Table Of ContentHeat 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.
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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
