Table Of ContentRADIATION GAS DYNAMICS
BY
SHIH-I PAl
RESEARCH PROFESSOR
INSTITUTE FOR FLUID DYNAMICS AND APPLIED MATHEMATICS
UNIVERSITY OF MARYLAND. COLLEGE PARK, MARYLAND, U.S.A.
WITH 76 FIGURES
1966
SPRINGER-VERLAG NEW YORK INC.
ALL RIGHTS INCLUDING TRANSLATION INTO OTHER LANGUAGES RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM
BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS. WITHOUT WRITTEN
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© 1966 BY SPRINGER-VERLAG I WIEN
Softcover reprint ofthe hardcover 1s t edition 1966
TITLE-NO. 9145
ISBN-13: 978-3-7091-5733-6 e-ISBN-13: 978-3-7091-5730-5
DOl: 10.1007/978-3-7091-5730-5
TO
ALICE, STEPHEN, SUE, ROBERT, LOU
Preface
When the temperature of a gas is not too high and the density of a gas is not
too low, the transfer of heat by radiation is usually negligibly small in comparison
with that by conduction and convection. However, in the hypersonic flow of
space flight, particularly in the re-entry of a space vehicle, and in the flow problem
involving nuclear reaction such as in the blast wave of nuclear bomb or in the
peaceful use of the controlled fusion reaction, the temperature of the gas may be
very high and the density of the gas may be very low. As a result, thermal
radiation becomes a very important mode of heat transfer. A complete analysis
of such high temperature flow fields should be based upon a study of the gasdynamic
field and the radiation field simultaneously. Hence during the last few years,
considerable efforts have been made to study such interaction problems between
gasdynamic field and radiation field and a new title, Radiation Gasdynamics,
has been suggested for this subject.
Even though radiative transfer has been studied for a long time by astro
physicists, the interaction between the radiation field and the gadsynamic field
has been only extensively studied recently. Since the knowledge of gasdynamics
and those of radiative transfer are reported usually in unrelated sources, it is desirable
to write a book which would furnish the readers some basic elements of both radative
transfer and gasdynamics and their interactions which would be very useful to
scientists and engineers who are interested in the high temperature flow problems,
and who are not familiar with both subjects. It is the author's hope that this
book will furnish the readers the basic elements of this new subject so that it is
useful for further study and research in this new field.
After the introduction of radiation gasdynamics, the author reviews the
fundamentals of radiative transfer in chapters n to IV and the gasdynamics in
chapters V and VI with special emphasics on the coupling terms between the
radiation terms and the gasdynamics. In chapter VII, the important parameters
of radiation gasdynamics are discussed.
In chapters VIn and IX, the flow problems of radiation gasdynamics based
on the continuum point of view are discussed with particular emphasis on the
wave motion, shock waves and heat transfer. In chapter X the kinetic theory
of radiating gas will be discussed. Because the photon gas moves with a speed
of light, relativistic effect must be considered. A brief discussion of relativistic
mechanics is included. In chapter X, we also treat some problems of rarefied
radiating gas, particularly the free molecule flow. Finally in chapter XI, the
radiative properties of high temperature gases will be briefly discussed with
special emphasis on the absorption coefficient of high temperature air and hydro
gen.
A large portion of the materials in this book was given by the author in a
seminar on Plasma Dynamics during the academic year 1962-1963 in the
VI Preface
Institute for Fluid Dynamics and Applied Mathematics, University of Maryland.
This seminar was joinly conducted by the author with Professors J. M. BURGERS
and T. D. WILKERSON. The author would like to express his appreciation to
Professors BURGERS and WILKERSON for many interesting discussions on this
subject; and to Professor M. H. MARTIN for his interest and encouragement. In
conclusion, the author takes this occasion to thank his wife, ALIOE YEN-LAN
WANG P AI, for her constant encouragement and help in the proof-reading during
the preparation of the manuscript.
College Park, Maryland, U. S. A.
January 15, 1966.
Shih-l Pai
Table of Contents
Page
Chapter I. Introduction................................................. 1
1. Radiation gasd ynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Thermal radiation effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3. Some thermal radiation phenomena ............................... 4
References ....................................................... 6
Chapter II. Fundamentals of Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1. Specific intensity ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2. The flux of radiation ............................................ 9
3. Energy density of radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4. The stress tensor of radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References ....................................................... 14
Chapter III. Equation 01 Transfer of Radiation. . . . . . . . . . . . . . . .. .. . . . . . . . . 15
1. Introduction .................................................... 15
2. Absorption coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3. Emission coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4. The equation of radiative transfer ................................ 21
5. A solution of the equation of radiative transfer .................... 23
References ....................................................... 23
Chapter IV. Radiative Equilibrium. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1. Introduction .................................................... 25
2. Kirchhoff's law of radiation ...................................... 25
3. Wien's displacement law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4. Planck's radiation law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5. Stefan-Boltzmann's law of radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32
6. Adiabatic changes in an inclosure containing matter and radiation. . . .. 32
7. Local thermodynamic equilibrium ............................ . . . .. 34
References ....................................................... 34
Chapter V. Fundamental Equations 01 Radiation Gasdynamics . . . . . . . . . . . . .. 35
1. Introduction .................................................... 35
2. Equation of state ............................................... 36
3. Equation of continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36
4. Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36
5. Equations of energy ............................................. 37
6. Equation of radiative transfer. . . . . . . . . . . . . . . . . . . .. . .. . . . . . . . . . . .. 37
7. General remarks on the fundamental equations. . . . . . . . . . . . . . . . . . . .. 38
8. Case of small mean free path of radiation . . . . . . . . . . . . . . . . . . . . . . . .. 39
9. Case of finite mean free path of radiation. . . . . . . . . .. . . . . . . . . . . . . . . . 41
10. One dimensional radiative transfer ................................ 45
11. The exponential integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49
Chapter VI. Boundary Conditions of Radiation Gasdynamics .. .. . . . .. . . . . . . . 51
1. Introduction .................................................... 51
2. Boundary conditions of gasdynamic field. . . . . . .. . . . . . . . . . . . . . . . . . . . 51
3. Boundary conditions of radiation field. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 52
4. Smooth surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53
VIII Table of Contents
Page
5. Rough surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6. Radiative transfer between two opaque parallel plates .............. 59
7. Emissivity of a constant temperature gas layer. . . . . . . . . . . . . . . . . . . . . 61
8. Radiation slip at finite mean free path of radiation ................ 63
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 64
Chapter VII. Similarity Parameters of Radiation Gasdynamics . . . . . . . . . . . . . . 66
1. Introduction .................................................... 66
2. Dimensional analysis and :7l-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3. Non-dimensional equations of radiation gasdynamics . . . . . . . . . . . . . . . . 69
4. Important parameters of radiation gasdynamics .................... 71
5. Some further remarks for the non-dimensional parameters. . . . . . . . . .. 75
References- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80
Chapter VIII. Waves and Shock Waves in Radiation Gasdynamics.......... 82
1. Introduction .................................................... 82
2. Wave of small amplitude in an optically thick medium............. 82
3. Wave of small amplitude in an radiating gas of finite mean free path
of radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92
4. Shock waves in an optically thick medium ........................ 98
5. Shock wave structure in an optically thick medium ................ 103
6. Shock wave in a medium of finite mean free path of radiation ..... 106
7. Flow field behind shock waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 112
References ....................................................... 120
Chapter IX. Heat Transfer in Radiation Gasdynamics ..................... 122
1. Introduction .................................................... 122
2. Radiative heat transfer in a non-absorbing medium ................ 123
3. Radiative heat transfer in an absorbing medium ................... 125
4. Heat transfer by simultaneous heat conduction and radiation in an ab-
sorbing medium ................................................. 126
5. Radiative processes in the atmosphere ............................. 130
6. Flow between two parallel plates in radiation magnetogasdynamics ... 133
7. Boundary layer flow in radiation gasdynamics ...................... 141
8. Stagnation point heat transfer in radiation gasdynamics ............ 149
9. Miscellaneous problems of heat transfer in radiation gasdynamics .... 161
References ....................................................... 162
Chapter X. Kinetic Theory of Radiating Gases ............................ 164
1. Introduction .................................................... 164
2. Molecular velocity and molecular distribution functions .............. 165
3. Relativistic mechanics ............................................ 166
4. Boltzmann aquation for material particles .......................... 169
5. Boltzmann equation for photons .................................. 174
6. Conservation equations ........................................... 175
7. Radiation stresses and radiation energy density .................... 180
8. Local thermodynamic equilibrium ................................. 184
9. Rarefied radiation gasdynamics ................................... 186
10. Free molecule flow .............................................. 191
References ........................................................ 196
Chapter XI. Radiative Properties of High Temperature Gases............... 197
1. Introduction .................................................... 197
2. Classical theory of absorption and emission of radiation ............. 198
3. The quantum theory of radiation ................................. 202
4. Spectroscopy of high temperature gas ............................. 206
5. The absorption coefficient of high temperature gases.... . . . . . . . . . . .. 208
6. Scattering coefficient of radiation ................................. 211
7. Planck and Rosseland mean absorption coefficients of air and hydrogen 214
8. Some experimental investigations of opacity of gases ............... 217
9. Non-equilibrium radiation ......................................... 217
References ....................................................... 218
A List of Important Symbols ....................................... 220
Author Index ........................................................ 224
Subject Index ....................................................... 226
Chapter I
Introduction
1. Radiation Gasdynamics. For the flow of a compressible fluid, we have
to study the fluid mechanics simultaneously with the heat transfer problem.
There are three basic modes of heat transfer:
(i) The transfer of heat by convection in fluids in a state of motion,
(ii) The transfer of heat by conduction in solids or fluids, and
(iii) The transfer of heat by radiation which takes place with no material
carrier.
In general, all these three modes of heat transfer occur simultaneously. If
the temperature is not too high and the density of the fluid is not too low, the
heat transfer by radiation is usually negligible in comparison with the heat
transfer by conduction or by convection. Hence in ordinary gasdynamics, the
thermal radiation effects are always neglected. In the present space age, we are
concerned with many technological developments in hypersonic flight, gas cooled
nuclear reactors, power plants for space exploration needs, fission and fusion
reactions in which the temperature is very high and the density is rather low.
As a result, the thermal radiation becomes an important mode of heat transfer.
A complete analysis of very high temperature flow field should be based upon
a study of both the gasdynamic field and the thermal radiation field simultane
ously. We use the term "Radiation Gasdynamics" for such a new branch offluid
mechanics (7, 11, 191).
The study of radiation in high temperature gases has been made by physicists
for a long time. At the turn of the present century, Planck found the correct
theory of radiation (12). It is not the intention of the author to discuss the
physics of radiation but only the influence of thermal radiation on the flow
field of high temperature gases, i.e., the radiative heat transfer. The radiative
heat transfer has been extensively studied by astrophysicists (2, 3, 5, 15, 17)
because the spectral distribution of the radiation from stars, planets etc. is the
main experimental verification of astrophysical analyses. However in most of
the radiative transfer problems studied by the astrophysicists, the interaction
between the gasdynamic field and the radiation is negligibly small. Hence we
may assume that the temperature distribution is independent of the radiative
heat transfer and we study the radiative transfer under the known distribution
of temperature. This procedure is not accurate if the rate of heat transfer by
radiation is of the same order of magnitude as those by convection and con
duction. On the other hand, if the rate of heat transfer by radiation is small,
1 The number refers the number of references at the end of each chapter.
Pai, Radiation Gas Dynamics
2 Introduction
we may estimate the thermal radiation from the temperature distribution in the
flow field without radiation effect. Since many basic concepts of radiative
transfer are not familiar to many engineers and scientists in the field of aero
dynamics, we shall discuss these fundamental concepts of radiation in chap
ters II to IV.
The modern trend of aerodynamics is toward high speed and high temperature
as well as low density and high altitude (1, 6). One of the extensive revicws
of the state of art of aerodynamics was given by the late Professor Thcodore
von K .. \RMAN in 1961 (18). Even as late as 1961, the thermal radiation effects
are not important in many practical problems of hypersonic flight. For an ICBM,
the maximum radiative heat transfer is only 1/10 of that of aerodynamic heat
transfer. Hencc the interaction between the aerodynamic field and the radiation
field is not important. As we sluiIl see later, at higher reentry speeds such as
that for a Mars probe (8), the interaction of aerodynamic field and radiation is no
longer negligible. It is the main purpose of this book to discuss the effects of
thermal radiation in a very high temperature gas flow.
2. Thermal radiation effects. There are three different thermal radiation
effccts on the flow field of a high temperature gas which arc
(i) Radiation stresses,
(ii) Radiation energy density and
(iii) Heat flux by radiation.
The exact expression for these radiation terms are very complicated and
will be derived in later chapters. In order to show the relative importance of heat
transfer by radiation and that by convection, wc consider the case of sufficient
opacity that the radiation can be considered as being trapped in the fluid and
it is close to the radiative equilibrium condition. Under this condition, the
radiation terms are as follows:
(i) Radiation pressure (cf. chapter II, § 4). The only components of the
radiation stresses which differ from zero are the radiation pressure PR which
may be expressed as
1
PR= --aRT4 (1.1)
3
where T is the temperature of the gas in OK and aR is known as the Stefan
Boltzmann constant which is 7.67 X 10-15 erg-cm-3 - °K-4. This radiation
pressurc should be added to the gas pressure in order to get the total pressure
at each point of the flow field.
(ii) Radiation energy density (cf. chapter II, § 3). The radiation energy den
sity per unit mass of the fluid is
(1.2)
where p is the density of the fluid. We should add ER/p to the internal energy
Um =: cvT of the fluid where Cv is the specific heat at constant volume of the
fluid.
In order to show the relative magnitude of radiation energy density and thc
internal energy, we calculatc these values for air at an altitude of 72 km or
Thermal radiation effects 3
45 miles. The density of the air behind a normal shock at this altitude in a
hypersonic flow is about p = 1.23 X 10-6 gr/cm3, i.e., 10-3 of the density at
the standard sea level value. We take Cv = 7 X 106 erg gr-OK which is the
value for a diatomic gas such as air. The radiation energy density and the internal
energy at various temperatures are shown in Table 1.
Table 1. Radiation energy density V8 internal energy
T OK............ 103 104 105 106
E.f1/p, erg gr ..... 6 X 103 6 X 107 6 X 1011 6 X 1015
cvT erg gr ....... 7 X 109 7 X 1010 7 X 1011 7 X 1012
It should be noted that the estimate given in table 1 may be called the astro
physical estimate in which if two quantities are within one or two order of magni
tude, they are not negligible with respect to one another. Only when the difference
between them is of several order of magnitude, the smaller one is negligible.
Hence from table 1, we see that when the temperature is less than 104 OK, the
radiation energy density is negligibly small in comparison with the internal
energy under this given density while for high temperature, the radiation energy
density may be of the same order of magnitude or even larger than the ordinary
internal energy of the gas. If we increase the density of the fluid, the temperature
above which the radiation energy becomes important increases too. It is easy
to show that whenever the radiation energy density is not negligible, the radiation
pressure is also not negligible. Most of the flow problems of current interest such
as reentry problem have the conditions that the maximum temperature is of
the order of 104 OK and the density of the fluid is higher than 1.23 X 10-6 gr. cc.
Hence the radiation prei:>sure and the radiation energy density are still negligible.
However, for the cases of higher temperature such as in the fusion research where
106 OK is a relative low temperature and for the cases of lower density such as
in the outer space, the radiation pressure and radiation energy density should
be considered in the analysis of the flow problem.
(iii) Flux of radiation. For the radiation equilibrium condition, the radiation
flux is given by the formula (cf. chapter II, § 2)
ER crT4
C
qR=----=- (1.3)
4 P P
where c is the velocity of light which is 3 X 1010 cm/sec. in vacuum. The factor
cr = c aR/4 = 5.75 X 10-5 erg-cm-2 - sec-1 - °K-O is also referred to as the
Stefan-Boltzmann constant which is used quite often in the literature to show
the radiative heat transfer. The term qR should be compared with the heat
flux by convection, qv, i.e.,
(1.4)
where U is a typical flow velocity. If we take the same conditions used in table 1
and assume a mean flow velocity U = 104 m/sec. which is an average speed
of a satellite, we obtain the values in table 2 as follows:
