Ane Books in South Asia (India, Pakistan, Sri Lanka, Bangladesh, Nepal and Bhutan) and Africa (all countries in the African subcontinent). ISBN of the Co-Publisher’s edition: 978-9-3854-6206-1 © The Author(s) 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Education in its real sense is the pursuit of truth. It is an endless journey through knowledge and enlightenment. —A.P.J. Abdul Kalam

To IIT Madras and its vibes

Preface

This book is an outcome of continuous and ongoing research by the authors in the area of radiative heat transfer, and gas and particle radiation applied to industrial and space applications. The computations related to radiative heat transfer are very relevant in applications such as iron and steel manufacturing industries, rocket exhaust designing, fire resistance testing, and atmospheric and solar applications. There exist a plethora of literature and texts on heat transfer modeling in industrial engineering applications, where the treatise has been both experimental and numerical. This said, more accurate, versatile, and at the same time, computationally efficient numerical solutions to practical heat transfer problems are critical. This is especially true in radiative heat transfer, where the experimental approach is rather non-economic due to large cost of fabrication, installation and operation, and severe constraint due to high operating temperatures. This makes the use of numerical tools in radiative heat transfer analysis of profound significance. The contents in this text mainly outline the process of conducting numerical simulations in radiative heat transfer for various industrial applications and the use of artificial intelligence in developing and testing fast heat flux predictor tools. The numerical simulations require the development of computational tools (or codes) to solve the governing radiative heat transfer equation. The development, testing of codes, and results of numerical simulations have emerged as a part of doctoral degree research conducted at IIT Madras during 2013–2018. The primary motive is to help the engineers and researchers understand the governing physics of radiative heat transfer in participating media, model a radiative heat transfer system numerically, develop more physics-informed approximations, and take steps toward developing their own codes for radiative heat transfer analysis. Keeping in view the complexity of the subject, the discussion is very basic to help even a budding engineer sail firmly through the domain of radiative heat transfer. Starting from the concepts of thermal radiation and reviewing the important approaches to solve the problem, a mathematical framework for the solution of governing radiative heat transfer equations with a separate and detailed treatment of gas and particle radiation is presented. The spectral

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line-based weighted sum of gray gases method along with its coupling with particle radiation framework using the Mie scattering algorithm is an important inclusion. Development of a radiative heat transfer computation tool using the finite volume method applied to a cylindrical, conical, and rectangular computational domain with application to combustion cylinders, rocket exhaust plume, rocket nozzles, fire resistance testing, and radiant furnace is presented. The discussion on the use of artificial intelligence concepts in speeding up the radiative heat transfer calculations in traditional industrial setting is unique, novel, and inline with the generation’s need. Use of such predictor tools in applications for solving practical problems of process parameter optimization in industry is also described in detail. The important MATLAB codes are provided in the end to help engineers begin developing their own radiative heat transfer solvers, and use them in practical heat transfer analysis. The book, therefore, serves as a comprehensive package by taking the readers from the basics of radiative heat transfer in participating media to equip them with their own codes to solve practical problems of their interest. During the course of completion of these research works, we acknowledge the kind support received from the Indian Space Research Organization (ISRO), Trivandrum, for sharing their technical expertise on the subject. Affectionate thanks are also due for our colleagues in the Heat Transfer Lab, IIT Madras, for their continuous motivation. The constant encouragement we received from our families has been remarkable in this journey. We wholeheartedly believe that this text would impart comprehensive knowledge on radiative heat transfer phenomenon in participating media and its application, usage of artificial intelligence, along with training the readers on solver development for these applications. Bengaluru, India Chennai, India Kancheepuram, India

Rahul Yadav C. Balaji S. P. Venkateshan

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Importance of Radiative Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Radiative Heat Transfer in Participating Medium . . . . . . . . . . . . . . . . . . . . 1.4 Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Properties of a Participating Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 5 8 9 10

2 Important Literatures on Radiative Heat Transfer . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Studies on Development of RTE Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Studies on Development of Band Models . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Studies on Inclusion of Particle Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Application-Based Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Relevance, Scope, and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 11 13 16 18 20 21 22

3 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Solution Methods for RTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Traditional Discrete Ordinates Method . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Estimation of Gas Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Full Spectrum Band Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The SLW Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Functional form of the ALBDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Formulation for Non-isothermal, Non-homogeneous Media . . . . 3.3.5 SLW-Gray Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 27 28 33 36 38 38 40 42 43

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3.4

Estimation of Particle Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Scattering by a Single Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Scattering by a Group of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Treatment of the Phase Function and Anisotropic Scattering . . . . 3.4.4 Calculation of Particle Properties in Conjunction with Band Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Modeling of Radiative Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44 45 45 47

4 Radiative Heat Transfer in Cylindrical Geometries . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Development of the FVM-SLW Method for a Cylindrical Geometry . . . 4.3 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Validation of the FVM-SLW Method for the Cylindrical Geometry . . . . 4.4.1 Validation with Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Validation for a Non-Gray Gas-Particle Mixture . . . . . . . . . . . . . . 4.4.3 Validation for Anisotropic Scattering . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Decision on the Number of Gray Gases . . . . . . . . . . . . . . . . . . . . . 4.5 Application to an Industrial Scale Delft Furnace . . . . . . . . . . . . . . . . . . . . 4.5.1 Effect of Gas Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Effect of Particle Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Application to a Rocket Plume Base Heating Problem . . . . . . . . . . . . . . . 4.6.1 Effect of Gas Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Effect of Particle Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 53 57 59 60 61 62 63 64 65 66 68 69 70 71 73 73

5 Radiative Heat Transfer in Conical Geometries . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 FVM-SLW Formulations for Body-Fitted Conical Geometries . . . . . . . . 5.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Validation with Absorbing Emitting and Scattering Medium . . . . 5.4 Application to a Conical Diffuser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Decision on the Minimum Number of Gray Gases . . . . . . . . . . . . 5.4.2 Effect of Gas Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Effect of Particle Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Effect of Cone Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Effect of Anisotropic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.6 Effect of Wall Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 75 79 79 79 81 82 83 84 85 87

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5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88 89 89

6 Radiative Heat Transfer in Three-Dimensional Geometries . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Formulations for a Three-Dimensional Rectangular Geometry . . . . . . . . 6.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Validation for a Three-Dimensional Furnace with Measured Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Validation for a Mixture of Non-gray Gases . . . . . . . . . . . . . . . . . . 6.3.3 Validation with Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Application to a Section of a Reheating Furnace . . . . . . . . . . . . . . . . . . . . 6.4.1 Effect of Particles and Anisotropic Scattering . . . . . . . . . . . . . . . . 6.4.2 Effect of Gas Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Effect of Roof Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 91 94

7 AI-Based Solution to Practical Radiant Heating Problems . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Need for a Fast Forward Prediction Model . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Development of ANN for a Two-Dimensional Rectangular Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Comparison of ANN with RTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Application of the Network to the Inverse Problem . . . . . . . . . . . . . . . . . . 7.6.1 Optimum Configurations for the Design Case . . . . . . . . . . . . . . . . 7.6.2 Validation of Optima Using Forward Model Calculations . . . . . . 7.7 Development of ANN for a Three-Dimensional Rectangular Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Comparison of ANN Prediction with RTE Solution . . . . . . . . . . . . . . . . . . 7.9 Application of the Network to the Inverse Problem . . . . . . . . . . . . . . . . . . 7.9.1 Genetic Algorithm (GA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.2 Optimal Configuration for Design Case . . . . . . . . . . . . . . . . . . . . . . 7.9.3 Validation of Optima with Forward Model Calculations . . . . . . . . 7.10 Validation of Optima with Exhaustive Search . . . . . . . . . . . . . . . . . . . . . . . 7.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94 95 95 99 101 103 104 105 106 106 109 109 110 110 112 117 117 120 121 125 127 129 130 132 135 135 136 137 137

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8 Conclusions and Future Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Overview of the Framework of A Generic RHT Solver . . . . . . . . . . . . . . 8.3 A Grand Overview of the Present Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 141 142 142 144

Appendix A: Formulation of Mie Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . .

145

Appendix B: Radiative Properties of Soot Based on Temperature and Type of Fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix C: MATLAB Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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About the Authors

Dr. Rahul Yadav completed his Ph.D. in Mechanical Engineering from IIT Madras in the year 2018 and served as an Institute Post Doctoral Fellow at IIT Kanpur for 2.5 years. He has more than 7 years of experience as a researcher in radiative heat transfer in participating media. He has around 7 publications to his credit in reputed international journals in topics related to computational radiative heat transfer and neural networks. He is a reviewer for several international journals. He has been awarded a pre-doctoral fellowship from IIT Madras and is also a recipient of Institute Post-Doctoral Fellowship from IIT Kanpur. Dr. C. Balaji is currently T. T. Narendran Chair Professor in the Department of Mechanical Engineering at the Indian Institute of Technology (IIT) Madras, India. He graduated in Mechanical Engineering from Guindy Engineering College, Chennai, in 1990 and obtained his M.Tech. (1992) and Ph.D. (1995) from IIT Madras in the area of heat transfer. His areas of interest include heat transfer, computational radiation, optimization, inverse problems, satellite meteorology, and atmospheric sciences. He has more than 200 international journal publications to his credit and has guided 30 students so far. Prof. Balaji has several awards to his credit and notable among them include Young Faculty Recognition Award of IIT Madras (2007) for excellence in teaching and research, K. N. Seetharamu Award and Medal for excellence in Heat Transfer Research (2008), Swarnajayanthi Fellowship Award of the Government of India (2008–2013), Tamil Nadu Scientist Award (2010) of the Government of Tamil Nadu, Marti Gurunath Award for excellence in teaching (2013), and Mid-Career Research Award (2015) both awarded by IIT Madras. He is Humboldt Fellow and Elected Fellow of the Indian National Academy of Engineering. Prof. Balaji has authored 8 books. Prof. S. P. Venkateshan obtained his Ph.D. from the Indian Institute of Science (IISc), Bangalore, in 1977. After spending three years at Yale University, he joined Indian Institute of Technology (IIT) Madras in 1982. He has been teaching subjects related to thermal Engineering to both UG and PG students for the past 32 years. He has published extensively and has more than 100 publications to his credit. The areas of his interest include xv

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interaction of natural convection with radiation, numerical and experimental heat transfer, heat transfer in space applications, and radiation heat transfer in participating media and instrumentation. Prof. Venkateshan has been Consultant to ISRO, DRDO, and BHEL in India and NASA in the USA. He has three patents to his credit in the area of instrumentation. He has also guided about 30 scholars toward the Ph.D. and a similar number of scholars toward the M.S. (by research) degree at IIT Madras.

1

Introduction

1.1

Thermal Radiation

Radiative heat transfer is essentially one of the fundamental processes in nature. The transfer of thermal energy from one system to the other which are not in close proximity to each other can take place due to radiation. Be it the travel of the sun’s energy to the earth, be it the cooking of food in microwave oven, or be it the heating of room with a fireplace, radiation heat transfer is involved in numerous activities of mankind and nature. According to Prevost’s law, any material at a temperature greater than 0 K emits radiation. So, noticeable or unnoticeable, there is always a transfer of radiation taking place in our surroundings. At the quantum level, a radiation beam can be understood as a group of photons having the same energy and traveling in the same direction. The radiation beam is electromagnetic in nature and is characterized by its associated wavelength or frequency. The wavelength tells how far the radiation wave can travel in a unit time, while frequency tells how many cycles the wave can complete with respect to the origin in a unit time. The wavelength and frequency are helpful in knowing the energy content of the traveling photons. The energy is given by the famous Planck’s relation E = hν, where h = 6.626 × 10−34 m2 kg/s is the Planck’s constant. The above is also the amount of energy required to dislodge the photons from a surface or body at a wavelength λ. Depending upon the wavelength or frequency, the radiation can be characterized as gamma rays, X-rays, ultraviolet, visible, infrared radiation, and microwaves. The wavelengthbased classification of these rays is depicted in Fig. 1.1. Each of these has its own importance and application. For example, X-rays can pass through materials which have low density and hence can be used to produce photographic images of the fractured bones. Radio waves due to their large wavelength are most suitable for communication purposes. However, it is the region of the spectrum that contains infrared, visible, and some part of ultraviolet waves which is of interest to the heat transfer engineers. This roughly falls in the wavelength range

© The Author(s) 2023 R. Yadav et al., Radiative Heat Transfer in Participating Media, https://doi.org/10.1007/978-3-030-99045-9_1

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1 Introduction

Fig. 1.1 Classification of electromagnetic radiation into different categories

0.1–100 µm. Radiation emitted at these wavelengths can cause a change in temperature of the interacting bodies and hence is frequently referred to as thermal radiation.

1.2

Importance of Radiative Heat Transfer

Radiation propagation is accompanied by emission, absorption, and scattering by a medium, surface, or a body which results in the change in energy of the propagating beam. The fundamental nature of radiative heat transfer makes it inherent in any physical process. Though, in some low-temperature applications where other modes of heat transfer also exist, there is a natural tendency to neglect radiation effects. However, in many engineering applications that involve surfaces and media at high temperatures, radiation effects become substantially important. For example, in the rocket base heating problem (see Fig. 1.2), the base wall of the rocket gets heated up due to radiation from the high-temperature exhaust that is mainly from the unburnt particles. In the testing of building materials for their endurance against the real-time fires, the specimens are subjected to artificial fires in the laboratory furnaces, the major portion of heating happens due to radiation from a hot medium. In heat treatment furnaces where the hot combustion gases are used to heat the specimens (see

1.2

Importance of Radiative Heat Transfer

3

Fig. 1.2 Hot exhaust plume emanating from the rocket nozzle

Fig. 1.3), radiation is the dominant mode of heat transfer. The collection of solar radiation by the collectors and its conversion into electrical energy involves the modeling of thermal radiation. Exchange of radiation by the earth’s surface and the outer space and the internal absorption and scattering by the earth’s atmosphere affects the overall energy balance of the earth. Outside of these, in several applications where the combustion of fuel causes the release of radiatively participating gases like H2 O, C O2 , C O, and C H4 , radiation from the surfaces and from the gases becomes the dominant mode of heat transfer. Even in a multimode heat transfer problem at low temperatures (say less than 100 ◦ C), if natural convection is present in conjunction with radiation with or without conduction, several studies in the past have shown that radiation will be the dominant mode of heat transfer [1, 2]. Thus, to appropriately take into account radiation contribution in these situations, modeling of the radiation exchange between the surfaces concerned and the medium involved is required to be done from its first principles, as many experiments are very hard to realize and are very expensive and time-consuming. It puts a strong reason for a detailed modeling and estimation of radiation in these situations. In the ensuing sections, the approach to analyze the problem of radiation heat transfer through an absorbing, emitting, and scattering gas volume is presented by deriving a set of mathematical equations from the first principles which govern the physics of the process.

4

1 Introduction

Fig. 1.3 A reheating furnace employed in the iron and steel industry

1.3

Radiative Heat Transfer in Participating Medium

Consider a beam of radiation emitted from a surface that passes through a medium which has the tendency to interact with the incoming radiation. These interactions can either augment or attenuate the energy content of the radiation beam (see Fig. 1.4). These interactions can be classified as follows: Absorption: The radiation beam loses energy due to absorption of the photons by the medium and the tendency of the medium to absorb depends on the thermodynamic state of the system such as pressure, temperature, and concentration of the absorbing medium as well as the energy content of incoming radiation. The extent of absorption by the medium is defined by a property called absorption coefficient, which provides the amount of radiation energy lost per unit length at a particular wavelength and direction due to absorption of the photons. Outgoing radiation

Scattering in other directions (out-scattering)

Incoming radiation

Emitted radiation

Absorbed radiation

Scattered radiation from other directions incoming to original direction (in-scattering)

Fig. 1.4 A depiction of the interaction of the participating medium with incoming radiation

1.4

Radiative Transfer Equation

5

Emission: The augmentation in the energy of the radiation beam may happen due to emission from the medium. Emission signifies that the bundle of photons emitted by the medium at a certain temperature that falls in the wavelength range of the incoming radiation will add up to the energy content of the beam. At the molecular level, the emission can happen in two ways. First, the atoms or molecules may lower their energy level and emit the photons spontaneously. This spontaneous emission is, in general, isotropic in nature. Second, the incoming radiation may trigger the atoms or molecules sitting at a lower energy level to emit the photons in its direction of travel, this phenomenon is called induced emission. However, at the same time, some photons of the radiation beam will get absorbed by these atoms. In actual sense, it is difficult to treat induced emission and absorption separately, therefore, the absorption coefficient what we define as an effective absorption coefficient takes into account both, the induced emission and absorption. The spontaneous emission at any wavelength can then be quantified by integrating the energy content of the photons emitted in all directions. Scattering: A single molecule or group of molecules, a single particle or group of particles, tends to alter the direction of travel of the incoming radiation beam. There is no net loss or gain of energy due to scattering. However, the energy gets redirected in different directions which makes it diminished in the direction of interest, this phenomenon is termed outscattering. At the same time, the energy content in the direction of travel may get augmented due to out-scattering of rays from other particles into the direction of interest, and this phenomenon is termed in-scattering. The extent of scattering is defined by the scattering coefficient, which, in a similar way to absorption coefficient signifies the loss of energy per unit length in the direction of travel due to scattering. When the scattering from one particle does not affect the scattering taking place from another particle, the phenomenon is called single scattering. Otherwise, the phenomenon is known as multiple scattering. As heat transfer problems of practical interests involve geometries that have a scale much larger than the length and distance scale of the particles and their group, multiple scattering effects die out. Generally, the particles are assumed to be spherical, since the change in the value of scattering coefficient due to irregularity in the topology of the particles when integrated over all particles is usually minimal.

1.4

Radiative Transfer Equation

The phenomena discussed in the above section occur simultaneously in the participating medium. A quantity called intensity is defined, to assess the net change in the energy content of the radiation beam. Intensity is specifically introduced in radiative heat transfer to handle spectral and directional effects. Since radiation energy varies with wavelength as well as with direction, the introduction of intensity helps one to take into account these effects. Intensity is defined as the radiant energy carried by a ray per unit area of the emitting surface normal to the direction of the travel per unit solid angle per unit wavelength λ over a small interval

6

1 Introduction

Out-scattered, σλ ′ ( )

Incoming,

′

( )

Outgoing,

′

( +

)

Absorbed, κλ ′ ( ) ds Fig. 1.5 A depiction of attenuation of radiation intensity due to a participating medium

dλ. Consider a beam of radiation entering into a participating medium with an intensity Iλ (s), and leaving with an intensity Iλ (s + ds), the change in intensity due to presence of participating medium is given by Beer Lambert’s law : Iλ (s + ds) = Iλ (s) − βλ Iλ (s)ds

(1.1)

where β is the total extinction coefficient, which is the sum of absorption coefficient and scattering coefficient and ds is the length of the gas column. The total loss and gain of the intensity as radiation travels through a small gas column of length ds and can be broken down as follows: (i) Attenuation: Attenuation in the radiation intensity may occur due to absorption and out scattering (see Fig. 1.5). Due to absorption: d Iλ (1.2) = −κλ Iλ (s) ds Due to out scattering: d Iλ (1.3) = −σλ Iλ (s) ds (ii) Augmentation: Augmentation in the radiation intensity can occur due to emission by the medium and by in-scattering (see Fig. 1.6). While emission is intrinsic to the medium, in-scattering depends upon other particles in the medium. Both of them increase the radiation intensity in the direction of calculation. Due to emission: d Iλ (1.4) = Jλ (s) ds

1.4

Radiative Transfer Equation

7

Fig. 1.6 A depiction of augmentation of radiation intensity due to a participating medium

Due to in-scattering: d Iλ σλ = ds 4π

 =4π

Iλ λ ( →  )d

(1.5)

where λ is the scattering phase function which accounts for the probability of a ray traveling in direction  to get scattered in the direction of calculation  . In Eq. 1.4, Jλ (s) is the net spectral emission from the medium at the wavelength λ in the direction of interest. Combining all the effects together for the net change in radiation intensity of the beam due to passage through a medium of length ds can be written as d Iλ σλ = −κλ Iλ (s) − σλ Iλ (s) + Jλ (s) + ds 4π

 =4π

Iλ λ ( →  )d

(1.6)

Consider a case where the medium is in local thermal equilibrium (LTE) with a hypothetical cavity surrounding it, then as per Kirchoff’s law, the spectral absorptivity of the medium is equal to its spectral emissivity. Therefore, it can be shown that net spectral emission (Jλ ) varies as a function of blackbody emission, i.e. (κλ Ib,λ ) [3]. Though derived for a special case, the Kirchoff law has broad applicability to many practical problems of interest. The Eq. 1.6 can now be written as, d Iλ σλ = −(κλ + σλ )Iλ (s) + κλ Ib,λ + ds 4π

 =4π

Iλ () λ ( →  )d

(1.7)

Equation 1.7 is one form of the radiative transfer equation or RTE. In a coordinate-free notation and considering that the medium may consist of particles as well, the compact form of the above RTE can be expressed as s · ∇ Iλ (r, sˆ ) = −(κg,λ + κ p,λ )Iλ (r, sˆ ) + Sr ,λ (r, sˆ )

(1.8)

8

1 Introduction

where Sr ,λ is the source term which accounts for total augmentation in the radiation intensity and is expressed as Sr ,λ (r, sˆ ) = κg,λ Ib,g,λ + κ p,λ Ib, p,λ +

σ p,λ 4π

 =4π

Iλ () λ ( →  )d

(1.9)

In Eq. 1.9, the subscript g denotes the gaseous properties and subscript p denotes the particle properties. Ib,g,λ and Ib, p,λ are the local blackbody intensity of the gases and the particles, respectively, at the location r.  and  signify the direction centered at the discrete solid angle subtended by the rays sˆ and sˆ , respectively. Equations 1.8 and 1.9 demonstrate that, at every location, the spectral radiative intensity is a function of the spectral intensity of all other directions. This demands the solution of the equation at all possible locations and directions. The equation has a derivative term with respect to location on the left-hand side and an integral term with respect to the directions on the right-hand side making the RTE an integro-differential equation, which is formidable to solve and requires special numerical methods for their solution.

1.5

Properties of a Participating Medium

The variables in the above Eqs. (1.8 and 1.9) which govern the attenuation or augmentation of radiation energy are the properties of the participating medium under consideration. Out of these, the absorption coefficient depends on the local thermodynamic state, while the scattering coefficient and phase function depend on the particle size and their distribution in the medium. Apart from this, they both vary with respect to the wavelength of the incoming radiation. It is this wavelength dependency which makes the calculation of the radiative properties very challenging. For example, Fig. 1.7 shows the variation of the absorption cross section with respect to the wavenumber for pure H2 O gas at 1000 K and 1 atm pressure calculated from the High-Temperature Molecular Spectroscopic Database (HITEMP) 2010 [4]. The absorption coefficient of the gas is directly proportional to the absorption cross section of its molecules. Abrupt changes in the values of the absorption coefficient can occur even if a small change in the wavelength (or wavenumber) is made, as seen in Fig. 1.7. This kind of behavior is very difficult to incorporate into mathematical calculations. The most sophisticated way to consider this would be to solve the RTE for each wavelength so-called Line by Line (LBL) calculations . Given that the number of contributing lines or wavenumbers may reach the order of several millions corresponding to the temperatures encountered in common combustion applications, this method is quite formidable even with high computational resources available. The other extreme and simplifying assumption would be to assign an averaged absorption coefficient for the complete spectrum, thereby removing its dependence on wavelength. This is popularly known as the gray assumption and despite it being crude and quite removed from reality, it is still in use in many applications

Organization of the Book

9

Absorption cross section, cm2/mol (log scale)

1.6

Wavenumber, ν in cm-1

Fig. 1.7 Variation of absorption cross section with wavenumber for pure H2 O at 1000 K and 1 atm, calculated from HITEMP2010 database

in view of its simplicity. Between these two extremes, for accurate and efficient modeling of the spectral nature of gases, several approximations in the form of band models are available. As far as particles are concerned, the variation of the spectral properties is not as pronounced as the absorption coefficient of the gases but still demands careful attention. The tendency of the particle to scatter in a certain direction depends on the size parameter of the particle given by 2πr /λ, where r is the radius of the particle. When this size parameter is of the order of unity, the corresponding scattering phenomenon is called Mie Scattering. The other two cases which are demarcated based on the size parameter are 2πr /λ  1 and 2πr /λ  1, which correspond to theories of Rayleigh Scattering and Geometric Optics, respectively. The detailed illustration of these theories of scattering can be found in Bohren and Huffman [5]. Majority of the heat transfer applications involve particles of the sizes such that their calculations can be considered under the Mie scattering regime [3]. For an accurate estimation of the particle spectral properties, Maxwell’s electromagnetic equation should be solved for each category of particles. A detailed discussion on the estimation of gas and particle radiative properties using the aforementioned models is presented in the subsequent chapters with an emphasis on inherent challenges and the associated approximations.

1.6

Organization of the Book

A quick background of thermal radiation in participating media has been presented in the first chapter, along with the basic formulation of the radiative transfer equation that arises in an

10

1 Introduction

emitting, absorbing, and scattering medium. In the second chapter, a review of the literature pertinent to the radiative heat transfer, gas, and particle radiation research and development is presented. A discussion on the application of these concepts to various industrial problems of interest is also provided. Chapter 3 shows the basic formulation of the finite volume method for solving the RTE. Additionally, gas radiation calculations are discussed elaborately in the context of the band models where the emphasis is given to the full spectrum band models. This is followed by a discussion on the modeling techniques for particle radiation and the Mie scattering formulation. Chapter 4 presents the general formulation for a cylindrical geometry. A detailed discussion on the strategy of code development and its validation with available experimental and numerical benchmarks is presented. The application of this formulation and code to a problem of open-end cylindrical furnace and rocket exhaust base heating is illustrated. Chapter 5 shows the extension of the code to conical geometries and corresponding validations. For showing the applicability of the code to practical problems, a case of rocket nozzle has been taken and analyzed for governing parameters. Chapter 6 provides the general formulation for three-dimensional rectangular geometries and presents the extension of the code and associated validations. This is followed by the application of the code to an industrial scale reheating furnace. The effect of medium and wall properties on the radiative heating of the specimen is then presented. Chapter 7 discusses the limitations of the present forward model in several industrial scale inverse problems where a need for a fast forward model can arise. This forms the basis for the idea of developing a surrogate model using the concepts of artificial intelligence (AI). The multi-layer perceptron-based approach to develop such a model is then presented in detail along with the required validations. The neural network model thus developed is then applied to the inverse problem of estimation of the best configuration of heaters in a radiant furnace, which is solved by treating it as an optimization problem. Optimum configurations of the heaters have been obtained using a genetic algorithm and a post-optimal sensitivity analysis has been performed. The representative codes to develop the radiation heat transfer solvers for the participating media are presented in the Appendix.

References 1. Balaji, C., & Venkateshan, S. P. (1993). Interaction of surface radiation with free convection in a square cavity. International Journal of Heat and Fluid Flow, 14(3), 260–267. 2. Balaji, C., & Venkateshan, S. P. (1996). Natural convection in L corners with surface radiation and conduction. Journal of Heat Transfer, 118(1), 222–225. 3. Modest, M. F. (2013). Radiative heat transfer. New York: Academic. 4. Rothman, L. S., Gordon, I. E., Barber, R. J., Dothe, H., Gamache, R. R., Goldman, A., Perevalov, V. I., Tashkun, S. A., & Tennyson, J. (2010). HITEMP, the high-temperature molecular spectroscopic database. Journal of Quantitative Spectroscopy and Radiative Transfer, 111(15), 2139–2150. 5. Bohren, C. F., & Huffman, D. R. (2008). Absorption and scattering of light by small particles. New York: Wiley.

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Important Literatures on Radiative Heat Transfer

2.1

Introduction

This chapter gives an overview of the research studies in the past that have made a significant contribution to the state of the art in radiative heat transfer. As already mentioned, due to the inherent complexities of the problem, landmark improvements have often taken place simultaneously. The first section presents the contributions to the development of an accurate and efficient solver for the radiative transfer equation. The second and third sections discuss the key approximations made and the corresponding algorithms developed to appropriately treat the spectral nature of the gas and particle properties. The last section describes the efforts made to solve problems of engineering and scientific interest using these algorithms and solvers.

2.2

Background

In Chap. 1, we have seen several practical examples in which radiative heating through a hot participating medium is important. The participating medium, in general, contains a mixture of gases such as H2 O, C O2 , and C O, and several types of particles such as alumina particles, unburnt coal, ash, and soot. These gases and particles exist in varied proportions and distributions within the medium. Additionally, these systems are also governed by hightemperature gradients in the medium. Figure 2.1 shows the typical effects to be considered in dealing with practical radiative heat transfer problems. To accurately estimate the radiative heating in these applications, the solution of a complex integro-differential Radiative Transfer Equation (RTE) is required. In view of the complexity associated with the RTE, the exact solution is only available for the simplest of the cases. In view of this, the popularity of numerical solutions for radiative heat transfer problems has increased in the past few decades. The properties of the medium such as absorption coefficient and scattering coeffi© The Author(s) 2023 R. Yadav et al., Radiative Heat Transfer in Participating Media, https://doi.org/10.1007/978-3-030-99045-9_2

11

12

2 Important Literatures on Radiative Heat Transfer

Radiation heat transfer in combustion applications

High temperature particulate medium

Particles

Gases High temperature gradients H2O, CO2,CO, HCl etc. With varied concentrations

Unburnt particles, soot With varied concentrations and distributions

Fig. 2.1 Typical effects to be considered in radiation problems of practical interest

cient serve as key inputs to these solution schemes. In general, the estimation of properties of gases and particles is non-correlated and hence they act as two independent modules attached to the same solver. This breaks down the problem into three separate areas of concern, viz., RTE solver, gas radiation, and particle radiation. Figure 2.2 shows a typical overview of the various facets of a radiation problem. Apart from the difficulty in solving the RTE, the key challenges, as seen in the figure are gas radiation and particle radiation. In what follows, the literature review pertaining to each of these facets of a radiation heat transfer problem is discussed. A very focused and to-the-point survey is included, in consonance with the goals of this book, which is to form a computational basis for the development of a standalone radiative heat transfer solver for the industrial scale problems. The most widely used solution methods for the numerical solution of RTE, i.e. discrete ordinates and finite volume methods are reviewed, along with their advantages, shortcomings, and applications. Similarly, the full spectrum-based treatise of the spectral behavior of participating specie is discussed. Even so, a discussion on narrowband and wideband models is beyond the scope of the book. Majority of the industrial scale heat transfer involving participating particles can be modeled using the Mie scattering algorithm, and therefore the literature pertaining to Rayleigh scattering and geometric optics algorithm is not included. Finally, studies focused on the assimilation of all the generalities of a radiatively participating medium in a single platform and its potential application in typical industrial problems are discussed.

2.3

Studies on Development of RTE Solver

13

Radiation Heat Transfer Calculation in participating medium Gas properties estimation (κλ ,ελ ) Band Models

Radiative Transfer Equation (RTE) Solver

Discrete Finite Monte Ordinates Volume Carlo Method Method Method

Narrow Wide Full band band spectrum

Particle properties estimation (σλ ) Scattering algorithms

Mie Geometric Rayleigh scattering scattering optics

Fig. 2.2 Overview of the various facets of the problem of radiative heating through hot participating medium. Three major sections are usually involved, viz., RTE solver, gas radiation, and particle radiation

2.3

Studies on Development of RTE Solver

In, the RTE , the appearance of an integral, and the differential term in the same equation makes its analytical treatment very formidable. Such equations are generally classified as integro-differential equations and the solutions are only obtainable for a few simple cases. For instance, the exact solutions to RTE corresponding to Cartesian geometry were first developed by Lockwood and Shah [1] for a rectangular enclosure with cold and black walls. The assumption of cold and black walls simplifies the original RTE to a great extent, hence, an exact solution for the same is achievable. However, a large class of problems in actual industrial scenarios seldom involve cold and black walls, rather the enclosure walls are nonisothermal and the emissivity is spectral dependent. Furthermore, the participating media is invariably always a non-homogeneous mixture of gases and particles, with highly non-linear spectral dependence. Therefore, the exact solutions to RTE may have little applicability in actual radiative heat transfer problems. Since the second half of the twentieth century witnessed a dramatic increase in the use of computers to solve a variety of engineering problems, attention started shifting towards developing improved numerical algorithms for solving more complex forms of RTE.

14

2 Important Literatures on Radiative Heat Transfer

In the context of numerical treatment of radiative heat transfer problems, the Discrete Ordinates Method (DOM) developed by Chandrasekhar [2] for stellar applications was first used by Carlson and Lathrop [3] in neutron transport applications. The original formulation pertained to the axisymmetric cylindrical geometry, where, the change in path of the traveling neutron due to curvature of the wall was treated by defining a finite difference approximation to the angular redistribution term. The proposed method made the numerical treatment of angular derivative term of the intensity quite easier and provided a base for studying the numerical solutions to radiative heat transfer problems. Following this, the DOM was successfully applied by Fiveland [4] for developing the first benchmark results of radiative heat fluxes at the wall due to a participating medium in a cylindrical geometry. The participating medium was considered to be homogeneously absorbing and emitting, surrounded by cold and black walls. These formulations under similar conditions but for Cartesian geometries were also extended by Fiveland [5]. The principle of DOM lies in performing a control volume-based discretization of the computational domain and solving the RTE at each node of the control volume and in a discrete set of predefined directions characterized by their direction cosines and weights, to obtain a spatial and directional radiation intensity field within the domain. Truelove [6] focused on the development of more accurate weights for the discrete ordinates by matching the first and the second range moments. These updated weights for the Sn quadrature were found to provide considerable improvement in the results when compared to analytical results. With the availability of standard directions and corresponding weights for DOM, its credibility in the problems of practical interest was highlighted by Jamaluddin and Smith [7], where a systematic comparison of the results obtained by DOM with those of experimental results of the measurement of wall heat flux due to radiation in an industrial scale Delft furnace was performed. This made a significant contribution to the state of the art as it cemented the DOM as a credible alternative to solve radiation problems in engineering applications. Among the other important numerical methods developed to solve RTE was the Finite Volume Method (FVM) formulated by Chai et al. [8] for a simple case of gray participating medium with black boundaries. This formulation was made more generalized for threedimensional geometries by Raithby and Chui [9]. The advantage it holds over the traditional DOM is that the elimination of the need of satisfying the half-range and full-range moments by solving the equation in discrete control angles. The directions are then identified with directional weights, which is a product of unit normal at the control volume face, directions cosines, and the discrete solid angle. This is a powerful intervention since the user has to no longer depend on the available discrete sets of directions and weights, but can choose his own set of directions depending upon the requirement of the problem. Though this hindrance of the traditional discrete ordinates was realized at an early stage, its inherent simplicity continued to attract researchers. Baek and Kim [10] modified the DOM in axisymmetric geometries to make it more flexible like FVM, where, instead of using a predefined set of directions, the directions were chosen based on the control angle discretization. However, the angular redistribution term

2.3

Studies on Development of RTE Solver

15

was still treated by a simple finite difference technique. This modified DOM or MDOM would break down to FVM when considered for Cartesian geometries, and hence both of them should be considered to be the same solvers, so long as cylindrical geometries are not under consideration. When the RTE is written in the axisymmetric geometries using cylindrical base vectors, complexities arise due to the appearance of the angular redistribution term [3]. Carlson and Lathrop used a recursive relation for the evaluation of angular redistribution coefficients following the divergenceless flow condition. In contrast to this, Chui et al. [11] introduced the finite volume method where the RTE was integrated both in the control volume and the control angle, and a novel mapping technique was developed to handle the intensities in angular directions. Baek and Kim [10] utilized the idea of control volume and control angle discretization similar to FVM but kept the treatment of angular redistribution same as that of Carlson and Lathrop. The DOM solved with this procedure provided better numerical treatment, ease of formulations, and solutions with higher accuracy. The agreement of the numerical solutions obtained using this method with the measured heat flux values for a Delft furnace [12] was found to be closer than the traditional DOM. The introduction of this modification made the traditional DOM more versatile, for example, Kim and Baek [13] extended it to conical geometries with participating media. In their work, the effect of participating medium in the conical enclosure was studied for estimating the heating of the curved wall. For the treatment of body-fitted geometries, the only requirement is that an accurate estimation of the directional weights should be performed, which depends on the unit normal to the control volume faces. The primary advantage of using numerical techniques with spatial control volume treatment is their easy coupling with other flow simulation codes. DOM and FVM both solve the RTE in spatial control volumes and differ only slightly in the treatment of angular terms. Radiative heat transfer phenomenon in several applications almost always occurs with fluid flow and serves as an additional module for the heat transfer calculations, where the divergence of radiation heat flux appears as the source term in the overall energy balance equation. In view of this, the radiation solver must have the capability to interact with Computational Fluid Dynamics (CFD) code back and forth. Thus, methods like DOM and FVM always enjoy an advantage of easy coupling with CFD codes over their other counterparts such as spherical harmonics and the Monte Carlo method. Although sufficiently accurate to the scale of engineering applications, DOM and FVM suffer from some inherent numerical issues which propagate to produce unphysical false scattering and ray effects in the solution. Unfortunately, such errors are correlated in nature, and trying to reduce one increases the other and vice versa. The phenomenon of false scattering and ray effects was studied by Raithby [14] and was further highlighted by Coelho [15] within the framework of DOM. These studies also concluded that these errors are inherent as they depend on the spatial and angular discretization and it is not possible to completely get rid of them. However, several significant modifications to the DOM and FVM to mitigate the above issues have been introduced by researchers [16]. Hunter and Guo [17], provided the relationship between these two errors and quantified their dependence

16

2 Important Literatures on Radiative Heat Transfer

on spatial and angular discretization. Several recent modifications to DOM and FVM are also aimed at improving the radiative heat transfer solutions by reducing one of these errors [18, 19]. Even so, the purpose of these is to progress towards more accuracy with better and generalized numerical handling, their performance in problems of engineering scale needs to be investigated. Among the various spatial marching schemes available, the importance of choosing the appropriate one in employing DOM and FVM was highlighted by Coelho [20]. Here, it was emphasized that the step scheme provided better stability with less computation time when compared to other marching schemes in the literature. The key developments in the state of the art in the quest to obtain an efficient solver for the RTE are elaborately reviewed by Coelho [21].

2.4

Studies on Development of Band Models

One of the most important properties of the participating gases required as an input to the RTE solver is the absorption coefficient. Some of the ray tracing methods may require net transmissivity in the direction of travel but the discussion on this is beyond the current scope. As discussed in Chap. 1, for the absorption coefficient, apart from its dependence on local thermodynamic state, its variation with respect to wavelength produces additional challenges in its numerical modeling. Thus, to accurately model the participating gas, the knowledge of gas radiative properties (mainly absorption coefficient for DOM and FVM based solution technique) at various wavelengths and their corresponding change with respect to change in local thermodynamic state should be available. In the late twentieth century, several researchers have focused their attention on developing a broad database which includes the gas radiative properties and other associated parameters for major participating gases by performing the spectroscopic calculations. One of the earliest databases made available was the one by McClatchey [22] for atmospheric radiative transfer applications. This was further extended to include more number of radiatively active gases and the updated version was released as High-Resolution Transmission Database (HITRAN) in 1992 [23]. Followed by this, HITRAN has also witnessed major updates in 1996 [24] and in 2008 [25]. The temperature range covered by these databases was within 300–1000 K. Since the participating gases at high temperature exhibit more strong and dense absorption characteristics, a need to extend this database to higher temperature ranges was realized and the corresponding updated version of HITRAN was released with the name HITEMP in 2010 [26]. The ideal solution to any radiative heat transfer problem needs the evaluation of radiation intensity and the wall fluxes at each wavelength of the spectrum for each of the participating specie. In this, the radiative properties of the medium at the respective wavelength are obtained from the aforementioned databases. Such a solution technique is called Line by Line or LBL approach. However, in most cases, it is physically unrealistic to perform the

2.4

Studies on Development of Band Models

17

RTE calculation at every wavelength of the spectrum, due to the presence of a significantly large number of absorption lines, and non-linear variation. Furthermore, the absorption line characteristics of the gas differ from each other and they are a strong function of gas temperature, pressure, and concentration. Therefore, the LBL is not suited for most engineering applications, as the computational time involved in solving even the simplest of the problems would be very large. This has led to the development of certain approximate models to take into account the spectral behavior of gases with relatively lesser computation time and acceptable accuracy. Such models are referred to as band models in literature and there has been a considerable improvement in their formulation and application ever since their conception. Band models tend to produce the effective absorption coefficient representative of a band or a small spectral interval. Depending upon the width of the band, these models are classified as narrow-, wide-, or full spectrum band models. Some of the widely used narrowband models include Statistical Narrowband (SNB), Correlated-k (CK) method, and combined SNB-CK method. Though their comparison with the LBL solution for benchmark problems shows a close agreement, their overall implementation in RTE solvers is not so straightforward. Additionally, the treatment of a non-homogeneous media is often complex. The narrowband models cover the spectral intervals in the segments of 25–400 cm−1 , and the integration points within the band increase substantially when more than one gas species is involved. In view of this, while narrowband models are a good choice for developing the benchmark results, their use in practical problems of interest is invariably not justified, unless a very detailed information on spectral variation of fluxes or intensities is required. The full spectrum band models, on the other hand, give the facility to solve the RTE at a fewer discrete points after a specific reordering of the spectrum and produce solutions with acceptable accuracy. Furthermore, in many applications of practical interest, it is the total radiative heat flux which is the desired quantity rather than the spectral flux. This makes the full spectrum band models an obvious choice to be applied in engineering heat transfer problems. The simplest assumption for a full spectrum model would be to take a single absorption coefficient for the whole spectrum. This is called the gray assumption and is a very crude approximation, which leads to inaccurate results with large deviations in many cases. One of the earliest full spectrum models developed was the Weighted Sum of Gray Gases method (WSGG) by Hottel [27]. In this method, the total emissivity of the gas column is represented in terms of individual gray gas emissivities. These emissivities are calculated based on the product of partial pressure path length and absorption coefficient. The term gray gas represents the zones in the spectrum with similar values of absorption coefficient. The weighting factor was also introduced which depends non-linearly on the medium temperature. This information about the total emissivity and total absorptivity of a gas column with major combustion gases such as H2 O and C O2 was made available in the form of charts. Smith et al. [28] developed a mathematical model where the weighting factors were evaluated from a non-linear regression fit to the emissivity and absorptivity data calculated

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2 Important Literatures on Radiative Heat Transfer

using the wideband models. This model and the associated coefficients gave very close results to the benchmark problems and are frequently used in many combustions and radiative heat transfer codes. Modest [29] showed that the traditional formulation of WSGG can be used with any arbitrary RTE solver. After the introduction of the HITRAN database in 1992, Dennison and Webb [30] extended the WSGG formulation, where the weighting coefficients were evaluated from LBL spectral information of the gases. This distribution function which was a hyperbolic tangent fit to the reordered absorption cross section derived from detailed absorption line data was called Absorption Line Blackbody Distribution Function (ALBDF) and the method was termed as spectral line-based weighted sum of gray gases method or SLW method in short. This method proved to be very convincing as it could handle both nonisothermal and non-homogeneous media by evaluating the ALBDF at local thermodynamic conditions. Solovjov [31] made the SLW model more generalized in terms of handling gas mixtures. Among the various strategies suggested, the multiplication approach, the convolution, and the hybrid approach were considered to be more accurate. In consonance with this, Modest [32] developed a full spectrum correlated-k (FSCK) distribution by reordering the absorption coefficient from the database. Efforts were also made to extend the model non-isothermal and non-homogeneous media. Among all the full spectrum models developed, Demarco et al. [33] made a fair comparison of them and suggested that the SLW model and FSCK model are the best choices among the others for accuracy and computation time. The versatility of SLW can be understood by its recent applications to a variety of complex radiation problems involving high temperature gradients and particles [34], non-gray walls [35], and combustion environments [36]. With the advent of the new and updated databases for spectral information, the band model parameters of SLW were also updated. For example, Kangwanpongpan [37] utilized the HITEMP2010 database to generate the absorption coefficients and weights for H2 O and C O2 , in the partial pressure ratio range of 0.125–4 and for temperature and pressure up to 2500 K and 60 bar, respectively. Recently, Pearson et al. [38] improved and extended the SLW model parameters so that the model is applicable to temperatures up to 4000 K and pressures up to 50 bar. The representation of the data has been made into easy-to-use correlations and look-up tables.

2.5

Studies on Inclusion of Particle Radiation

The presence of particles in the participating medium alters the direction of travel of the radiation beam and causes the redistribution of the radiation energy, which is generally known as scattering . As discussed in Chap. 1, in the practical cases of interest, the particle radiation interaction fall under Mie Scattering regime. To estimate the extent of scattering by a group of particles or the effective scattering coefficient, the interaction of electromagnetic waves with the dielectric field of particles should be modeled. This requires a complete solution

2.5

Studies on Inclusion of Particle Radiation

19

to Maxwell’s equations [39]. For a spherical particle, this formulation was generated first by Lorenz, and then independently by Mie, hence this theory is also popularly known as Lorenz-Mie scattering theory. With the help of this, knowing the size of the particle and incoming wavelength, the parameters that govern the scattering phenomenon such as scattering coefficient, asymmetry parameter, and phase function can be calculated. Although very generic, the engineering implementation of the Mie scattering theory is quite scarce in literature. However, a recent treatise on Mie scattering by Hergert and Wriedt [40] summarizes the basis of the theory and its important applications. Many of the earliest studies on radiative heat transfer with an absorbing-emitting medium did not consider the influence of the particle type and its scattering characteristics to a significant extent. The inefficiency of certain band models to incorporate the spectral nature of the particle radiation brought in additional challenges in the modeling of the particle radiation in practical problems. Among some of the early studies that laid emphasis on particle radiation for radiative heat flux estimation, was the one by Jendoubi et al. [41]. Jendoubi et al. [41] solved the problem of particle radiation in a cylindrical enclosure by using a Legendre polynomial expansion of the phase function. The medium comprising particles and gases was assumed to be gray and isothermal. Yu et al. [42] proposed benchmark solutions of radiative heating by a mixture of gases (H2 O and C O2 ) and alumina particles in a cylindrical geometry. The medium was assumed isothermal throughout. Their study, however, did not consider the effect of anisotropic scattering. Additionally, the particle properties were calculated from the traditional approach of determining the mean emissivity of the particle cloud. More accurate benchmark results with a spatially non-uniform temperature field were developed by Perez et al. [43]. The sophisticated formulations of the SNB-CK band model were coupled with the traditional DOM in axisymmetric cylindrical geometries. The participating medium was made more generalized by the inclusion of non-gray soot particles. In many of the studies that consider two-phase mixture of gases and particles, a general practice is to assume thermal equilibrium between them, i.e. the gases and particles are considered to be at same temperature at any given location in the domain. However, some of the studies like Dombrovsky and Baillis [44], and Dombrovsky [45] presented a formulation for taking into account thermal non-equilibrium between gases and particles. A more general numerical treatment of two-phase flow over a blunt body without thermal equilibrium was also presented recently by Dombrovsky et al. [46]. Furthermore, the importance of scattering by particles within an absorbing media was highlighted, by solving the electromagnetic equations for the particles from first principles [47]. In some recent studies, Trivic [48] considered a generalized three-dimensional rectangular furnace commonly encountered in industrial applications such as a coal-fired furnace or a boiler. In this work, different kinds of particles were considered in the furnace and the relative contribution of each to the radiative heating was assessed. Gas radiation was modeled using the WSGG method. To incorporate the particle spectral nature, a new method following the equal distribution of blackbody radiation was introduced. This approach helps in the

20

2 Important Literatures on Radiative Heat Transfer

evaluation of the particle properties at mean wavelength and at local temperature, which is a better approximation instead of using a gray and isothermal assumption for particles.

2.6

Application-Based Studies

Along with the generation and modification of new and versatile methods of solving RTE, consideration of spectral nature of gases and particles, attempts that have been made to solve practical problems of interest considering the aforementioned generalities are discussed here. One of the prominent industrial problems with gas particle hot dispersed medium is the estimation of radiation heating from the rocket exhaust. The exhaust plumes of solid rocket motor contain radiatively participating gases and particles in significant concentrations, along with high-temperature gradients within the plume. To accurately evaluate the radiative heating at the base of the rocket, the complete solution of RTE in an absorbing, emitting, and scattering media should be performed. Everson and Nelson [49] used the reverse Monte Carlo method to estimate radiation from rocket exhaust plume. Both the exhaust plume of solid as well as the liquid propellant-based rocket were studied. However, the effect of gas was not taken into account. Cai et al. [50] calculated the radiative heating at the base of the rockets due to high-temperature exhausts using an exponential wideband model for gas radiation. It was emphasized that the effect of gas emission on the spectral fluxes at the rocket base can be significant in the ranges where the gas participation is dominant. Using a statistical narrowband model and particle properties at a defined mean temperature, Duval et al. [51] highlighted the importance of radiation from alumina particles in solid rocket motor plumes. In general, the complete solution of flow and heat transfer in practical situations involve the coupling of the CFD codes with the radiation solvers. The temperature, pressure, and species concentration field obtainable from the CFD codes serve as an input to the radiation solvers and provide with the divergence of radiative heat fluxes in the control volumes of the domain. This divergence of radiative heat flux serves as a source term for the CFD simulators for obtaining solutions at further time steps. Such coupled simulations also need a detailed modeling of the turbulence-radiation interaction for an accurate evaluation of temperature, pressure, and velocities [52]. Other important applications, where the gases and particles are present in considerable fractions with high temperatures, are industrial furnaces, boilers, reheaters, and combustion chambers. Harish and Dutta [53] performed an evaluation of the temperature distribution of the gas, the roof, and the billet surface by incorporating the finite volume method with WSGG approach in a pusher type reheating furnace. The effect of parameters such as billet speed, residence time, and the load emissivity on the temperature distribution was studied. Yuen et al. [54] coupled the zonal method with a neural network-based total gas emissivity model. This approach was applied to estimate the heating of the specimens in a fire resistance test furnace. The effect of the wall temperature lag with respect to the gas was found to be critical in such applications. Yuen et al. [55] performed the evaluation of the radiation heating

2.7

Relevance, Scope, and Challenges

21

due to a non-isothermal and inhomogeneous medium by using a neural network-based gas emissivity model. The importance of enclosure geometry and the accurate estimation of the radiative properties of the medium was highlighted. Additionally, it was concluded that the high non-linearity of the problem demands careful attention and the approximate models such as a neural network fit may lead to significant errors in some cases. In particular to industrial furnaces, where radiative heat transfer has a significant role to play, it is often required to tune the wall insulations, furnace geometry, and operating conditions in order to achieve the desired performance. However, such problems are illconditioned, since more than one boundary condition need to be specified at the desired location, while at the other locations, the conditions are not known. Sarvari et al. [56] formulated the problem of estimation of effective distribution of heat input in order to obtain the desired heat flux at the specimen surface in a three-dimensional furnace. The relevance of the inverse modeling in determining the heat load to the radiant furnaces in industries was studied by Daun and Howell [57]. A detailed methodology and the solution procedure were discussed by the authors. The above two studies helped in implementing the inverse solution technique to many important problems in industry. It is pertinent to mention that, prior to this, in most cases, the solution to such problems was obtained by using trial and error. A similar study as in Sarvari et al. [56] with homogeneous participating medium in radiative equilibrium was performed by Amiri et al. [58]. However, in the latter case, the inverse problem was solved as an optimization problem by minimizing the error between the desired and the estimated values using micro-genetic algorithm. Some of the recent studies have emphasized that a neural network-based surrogate can be a possible alternative to rather time-consuming RTE computations in such inverse problems [59, 60].

2.7

Relevance, Scope, and Challenges

From the review of literature, it can be understood that major industrial applications involving heat transfer from hot combustion media contributes significantly to the total heat transfer. An accurate numerical modeling of radiative heat transfer in these applications is necessary for the efficient design of the furnaces and other process parameters. Additionally, it is also observed that state of the art of radiative heat transfer modeling in real-time practical problems, has witnessed a part-by-part development in the past few decades where the primary focus has been on the generation of new and modification of the existing solution schemes and algorithms. It is only during the last decade, that the application of the models to problems of real practical interest has been gaining attention. However advanced they may be, the engineering implementation of available sophisticated models is surprisingly rare. Even with the recent advent of computing advancements, many industrial scale problems are still preferred to be solved using simplified emissivity correlations, during the course of which the exact determination and the influence of the governing parameters are consistently compromised. Additionally, depending upon the needs of the problem, several approximations

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2 Important Literatures on Radiative Heat Transfer

in the solution procedures have also been made by ignoring one or the other attributes of the participating medium. For example, in studies pertaining to rocket base heating problems, gas radiation is either ignored or solved using simplified gray assumptions. Similarly, in the applications of furnace radiation heating, the anisotropic behavior of the particles has been compromised by assuming isotropic scattering. Though these approaches may work in certain specific situations, a systematic investigation of all the primary influences in a hightemperature participating medium pertaining to both direct and inverse radiation problems is necessary. In consideration of this, the present book comprehensively accomplishes the following goals: 1. To present a step-by-step development of a generic radiative heat transfer model by taking into account all the generalities usually encountered in the practical problems. This model should be capable of serving as an efficient tool to estimate the contribution of the different aspects of the problem such as geometry, temperature, concentration variation of the gases and particles, particle distribution, and anisotropic scattering. 2. To demonstrate the validation and implementation of the developed code for industry scale problems and analyze the results thereof. 3. To incubate the solver with smart computing capabilities such as neural networks for fast and efficient prediction in inverse problems of interest. 4. To use the neural network-based smart predictor tools in conjunction with an evolutionary optimization technique like genetic algorithm for the problem of inverse estimation of heater power and heater locations for industrial scale furnaces. In consideration of the above, this book first presents the development of a comprehensive fast radiative heat transfer package in MATLAB framework. The potential use of this package is in the estimation of radiative heating in all its generalities without making limiting assumptions. This can be used as an independent as well as a coupled solver, in case the radiation contribution has to be determined in conjunction with fluid flow simulations. In the second part of the book, the results of this solver are used to develop a neural network-based architecture, which can reduce the computational efforts involved in solving the RTE. In the final part of the work, the neural network is used to solve the inverse problem of the design of a radiant heating furnace.

References 1. Lockwood, F. C., & Shah, N. G. (1981). A new radiation solution method for incorporation in general combustion prediction procedures. In Symposium (International) on Combustion (pp. 1405–1414). 2. Chandrasekhar, S. (1960). Radiative transfer. New York: Dover Publications. 3. Carlson, B. G., & Lathrop, K. D. (1968). In H. Greenspan, C. Kelber, & D. Okrent (Eds.) Transport theory-the method of discrete-ordinates in computing methods in reactor physics. New York: Gordon and Breach.

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4. Fiveland, W. A. (1982). A discrete ordinates method for predicting radiative heat transfer in axisymmetric enclosures. Journal of Heat Transfer, 82, 1–8. 5. Fiveland, W. A. (1984). Discrete-ordinates solutions of the radiative transport equation for rectangular enclosures. Journal of Heat Transfer, 106(4), 699–706. 6. Truelove, J. S. (1987). Discrete-ordinate solutions of the radiation transport equation. Journal of Heat Transfer, 109(4). 7. Jamaluddin, A. S., & Smith, P. J. (1988). Predicting radiative transfer in rectangular enclosures using the discrete ordinates method. Combustion Science and Technology, 59(4–6), 321–340. 8. Chai, J. C., Lee, H. S., & Patankar, S. V. (1994). Finite volume method for radiation heat transfer. Journal of Thermophysics and Heat Hransfer, 8(3), 419–425. 9. Raithby, G. D., & Chui, E. H. (1990). A finite-volume method for predicting a radiant heat transfer in enclosures with participating media. Journal of Heat Transfer, 112, 415–423. 10. Baek, S. W., & Kim, M. Y. (1997). Modification of the discrete-ordinates method in an axisymmetric cylindrical geometry. Numerical Heat Transfer, 31(3), 313–326. 11. Chui, E. H., Raithby, G. D., & Hughes, P. M. J. (1992). Prediction of radiative transfer in cylindrical enclosures with the finite volume method. Journal of Thermophysics and Heat Transfer, 6(4), 605–611. 12. Wu, H. L., & Fricker, N. (1971). An investigation of the behaviour swirling jet flames in a narrow cylindrical furnace. In 2nd Members Conference. Industrial Flame Research Foundation Ijmuiden. 13. Kim, M. Y., & Baek, S. W. (1998). Radiative heat transfer in a body-fitted axisymmetric cylindrical enclosure. Journal of Thermophysics and Heat Transfer, 12(4), 596–599. 14. Raithby, G. D. (1999). Evaluation of discretization errors in finite-volume radiant heat transfer predictions. Numerical Heat Transfer: Part B: Fundamentals, 36(3), 241–264. 15. Coelho, P. J. (2002). The role of ray effects and false scattering on the accuracy of the standard and modified discrete ordinates methods. Journal of Quantitative Spectroscopy and Radiative Transfer, 73(2), 231–238. 16. Ramankutty, M. A., & Crosbie, A. L. (1997). Modified discrete ordinates solution of radiative transfer in two-dimensional rectangular enclosures. Journal of Quantitative Spectroscopy and Radiative Transfer, 57(1), 107–140. 17. Hunter, B., & Guo, Z. (2015). Numerical smearing, ray effect, and angular false scattering in radiation transfer computation. International Journal of Heat and Mass Transfer, 81, 63–74. 18. Guedri, K., & Al-Ghamdi, A. S. (2018). Improved Finite Volume Method for Three-Dimensional Radiative Heat Transfer in Complex Enclosures Containing Homogenous and Inhomogeneous Participating Media. Heat Transfer Engineering, 39(15), 1364–1376. 19. Zhou, R. R., & Li, B. W. (2019). The modified discrete ordinates method for radiative heat transfer in two-dimensional cylindrical medium. International Journal of Heat and Mass Transfer, 139, 1018–1030. 20. Coelho, P. J. (2008). A comparison of spatial discretization schemes for differential solution methods of the radiative transfer equation. Journal of Quantitative Spectroscopy and Radiative Transfer, 109(2), 189–200. 21. Coelho, P. J. (2014). Advances in the discrete ordinates and finite volume methods for the solution of radiative heat transfer problems in participating media. Journal of Quantitative Spectroscopy and Radiative Transfer, 145, 121–146. 22. McClatchey, R. A., Benedict, W. S., Clough, S. A., Burch, D. E., & Calfee, R. F. (1973). AFCRL atmospheric absorption line parameters compilation. 23. Rothman, L. S., Gamache, R. R., Tipping, R. H., Rinsland, C. P., Smith, M. A. H., Benner, D. C., Devi, V. M., Flaud, J.-M., Camy-Peyret, C., Perrin, A., et al. (1992). The HITRAN molec-

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24.

25.

26.

27. 28. 29. 30. 31. 32. 33. 34.

35.

36.

37.

38.

39. 40.

2 Important Literatures on Radiative Heat Transfer ular database: editions of 1991 and 1992. Journal of Quantitative Spectroscopy and Radiative Transfer, 48(5–6), 469–507. Rothman, L. S., Rinsland, C. P., Goldman, A., Massie, S. T., Edwards, D. P., Flaud, J. M., Perrin, A., Camy-Peyret, C., Dana, V., Mandin, J.-Y., et al. (1998). The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition. Journal of Quantitative Spectroscopy and Radiative Transfer, 60(5), 665–710. Rothman, L. S., Gordon, I. E., Barbe, A., Benner, D. C., Bernath, P. F., Birk, M., Boudon, V., Brown, L. R., Campargue, A., Champion, J.-P., et al. (2009). The HITRAN 2008 molecular spectroscopic database. Journal of Quantitative Spectroscopy and Radiative Transfer, 110(9–10), 533–572. Rothman, L. S., Gordon, I. E., Barber, R. J., Dothe, H., Gamache, R. R., Goldman, A., Perevalov, V. I., Tashkun, S. A., & Tennyson, J. (2010). HITEMP, the high-temperature molecular spectroscopic database. Journal of Quantitative Spectroscopy and Radiative Transfer, 111(15), 2139– 2150. Hottel, H. C., & Sarofim, A. F. (1967). Radiative Transfer. New York: McGraw-Hill. Smith, T. F., Shen, Z. F., & Friedman, J. N. (1982). Evaluation of coefficients for the weighted sum of gray gases model. Journal of Heat Transfer, 104(4), 602–608. Modest, M. F. (1991). The weighted-sum-of-gray-gases model for arbitrary solution methods in radiative transfer. Journal of Heat Transfer, 113(3), 650–656. Denison, M. K., & Webb, B. W. (1993). A spectral line-based weighted-sum-of-gray-gases model for arbitrary RTE solvers. Journal of Heat Transfer, 115(4), 1004–1012. Solovjov, V. P., & Webb, B. W. (2000). SLW modeling of radiative transfer in multicomponent gas mixtures. Journal of Quantitative Spectroscopy and Radiative Transfer, 65(4), 655–672. Modest, M. F., & Zhang, H. (2002). The Full-Spectrum Correlated-K Distribution for thermal radiation from molecular gas-particulate mixtures. Journal of Heat Transfer, 124(1), 30–38. Demarco, R., Consalvi, J. L., Fuentes, A., & Melis, S. (2011). Assessment of radiative property models in non-gray sooting media. International Journal of Thermal Sciences, 50(9), 1672–1684. Yadav, R., Chakravarthy, B., & Venkateshan, S. P. (2017). Implementation of SLW model in the radiative heat transfer problems with particles and high temperature gradients. International Journal of Numerical Methods for Heat & Fluid Flow, 27(5), 1128–1141. da Silva, R. M., da Fonseca, R. J. C., & França, F. H. R. (2019). Radiative transfer prediction in participating medium bounded with nongray walls using the SLW model. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 41(12), 568. Badger, J., Webb, B. W., & Solovjov, V. P. (2019). An exploration of advanced SLW modeling approaches in comprehensive combustion predictions. Combustion Science and Technology. https://doi.org/10.1080/00102202.2019.1678907. Kangwanpongpan, T., França, F. H. R., da Silva, R. C., Schneider, P. S., & Krautz, H. J. (2012). New correlations for the weighted-sum-of-gray-gases model in oxy-fuel conditions based on HITEMP 2010 database. International Journal of Heat and Mass Transfer, 55(25–26), 7419– 7433. Pearson, J. T., Webb, B. W., Solovjov, V. P., & Ma, J. (2014). Efficient representation of the absorption line blackbody distribution function for H2 O, C O2 , and C O at variable temperature, mole fraction, and total pressure. Journal of Quantitative Spectroscopy and Radiative Transfer, 138, 82–96. Bohren, C. F., & Huffman, D. R. (2008). Absorption and scattering of light by small particles. New York: Wiley. Hergert, W., & Wriedt, T. (2012). The Mie theory: basics and applications (Vol. 169). Berlin Heidelberg: Springer.

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41. Jendoubi, S., Lee, H. S., & Kim, R.-K. (1993). Discrete ordinates solutions for radiatively participating media in a cylindrical enclosure. Journal of Thermophysics and Heat Transfer, 7(2), 213–219. 42. Yu, M. J., Baek, S. W., & Park, J. H. (2000). An extension of the weighted sum of gray gases non-gray gas radiation model to a two phase mixture of non-gray gas with particles. International Journal of Heat and Mass Transfer, 43(10), 1699–1713. 43. Perez, P., El Hafi, M., Coelho, P. J., & Fournier, R. (2005). Accurate solutions for radiative heat transfer in two-dimensional axisymmetric enclosures with gas radiation and reflective surfaces. Numerical Heat Transfer Part B-Fundamentals, 47, 39–63. 44. Dombrovsky, L. A., & Baillis, D. (2010). Thermal radiation in disperse systems: an engineering approach. Begell House New York. 45. Dombrovsky, L. A. (2012). The use of transport approximation and diffusion-based models in radiative transfer calculations. Computational Thermal Sciences, 4(4), 297–315. 46. Dombrovsky, L. A., Reviznikov, D. L., & Sposobin, A. V. (2016). Radiative heat transfer from supersonic flow with suspended particles to a blunt body. International Journal of Heat and Mass Transfer, 93, 853–861. 47. Mishchenko, M. I., & Dlugach, J. M. (2018). Scattering and extinction by spherical particles immersed in an absorbing host medium. Journal of Quantitative Spectroscopy and Radiative Transfer, 211, 179–187. 48. Trivic, D. N. (2014). 3-D radiation modeling of nongray gases-particles mixture by two different numerical methods. International Journal of Heat and Mass Transfer, 70, 298–312. 49. Everson, J., & Nelson, H. (1993). Development and application of a reverse Monte Carlo radiative transfer code for rocket plume base heating. In 31st Aerospace Sciences Meeting (pp. 138). 50. Cai, G., Zhu, D., & Zhang, X. (2007). Numerical simulation of the infrared radiative signatures of liquid and solid rocket plumes. Aerospace Science and Technology, 11(6), 473–480. 51. Duval, R., Soufiani, A., & Taine, J. (2004). Coupled radiation and turbulent multiphase flow in an aluminised solid propellant rocket engine. Journal of Quantitative Spectroscopy and Radiative Transfer, 84(4), 513–526. 52. Coelho, P. J. (2012). Turbulence-radiation interaction: From theory to application in numerical simulations. Journal of Heat Transfer, 134(3), 031001. 53. Harish, J., & Dutta, P. (2005). Heat transfer analysis of pusher type reheat furnace. Ironmaking and Steelmaking, 32(2), 151–158. 54. Yuen, W. W., Tam, W. C., & Chow, W. K. (2014). Assessment of radiative heat transfer characteristics of a combustion mixture in a three-dimensional enclosure using RAD-NETT (with application to a fire resistance test furnace). International Journal of Heat and Mass Transfer, 68, 383–390. 55. Yuen, W. W., Chow, C. L., & Tam, W. C. (2016). Analysis of radiative heat transfer in inhomogeneous nonisothermal media using neural networks. Journal of Thermophysics and Heat Transfer, 30(4), 897–911. 56. Sarvari, S. M. H., Mansouri, S. H., & Howell, J. R. (2003). Inverse design of three-dimensional enclosures with transparent and absorbing-emitting medial using an optimization technique. International Communications in Heat and Mass Transfer, 30(2), 149–162. 57. Daun, K. J., & Howell, J. R. (2005). Inverse design methods for radiative transfer systems. Journal of Quantitative Spectroscopy and Radiative Transfer, 93(1–3), 43–60. 58. Amiri, H., Mansouri, S. H., Safavinejad, A., & Coelho, P. J. (2011). The optimal number and location of discrete radiant heaters in enclosures with the participating media using the micro genetic algorithm. Numerical Heat Transfer, Part A: Applications, 60(5), 461–483.

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59. Yadav, R., Balaji, C., & Venkateshan, S. P. (2019). Inverse estimation of number and location of discrete heaters in radiant furnaces using artificial neural networks and genetic algorithm. Journal of Quantitative Spectroscopy and Radiative Transfer, 226, 127–137. 60. Yadav, R., Tripathi, S., Asati, S., & Das, M. K. (2020). A combined neural network and simulated annealing based inverse technique to optimize the heat source control parameters in heat treatment furnaces. Inverse Problems in Science and Engineering, 28(9), 1265–1286.

3

Mathematical Formulation

3.1

Introduction

This chapter discusses the governing radiative transfer equation (RTE), its discretization, and the solution procedure. Information on the gas and particle spectral nature has been provided along with a discussion on the full spectrum band models. The key concepts, theories, and approximations that are pertinent to the development of a comprehensive industrial scale radiative heat transfer solver have been highlighted.

3.2

Solution Methods for RTE

A number of methods are available to solve the radiative transfer equation (RTE) and these include zonal method, P-N method, spherical harmonics, Monte Carlo method, finite volume, and discrete ordinates. Over the years, the DOM and FVM have received considerable attention due to their versatility and applicability to a wide class of problems. DOM and FVM are more or less the same in all aspects, except in the consideration of discrete directions. In DOM, the intensity is solved for each discrete direction spanning over the entire sphere, and each spatial control volume. Instead of the discrete directions, the FVM solves the intensity by integrating it inside a control volume and control angle. Both methods lend themselves to coupling with CFD solvers and are very useful when one is looking for surface-based properties such as wall fluxes and irradiation at many locations in the domain. On the other hand, the Monte Carlo methods are statistical methods, which trace the path of every emitted photon, until it gets absorbed or scattered. Monte Carlo methods are very accurate but are also time-consuming and are advantageous only if a few locations need to be monitored in problems involving gas radiation problems.

© The Author(s) 2023 R. Yadav et al., Radiative Heat Transfer in Participating Media, https://doi.org/10.1007/978-3-030-99045-9_3

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3.2.1

3 Mathematical Formulation

Traditional Discrete Ordinates Method

Consider the vectorized form of the radiative transfer Eq. (1.8), as derived in Chap. 1. In order to focus on the geometrical aspect of the equation, the RTE is rewritten without the spectral dependency and considering the total extinction coefficient as  σp   ˆ ˆ sˆ · ∇ I ( r , s ) = −β I ( r , s ) + κ Ib + I ( r , sˆ )( →  )d (3.1) 4π =4π where r and sˆ represent the spatial location and unit direction vector of the intensity I . The prime in the superscript represents the direction of calculation. In the above equation, r and sˆ depend on the coordinate system and may vary with respect to the choice of the same. A set of base vectors now needs to be defined in order to monitor the directions. Cartesian base vectors are chosen for Cartesian coordinates, cylindrical for cylindrical, and spherical for spherical coordinates. However, the use of cylindrical and spherical base vectors is not so straightforward due to the fact that in Cartesian coordinates, each predefined set of values of (θ, φ) represents the same direction everywhere, or in other words, every direction has its unique (θ, φ) value throughout the domain. However, this is not the case with other (curvilinear) coordinate systems. The change in the location in such coordinate systems is accompanied by a change in the alignment of the base vectors and the direction of travel can no longer be considered a straight line. This gives rise to the so-called angular redistribution term in curvilinear coordinates. More discussion on angular redistribution term and its treatment is provided in Chap. exrefchapter:cylinder, where cylindrical geometries are examined. For the Cartesian coordinate system, consider a ray emanating from location r in the direction sˆ . As can be seen from Fig. 3.1, there exists a single direction represented by the unit vector. In this case, the value of the unit vector defined with respect to the angle it makes from horizontal and vertical axes is given as sˆ = (sinθ cosφ)iˆ + (sinθ sinφ) jˆ + (cosθ )kˆ

(3.2)

ˆ j, ˆ and kˆ are the unit vectors in the x, y, and z directions, respectively, and (sinθ cosφ), where i, (sinθ sinφ), and cosθ are the direction cosines. Let them be termed as μ,ζ , and η in the same order. Radiation emanating from any point of interest propagates into the sphere of solid angle 4π . Let this solid angle of 4π around the point of interest be divided into N discrete directions. Each of these directions is defined by its own set of (θ, φ) or more precisely by its direction cosines. Depending upon the requirement of the problem, weights to these directions are assigned, for example, in problems with collimated radiation, a specific region of the sphere is of interest. These weights should hence be controllable by the user. Additionally, from the definition of a weight, it is apparent that

3.2

Solution Methods for RTE

29

Fig. 3.1 A depiction of direction in the Cartesian coordinate system

N 

wn = 4π

(3.3)

n=1

Consider that such a set of discrete directions along with its corresponding weights is available. The RTE can then be written for each of these direction as μ

n d I

n

dx

+ζ

n d I

n

dy

+η

n d I

n

dz

= −β I

N σ p  n n nn  + κ Ib + I w  4π

n

(3.4)

n=1

The last two terms on the right-hand side may be considered as the source term as they contribute to the increase in the intensity value due to emission and in-scattering. Hence,  Srn

N σ p  n n nn  = κ Ib + I w  4π

(3.5)

n=1



where nn is the phase function or the probability that the intensity in direction n will get scattered to the direction of calculation n  . From the preceding discussion, it follows that, on discrete directions, the following holds for any phase function: N 1  n nn  w  =1 4π

(3.6)

n=1

This expression is often used to normalize the phase function. A preliminary discussion on phase function was provided in Chap. 1, and more details are given in a subsequent section discussing particle radiation. The discrete direction radiative transfer Eq. (3.4), after the inclusion of source term becomes μn









n n dIn  dI  dI   + ζn + ηn = −β I n + Srn dx dy dz

(3.7)

30

3 Mathematical Formulation

This is a first-order ordinary differential equation and hence requires one boundary condition to close the problem mathematically. It is always assumed that the traveling radiation emitted by any source will encounter a wall or a surface during its course of travel. This is logical following the fact that till the radiation beam interacts with any surface or matter, the changes in its physical properties are difficult to monitor. Thus, the radiation hitting the wall can be allowed to get absorbed, transmitted, or reflected depending upon the nature of the wall. Considering an opaque wall with no transmission, the intensity directed back from the wall in the direction of calculation can be written as 

I n = wall Ib,wall +

N (1 − wall )  n n n w μ I π n

(3.8)

μ 0)

qr ,t =



n,(nrˆ,i ·ˆs >0)

wn ηn Irn

(3.17)

3.2

Solution Methods for RTE

33

Similar expressions can be derived for the west and bottom walls, except that the summation has to be followed for the directions where nrˆ,i · sˆ < 0. Despite its elegance, one can see that in terms of directional dependence, the traditional discrete ordinates method is rather crude. One has to depend on the predefined set of directions which have been evaluated after satisfying all the moment requirements. In many cases, the set of directions developed or available may not satisfy all the moments, or the generation of the directions itself may look like an additional task. Moreover, based on the need of the problem, there may be cases where a certain direction (or a range of θ and φ) must be carefully examined. In these cases, the possibility that the predefined set of directions not being able to capture the intensity variation in that range arises and so one may have to choose a sufficiently large number of directions which would incur an increase in the computational cost. Therefore, it can be understood that the analyst should have a control over the directions as demanded by the problem and the pre-chosen directions may not always work.

3.2.2

Finite Volume Method

In the finite volume method, the directions are defined based on the control angle, which is similar to a control volume but in angular domain. Following this, all the required moments are automatically satisfied. The discrete direction is assumed to be centered at this control angle and the intensity value over this control angle is assumed constant. Figure 3.3 shows a typical control angle in the direction of calculation  and is defined as d = sinθ dθ dφ

(3.18)

The basic radiative transfer equation (given already as 1.8 in Chap. 1) is sˆ · ∇ I ( r , sˆ ) = −β I ( r , sˆ ) + Sr ( r , sˆ )

(3.19)

The above equation is integrated over the control angle (Fig. 3.3a) and control volume (Fig. 3.3b) as        ˆ ˆ s · ∇ I ( r , s )d xdzd = (−β I ( r , sˆ ) + Sr ( r , sˆ ))d xdzd (3.20)





V

V

On applying the Gauss divergence theorem and writing the volume integrals in terms of surface integral, one gets    sˆ · nˆ I  d Ad = (−β p I p V P + (Sr ) P V P )d (3.21)



A



where A is the area of the surface element bounding the control volume. Since in the finite volume method the intensity is assumed constant over the control angle, the above

34

3 Mathematical Formulation

Fig. 3.3 A typical control angle and control volume in the finite volume method

d ' z

d

d y (a) A control angle in finite volume method.

Discrete solid angle of calculation, dΩ’

t e

w P b

(b) A control volume.

x

3.2

Solution Methods for RTE

35

expression can be rewritten as   I d A (sˆ · nˆ )d = −β p I p V P  + (Sr ) P V P 



A

(3.22)

or, 

Ii Ai

i=e,w,t,b





(sˆ · nˆ )d = −β p I p V P  + (Sr ) P V P 

(3.23)

where the term inside the integral on the left-hand side is called the directional weight of the direction under consideration. For the particular case of orthogonal grids, these weights for the control volume faces in the positive x and y directions can readily be calculated as   Dcx = (sinθ cosφ)d (3.24)



 Dcz =





(cosθ )d

(3.25)

Following this, Eq. 3.23 can be written for a ray emanating from the bottom left corner of the domain as     Ie Dce

Ae − Iw Dcw

Aw + It Dct

At − Ib Dcb

Ab

= −β p I p V P  + (Sr ) P V P 

(3.26)

Using the step scheme of spatial differencing, as presented in Eq. 3.12, the above expression can be simplified as   I p (Dce

Ae + Dct

At +β p V P  )  = (Iw Dcw

Aw + Ib Dcb Ab ) + (Sr ) P V P 

or, I P =

 A + I  D  A ) + (S  ) V  (Iw Dcw w b P r P b cb    (Dce Ae + Dct At + β p V P  )

(3.27)

(3.28)

Equation 3.28 is the explicit equation of nodal intensity in terms of known parameters and should be solved for each direction at each location similar to Eq. 3.14. To assign the nodal intensities to the facial intensities, a more compact and general representation of step scheme is given as   Ii = I P max(Dci , 0) − I I max(−Dci , 0) (3.29) where i corresponds to the east, west, top, and bottom faces, as the case may be and I in the subscript corresponds to the respective adjacent node. The wall boundary conditions can be updated by evaluating the total radiosity from the wall. The expression for the radiation intensity emanating from the west and bottom wall are written as

36

3 Mathematical Formulation

Ii = i,wall Ib,i,wall +

(1 − i,wall )    Ii |Dci | π 

(3.30)

Dci 0

The solution procedure remains similar to the one employed for solving the RTE with the discrete ordinate method. An intensity field is assumed and the solution is started from the walls of known boundary condition. The intensities at the nodes are calculated using Eq. 3.28. The nodal intensities are used to calculate the facial intensities following the step scheme Eq. 3.29. After completing the first sweep of the domain, the wall boundary conditions are updated with Eqs. 3.30 and 3.31. The solution is again sought for this updated radiosity from the wall and the solution progresses iteratively. The step-by-step solution procedure is discussed in the subsequent chapters for the respective geometries under considerations. After the intensity field has converged in the domain for all the locations and all the directions, the fluxes at the location r on the east and the top walls, are evaluated, respectively, as    qr ,e = Dce Ie,r (3.32) all dir ections    qr ,t = Dct It,r (3.33) all dir ections

Similar expressions can be used for evaluating the fluxes at the bottom and the west walls by using the appropriate directional weights. Problem-specific parts of the solution procedure are elucidated for the respective geometries in the corresponding chapters. The next section discusses the estimation of gas and particle radiative properties, which serve as critical inputs to the RTE solver.

3.3

Estimation of Gas Properties

When a traveling photon interacts with the gas molecule, it can be either absorbed or scattered. The scale of scattering by gas molecules is, in general, minimal and can be neglected for heat transfer applications. The absorption of the traveling photos may result in an increase in the energy level. The gas molecule, on the other hand, itself may lower its energy level by emitting a photon. As discussed in Chap. 1, these interactions are sensitive to the wavelength of the interacting photon. On looking closely, it is observed that the increase/decrease in energy levels can occur in the following three ways: 1. Bound-Bound transitions. 2. Bound-Free transitions.

3.3

Estimation of Gas Properties

37

3. Free-Free transitions. A bound-bound transition is the change that occurs within the fixed molecular states, since this discrete change in energy must be quantized, the transitions associated with boundbound type result in single or discrete absorption line in the spectrum. In other words, this fixed amount of energy change can be expressed by the particular frequency or wavelength of the photon. This phenomenon occurs at moderate temperatures and is associated with the wavelengths in the near infrared region. The amount of energy change involved in boundbound transitions can alter only the translational, rotational, and vibrational states of the molecules. On the other hand, wherever a free transition is involved, the energy levels can no more be considered as quantized as they can have values greater than the minimum ionization or dissociation energy of the molecule. In other words, once the molecule has acquired a free state from a bound one (which generally happens due to ionization), it is not possible to assign any fixed wavelength to the energy change that has occurred and hence they give continuous participation throughout the spectrum. However, these phenomena happen at very high temperatures and at very low wavelengths of the interacting photon. For the range of temperatures normally encountered in industrial applications, the changes in the translational, vibrational, and rotational energy levels are of significant interest. This implies that a knowledge of the values of spectral parameters associated with such changes is required. In view of this, during the last few decades, a section of the researchers have focused on the generation, compilation, and modification of databases for major participating species. HITRAN (1992) [1] is one such elaborate database for the LBL spectrum information of combustion gases. At high temperatures, generally encountered in combustion applications, not only is the participation of the gases more pronounced but also the overlap between the spectral lines of the participating gases increases. In view of this, the HITRAN database was modified for high-temperature applications and was released under the name HITEMP [2]. This database till date is the most detailed and accurate one for the calculation of radiative properties of gases in combustion applications, as already mentioned. However, the spectral information of the gases may include millions of absorption lines within the range of interest of spectra and keeping in mind the complexity associated with the solution of multidimensional RTE for each location, the idea of seeking a solution for each spectral line may have to be abandoned for problems of engineering interest. This logically leads us to designing and implementing the modeling strategies to consider the significant variations in certain ranges of the spectrum which are also known as bands. Depending upon the spectral width, such models can be divided into narrow-, wide-, and full spectrum band models. In several engineering applications, the radiative heat transfer is usually considered in conjunction with other modes of heat transfe,r and a knowledge of spectral fluxes is seldom desired (except in a few cases). In view of this, the use of narrow- and wideband models in practical applications is limited. Full spectrum band models, on the other hand, provide a fast and accurate means of consideration of gas spectral nature by solving the RTE at a few

38

3 Mathematical Formulation

discrete locations in the spectra. This makes them a viable alternative for consideration in industrial scale radiation heat transfer solvers.

3.3.1

Full Spectrum Band Models

The initial versions of full spectrum band model proposed that the total emissivity (and also absorptivity) of a gas column of path length L, can be expressed as a weighted sum of gray gas emissivities as  = a j (1 − ex p(−κ j L)) (3.34) j

where κ j is the temperature-independent gray gas absorption coefficient and a j is the temperature-dependent weighting factor. There are several choices available in terms of the number of gray gases ( j  s) and the expression for the a j . In most cases, a polynomial representation is preferred. The most popular one comes from the work of [3], where the coefficients are expressed as the fourth-degree polynomial in temperature for partial pressure ratios of 1 and 2 for H2 O and C O2 gas. A possible physical interpretation of a weight could be the fraction of blackbody energy which falls in the spectrum where the absorption coefficients hold the given value. Consequently, for high absorption coefficient values, the weights will be lower, as they will accommodate very less part of the blackbody energy, and vice versa. The expressions for a j in the work of [3] were derived from the least square fit to the emissivity data calculated from a wideband model. Dennison and Webb [4] argued that a detailed spectral database like HITRAN 1992 can be a better choice for the development of a precise expressions for the weights. This initiated the development of a series of models which fall under the name spectral line-based weighted sum of gray gases method or SLWSGG method or simply SLW method.

3.3.2

The SLW Model

The SLW model was developed with the idea of providing a more general and accurate treatment to the standard WSGG method, by evaluating the weight with the help of detailed line by line spectra. Even so initially, an approach similar to the one used by [3] was employed to estimate the gray gas absorption coefficients and weights, just for the purpose of minimizing the error between emissivity values obtained from Eq. 3.34 and those calculated directly from the database. However, following the physical definition of the weight factor, the need for more general functional representation was realized. Such a function was developed by [5] as  1 F(C) = E b,η (T )dη (3.35) E b (T ) ηi :Cη 1 would be treated by geometric optics. In most of the engineering problems, particles in the range of 0.1–100 μm are encountered and scattering would be largely governed by the Mie scattering theory. Mie scattering theory is well documented in literature and a brief introduction has been provided in Appendix A. However, for the sake of completeness, certain key concepts are discussed below.

3.4

Estimation of Particle Properties

3.4.1

45

Scattering by a Single Particle

Broadly speaking, apart from physical interaction of radiation with the particle which can cause reflection or refraction of the ray, some bending in the radiation beam is also possible due to the interaction of electromagnetic field of the particle with the beam. This, in turn, would depend upon the type and size of the particle and the separation distance between the different particles. Again, for most of the engineering applications, only spherical particles are considered as irregularities in the topology of the individual particles when integrated over all particles usually die down. Moreover, scattering due to a category of particle does not affect the other particles, i.e. the single scattering assumption is made. The above is reasonable in view of the geometrical scale of the industrial problems. The amount of energy scattered by a particle in a particular direction would depend upon the phase function () value of the particle in that direction. In view of this, it is clear that the scattering of radiation by particles or a group of particles in different directions is not uniform, i.e. there is anisotropy in scattering. Figure 3.4 shows two cases (a) when energy is distributed evenly in all directions which is known as isotropic scattering and (b) when there is unevenness in the distribution of energy in different directions is termed anisotropic scattering. The scattering phase function (, see Eq. 1.5) depends upon the scattering angle () which is the angle between the incoming radiation and the scattered radiation. In the Mie scattering theory, the phase function is represented as () = 2

i1 + i2 χ Q 2sca

(3.55)

Here, the values i 1 , i 2 , and the Q sca can readily be calculated from the Mie theory (see Appendix A). Q sca is the scattering efficiency of the particle which is a non-dimensional ratio of the scattering cross section to total projected area of the sphere. For the total extinction of the energy through the particle in all possible ways, the terms extinction efficiency is used Q ext , which is given by the sum of absorption efficiency and scattering efficiency as Q ext = Q sca + Q abs

3.4.2

(3.56)

Scattering by a Group of Particles

In almost all the practical situations, the particles happen to be present in clusters. These clusters can have particles of different sizes. The scattering algorithm should then determine the absorption and scattering efficiencies of each of these particle groups. Say for a group of particles of non-uniform size N (a), where a is the radius of the particle, the scattering coefficient and asymmetry parameter are calculated as

46

3 Mathematical Formulation

Fig. 3.4 A depiction of a Isotropic scattering, and b Anisotropic scattering

Outgoing radiation (a)

Incoming radiation

Outgoing radiation (b)

Incoming radiation

 σsλ = gλ =

1 σsλ

∞

 0∞

Q sca πa 2 N (a)da

(3.57)

Q sca g(a)πa 2 N (a)da

(3.58)

0

where Q sca is the scattering efficiency of the individual group of particles having a radius a. The subscript λ denotes that these properties are wavelength-dependent and hence should be evaluated at each spectral location or for each gray gas in conjunction with the gas band model. In a similar way, using Eq. 3.56, the absorption efficiency can be calculated and the representative absorption coefficient for the cluster of particles is calculated

3.4

Estimation of Particle Properties

47

 p,λ

∞

=

Q abs πa 2 N (a)da

(3.59)

0

In actual practice, the total number of particles or the particle number density (N (a)) inside the domain may not be directly known but rather the mass fraction of the particles is known from the combustion and flow field data. A mass distribution function ( f M (a)) can then be defined and the number density function ( f N (a)) is calculated from it. A relationship between the mass density and number density functions can be written as f M (a) =

N0 m f N (a) M0

(3.60)

where m is the mass of a single particle, M0 and N0 are the total mass and the total number of particles, respectively, and f M (a) is the mass distribution of the particles. For closure of the formulation, the particle mass distribution must be appropriately defined. There are several options available like uniform distribution, multimodal, normal distribution, etc. However, a Rossin-Rammler distribution [11] is more general as it depends on two parameters, namely the mean diameter D0 and the extent of spread nr . The equation for cumulative mass fraction is then given as D w = exp[(− )nr ] (3.61) D0 This gives the mass fraction of the particles having a diameter greater than D. D0 defines the peak of the center of the spread and nr gives the extent of the spread of mass fraction in the particle range. The use of Rossin Rammler mass fraction distribution gives the facility to choose the effective range of particles particular to the problem under consideration. For example, in a rocket exhaust the particles near the centerline of the plume mostly the heavier particles are present that have a diameter in the range 5–15 μm, while at the periphery of the plume lighter particles in the range 0.5–5 μm are present. Thus, the use of such a distribution function allows a more general treatment of the particle distribution depending upon the problem.

3.4.3

Treatment of the Phase Function and Anisotropic Scattering

From Eq. 3.55, it is evident that the phase function depends upon the scattering angle. In simple Cartesian coordinates, for radiation coming from the direction sˆ , getting scattered in a direction sˆ the scattering angle can be expressed as cos()sˆ,sˆ = μ · μ + ζ · ζ  + η · η

(3.62)

which on simplification becomes, cos()sˆ,sˆ = cosθ cosθ  + sinθ sinθ  cos(φ − φ  )

(3.63)

48

3 Mathematical Formulation

Upon knowing the scattering angle between the incident and the scattered ray, the phase function corresponding to the present direction needs to be evaluated. There are several expressions available to approximate the phase function, and the most widely used ones include the Delta-Eddington phase function, Henyey-Greenstein phase function, and the phase function represented by Legendre polynomial expansion. Among these, the Legendre polynomial expansion is considered to be the most accurate. However, as the anisotropy increases with more peaks in forward direction, this method may become very timeconsuming and computationally expensive with the number of terms required to truncate the series becoming higher in these cases. Other approximate representations like the HenyeyGreenstein phase function prove to be accurate in modeling forward scattering and are widely used in atmospheric applications. The estimated set of phase function values should be normalized following Eq. 3.6, where, within the framework of the discrete ordinates method, the phase function is multiplied with the discrete direction weight and in the finite volume method, with the discrete solid angle. In recent years, the transport and diffusion-based models have gained popularity among the researchers since they offer simple yet reasonably accurate ways of modeling complex radiation phenomenon. In the modeling of anisotropic scattering, transport approximation may prove to be the best choice in terms of accuracy and computational time. The initial version of this approximation was introduced by [12] in neutron transport application. The simplicity of the model attracted the attention of researchers and with time, it was put to use in several engineering applications. Some of the recent researchers have tested the formulation against the other available models and a good agreement was found in both the forward and backward scattering problems. Thus, it is advantageous to use transport approximation for multidimensional engineering problems where computational time can be a critical concern. Under the transport approximation, the scattering coefficient (σs ) obtainable from the Miescattering formulations can be modified with the help of asymmetry parameter (g) as σtrans = σs (1 − g)

(3.64)

In the above equation, σtrans is the modified scattering coefficient defined on the basis of the transport approximation. In the original RTE, the regular scattering coefficient can then be replaced by this modified scattering coefficient and the equation can be solved similarly to the case of isotropic scattering (i.e. considering  = 1). This approach is seen to give considerable simplicity yet accurate results in multidimensional radiative heat transfer problems particularly in industrial applications, as will be discussed in the subsequent chapters.

3.4.4

Calculation of Particle Properties in Conjunction with Band Models

The radiative properties of the gas and particles show uncorrelated behavior, thereby making their coupling a complex task. However, the case of narrowband models is an exception, where the particle properties may be calculated at the representative wavelength of the band.

3.5

Modeling of Radiative Equilibrium

49

In case of all the full spectrum band models, it remains extremely difficult to incorporate exact particle spectral nature since the information about the relevant wavelength bands gets overwritten by some other explicit parameter such as the absorption cross section. Most of the studies in the past have considered particle nature to be gray. For example, the effect of particles in problems involving cylindrical furnaces and rocket base heating was studied by [13, 14], respectively. One of the recent methods of particle properties estimation as, suggested by [15], for the finite volume-based solvers is to first evaluate the mean temperature of the domain and mean wavelength following the equal energy relation for the blackbody radiation [16]: λm Tm = 4107 and,

 Tm =

4

Nv ij

(μm K )

(3.65)

Vi j Ti4j

(3.66) V where Tm is the fourth-power averaged temperature of the domain and λm is the mean wavelength. The particle properties can then be estimated at this mean wavelength and local temperature using the Lorentz-Mie scattering algorithm [17]. While it can be argued that Wien’s displacement law could be a possible variant of Eq. (3.65) since one requires the mean wavelength and not the wavelength corresponding to maximum emission, the wavelength at which the distribution of blackbody energy is equal on either side of the spectrum needs to be used. So, if Fb (λ, T ) is the blackbody radiation energy distribution function, then λ f = 0∞ 0

and,

Fb (λ, T )dλ Fb (λ, T )dλ

f (λT = 0 to 4107) = 0.5 f (λT = 4107 to ∞) = 0.5

(3.67)

(3.68)

where f is the fractional blackbody energy at a temperature T and wavelength λ. In other words, 50% of the total blackbody energy is found under λT = 4107μm. This has been demonstrated in Fig. 3.5 by plotting the universal blackbody function with respect to λT . The two vertical lines signify the equation of Wien’s law and the Eq. 3.65. The corresponding fraction of energy starting from the origin has been mentioned at the top axis.

3.5

Modeling of Radiative Equilibrium

In many practical situations, one may encounter situations of radiative equilibrium. Under the conditions of radiative equilibrium, there is no net gain or loss of heat from the participating medium, i.e. the divergence of radiative heat flux is zero at each control volume. The gas

50

3 Mathematical Formulation

Fig. 3.5 Variation of the universal blackbody distribution function with (λT ) product

adjusts itself to such a temperature that whatever heat is received by the gas volume is transferred to the neighboring control volume. Mathematically, this can be represented as  ∇ ·q = κ j (4πa j Ib − G j ) = 0 (3.69) j

where G j is the irradiation at the control volume node for the j th gray gas and Ib is the blackbody intensity at the local temperature. On further simplifying Eq. 3.69, one gets ⎛ ⎞    κ j ⎝4π Ib aj − G j⎠ = 0 (3.70) j

j

j

or, Ib = and,

 T =

G 4π

π Ib σ

(3.71) 1 4

(3.72)

The above expression can be used to calculate the unknown equilibrium temperature of the gas. In such cases, the temperature field is assumed first and the intensity field is generated from the same, after following the usual solution procedure. The converged intensity field is used to calculate the new temperature field and the procedure is repeated iteratively till the

References

51

difference between temperature fields in two successive iterations is negligible. The presence of radiative equilibrium makes the solution of radiative heat transfer problems complex and time-consuming.

3.6

Closure

This chapter has discussed the mathematical formulation and the solution procedure employed to solve the RTE with complexities such as spectral variation of properties of gases and particles, anisotropic scattering, and radiative equilibrium. An elaborate discussion on the phenomenon involved and the rationale behind choosing the particular approximation or method was also presented. The next chapter presents the results of the solution to the RTE in cylindrical geometries.

References 1. Rothman, L. S., Gamache, R. R., Tipping, R. H., Rinsland, C. P., Smith, M. A. H., Benner, D. C., Devi, V. M., Flaud, J.-M., Camy-Peyret, C., Perrin, A., et al. (1992). The HITRAN molecular database: editions of 1991 and 1992. Journal of Quantitative Spectroscopy and Radiative Transfer, 48(5–6), 469–507. 2. Rothman, L. S., Gordon, I. E., Barber, R. J., Dothe, H., Gamache, R. R., Goldman, A., Perevalov, V. I., Tashkun, S. A., & Tennyson, J. (2010). HITEMP, the high-temperature molecular spectroscopic database. Journal of Quantitative Spectroscopy and Radiative Transfer, 111(15), 2139– 2150. 3. Smith, T. F., Shen, Z. F., & Friedman, J. N. (1982). Evaluation of coefficients for the weighted sum of gray gases model. Journal of Heat Transfer, 104(4), 602–608. 4. Denison, M. K., & Webb, B. W. (1993). A spectral line-based weighted-sum-of-gray-gases model for arbitrary RTE solvers. Journal of Heat Transfer, 115(4), 1004–1012. 5. Denison, M., & Webb, B. W. (1993). An absorption-line blackbody distribution function for efficient calculation of total gas radiative transfer. Journal of Quantitative Spectroscopy and Radiative Transfer, 50(5), 499–510. 6. Pearson, J. T., Webb, B. W., Solovjov, V. P., & Ma, J. (2014). Efficient representation of the absorption line blackbody distribution function for H2 O, C O2 , and C O at variable temperature, mole fraction, and total pressure. Journal of Quantitative Spectroscopy and Radiative Transfer, 138, 82–96. 7. Solovjov, V. P., & Webb, B. W. (2000). SLW modeling of radiative transfer in multicomponent gas mixtures. Journal of Quantitative Spectroscopy and Radiative Transfer, 65(4), 655–672. 8. Solovjov, V. P., & Webb, B. W. (1998). Radiative transfer model parameters for carbon monoxide at high temperature. In: Heat Transfer 1998, Proceedings of the 11th IHTC, August 23–28, 1998 (Vol. 7, pp. 445–450). Kyongju, Korea. 9. Solovjov, V. P., Andre, F., Lemonnier, D., & Webb, B. W. (2017). The rank correlated slw model of gas radiation in non-uniform media. Journal of Quantitative Spectroscopy and Radiative Transfer, 197, 26–44.

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3 Mathematical Formulation

10. Coelho, P. J. (2002). Numerical simulation of radiative heat transfer from nongray gases in threedimensional enclosures. Journal of Quantitative Spectroscopy and Radiative Transfer, 74(3), 307–328. 11. Rosin, E., & Rammler, P. (1933). The laws governing the fineness of powdered coal. Journal of the Institute of Fuel, 7, 29–36. 12. Davison, B. (1957). Neutron transport theory. London: Oxford University Press. 13. Yu, M. J., Baek, S. W., & Park, J. H. (2000). An extension of the weighted sum of gray gases non-gray gas radiation model to a two phase mixture of non-gray gas with particles. International Journal of Heat and Mass Transfer, 43(10), 1699–1713. 14. Kim, M. Y., Yu, M. J., Cho, J. H., & Baek, S. W. (2008). Influence of particles on radiative base heating from the rocket exhaust plume. Journal of Spacecraft and Rockets, 45(3), 454–458. 15. Trivic, D. N. (2014). 3-D radiation modeling of nongray gases-particles mixture by two different numerical methods. International Journal of Heat and Mass Transfer, 70, 298–312. 16. Modest, M. F. (2013). Radiative heat transfer. New York: Academic Press. 17. Bohren, C. F., & Huffman, D. R. (2008). Absorption and scattering of light by small particles. New York: Wiley.

4

Radiative Heat Transfer in Cylindrical Geometries

4.1

Introduction

This chapter presents the development of the FVM formulation coupled with the SLW model for gas radiation and the Mie Scattering theory for particle radiation, which forms an integral part of the radiative heat transfer solver package in cylindrical coordinates. To keep it consistent and understandable with respect to the methods used, this formulation will be named FVM-SLW henceforth. The discretization of the governing equation is presented along with the detailed solution methodology. The in-house developed code is validated against available experimental and numerical results both for gray and non-gray media. The code is then applied to two important cases of practical interest, (i) a case of an industrial scale Delft furnace and (ii) a case of rocket base heating due to high-temperature exhaust. Real-time scenarios in terms of gas temperature profile, geometry, wall boundary conditions, gas and particle concentrations, etc., have been considered and the radiative heat flux at the walls has been estimated. Finally, a parametric study has been performed to assess the importance and contribution of the above-mentioned quantities to the radiative heat transfer.

4.2

Development of the FVM-SLW Method for a Cylindrical Geometry

The basic vectorized form of RTE written for a jth gray gas as per SLW model in cylindrical coordinates can be written as μ ∂ 1 ∂ ∂ [r I j ] − [ζ I j ] + η [I j ] = −(κg, j + κ p, j + σs, j )I j r ∂r r ∂φ ∂z  σs, j + κg, j Ib,g, j + κ p, j Ib, p, j + I j ( →  )d 4π =4π © The Author(s) 2023 R. Yadav et al., Radiative Heat Transfer in Participating Media, https://doi.org/10.1007/978-3-030-99045-9_4

(4.1) 53

54

4 Radiative Heat Transfer in Cylindrical Geometries

Fig. 4.1 A typical cylindrical enclosure and axisymmetric plane of calculation

z Plane of calculation Δz

r Δr

where μ, ζ, and η are the direction cosines in r, φ, and z directions, respectively. Consider a typical cylindrical geometry as shown in Fig. 4.1. For a finite plane bounded by r and z axes, let the total number of spatial discretizations along these axes be termed as Nr and Nz , respectively. Let the spherical domain around the point of interest also be divided into Nθ and Nφ number of discrete solid angles, as shown in Fig. 4.2a. Considering hemispherical symmetry, it is sufficient to solve the RTE in one hemisphere and then multiply the scalar results by a factor of 2 in order to obtain results for a complete sphere. Thus, only the hemispherical region will be considered bounded by a polar angle ranging from 0 to π/2 and azimuthal angle ranging from 0 to π. Let each of the discrete directions be represented by a unique value argument (m, n) which signifies the mth discretization in the polar and nth discretization in azimuthal angular coordinates. Equation 4.1 can now be rewritten for the direction of calculation m  n  as  

μm n ∂ 1 ∂ m n m n ∂  n  n  n   ]− I j ] + η [I m ] = −β j I m + Srm, jn [r I m [ζ j j j r ∂r r ∂φ ∂z

(4.2)

Where,  

Srm, jn = κg, j Ib,g, j + κ p, j Ib, p, j +

σs, j 4π



 

mn =4π

I j  j (mn → m n )dmn

(4.3)

The expression for the discrete solid angle for a direction mn is given by  mn =

φn+ φn−



θm+

θm−

sinθdθdφ

(4.4)

4.2

Development of the FVM-SLW Method for a Cylindrical Geometry

55

Fig. 4.2 A typical control angle and control volume in the direction and location of calculation respectively

As can be seen in Fig. 4.1, it is convenient to consider that the axisymmeteric solution exists at the plane of calculation. This helps to reduce the calculations for a three-dimensional geometry to that for a two-dimensional one considering no variation of the properties across the axis of the plane. Such an assumption helps in considerable savings in computational time and effort. In the recent past, several researchers have successfully applied the above idea to several flow and heat transfer problems in pipes, containers, nozzles, etc. Under the

56

4 Radiative Heat Transfer in Cylindrical Geometries

assumptions of axisymmetric conditions in the cylindrical geometry, the variation of the parameters along the azimuthal direction can be considered constant. Hence, a solution to Eq. 4.1 need to be found only in two directions, namely, r and z. However, as the radiative heat transfer involves the propagation of intensity in a straight line at any direction within the hypothetical sphere around the point of interest, the change in its direction due to the curvilinear path should be accounted for. This happens because the direction of the traveling photon keeps changing in the curvilinear coordinates even though it physically does not change the direction. This is called angular redistribution. In view of this, the intensity variation in the spatial azimuthal direction cannot be ignored even though the axisymmetric assumption is made. The derivative of radiative intensity with respect to the spatial azimuthal direction can be written as a finite difference approximation of the surrounding angular face intensities as   1   1 αm  n  + 1 I m n + 2 − αm  n  − 1 I m n − 2 ∂   2 2 (4.5) [ζ Iλ ] =   ∂φ m n where αm  n  + 1 and αm  n  − 1 are called the coefficients of angular redistribution term for the 2 2 direction m  n  which are nothing but the angular weights of the direction to account for the variation of intensity in the azimuthal direction. Following the conventional artifice of [1], the expression for angular redistribution is given as αm  n  − 1 − αm  n  + 1 = − 2

2

rp V



 

mn Ai Dci

(4.6)

i=e,w,n,n

where r p is the distance of the control volume node from the r axis. If all the control volume faces are orthogonal to the coordinate axes, the right-hand side of equation 4.6 is simply equal mn . The formulations of FVM as represented in chapter 3 for the Cartesian geometry, to Dcr can be extended to cylindrical geometries as well. Equation 4.2 is integrated over control   volume V and control angle m n to obtain 

 m  n





=

 

V m

(

μm n ∂ 1 ∂ m n m n  n ]− Ij ] [r I m [η j r ∂r r ∂φ   n     (−β j I m + Srm, jn )V m n j

 n

+ζ

∂ m n   [I j ])V m n ∂z (4.7)

V

On applying the Gauss divergence theorem and from the definitions of directional weights, this can be written as  

 

 

 

 

 

 

 

n mn mn mn mn mn mn mn Im j,e Dce Ae − I j,w Dcw Aw + I j,t Dct At − I j,b Dcb Ab  

 

 

 

n mn = −β p I m + (Sr , j )mP n V P m n j, p V P 

(4.8)

where, for the orthogonal grids, the directional weights for the r and z directions are given as

4.3

Solution Procedure

m n Dcr

=

m n Dcz

57

m n Dce

=

m n Dct

= =

m n −Dcw

m n −Dcb

 =

φn

φn

 =

+

−

φn

φn



θm θm

+

−



+

−

θm θm

sin 2 θcosφdθdφ

(4.9)

sinθcosθdθdφ

(4.10)

+

−

Following the step scheme of spatial marching given in 3.29, the generalized expression for the intensity at the control volume node P can be written as  

 

 

 

 

 

 

 

 

 

 

n mn mn mn mn mn mn mn mn mn am I j,S + (Sr , j )mP n P I j,P = a E I j,E + aW I j,W + a N I j,N + a S

(4.11)

where  

 

n mn am = max(−Ai Dci , 0) I

 

n am = P



 

max(Ai Dci , 0) + β j,P V m n +

i=e,w,n,s

(4.12)

V α   1 rP m n − 2

(4.13)

The source term in Eq. 4.11 is written as  

(Sr , j )mP n =ag, j (Tg )κg, j Ib,g + a p, j (T p )κ p, j Ib, p M N σs, j   mn I j mn→m  n  mn 4π

+ (4.14)

m=1 n=1

4.3

Solution Procedure

The step-by-step solution methodology for obtaining the solutions to the RTE is as follows: 1. Discretize the spatial domain into Nv control volumes and the angular domain into Nθ × Nφ directions. 2. Calculate discrete values of θ, φ, Dcr , and Dcz in all the directions. 3. Evaluate the gray gas weights for the walls keeping the wall temperature as the blackbody source temperature in Eq. 3.41. 4. Initialize the values of intensities from the wall depending upon the known temperature and emissivity of the wall. 5. Initialize the source term by evaluating the local values of the source term considering volumetric emission from the control volume and no scattering. 6. Starting from the walls with known boundary conditions solve for the nodal intensities using the Eq. 4.11. The functions or subroutines for evaluating the local absorption coef-

58

7.

8.

9.

10.

4 Radiative Heat Transfer in Cylindrical Geometries

ficient and gray gas weights for gases and modified scattering coefficient for the particles to be called upon in each control volume. Upon knowing the properties of the medium such as κg, j , κ p, j , and σs, j , solve Eq. 4.11 to obtain the local intensity in the direction m  n  . Use the step scheme (Eq. 3.29) to calculate the facial intensities from the nodal intensities. Decide the marching order in the spatial domain by the value of directional weights (Eqs. 4.9 and 4.10). The possible marching sequence when a solution is started from the top right corner of the domain is depicted in Fig. 4.3. Finally, obtain the values of intensity at all the nodal locations and faces in corresponding directions. After a sweep for all the locations satisfying the present criteria of directional weights is complete, other corners with bounding walls of known boundary conditions or updated after the first sweep is selected and the solution is obtained at all the locations. Similarly, obtain the solutions for other two sets of directional weights starting from the appropriate corner. This suggests that there will be four sweeps of the domain to cover all directions. Once the intensity values at each location and each direction is obtained, update the boundary conditions at the wall, and the source term following Eqs. 3.30, 3.31, and 4.14, respectively. In case an axis is encountered at the end of the sweep, update the axis boundary condition using the following equation:  

n Im = I mn j , j

at r = 0,

 

mn mn |Dcr | = |Dcr |

(4.15)

11. Repeat steps 6–10 until the intensity field is converged. Define a suitable convergence criterion to monitor the same. Here, a root mean square value is calculated as SS E =

Nv  MN 

 

 

n mn 2 (I m j,P,new − I j,P,old )

(4.16)

P=1 m  n  =1

or,

 RMSE =

SS E Nv × Nθ × Nφ

(4.17)

For the solution to stop, R M S E ≤ 1 × 10−6 . 12. Save the converged intensity field for the jth gray gas and repeat steps 6–11 for all the gray gases and finally calculate total intensity as  I = Ij (4.18) j

13. Following this, use the total intensity values to calculate the fluxes at the desired locations on the wall using Eqs. 3.32 and 3.33.

4.4 Validation of the FVM-SLW Method for the Cylindrical Geometry

z

59

z

r

r

(b). 2nd sweep, for directions:

(a). 1st sweep, for directions: ′ ′

< 0,

′ ′

′ ′

0,

′ ′

0

r (d). 4th sweep, for directions: ′ ′

> 0,

′ ′

>0

Fig. 4.3 Marching order in the spatial domain based on the values of directional weights

4.4

Validation of the FVM-SLW Method for the Cylindrical Geometry

The in-house code based on the general formulations provided in Chap. 3 is now validated with different benchmark cases in the literature for a cylindrical geometry. In order to obtain confidence in the solution methodology before applying it to problems of practical interest, validation has been done with both experimental and numerical benchmark cases.

60

4.4.1

4 Radiative Heat Transfer in Cylindrical Geometries

Validation with Experimental Results

For, the purpose of validation of the FVM-SLW formulation, the classic benchmark case of a cylindrical furnace is considered. Measured temperature and heat flux data for this are available from literature [2] and these have been extensively used by researchers to validate their radiative heat transfer codes, for example, [3] solved it using S8 DOM. A Delft furnace with radius, r = 0.45 m in axial length z = 5 m with hot combustion gas inside, and the measured temperatures distribution as given in [2] is considered. All the walls have a temperature of 425 K and an emissivity of 0.8. The furnace has blackbody inlet and outlet temperatures kept at 425 K and 300 K, respectively. The medium is considered to be gray and represented by a constant absorption coefficient of 0.3 m−1 . A spatial grid of Nr × Nz = 3 × 17 and angular grid of Nθ × Nφ = 10 × 8 are selected to solve the problem. The values of the incident radiative heat flux are plotted against the location at the side wall and are shown in Fig. 4.4. From the figure, it is seen that FVM-SLW code agrees well with the experimental data with a little deviation seen far away from the inlet. This may be due to the assumptions like zero scattering and constant absorption coefficient.

Fig. 4.4 Validation of methodology for an axisymmetric furnace with a gray participating medium— comparison of local radiative heat flux calculated at the furnace wall with experimental data

4.4 Validation of the FVM-SLW Method for the Cylindrical Geometry Fig. 4.5 A typical cylindrical furnace geometry

61

r Furnace wall Qradiation

Hot gases and particles z

Qradiation

4.4.2

Validation for a Non-Gray Gas-Particle Mixture

The FVM-SLW formulation is next tested against a non-gray participating medium with soot particles inside the enclosure (Fig. 4.5). A cylindrical enclosure with diameter and height 0.6 m and 1.2 m, respectively, has been taken [4]. This case is quite general as it includes the non-gray gas mixture with a non-black wall and also particles. The temperature of the medium varies radially as well as axially. The side wall at r = R has an emissivity of 0.8, while all the other walls are black. The gas mixture contains 20% H2 O and 10% C O2 (mole fraction basis), respectively. The temperature in the medium is given by the following expression:    z r T (z, r ) = 800 + 1200 1 − (4.19) R Z ax where Z ax is the total axial length of enclosure. The enclosure walls are at 800 K, except the wall at z = Z ax , which is at 300 K. The soot volume fraction is f v = 10−6 . In the context of the SLW model and under the limitation of full spectrum band models, only gray soot has been considered here. The soot is assumed to be generated from the combustion of propane-air as fuel. The corresponding properties are taken from [6] (see Appendix). The net radiation heat flux has been determined on the enclosure wall (at r = R) by using a coarse (8 × 6) and a fine (20 × 16) angular discretization in the FVM [5]. The results are compared with the benchmark results and are presented in Fig. 4.6. From the figure, it is clear that the results obtained using the present FVM-SLW formulation are in good agreement with benchmark results for both coarse and fine angular grids. The maximum errors in radiation heat flux are found to be less than 10% and these may be due to the assumption of gray soot unlike in the benchmarks where a non-gray formulation for the soot was adopted [4].

62

4 Radiative Heat Transfer in Cylindrical Geometries

Fig. 4.6 Validation of methodology for an axisymmetric furnace with a non-gray participating medium—comparison of net radiation heat flux along the side wall calculated with different solution methods

4.4.3

Validation for Anisotropic Scattering

The transport approximation invoked in the present formulation in order to monitor the anisotropic behavior of the particle must be tested against the other representations of anisotropic phase function. Exact solutions do not exist for such cases and the only way to compare a model is to assess its closeness in prediction when compared to other numerical models. However, wherever possible a superior model must be chosen for the comparison. In the present case, a Legendre polynomial expansion of the phase function in order to account for anisotropic scattering has been selected. A plume geometry as shown in Fig. 4.7, with particles that cause anisotropic scattering is considered, similar to the case solved by [7]. The phenomenon-of both plume emission and searchlight emission has been considered. The searchlight emission corresponds to the emission from the hot gases and particles inside the rocket nozzle that gets scattered back to the rocket base due to the presence of particles in the plume. Hence, it is apparent that the radiative heating to the base is sensitive to the particles present in the plume. The effects of the anisotropic nature of particles on the base wall heating have been studied by keeping the plume inlet as hot as reference temperature Tr , while the plume body is assumed to be cold and isothermal, in order to clearly bring out

4.4 Validation of the FVM-SLW Method for the Cylindrical Geometry Fig. 4.7 A typical plume geometry

63

Base wall

r

Qradiation

Rexit

Plume

Hot gases and particles Zplume Qradiation

the effect of particle scattering. The other medium properties are defined in terms of optical depth τ0 = β Rexit = 0.5 and the single scattering albedo ω = 0.9. The non-dimensional radiative heat flux at the base wall of the rocket, due to different levels of anisotropic scattering such as forward and backward scattering governed by the value of the asymmetry parameter has been plotted in Fig. 4.8 and is compared with the solution obtained with the Legendre polynomial expansion as reported in [7]. It is seen that the transport approximation provides good agreement with literature and hence can be used as a reliable tool to model the anisotropic scattering in such multidimensional radiative heat transfer problems.

4.4.4

Decision on the Number of Gray Gases

Before applying the FVM-SLW code to the cases of industrial relevance, a decision on a minimum number of gray gas discretization must be made. This is to ensure that the results obtained are invariant with respect to the number of gray gases considered in the SLW model. This number may depend upon the geometry and the operating conditions and so has to be determined separately for particular cases being investigated. For this purpose, a plume geometry, as shown in Fig. 4.7, is considered. The plume is assumed to be isothermal (at Tr e f ), non-scattering with an absorbing and emitting hot mixture of H2 O, C O2 and C O. The gas mole fractions are chosen to mimic values that are normally encountered in solid rocket motor exhausts. The case is solved with no particles in order to assess the efficacy of gas radiation modeling. The estimated heat flux at the base wall is non-dimensionalized as qest /(σTr4e f ) and this value has been plotted against the non-dimensional location (r /Rexit ) at the base wall and the corresponding results are shown in Fig. 4.9. It can be seen that when

64

4 Radiative Heat Transfer in Cylindrical Geometries

Fig. 4.8 Comparison of transport approximation in the case of forward and backward scattering in an isothermal homogeneous plume with τ0 = 0.5 and ω = 0.9

more than five gray gases are considered in the SLW model, there is no significant change in the results. In the present study, seven gray gases have been chosen in the SLW model for cylindrical geometries.

4.5

Application to an Industrial Scale Delft Furnace

The FVM-SLW code developed has then been applied to a practical case of a narrow cylinder Delft furnace ([2]) with a non-swirling gas flame. The shape of the furnace corresponds to the one shown in Fig. 4.5. The diameter of the furnace is 3 m and the axial length is 5 m. The furnace wall has an emissivity of 0.8 and is maintained at a temperature of 400 K by the circulation of a cooling fluid. All the other walls are black. The temperature of the other bounding wall is 400 K.Following the experimentally measured temperatures of [2], a fourth-order temperature profile has been generated to model the axial temperature variation. This turns out to be T (z) = −7.382z 4 + 88.63z 3 − 354.2z 2 + 394.5z + 1486

(4.20)

4.5

Application to an Industrial Scale Delft Furnace

1

2

3

4

65

5

6

7

8

0.06

0.06 3 gray gases 5 gray gases 7 gray gases 10 gray gaes 20 gray gases

Non dimensional radiation heat flux, qest /σT4

0.05

0.05

0.04

0.04

0.03

0.03

YH2O = 0.11 YCO2 = 0.015 YCO = 0.23

0.02

0.02

0.01

0.01

0

0 1

2

3

4

5

6

7

8

r/Rexit Fig. 4.9 Effect of number of gray gases on the non-dimensional heat flux at the base wall of solid rocket motor from an isothermal non-scattering plume with τ0 = 0.5

The temperature profile and level of temperatures can also be assessed from Fig. 4.10. The following two cases have been considered for investigation:

4.5.1

Effect of Gas Concentration

A total of three participating gases that are commonly encountered in the gas flames, viz., H2 O, C O2 , and C O have been considered. The baseline values for the mole fractions are taken to be Y H2 O = 0.2, YC O2 = 0.12, and YC O = 0.08. The effect of doubling any one gas specie concentration while keeping the other two fixed has been studied. This one helps in analyzing the effect of a particular gas on the radiation heating of the wall. The incident radiation heat flux values estimated at the furnace wall are plotted against the wall location and shown in Fig. 4.11. From the figure, it is observed that doubling the concentration of C O and C O2 has little effect on the radiation heat fluxes as compared to H2 O. There is a 27.88% increase in the flux values on doubling H2 O concentration, while only 5.6 and 0.06% increase is seen when C O2 and C O concentrations are doubled, respectively.

66

4 Radiative Heat Transfer in Cylindrical Geometries

0

1

2

3

4

5

1,700

1,700 Furnace medium Temperature

1,600

1,500

1,500

Temperature in K

1,600

1,400

1,400

1,300

1,300

1,200

1,200

1,100

1,100 1,000

1,000 0

1

2

3

4

5

Axial distance, z in m

Fig. 4.10 A fourth-order polynomial fit to account for the axial variation of temperature inside the furnace

Larger absorption region of H2 O in the complete spectrum provides high emission and consequently more energy reaches the furnace wall. Outside of this, it is also observed that the peak of the heat flux occurs near the zones of maximum temperature in the furnace as expected.

4.5.2

Effect of Particle Concentration

In order to assess the effect of particle concentration on radiative heating of the furnace wall, four separate bins of alumina particles of sizes 50, 60, 80, and 100 µm are considered with a uniform distribution. The complex refractive index of alumina is taken from [8] (see also Appendix A) and the concentration of particles is considered to vary from 0.1 to 0.3 kg/m3 . For each bin, the representative particle properties such as absorption coefficient and scattering coefficient are determined using the Mie scattering theory. The gas concentration in the medium remains similar to the baseline case as mentioned in the previous section. The temperature profile is considered to take a form similar to Eq. 4.20. The change in the value of radiative heat flux at the furnace wall has been estimated and plotted against the

4.5

Application to an Industrial Scale Delft Furnace

0

1

2

67

3

4

5 140

Radiation heat flux at furnace wall, kW/m 2

140 a,b,c a,b,2c a,2b,c 2a,b,c

a = YH2O b = YCO2 c = YCO

120

120

100

100

80

80

60

60

40

40

20

20 0

0 0

1

2

3

4

5

Axial distance, z in m Fig. 4.11 Effect of concentration of the gas species on the radiative heating of the furnace wall. The temperature of the gas is axially varying. (Baseline values: a = 0.2, b = 0.12, c = 0.08)

location at the furnace wall and is shown in Fig. 4.12. From the figure, it can be seen that in the presence of particles, the pronounced effect of gas radiation as seen in the previous case diminishes substantially. Flux values increase by 1.34 times when the particle concentration is 0.1 kg/m3 and 1.8 times with 0.3 kg/m3 with respect to the no particles case. Moreover, as compared to the previous case, there is only a 16% increase in the flux values on doubling H2 O concentration with 0.1 kg/m3 particle loading which further drops to 8.45% with 0.3 kg/m3 loading. This happens due to a large number of particles present in the domain tending to scatter more and hence more and more radiation is directed towards the wall. Thus, as the particle loading increases, the individual contribution of gases towards the wall heating reduces and the radiation heating is mainly due to particles. The present formulation helps to adequately quantify these effects and this in turn can serve as a critical input for furnace designers.

68

4 Radiative Heat Transfer in Cylindrical Geometries

0.3 kg/m3 0.1 kg/m3

Fig. 4.12 Effect of concentration of the particles on the radiative heating of the furnace wall. Temperature of the gas is axially varying. (Baseline values: a = 0.2, b = 0.12, c = 0.08)

4.6

Application to a Rocket Plume Base Heating Problem

The exhaust plumes of solid rocket motors contain hot gases and particles in significant proportions. Correct information of fluxes at the base wall is required to design the various stage of these rocket motors. The temperatures in the exhaust of the rocket are around 2500 K and so radiation is the dominant mode of heat transfer. In order to accurately predict the heating of the base wall, an accurate estimation of the gas and particle radiative properties within the high-temperature gradient fields must be made. In general, several ground tests are performed in order to measure the heating levels due to rocket exhaust plumes at the offset locations on the rocket base. However, for performing parametric studies on the effect of gas and particle properties in such problems, a generic numerical model is required. The present FVM-SLW formulation provides an efficient means for the calculation of radiation heating in such cases. Outside of this, the FVM-SLW model can also serve as a radiation subroutine and can be incorporated with flow solvers. In most cases of rocket base heating,

4.6

Application to a Rocket Plume Base Heating Problem

69

Table 4.1 Coefficients used in Eq. 4.21 a

b

c

1

1616

23.3

5.778

2

2178

39.06

11.96

3

2064

9.719

11.1

4

4097

–20.64

18.12

the radiation part has to be solved simultaneously with the flow simulations, where the divergence of radiation heat flux term needs to be supplied in the energy equation. The present formulation is applied to a basic rocket plume geometry as shown in Fig. 4.7. The radius of the plume exit is 1.5 m and the axial length is 25 m, consistent with the first level firing of solid propellant rockets. The complete computational domain is taken as 12 m in the radial and 25 m in axial direction to cover the complete base wall. To account for the real-time temperature variation a four-term Gaussian profile that varies axially in the plume is considered. Considering the plume to be a narrow cylinder, the radial variation of temperature is not taken into account. The expression for the temperature variation is provided as T (z) = a1 e

(

z−b1 2 c1 )

+ a2 e

(

z−b2 2 c2 )

+ a3 e

(

z−b3 2 c3 )

+ a4 e

(

z−b4 2 c4 )

(4.21)

The corresponding coefficients appearing in above equation are presented in Table 4.1 and the variation of temperature with axial distance is shown in Fig. 4.13. The temperature outside the plume domain is considered to be at the ambient and the plume interface is treated as transparent. The following two effects are studied (i) Gas concentration and (ii) Particle concentration.

4.6.1

Effect of Gas Concentration

For studying the effect of gas concentration in the present case, a gas mixture of H2 O, C O2 , and C O has been considered. The baseline proportions of the gases are close to those ones that is commonly encountered in this application, namely, a = Y H2 O = 0.12, b = YC O2 = 0.01, and c = YC O = 0.23. In the solid rocket motor exhaust, the presence of particles is inevitable, due to the burning of solid propellant in the combustion chamber. Some unburnt particles in varied dimensions are always present in the exhausts. Following this, four separate bins of particles with diameters 2, 4, 8, and 12 µm with uniform distribution have been considered. The mass fraction of the particles is 25% with a total particle loading 0.1 kg/m3 , which is a moderate value for particle loading in this application. Under these conditions, the individual gas concentration is halved from the baseline value and its effect

70

4 Radiative Heat Transfer in Cylindrical Geometries

Fig. 4.13 A four-term Gaussian profile to account for the axial variation of temperature inside the plume

on the base heating is monitored. The estimated fluxes have been non-dimensionalized with respect to the blackbody emission corresponding to the mean temperature in the domain (Eq. 3.66). These values have been plotted against the non-dimensionalized location at the base wall and are presented in Fig. 4.14. From the figure, it can be observed that even if the particle loading is taken to be moderate in solid rocket exhaust conditions there is negligible or no effect of changing the gas concentration on the rocket base heating. This clearly implies that the major part of the radiation comes from the particles in the present case.

4.6.2

Effect of Particle Concentration

Considering the conditions similar to those in the above section, but keeping the gas concentration as constant at the baseline values viz. Y H2 O = 0.12, YC O2 = 0.01, YC O = 0.23, the effect of changing the particle concentration on the base heating has been examined. This has been accomplished by taking a very low (0.01 kg/m3 ), a moderate (0.1 kg/m3 ) and a high value (0.3 kg/m3 ) of particle concentration. The non-dimensional radiative heat flux is plotted against the location at the base wall and is shown in Fig. 4.15. The strong effect of

4.7

Conclusions

71

1

2

3

4

5

6

7

8

Non dimensional RHF at base wall, qest/σTm4

9 0.6

0.6 a,b,c a,b,c/2 a,b/2,c a/2,b,c

a = YH2O b = YCO2 c = YCO

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1

2

3

4

5 r/ R exit

6

7

8

0 9

Fig. 4.14 Effect of concentration of the gas on the radiative heating of the base wall. Temperature of the plume is axially varying. (Baseline values: a = 0.12, b = 0.01, c = 0.23)

the concentration of alumina particles on the radiative heating of the rocket base can be seen. The maximum heat fluxes are almost 50% higher than the low particle loading case, and almost 20% higher than the moderate loading case. Thus, an accurate estimation of particle properties should be the primary focus of such problems. Moreover, the exact information about the particle loading is also necessary which usually comes from a detailed numerical modeling of highly turbulent two-phase flow of the gases and particles in the exhausts. In view of the results obtained from the present FVM-SLW formulation, it is clear that for problems involving rocket base heating through solid propellant exhausts, a simplified gas radiation model like the weighted sum of gray gases method can be employed rather than the complex SLW or similar spectral database models in order to save computational resources without a significant loss of accuracy.

4.7

Conclusions

The formulation of FVM-SLW and a step-by-step solution procedure for a two-dimensional axisymmetric geometry have been presented. The formulation was validated against the var-

72

4 Radiative Heat Transfer in Cylindrical Geometries

Fig.4.15 Effect of concentration of the particles on the radiative heating of the base wall. Temperature of the plume varies axially

ious experimental and numerical benchmark cases in the literature. Using this formulation, radiative heat transfer analysis was performed on two practical cases of interest, namely (i) a narrow cylinder Delft furnace and (ii) a rocket exhaust plume. Results obtained for the case of Delft furnace show a strong contribution of H2 O gas on the radiative heating of the furnace wall (about 27.9% increase on doubling the concentration from the baseline case) when compared to other gases. In the case of rocket exhaust heating, the variation of concentration of gases seems to have a negligible effect on the base heating. Particle radiation is observed to be dominant in this application, where the maximum particle loading gives three times more heating than the no particles case. For both problems, the presence of particles is seen to diminish the gas effects. Even so, gas still has a substantial share in the total heat transfer.

References

4.8

73

Closure

In this chapter, the development and application of the FVM-SLW approach in cylindrical coordinates was discussed. The generic formulation was applied to the practical cases of interest like a narrow wall non-swirling flame furnace and the exhaust plume of solid rocket motor. The practical conditions prevailing in these cases were considered such as temperature, mole fractions, and wall boundary conditions. The assessment of the effect of gas and particle concentrations was performed thereafter. In the next chapter, the results of studies on modification of the FVM-SLW to incorporate complex geometries with curved inclined walls and the relevant practical cases of interest in the same will be presented.1

References 1. Carlson, B. G., & Lathrop, K. D. (1968). Transport Theory-The Method of Discrete-Ordinates in Computing Methods in Reactor Physics, H. Greenspan, C. Kelber, & D. Okrent (Eds.). Gordon and Breach, New York. 2. Wu, H. L., & Fricker, N. (1971). An investigation of the behaviour swirling jet flames in a narrow cylindrical furnace. In 2nd members conference Industrial Flame Research Foundation Ijmuiden. 3. Jamaluddin, A. S., & Smith, P. J. (1988). Predicting radiative transfer in rectangular enclosures using the discrete ordinates method. Combustion Science and Technology, 59(4–6), 321–340. 4. Perez, P., El Hafi, M., Coelho, P. J., & Fournier, R. (2005). Accurate solutions for radiative heat transfer in two-dimensional axisymmetric enclosures with gas radiation and reflective surfaces. Numerical Heat Transfer Part B-Fundamentals, 47, 39–63. 5. Baek, S. W., & Kim, M. Y. (1997). Modification of the discrete-ordinates method in an axisymmetric cylindrical geometry. Numerical Heat Transfer, 31(3), 313–326. 6. Modest, M. F. (2013). Radiative heat transfer. New York: Academic. 7. Baek, S. W., & Kim, M. Y. (1997). Analysis of radiative heating of a rocket plume base with the finite-volume method. International Journal of Heat and Mass Transfer, 40(7), 1501–1508. 8. Dombrovsky, L. A. (1996). Radiation heat transfer in disperse systems. New York: Begell House.

1 Contents of this chapter have been published in Rahul Yadav, C. Balaji and S. P. Venkate-

shan, Implementation of SLW model in the radiative heat transfer problems with particles and hightemperature gradients, International Journal of Numerical Methods for Heat and Fluid Flow, 27-5 (2017), pp. 1128-1141. (doi: https://doi.org/10.1108/HFF-03-2016-0095).

5

Radiative Heat Transfer in Conical Geometries

5.1

Introduction

This chapter presents the extension of the finite volume method to generic body-fitted axisymmetric geometries to model the radiative heat transfer from hot participating media in some of the important industrial applications such as a conical nozzle, combustors, or diffusers. First, the modifications in the FVM-SLW to model such a case are presented, followed by their validation. Further, a practical case of conical diffuser with a realistic temperature profile and boundary conditions has been taken up and the results of a parametric study thereof are presented.

5.2

FVM-SLW Formulations for Body-Fitted Conical Geometries

Radiative heat transfer problem in conical enclosures can be observed in applications such as combustors, gas turbine diffusers, and rocket nozzles. These usually involve a conical enclosure where the hot combustion gas is expanded or compressed depending upon the requirement. In these geometries, the boundaries often do not align with the coordinate axes, and in view of this, some special treatment while performing radiation heat transfer calculations is required. Several methods are available to handle complex geometries in radiation modeling. However, when only one or a few of the boundaries are misaligned, a body-fitted meshing approach is efficient. In this approach, the spatial discretization of the domain follows the topology of the misaligned boundary, and the control parameters such as volume and surface area are calculated accordingly. Figure 5.1 shows a conical diffusertype geometry, with the axisymmetric plane, the properties over which are assumed to be constant over the azimuthal (φ) direction.

© The Author(s) 2023 R. Yadav et al., Radiative Heat Transfer in Participating Media, https://doi.org/10.1007/978-3-030-99045-9_5

75

76

5 Radiative Heat Transfer in Conical Geometries

Plane of calculation r

Zax

z

Fig. 5.1 A typical conical geometry with body-fitted mesh

The finite volume formulation for geometries of this type remains similar to the one discussed in the previous chapter except for a few differences. The normal vector to the inclined face is no more aligned with the base vectors in these geometries. In view of this, two components of the normal vector will appear. Let the unit normal to the face be denoted by nˆi and its vertical and components be denoted by n i,r and n i,z , respectively. Consider a control volume as shown in Fig. 5.2. The expression for the unit normal at the inclined face is given as nˆi = n i,r iˆ + n i,z kˆ (5.1) In terms of the dimensions of the control volume with respect to the coordinate axes, these components can be written as z n i,r =  2 (z) + (r2 − r1 )2 r2 − r1 n i,z =  (z)2 + (r2 − r1 )2

(5.2)

Recalling Eqs. 4.9 and 4.10 from Chapter 4, it is evident that an appropriate evaluation of the directional weights in the finite volume method is dependent on the unit outward normal to the face. The sign of the directional weights is critical in deciding the marching order. When the control volume faces are orthogonally aligned with the coordinate axes, every combination of the r and z directional weights (or more specifically every quadrant)

5.2

FVM-SLW Formulations for Body-Fitted Conical Geometries

77

Fig. 5.2 A typical control volume of geometry under consideration

has an equal number of directions. This is not the case with complex geometries. For the present conical geometry under consideration, it can be seen from Fig. 5.2 that due to the inclined wall, at some locations the directions may not participate in the radiation from the control volume as they do not enter through the inclined face. This, in fact, may reduce the number of directions for a particular set of directional weights. Similarly, on the other hand, few directions which were not contributing previously may start entering into the control volume due to skewness of the boundary face, thus increasing the number of directions for that set of directional weights. In any case, for the case of the inclined wall geometry, the evaluation of directional weights must be made carefully. For understanding this, the appropriate expression for the evaluation of directional weights must be given. For example, for a face i, for a given direction sˆ , the directional weights are given as   Dci = (sˆ · nˆi )d (5.3) 

where following the Cartesian base vectors as discussed in precious chapters, the expression for sˆ for a direction in the axisymmetric plane (r,z) is given as sˆ = (sinθ cosφ)iˆ + (cosθ )kˆ

(5.4)

 becomes Substituting for sˆ in the Eq. 5.3 and using Eqs. 5.2 and 5.1, the expression for Dci  z  Dci = ( (sinθ cosφ)+ 2   (z) + (r2 − r1 )2 (5.5) r2 − r1   (cosθ ))d (z)2 + (r2 − r1 )2

78

5 Radiative Heat Transfer in Conical Geometries

For a discretized set of Nθ × Nφ directions, the individual directional weights can be evaluated as  φ n +  θ m  + z m n Dci = ( (sin 2 θ cosφ)   2 φn − θm − (z) + (r2 − r1 )2 (5.6) r2 − r1 + (cosθ ))dθ dφ (z)2 + (r2 − r1 )2 The set of directional weights for the inclined faces of the control volume can be calculated from the above expression and the marching order can be determined. Additionally, it can m  n  , the evaluation of the be observed that, due to a change in the process of evaluation of Dci angular redistribution term depends upon the location now. The angular redistribution term is now given by rp  m n αm  n  +1/2 − αm  n  −1/2 = Ai Dci (5.7) V i

where r p is the centroid of the control volume. The expressions for control volume and the facial area of the inclined surface can be calculated from solid geometry as π z 2 (r1 + r22 + r1r2 ) 3  Ai = π(r1 + r2 ) (z)2 + (r2 − r1 )2 Vi =

(5.8)

In orthogonal geometries, as the faces align with base vector directions, the discretized control angle is hosted by a single direction everywhere in the domain, but this is not the case when inclined walls are encountered. As can be seen from Fig. 5.2, there may appear some control angle overlap at the inclined faces. The treatment of this control angle overlap has been addressed by many researchers. In this study, a bold approximation [1] is employed, where the decision on the directions to be considered as incoming or outgoing from the control volume is determined by the following expression:  

 

 

 

 

 

mn mn mn Iim n Dci = I Pm n max(Dci , 0) − I Im n max(−Dci , 0)

(5.9)

The above equation represents that at the face of a control volume, a ray will be treated as totally incoming, even though some part of its control angle lies outside of the control volume [1]. The usual step scheme of assignment of nodal intensities to the facial intensities can be replaced by this expression and the usual solution procedure can be followed. Note that this does not produce any effect on the actual governing equation (4.11) and is merely a tool to take care of the contribution of a certain direction in the control volume. The solution procedure for RTE and the angular discretization remain similar to the one explained in Sect. 4.3.

5.4

Application to a Conical Diffuser

5.3

79

Validation

Before employing the formulation for an industrial problem, the methodology explained above has been first validated against benchmark cases available in the literature. A case corresponding to an absorbing, emitting, and scattering medium in a conical enclosure has been considered.

5.3.1

Validation with Absorbing Emitting and Scattering Medium

The code developed has already been tested for the axisymmetric geometry in the previous chapters under generalized conditions and hence only the additional formulation of the bodyfitted geometries has been validated here. The FVM-SLW code is applied to the benchmark case of a truncated cone enclosure with an absorbing and emitting medium [2]. The short radius of the cone is 1 m and the axial length (Z ax ) is 2 m. The side wall is inclined at an angle of 30◦ with respect to the horizontal. The total extinction coefficient of the medium is taken as β = κ + σs = 1 m−1 . The medium is cold with the scattering albedo (ω) varying from 0 to 1. The bottom wall is heated and is at a temperature Tr e f = 100 K. The nondimensional radiative heat flux (q R /σ Tr4e f ) has been calculated and plotted against the wall location. Figure 5.3 shows the results for the non-dimensional heat flux at the side wall for various values of the scattering albedo. It is seen from the figure that the radiation heating to the wall decreases towards the larger end of the enclosure. Additionally, as the albedo (ω) is increased, more radiation reaches the wall which happens because of increased scattering. The figure confirms a good agreement between the results of the present study with those of reference [2].

5.4

Application to a Conical Diffuser

A common type of conical geometry used in the industrial and space applications is shown in Fig. 5.4. This geometry finds its typical use in rocket nozzles, where hot combustion gases from the throat are expanded and released into the ambient as a plume. For designers of such a furnace or nozzle, information about the heating of the enclosure wall is of primary importance. An accurate estimation of the fluxes at the walls or the vicinity of the fired hot medium is essential for the design of thermal protection systems. The only wall in the geometry is the inclined bounding wall. The aim is to estimate the radiative heat load to the bounding wall due to a mixture of hot gases and particles in the medium. The inlet and outlet sections are non-emitting but reflecting surfaces. In order to make the problem relevant to actual applications, a radially and axially varying synthetic temperature field, as given by Eq. 5.10, is assumed. The corresponding coefficients appearing in these equations are provided in Table 5.1. For a better understanding of the

80

5 Radiative Heat Transfer in Conical Geometries

Fig. 5.3 Validation of methodology for a conical diffuser with gray participating medium— comparison of local, dimensionless radiative heat flux across the side wall obtained in the present study with those of reference [2]. The results of the present study and those of reference [2] are shown for different values of ω ranging from 0 to 1 Fig. 5.4 A typical conical enclosure under consideration

r

Qradiation Cone angle

r2

Hot gases and particles

r1

z Zax

5.4

Application to a Conical Diffuser

81

Table 5.1 Coefficients of the temperature profile Coeffs

1

2

3

4

5

p

–9.408

111.8

–447.9

543.8

1415

C

0.1111

–0.352

1.056

–

–

level of temperatures achieved within the enclosure using this profile is also shown in the figure as a contour plot (Fig. 5.5).       r 2 r + C3 T (r , z) = Tmax C1 + C2 (5.10) Rz Rz In Eq. 5.10, Rz is the radius of the diffuser at any z location. Tmax is the maximum temperature in the domain which occurs at the axis, the variation of Tmax with respect to the axial distance is represented by a fourth-order polynomial, which is given as Tmax = p1 z 4 + p2 z 3 + p3 z 2 + p4 z + p5

(5.11)

This type of variation in temperatures is commonly seen in applications involving conical diffusers of the gas turbine engines equipped with afterburners and in the rocket nozzles employed in high-altitude operations, where the hot combustion gases from the combustor enter the nozzle and are expanded to provide the necessary thrust. The latter application is also dominated by the presence of alumina particles of various sizes. Knowledge of radiative heating due to individual gases and particles under these conditions is of primary interest to rocket designers. For a detailed parametric study, the conical enclosure given in Fig. 5.4 has been considered. The short radius of the cone is 2 m and the axial length is 4 m and inclination of the side wall with respect to the horizontal is 20◦ . As the emphasis is on the quantification of the radiative heat load from the medium to the wall, the bounding wall is assumed to be cold. However, an analysis of the effect of wall emission has been made. The emissivity of the wall is set as 0.8 for the baseline case. The medium is assumed to comprise three gases, viz., H2 O, C O2 , and C O with varying alumina concentration. In what follows, the results of a study on the effect of (i) number of gray gases considered, (ii) gas concentration, (iii) particle concentration, (iv) anisotropic scattering, and (iv) wall emission on the radiative heating of the bounding wall are presented.

5.4.1

Decision on the Minimum Number of Gray Gases

Similar to the case discussed in the previous chapter, a decision on the minimum number of gray gases required to achieve the solution that remains invariant on further increasing the number of gray gases in the SLW model needs to be made first. For this purpose, the baseline

82

5 Radiative Heat Transfer in Conical Geometries

Fig. 5.5 Radially and axially varying temperature profiles corresponding to Eqs. 5.10 and 5.11

case is Y H 2O = a = 0.2, YC O2 = b = 0.12, and YC O = c = 0.08. In order to clearly bring out the effect of gas radiation modeling the particle concentration is kept as low as 0.0001 kg/m3 . Five particle bins of diameter 50, 60, 70, 80, and 100 µm with uniform distribution have been considered. The temperature-dependent complex index of refraction has been calculated using the procedure given in [3] (see Appendix A). A spatial discretization of 12 × 24 is chosen and an angular discretization of 8 × 6 is considered in the FVM. Further refinement in the grids did not have a significant effect on the solutions. Variation of the incident radiation heat flux plotted along with the wall location for different numbers of gray gases in the SLW model is shown in Fig. 5.6. One can see that the solutions are invariant beyond 5–7 gray gases. In view of this, a total of five gray gases have been considered for all the cases discussed hereafter.

5.4.2

Effect of Gas Concentration

For studying the effect of gas concentration, the baseline values as mentioned in the above section have been chosen. The particle loading is set to 0.0001 kg/m3 . In order to assess the contribution of individual gas species into the radiative heating, the value of the gas

5.4

Application to a Conical Diffuser

83

Fig. 5.6 Sensitivity of axial variation of local radiative heat flux with the number of gray gases to be considered in the SLW model

concentration is doubled for one gas at a time keeping those of others fixed. The variation of radiation heat flux along the wall is shown in Fig. 5.7. The flux values are seen to be maximum at the zones of maximum temperature. Additionally, C O2 and C O concentration seem to have little effect on the radiative heating when compared to the concentration of H2 O. The radiation heating is mainly dominated by H2 O. The maximum flux value increases by about 19.12% when the concentration of H2 O is doubled from its baseline value. This is mainly because of the fact that H2 O has many participating bands in the full spectral range, as compared to C O2 and C O, at high temperatures and thereby emits more radiation.

5.4.3

Effect of Particle Concentration

The conical diffuser geometry, as shown in Fig. 5.4, is now studied to assess the effect of the particle concentration on radiative heating at the bounding wall. For this purpose, the particle concentration is chosen to a low (0.01), moderate (0.1), and a high (0.3) value (all in kg/m3 ), respectively. The particles are distributed uniformly in the bins of diameters 50, 60, 70, 80, and 100 µm. The gas concentrations are varied in a manner similar to what was done in the previous subsection for each value of the particle concentration. The corresponding changes in the values of incident heat flux at the wall are shown in Fig. 5.8. From the figure,

84

5 Radiative Heat Transfer in Conical Geometries

Fig. 5.7 Effect of concentration of the participating gases on the axial variation of the radiative heat flux at the curved wall. Baseline values: Y H 2O = a = 0.2, YC O2 = b = 0.12, and YC O = c = 0.08

it can be observed that the effect of gas concentration seems to diminish as the particle concentration is increased. When the particle concentration takes a high value of 0.3 kg/m3 , very little change in the fluxes is observed even on doubling the gas concentration. The increase in maximum flux values on doubling the H2 O concentration which was 19.12% for the previous case with no particles reduces to 10.7% with 0.1 kg/m3 particle loading and further to 4.73% with 0.3 kg/m3 particle loading. This demonstrates the importance of an accurate estimation of particle radiation for correctly predicting radiative heat transfer in such applications.

5.4.4

Effect of Cone Angle

The effect of cone angle on the radiative heating to the curved surface in the conical diffuser geometry with radially and axially varying temperature field has been examined. Three values of the cone angles, namely, 0◦ , 10◦ , and 20◦ have been chosen. Variation of the incident radiative heat flux along the wall length for three cone angles is shown in Fig. 5.9. As the cone angle increases, the peak of the heating is seen to shift more and more towards the diffuser inlet. Furthermore, the flux values reduce towards the exit of the diffuser as the

5.4

Application to a Conical Diffuser

85

Fig. 5.8 Effect of concentration of the particles on the axial variation of the radiative heat flux at the curved wall. Baseline values: Y H 2O = a = 0.2, YC O2 = b = 0.12, and YC O = c = 0.08

cone angle increases. These results suggest that in the design stage of such diffuser, the peak heating near the inlet section of the diffuser needs to be factored in.

5.4.5

Effect of Anisotropic Scattering

As discussed in the Sect. 3.4, most of the scattering phenomena occurring in nature are in general anisotropic in nature. Since an accurate accounting of anisotropic scattering may involve considerable effort and is also computationally intensive, many times an assumption of isotropic scattering is made to simplify the problem. Even so, the errors involved in making such an assumption must be quantified. In the present case, with the help of transport approximation, an attempt has been made to estimate the effect of anisotropic scattering radiation heating. It is instructive to mention here that, though the transport approximation is a simplified expression to model the anisotropic behavior, it gives considerable accuracy with significant savings in the computation time. For example, [4] successfully applied the transport approximation in estimating radiative heat transfer from a two-phase mixture gen-

86

5 Radiative Heat Transfer in Conical Geometries

Fig. 5.9 Effect of the diffuser cone angle on the axial variation of the radiative heat flux at the curved wall

erally encountered in jet exhausts. In the study, it was seen that the transport approximation provided appreciable agreement in the estimation of fluxes when compared to other formulations such as the Henyey-Greenstein phase function. Moreover, as already presented in Sect. 4.4.3 of Chap. 4, a comparison of the transport approximation with the Legendre polynomial representation of the phase function has been made and a close agreement between the two was observed both in highly forward as well as backward scattering cases. Hence, it is reasonable to use the transport approximation for multidimensional practical problems having computational time as a primary concern. In the problem under consideration, a high value of particle concentration is selected i.e. 0.3 kg/m3 with uniform distribution, in order to assess the effect of anisotropic behavior of the particles. The radiative transfer equation for the conical geometry as shown in Fig. 5.4 with prescribed temperature variation (Eq. 5.10) was solved first for the anisotropic scattering and then isotropic scattering by simply setting the asymmetry parameter g to zero. The radiative heat fluxes plotted against the location at the inclined wall for these cases are presented in Fig. 5.10. From the figure, it is clear that the isotropic assumption leads to an error of around 15%. This is possible due to the fact that with an increase in the anisotropy more radiation is scattered towards the wall. However,

5.4

Application to a Conical Diffuser

87

Fig. 5.10 Effect of anisotropic scattering of particles on the axial variation of the radiative heat flux at the curved wall - comparison with isotropic scattering case

the actual effect of anisotropy is sensitive to the size of the particles present in the domain and the geometry of the problem.

5.4.6

Effect of Wall Emission

Finally, an assessment of the effect of wall emission has been made by changing the emissivity of the wall from 0.6 to 1. In this case, the total heat flux, which is the difference between the sum of emission and reflection fluxes minus incident fluxes, has been estimated. The temperature of the wall is taken to be equal to the mean temperature of the domain and the reflection is diffuse. The cone angle is set to 20◦ . The total heat fluxes at the wall have been plotted against the wall location and are shown in Fig. 5.11. From the figure, it can be seen that, as the emissivity of the wall increases, radiation heat transfer increases sharply even towards the exit of the diffuser. However, at the inlet, due to the high value of the incident and reflection fluxes, the total heat flux is less and is in the wall (heat flux is negative). This effect dies out as one moves away from the inlet as the incident fluxes continuously reduce (as can be seen from previous figures) after a certain distance from the inlet and the emission from the wall totally overcomes the incident radiation from the gases. The net flux becomes

88

5 Radiative Heat Transfer in Conical Geometries

Fig. 5.11 Effect of wall emission on the axial variation of the total radiative heat flux at the curved wall of the diffuser

positive beyond z/Z ax = 0.4 and is getting dissipated into the domain (heat flux is positive). In consideration of the above, it is clear that the heating of the walls is predominant in the inlet section and hence may require careful consideration during the design stage of the diffuser.

5.5

Conclusions

The methodology of FVM-SLW has been extended to two-dimensional axisymmetric complex geometries that resemble a conical enclosure. The modifications required to incorporate such type of geometries with the help of body-fitted mesh was presented. The code developed with the necessary modifications on the basis of this has been validated against the available benchmarks in the literature. A practical case of diffuser section of a rocket nozzle was considered with three participating gases viz. H2 O, C O2 , and C O with particles that cause anisotropic scattering along with high-temperature gradients. Several critical parameters like gas concentration, particle concentration, wall emissivity, and cone angle were varied and their effect on radiative heating at the wall has been evaluated. From the results obtained, it was seen that in the absence of particles, the radiative heating is mainly domi-

References

89

nated by H2 O rather than C O2 and C O, due to its large number of participating bands over the spectrum. The particle concentration was seen to dominate the radiative heating due to scattering, even if the concentration of the gas species is doubled (the H2 O gas contribution decreases by 9% for 0.1 kg/m3 particle loading compared to the no particle case). The study also showed that as the cone angle of the diffuser is increased, the peak value of the radiative heat flux shifts more towards the inlet and reduces significantly towards the exit. A simple assumption of isotropic scattering resulted in a 15% reduction in the flux values compared to the anisotropic case. The heating of the bounding wall is critical at the inlet section and reduces with decreasing emissivity of the wall.1

5.6

Closure

Modifications required to incorporate the inclined wall geometries resembling a rocket nozzle or gas turbine diffuser were presented in this chapter. After validation with the benchmarks, a practical case of conical diffuser geometry was taken up with the real-time conditions of a varying temperature profile, gas-particle mixture, and boundary conditions. The assessment and quantification of the influence of the medium in terms of gas and particle concentration, diffuser geometry, anisotropic scattering, and wall emission were presented. In the next chapter, results of the FVM-SLW model when applied to a problem of radiation in a three-dimensional rectangular geometry in presence of a participating media are presented.

References 1. Moder, J. P., Chai, J. C., Parthasarathy, G., Lee, H. S., & Patankar, S. V. (1996). Nonaxisymmetric radiative transfer in cylindrical enclosures. Numerical Heat Transfer, 30(4), 437–452. 2. Kim, M. Y., & Baek, S. W. (2005). Modeling of radiative heat transfer in an axisymmetric cylindrical enclosure with participating medium. Journal of Quantitative Spectroscopy and Radiative Transfer, 90(3), 377–388. 3. Dombrovsky, L. A. (1996). Radiation heat transfer in disperse systems. New York: Begell House. 4. Dombrovsky, L. A. (2012). The use of transport approximation and diffusion-based models in radiative transfer calculations. Computational Thermal Sciences, 4(4), 297–315.

1 Contents of this chapter have been published in Rahul Yadav, C. Balaji and S. P. Venkate-

shan, Analysis of radiative transfer in body-fitted axisymmetric geometries with band models and anisotropic scattering, Computational Thermal Sciences, 11 (1–2), 2019, pp. 161–176. (doi: https:// doi.org/10.1615/ComputThermalScien.2018021506).

6

Radiative Heat Transfer in Three-Dimensional Geometries

6.1

Introduction

In this chapter, the framework of finite volume and spectral line-based weighted sum of gray gases method has been extended to three-dimensional geometries (in Cartesian coordinates). The generic methodology discussed earlier is applied to three-dimensional rectangular enclosures of practical interest. A set of modifications that needs to be incorporated in the solver are discussed, and the validation cases are illustrated. This is followed by the application of the solver to industrial scale coal fired reheating furnaces.

6.2

Formulations for a Three-Dimensional Rectangular Geometry

Many furnaces and enclosures applied in industry can be approximated as three-dimensional rectangular geometries, as shown in Fig. 6.1. Some of the typical examples are fire resistance test furnaces, reheating furnaces, and the enclosures which host specimens for specific heat treatment. Starting from the coordinate free form of the RTE again, sˆ · ∇ I ( r , sˆ ) = −β I ( r , sˆ ) + Sr ( r , sˆ )

(6.1)

The base vectors and the geometry share the same coordinate system and so the formulation is straightforward. Under the finite volume method, Eq. (6.1) is integrated over a control volume and control angle as     sˆ · ∇ I ( r , sˆ )d xd ydzd = (−β I ( r , sˆ )  V

 V

r , sˆ ))d xd ydzd + Sr (

© The Author(s) 2023 R. Yadav et al., Radiative Heat Transfer in Participating Media, https://doi.org/10.1007/978-3-030-99045-9_6

(6.2)

91

92

6 Radiative Heat Transfer in Three-Dimensional Geometries

y

Hot gases

x

z specimen surface Fig. 6.1 A typical rectangular furnace

The intensity within a control angle remains constant but its direction is allowed to vary. Eq. (6.2) is simplified using the Gauss divergence theorem to convert the volume integral to an area integral. Upon doing this, one can write Eq. (6.2) for a direction m  n  and from a discretized set of Nθ × Nφ number of control angles as nb MN  

 

 

mn  + (Sr , j )m n )V  mn = (−β j,P I j,P I j,i Ai Dci P

(6.3)

m  n  =1 i=1

where nb denotes the number of neighboring boundaries or the control volume faces. The directional weights for each of these directions can be calculated as m n

Dcx

n φ 

Dcy

+

(sin 2 θ cosφ)dθ dφ −

n φ 

θm

+

−

θm

+

=

(sin 2 θ sinφ)dθ dφ φ

n −

θ

n +

 

θm

= φn

m n

+

φ 

(6.4)

m −

θm

+

mn = Dcz

(sinθ cosθ )dθ dφ φn

−

θm

−

In a three-dimensional rectangular enclosure, six faces for each control volume node are present. Equation 6.3 can be further simplified and written in terms of the known upstream face intensities as

6.2

Formulations for a Three-Dimensional Rectangular Geometry  

 

 

 

 

 

 

93

 

 

 

n mn mn mn mn mn mn mn mn mn am I j,S P I j,P = a E I j,E + aW I j,W + a N I j,N + a S  

 

 

 

 

n mn mn mn + aTm n I m j,T + a B I j,B + (Sr , j ) P

(6.5)

In Eq. (6.5), the subscript j signifies that the equation has to be solved for a finite number of gray bands or gray gases, as used in the WSGG and SLW methods. S j, p denotes the source term at the control node P. The coefficients appearing in Eq. (6.5) are given by  

 

n mn am = max(−Ai Dci , 0) I

 

n am = p



 

(6.6)  

mn max(−Ai Dci , 0) + (κg, j + κs, j )V m n

(6.7)

i=e,w,n,s,t,b

The source term appearing in Eq. (6.5) can be written as  

(Sr , j )mP n = a j κg, j Ib,g + a j κ p, j Ib, p +

M N σs, j   mn I j mn→m  n  mn 4π

(6.8)

m=1 n=1

where a j are the weights corresponding to the jth gray band and can be obtained by the application of the band model. The discretization of control volumes, control angles, spatial differencing and marching procedure are all done in an exactly similar fashion to those that have been mentioned in Chap. 4. However, in the present case the marching has to start from each of the 8 corners, as given in Fig. 6.2, one by one, for the set of directions satisfying the criteria of directional weights as presented in Table 6.1 in order to march in those directions. The corner with known boundary conditions can be chosen and the solution is sought at each spatial node and each direction, satisfying the necessary directional weight requirement in the octant. In a similar way, other corners are chosen one after the other, and the solution is sought for other sets of directions. This implies that in order to cover all the directions in the complete spherical domain, one has to traverse the complete spatial domain 8 times. Once the intensity field for the complete domain in all directions has reached the convergence criterion (as stated in Eq. (4.17)), the radiative heat flux at a location in any direction (i = x, y or z) can be estimated as qi =

N GG+1 MN   j=1

m  n  =1

 

 

mn mn I p, j Dci

(6.9)

94

6 Radiative Heat Transfer in Three-Dimensional Geometries

Fig. 6.2 Discrete directions in the respective octants

IV

III

VIII

VII II

I V y

VI

z x

Table 6.1 Set of direction cosines in the various octants  

 

 

Octant

mn Dcx

mn Dcy

mn Dcz

I

>0

>0

0 )

382 383

b = b +1;

384 385 386 387 388

Dce Dcw Dcn Dcs

= = = =

Dcx ( m , n ) ; −Dcx ( m , n ) ; Dcy ( m , n ) ; −Dcy ( m , n ) ;

389 390

d s a a ( b ) = dsa (m, n ) ;

391 392

393

394

dr2 = ( ( ka + sigm_s ) ∗ d e l _ v ( y , x ) ∗ dsa (m ,n) ) ; d r 1 = max ( Ae ∗ Dce , 0 ) + max ( Aw∗Dcw , 0 ) + max ( An ∗ Dcn , 0 ) + max ( As ∗ Dcs , 0 ) ; ap = dr1 + dr2 ;

395 396

bp = ( S r ( y , x , 1 ) ∗ d e l _ v ( y , x ) ∗ d s a (m, n ) );

397 398 399 400 401

aE aW aN aS

= = = =

max ( − Ae ∗ Dce , 0 ) max ( −Aw∗Dcw , 0 ) max ( −An ∗ Dcn , 0 ) max ( − As ∗ Dcs , 0 )

; ; ; ;

402 403

I p a = ( ( aE ∗ I e ( y , x , b ) ) + ( aW∗ Iw ( y , x , b ) ) + ( aN ∗ I n ( y , x , b ) ) + ( a S ∗ I s ( y , x , b ) ) + bp ) / ap ;

404 405

Ip ( y , x , b ) = Ipa ;

406 407

if

408 409

( I p ( y , x , b ) 0 posx = posx +1; q r x _ p o s ( : , : , p o s x ) = I p ( : , : , b ) . ∗ ( ( Dcxx ( b ) ) ) ; end

505

i f Dcxx ( b ) 0 posy = posy +1; q r y _ p o s ( : , : , p o s y ) = I p ( : , : , b ) . ∗ ( ( Dcyy ( b ) ) ) ; end

511 512 513 514 515

i f Dcyy ( b ) 0 && Dcy ( m , n ) > 0 ) b = b +1; Dcxx ( b ) = Dcx ( m , n ) ; Dcyy ( b ) = Dcy ( m , n ) ; end end end

155 156 157

158 159

%r e a d i n g p e a r s o n c o e f f i c i e n t s from P e a r s o n d a t a b a s e [ coeff ,~] = xlsread ( ’ corr_coeff_pearson . xlsx ’ , ’ Correlation Coefficients ’) ; c o e f f ( isnan ( c o e f f ) ) =0; NGG = 2 0 ;

160 161 162 163

maxitr = 1000; k = N t h e t a ∗ Nphi ; P = 1;

%maximum n u m b e r o f i t e r a t i o n s %t o t a l number of d i r e c t i o n s

164 165

for

cse = 1:1: cse_tot

166 167 168

d i s p ( ’ c a s e no ’ ) ; disp ( cse ) ;

169 170

nh = n h _ b i ( c s e , : ) ;

171 172 173

Nh = l e n g t h ( n h ( n h ~ = 0 ) ) ; % t o t a l n u m b e r o f ON h e a t e r s x h = xL / Nhn ; % s i z e o f e a c h h e a t e r

174 175 176 177 178 179

i f Nh ~=0 q h = Qh ( c s e ) / ( Nh ∗ x h ∗ 1 ) ; %p o w e r t o else qh = 0 ; end

each

heater

180 181 182

nhx = xh / d e l _ x ; h t _ a r y = z e r o s ( 1 , Nx ) ;

183 184

dx = xd / d e l _ x ;

185 186 187 188

%m a k i n g g r i d l i n e s t o f a l l f o r i = 1 : 1 : l e n g t h ( nh ) i f n h ( i ) >=1

over

the

edge of

the

heaters

Appendix C: MATLAB Codes c n t = ( i ∗ n h x ) −( nhx −1) ; f o r j = 1 : 1 : nhx ht_ary (1 , cnt ) = 1; cnt = cnt +1; end

189 190 191 192 193

end

194 195

175

end

196 197 198 199 200 201

eps_ew = 0 ; % e m i s s i v i t y o f r i g h t w a l l f o r a d i a b e t i c , no e m i s s i o n eps_ww = 0 ; % e m i s s i v i t y o f l e f t w a l l f o r a d i a b e t i c , n o e m i s s i o n e p s _ s w = 0 ; % e m i s s i v i t y o f b o t t o m w a l l f o r a d i a b e t i c , no e m i s s i o n eps_nw = 1 ; %e m i s s i v i t y of h e a t e r wall eps_d = epsd ( cse ) ; %e m i s s i v i t y of h e a t e r wall

202 203 204 205 206 207 208

%r e f l e c t i v i t y factor f 1 _ e = (1 − e p s _ e w ) / ( p i f 1 _ w = (1 − eps_ww ) / ( p i f 1 _ n = (1 − e p s _ n w ) / ( p i f 1 _ s = (1 − e p s _ s w ) / ( p i f 1 _ d = (1 − e p s _ d ) / ( p i )

for ); ); ); ); ;

(1 − wall

walls

emissivity ) / pi

209 210 211

212 213 214 215 216

Tnw = z e r o s ( 1 , c o l s ) ; Tew = 0 ; %T e m p e r a t u r e o f r i g h t s i d e w a l l d o e s n ’ t m a t t e r f o r adiabetic Tww = 0 ; %T e m p e r a t u r e o f l e f t w a l l d o e s n ’ t m a t t e r f o r a d i a b e t i c Tnw = 0 ; %T e m p e r a t u r e o f t o p w a l l d o e s n ’ t m a t t e r f o r a d i a b e t i c Tsw = 0 ; %T e m p e r a t u r e o f t o p w a l l d o e s n ’ t m a t t e r f o r a d i a b e t i c T d e s = Td ( c s e ) ; %T e m p e r a t u r e o f t h e d e s i g n s u r f a c e Tg = 4 0 0 ; %T e m p e a r t u r e o f t h e medium , i n i t i a l g u e s s

217 218 219 220 221 222

%i n i t i a l i s i n g I e = z e r o s ( rows Iw = z e r o s ( r o w s I n = z e r o s ( rows I s = z e r o s ( rows

, , , ,

qrx_pos qrx_neg qry_pos qry_neg

= = = =

zeros zeros zeros zeros

( rows ( rows ( rows ( rows

, , , ,

cols cols cols cols

qew_inc qww_inc qnw_inc qsw_inc

= = = =

zeros zeros zeros zeros

( ( ( (

) ) ) )

; ; ; ;

cols cols cols cols

,k) ,k) ,k) ,k)

; ; ; ;

223 224 225 226 227

, k /2) , k /2) , k /2) , k /2)

228 229 230 231 232

rows rows cols cols

233 234

I p _ c h k = z e r o s ( rows , c o l s , k ) ;

235 236 237 238

I p = z e r o s ( rows , c o l s , k ) ; I p _ o l d = z e r o s ( rows , c o l s , k ) ; I p _ d s a 1 = z e r o s ( rows , c o l s , k ) ;

239 240

I b _ g = ( s i g m a ∗ ( Tg ^ 4 ) ) / p i ;

; ; ; ;

176

Appendix C: MATLAB Codes

241 242 243 244 245

% s e t t i n g up t h e t e m p e r a t u r e Tx = z e r o s ( 1 , Nx ) ; Txy = z e r o s ( r o w s , c o l s ) ; Txy ( : , : ) = Tg ;

field

246 247

Y = [ Yh2o ( c s e ) , Yco2 ( c s e ) , 0 . 0 0 0 0 1 ] ;

248 249

Sr = z e r o s ( rows , c o l s , 1 ) ;

250 251 252 253 254 255 256

%f o r no p a r t i c i p a t i o n o f p a r t i c l e s o r t h e f u n c t i o n %a b s _ s c a t _ f r o m _ m i e c a n be c a l l e d upon f o r p a r t i c l e % r a d i a t i v e p r o p e r t i e s b a s e d on l o c a l t h e r m o d y n a m i c %c o n d i t i o n s . kp_abs = 0; sigm_s = 0;

257 258

Gr_lm = z e r o s ( rows , c o l s ) ;

259 260

kxy = z e r o s ( rows , c o l s ) ;

261 262 263

tmax = 5 0 ; %maximum t i m e T x y _ o l d = Txy ;

steps

for

radiative

equilbrium

264 265

for

t = 1 : 1 : tmax

266 267 268 269 270

Gr_lm_sum Ib_lm_sum Ip_lm_sum Gr_lm ( : , : )

= 0; = 0; = 0; = 0;

271 272

T r e f = ( sum ( sum ( Txy ) ) ) / ( r o w s ∗ c o l s ) ;

273 274 275

f o r y = 1 : 1 : rows for x = 1:1: cols

276

Tg = Txy ( y , x ) ;

277 278

add = 0 ;

279 280 281

[ aj ,

k j , ~ , ~ , ~ ] = g a s _ m i x _ m u l t i p l i c a t n 2 ( Tg , Tew , Tww , Tsw , P , Y , NGG , c o e f f ) ;

for

j = 1:1: length ( kj ) a d d = a d d + ( a j ( 1 , j ) ∗(1 − e x p ( − k j ( 1 , j ) ∗Lm ) ) ) ;

282 283 284

end k j = ( − l o g (1 − a d d ) ) / Lm ; kxy ( y , x ) = k j ; aj = 1;

285 286 287 288

end

289 290

end

291 292

aj_n = aj ;

a j _ e = a j ; aj_w = a j ;

aj_s = aj ;

Appendix C: MATLAB Codes

177

293 294

f o r lm = 1 : 1 : l e n g t h ( a j )

295 296 297 298 299 300 301

%b o u n d a r y c o n d i t i o n s for x = 1:1: cols i f h t _ a r y ( x ) ==1 I n ( rows , x , : ) = ( qh ) / p i ; %h e a t e r end end

wall

302 303 304 305

306 307 308 309

for x = 1:1: cols i f x c ( x ) >= x d && x c ( x ) < = ( xL−x d ) I s ( 1 , x , : ) = ( a j _ s ( 1 , lm ) ∗ e p s _ d ∗ s i g m a ∗ ( T d e s ^ 4 ) ) / p i ; design wall else I s ( 1 , x , : ) = ( a j _ s ( 1 , lm ) ∗ e p s _ s w ∗ s i g m a ∗ ( Tsw ^ 4 ) ) / p i ; end end

%

310 311 312

I e ( : , c o l s , : ) = ( a j _ e ( 1 , lm ) ∗ e p s _ e w ∗ s i g m a ∗ ( Tew ^ 4 ) ) / p i ; Iw ( : , 1 , : ) = ( a j _ w ( 1 , lm ) ∗ eps_ww ∗ s i g m a ∗ ( Tww ^ 4 ) ) / p i ;

313 314 315 316

%SOURCE TERM UPDATION f o r y = 1 : 1 : rows for x = 1:1: cols

317

Tg = Txy ( y , x ) ; ka = kxy ( y , x ) ; Sr ( y , x , 1 ) = 0; I b _ g _ f u l l = ( s i g m a ∗ ( Tg ^ 4 ) ) / p i ; I b _ g = a j ( 1 , lm ) ∗ I b _ g _ f u l l ;

318 319 320 321 322 323

Sr ( y , x , 1 ) = Sr ( y , x , 1 ) +( f ∗ ka ∗ I b _ g ) ; %u p d a t i n g source term for i s o t r o p i c s c a t t e r i n g

324

end

325

end

326 327 328

for

i t r = 1:1: maxitr

329 330

331 332 333 334 335

if

( i t r >1) %u p d a t i n g b o u n d a r y and r e f l e c t i v i t y %w a l l f l u x e s m a t r i c e s qrx_pos ( : , : , : ) = 0; qrx_neg ( : , : , : ) = 0; qry_pos ( : , : , : ) = 0; qry_neg ( : , : , : ) = 0;

conditions

336 337 338 339 340

posx negx posy negy

= = = =

0; 0; 0; 0;

341 342

%e s t i m a t i o n

of

incident

fluxes

for

wall

emission

178

343 344 345 346 347

Appendix C: MATLAB Codes for b = 1:1: k i f Dcxx ( b ) >0 posx = posx +1; q r x _ p o s ( : , : , p o s x ) = I p ( : , : , b ) . ∗ ( ( Dcxx ( b ) ) ) ; end

348

i f Dcxx ( b ) 0 posy = posy +1; q r y _ p o s ( : , : , p o s y ) = I p ( : , : , b ) . ∗ ( ( Dcyy ( b ) ) ) ; end

354 355 356 357 358

i f Dcyy ( b ) = x d && x c ( x ) < = ( xL−x d ) I s ( 1 , x , : ) = ( ( a j _ s ( 1 , lm ) ∗ e p s _ d ∗ s i g m a ∗ ( T d e s ^ 4 ) ) / p i ) + ( f 1 _ d ∗ q s w _ i n c ( x ) ) ; %d e s i g n w a l l else I s ( 1 , x , : ) = ( ( a j _ s ( 1 , lm ) ∗ e p s _ s w ∗ s i g m a ∗ ( Tsw ^ 4 ) ) / pi ) +( f 1_s ∗ qsw_inc ( x ) ) ; end end

395 396 397

398 399

400 401 402 403 404 405 406

end %s t a r t of c a l c u l a t i o n of r a d i a t i o n i n t e n s i t i e s f o r y = rows : −1:1 %t h i r d q u a d r a n t f o r x = c o l s : −1:1

407 408

Tg = Txy ( y , x ) ;

409 410

ka = kxy ( y , x ) ;

411 412

b = 0;

413 414 415

for m = 1:1: Ntheta f o r n = Nphi : −1:1

416 417

if

( Dcx ( m , n ) 1) I e ( y , x −1 , b ) = Iw ( y , x , b ) ;

Appendix C: MATLAB Codes

181

483 484

b = k /4;

485 486 487

for m = 1:1: Ntheta f o r n = Nphi : −1:1

488 489

if

( Dcx ( m , n ) >0 && Dcy ( m , n ) < 0 )

490 491

b = b +1;

492 493 494 495 496

Dce Dcw Dcn Dcs

= = = =

Dcx ( m , n ) ; −Dcx ( m , n ) ; Dcy ( m , n ) ; −Dcy ( m , n ) ;

497 498

d s a a ( b ) = dsa (m, n ) ;

499 500

501

502

dr2 = ( ( ka + sigm_s ) ∗ d e l _ v ( y , x ) ∗ dsa (m ,n) ) ; d r 1 = max ( Ae ∗ Dce , 0 ) + max ( Aw∗Dcw , 0 ) + max ( An ∗ Dcn , 0 ) + max ( As ∗ Dcs , 0 ) ; ap = dr1 + dr2 ;

503 504

bp = ( S r ( y , x , 1 ) ∗ d e l _ v ( y , x ) ∗ d s a (m, n ) );

505 506 507 508 509

aE aW aN aS

= = = =

max ( − Ae ∗ Dce , 0 ) max ( −Aw∗Dcw , 0 ) max ( −An ∗ Dcn , 0 ) max ( − As ∗ Dcs , 0 )

; ; ; ;

510 511

I p a = ( ( aE ∗ I e ( y , x , b ) ) + ( aW∗ Iw ( y , x , b ) ) + ( aN ∗ I n ( y , x , b ) ) + ( a S ∗ I s ( y , x , b ) ) + bp ) / ap ;

512 513

Ip ( y , x , b ) = Ipa ;

514 515

if

516 517

( I p ( y , x , b ) 1) I s ( y , x , b ) = ( I p ( y , x , b ) −( I n ( y , x , b ) ∗(1 − f ) ) ) / ( f ) ; i f I s ( y , x , b ) 0 )

634 635

b = b +1;

636 637 638 639 640

Dce Dcw Dcn Dcs

= = = =

Dcx ( m , n ) ; −Dcx ( m , n ) ; Dcy ( m , n ) ; −Dcy ( m , n ) ;

641 642

d s a a ( b ) = dsa (m, n ) ;

643 644

645

646

dr2 = ( ( ka + sigm_s ) ∗ d e l _ v ( y , x ) ∗ dsa (m ,n) ) ; d r 1 = max ( Ae ∗ Dce , 0 ) + max ( Aw∗Dcw , 0 ) + max ( An ∗ Dcn , 0 ) + max ( As ∗ Dcs , 0 ) ; ap = dr1 + dr2 ;

647 648

bp = ( S r ( y , x , 1 ) ∗ d e l _ v ( y , x ) ∗ d s a (m, n ) );

649 650 651 652 653

aE aW aN aS

= = = =

max ( − Ae ∗ Dce , 0 ) max ( −Aw∗Dcw , 0 ) max ( −An ∗ Dcn , 0 ) max ( − As ∗ Dcs , 0 )

; ; ; ;

654 655

I p a = ( ( aE ∗ I e ( y , x , b ) ) + ( aW∗ Iw ( y , x , b ) ) + ( aN ∗ I n ( y , x , b ) ) + ( a S ∗ I s ( y , x , b ) ) + bp ) / ap ;

656 657

Ip ( y , x , b ) = Ipa ;

658 659

if

660 661

( I p ( y , x , b )