Table Of Content

Heat Transfer 2 Mathematical and Mechanical Engineering Set coordinated by Abdelkhalak El Hami Volume 10 Heat Transfer 2 Radiative Transfer Michel Ledoux Abdelkhalak El Hami First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com © ISTE Ltd 2021 The rights of Michel Ledoux and Abdelkhalak El Hami to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2020949611 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-517-6 Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Chapter 1. General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Propagation of a sinusoidal electromagnetic wave . . . . . . . . . . . . . 1 1.2.1. Frequencies and wavelengths . . . . . . . . . . . . . . . . . . . . . . . 1 1.2.2. Radiation spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3. The concept of photometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1. Geometric parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2. Radiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3. Bouguer–Lambert law . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.4. Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.5. Lambert’s law – a surface’s emissivity . . . . . . . . . . . . . . . . . 14 Chapter 2. Calculating Luminances . . . . . . . . . . . . . . . . . . . . . . 17 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2. The black body: concept, luminance, Planck’s law and approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1. Paradoxically, the black body is defined with reference to its absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.2. Black body luminance . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.3. Emittance from the black body . . . . . . . . . . . . . . . . . . . . . . 22 2.2.4. Approximations of the luminance of the black body . . . . . . . . . 23 2.2.5. Writing the luminance in terms of frequency . . . . . . . . . . . . . 25 2.3. Stefan–Boltzmann law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1. Establishing the law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 vi Heat Transfer 2 2.3.2. A direct application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4. Wien’s laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.1. Wien’s displacement law . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.2. Wien’s second law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.3. Greenhouse effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.5. Fraction of the total emittance of a black body radiated in a spectral band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5.1. An important tool: G0−λT functions . . . . . . . . . . . . . . . . . . . 38 2.5.2. An application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.6. Emissivity of any body: a general case of a non-black body . . . . . . . 43 2.6.1. Definition of monochromatic emissivity . . . . . . . . . . . . . . . . 43 2.6.2. Definition of global emissivity: a tricky concept . . . . . . . . . . . 45 2.6.3. Emissivity of a gray body: a particular case . . . . . . . . . . . . . . 45 2.7. Simple applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Chapter 3. Emission and Absorption . . . . . . . . . . . . . . . . . . . . . . 53 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2. Absorption, reflection, transmission . . . . . . . . . . . . . . . . . . . . . 53 3.3. Kirchhoff’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4. Recap on the global absorption coefficient . . . . . . . . . . . . . . . . . 57 3.4.1. General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4.2. Case of the gray body . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5. General case: multiple transfers . . . . . . . . . . . . . . . . . . . . . . . . 59 3.6. Absorption: the Beer–Lambert law . . . . . . . . . . . . . . . . . . . . . . 61 3.6.1. Radiation transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.6.2. Beer’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Chapter 4. Radiation Exchanges Between Surfaces . . . . . . . . . . . 65 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2. Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3. The case of total influence . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3.1. The case of two parallel plates. Lambert’s law . . . . . . . . . . . . 66 4.3.2. Total influence between two black body surfaces, of temperatures T and T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 w a 4.3.3. Total influence between two surfaces . . . . . . . . . . . . . . . . . . 68 4.3.4. Total influence between two surfaces . . . . . . . . . . . . . . . . . . 69 4.3.5. Wall in total influence in an enclosure . . . . . . . . . . . . . . . . . 71 4.3.6. Important note on the “thermal balance” . . . . . . . . . . . . . . . . 72 4.3.7. A practical approximation . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.8. Complex system of radiant finished surfaces: geometric form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3.9. Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Contents vii Chapter 5. Analytic Applications . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2. Radiators, convectors and radiating fins . . . . . . . . . . . . . . . . . . . 89 5.3. Radiation and oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.4. Radiation and metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.4.1. Measuring a thermal conductibility . . . . . . . . . . . . . . . . . . . 137 5.5. General problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Chapter 6. Modeling and Simulations under ANSYS . . . . . . . . . . . 169 6.1. Conduction, convection and radiation . . . . . . . . . . . . . . . . . . . . 169 6.2. Conduction and convection using ANSYS software . . . . . . . . . . . . 172 6.2.1. Representation of the temperature field . . . . . . . . . . . . . . . . . 174 6.3. Radiation using ANSYS software . . . . . . . . . . . . . . . . . . . . . . 175 6.4. Examples of modeling and analysis with ANSYS . . . . . . . . . . . . . 177 6.4.1. Simple thermal conduction . . . . . . . . . . . . . . . . . . . . . . . . 177 6.4.2. Mixing conduction/convection/isolation . . . . . . . . . . . . . . . . 180 6.4.3. Transient thermal conduction . . . . . . . . . . . . . . . . . . . . . . . 182 6.4.4. Study of thermal transfers from a brick wall and a cement wall (application to an oven) . . . . . . . . . . . . . . . . . . . . . 186 6.4.5. Study of stationary thermal conduction in a reservoir intersected by a tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.4.6. Stationary thermal conduction on a cylinder . . . . . . . . . . . . . . 196 6.4.7. Cooling of a puck in transitory thermal . . . . . . . . . . . . . . . . . 199 6.4.8. Study of a heat exchanger . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.5. Study of a thermal exchanger on ANSYS . . . . . . . . . . . . . . . . . . 204 6.5.1. Effectiveness of the PCM . . . . . . . . . . . . . . . . . . . . . . . . . 204 6.5.2. Parameterizing the analysis . . . . . . . . . . . . . . . . . . . . . . . . 204 6.5.3. Thermal exchanger without an PCM . . . . . . . . . . . . . . . . . . 207 6.5.4. Thermal exchanger with hydrated salt . . . . . . . . . . . . . . . . . . 207 6.5.5. Thermal exchanger with paraffin . . . . . . . . . . . . . . . . . . . . . 210 6.5.6. Influence of heat flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.5.7. Comparing PCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Appendix. G0−λT Function Table . . . . . . . . . . . . . . . . . . . . . . . . . . 217 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Preface Thermal science is to thermodynamics as decree is to law. It answers the following question – which all good leaders must (or should) ask themselves whenever they have an “idea”: “How would this work in practice?”. In a way, thermal science “implements” thermodynamics, of which it is a branch. A thermodynamics specialist is a kind of energy economist. Applying the first principle, they create a “grocery store”. With the second principle, they talk about the quality of their products. I add or remove heat from a source or work from a system. And the temperature, among other things, defines the quality of the energy for me. But by what means do I take or do I give? Even calculations of elementary reversible transformations do not tell us by what process heat passes from a source to a system. Thermal science specifies how, but “evacuates” work. If in a given problem related to, for example, a convector where an electrical energy (therefore in the “work” category) appears, it is immediately dissipated into heat by the Joule effect. Three heat transfer modes can be identified: conduction and radiation – which can be seen separately, although they are often paired up – and convection, which is by nature an interaction of fluid mechanics and conduction. Dividing the study of thermal science into three is the result of logic. Presenting this work in three volumes is somewhat arbitrary; in our opinion, x Heat Transfer 2 however, this split was necessary in order to keep the volumes in the collection a reasonable size. This is Volume 2 of a collection of problems on thermal transfer, dedicated to radiation and digital approaches to transfer. Even though it is primarily a collection of exercises, a great deal of attention is focused on lessons. For the most part, the work is a first introduction to the thermal calculation of practical devices, which may be enough in itself. For subsequent calculations, the reader will still have to turn to specialist works or encyclopedias available in the field of thermics. In Chapter 1, after a brief historical background, we summarize the vital notions of electromagnetic radiation and how they are written. The emphasis in this book is on the aspect of energy: the notions of photometry prove indispensable at this stage of exploration. At the heart of studying radiation, Chapter 2 focuses on calculating luminances, relying on black body laws (Planck’s law; Rayleigh–Jeans and Wien approximations) and its derivatives: Stefan–Boltzmann laws and Wien laws. For evaluating a fraction of total emittance radiated in a spectral band, the G function proves vital. λ,λ 1 2 Chapter 3 tackles these interactions between a light flux and a material medium, a fundamental subject in any practical calculation of radiation: the phenomena of emission, absorption, transmission, etc., as well as the Kirchhoff law, emissivity, absorption coefficient, etc. Chapter 4 presents the general notions on reciprocal radiation from several surfaces. We distinguish total influence and reciprocal radiation from finite surfaces. This subject is central in particular to the calculation of ovens. Here, we should restrict ourselves to notions, returning to specialist works for application by professionals working with ovens. Solving a radiation problem often involves knowing and understanding the essence of the corresponding lessons. It is therefore not so easy to produce (interesting) problems that are limited to only being a paragraph long. This is why we were led to focus in Chapter 5 on the essence of exercises dedicated to radiation and coupled transfers. Preface xi In a domain where digital methods are becoming the rule for complex situations, it seems important to reserve a particular place for “monitoring” analytical approaches. In addition to their usefulness in understanding this domain, they offer the reader a precious tool for making calculations “on the back of an envelope”. Finally, Chapter 6 introduces the reader to a digital approach for different transfer modes. The general problem of modeling is tackled here and examples of processing using ANSYS are presented. The Appendix covers the tabulations of G functions whose practical O−λT importance emerges from the problems. December 2020