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RATIONAL AND PRACTICAL
NONEQUILIBRIUM THERMODYNAMICS

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSlTY
SARA ERB INGLE
1971

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This is to certify that the
thesis entitled
RATIONAL AND PRACTICAL

NONEQUILIBRIUM THERMODYNAMICS

presented by

Sara Erb Ingle

has been accepted towards fulfillment
of the requirements for

 

 

 

Ph. D. degree in Chemistry
Major professor
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ABSTRACT

RATIONAL AND PRACTICAL
NONEQUILIBRIUM THERMODYNAMICS

“By
Sara ErbIngle

A fundamental theory of nonequilibrium thermody-
namics based upon rational mechanics is presented, is
compared with traditional nonequilibrium thermodynamics,
and is applied to two practical problems: heat conduc-
tion of pure fluids and the Dufour effect in fluid mix-
tures.

The rational mechanical theory of nonequilibrium
thermodynamics is developed logically and systematically
from the conservation laws of mechanics and the Second
Law of thermodynamics. Full account is taken of the
kinetic energy of diffusion, and a variety of sets of
independent variables is investigated. For the case of
linear reSponse functions, a full, illuminating comparison
is made between the rational approach and the traditional
approach. It is found that the two approaches give simi-
lar, but not identical, results.

Equations directly applicable to experimental

determination of thermal conductivity are obtained from

Sara Erb Ingle

macroscopic transport equations by means of a well—defined,
self-consistent perturbation scheme. The barycentric ve-
locity and variability of all properties are included ex-
plicitly.

By introduction of a second perturbation scheme,
the transport equations for the Dufour effect, which is
the deve10pment of a temperature gradient due to diffusion,
have been solved for geometrically well-defined cells
which have either all adiabatic walls or adiabatic lateral
walls and diathermic ends. The heat of mixing and varia-
tion of other physical properties are fully included.
For typical nonelectrolytes, the temperature difference
produced by the Dufour effect could be as large as 0.2
degrees for a diffusing mixture with initial mass frac-
tion difference of 0.8. These results can be used (1)
to design experiments to test the Onsager heat-matter
reciprocal relation and (2) to avoid undesired tempera-

ture variations in diffusion experiments.

RATIONAL AND PRACTICAL

NONEQUI LI BRI UM THE RMODYNAMI CS

BY

SARA ERB INGLE

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1971

To Jim

ii

ACKNOWLEDGMENTS

I wish to thank the National Science Foundation
for the Predoctoral Fellowships which I held throughout
my graduate career. I would also like to thank Michigan
State University for a First-Year Fellowship.

I am deeply grateful to my research director,
Dr. Frederick Horne, for his consultation and direction,
for his provoking research ideas, and for keeping my
weight down by his being so hard to find.

I appreciate the helpful suggestions and criti-
cisms of Dr. James Dye, who served as my second reader.

My thanks are due to Mr. James Olson and Mr.
Ping Lee for their helpful discussions of research. I
also thank Mr. William Waller and Mr. Richard Vandlen
for their help with my computer problems.

Finally, I would thank the entire department for
putting up with "Grumpy Old Sally," and particularly my

husband Jim who has borne the brunt of my frustrations.

iii

TABLE OF CONTENTS

Page
LISTOFTABLES......"............ vi
LIST OF FIGURES O O I I O O O O O O O I O O O O O O Vii
LIST OF SYMBOLS FOR CHAPTER II . . . . . . . . . . X
Chapter
I 0 INTRODUCTION 0 O O O O O O I O O O O O O O I 1
II 0 RATIONAL MECHANICS O O O O O O O O O O O O O
1. Introduction . . . . . . . . . . . . . .
2. Thermodynamic Process and
the Equations of Balance . . . . . . . . 12
3. Constitutive Assumptions . . . . . . . . 18
4. Ordinary Binary Fluid Mixtures . 23
5. Transport Equations in the Independent
Variables T, p, W1 . . . . . . . . . . . 32
6. A New Rational Mechanical Derivation . . 38
7. Comparison of Several Theories
for Binary Mixtures of Fluids. . . . . . 57
8. Expressions for EntrOpy Production and
the Entropy Balance Equation . . . . . . 62
9. Conclusions and Discussion . . . . . . . 71
III. THERMAL CONDUCTIVITY. . . . . . . . . . . . 77
1. Introduction . . . . . . . . . . . . . . 77
2. Transport Equations. . . . . . . . . . . 81
3. Approximation Methods. . . . . . . . . . 83
4. Practical Results. . . . . . . . . . . . 91
IV. THE DUFOUR EFFECT . . . . . . . . . . . . . 102
1. Introduction . . . . . . . . . . . . . . 102
2. Transport Equations. . . . . . . . . . . 106
3. Perturbation Scheme. . . . . . . . . . . 109
4. Zeroth-Order Solutions
for an Adiabatic System. . . . . . . . . 112
5. Higher Order Solutions
for an Adiabatic System. . . . . . . . . 116
6. Solutions for a Cell with
Diathermic Ends. . . . . . . . . . . . . 122
7. Calculated Temperature Variations. . . . 123
8. Discussion . . . . . . . . . . . . . . . 145

iv

Page
BIBLIOGRAPHY O O O O O O O O O O O O O O O O O O O 158
APPENDIX A. Some Equations of Thermostatics . . . 161

APPENDIX B. A Rational Mechanical Development
Of the TIP Theory I O C O O O O O O O 162

APPENDIX C. The Temperature Solution T1 for
Thermal Conductivity in a
Pure Fluid I O O O O C O O O O O O O 166

APPENDIX D. The Temperature Solutions T55 for
the Dufour Effect in Adiabatic Cells
and Cells with Diathermic Ends. . . . 168

APPENDIX E. The Temperature Equations and
Solution for T555 for the Dufour
Effect in Adiabatic Cells . . . . . . 174

APPENDIX F. The Relationship of Heat of
MiXing to 3(Hl - H2)/3W1. o o o o o o 178

LIST OF TABLES
Table Page
1. Approximate physical properties of CCl4. . . 95

2. Typical values of physical properties for
a binary mixture of organic liquids. . . . . 124

vi

LIST OF FIGURES
Figure Page

1. Plot of To versus (z/a) for CCl4 at 30 s,
60 s, 90 s, 120 s, and 150 s . . . . . . . . 97

2. Plot of T1 versus (z/a) for CCl4 at 30 s,
60 s, 90 s, 120 s, and 150 s . . . . . . . . 100

3. Temperature variations in an adiabatic

cell for Afifi = 0.0, pDQi = 3 x 10‘2J

-13-1, K = 2 x lO-lJ m-ls—lK-l, and
mi = 0.8 at (t/e) = 0.01, 0.05, 0.09,
0.13, and 0.17 . . . . . . . . . . . . . . . 129

m

4. Temperature variations in an adiabatic

cell for 085 = 0.0, pDQi = 3 x 10‘20

‘1s-1, K = l x lO-lJ m-ls-lK-l, and
mi = 0.8 at (t/e) = 0.01, 0.05, 0.09,
0.13, and 0.17 . . . . . . . . . . . . . . . 132

m

5. Temperature variations in an adiabatic

cell for Afih = - 6.5 x 1030 kg‘l, 9991

= 3 x lO-ZJ m—ls-l, K = 2 x lO-lJ m-ls

K'l, and mi = 0.8 at (t/e) = 0.01, 0.05,

0.09, 0.13, and 0.17 . . . . . . . . . . . . 134

-1

6. Temperature variations in an adiabatic

cell for (t/e) = 0.1, 600*l = 3 x lo'zJ

m-ls-l, K = 2 x lO-lJ m-ls_1K-l, and w:
= 0.8 with Aim = - 19.5 x 103, - 6.5 x 103,

0.0, 6.5 x 1033 kg"1 . . . . . . . . . . . . 137

vii

Figure

7.

10.

11.

12.

Temperature variations in an adiabatic

cell for (t/0) = 0.1, Afih = - 6.5 x 1030
kg-l, K = 2 x lo-lJ m-ls_lK-l, and mi = 0.8
with “”01 = 6 x 10'2, 3 x 10‘2, 3 x 10‘3,
and -3 XlO-ZJ mnls"l . . . . . . . . . . .
Temperature variations in an adiabatic

cell for (t/G) = 0.1, AHm = 0.0, K =

2 x 10-1J m 1s T1 1, andm pDQi = 3 x lO-ZJ

m‘ls"1 with w; = 0.8, 0.5, 0.2, 0.05, and

0.01 . . . . . . . . . . . . . . . . . . .

Temperature variations in a cell with
diathermic ends for A3 = 0.0, pDQ* =

-2 -1 -1 m -1 .. -1 -l
3 x 10 J m s , K = 2 x 10 J m s K ,
and mi = 0.8 at (t/e) = 0.05, 0.09, 0.13,

and 0.17 C O O O O O O O O C O I O O O O 0

Temperature variations in a cell with
diathermic ends for AHm = 0.0, pDQ* =
3 x 10 2J m 1s 1, K = l x lo—lJ m‘ s-lK-l,
and mi = 0.8 at (t/e) 0.05, 0.09, 0.13,

and 0.17 . . . . . . . . . . . . . . . . .

Temperature variations in a cell with

diathermic ends for Afifi = - 6. 5 x 103J
kg-l, 000i = 3 x 10 2J m 1s 1, K =

2 x 10 1J m , and wl = 0.8 at

(t/O) = 0.05, 0.09, 0.13, and 0.17 . . . .

-ls- -1K -1

Temperature variations in a cell with

diathermic ends for (t/G) = 0.1, Afih
= - 6.5 x 1030 kg‘l, K = 2 x 10‘10 m
s‘lx'l, and w? = 0.8 with AER = - 19.5
x 103, - 6.5 x 103, 0.0, and 6.5 x 103 . .

-1

viii

Page

139

142

144

147

149

151

Figure Page

13. Temperature variations in a cell with

diathermic ends for (t/G) = 0.1, Afih

= - 6.5 X 103J kg-l, K = 2 x lO-IJ
-1 -l -l o .

m s K , and ml = 0.8 With pDQi =
6 x 10'2, 3 x 10'2, 3 x 10'3, and

— 3 x lO-ZJ m-ls-l . O O C O C C C C C C O O 153

14. Temperature variations in a cell with
diathermic ends for (t/O) = 0.1, Afih

= — 6.5 x 1030 kg‘l, K = 2 x 10‘10
m-ls-lK-l, and pDQi = 3 x 10-2J m-ls-l,
with 03 = 0.8, 0.5, 0.2, 0.05, and 0.01. . . 155

ix

LIST OF SYMBOLS FOR CHAPTER II

Roman miniscules

b: -- external specific forces acting on component a
(2.5)*

cu -— number density of component a (2.19)

d:. -- symmetric part of the gradient of the component

3 velocity (3.3)

fi -- total entropy flux (3.2)

j: -- diffusion flux of component a relative to the
barycentric velocity (5.2)

ki -— entrapy flux not due to heat flow (3.2)

mi -- specific interaction force exerted on component 1
by component 2 (6.7)

m: -- Specific interaction forces acting on component a
due to interaction with other components (2.5)

p -- a constitutive coefficient (6.47)

qk -- a heat flux (Bartelt's) (2.8)

qfi —- heat flux for total energy balance equation (2.10)

r -- Specific radiative heat supply (2.8)

ui -— diffusion velocity of component 1 relative to
component 2 (6.2)

u? -- diffusion velocity of component a relative to the
barycentric velocity (2.1)

vk -- barycentric velocity (2.2)

v: -- velocity of component a (2.2)

 

*These numbers refer to the equations in Chapter II
where the symbols first appear.

X

w -- mass fraction of component a (5.2)

Xk -- space coordinate (2.1)

Roman majiscules

AI -- specific internal Helmholtz free energy (3.11)
Cb -- specific heat capacity at constant pressure (5.21)
E§ -- specific heat capacity at constant volume (5.20)
ny -- constitutive coefficient (4.4), (4.6)
ny -- constitutive coefficient (6.44), (6.45)
Dy -- combinations of constitutive coefficients (5.12)
E -- total energy (2.9)
EB -- internal energy (total energy less bulk kinetic
energy) (6.8)
EI -- specific internal energy (total energy less
kinetic energy) (2.8)
E -- partial specific internal energy of component 0
°‘ (5.21)
*
Fél) -- interaction forces of Bearman and Kirkwood
°‘ (2.18)
fig -- partial specific enthalpy of component a (5.22)
K -- a constitutive coefficient (4.9)
Ma -- molecular weight of component (2.17)

P -- pressure (4.22)

P' —- (= P) a constitutive coefficient (6.46)

Qi -- a generalized heat flux (7.2)

S —- specific internal entropy (3.9)

Sa -- partial specific density of component (5.14)
T -- temperature

Tij -— total stress tensor (2.10)

xi

Tij -- sum of the partial stress tensors (6.7)
V -- specific volume (4.38)

V -- specific scalar potentials acting on component 0

°‘ (2.14)
Va -- partial specific volume of component a (5.5)
XA —- a variable which must be zero at equilibrium
(4.17)
X? -- external forces of Bearman and Kirkwood (2.17)
Zi -- any force in the entropy production which in-

cludes the gradient of the chemical potential
difference (7.1)

Greek miniscules

ax —- viscosity coefficients (6.47)

8 -- thermal expansivity (5.5)
8' -- isothermal compressibility (5.5)

0 -- Kronecker delta (3.15)

ij
n -- shear viscosity coefficient (6.46)

”GB -- viscosity coefficients (4.8)

u -- viscosity coefficient (4.7)

u' -- a viscosity coefficient (6.48)

“a -- specific chemical potential of component a (4.38)

u' -- chemical potential of component a plus the exter-

nal potentials (5.12)

”11 -- derivative of the chemical potential with re-
spect to wl (5.17)

v -- number of components

Egj -- symmetric part of the partial stress tensor (3.1)

Na -- constitutive coefficients (4.8)

xii

p -- density of component a (2.1)

a
p -- density

a

Gji -- partial stress tensor of component a (2.5)

BKoqi -- partial stress tensors of Bearman and Kirkwood
3 (2.16)

Tij -- a difference of partial stress tensors (6.10)

ng -- antisymmetric part of the partial stress tensor

of component (2.6)

0 -- the coefficient of bulk viscosity (6.46)

¢aB -- viscosity coefficients (4.8)

Xx -- viscosity coefficients (6.46)

mi. -- linear combination of the antisymmetric parts
3 of the gradients of component velocity (3.3)

Greek majiscules

Ei -- the force of rational mechanics and of Bearman
and Kirkwood which corresponds to the diffusion

flux (7.4)
nij -— viscous pressure tensor (5.5)
0 -- rate of specific entropy production (3.9)
Qxy -- phenomenological coefficients (7.1)

xiii

CHAPTER I
INTRODUCTION

Few chemists have an academic knowledge of the
field of nonequilibrium thermodynamics, yet we encounter
it in almost every chemical phenomenon we study. The
treatment of processes as equilibrium phenomena is often
reasonable, but the validity of such assumptions can be
proven only by exploring the phenomena from nonequilibrium
vieWpoints. Moreover, the vast majority of physical phe-
nomena are not at equilibrium. In order ultimately to
understand such important phenomena as metabolism, at one
extreme, and spectrosc0py, at the other, we must treat
them as dynamic nonequilibrium phenomena.

To these ends we use rational mechanics to examine
the theory of nonequilibrium thermodynamics, and we com-
pare this theory to the traditional macroscopic nonequi-
librium theory which has been used in practical applica-
tions for many years. In the development either of an
equilibrium or of a nonequilibrium theory, assumptions
are made in order that the theory be viable; however, it
is important that one knows the limitations of his theory

as determined by these assumptions. Moreover, we make

actual application of the full theory to two problems of
considerable practical importance: namely, we deve10p
complete phenomenological theories for thermal conduction
experiments and for the Dufour effect.

The rational mechanics approach to nonequilibrium
thermodynamics consists in the logical mathematical deriva-
tion of the consequences of the laws of physics and the
Second Law of Thermodynamics when these laws are applied
to some well-defined physical system. We restrict our
work to fluids and, in general, define a rather simplistic
(to the mathematical mind) system. Yet what we are con-
cerned with as chemists is what we can measure; regrettably,
the translation of a mathematical truth into an expression
for an experimental observable is no easy process. If we
had a handy entrepy meter, nonequilibrium thermodynamics
might not be so formidable, but even this quantity, funda-
mental to the definition of equilibrium and nonequilibrium,
must be translated into observable quantities. We cannot
deny that our mathematical statements are true; what we
seek is to know what they can tell us about the world we
observe.

Nonequilibrium thermodynamics in the form of a
theory to fit empirical observations has been used since
the Nineteenth Century, but its recent growth is due to
the pioneer work of Onsager in 1931. A coherent, practical

theory has been developed by Prigogine (1947, 1955),

Kirkwood and Crawford (1952), de Groot (1945, 1955), Meix-
ner and Reik (1959), Fitts (1962), and de Groot and Mazur
(1962). This theory has been the foundation of modern
applications in nonequilibrium thermodynamics. More re-
cent still is the theory of thermodynamics and rational
mechanics which has been develOped by mathematicians of
the first rank. Much work is still being done in the
field of continuum mechanics, but among its greatest con-
tributions we can name the works of Truesdell (1957),
Truesdell and Toupin (1960), and Coleman and Noll (1963).
Our second chapter deals with rational mechanical
theories of binary fluid mixtures. Though there exists a
great body of works on such fluids, the significance of
the theory to the exPerimentalist has not been explored
and an explicit comparison of the rational mechanical
theories to the practical theory has not been made. Such
extensions and comparisons are the subject of Chapter II.
The third and fourth chapters rest heavily on the
practical theory. Both chapters are eXperimentally ori-
ented in that they concern the solution of nonequilibrium
prdblems under particular initial and boundary conditions
and, therefore, experiments could be designed using the
results here. In Chapter III we present the theory of
thermal conductivity for a pure fluid in a flat plate or

thermal diffusion cell to which a temperature gradient is

applied. A self-consistent, well-ordered perturbation

scheme enables us to account for variations of thermodynamic
quantities in Space and time and to predict accurately tem-
perature variations in the cell. Hopefully, these results
will allow experimentalists to improve their accuracy.
Chapter IV is a theoretical investigation of the
Dufour effect in liquids. This phenomenon is the produc-
tion of a temperature gradient in response to a concentra—
tion gradient. The effect is quite small in liquids and
is complicated by heat of mixing, and, thus, it has been
ignored for many years. Our theory provides a quantita-
tive prediction of the temperature variations caused by
the Dufour effect and serves as a guide to the experi—
mentalist in measuring the effect. Philosophically, the
reason for measuring it is that such an experiment would
test the previously untested heat-matter Onsager reciprocal
relation. The most important practical usefulness of a
full theory of the Dufour effect is that it gives guide-
lines to the diffusion experimentalist who wishes to avoid

temperature effects.

CHAPTER II

RATIONAL NONEQUILIBRIUM THERMODYNAMICS

1. Introduction

 

In recent years many investigations of the thermo-
dynamics of binary mixtures of nonequilibrium fluids have
been made based on the works of Coleman and Noll (1963),
Coleman and Mizell (1963, 1964), Truesdell (1957), and
Truesdell and Toupin (1960). Typical of these are the
works of Muller (1967, 1968), Bartelt (1968), Doria (1969),
Bartelt and Horne (1970), and Truesdell (1969). The logi-
cal superiority of these methods, often referred to as
rational mechanics, over the traditional heuristic ap-
proach called "thermodynamics of irreversible processes"
(TIP) cannot be questioned, but it is also true that the
TIP theory as exemplified by Fitts (1962), de Groot and
Mazur (1962), Gyarmati (1970), and Haase (1969) has been
most useful and has enjoyed wide, successful application.
A full comparison of TIP results with those derived from
rational mechanics is clearly desirable, although most of
the eXpounders of rational mechanics disdain to make such
a comparison. We show that the two approaches lead to

very similar results and that the practitioners of TIP,

with their practicality and intuition, have arrived at
almost the same results as the rational mechanists, with
their mathematical rigor and integrity. In the course
of making the comparison, we also illuminate the meaning
and significance of various parts of both theories.

The essential difference between the rational
mechanics approach and the TIP approach is implied in the
word "rational." The rational mechanics approach is based
on known balance (or conservation) equations and the re-
sults of these laws are logically and systematically de-
rived. There is no point at which the rational mechanist
can interject his intuition as to how the physical system
being described ought to behave; this is, in fact, why
rational mechanists abhor the practical theory while, at
the same time, the users of TIP have difficulty in grasp-
ing the physical significance of rational mechanics.

The assumption essential to rational mechanics is
the Second Law of Thermodynamics. Further assumptions,
called constitutive assumptions, are made after a set of
variables has been chosen for the physical system. These
constitutive assumptions are equations which mathematically
describe how dependent quantities respond to the selected
independent variables. Initially, all constitutive assump-
tions obey the principle of equipresence; i;g;, every con-
stitutive equation depends on the same independent variables

unless this dependence contradicts the symmetry of the

material, the entropy inequality, or fundamental mathe-
matics. Through such contradictions the response functions
are limited to dependence only on certain independent
variables or combinations thereof.

The theory of TIP likewise depends on balance
equations and on the Second Law, but the phenomenological
equations, which correspond to the constitutive equations,
are not rigorously developed. The entropy production ob—
tained is said to be a sum of products of "fluxes" and
"forces." Just what is a flux and what is a force is de-
cided by intuition. Moreover, the fluxes are always
written only as linear functions of the forces in the
phenomenological equations. All thermodynamic functions
are taken to be the same functions of a set of state
variables as they are in thermostatics.

The work of Muller (1967, 1968) was the first suc-
cessful formulation of a theory for binary fluids through
a rational mechanical development. It was from this theory
that both the theory of Doria (1969) and that of Bartelt
(1968) and Bartelt and Horne (1970) were developed. The
principles layed down by Coleman and Noll and extended to
mixtures by Mfiller are used by both. The work of Bartelt
and Horne is particularly valuable because through care-
ful selection of variables it attained a bilinear form
for the entropy production and because it gives more

practical equations (i.e., transport equations) than does

the more general theory of Doria. Bartelt and Horne pre-
sented a linear theory, in that the constitutive equations
are assumed to be linear functions of those independent
variables which are not scalar state variables, whereas
the theory of Doria is not restricted to the linear case.
Bartelt did consider general non-linear constitutive
equations, but the consequences of the Clausius-Duhem
inequality were not fully taken into account in these
general equations.

The choice of independent variables greatly af-
fects the appearance of any rational mechanical theory;
the choices even lead to some contradictory results in
the limit of the linear theory. No matter how rigorously
the linear approximation is made with respect to one set
of variables, it is still an approximation, and an equally
rigorous linearization with respect to another set of
variables is not the same approximation. There is an
Obvious similarity between, on the one hand, the multitude
of sets of conjugate fluxes and forces which TIP boasts
and, on the other hand, the multitude of sets of indepen—
dent variables with respect to which the rational mecha-
nist can linearize his equations. The variables in
each theory are selected by trial and error in order to
attain the result in its best possible form for some con-
text. The choices of TIP have traditionally been experi-

mentally motivated; those of rational mechanics are

mathematically motivated. The mere choice of variables
even before linearization can cause varying results from
the Clausius-Duhem inequality (Truesdell, 1969) if an
insufficient number of variables is used.

The variables of Bartelt and Horne give a concise
bilinear form for the entropy production, but, unfortunately,
they do not lead to linear constitutive equations which are
fully comparable to those found in TIP. The chief diffi-
culty appears in attempting to relate the nine viscosity
coefficients of the rational mechanics approach to the
two more familiar coefficients of bulk and shear viscosity.

Doria's variables allow him to give rigorous mathe-
matical interpretations of the Clausius-Duhem inequality
together with proofs that some constitutive relations are
independent of certain independent variables. When Doria's
constitutive equations are linearized, however, one does
not obtain a bilinear form for the entropy production.

Many restrictions can, nevertheless, be obtained from the
general entropy inequality.

We present here a new rational mechanical develop-
ment of the thermodynamics of nonequilibrium binary fluid
mixtures. It is based on a set of variables which is a
mixture of those used by Bartelt and by Doria. Unlike
the work of Bartelt, the independence of the internal
Helmholtz free energy of the component density gradients

is proven in general rather than only in the linear case.

10

Our development is superior to others in that more of our
phenomenological coefficients can be given physical meaning,
and the nature of their source can be better understood.
Two of the nine viscosity coefficients are the recognized
shear and bulk viscosity coefficients. The entropy pro-
duction, though not bilinear in form, is readily comparable
to the entrOpy production of TIP.

In addition, this work serves as clarification and
extension of previous work by other authors. The implica-
tions of the transport theory of Bartelt and Horne are
examined, the great similarity of Bearman and Kirkwood's
(1958) microscoPic theory to the theories of rational
mechanics is demonstrated, the assumptions which must
necessarily be made in a rational mechanical development
in order to arrive at the results of TIP are illustrated,
the limitations of the postulate of local state1 are ex-
plored, and the similarities as well as the irreconcili-
able disparities among several approaches to nonequilib-

rium fluid mixture theory are discussed.

 

1"For a system in which irreversible processes are
taking place, all thermodynamic functions of state exist
for each element of the system. These thermodynamic quan-
tities for the nonequilibrium system are the same functions
of the local state variables as the corresponding equilib-
rium thermodynamic quantities."--Fitts (1962). This is
usually called the postulate of local equilibrium, which
is a misnomer. Kestin (1966) introduced the more apt ter-
minology, "postulate of local state."

11

Particular attention is given to the kinetic energy
of diffusion2 and its consequences. Rational mechanics has
generally included the kinetic energy of diffusion, but has
also failed to give transport equations which are as prac-
tical as those of TIP. The enigma of the contribution of
kinetic energy of diffusion has been considered only by
Gyarmati whose work is extremely lucid, but unviable. To
avoid the problems created by inclusion or exclusion of the
kinetic energy of diffusion he defines two total energies,
one from which one subtracts the bulk kinetic energy, the
other from which he subtracts the bulk and diffusion ki-
netic energies, thus arriving at the same internal energy
in both cases. Both total energies are subject to the
conservation of energy equation although they differ by
the kinetic energy of diffusion. This implies that the
kinetic energy of diffusion is conservative, which is
clearly an incorrect assumption. We feel that the logical
way to approach the subject is to define one total energy
and two internal energies, one of which includes the ki-

netic energy of diffusion.

 

2

v

to be 2 (l/2)mdy:, where ma is the mass of component a,
a=l

Ya is its velocity, and v is the total number of components.

Others, particularly practitioners of TIP, take the kinetic

energy of the volume element to be (l/2)my , where m is the

mass of the volume element and y is the velocity of the cen-

We take the kinetic energy of the volume element

v
ter of mass, mv = Z maya. The difference between the two
. V “:12 2 v 2 .
energies, E (1/2)ma(ya — y ) = 2 (1/2)ma(ya - y) , is the

a=1 a=l

kinetic energy of diffusion.

12

It is the definition of two internal energies which
presents a dilemma in the postulate of local state. To
which internal energy does a Gibbsian equation apply? Cer-
tainly the energy internal to the macroscopic volume ele-
ment under discussion includes the kinetic energy of dif-
fusion. However, we know that the internal energy of the
volume element changes due to work done on or by the volume
element and to heat transferred to or from the body. This,
too, would imply that internal energy includes the kinetic
energy of diffusion. The internal energy is a function of
three state variables (ng;, temperature, pressure, and
composition) in a binary fluid mixture; if kinetic energy
is part of this energy, then it, too, is a function of
three state variables, which seems unlikely. The postulate
of local state obviously deserves the further investigation

which we present here.

2. Thermodynamic Process and the Equations of Balance

 

In order to describe a thermodynamic process in
any system, it is necessary to know how the several func-
tions which are characteristic of the system change spa-
tially and temporally and to know the balance equations
which the system obeys. Bartelt (1968), whose notation
is slightly different from ours, gave as the generaliza-

tion for mixtures of the Coleman and Noll (1963)

13

definition: A process is a thermodynamic process for a
v—component mixture if it can be described by the follow-
ing set of 50 + 6 functionsB:

i. the component densities, pa

0 0 I O a
11. the component veloc1t1es, Vi

iii. the specific interaction forces, mg, acting on
component a due to interaction with other com-
ponents

iv. the specific external forces, bi, acting on
component a
v. the partial stress tensors, ogj
vi. the specific internal energy, EI
vii. the temperature, T
viii. the specific radiative heat supply, r
ix. the heat flux, qi
x. the specific entrOpy, S
xi. the entropy flux, fi'
These functions must satisfy the following 20 + 2 balance

equations for the system:

1. the balance of mass for each component4,

 

3Cartesian tensor notation is employed throughout
this chapter. Vectors are denoted by a single subscript,
v-, while tensors are doubly subscripted, tij' The summa-
t1on convention on indices is also used, i;g,,

3

v.T.. = v.T...
1 1] i=1 1 13

4The local time derivative (a/at) is related to the
substantial time derivative (d/dt) by:
(d/dt) = (B/at) + vi(3/3xi).

14

a -
dpa/dt + apauk/axk + pa(3vk/3xk) - 0, (2.1)
where u: is the diffusion velocity of component a defined
by
c _ a _
ui — V1 V1' (2.2)

where vi is the barycentric velocitys,

__ a
ovi - Zpavi. (2.3)

o = 200‘. loan: = 0. (2.4)

2. the balance of each vector component of linear

momentum for each component,

a a _ . a a
Boavi/at + 8(9 v - oji)/8xj - pmii-pabi, (2.5)

c a
a j 1

.v.
3. the balance of angular momentum for the mixture,

0 —
ZTij — 0, (2.6)

where sz is the antisymmetric part of the partial stress

a

 

13
a a a
Tij — (1/2)(Oij - Oji)’ (2.7)
5 v
2 = 2 , where v is the total number of components
c=1

in the system.

15
4. the balance of internal energy,
p(dEI/dt) + aqk/Bxk = pr — pimfiu:
+ Zo:j(3vg/3xj). (2.8)

Equation (2.8) results from the equation of conservation

of total energy after the balance equations for potential
energy and kinetic energy have been subtracted. The "total
energy" of Truesdell or Bartelt and that considered here

includes only the internal and kinetic energies:

01

_ a
E — EI + Zpavivi/Zp. (2.9)

The balance equation of this "total energy" is
_ a
p(dE/dt) + 8(q3 - viTij)/8xj — pr + Zpavzbi, (2.10)

where q; is a heat flux and Tij is the total stress tensor

defined by

T.. _ a _ a a

13 — 2(oij pauiuj)° (2.11)
.This total stress tensor is identical to the total stress
tensor of the microscopic theory of Bearman and Kirkwood
(1958). The balance of kinetic energy is derived from

(2.4) with the help of (2.3), with the result

(1 a. co: 0).
9d()oavgvi/29)/dt = 3[Z(oji - paujuiwi

+ (1/2)zpaugu:u:]/3xj - §c§i(avg/axj)

co as
+ pimiui + {pubivi (2.12)

16

Subtraction of (2.11) from (2.10) gives (2.8), wherein the

two heat fluxes are related by:

= * a _ a a a a a a
qj qj + 2(Oji pcujui)ui + (l/2)Xpaujui i‘ (2.13)

Only conservative external forces are considered

here; thus, in terms of a specific scalar potential, Va:
b“ = - av /8x (2 14)
i a i' '

The definition of internal energy is such that the
internal energy does not include any external force terms.
Inclusion of the conservative external forces is achieved
through addition of terms to the chemical potential of each
component. If terms for kinetic energy or external fields
are included explicitly or implicitly in definitions of
partial specific energies, it must be realized that the
sum of these energies will not give the internal energy
alone, but rather the internal energy plus all or some
portion of the kinetic energy and of the potential energy.

The sum of the momentum balance equations (2.5)
gives the total momentum balance equation which does not
depend on the component interaction forces, mg. This

means that the interaction forces are not all independent:
{m9 = 0. (2.15)

The validity of these balance equations is generally

accepted. One may work either with internal energy including

17

or not including the kinetic energy of diffusion so long
as he uses the balance equations appropriate to his choice.
The greatest confusion arises in reviewing TIP, where the
kinetic energy of diffusion is generally included in the
internal energy and, therefore, is implicit in all partial
quantities. Likewise, if one adopts the Gibbsian equations
for enthalpy and the free energies, he must specify to
which "internal" thermodynamic function he is applying
them.

To specify a thermodynamic process only the 40 + 5

a a 0

functions, 0 , v., m.,o:., E

0i 1 1 13 I' T, 91! Sr and f1, must be

specified while the v + 1 functions, b: and r, can be ob-
tained in terms of the other functions by use of the
balance equations (2.5) and (2.8).

We now make several identifications between the
present macrosc0pic theory and the microsc0pic theory of
Bearman and Kirkwood. The partial stress tensors of Bear-

BK a

man and Kirkwood, o ., are related to the macroscopic

in
partial stress tensors by

013' = Oij ‘ pcui

BK “ a a 9. (2.16)
J
The microsc0pic theory excludes radiative energy source
terms which are included in the macrosc0pic theory. The

. . a
conservative forces of Bearman and K1rkwood,Xi, are re-

lated to those of Bartelt by

(1"- =
xi - 0(3Va/3xi) M b.

a 1’ (2.17)

18

where Ma is the molecular weight. Likewise, for interaction

forces, the relationship is

(1)* _ a
Fic - (p/pa)Mami. (2.18)

Thus, the equation of Kirkwood and Bearman,

apavi/at — 8i cji pa(vjvi Vjvi vjviH/Bxj
-(1)* a
+ Ca ia + caxi, (2.19)

where ca is the number density of component a, is identical
to (2.5) when the identities, (2.16), (2.17), and (2.18)

are used.

3. Constitutive Assumptions

A material is defined by a constitutive assumption
which is a restriction on the processes admissible in a
body of that material. A set of constitutive equations
completely describes the response of the system to varia-
tions of the independent variables. A mixture of visco-
elastic materials susceptible to diffusion and heat con-

duction is completely determined by the values of 3v + 4

. a a a .
functions. AI, S, qi' ki’ gij, Tij, and mi, where AI is

the specific internal Helmholtz free energy, and gij is

the symmetric part of the stress tensOr, 09.,

13

a _ a a
gij - l/2(oij + oji). (3.1)

19

The flux, k1, is the difference between the total entropy

flux and the entrOpy flux due to heat flow,

ki = fi - qi/T. (3.2)

The 30 + 4 functions selected here could be re-
placed by any equivalent set of 3v + 4 functions. The
choice made in this section is motivated by those terms
which appear in the balance equations of Section 2 and in
the entrOpy balance equation.

According to Bartelt's definition, a binary mixture
of fluids is one whose constitutive functionals are func-

tions of the independent variables

91.02.T,(301/3xi).(apz/Bxi),u1.dij.dij.wij. (3.3)

where dgj and wij arise from the separation of avg/3x

into its symmetric and antisymmetric components. The

i

symmetric tensor is:

a _ a a
dij - (1/2)(8vj/8xi + avi/ij) (3.4)

and the antisymmetric tensor is given by

mi]. = [9/2(p - pmnuavg/axi - avg/3x3.)

- (avj/Bxi - aviaxj)

\)
B _ B
+ (l/p)8£1[(apB/3xi)uj (BOB/ij)ui]} (3.5)

20

or by

“ij = (1/2)[3(pu%/02)/3Xi - 3(oui/02)/3xj]. (3.6)

The terms added to the antisymmetric components of avg/3xi
change neither its symmetry nor its objectivity. This very
complicated definition leads to a very simple relationship

between wij and w?., which are not independent since

13
w.. = mi. = - w?., (3.7)
13 1] 13
and hence,
l _ l _ l
wij mi] — (1/2)(8vj/8xi BVi/axj)

- (1/2)(3v§/axi - avg/axj) = Zwij.(3.8)

Thus, in the difference between the antisymmetric tensors
(which appears frequently in subsequent sections) the terms
which were added to the antisymmetric component of avg/3xi
have cancelled out. The variables of (3.3) satisfy the
principle of objectivity, which means that they are inde-
pendent of any reference frame. Extreme caution-must be
used in the manipulation of equations which contain these
variables, for, while avg/axj is objective, v: is not be-
cause an observed velocity depends on the observer's ref-
erence frame. We want to keep our equations in terms of
objective expressions.

The entrOpy and entropy flux are related by the

balance equation

21
p(dS/dt) = - afi/Bxi + p0 + pr/T, (3.9)

where 0 is the rate of specific entropy production not due
to external radiation sources. The function, 0, can be
obtained for any specified thermodynamic process from the
constitutive equations for S and fi and from the energy
balance equation (2.8) used in conjunction with other
constitutive relations.

The second law of thermodynamics here becomes the

mathematical statement
9 3 0, (3.10)

which is known as the Clausius-Duhem inequality or the
entrOpy inequality. This inequality must hold for every
admissible thermodynamic process.

The internal Helmholtz free energy, A , is defined

I
by

A = E - TS. (3.11)

Substitution of this expression into (3.9) and use of

(3.2) and (2.8) gives

pT¢ = - pdAI/dt - pS(dT/dt) + T(8kj/3xj)
0(0) 0.0.
(qj/T)(8T/8xj) pijuj + {dijgij

2w..T... (3.12)
1] 31

22

Equations (2.7), (3.1), (3.4), and (3.7) have been used
to obtain the last two terms.

Both of the derivatives,dAI/dt and aki/Bxi, can
be expanded in terms of the independent variables. After
this procedure all terms which do not satisfy the second
law postulate are dropped, i;g;, we drop each term in
which the independence of some particular variable allows
the term to be either positive or negative when all other
independent variables are set equal to zero. These re-
strictions reduce the constitutive equation for A to a

I
function of five variables:

AI = AI (plrpzIapl/axilapz/axilT)0 (3.13)

In particular, AI is not a function of the diffusion
velocity, ui. Another requirement of the Clausius-Duhem

inequality is
S = - aAI/aT (3.14)

as is found in thermostatics. (Some equations of thermo—
statics are given in Appendix A.)
Chain-rule expansion of dAI/dt and Ski/8x1, intro-

duction of the conservation of mass (2.1) in the form

_ _ (I _ 0:

expansion of Bug/8xi in the manner

23

V
8 9 a . = - 8. + 6?. - 1 d9.
“3/ x1 (0 pa)wlj/0 13 ( /p)8£198 13

V
_ 8
(l/p)8£l(3pB/3xi)uj, (3.16)

and use of equations (2.4) and (2.6), and of the restric-
tions for the Clausius-Duhem inequality permit (3.12) to

be written as

 

 

 

 

 

 

 

pT¢ = - 9 TL (3A1 Sol - _£_3AI 8oz p 1
D1 D1 5xi p2 3p2 5xi 1 1
AI 82p 8k 829a
2" T’pa ataxl + ET[ 89a ]'8‘_x 3x
8 x. a 5x.
1 J
+ T 3ki _ Si 3T + T 3ki 396
T T xi pa 5x1
2 3k. 3k
- E—-m1ul + TEE—w.-L w.. + T ——J--dl
Q2 1 1 p Sui 31 aui 13
3A p 8k.
I a _g 1 a + a a
+ 29 3o padijdl] ZTgl anl dij 28Iijdij
1
-23k iapl-i—apzpul+2'rlw
p Bui p1 5xj 92 5x3 .1 i 1] )1
3kg ad“ 8kz 801
+ T a 8x + T EETT'_§§1‘ (3'17)
adij 13 1

4. Ordinary Binary Fluid Mixtures

 

An ordinary binary fluid mixture is defined as one
whose constitutive assumptions are linear in the indepen-

dent variables

24

1

1 2
(Sol/8x1),(8p2/3xi),ui,dij,dij,wij,3T/axi. (4.1)

This definition includes all the variables given in (3.3)
except those which are scalars. The constitutiye func-
tionals are restricted from being dependent on tensor or
vector products of the independent variables. A tensor
of one rank can only be a linear function of tensors of
the same rank. Thus, vectors will be linear in the in-
dependent variables which are vectors and will have scalar
coefficients which are functions of the independent scalar
variables. Similarly, tensors of one symmetry will be
functions of the variables which are tensors of the same
symmetry and will have scalar coefficients. Scalars can
only be functions of scalars.

Applying the principle of equipresence and elimi-
nating immediately from each constitutive relation all
variables which are not of the same tensorial rank or

symmetry, we write the constitutive equations

AI = AI(pl,p2,T) (4.2)
S = S(plppsz) (4.3)

qi = - CqT(3T/3xi) - qu(8pl/8xi) - Cq2(302/8xi)
- C u; (4.4)

25

H

mi = - CmT(8T/3xi) - le(apl/8xi) - Cm2(8p2/8xi)
l
- Cmuui (4.5)

k. = CkT(8T/3xi) + Ckl(3p1/8xi) + Ck2(8pa/8xi)

l
1
+ Ckuui (4.6)
1——
Tij - uwij (4.7)
59...-“ +§¢ Ba +§2 {d8} (48)
ij 0 ij 8:1 aBdkk ij 8:1 ”as ij ' °

where dfik is the trace of dgj and {dgj} is the traceless
part of dgj' Separation of dij into these two parts is
significant physically in that the trace of dgj is re-
lated to dilatation and the traceless part of dgj is re-
lated to shear. Inclusion of the first term on the right
hand side of (4.8) is required because 6ij is the ever-

present unit tensor and, since it is symmetric, the sym-

metric tensor Efj depends on it; — "a is a scalar co-
efficient.

The Clausius-Duhem inequality when applied to
(3.17) with the linear constitutive equations for ki and
AI requires that ki not be a function of the density
gradients or temperature gradient; therefore, (4.6) be-

come 8

_ 1 _
k- — Kuir Cku —VK. (4.9)

Insertion of the constitutive equations (4.4),

(4.5), (4.7),

26

entropy production equation

0T9

where

U
l

’11
ll

:13
ll

_ l 1
- D + Y + Fp1(8pl/axi)ui + Gpl(3p2/3xi)ui

C l(aT/axiHapl/Bxi) + qu

+ Hd}. + 18?.
ii 11

(CqT/T)(3T/3Xi)(3T/3xi) + [(T/ol)(3K/3T)

2 1
+ Cqu/plT + p CmT/plpzlplui(8T/3xi)

+ (02/02)cm unilu i

2
2 21¢aBd££dnn 2

0:18 =

+ Zuwijw ji

2

Z 20

a=l B=1

a B

(D/pl)(3AI/Bpl) + (T/pl)(3K/301)

2
+ 9 le/plo2 - TK/oo1

- (9/92)(8AI/392) + (T/pl)(3K/392)

2
+ o C mz/oloz + TK/oo2

TKoz/o + 991(3AI/391)

Tr1

(4.8), and (4.9) into (3.17) leads to the

(8T/8xi)(302/3xi)

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

(4.15)

27
I = - TKpZ/p + ppz(8AI/8p2)--1r2 (4.16)

The entropy production (4.10) is a minimum at
equilibrium. From equations (4.10), (4.11), and (4.12)
we see that it can be written as a function of twenty-one

variables, XA'

_ 1 2 1 1
XA - (aT/axi'dij'dij'wij'ui)’ (4.17)

and the condition for equilibrium is that every XA be
equal to zero. As Bartelt and Horne (1970) have noted,

vanishing of density gradients and the barycentric velocity

is not required for equilibrium.

The minimum entropy requirement can be eXpressed by

the mathematical conditions
(30/8XA)0 = 0 (4.18)

2
(3 ¢/aanxB)o > 0, (4.19)

where the subscript 0 indicates that the derivative func—
tion is to be evaluated at equilibrium. Applications of
(4.18) and (4.19) to (4.10) impose certain restraints on
the coefficients of the bilinear form of the entropy pro-

duction.

Two of these restrictions are

H = 0 or HI = 001(3AI/3p1) - Tsz/o (4.20)

and

28

I = 0 or n2 = 092(8AI/8p2) + TKpZ/p. (4.21)
From (4.20) and (4.21) one can identify the sum of VI and
n2 as the pressure, P,

W1 + “2 = 991(3AI/391) + 002(3AI/302) = P. (4.22)

where the thermostatic definition of pressure has been used.
The coefficients ml and 82 thus have characteristics of par-
tial pressures.

Other results include

cql = cqz = o (4.23)
CqT : 0 (4.24)
n 3 0 (4.25)
911 1 0 (4.26)
022 1 0 (4.27)
411422 — (1/4) (412 + 621% _>_ 0 (4.28)
011 3 0 (4.29)
022 1 0 (4.30)
011022 - (1/4)(012+n21)2 3, o (4.31)
Cmu 1 0 (4.32)

2
F = 0 or p le/plpz = TK/Di - (T/ol)(3K/3Pl)

2
— fil/pl (4.33)

29

G = 0 or pzcmz/plo2 = — (T/pl)(aK/apz) + ”2/63 (4.34)
pZCqTCmu/pzT - (1/4)[T(8K/8T) + Cqu/T
+ pzcmT/pzlz 3 0. (4.35)

These restrictions reduce the entropy production (4.1) to

the form
pT0 = D + Y. (4.36)

Moreover, (4.23) shortens the constitutive equation
(4.4) to

_ _ l _
qi — cquui ch(eT/axi). (4.37)

Equations (4.33) and (4.34) make it possible to eliminate
the coefficients le and sz from (4.5). The complete set
of constitutive relations is then (4.2), (4.3), (4.7), (4.8),
(4.9), (4.37), and (4.5) with (4.33) and (4.34). The ap-
parent lack of symmetry with respect to the two components

in equations such as (4.13), (4.14), (4.33), and (4.34) is

due to the non-independence of the variables u: and u? and

1
can be eliminated by retaining both variables.

From the Gibbsian equation of thermostatics for
two components

dAI = PdV - SdT + (pl - u2)dwl, (4.38)

where “a is the Specific chemical potential of component a

and V is the Specific volume,

V = l/p, (4.39)

30
the Euler equation

Ewan“ = AI + P/p (4.40)

can be derived. Equations (4.38) and (4.40) yield a

thermostatic identity,

pa = AI + p(8AI/8pa) DB’T (4.41)

We assume that this is the definition of chemical potential

in nonequilibrium processes. Using (4.20), (4.21), and

(4.41), we find
p1 — p2 = fll/pl — “2/02 - TK/Ol, (4.42)

which will be most useful later.
The constitutive equation for mi is particularly

interesting. By (2.5) and (2.15)

1 _ 1 _ 1 _ 2
pmi _ dolui/dt [(02/9)(30ji/3xj) (91/9)(30ji/3xj)]

l 2
+ Ji - (0102/0)(bi - bi), (4.43)

where Ji consists entirely of terms of the order of velocity

squared,

1 l
= , 8 8 , . 3 . 3 .
Ji plul( Vj/ x3) + plu3( vl/ x3)

2 l l
- ((01 - 02)/0102](301ujui/3xj)

1 1
(olozuiuj/O)[(1/ol)(391/3xj)

2 3
- (01/02)(302/3xj)]. (4.44)

31

Equation (4.43) does not define mi, as we have no con-
stitutive equation for the external forces, but in the
event that there were no external forces it would be a
defining relationship. The difficulty in carrying out an
explicit analysis under the restriction that there be no
external forces lies in the fact that while the constitu-
tive equations are required to be objective, the balance
equations are not. The balance equations obey the princi-
ple of the invariance of work under changes of frame. The
external forces which appear in the momentum equations are,
in fact, only apparent external forces. True forces are
defined in an inertial frame, and the momentum balance
equations hold true in those inertial frames where real
forces are exerted. A detailed discussion of inertial
frames, forces, and apparent forces is given by Truesdell
and Toupin (1960). Because we have written a general
formulation which holds in all frames, it is impossible

to require that the external forces be objective or, in
fact, simultaneously zero in all reference frames.

The right hand Side of (4.43) appears to be non-
linear according to the definitions of this section. Con—
stitutive relations could be introduced for the stress
tensors, and this would result in inclusion of second
derivatives of the component velocities. The terms con-
taining the barycentric velocity are even more unusual

since they lead to terms containing three or more variables

32

when the gradient of the barycentric velocity is expressed
in terms of the objective independent variables chosen in
Section 3. Comparing (4.43) with the constitutive equation
for mi, (4.5), we see that the linearity of mi there is in
distinct contrast to the right hand Side of (4.44). The
first term in (4.43) is called an inertial term and all

the terms in Ji are called viscous terms or non-linear
terms. Clearly, no one-to-one correlation can be made be-
tween (4.43) and (4.5), but (4.43) places restrictions on

m; which will be utilized later.

5. Transport Equations in the Independent Variables T, 0, W1

 

The transport equations given in this section follow
algebraically from the equations of the previous sections.
The independent thermodynamic variables considered in this
section are the more commonly used p, WI, and T rather than
pl, 92, and T. These transport equations are equivalent to
the 20 + 2 balance equations of Section 2, where in the
binary case we have 0 equal to 2.

The balance of mass equations (2.1) are replaced by
and
.l
p(dW1/dt) = " BJi/BXi: (5.2)

where j: is the diffusion flux,

33

.1
3i = pluil. (5.3)

The balance equation for linear momentum of the
system as a whole is found by summing equations (2.5) over

a to obtain

p(dvi/dt) - 82(ogi - paugu§)/8xj = {pab:. (5.4)
The form
0(dvi/dt) = - (B/B')(8T/3xi) - (l/DB')(80/3Xi)

+ Zpabg + BHji/axj (5.5)

is obtained by simultaneous addition of aP/axi and sub-

traction of its equivalent form,

aP/axi (B/B')(8T/3xi) + (l/OB')(30/3Xi)

+ [0(\7l - V2)/B'](3wl/8xi). (5.6)

The viscous pressure tensor, Hij' is defined as

_ a _ o a =
H.. — {(c.. p uiuj + "65ij) Tij + Pdij. (5.7)

Here 8 represents the thermal expansivity,

B = - (l/p)(39/3T)P'Wl' (5.8)

8' is the isothermal compressibility,

34
' =

B (1/0)(89/3P)T’w1. (5.9)
and Va is the partial specific volume of component c.

There exists another momentum balance equation
corresponding to the second of the two balance equations
introduced initially in (2.5). It can be termed the
balance equation for diffusion momentum, dji/dt. One

form of this equation is found by rearrangement of (4.43).

 

d.1
31 = 33 3 01 -‘:l a 02 - J + m
at p 5xj ji p 5x3. ji 1 p
9 p
+ 1:2 (b: - bi). (5.10)

In terms of the variables of this section, Ji [see (4.44)]

has the form

 

 

. 1
J = _ (pl - 02) Bjjji + 8v jl + 31 3vi
l 0102 8xj xj 1 j ij
+ j 31 (92 " 91) a
i x.
J 09192 3
8w
2 2 1
+ 2 (92 ‘ 0291 + 91) SET ' (5°11)
9192 3

If the constitutive relation for mi, (4.5), is introduced

in (5.10) with the use of (4.33) and (4.34), the balance

equation is

35

 

 

 

1
dj.
9 1 = _ .1 _ _g$ _ ._ .
plpz at Du3i DT axi 8T(“1 “2)/3xi
1
+.;L 3(0ii + nldij)
pl ij
2
a .. + ..
- 41 (031 "2 11) — —Jl— J . (5.12)
02 3"j 0192 1

The coefficients Du and D in (5.12) are functions of T,

T
p, and W1' only:

_ 2 2
Du - (p /0102)Cmu (5.13)
DT = (02/0102)CmT + K/pl + (T/pl)(8K/3T)

- (§1"§2)° (5.14)

The chemical potential, ué, includes external potentials,

which have been introduced by means of (2.14),
0' = u + V . (5.15)

The gradient of the chemical potential difference, which

appeared in the form of (4.42), has been replaced by
_ I ___. _ u
3(ui uZ)/8xi 8T(ui “2)/axi
— (Sl - Sz)(3T/3xi), (5.16)

where Sa is the partial specific entropy of component a

and where

 

 

 

 

 

I
Bxi 8 3x1
— ._ 2
1
B 02 8x
(V’ - V )
l 2 8p
+ 08‘ Bxi (5.17)
with
“11 = (8u1/3w1)T,P. (5.18)

The conservation of moment of momentum or angular
momentum is Simply the requirement that the total stress

tensor be symmetric,

a a a _ a a o
{(6ij + pauiuj) — 2(6ji + paujui). (5.19)

The balance of energy can be replaced by a tem-
perature equation by means of identities from thermostatics.
Either of the following temperature equations containing

heat capacities can be used:

 

— dT _ dS 8T g3
pCv—E-pT—Tt-+——TOB t
_ 8(V-V) dw
- pT (s — § ) - l. 2 1,
1 2 8 St

(5.20)

where C; is the specific heat capacity at constant volume,

or

dB

91%“ (0'6P - PB) 3% - (TB - PB') 3%
dw

where C? is the specific heat capacity at constant pressure

and Ed is the Specific internal energy,

Ea = Ha - PVa == pa + TSa - PVa (5.22)

The balance of entropy, (3.9), can be used in (5.20) with
any choice for the entropy production. The internal energy
balance equation (2.8) can be used in (5.21). Comparison
of the resulting two final equations is easily effected

with the help of the thermostatic relationship

___ 2 '
CP - Cv + B T/oB . (5.23)

The results of this comparison could be taken to be a
measurement of the accuracy of representing the quantities
which appear in the entropy production equation by con~
stitutive equations, Since (5.21) is exact while (5.20)
contains the entropy production which is inevitably ap-
proximated to some extent by the linearity of the con-
stitutive relations.

There is also the possibility, however, that the
thermostatic equations (5.20) and (5.21) are not applicable
in the nonequilibrium case. Equation (5.21) is indeed an
equation for internal energy but in thermostatics this im-

plies the internal equilibrium energy of some volume

38

element. In the nonequilibrium case there are three ways
in which we could interpret this. Firstly, the internal
energy of (5.21) may be only the equilibrium part of the
total internal energy which in the nonequilibrium case
contains both equilibrium and nonequilibrium parts.
Secondly, the internal energy of (5.21) might be the
nonequilibrium internal energy plus the kinetic energy

of diffusion, Since the sum of these two energies is the
total energy internal to the volume element being de-
scribed. Finally, the energy of (5.21) could be the non-
equilibrium internal energy, which would be consistent
with our other usage of the symbol EI in this chapter.

If the first interpretation were true, an arbitrary sepa-
ration of thermodynamic quantities into equilibrium and
nonequilibrium parts would have to be made. In the second
interpretation the partial specific thermodynamic quanti-
ties would be defined as previously with additional terms
for the component kinetic energy of diffusion. Ramifica-
tions of these possibilities will be discussed in Section
8 after a survey of the possible forms of the entropy

production which can be used in (5.20).

6. A New Rational Mechanical Derivation

 

The theory of mixtures presented in Sections 2
through 5 is a typical example of a rational mechanical

develOpment. It is only one of many theories which can

39

be derived. The particular advantage of that theory is
the very concise bilinear form of its entrOpy production.
In this section we derive another theory in a similar
manner, but in this instance we obtain more familiar vis-
cosity coefficients while the entrOpy production equation
becomes much more complicated.

The Sv + 6 functions which define a thermodynamic
process in mixtures need not be those of Section 2, but
can be any set of equivalent functions; and, likewise,
the balance equations may be written in many forms. In

this section we choose to let our independent variables

 

 

be
891 8p2 8T avi Bu. sym (6 l)
D ID ITIu I ”1"“! I_I I“): or o
l 2 i axi axi 5xi axj xi 1]
where
_ 1 _ 2 _ l _ _ 2
One can readily verify that
asym _

These variables are objective and more nearly symmetric
with respect to the two components than the variables of
Sections 2 through 4.

We have adOpted here the diffusion velocity and

tensors of Doria while retaining the component densities

40

and density gradients of Bartelt (3.3) and Mfiller. Doria
used p and W1 and their derivatives in place of the com-
ponent densities and their derivatives in (6.1). Mfiller's
variables differ from Bartelt's in that he used the dif-
fusion velocity, ui, and an antisymmetric strain tensor,
Zwij.

The balance equations, written in terms of these

variables when possible, are:

 

 

2 2
dpl — _ 3% u 8pl _ p1 u 8p2 - plp2 8u. - 9 8V]
at - 8 _2 x. x. l x.
9 3 x3 9 j a J p a J a J
(6.4)
d92 93 391 9% 392 + 9192 Buj Vj
at = _2 u. x. + _2 u. x. x. - p2 x.
9 3 l 9 3 l p a J a J
(6.5)
dv
1 _ 8 l 2
p182 dul 0192 3V1 91 2 ani
T-at+Tuj63:.-+ 2‘92‘91’“j6‘i‘
J 9 j
2 a 2 a
_ 9192, pl + 9192 u u 92
3 1 8x 3 1 x
9 3 J 9 3 a J
31-- 9 39 9 89
= ._l£'- l'T' _1£.+ .3 T' __l
~ x. _2 i x. 2 i' x.
3 J 9 j J 9 3 3
D192 2
____ 1
+ 9mi + 9 (bi bi) (6.7)

41

dB 8v. 8q$ p p
B _ 1 _ l 2 l _ 2
p at _ 8xj Tij xj + prl+ p ui(bi bi)° (6'8)

 

The total stress tensor, Tij’ is that defined in Section 2,

and Tij is defined by

I
Q
+
Q

' = = aa-
Tij Tij + (p102/9)uiuj Tij + {pauiuj

(6.9)

The second stress tensor, Tij, arises in the rearrangement

of the momentum balance equations and has the definition

Tij = (92/0)9ij - (pl/9)0ij. (6.10)

This stress is a measure of the difference between the

component stresses per unit mass. By writing

02

1 ' 1 I
" (pl/p)( ij ij

13 ij ), (6.11)

we see that Tij represents the difference between com-
ponent 1's contribution to the stress and the ideal con—
tribution of component 1 to the total stress if the stress
of components 1 and 2 were the same per unit mass. The
stress tensor is, of course, also just the negative of

the difference between component 2's contribution and the
ideal contribution to the stress. The specific inter-
action force m1 is the same as m: of Sections 2 through 5.

The energy which appears in (6.8) is the total energy de-

fined in (2.9), less the bulk kinetic energy, so

42

_ 2
EB - EI + (plpz/Zp )uiui. (6.12)

In a manner completely analogous to that of Section
3, the entropy production can be obtained in terms of the

set of variables (6.1). It has the form

pT0 = p(dAI/dt) - pS(dT/dt) + T(8kj/8xj)

(qj/T)(8T/8xj) - pm.u. + T

+

sym

- (pz/pz) Tijuj(8pl/8xi)

+ (pl/p2)Tijuj(8p2/8xi). (6.13)

The heat flux, qj, is still that of Bartelt, but it can

now be written as

— 2 —
qj - q; + uiTij + (0102/20 )(pl pz)ukukuj. (6.14)

After examining the balance equations and the
entropy production, we choose to write constitutive equa-
tions for AI, S, Tij'3Tij’ qj, kj' and mj. The derivatives
dAI/dt and akj/axj in (6.13) must be expanded in terms of

the independent variables, so that we then have

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dAI 3A1 dpa 8A1 d 3904
' p t ‘ ‘ 92 Spa dt ’ Z 8pc )8? 8x.
8
5x1
_ aAI dT _ aAI d 8T _ ”‘1 cm1
"‘TT‘8E “’3 8T 8? 83;“ p 86; ‘8?
8x1
8vi
. r)
8AI dwii _ 8AI x.
awij dt 9 vi 8t "
8 5——
x.
3
8A 8u
_ I ‘41 i
8 8x.
3
and
2
T8ki = 2T‘8ki)8pa + 2T 8ki\ 8 pa + T 8ki 8T
8x. 8p 8x. 48p )8x.8x. 8T 8x.
1 c 1 a 1 j 1
8x.
3
+ T 8k; a2T + T 8ki 8u. sym + T 8ki w
8T 8x.8x. 8n. x. 8n. ij
%8x. 1 3 3 1 J
3
8k 82u. 8k 82v 8k 8w..
+ T l) 1 + T x \ i + T g 11
8ui 8xj8x2 8vij8x£8xj Bwij 8x2 °
8xj *8xj
(6.16)

The coefficient of any derivative of an independent
variable in (6.13) must be zero since that derivative could

be either positive or negative. However, several of the ‘

44

variable derivatives in (6.15) and (6.16) are not inde-
pendent. The time derivatives of component densities and
of component density gradients are given by the balance

of mass equations. For dpl/dt and dpz/dt we use (6.4) and
(6.5), while for the time derivatives of the density gra-

dients, we use

d(8pa/8xj)/dt = 8(dpa/dt)/8xj - (8vi/8xj)(8pa/8xi)

 

 

 

 

 

 

 

 

 

 

 

 

(6.17)
as well as (6.4) and (6.5) to write
30 Eu oz an 02 $0 29 9 Bo Bo
d 1_ i _g 1 + _1 2 _ 1 2 1 2
a? 8x. 5x. 2 8x. 2 8x. 5 L‘li x. x
j 3 p 1 p 1 p 1 3
2 2 2 2
_ 29192u 392 391 o2u 3 p1 o1u 3 92
p3 u15xl 8x3 pZu i8 x. 8xj p2 ui8xi8xj
2
p2 ap1 ap1 2‘)1 3p2 3‘)2 301 3V1
+ 2—3 18x. 8x + §Ji8x 8x _ 28x. 8x
p 1 p i j 1
2 2 2
- 8 vi _ p1 8p2 8ui - p2 8pl 8ui
p15x.8x. _2 8x. x. .2'8x. 8x.
3 1 O 3 1 D J 1
2
0192 3 u
+, p 3x.“ (6.18)
1
d 3‘)2 _ aui fig 301 + p1 3p2)+ 20192 ap2 301
‘3? 8xj 8xj p2 8xi 02 8x1 3 18xi 8x

45

2p192 8p1 3p2 Di 82‘)2

+ u. + —u.
3 15x 5x. 2 15x.5x.
o i J o 1 3

 

9% 32.1 29E 902 391
+ —2ui5x.5x. - 3ui5x. 8x.
0 l J O l J
2 2
292, 8pl 8p2 28p2 8vi 8 vi

 

 

34i8x. 8x. - 8x. 8x. - p28x.5x.
3 1 J 1

 

o 1 J
2 2 2
+ p1 ap2 aui + fig 801 aui _ 0192 a “i
-2'8x 8x 5x. 8x. p 8x.8x. ’
D 3 i D J 1 1 3
(6.19)

The strain tensor derivatives are not all independent either

since one can readily show that

= sym _ sym
Bwij/sz 8(8ui/8x2) /8xj 8(3u2/8xj) /3xi.

 

 

 

 

 

 

 

 

 

 

 

(6.20)
Using this identity we write
sym
8kg 8 8ni + 8k£ 8w.i
8u. sym 8x 8x. 8m.. x
3 1 2 3 13 2
8x.
3
31:1 8k. 8ki 3 ani sym
= T111 sym + Flea“ + 801?]. 3x1 8x3. (6'21)
8 5;;

Recognizing that these substitutions must be made in (6.13),
we invoke the Clausius-Duhem inequality in order to set
equal to zero all those coefficients of derivatives of

independent variables which can be positive or negative

46

while all other variables can be set to zero. Thus, the
coefficients of dT/dt, d(8T/8xi)/dt, dui/dt, d(8vi/8xj)/dt,
d(8ui/8xj)8ym/dt, dwij/dt, 82ui/8x18xj, 82T/8xi8xj, and
82vi/8xj8x£ must be set equal to zero.

An immediate consequence of this is that AI’ as
before, cannot be a function of 8T/8xi, ui, 8vi/8xj,
(8ui/8xj)sym, or wij' The normal thermostatic relation-

ship between entropy and the Helmholtz free energy is

found as before,

These results reduce the entropy production to the form

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

%3 8pm dt 8xi 89a 8x1
8x1
8k. 2 sym
1 8 pa 8ki 8T 8k£ 8ui
+ 2T 8p 8x 8x + T 8T 8x + T8u 8x
a i j i 1 2
8x.
3
_ sym
+ 8k£ T 8ki + 8k 8ki 8 8ui
8u.w£1 u. 8w . 8w . 8 8x.
1 43 1 21 £3 2 3
L 8x.
3
1r 8xj 9 3 3 2 lj i8x 2 ij 18x
8vi 8ui sym

47

If we consider the coefficient of 82vi/8xj8x ,

we obtain the following restriction:

 

 

 

 

 

p1 6ij + 929—27334”
8x fiw) 8x 8x: (6. 24)
This can be integrated at once to give
8AI 8AI 8v.
_ _ .______ 1
k2 - N1 p[p133:l + p23 3:: J 8x , (6.25)

 

 

 

 

where N1 is independent of 8vi/8xj. Now, if this result
is substituted into the coefficients of 82p1/8xi8xl and
82p2/8x18x1 which themselves must be equal to zero, two

further restrictions ensue,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 2 2 2
838A1u+p£ 811111-12- 89.1‘1111318111In
p a(501,1 o a 501 j o a 392 o a 592 3'
x3 x1 8xj 8xi
8Ni 3N.
+ T—s—p-f- + T—Tai- + T[291‘:3;—:[3:1
8 5;; 8 5;;
82% 8215.1:L 8vk
301? + p92 301 302 8xk = 0, (6.26)
%( 8 §—— 8 -—-
xj 8xi

 

 

 

 

 

 

and

48

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 D2 2
31$. ___.____ -3111,
D 8 8p1 1 93(8913 p 8p2 1
8x. x.
J 8 J
2 2
_£1 8AIu + 8N].- 3N. 8
p 392 3 392 392 0013 301 3 302
8x1 8xj 8x1 8x1 axj
82AI 82AI 8vk
+ 2 = . 6.27
pp1a 8p1 8p2 pp2a 8p2 892 8xk ( )
8xj 8xi 8xi 8xj

 

 

 

 

 

 

 

 

The coefficient of 8vk/8xk must be equal to zero in each

equation so that from (6.26) we can write

 

 

 

 

 

891) pp1 8p1 pp2 (892) 891 ppl 891
8 -—— 8 ——- 8 -—- -——- ———
8xj 8xi 8xi 8xi axj

8AI

+ pp2 ”1.302 (6.28)
a .
8x.
3

 

 

An equation of this form has the general solution

8AI 8AI .8p1 ;
8 -——) 8 -—-
8x1 8x1

 

The function Qij is antisymmetric and neither Qi' nor fi

can be a function of 891/8xj. They must, therefore, be

functions of 8p2/8xj only. By the objectivity of the

49

scalar, AI' Qij must be zero. From the coefficient of
8vk/8xk in (6.27) we obtain a similar expression in terms

of 8p2/8xj, and the two results imply that

8AI

0275—: " 01W. . (6.30)
8

 

 

8x.
3

Returning to (6.26), we see that all those terms
but the last one, the one in 8vk/8xk, must be zero in-
dependently. We rearrange this equation using (6.30) to

a form similar to that of (6.28) and then using the general

solution we find

8p2
'I'Ni - AIplui — Qij'a'}; + F1 (6.31)

where Qij and Fi are independent of 8p2/8xj and Qij is
antisymmetric. Considering the terms in (6.26) which are
independent of 8vk/8xk and employing the results (6.30)

and (6.31), we write

. + F.
8(391 AUI i Qikaxk+ i

 

 

39
a —
8Xi

We multiply both sides of the equation by (8p2/8xi)(8p2/8xj)

and take advantage of the antisymmetry of Qij to arrive at

50

 

 

 

 

 

 

 

 

a A u 3p2 302 + F1 3:2 392

8p1 I 8x. 8x. fi8xj
8 8x.
3

8p 8p 8p 8p
' a: AIuj 8x? EEE Fj 8x2 8x2 (6'33)
8 l j i J i
8xi

 

 

Because both sides of (6.33) are scalars, this statement

is equivalent to

 

 

 

 

 

 

 

3 3p2 Bp2 8p2 ap2
—-5—p—1—- AIuj axj 3X1 + Fj 5-;- Fit—i- - 0. (6.34)
8 ——— 3
8x.
1
Therefore,
8AI 8F.
0 = - 6.35
—5-—(u 91 3 —1-3 301 . < )
axi 8x1

 

 

 

and AI cannot be a function of (8p1/8xj)(8p2/8xj). Thus,
AI can be an objective function only of (8p1/8xi)2 and

(8p2/8xi)2. But we know that

p2 8AI 8F. (8p2
E: .53;._ - _—3Bi—.¢ f 5;; , (6.36)
is; W:-

 

 

 

so AI can be at most only first order in 8p2/8xj and this
is impossible. The only remaining possibility is that AI
is independent of both 8p1/8xi and 8p2/8xi, so we have

shown that

51

AI = A1(pl'°2'T)' (6.37)

This result is very desirable in that we would hope that
our thermodynamic variables are functions only of the
state variables, p1, p2,fm in the nonequilibrium case as
they are in the equilibrium situation. This we have shown
to be true of the Helmholtz free energy.

The result of equation (6.37) greatly simplifies
the entropy production equation so that we now have the

form

8ki 8p1 8ki 3p2 8k

 

 

 

 

 

 

_ __i_ .312 - 31 ET
0T4) — T871 87:1 + T872 8—xi + T 8T 3‘1 T 8?].
8k.
- pmiui + T “1 + Tij wji
a a 8k. 8u. sym
l 2 8pl 8p2 13 8n. 1) 8x.
8A 8A 8v
I I . 1
+ p(918p1 p28p2) 613 Tij8xj
2
+ 3; 3A1 _ 3A1 5 p2.1.. “391
o 591 592 13 2 13 15xj
2
pl ‘8AI 3AI pl 302
+ —— - 6.. + ——T!. u. > 0. 6.38
p 53:' 83; 1] D2 13 i8§§ — ( )

 

Among the restrictions which are obtained in arriving at

this form are:

52
8ki/8(8pa/8xj) + 8kj/8(8pa/8xi) = 0 . (6.39)
8k2/8(8Vi/8xj) = 0 (6.40)
T[8k£/8(8uj/8xi)4-8kj/8wzi + 8ki/8wkj] = 0. (6.41)

Although further restrictions on the general forms
of the response functions could be derived, their physical
significance becomes less and less clear. For this reason
we now restrict ourselves to a linear theory. The omission
of terms which are of second or higher order in the inde-
pendent variables will naturally introduce approximation
into a theory which until this point has had none. (Re-
call that this section is not merely a continuation of
the previous one.) The error due to these omissions is
minimized by retaining explicit second order dependence
of the total stress tensor and of the energy, E, and by
using linear functions to represent only those parts of
E and Tij which do not have explicit second order depen-
dence.

As in Section 4, we can immediately limit the con-
stitutive equations to functions of tensors of the same
rank. We also have the restriction (6.39) and the inde-

pendence of A from density gradients which say that ki

I
cannot be a function of 8pl/8xj, 8p2/8xj, or ui, so the

constitutive equations take the forms

AI = AI(ol.pz,T) (6.42)

53

k“ = K’”' 6 43
J J ( )
39 8p
= - ' _ I 3T _ | l ' 2
qj Cquu j CqT8xj qu8xj qzaxj (6.44)
39 8p
= _ I _ | 3T - ' 1 _ ' 2
“‘3' Cmuu j CmTax. Cm18x.cm2—3x. (6.45)
3 j 3
I = _ |
Tij = - p513“ + “1577513' + “253?" + “3W5”
n j n
8ui
+ “fa—“x (6.47)
3'
asym = ,
Tij “ wij (6.48)

The primes are to indicate that the coefficients may not
necessarily have the same dependence on pl, p2, and T as
similar coefficients of Section 4. These explicit func-
tions can be placed in the entropy production equation
which again becomes a function of twenty-one variables,
all of which must be zero at equilibrium. The equations

(4.18) and (4.19) still apply for the variables

sym
uj,8T/8xj,8vi/8xj,(8ui/8xj) ’wji'

The final form of the entropy production is then

_ . 8K' qu . 8T , 8T 8T
pT¢ Cm mu “in + T‘Sfi + T + CmT ujgig + qT8§3 5;;
8v 8v 8vu 8v.
k 2 1
+ ¢___.___ + 2n——— ——1 + L {5-49)
8xk 8x2 8xj 8xi '

54

 

 

 

 

 

 

 

 

 

  

 

where
8n 8v 8ui 8v. 8 8u
_ k 2 “k 9.
1" (X1+°‘1’:—x;53:+ (X2+°‘2"5x—j xi+°‘3'5§;'a§;
+ SS8uk + u w - E£¢3Vku 891
0L45); 5xi wij ji p2 8xk 15xi
+ 91¢ 8v vku 892 -‘E% 2n8viu 8p1+plzn8viu 8oz
02 MR: 1‘53}: 92 ”5323’ 183?? ’7 83?? 1332’.
__ ‘32 a“): ap1 + 01 auku 3p2 3p1
ple‘r‘“ 2‘53?— pleaxk 2sz "7x2;— “1‘5“
9 an. S 89
+ —ix 1 u 2 (6 50)
2 2 5x. 18x ' '
o J j
The positive-definiteness of entropy production, (4.18)
and (4.19), has also generated the requirements
qu = qu = 0 (6.51)
“T393A13?92 8K'
+—7P p'q-T g—— + ple = 0 (6.52)
D2 0
AI AI 1 8K'
T‘s—pl - r02 - :2- P' +T r- + ‘3sz - 0 (6.53)
P' - 3A1 + 8A1 (6 54)
’ p pl‘a'a- 9283‘ '
3A1 p192
_ I _. I _
p-TK +01 ZW- 1:72 -TK + (111 112).

(6.

55)

55

Again we have identified the thermostatic pressure, P', in
(6.46) and we can use (4.41) for the difference in chemical
potential. Equation (6.55) gives us some physical feeling
for the coefficient K'. It is dependent on the difference
between the specific partial pressures (equilibrium parts
of the specific stress tensors) and on the chemical poten-
tial difference. Using (6.55), (6.52), and (6.53) we can

0 . | I
solve for the coefficients le and sz,

0C1;1 = - (oz/02)P - Sp/apl + (plpz/p)8(ul - u2)/Bol
(6.56)

ocfiz = (pl/02)P - 3P/302 + (ploz/p)3(u1 - u2)/392.
(6.57)

Many other restrictions similar to those imposed on the
coefficients of Section 4 can also be found.

At this point we have only fourteen unidentified
coefficients in the constitutive equations. Nine of these
are viscosity coefficients of which we readily recognize o
and n as the coefficients of bulk and shear viscosity,
respectively. We have some feeling for p, but Cfiu, CfiT,

C' , and C' remain unidentified.

qu qT
By using (6.56) and (6.57) in the constitutive
equation for internal force and by making transformations

analogous to those in Section 3, we can write

56

 

 

 

 

 

 

 

p192 aT(”1 ‘ “2) p192 — —
= — V + - ' _
pmi p 8x1 [ pCmT + p (51 82)
+ 9192 a(“1'“2) a'r . p2 P+ 3p 3‘)1
8F 8):. + ’2' 8p 8x.
1 p l 1
01 3p ap2
+ - — P + - DC. “to (6058)
p2 8p2 8xi mu 1

 

When this equation is put in the balance of diffusion
momentum equation, the result is again identical to that
of Bearman and Kirkwood except that the coefficients are

more clearly identifiable. One form of this equation is

 

 

 

 

 

3(u -u )
. .1 _ _ p192 — _ — D192 1 2
pcmuji - [ 9931 + p (51 82) + p 8T
3p 31 D2 3 1
+ 571,-] 3x + —5- T'xjwji + 111613)
pl 3 2 0102 3(u1 pi)
- 77'8x3(°ji + Tr2631 p 8x1
1
+ dji + D2 D2 3 (.ljl)
at p p2 8xj 3j i
.1.1 p ap1 p1 3p2 13V:
- Ji3j( 2 8x. - 2 8x + ji x.
091 3 092 3 J
18vi
+ Jjgig o (6.59)

This theory does not vary radically from that of
Sections 2 through 5, but it is decidedly different from

it. The entropy production (6.49) contains five terms

57

which are predicted by TIP and also those terms in L,
(6.50), which would typically be called viscous terms.
The constitutive equations are very similar to those of
Section 4 but not interchangeable with them.

7. Comparison of Several Theories for Binary Mixtures of
(Fluids

There have been many approaches to the thermo-
dynamics of irreversible processes in binary fluid mix-
tures. Both of the rational mechanical theories presented
here compare readily with the microscopically grounded
theory of Bearman and Kirkwood and with the theories of
Pitts, de Groot and Mazur, etc., for nonequilibrium thermo-
dynamics. All of these are so-called "linear theories"
with functions being represented in a linear manner such
as that presented in Section 4. However, some also con-
tain "linearization" of other forms including dropping
of terms in any equation in which such terms appear to be
non-linear.

The theory of Bearman and Kirkwood is based upon
the same balance equations as those of Section 2, includ-
ing partial stress tensors which are defined in a slightly
different manner as discussed in Section 2. The TIP
theories do not treat partial stress tensors and only
rarely include any terms arising from kinetic energy of
diffusion. The exact limitations of the balance equations

used in TIP and their differences from those used in

58

rational mechanics can best be seen by studying the
balance equations of Section 6 which are equivalent to
those of Section 2. The theories of TIP neglect entirely
balance of diffusion momentum (6.7) and all terms which
are introduced in this equation and not in any other
balance equations, iifiiv interaction forces and partial
stress tensors. The energy, EB, of the balance equation
(6.8) is then taken as the thermodynamic internal energy
since there is no longer a substitution to be made for

the external forces of (6.8) which would introduce kinetic
energy of diffusion. It is, thus, obvious that TIP and
rational mechanics will not yield the same entropy pro-
duction equation. It is interesting to note that the re-
sults of a rational mechanical approach to the problem as
defined by the TIP balance equations leads precisely to
the usual TIP result. This development is given in Appen-
dix B.

It is also possible to show that rational mechanics
and TIP are the same except for kinetic energy of diffusion
terms. We do this by rearranging the equations of the
rational mechanical theory of Sections 2 through 5. In
both microscopic and macroscopic TIP theory the diffusion
flux, j;, as well as the heat flux, 01' is written as a

1.

linear function of two forces:

2. (7.1)

82.nT/8xi + 911 1

_ Q
10

I
U.
P
l

i (20089.nT/8xi + 9012i (7.2)

I
0
ll

59

where the flag are Onsager or phenomenological coefficients.
In microscopic theory the force, 21, generally takes the

form

zi = 8T(u1 - u2)/8xi. (7.3)

(De Groot and Mazur, by inclusion of kinetic energy of
diffusion in their energy, E, admit the possibility of an
inertial term, but not of viscous terms, in Zi but do not
generally include it in their work.)

Equation (5.12) can be solved for j: to yield

- j: = (DT/Du)T<aznT/axi) + ai/nu (7.4)

where

a = I _ I _ I
“i 3T(“1 u2)/3xi (l/pl)a(oji + nldji)/3xj

+ (1/92)8(o§i + "25ji) + (p/plpz)d§i/dt
+ (o/olpz)Ji.

In the microsc0pic theory the force, 21, is identical to

the Ei given in (7.5). The work here is the first success-

ful reproduction of that force with a macrosc0pic theOry.
At this point we note from (7.1) and (7.4) the

coefficient identities

Q = TDT/D (7.6)

a = l/D . (7.7)

60

In order to determine the heat flux, Qi' which
corresponds for rational mechanics to the heat flux in
the entropy production of TIP, we turn to the entropy
production in (4.36) and specifically to the term D given
by (4.11). Each time u: appears, it can be replaced by
ji/pl. In both the TIP and the microscopic theories, the
entropy production is a bilinear form of independent fluxes
and their conjugate forces, where the vectorial fluxes and

forces are those of equations (7.1) and (7.2), so that

pT¢ = - jizi - Qi(8£nT/8xi). . (7.8)

However, in order to obtain this relationship in the micro-
scopic theory, all of the equations of Bearman and Kirkwood
are linearized to the extent that Bi of (7.5) now contains
only the one term in (7.3). All viscous and inertial terms
are dropped. The assumptions by Fitts that Si can replace
8T(u1 — u2)/8xi directly in the entrOpy production with no
change in the non-vectorial terms is incorrect.

In order to determine what Qi represents in the
rational mechanical theory in terms of the work described
here, we rearrange our entropy production. Since D of
(4.36) contains all of the bi-vectorial terms of the
entropy production, we can equate it with the first two

terms of (7.8). In this manner we find for the general

Qi denoted in this theory as qi,

._ _ —_—' .1
- qi - CqT(aT/3xi) + [Cqu/pl KT/p1 + T(s1 52)]3.

i
(7.9)

61

Introducing (7.4) into (7.9) and equating coeffi-

cients with (7.2), we find

(200 = chT + T(DT/Du) [cqu/p1 - KT/pl + '1'(§1 - “8'2”
(7.10)

901 = [Cqu/pl - KT/pl + T(§1 - Sé)]/Du. (7.11)

By application of equations (3.2), (4.9), and (4.37) to
(7.9), the new heat flux is seen to be the difference be-
tween the entropy flux and the entropy of the components

carried by the diffusion flux:

I_ _ -__~1_ _ -_—.l
qi ‘ Tfi T‘sl 52’31 ‘ qi + Tki T(S1 52’31'
(7.12)
This agrees with the identity of qi in TIP from the work
of Kirkwood and Crawford.

The non-vectorial terms of pT¢ in TIP are the form

(Tij — Péij)(8vi/8xj), (7.13)

where this term can be split into symmetric second order
tensor and scalar parts. The phenomenological relations

of TIP include (7.1) and (7.2) with (7.3) as well as

{Tij} = - 2n{8vj/8xi} (7.14)

Tijéij - 3P = - ¢(8vi/8xi). (7.15)

The general heat flux, Qi' in terms of TIP is

Qi = q; - j§<fil - fiz), (7.15)

62

so that
pT¢ = - { .- “1(fi -fi‘)}aznT/ax - '13 < - )/8x.
qi 31 1 2 i 31 T 111 “2
- (Tij - Pdij)(8vi/8xj) (7.17)

is the entropy production of TIP.

8. Expressions for the Entropy Production and the Entropy
Balancei Equation

 

In this section we first discuss five possible forms
that the entropy production and entropy balance equations
can assume and then discuss the similarities and discrep-
ancies of these forms.

a) The final expression which we obtained for the
eaItropy production in Section 4 by using all of the con-

stitutive equations including that for mi gives us

:3ch = D + Y, (8.1)
where
2 a
Y = ail Bil¢asda£1dnn'+a:1 Bilnw dij}{dij} + zuwijwji
(8.2)
D .—

(ch/T)<aT/ax,)(aT/ax£) + [<Tol)<aK/8T)
+ (oz/p1p2)cmT1plufi(aT/ax£> + (p2/9192)lepluiu%.
(3.3)

flhe expression for D can be modified by replacing the con-

stitutive equation for qi by qi itself to yield

63

D = - (qi/T)8T/8xi + [(T/pl)8K/8T

+ (92/9192)CmT]plu:(8T/8x£)

+ (92 /9192)lepl%

where

.= 2+
q: q:

The entropy balance

pT(dS/dt)

then has the form

T(dS/dt)

1u 1 (8.4)

a _ a a a

2(Oji paujui1ui + (l/2)Epaug uiui. (8.5)
equation

= - T(8fk/8xk) + pT¢ + pr (8.6)

+

+

T(8fk/8xk) - qi(8£nT/axi) + pr

[(T/ol)3K/3T + (oz/ploz)CmT]plui(3T/ax£)

2 1 l
(p /0192)Cmuplu£u2 + Y'

b) A second form of the entropy production is ob-

tained as in Section 7 by introduction of a new heat flux

into the linear equations. Again we have used the con-

stitutive equation for mi. Thus,

pT¢ = F + Y,

where

F = - qi(8£nT/8xi) - j

(8.8)

(8.9)

(1]

64

— — .1
Tfi - T(S1 - 82)]i

.Q
P. -
I)

q; + 2(agj - p uguq

a a a
a l J)uj + (1/2)Zpauiuju

0!.
j
— — I 1
+ Tki - T(sl - $2)ji. (8.10)

The entropy production is then

pT(dS/dt) = - T(8fk/8xk) — jisi

- qi(8£nT/axi) + Y + pr. (8.11)

c) A third form of the entropy production is ob-
tained by returning to the expression for entropy produc-
tion before the introduction of the constitutive equations.
Equation (3.17) for pT¢ is rewritten using the restrictions
from the constitutive equations and (4.41), (4.15), and

(4.16):

_ _ _ .1
pT¢ - fj(3T/3Xj) [3(U1 H2)/8lejj

a _ a
X(pmi and/axi)ui

+ 2(ogj + naaij)(8vg/8xj). (8.12)

For ZmEug we multiply (4.43) by pUi/pz and rearrange this

eXpression to find
a a _ a a _ a a _ a a
pZmiui - pd(Zpauiui/2p)dt 8{2ui(oij pauiujH/axj

— 8(2paugugug/2)8Xj - )(ogj - paugug)(avi/axj)

+ 2°:j(3V:/3xj) - {paugbg. (8.13)

65

This uses the second of the two expressions for m: which
were discussed in Section 4. Substituting (8.13) and
(8.12) into (8.6) and rearranging, we arrive at an entrOpy
balance equation

1
j

.1
(q; — jj(Hl H2)}89.nT/8xj + Hij(8vi/8xj)

ac __. ,_,
T<dS/dt) + od<Zoauiui/2p)/dt - a 3T(u1 u2)/3xj

+ pr - T8{%[q; - j%(ul- u2)]}/3xj. (8.14)

In (8.14) we no longer have the balance equation
for entrOpy alone, but for entropy plus kinetic energy of
diffusion. The flux in this balance equation is q; -

- j;(§i - 52), which is related to the heat flux, q;

j%(u1 - 82), of this equation by

q; - j%(fil - 82) = q; — j%(ul - uz) - T(§l -‘§2)j%.
(8.15)

which is the same difference between the two entropy fluxes

as that given in (8.10). By inclusion of all constitutive

equations except that for mi, we have succeeded in changing

the form of the entropy balance equation radically. In

particular, the tensorial term, Y, has been replaced by

the single term, Hij(8vi/8xj).

d) The expression for entropy production given in
Section 6, (6.42), and the concommittant entropy balance

equation are not readily comparable to others discussed

because the entropy production is not a bilinear form.

It can be written

66

pT¢ = qj(8T/8xj) + {Cémuj + [T(8K'/8T)](8T/8xj)}uj

+ ¢(8vk/8xk)(8v£/8x£) + 2n(8vi/8xj)(8vj/8xi) + L.
(8.16)

All non-linear terms are in L.

Because the expression for the balance of momentum
of diffusion is the same as that of Bearman and Kirkwood,
the entropy production equation will reduce in the same
way as that of the Bartelt-Horne theory to the equation
of section c with slightly modified coefficients. One
could also note at this point that it would also be en-
tirely possible to obtain a bilinear entropy production
for this theory if, instead of writing a linear constitu-
tive equation for mi, one represented the expression mi
+ Tiju.

J
This may seem to be an arbitrary choice, and indeed it

(pl - p2)/p by a linear constitutive equation.

is--representing the product of an independent variable
and a dependent tensor by a linear constitutive equation.
However, this choice is no more arbitrary than the defi-
nition for heat flux, (2.14), used in both theories. This
choice for a constitutive relation to represent a term
which is poorly understood in the first place would hardly
make the theory more obscure.

e) The traditional fonm of entropy balance equa-

tion in TIP is

67

pT(dS/dt) = - Ta{%.-[q3: - 3'31(ul - “2”}
_ * _ .1 — _ —

- 3%[8T(ui - u5)/axj1 - nij(avi/axj).
(8.17)
All of the quantities appearing in (8.17) have the same
definitions as those in this paper, assuming that as dis-
cussed in Section 5 the thermodynamic quantities are based
on the same thermodynamic energy.

By comparing the five forms of the entropy pro-
duction we see first that all forms except that of d are
bilinear. Success in attaining bilinear forms is due to
proper choices of the functions to be represented by lin-
ear constitutive equations. The second and third forms,
(8.8) and (8.12), are very different. The third form,
which does not use the constitutive equation for mi is
identical to the classical form of TIP, (8.17), except
for the term for the kinetic energy of diffusion
which appears in the entropy balance equation. From
this comparison one can readily see that the only con-
tribution of kinetic energy of diffusion to the entrOpy
production is in a single term,_so that if one wants to
exclude kinetic energy of diffusion systematically, then
only this term need be dropped and the form will be re-
duced to that found from TIP. The fourth form is also

very similar to the TIP theory in that the tensorial

68

terms are the same and there is no inertial term. The
entropy flux, heat flux, and conjugate force of uj are
present in both but are different, and the non—linear
terms do not appear in TIP..

Having reviewed our entropy production equations,
we can measure the success of the Bartelt-Horne theory in
approximating reality by re-examining the two temperature
equations of Section 5. By putting (8.14) into (5.20),

with some rearrangements, we have

pE§(dT/dt) + pd(2pau:u:/2p)/dt =

01 — '—
- 3(q3 - ]j(H1 - H2)}/axj + pr

+ Hij(3vi/8xj) + (pBT/B')(Vl - V2)(dw1/dt).
(8.18)

To obtain the temperature and pressure equation we use
(5.21). There is considerable controversy over the in-
clusion or exclusion of kinetic energy of diffusion in
such an equation. If this kinetic energy were included,
as Truesdell has suggested it should be, the equation

would have the form

ac .—
0(dEI/dt) + pd()oauiui/2p)/dt = (ocp - PB)(dT/dt)

- (TB - pB')(dP/dt)

2 + (1/2)u1u§ - (1/2)u§u§1(dwl/dt).

+ p[§1 -
(8.19)

69

In either case the disputed terms are sometimes small and
are usually neglected justifiably or not. The assumption
of the validity of the Gibbs equation for nonequilibrium
processes is part of the principle of local state. The
principle applies to a small volume element as a whole,
the internal energy of which includes kinetic energy of
diffusion. With the adoption of this principle, however,
the state of the volume element is described by three
state functions (wl, p, and T) and cannot be a function

of ui, so that we have the relationship
a a _ _ a a .a
pd(2pauiui/2p)dt - )(1/2)uiui(an/axj) (8.20)

under the postulate of local state, and equation (8.19)
reduces to (5.21). Thus, if (8.20) holds true, (5.21)
and (8.19) are precisely the same equation. However,

what (8.20) implies is that
a .
dui/dt = 0, (8.21)

which seems unlikely for it implies the cancellation of the
contributions of partial stress tensors and interaction
forces in the diffusion momentum balance equation. TIP
never refers to the diffusion momentum balance equation;
only the barycentric momentum is considered.

However, if the energy balance equation (2.8) is
used in (5.21) with the time derivative of P analogous to

(5.6), we have

70

pC§(dT/dt) + pd(zpau:uz/2p)/dt =

I 1 — —
- 8 8 8x. + .8 H' - H' 8 .
qj/ J 33 ( 1 2)/ x3

+ Hij(3vi/3xj)+-(BT/oB')(do/dt)

+ pT(Vi - Vé)(8/B')(dwl/dt), (8.22)

where we have used (5.23). Comparison of (8.22) with
(8.18) shows that they are identical. To the extent that
the principle of local state holds, our theory is valid,
but if the principle of local state were really true,
then there would be no need for improvement of the equa-
tions of TIP.

The only modification which we have introduced in
the temperature equation is the time derivative of the
kinetic energy of diffusion. It is this term which is
responsible for introducing internal forces and which
necessitates partial stress tensors, both of which (in-
teraction forces and partial stress tensors) are rou-
tinely included in the development by rational mechanics.
These quantities are also the most difficult to express
by constitutive relations because we have little experi-
mental feeling for them.

Although we have managed to make several rational
mechanical forms of the entropy balance equation resemble
the TIP form of the balance equation, we have lost sight
of the Clausius-Duhem inequality. Because we defined our

entropy flux at the outset of our theories, we can only

71

apply the Clausius-Duhem inequality to those terms of the
entropy balance equation which are not the gradient of the
entropy flux. We cannot redefine entropy flux in midstream,
as is done in TIP, for the purpose of application of the
Clausius—Duhem inequality. Thus, the inequality applies

to (8.1), (8.8), and (8.16) but not to the terms outside

of the flux gradient in (8.14). This is a difficulty in
the TIP theory. According to Appendix B, all terms in

the right hand side of (8.18) except - 8(q35/T)/8xj are in
the entropy production, and therefore greater than or equal
to zero. In normal TIP theory the entrOpy production is
considered to be given by those terms which are not the
gradient of a flux. When terms which are not included

in the entropy flux as defined initially are put in the
entropy production, it no longer will be a bilinear form

composed of the products of fluxes and forces.

9. Conclusions and Discussion

In the preceding sections we have investigated
the progress of rational mechanics in describing trans-
port in fluid mixtures. Hopefully, we have dispelled
some myths about the irreconcilability of TIP and rational
mechanics. At the same time we have laid no claim to veri-
fication or denunciation of Onsager's microscopic theory.
The necessary assumptions in defining a linear theory have

been given. The real danger in any theory is inconsistency

72

in the linearization scheme. It is the theories which
have tried, more or less randomly, to take into account
some degree of non-linearity that have remained farthest
from internal consistency and applicability.

Now that we have defined a linear theory formula-
tion and have given several examples of such, we must ask
to what extent are such theories valid. The simple re-
lations of TIP have been known for years due to the fact
that they are empirically motivated and are adequate for
the description of any experimental system now available
(excepting those in which chemical reactions occur). That
the linear relations represent ultimate truth is not only
doubtful, but impossible. So long as a sufficient number
of variables are introduced the rational mechanical theo-
ries are correct until they are linearized, but from this
point on they can only approximate the truth and frequently
they contradict one another. One knows that any properly
chosen set of independent variables must be equivalent to
any other such set in generating a general theory; but,
when one writes a linear theory in terms of one set of
variables and then tries to change his set of variables,
he finds he no longer has a linear theory and in fact is
lucky if his transformed theory still contains only ob-
jective terms. Perhaps one linear theory presents a de-

scription very near to reality, but no one knows which

theory it might be.

73

Another problem arises in Section 6, which treats
the only theory which was presented with a non-bilinear
form for the entr0py production. There is no apparent
reason for the entropy production necessarily to be bi-
linear in the rational mechanical case. However, in
examining the restrictions placed by (4.18) and (4.19)
on the coefficients of terms in the entropy production,

we find relationships such as

pcfiu - (¢/4)[(ol/p)(ap1/axi) - (oz/o)(302/8xi)12 > o.
(9.1)

If the coefficients ¢ and Cfiu depend only on p1, p2, and
T, it would seem that this expression places a restriction
on among the variables p1, p2, T, 8p1/8xi, and 8p2/8xi.
At what point did our variables lose their independence?

We have assumed that thermodynamic quantities are
the same functions of state variables in nonequilibrium
thermodynamics as they are in thermostatics. If this
postulate is not true, we have a whole new problem. Trues-
dell actually gives a chemical potential tensor for each
component. Of course, we can define any combination of
terms by any name we desire.

An interesting point to investigate will be the
nine viscosity coefficients that appear in rational
mechanical theories. Only two coefficients have ever

been measured. The problem of actually measuring stress

74

in a fluid due to a diffusion velocity gradient seems in-
surmountable and much work remains to be done in this area.

Certainly there are many implications of rational
mechanical theory which remain to be investigated. How-
ever, the theory needs also to be extended, particularly
to include second order terms in the constitutive equations.
This is an immense problem which would more than quadruple
the number of terms in each constitutive equation. It
would produce a more realistic theory, but attainment of
relationships between constitutive coefficients would be
much more difficult since (4.18) and (4.19) tell us nothing
of terms with more than two of the 21 nonequilibrium vari-
ables. One would also be tempted to try to formulate a
theory by adding only those second order terms which he
feels are likely to be important. This would be highly
subjective, however, for who among us knows how inter-
action forces depend on a velocity gradient dotted into
a temperature gradient and so forth? Nevertheless, it
will not be possible, except by experiment, to assess
the accuracy of the linear theory until a second-order
theory is formulated and the contributions of terms added
are evaluated.

We have seen that terms which are second order in
u: have caused the greatest problems in comparing the
rational mechanics theories to the theories of TIP.

Ideally, one would minimize these contributions in an

75

experimental situation. This, however, implies that the
diffusion flux itself will be minimized and that we are
approaching the study of a binary mixture behaving as a
pure fluid in nonequilibrium. We would be more certain
that our experiments were correct as evaluated by the
theory of rational mechanics or TIP, but problems of
diffusion would be unanswered.

If a complete non-linear treatment of rational
mechanics is done, many, many more terms which are second
order in u: will no doubt arise. It may well be that
these terms in combination with the terms which come out
of the linear theory will vanish, or will at least form
clearly nonlinear combinations. The foundations of TIP
are the same as those of rational mechanics except that
TIP does not treat the balance of diffusion momentum, and
we have established that the theories are identical ex-
cept for the second order terms in diffusion velocity.

In thermostatic derivations, a change of state
variables gives formal results different from the original
equations, but each set of results equally well describes
reality and gives the same calculable results. One's
choice of state variables is usually directed toward giv-
ing the simplest formulation for the problem at hand. In
TIP one can change his reference frame (reference velocity
to which component velocities are compared) and derive

several formally different theories. A comparison of the

76

entropy productions of these theories allows one to obtain
readily relationships among the various phenomenological
coefficients because the entropy production is invariant.
We would suggest then that, since all the rational me-
chanics theories have been developed from prOper sets of
Objective independent variables applied to the same bal-
ance equations and since all the theories have been
linearized in exactly the same rigorous manner, the en-
tropy productions of the theories should be equivalent
and any formal difference occurring between them can be
equated to zero. This procedure would hopefully give us
new insights into the meaning of the constitutive co-
efficients or at least giVe us relationships among the

coefficients of various theories.

CHAPTER III
THERMAL CONDUCTIVITY

1. Introduction

A great deal of research on thermal conductivity
in fluids has been done over many years, particularly be-
cause of the diverse technological applications of heat
conducting liquids. A review of theories of conductivity,
all of which are microscopic, is given by McLaughlin
(1964). However, a fully rigorous mathematical treatment
of thermal conductivity as a macroscopic process has not
been presented in the literature. Solutions of the macro-
scopic heat transport equations for liquids have, in gen-
eral, been found by assuming that physical properties
such as heat capacity, density, and thermal conductivity
are constants. These solutions then resemble solutions
for heat conduction in solids as discussed by Carslaw and
Jaeger (1959).

Many experimental techniques have been developed
for measuring thermal conductivity in liquids, but most
of these depend on calorimetry measurements and measure-
ments of heat flow and, therefore, either require calibra-

tion or are subject to large experimental error.

77

78

These problems are emphasized in a recent review (Tree and
Leidenfrost, 1969) of the methods that have been used to
measure the thermal conductivities of carbon tetrachloride
and toluene. The values measured by these methods show

a variation of more than ten percent for each of the
fluids. Such findings suggest that one needs a better
theoretical treatment for liquids and more reliable ex-
periments.

We have completed a theoretical investigation of
heat conduction in a flat plate cell (thermal diffusion
cell). The theory does not treat coefficients in the
thermal conductivity equation as constants, and thus an
analysis of the error introduced by such an assumption
is possible through comparison with this rigorous theory.
The attainment of general macrosc0pic solutions for tem-
perature as a function of thermal conductivity makes
possible the design of new time dependent experiments
based on temperature measurements rather than on measured
heat flow. The time needed for a liquid to adjust to new
temperature boundary conditions is a relaxation time de-
pendent directly on the liquid's thermal conductivity.
Experiments which measure these relaxation times can be
done with greater accuracy than those which depend on
calorimetry and heat flow.

In order to determine the thermal conductivity

of a pure fluid, one needs a completely general

79

description of temperature variation as a function of
space, time, and thermal conductivity in a cell. Then,
after determining the temperature or its gradient at
specific positions and times, one can fit the general
temperature solution to the measured temperatures by
varying the value of the thermal conductivity. If one's
solution is correct and his experiments accurate, the
value which gives the best fit will be the best experi-
mental estimate of the true thermal conductivity.

Horne and Anderson (1970) have recently developed
such a temperature solution for binary mixtures of fluids
in a pure thermal diffusion cell. Using a perturbation
scheme, they solved the transport equations for both the
heat conduction and the mass fraction as functions of
position and time under the initial and boundary condi-
tions of the pure thermal diffusion cell. Their solutions
are superior to those of others in that Horne and Anderson
account for convective heat and mass transfer, the depen-
dence of density and other properties upon composition and
temperature, and warming-up effects. Their perturbation
scheme is readily applicable to many problems, and we use
it here to solve the transport equation for heat conduction
in a pure fluid. The transport equations for a pure fluid
are actually simpler than those for mixtures, but the

solutions of interest in such a case must be known at

very short times, during which the temperature gradient

80

is being established in the cell, whereas the equations of
Horne and Anderson are primarily intended to describe dif-
fusion effects after this warming-up period is over.

The classical thermal diffusion cell is a rectangu-
lar parallelepiped bounded above and below by metal plates
in contact with heat reservoirs which can be maintained
at any temperatures. The area of the plates is made much
greater than the distance between them in order to mini-
mize edge effects. The cell often has glass walls so that
refractive index changes resulting from temperature changes
in the fluid can be measured. After the plates, reservoirs,
and fluid in the cell have equilibrated at some temperature,
a positive temperature gradient is applied to the cell by
simultaneous heating of the upper plate and cooling of the
lower plate through their respective reservoirs. This
temperature gradient produces a negative density gradient
without convection (for the normal fluid, whose thermal
expansivity is positive). The establishment of the den-
sity gradient occurs within the time necessary to reach
the temperature steady state in the cell. This is a much
shorter time than that within which a density steady-
state is established in a pure thermal diffusion experi-
ment.

As in the work of Horne and Anderson, we use
empirical parameters to characterize the times needed

for the plate temperatures to reach their steady values.

81

Horne and Anderson assumed in their solutions that the
upper and lower plate parameters are equal, while our
solutions include a different parameter for each plate.

In Section 2 we present the complete set of prac-
tical macrosc0pic transport equations for a pure fluid and
the conditions which the solutions of these equations must
satisfy in a typical cell. In Section 3 we introduce the
approximations which must necessarily be made in order to
solve the equations of Section 2. The solutions of the
temperature equations are presented and analyzed in Sec-

tion 4.

2. Transport Equations

 

Our fundamental transport equations correspond to
equations (II.5.1), (II.5.5), and (II.8.22) with (II.5.23).
These equations are greatly simplified when the fluid in
question is assumed to be a one component liquid subject
to only a gravitational external force. The above equa-
tions under these conditions are also those of TIP (Horne
and Bearman, 1967). We assume that the temperature gra-
dient is applied only in the vertical direction with no
horizontal effects and that the cell walls are adiabatic
so that there is no horizontal heat flux through the walls.
Heat and matter flow only in the vertical direction and

there is no steady state convection.

82

The hydrodynamic equation of continuity of mass is

(dp/dt) + p(8V/8z) = 0, (2.1)

where p is the density, v is the barycentric velocity, t
the time, and z, the vertical position coordinate, and
where the substantial time derivative, d/dt, is related

to the local time derivative by

(d/dt) = (8/8t) + v(8/8z) = 0. (2.2)

For the equation of energy transport we have

pEé(dT/dt) - TB(dP/dt) = ¢1 - (8q/8z), (2.3)

where E? is the specific heat capacity at constant pres-
sure, T is the temperature, 8 is the thermal expansivity,
¢l is the entrOpy source term for bulk flow, and q is the
heat flux.

The linear phenomenological equation for the heat

flux is

-q = K(3T/32). (2.4)

where K is the thermal conductivity of the fluid.

The initial conditions in the cell are

V(z,0) = 0, T(z,0) = TM, (2.5)

and the boundary conditions at all times are

83

V(a/2It) 0 = V(-a/21t)

T(a/2,t)

¢U(t) T(-a/2,t) = ¢L(t), (2.6)

where g.is the cell height.
Warming-up effects are taken into account by the

time dependent functions for plate temperatures, 8 and °L'

U
One can empirically determine the time required for a plate
to achieve its steady state. It will generally depend on
the plate material and thickness, on the reservoir heat
capacity, and on the means of supplying heat to and re-

moving it from the reservoirs. A typical functional form

for the plate temperatures is

¢U(t) = TM + (TU-TM)[1 - exp(-t/YU)]

¢L(t) = TM + (TL- TM) [1 - exp(-t/‘YL)]) (2.7)

where TU and TL are the temperatures applied to the upper
and lower plates, respectively,and YU and YL are the
characteristic relaxation times of the upper and lower
plates. These times are best obtained experimentally.
Typically (Anderson, 1968), YU and YL are of the order of

40 to 60 seconds.

3. Approximation Methods.

In order to solve the transport equations given
above we must make approximations on three levels: (i)

the equations are simplified by suppressing demonstrably

84

negligible terms; (ii) a self-consistent, well-ordered
perturbation scheme is introduced to take into account
variable coefficients; and (iii) a Fourier sine or cosine
solution is obtained for each perturbation equation. It
is essential to recognize that at no point do we assume
that coefficients are constant or that the barycentric
velocity is zero.

Simplification of Equations.--We can neglect

 

terms in the transport equations (2.1), (2.3), and (2.4)
if they are very small compared with other terms. Ob-
viously equations (2.1) and (2.4) cannot be simplified;
but in equation (2.3) we examine the pressure and entropy
source terms. The magnitude of these-terms in a pure
fluid is comparable to their magnitude in a mixture, and
the magnitude of the heat flux in a pure fluid is compa-
rable to its magnitude in a mixture, when both the pure
fluid and mixture are subjected to the same experimental
conditions. For this reason, the arguments of Horne and
Anderson for the suppression of these two terms are valid
for pure fluids as well as for mixtures. The time depen-
dence of pressure is neglected because we assume that the
steady-state pressure distribution in the cell is attained
instantaneously. Horne and Anderson found that ¢1 is at

most lo‘lsJ'm-Bs-l, whereas aq/Bz is on the order of 25J

-38-1

m , so ¢1 in (2.3) is certainly negligible in com-

parison to the heat flux gradient.

85

These approximations simplify our energy trans-

port equation (2.3) so that with (2.4) we have

pEé(8T/8t) = - pCév(8T/8z)4-8[K(8T/8z)]/8z. (3.1)
By equation (2.2) we have for (2.1)

(8v/8z) =.- (82np/8t) - v(8£np/8z), (3.2)

where density derivatives are simply related to tempera-

ture derivatives by the chain rule,
dznp =-BdT, (3.3)

when pressure derivatives are neglected as assumed above.

Perturbation Scheme.--The perturbation scheme of
Horne and Anderson is used to take the non-constancy of
coefficients in the transport equations into account. It
is certain that the coefficients vary with temperature
(and pressure) in the cell, but it is also true that fairly
reasonable solutions to the transport equations have been
obtained using constant coefficients. Following this
reasoning, we treat every coefficient as a constant plus
perturbation terms which depend on temperature fluctua-
tions. This scheme modifies the usual solutions of the
energy equation and also allows the density of the system
to vary, thus permitting the barycentric velocity to

change.

86

The most general expression for the representation
of a coefficient which varies with temperature (or other
state variables) about some value for a fixed set of state
variables is the Taylor series; therefore, we write each
coefficient, L, as a Taylor series with successive higher
order derivatives being preceded by higher orders of the

ordering parameter, 6:

1’3 = Lo + em - TM)LT+ 52(1/2)(T - TM)2LTT + NJ),
(3.4)
where
_ _ _ 2 2
Lo - L(TM) LT - (8L/8T)TM LTT — (8 L/8T )TM.
(3.5)

When a = l, L = L and we have the exact Taylor series ex-
pression for L (neglecting pressure dependence in accor-
dance with our previous assumption). The increasing
orders of 8 indicate the decreasing contribution of those
terms to L. Accordingly we write our solutions as per-

turbations in s also so that

A _ 2 3
T — TM + To + 5T1 + a T2 + O(e ),

9 = v + evl + 52v + C(63), (3.6)

0 2

where T = T and G = v if e = 1. All of the solutions Tn
and vn are functions of space and time. The total solu-

tions are actually Taylor series in a about a a 0.

87

The parameter s is a dummy device used to order subsequent
manipulation of equations in accordance with the tempera-
ture derivatives of the coefficients. Thus, the zeroth-
order problem and solution involve no temperature deriva-
tives, the first-order problem involves first derivatives,
the second-order problem and solution involve second de-
rivatives and products of first derivatives, and so forth.
As in any perturbation method, one cannot be certain a
priori whether the solutions (3.6) converge when a a l.
A reasonable procedure is to find the solutions through
the second-order in e and to compare the zeroth-, first—,
and second-order solutions. If the importance of the re-
sults diminishes relatively rapidly with increasing order,
then one may stop. If, on the other hand, some higher-
order result appears to be more important than a corre—
sponding zeroth-order result, one must adjust his per-
turbation scheme so that the offender is included at
zeroth order. The scheme proposed in equations (3.4)-
(3.6) works well for thermal conduction in a pure fluid.
The first application of equation (3.6) is to

equation (3.4), which becomes

" _ 2 2 3
L _ Lo + eToLT-l-e [(1/2)TOLTT+T1LT] + 0(6 ). (3.7)

wherein all of the subscripted L's are constants as de-
fined in (3.5) and all of the T's vary Spatially and

temporally.

 

 

 

 

88

The second application of the perturbation scheme

is to the velocity equation (3.2). The initial and bound-

ary conditions at each order of e are, by equations (2.5)

and (2.6),

vn(z,0) = 0 vn(ia/2,t) = 0, n = 0,1,.... (3.8)

Putting L = lnp in (3.3) and (3.7) we have for the zeroth-

order velocity

8v
0 (3.9)

.._..._.=O'

8t

which by (3.9) implies

V0(z,t) = 0. (3.10)

By equations (3.2), (3.3), (3.7), and (3.10), the

velocity equations for higher powers of s’are all of the
type
(8vn/82) = 80(8Tn_1/8t) + Vh(z,t), n = 1,2,...
(3.11)
where for the first and second powers of e
Vl(z,t) = 0,
V2(z,t) = TOBT(8To/8t) + v180(8T0/az). (3.12)

The result of our perturbation scheme is thus a differ-

ential equation for each order whose inhomogeneous part

contains only terms of lower order.

89

Substitution of equations (3.6), (3.7), and (3.10)
into equation (3.1) and separation of the results accord-
ing to the power n of 6 yields the following temperature

equations:

1 2 2
(8 Tn/az ) = Un(z,t) n = 0,1,...

(8Tn/8t) - KO(DCP)O
(3.13)

with

U0(z,t) = 0
_ - -l
Ul(z,t) — (pCp)o {8[TOKT(8TO/8z)]/8z}
— - -1
- (pCp)T(pCp)o TO(8TO/8t) - v1(8To/82).
(3.14)

The similar but very long expression for U2(z,t) is omitted
for brevity. The initial and boundary conditions for each

power n of e are, by equations (2.5), (2.6), and (3.6),

Tn(2,0) = 0 n = 0,1,..., Tn(ia/2’t) = o n = 1,2,...,

T0(a/2.t) = ¢U(t) - TM. T0(-a/2,t) = ¢L(t) - TM.
(3.15)

Fourier Transforms.--Inspection of equations (3.11)
--(3.15) reveals that the equations must be solved in the
progression To, v1, T1, v2, T2, etc., and that the tem-
perature equations are all of one type. For this reason
the method of solution will be the same, and we use the

method of Fourier transforms for the temperature solutions

90

as did Horne and Anderson. Whether we choose a sine or
cosine transform solution is dictated by our boundary con-
ditions, 122;! whether we know the solution or its Spatial
derivative as a function of time at the boundary. In the
first instance, we use a sine transform; in the latter, a
cosine. Now, for our temperature equations, we know the
behaviour with time of the temperature at the walls rather
than the behaviour of its derivative, so we can write for

every order, T ,

n
Tn = Z gnm(t)sin(mwx/a), n = 0,1,..., (3.16)
m=l
where
x = z + a/2 (3.17)
and where
a
g (t) = (2/a) ] T (x,t)sin(mnx/a)dx, m = 1,2,...
nm 0 n
n_= 0,1,2,...
(3.18)

We multiply each term of equation (3.13) by (2/a)sin(mwx/a)
and integrate each of them over x from 0 to a. This pro-

cedure gives us

a (twat - ( '6 )'1(2/ > [a (32 T (tn/3x2} 1 (mvrx/a)dx
gnm Ko 0 p o a 0 1'n s n

a
= (2/a)(pE ) 1 f u (x,t)sin(mnx/a)dx, n = 0,1,...,
p o 0 n

I“= 1,2,... 0

(3.19)

91

The second term is readily integrated by parts using the

boundary conditions (3.15) to yield

2
dgnm(t)/dt + (m /T)gnm(t) = ¢Om(t)60n

+ (2/a) [a Un(x,t)sin(mwx/a)dx, n = 0,1,...,
0
m = 1,2,..., (3.20)
where 50n is a Kronecker delta, where
- - - m _ - -
¢0m1t1 - (2m/1rT){( 1) [¢U(t) TM] [¢L(t) TMJ}.
(3.21)
and where T is a thermal relaxation time,
_ 2 — 2
T — a (pCp)o/w KO. (3.22)

The differential equation for each gnm(t) is easily
solved, in principle, by means of the integrating factor,
exp(m2t/T). The initial conditions are readily found by
inserting the intial conditions (3.15) in equation (3.18).

Thus we have

0,1,...,

3
ll

gnm(o) = OI n 1,2,... 0 (3.23)

4. Practical Results

 

We can now find our temperature solutions to any
desired degree of accuracy by solving equation (3.2) re-
peatedly for successively higher values of n. We solve

the first two orders rigorously and estimate the

92

contribution of the solution which is second-order in e
from its steady state solution.
By equations (3.16)--(3.23) we find the solution

for zero-order temperature.

T0 = (z/a +‘l/2)(TU r TM)[1 — exp(-t/YU)]

+ (z/a + 1/2)(TM - TL)[l - exp(-t/yL)]

+ 21 (-l)n(nn)-1{[4n2(YU/T)'1]-1(TU-TM1
n:

x [exp(-t/YU) - exP(-4n2t/T)]

+ [4n2(YL/T)—1]'1(TM- TL) [exP(--t/YL) - exp(-4n2t/T) 1}

x sin(2nnz/a)

so

+ g (2/n)(-1)1(22+1)‘1 [(22+1)2(yU/T)-1]-1(TU - T

M)

x {epr-(2(+1)2t/rl«-eXP(-t/YU>}

- [(22+1)2(YL/r)-11‘1(TM - TL)

x {exp[-(2£+l)2t/T] - exp(-t/YL)})008[2£+1)fl22/a]. (4.1)

If the initial temperature, TM, is the average of the

applied temperatures, T and T

U L' the form of this equa-

tion is reduced to

T0 = AT(z/a)[2 - exp(-t/YU) - exp(-t/YL)]

+ (AT/4)[eXP(-t/YL) - exp(-t/YU)]

+ (AT/2n) 2 (-1)nn‘1{[4n2(Y_/T)—11‘1
n=l U’1

93

X

[exp(-t/YU) - exp(-4n2t/T)]

+

[4n2(YL/T)-1]-1[exp(-t/YL) - exp(-4n2t/T)]}

X

sin(2nnz/a)

as

+ (AT/n) 2 (-1)1(21+1)‘1 [(21+1)2(yU/T)-11'
(=0

1

X

{epr-(22+l)zt/Tl-exp(-t/YU)}

- [(22+1)2(YL/T)-l]{exp[-(2£+l)2t/1]-exp(-t/YL)}

X

cos[(2£+l)wz/a] (4.2)
where

AT = TU - TL. (4.3)

If the plate relaxation times are the same, but the initial
temperature is not the same as the average of the applied

temperatures, we have the reduced form

To = (Z/a)(TU - TL)[1 - exp (-t/Y)]
+ l/2(TU + TL - 21M)[1 - exp(-t/Y)]

(-1)“(1rn)"1('r 1

- TL)[4n2(v/r)-11‘
1

+ U

IIM8

n

X

[exp(-t/Y) - exp(-4n2t/t)]sin(2nnz/a)

co

2 (2/11)(—1)1(22+1)('rU + T

+

2 —l
L - ZTM)[(2£+1) (Y/T)-1]

X

{expl-(2£+l)2t/r] - exp(-t/Y)}cos[(2£+l)nz/a], (4.4)

94

where
Y = YU = YL. (4.5)

Finally, if both plates have the same relaxation time and
the initial temperature is the average of the applied tem-

peratures, we have the greatly simplified form

T = AT{(z/a)[l - exp(-t/Y)] + 2 (-1)n(n/n)[4n2(y/r)—1l'1
1

0 n:
X [exp(-t/Y) - exp(-4n2t/T)]sin(2nnz/a) . (4.6)

The complex forms of equations (4.1), (4.2), (4.4),
and (4.6) might discourage one from attempting numerical
evaluation for specific experiments. The series converge
rapidly for times of ten seconds or more after the applica-
tion of the temperature gradient and could be evaluated on
a calculator, but evaluation by a digital computer is faster
and can be carried further with a greater degree of accuracy.
At short times the series term of (4.6) is nearly cancelled
by the leading term, thus it needs to be known to a very
high degree of accuracy. All evaluation of such equations
has been done on an CDC 6500 computer. The series have been
calculated until changing their length by ten percent fails
to change their value by approximately 0.1% or more (212;!
100 terms of a series are calculated and summed and then
flhe next ten terms are calculated and summed; if these ten

(xenms do not change the initial sum of 100 terms by more

"I

a—u- ‘u

95

than 0.1% of its original value, the series is truncated
after these 110 terms and all further terms are assumed
negligible). In higher order solutions where multiple
series appear, this criterion is applied to each of them.
A plot of T0 versus cell position at various times
as calculated from equation (4.6) is shown in Figure l.
The approximate parameters for carbon tetrachloride shown
in Table l have been used. The solution is antisymmetric

about 2 = 0.

Table l--Approximate physical properties of CC14

j.

e=1.2 x 10‘3K'1

I< =1.1 x lO-J'J' m'1s‘1x‘1

K-1(3K/3T)=- 1.2 x 10'3K’1

p'c‘ = 1.4 x 106.1 m‘3K‘1

p
(pEé)-l[8(pE§)/8T] = 7.1 x 10‘1K"1

 

In every case, our temperature equation depends
only on time, position, length of the cell, temperatures
of the plates, relaxation times of the plates, heat ca-
pacity of the fluid at its initial temperature, and the
thermal conductivity of the fluid at its initial tempera-
ture. The experimentalist can determine all of these

factors prior to and during his experiment, with the

96

Figure l--Plot of T0 versus (z/a) for CCl4; a = 303,
b = 605, c = 90s, d = 1205, e = 1508.

 

 

97

GdCb

0.4 )-

—0.4 -

 

- 0.2 0.0 0.2 0.4

— 0.4

2/8

Figure l

98

exceptions of the heat capacity and of the thermal con-
ductivity. If, however, the heat capacity of the fluid
is known from other experiments, the thermal conductivity
can be determined by monitoring the temperature of the
fluid (or some parameter which is simply related to tem-
perature) and fitting it to whichever of equations (4.1),
(4.2), (4.6), or (4.4) is applicable by variation of the
thermalifionductivity. Such an application of these equa-
tions has been made by Olson (1972).

In order to make such measurements, however, we
have assumed that only To in the perturbation solution
T - TM = TO + T1 + T2 +... is important. To test this
hypothesis the solution for T1 is found and evaluated for
typical parameters in order to determine its magnitude.
We have a complete solution T1 for y =4YU =4YL and T +

U

T = 2T Minor variations from these conditions do not

L M'
affect the magnitude of T1. The Fourier solution of (3.20)
with n = l and subject to the above conditions is very com-
plex, involving triple sums of infinite series. It is
given in full detail in Appendix C. This solution has
been evaluated for typical values for carbon tetrachloride.
The results are shown in Figure 2. It is symmetric about
2 = 0; therefore, its derivative is zero at z = 0 for all
t and will add nothing to the temperature derivative moni-

tored at z = 0. T1 is zero initially, rises to a maximum

of 5 x lO'SK at a time of approximately one-half T, and

99

Figure 2--Plot of T1 versus (z/a) for CCl4; a = 303,
b = 605, c = 90s, d = 1203, e = 1503.

100

4.0 -

3: $2 x c

\,

J

 

 

0.0 0.2 0.4
2 /a

—— 0.2

—— 0.4

Figure 2

101

then falls steadily, going negative until the steady-state

solution,
T1‘”’ = (1/2)(AT)1(KT/Ko)[(1/4)-(z/a)21. (4.7)

is achieved. The steady-state value in carbon tetrachloride
at z = 0 is approximately - 1.6 x 10-4K. The behaviour of
T1 at t < 21 is shown in Figure 2 and may be compared to

the values of To shown in Figure 1.

We can also investigate the behaviour of T It

2.
too will have a maximum absolute value in the steady state

where

12(w) = (1/2)(AT)3[(KT/Ko)2 - (KTT/3K011

x ((z/a)3 - (z/4a)1. (4.8)

This value has been shown to be small everywhere by Horne
and Anderson, and it is zero at z = 0 while its gradient
is on the order of lO-GK.

Comparison of the values of T0' T1, and T2 shows
that, indeed, the solution To gives a very good approxi-
1mation to the total temperature solution particularly at
short times, and thus, that the thermal conductivity as
fitted by equation (4.1), (4.2), (4.4), or (4.6) should
be a very good approximant to the true thermal conduc-
tivity. The estimates above indicate that the theoreti-

cal error caused by truncation of the temperature solution

after To is probably less than experimental error.

CHAPTER IV
THE. DUFOUR EFFECT

1. Introduction

When a concentration gradient is imposed on an
originally isothermal fluid mixture, the phenomenological
equation for the heat flux predicts that a temperature
gradient will develop as diffusion occurs. This phenome-
non is called the Dufour effect, after its discoverer
(1873), or the diffusion thermoeffect, to indicate that
it is the reverse of thermal diffusion. The effect is of
interest in liquids for three important reasons: (1) be-
cause it can be used to verify the heat-matter Onsager
reciprocal relations, (2) because the temperature varia-
tions could cause complications in diffusion experiments,
and (3) because it has never been unambiguously observed.
The purpose of this work is to predict theoretically the
magnitude of the Dufour effect when liquid mixtures are
allowed to diffuse into one another and to determine
where the Dufour effect can be best measured in a geo-
metrically well-defined cell.

Interest in the Dufour effect rose early in World

War II when Clusius and Waldmann (1942) and Waldmann

102

103

(1939, 1943, 1946, 1947a, 1947b, 1949), who were inter-
ested in determining thermal diffusion factors for gases,
developed kinetic and phenomenological theories and ex-
periments to measure the Dufour effect in gases. The
equations of the phenomenological theory were solved for
adiabatic diffusion and diffusion between cylindrical
chambers with diathermic walls. Waldmann's final results
depend heavily on the ideal gas law, and they therefore
cannot be extended to liquids. The work also requires
constancy of thermal conductivity and the heat of trans-
port within each chamber of the experimental apparatus,
and the solutions are based on the assumption that the
temperature at the boundary of the two chambers does not
change during the courSe of the experiment. Miller (1949),
noting the great fluctuations of the temperature at this
boundary, suggested that they might be due to heat of
mixing as well as to variations in density and Specific
heat. He did not, however, develop any theory to take
these into account. Further kinetic theory has been
develOped by Grew and Ibbs (1952), and the phenomeno-
logical theory is presented by Meixner (1943) who assumed
that the reciprocal relations hold.

Waldmann (1947a) used his phenomenological theory
to estimate, for the liquid mixture hexane-octane, that
the variation of temperature in one chamber would be a

maximum of 0.04 K. He neglected heat of mixing. The only

104

investigations of the Dufour effect in liquids have been
done by Rastogi and Madan (1965) and Rastogi and Yadava
(1969, 1970). Their work is primarily experimental and
their results are based on ill-founded mathematical as-
sumptions about the behaviour of their system. We shall
discuss their work in more detail at the end of this chap-
ter.

Thus, for many years the Dufour effect in liquids
was ignored because it was felt to be too small to measure.
Its presence in liquids is masked by the heat of mixing
which is not present (at least to any great extent) in
gases. Any measurement of the Dufour effect is complicated
by the fact that no steady-state other than the equilibrium
state is reached by a system which initially has a con-
centration gradient, and thus the measurement of the tem-
perature gradient arising from the Dufour effect must be
time dependent (Fitts, 1962).

The work reported here is a theoretical examina-
tion of the temperature effects which result when two
binary fluids of different composition are allowed to mix.
These are the effects predicted by the macroscopic trans-
port equations, which include in the temperature equation
terms for the transported enthalpy. These are the terms
that take into account the heat capacities of the two
fluids and their heat of mixing. Terms which are demon-

strably small are omitted from the transport equations,

105

and transport coefficients are allowed to vary with the
thermodynamic variables of the system by expressing them
in Taylor series in composition and temperature.

Two ideal experiments which could be approximated
in the laboratory are defined. The results of any real
experiment should extrapolate to the results given here.
We determine optimal conditions for (1) measurement of
and (2) neglect of the Dufour effect.

We define our first system to be an adiabatic
cylinder or rectangular tube of length a. The value of a
will be chosen to get maximum temperature variations as
predicted by the solutions of the transport equations
with consideration of experimental practicality. The
length would probably never exceed ten centimeters. The
cell is divided into an upper and a lower chamber, each
of which contains a mixture of the same two liquids but
in different proportions. In the extreme case each half
could contain a pure liquid. The more dense liquid is
placed in the lower chamber and the boundary between the
two mixtures is assumed to be infinitely sharp initially.
The two mixtures, originally at the same temperature, are
allowed to diffuse into one another. Diffusion occurs
only in the vertical direction and causes vertical tem-
perature variations in the cell. At infinite time the
composition in the cell will be uniform (except for sedi-

mentation effects) and the temperature will again be

106

uniform throughout the cell, though not necessarily at the
initial temperature.

The second system is identical to the first except
that the ends of the cylinder are assumed to be diathermic.
The lateral walls of the cylinder are still adiabatic, or
the cylinder is of sufficient diameter that variations in
edge temperature will be negligible,and diffusion will

hence occur only in the vertical direction.

2. Transport quations

 

The equation of energy transport in a binary liquid

is Lg£.(II.8.22) with (11.5.6)]

pCé(dT/dt) - TB(dp/dt) = ¢1 - (8q/8z)
- j1[a(fil-fiz)/321. (2.1)

where C? is the Specific heat capacity, p the density, T
the temperature, 8 the thermal expansivity, p the pressure,
01 the entropy source term for bulk flow, q the heat flux,
j1 the diffusion flux of component 1 relative to the bary-
centric velocity, and ii the partial specific enthalpy of
component i. The time variable is t, and z iS the vertical
position coordinate (the initial boundary is at z = 0, the
ends of the cell at i a/2).

The conservation of mass equations for the fluid

are [g£. (II.5.l) and (11.5.2)]

(dp/dt) = - p(8v/8z) (2.2)

107
and
p(8w1/8t) + (8j1/8z) + pv(8wl/8z) = 0, (2.3)

where v is the barycentric velocity and W1 is the mass
fraction of component 1.
The linear phenomenological equations for the

fluxes of heat and mass are (Horne and Bearman, 1967)

-q = K(3T/32) + 0DQi(8wl/Bz) + DDQiSIW1W2(8p/8z)
(2.4)

and

= pD(8wl/8z) - pDa w w T-1(8T/8z)

‘31 112

+ pDSlw1w2(8p/8z), (2.5)

where K is the thermal conductivity of the mixture when
there is no concentration gradient (at infinte time), D
is the mutual diffusion coefficient, Qi is the heat of
transport of component 1, al is the thermal diffusion
factor of component.1,and S1 is the sedimentation coef-
ficient of component 1.

The solutions to the transport equations for con-
centration and temperature must fit the initial and boundary

conditions of the systems described. Initially we have

w1(z<0,0) = w1A +

wlA '

NEH NIH
ego 15%

v(z,0) = 0, wl(z>0,0) , T(z,0) = T ,

w1(Q,0) = w (2.6)

lA'

108

where w1 is the mean of the mass fractions in the two

A
chambers of the cell and 1 (1/2)w$ are ‘thé‘initial diaplace—
ments of wl fromw1A in each half of the cell. At the
ends of the cell we have no matter flux and either no
heat flux in the adiabatic case or constant temperature
in the diathermic case, so

q(ia/2,t) = 0,

(adiabatic)

v(ia/2,t) = 0, j1(ia/2,t) = 0,

T(ia/2,t) = T .

(diathermic)M (2.7)

In order to solve the problems which have been
posed, we assume that the equations above are correct
and can be applied to the idealized experiments described
above. We must make further assumptions about terms which
can be shown to be essentially constant in order to solve
these equations. By using the same arguments as Horne
and Anderson (1970) as our first approximation, we neglect

pressure terms where now we consider I8T/8zl 5 1 deg cm-1

and |8wl/82| 5 10 cm-l. The size of the two derivatives
will diminish simultaneously. Secondly, we neglect the
thermal diffusion term in equation (2.5) because [alwlwz/TI

3 l, and magnitude of the thermal

is on the order of 10- deg-
diffusion term is therefore at most 0.01% of the diffusion
term. Finally, we neglect the entropy source term of
equation (2.1) since (Horne and Anderson, 1970) it is

proportional to the square of the velocity gradient,

109

(8v/az)2, which we assume to be unmeasurable in a dif-
fusion experiment. Subsequent results verify this assump-
tion '.

The equations remaining are then:
-q = K(8T/8Z) + pDQi(8w1/8z), (2.8)

pCp(8T/8t) = pD[8 (Fl-8'2) /8z] (8wl/8z)+8 [pDQi(8wl/8z) ]/8z

- pCév(8T/az) + 8[K(8T/8z)]/8z, (2.9)
p(8w1/8t) = 8[pD(8w1/8z)]/8z - pv(8w1/8z), (2.10)
(8v/8z) = -(82np/8t) - v(8£np/8z). (2.11)

3. Perturbation Scheme

The coefficients in equations (2.9)--(2.ll) are
functions of wl, T, and p. In order to solve these equa-
tions we must devise a perturbation scheme which allows
us to take into account the nonconstancy of the coef-
ficients in Space and time. For this purpose we use the
perturbation scheme of Horne and Anderson, introduced in
Chapter III of this thesis. Here, we expand the coef-
ficients in w1 and T about w1A and TM. Variations with
pressure are neglected in accordance with our previous
assumption, and successive derivatives with respect to

w1 and T are multiplied by higher powers of the ordering

parameter, 8. Thus, we have for any coefficient, L,

110

L = Lo + e[(T-TM)LT + (wl-wlA)Lw]

+ ez[(1/2)(T-TM)2LTT+(T-TM)(wl-wlA)LwT+(l/2)(wl-wlA)2wa]

+ C(63), (3.1)
where

L = L(T ,w ) L = (8L/8w) L = (8L/8T)
o M 1A w l TM,w1A T TM,w1A

as) L = (82L/8T2)T

2
L = (8 L/8w

2)
l TM'wlA wT

L = (82L/8w

WW 1

M’wlA
(3.2)
The first and third terms on the right hand side
of equation (2.9) arise because we are studying mixtures.
Without these terms equation (2.9) would simply be the
thermal conductivity equation for a pure fluid. Because
we do not expect thermal conductivity in a mixture to dif-
fer very much from that in a pure fluid, we can treat the
two terms which arise in the thermal conductivity of mix-
tures as perturbations on the pure thermal conductivity
equation. We recognize these perturbations by multiply-
ing each of the two terms by a second perturbation pa-

rameter, 6. Thus, we have

pCé(8T/8t) = 0pD[8(fii-fié)/8z] (8w1/8z) - pC§v(8T/82)

+ 8[K(8T/8z)]/8z + 68[pDQi(8wl/8z)]/8z. (3.3)

Our solutions for T, wl, and v are the perturba—

tion solutions

111

2

A 2
T=T+T+T+ T
o 6 6 T + 6Te + 86 86 + 56 T8

M 6 66 66

+ 0(53) + 0(52)

A - _ 2 2
W1 - w1A - w + 60)6 + 6 (066 + ewe + £60086 + 86 w

o 866

+ 0(53) + 0(22)
v=v +6v +62v +ev +e6v +e6zv
o 6 66 6 86 £66
+ 0(53) + C(82) , (3.4)

Each term in equations (3.4) (except w1 and TM) is a

A
function of space and time. When 8 = 1 and 6 = l, T = T,
81 = wl, and G = v.

The second perturbation parameter 6 has the effect
of making all mass fraction derivatives appear in lower
order equations than their counterpart temperature deriva-
tives. We select this scheme because coefficient varia-
tions with temperature are in general considerably smaller
than variations with mass fraction and because we are
dealing with small temperature variations and large mass
fraction variations. In Chapter III we investigated the
behaviour of zeroth-, first-, and second-order solutions
in e and found for the case of thermal conductivity that
the magnitude of these solutions decreased rapidly with
increasing order. We assume that the behaviour here will
be similar and terms of order 82 should be very small in-

deed. Terms of order a Should be smaller than terms of

zeroth-order in 6, although inversion of these

112

contributions will occur in a system with very large heat

of mixing and small heat of transport. We have not pre-

viously investigated the behaviour of terms which are

zeroth-, first-, and second-order in 6 so we investigate
2

our temperature solutions to order 6 .

We first apply (3.4) to (3.2) to obtain

A: +
L Lo €(TOLT + woLw) + €6(Dw(05 + DTT6)

2 3 2
+ 56 (Dwu)66 + DTT66) + O(6 ) + O(e ). (3.5)

By inserting equations (3.4) into equations (2.9), (2.11),
and (3.3) and by using equation (3.5), we obtain a set of
differential equations which can readily be solved for
each order of e and 6. Very good approximations to T, wl,
and v can be found by solving for the first few terms of
equations (3.4), provided that the series is convergent.
We are, of course, primarily interested in the equation
for T, but it is necessary to have lower order terms in

w1 and v in order to evaluate terms in T.

4. Zeroth-Order Solutions for an Adiabatic System

 

Before stating general equations for all orders
of e and 6, we solve our zeroth-order equations for v0,
To' and mo. These results can then be used to simplify
the general equations.

The initial and boundary conditions for the bary-

centric velocity are

113
vn(z,0) = 0, vh(ia/2,t) = 0, n = 0,6,€,... . (4.1)
The zeroth-order equation for velocity is
(8vo/8z) = 0, (4.2)
so from equations (4.1) and (4.2) we find
vo(z,t) = 0. (4.3)

The zeroth-order mass fraction equation has the

form
(80) /8t) - 0 (320) /8z2) = 0. (4.4)
O O O

The solution of this equation can be obtained through
Fourier transform methods as discussed in Chapter III.

We first change our space variable to x such that
x =12 + (a/2), (4.5)

Our boundary conditions dictate that a cosine transform
be used because we know the gradient of the mass fraction
at the boundaries from equations (2.7) and (2.8) (after

neglecting the last two terms):

I
o
s

(8wl/8x)o't = (8wl/8x)a't - (4.6)

so that

II
0
s

n = o,6,e,... .
(4.7)

(8wn/8x)o,t (800n/8x)a’t

114

We assume wo has the form

00

wo(x,t) = (l/2)foo(t) + m£1 fom(t)cos(mnx/a), (4.8)

where

a
fom(t) = (2/a) é wo(x,t)cos(mnx/a)dx, m = 0,1,2,...
(4.9)

The Fourier transform of equation (4.4) using (4.8) and

(4.9) is

[8fom(t)/8t] + (m2/0)fom(t) = o, (4.10)
where

0 = az/(NZDO). » (4.11)

This equation (or any similar inhomogeneous equation) can
readily be solved using an integrating factor, exp(mzt/O),

and we find
- _ 2
fom(t) - A exp( m t/O) (4.12)

where A is a constant not yet determined. Our initial

conditions on fom are found by requiring that
wo(x,0) = w1(x,0) - wlA' (4.13)
and leaving

(4.14)

II
0
s
5
II
0‘)
‘
m
s
I

wn(x,0)

115
Applying the initial conditions on mass fraction, we find

0 m is even

f0111(0) =

(m-l)/2

2w§(nm)’1(-1) m is odd. (4.15)

Thus, we have

1

wo(x,t) = Zwifl-l 2 (-l)£(2£+l)_ exp[-(22+1)2t/0]cos[(22+l)nx/a].

2 0
(4.16)

as our zeroth-order solution for composition.
Our zeroth-order equation for temperature has the

same form,

1

— - 2 2 _
(8To/8t) - (<0(pCp)O (8 TO/8z ) - 0. (4.17)

A solution is again found using a finite Fourier cosine
transform:

To = (l/2)goo(t) + mil gom(t)cos(mnx/a) (4.18)

because equations (2.7) and (2.4) (consistently consider-
ing the heat flux in a mixture as a perturbation of the

heat flux in a pure fluid) we have
[K(8T/8z) = - 6pDQi(8wl/8z)]z=ia/2, (4.19)
so that in the zeroth-order case we have from (4.7)

(8To/8x) = (8To/8x)a't = 0. (4.20)

0,t

116
The time contribution to the solution is
gom(t) = B exp(-mZt/T), (4.21)
where
r = (pE§)oa2/(Kon2). (4.22)

However, we have B = 0 because

T(x,0) = TM (4.23)
from (2.6), so

Tn(x,0) = 0, n = o,6,e,... . (4.24)
Thus,

To(x,t) = 0. (4.25)

5. Higher Order Solutions for an Adiabatic System

 

We use equations (4.3) and (4.25) to simplify the
differential equations for higher order terms. The ve-

locity equations have the form
(8vn/8z) = Vn(z,t), n = 5,8 (5.1)

where the Vn(z,t) are explicit functions determined from

previous solutions:

V6(z't) 0, V = 0,

66

V€(Z)t) - (inp)w(8wo/8t)r

117

V€6(z,t) = - (£np)w(8w6/8t) - (inp)T(8T6/8t) - v5(£np)w(8wo/8z),
V56612’t) = - (£np)w(8w56/8t) - (£np)T(8T66/8t)
- v5(£np)w(8w6/8z) - v6(2np)T(8T6/8z). (5.2)

We immediately see from (5.2) and (4.1) that

v6 = 0, = 0. (5.3)

V55

The equations for mass fraction have the form
(8w /8t) - D (326 /8z2) = w (z t) n = 5 e (5 4)
n o n n ’ ' "'°’ '
where

W6(z,t) = - V5(8wo/82),
. W65(z,t) = - v55(8wo/8z) - v6(8w6/8z)
W€(z,t) = [Do(1np)w + Dw](8wo/8z)2 + wao(82wo/8zz)
- v€(8wo/8z),
W86(z't) = [2Do(£np)w t 2Dw](8wo/8z)(8w6/8z)
+ [Do(£np)T + DT](8T6/8z)(8wo/8z)
+ wao(82w5/8z2) + wa6(82wo/8zz)
+ DTT6(82wo/8zz)
— v€6(8wo/8z) - v6(8w€/8z) - v€(8w5/8z),
W566 = [(2np)TDO+DT](8T66/8z)(8wo/8z)

+ 2[(2,np)wDo + Dw](8w66/8z)(8wo/8z)

+ [(2np)TDO+DT](8T5/8z)(8w5/8z)

118

+ [(2np)wDo+Dw](8w6/8z)2

- v€56(8wo/8z).

- v€(8w66/8z) - v€5(8w6/8z)
(5.5)

The temperature equations are very similar with

Un(z,t), n = 6,6,...,

- 2 2
(pCp)o(8Tn/8t) - Ko(8 Tn/8z )
(5.6)

where
U5(z,t) = (pDQi)o(82wo/8z2),
U55(z,t) (05§)ov5(8T5/32) + (0DQi)o(3w5/32).
U€(z,t) = 0

085(z.t) = - (pap)wwo(aT5/at)+((p0)o(fil-fiz)w + (puczgpwl(awe/32)2

- (pC§)ove(8T5/8z) (pCé)ov6(8Te/8z) + Kw(8wo/8z)(8T6/8z)

+ wao(82T6/8zz) + (pDQi)wwo(82wo/8zz) + (pDQi)o(82w€/822L

The similar but very long expression for 0566 is given in

Appendix E.
The initial conditions for equations (5.4) and

(5.6) are given by equations (4.13) and (4.23), respec-

tively. It is obvious that the equations must be solved

in a certain order, although there is some flexibility.
We choose to solve them in the order T5, v5, w6' T55,
Te’ Ve’ we' Te6' v86' ”66' T666’ v666’ we66'

V85' 856'
The order in which the solutions are found is immaterial

119

unless an unknown function is required in the inhomogeneous
part of some equation, in which case solution is impossible.

Obvious solutions of the above equations include
V5 =85 ”1'55 =V5<s =w55=Te=veaa =“eaa ‘0'

These results greatly simplify the expressions (5.1), (5.3),
and (5.5). In order to have a complete set of equations
through second order we would actually have to solve the

velocity expressions (5.1) for vE and v85 with

Vs = - (inp)w (8wo/8t)

V65 = — (inp)T (8T6/8t), (5.9)

the mass fraction equations (5.4) for we and “66 where

w8 = [Do(£np)w+Dw](8wo/8z)2 + wao(82wo/8z2)-v6(8wo/8z),

2 2
W56 — [Do(£np)T+DT](8T5/8z)(8wo/8z) + DTT5(8 wo/8z ), (5.10)
and the temperature equations (5.6) for T5 and T26 where
2
(3200/82 ).
U80 = "' (pCp)wwo(8T6/8t)

+ ((pn)o(fii-fiz)w-+(p003)w1(awo/az)2

(05)v

P o e(8T6/8z) + Kw(8wO/az)(aT6/3z)

+

(0DQi)wwo(32wO/322)

+ wao(32T6/322) + (0003 (azwo/azz). (5-11)

)0

120

The solutions of equations (5.1) have been obtained

using the boundary conditions (4.1). They are

v6 = Do(£n0)w2wia-1££0(-l)2{exP[-(2£+l)2t/0]}

x Sin[(22+l)nx/a],

v68 = (000;)0(2np)T(pE§>;1(1-(r/e)1'1zwia‘1

X

2

(-1)1+1{exp[-(22+1)2t/01}sin[(22+1)nx/a].

"MB

0
(5.12)

The equations (5.4) are solved by Fourier cosine

transforms as described previously. The result for we is

we = (2w22/02)<(£n0)w X (22+1)‘2{1-exp[-2(22+1)2(t/0)1}
(=0
+ Z X (24+l)-1(-1)n+1 {(2np) -n(D /D )(22+1—2n)‘
n=1 i=0 W w o

1}

1

X

[22+l-2nl-

X

(exp(-l(21+1)2+(22+1-2n)21(t/0)}-exp[-4n2(t/0)fl
1

+

{(2np)w+n(Dw/Do)(22+1+2n)‘1}[22+1+2n1‘

X

(exp{-[(22+1)2+(2£+l+2n)2](t/0)}-exp[-4n2(t/0)]n

X

cos (2n1rx/a)} . (5. 13)

The result for was is similar to that for we but smaller in
magnitude though much greater in length. Like we it is an
even function. This result is not given here as it is not
necessary for determining the temperature solution through

first order in e and second-order in 6.

121

The temperature equations which do not have trivial
solutions as shown in (5.8) are better expressed in the form
(p6 )(8T /8t)-K (321 /8z2)
p>o j6 o j6
2 2
= *
(pDQl)o(8 wj/8z ) + Yj(z,t)
j = 0,8 (5.14)
where

0)

Yo(z,t)

Y€(z.t) (pE§)wwo(aT5/at)

+ [(0D)o(fii-fié)w+(0voi)wl(awe/az)2

(pCé)ov€(8T5/8z) + Kw(8wO/8z)(8T6/8z)

+ wao(82T6/8zz) + (pDQi)wwo(82wo/822).
(5.15)

The equation for T€56 is given in Appendix E along with
its solution. The boundary conditions for each order are

found from (4.18):

[Ko(3Tj5/32) + (DD011018wj/3211ia/2,t = Cj(ia/2.t). j = 0.8

with

I
o

Co(ia/2,t)

c€(:a/2.t) 119D011ww018wo/az)1ia/2,t' (5.17)

122
The Fourier cosine transform of equation (5.14) is
atg. (t)1/at + (mz/r)g. (t) = c (t)(p6 )‘1
36m 36m jnl p o

2 3 - '1 2 a
+ (Zn /a )(pDQ11019Cp10 m g chos(mflx/a)dx

+ (2/a)(pC )-1 [3 Y.(x,t)cos(mnx/a)dx, (5.18)
p o 0 3

where
= - m - -
ij(t) (2/a)[( 1) Cj(a/2,t) Cj( a/2,t)]. (5.19)

The temperature solution T6 is then given by

15 = (zwi/n)(p00;)o K;1(1-(r/0)1'1

x Z (-1)1+1<22+1)‘1{exp[-(21+1)2t/0]-exp[-(22+1)2t/r]}
2=o

X coS[(22+l)nx/a]. (5.20)
and the expression for T56 is given in Appendix C.

6. Solutions for a Cell with Diathermic Ends

The temperature solutions for the cell whose ends
are diathermic differ radically from the adiabatic solu-
tions of Sections 4 and 5. However, the equations to be
solved and the initial conditions are the same; the only
difference is in the boundary conditions. This, fortu-
nately, causes all those solutions which were zero in the
adiabatic case to again be zero in the diathermic case;

i’namely, equation (5.8) is true in the diathermic case

also.

123

Our solutions are found in a manner similar to
that described in Sections 4 and 5. The boundary condi-

tions for temperature now are
Tn(ia/2,t) = 0, n = 6,66,..., (6.1)

and all of the temperature solutions have the form

00

Z 9nm(t)sin(mnx/a). n = 5,e5,... . (6.2)

T (x,t)
n m=l

For order 6 we have

T6 = <8wi/"2)(°DQE’OK81 g 2 (-1)1+1(21+1)2mt4m2-(21+1)21'
m=l 2—0

1

x [4m2-(r/0)(22+1)21'1{exp[-(22+1)2t/01-exp[-(22+1)Zt/r]}

X Sin(2mnx/a) (6.3)

and T86 is given in Appendix D. We expect T855 to be very

small as it is in the adiabatic case.

7. Calculated Temperature Variations

In order to examine the behaviour of the tempera-
ture solutions, we assign typical values for organic
liquids to all of the coefficients and calculate the
values of the solutions at various times during the ex-
periment and at various positions in the cell. The ther-
mal conductivity, heat of mixing, heat of transport, and
initial concentrations are varied in order to demonstrate

their respective contributions to the temperature

124

distribution Table 2 gives the values which have been as-
signed to those coefficients which we have not varied in
our calculations. These coefficients would not be correct
for all systems, but they represent typical behaviour in

any system at relatively short times (t < 20).

Table 2--Typical values of physical properties for a
binary mixture of organic liquids

 

- 3 -3 -
O - 0.8 x 10 kg m Dw/Do — 0.1
(inp)w = 0.1 (0DQi)w/(0D0i)o = 0.1

_ _ -3 _ -3

()an)T - 1.0 x 10 (DDQI1T/(OD0110 - 2.0><10
—- _ 6 -1 -3 _ _
pCP - 1.4 x 10 J K m Ko/Kw - 0.1

_ _. _ _ _ -3
(pCP)W/(pCp)o — 0.1 Ko/KT — 2.0 x 10
(pC)/(OC) =-10><1o‘3 (H—fi') =20x1020k‘1x'1

p T p o ' , 1 2 T ’ g
-9 2 -1

D = 1.4 X 10 m s

 

The coefficients (Hi - fiZ’w have been calculated
from typical heats of mixing (Rock, 1969; Lewis, Randall,
Pitzer, and Brewer, 1961) as is shown in Appendix F.
Heats of transport have never been measured in liquids,
but by assuming that the heat matter Onsager reciprocal

relation is valid so that (Horne and Bearman, 1967)

125
Qi = " alW1(3)Jl/3W1)T,Pp (701)

we have calculated values from thermal diffusion factors.
Thermal conductivity does not vary radically among systems
which could be studied, but, because it has a marked ef-
fect on the speed with which temperature variations in the
cell relax, we consider two thermal conductivities.

In all cases the results are presented in reduced
time (t/O) and reduced cell coordinate (z/a). Because the
length of the cell enters into the solutions only through
these two factors, the results are the same for any length
cell. The factor I depends on 3 also, but it can be re-
placed in all solutions by (T/O)0. The factor (1/0) is
independent of a. From the definition of 0(4.ll) we see
that it is proportional to a2 and, thus, for longer
cells much larger absolute times will be needed to reach
the temperature distributions which are rapidly attained
in Shorter cells. This fact will cause problems for the
experimentalist in that, since his cell cannot be truly
adiabatic, deviations from predicted behaviour due to non-
adiabaticity will be larger at longer times. For this
reason short cells should be used. In a 3 cm cell with D
of Table 2, 0 is 108 minutes; for a 10 cm cell, 0 is 1200
minutes.

All computations have been done on a, CDC 6500
computer with series truncation criteria similar to those

discussed in Chapter III. More terms are needed at Shorter

126

times, and with double and triple sums the computational
cost becomes quite high at these very short times (t/O i
0.01). Computation at forty—nine points in the cell was
done simultaneously for each t/O because the time depen-
dent parts for every point z/a are identical.

The results presented here have been determined

from

T = T6 + TeG' (7-2)
Since T855 is very small in the adiabatic cell, with
-2 -1 -1 — - 4 '1
pDQi = 3 x 10 J m S , (Hl - H2)w =.5 x 10 J kg ,
K = 2 x lo-lJ m-ls-lK-l, and w0 = 0.8, equation (E.4) of

1
Appendix E yields a maximum value of 4.32 x 10-4K at
t/O = 0.09. The value becomes more negative as t/O
increases and asymptotically approaches its equilibrium
value. The T566 term contributes less than 1.0% to the
total temperature perturbation and is symmetric about
z = 0, so that it does not affect the temperature dif-
ference between any two symmetrical points in the cell.
Because T266 is so small, we have omitted it from further
work without thereby introducing any measurable error.

The symmetries1of the solutions T and Te make

6 6

it possible to observe the Dufour effect. T6' which is

due solely to heat of transport, is antisymmetric about

2 = 0. The T6 solution is symmetric about 2 = 0. The

6
'value of this solution depends on the variations of

127

physical properties and on the heat of mixing. It is
generally small except for the contribution of a large
absolute heat of mixing. However, if one is interested
in measuring the Dufour effect, he must measure the tem-
perature difference in the cell. The solution, TeG' be-
cause of its symmetry adds nothing to the temperature dif-
ference which is measured at two points symmetric about
2 = 0. These symmetries can be seen in the next twelve
figures. Note that the ordinate scale of these figures
changes from one to another.

Figure 3 shows calculated temperature distribu-
tions in an adiabatic cell with no heat of mixing,

pDQi = 3 x lO-ZJ mfils-1

(a1 = - 0.9), and thermal con-
ductivity of 2 x 10-1J mmlsle-1 at various times. The
most important feature of this plot is that a temperature
difference of approximately 0.12K exists between the
right and left hand sides of the cell. This difference
is due almost entirely to the Dufour effect, and we thus
predict that the Dufour effect in liquid mixtures is
measurable. The slight deviation of the curves from true
antisymmetry about 2 = 0 is due to our inclusion of Spatial
variations of coefficients in the cell. We observe that
the temperature fluctuations arise in the center of the
cell where there is a concentration gradient and that the

temperature throughout the cell changes as heat is con-

ducted and convected. We also see that the temperature

 

128

Figure 3—-Temperature variations in an adiabatic cell
for 11111“ = 0.0, pDQ1= 3 x 10-2J m-ls_1, K =
2 x 10‘1J m'1s‘1K-1, and w: = 0.8 at (t/e) =
0.01(ooooo),(t/0)=0.05( ),
(t/e) = 0.09 ( —-), (t/O) = 0.13

( ————— ), and (t/0) = 0.17 (——-----).

 

 

 

0.12 1—

0.08 *-

 

129

 

 

0.04

8
l-g 0.0

I
L'.

-0.04

-0.08 *-

- 0.12 *-

l l I l J
- 0.4 - 0 2 0.0 0.2 0.4
2 /a

Figure 3

130

difference initially rises and then falls again, with the
maximum difference being observed at 0.05-0.09 t/O. The
fact that the temperature is quite stable for z > .35 and
z < -.35 for these times Should enable the experimentalist
to make accurate temperature measurements well inside the
cell; 112;! end effects will not matter. We note that,
Since 82.np/8w1 is positive, the end of the cell which
originally has the greater density is that which has the
greater concentration of substance 1, i;g;, the end for
which z < 0. The temperature gradient arises to oppose
this density gradient so that the end for which 2 > 0 is
the warmer end. This is what would be ordinarily expected
although a negative heat of transport would reverse the
effect.

The fourth figure shows temperature distributions
in a system identical to that considered in the first
figure except for the thermal conductivity. The lower
thermal conductivity produces much larger temperature ef-
fects because heat is not conducted so rapidly. Measure-
ments will be easier for systems with low thermal conduc-
tivities.

We have again studied the first system in Figure
5 where in this instance (Hi - E21w = 5.0 x 104 J kg_1.
which corresponds to a heat of mixing (Afih) of - 6.5 x 103

J kg-l. We see that this causes a general rise in the tem-

perature of the entire cell. The temperature difference

 

131

Figure 4--Temperature variations in an adiabatic cell
for AH = 0.0, pDQ* = 3 x 10‘2J m'1s'1, K =
91 -l 1 1--l o
l x 10 J m s' K , and wl = 0.8 at (t/0)
0.01 (00 o co ), (t/G) = 0.05 (—————)
(t/O) = 0.09 ( —), (t/O) = 0.13

( ______ ), and (t/0) ='0.l7 (—-—---—-—-).

 

I

132

/’::€::=-.-

 

 

 

_ _ _ _ _ _ _
2 8 4 O 4 8 2
m. 0. O 0. O 0. .I
0 O O 0 n_u n_u

_

AXVEEhIC

0.0 0.2 0.4
2 /a

-0.2

-0.4

Figure 4

 

133

Figure 5--Temperature variations in an adiabatic cell

1

for AH = - 6.5 X 103 J kg- , pDQi = 3 X
J m_15—11K = 2 X lO-lJ'm_ls_1K-l, and mi

0.8 at (t/O) = 0.01(OO 00.),

10‘2

(t/O) = 0.05

 

('———-—————-)p (t/G) = 0.09 ( —~
'-'-' 0.13 ( _____ )r and (t/e) =
_ ).

 

0.17

),

(t/G)

(T-Tm)/(K)

1.2

0.8

0.4

134

 

 

/
/
— /
—— /
’ ——————————
w
a
I
’1
L— -"”
/./-~-‘-‘
’/
/
-/‘
-——--—-'—
h-
o...
O
C .0.
O
.0 '0.
‘0. ‘..'
‘..". "0¢boo
..

 

Figure 5

135

due to the Dufour effect is remarkably constant, however,
at about 0.12K for t/G = 0.05-0.13. Because the overall
heating of the cell will cause slight errors in constants
used even though the constants can be calculated correctly
at any temperature, it would be best to measure the tem-
perature difference as soon as it is well established

(t/e = 0.09) and before it begins to decay.

Our next three figures are also for the adiabatic
cell, but the effects of varying various constants are
illustrated at a fixed time (t/e = 0.1). Figure 6 illus-
trates the variations of the temperature distribution
with (El — fié)w or heat of mixing. Values used for
(ii - fié)w are 1.5 x 105, 5.0 x 104, o, and - 5.0 x 104J
kg-'l which represent heats of mixing ranging from

3 1 l

-19.5 x10 to 6.5 x 103 J kg- . We see that large

J kg-
heats of mixing cause much greater effects on the tem-
perature than do heats of transport.

Illustrated in Figure 7 is the variation of the

temperature difference with heat of transport. Values

2 2 3 2

for pDQi are 6 x 107 , 3'x 10‘ , 3‘x 10‘ , and - 3 x 10‘

J mfls-a. The small heat of transport is swamped by the
heat of mixing and has a temperature difference of 0.012K.
In every case we have seen that the temperature differ-

ence at its maximum is directly proportional to the heat

of transport.

 

136

Figure 6—-Temperature variations in an adiabatic cell
for (t/G) = 0.1, pDQ* = 3 x lO-ZJ m-ls_l,
_ -l -d.-l -l o _ .
K - 2 x 10 J m s K , and ml - 0.8 With

Afim= - 19.5 x 103 (——-—-), - 6.5
103 ( —————— ),o.0(————),and
6.5 x 103 J kg’l (—————).

137

 

 

 

 

 

, / 4m
, ,
.
_ I .2
. o.
. _ — _
_
r. n _ 4
_ .0.
_ : _ _ __ _ __ _ _
4. 0. AN 2 8. 4. 0. 4. 8.
0‘ 0‘ 1| 1| AU AU nu JV mu

3: \ “Eh ..t

0.4

0.2

z/a

Figure 6

 

.
J
mum-"I.
.__

138

Figure 7-—Temperature variations in an adiabatic cell

for (t/e) = 0.1, Aim = - 6.5 x 103J kg'l,
K = 2 x 10‘1J m‘ls'lx'l, and w? = 0.8 with
'2 ( ), 3 x 10’2

(-—-—-——-—-), and

 

 

 

(T - Tm)/(K)

0.9

 

139.

 

Figure 7

 

140

The last graph for an adiabatic system, Figure 8,
shows how greatly temperature variations in the cell are
reduced as the two initial mixtures become more similar.
An 0.075K difference can still be seen with an initial
difference of 0.5 in the weight fractions. The heat of
mixing effect has been greatly reduced, however, as it
is proportional to wiz. Some sacrifice of temperature
difference for suppression of the heat of mixing might
be advisable in experimental systems with large heats of
mixing. Interesting to note are the very small variations
of temperature (0.005K) in the region of mi = 0.05—0.01.
Such variations would not affect diffusion measurements
at these relative concentrations.

Our final six figures deal with a cell whose ends
are diathermic. These figures present a much greater
challenge to the eXperimentalist because the temperature
fluctuations are much smaller and there is no temporal
region in which the temperature distribution remains
relatively constant. These six figures correSpond to the
Figures 3-8 except that Figure 14 has a heat of mixing.
Ianigure 9 we see that our temperature difference falls
rapidly with increasing time. At t/O = 0.05 the differ—
ence is 0.07K and at t/0 =_0.17 it has fallen to 0.04K.
The maxima occur between 0.15 and 0.25 z/a and the minima
between - 0.15 and - 0-25 z/a. The curves are again not

symmetric, but they are more so than in the adiabatic

 

141

Figure 8--Temperature variations in an adiabatic cell
for (t/O) = 0.1, Afih = 0.0 J kg-l:

 

K = 2 x 10'1J m‘ls‘lx'l, and one; = 3 x 10'
J m-ls-lwith w‘1’= 0.3 (o o o o u), 0.5
< -—-), 0.2 (--————-), 0.05

(————--——— ), and 0.01 (-————-—-—-).

2

0.12

0.08

0.04

(T- Tm)/ (K)
.0
o

I
.0
C)
a.

-0.08

-O.'|2

 

00.00.00.

+—

 

Figure 8

f
"
----_--'

142

 

0.0 0.2 0.4

 

143

Figure 9--Temperature variations in a cell with diathermic

ends for Afih = 0.0, pDQi = 3 X 10.2 J m-ls-l

I
K = 2 x 10-1 J m'ls‘lx-l, and mi = 0.8 at (t/G)
= 0.05 (———-*-———-), (t/e) = 0.09 ( ______ ),

(t/e) = 0.13 (————
(-————-———).

 

), and (t/O) = 0.17

144

0.06 f"

 

 

 

_\
0.03 .— / - _\\
3
E
I
t:
-0.03 P'— \ ‘_/l
-0.06 ‘-
J J l L 1
-0 4 '0 2 0.0 0.2 04

Figure 9

145

case. Halving the thermal conductivity again almost
doubles the observable temperature difference as seen by
comparison of Figures 9 and 10. The introduction of a heat
of mixing in Figure 11 changes the temperature distribu-
tions little from the original curves of Figure 9. The
curves are raised only a few hundredths of a degree due

to rapid replacement of the heat of mixing through the
end walls so that the overall temperature of the cell can-
not rise appreciably. Figure 12 better emphasizes the
greatly reduced effect of the heat of mixing, especially
compared to Figure 6. That the diffusionist needs worry
even less about difficulties arising from temperature
fluctuations in a diathermic cell than about those in an

adiabatic cell is attested to by Figure 14.

8. Discussion

 

We have attained temperature solutions both for
adiabatic cells and for cells with diathermic ends when
initially a sharp concentrations boundary exists in the
center of an initially isothermal cell. From these solu-
tions we find that the Dufour effect is measurable, that
an adiabatic cell produces larger temperature differences
than a cell with diathermic ends, that high heats of mix-
ing do not cause significant variations in temperature
changes due to the Dufour effect in a cell with diathermic

ends, that low thermal conductivities and low heats of

 

146

Figure lO--Temperature variations in a cell with diathermic

ends for Afih = 0.0, pDQ* = 3 x 10—2 J m-ls-l,

K = l x 10-1 J m_ls-1K‘}, and w: a 0.8 at (t/O)
= 0.05 (———---———-), (t/G) = 0.09 (—~—-—-—-—-)p
(t/@) = 0.13 (‘—__‘ —_—-), and (t/e) = 0.17
(————-—-).

 

(T-Tm)/(K)

0.06

0.03

0.0

-0.3

 

147

 

Figure l0

I/\
\
£7“.
_ M
'l
.I/
,’
I
//
I’l’
__ \ I,"
0‘ I
\K, /, ,/
\ l
__ \:~ /
1 V , 1 I
-0.4 -0.2 0.0 0.2 0.4
z/a

 

148

Figure ll--Temperature variations in a cell with diathermic

 

ends for Afih = - 6.5 X 103 J kg-l, pDQi = 3 x
10.2 J mfls-l, K = 2 x 10-1 J m-ls-lK-l, and
w‘1’= 0.8 at (t/O) = 0.05 ( — ). mm» =
0.09 ( —————— ), (t/e) = 0.13 (———— ).

and (t/O) = 0.7 (-—————-—-).

149

 

 

0.6 r

 

332:; :5

-0.03 '—

-0.06 -

0.4

0.2

0.0
z/a

-0.2

-0.4

Figure 11

‘. 'hffltcfl

 

Figure 12-—Temperature variations in a cell with diathermic

ends for (t/O) =

K = 2 x 10-
AK
m
6.5 x 10
guishable.

3

1

J kg—

J m-
— 19.5 x 103, - 5,5 x 103, 0.0, and

l

0

1s-

150

.1, pDQi = 3 x 10"2 J m'ls‘l,

1K‘1, and mi = 0.8 with

The curves are indistin-

 

151

0.0 0.2 0.4
z/a

- 0.2

-0.4

 

 

0.03 ~—

0.02 —-

0.0] ._

0
D

0
CSIEFIC

'-C)CH P—
-0.02 .—
‘0 03

Figure 12

 

152

Figure 13--Temperature variations in a cell with diathermic

 

 

ends for (t/O) = 0.1, Afih = - 6.5 x 103 J kg'l,
K = 2 X 10”1 J m-ls-lK-l, and mi = 0.8 with
pDQg=6x10'2( —),3><1o‘2

(——— ),3x10‘3(-———).and-3x

10-2 J m-ls-l ( ------ ).

0.6

0.3

- 0.03

- 0.06

153

 

 

 

 

 

Figure 13

g ,/ ‘\\
' \\ ’ ‘ ’
\ \\ / \
_ \ r“ /
\ /
\ /
\/
L...
I I l l 1
-04 -O.2 00 0.2 0.4

 

154

Figure l4--Temperature variations in a cell with diathernnil:

ends for (t/G) = 0.1, Ail-m = - 6.5 X 103 J kg-J':
K = 2 X lO-1 J m-ls-lK-l, and pDQi = 6 X 10"2
J'm-ls-l with w‘1’= 0.8 (o o o ...), 0.5
(—-——-),0.2( ------ )p0.05 (—_—)I
and 0.01 (——————).

155

 

 

 

 

oooouvsg 1A
0 no. \ xx \ 0
\ \ _
~
. _ Ila/m
/ ., 1 o
OOOOOI/ z/I
ooooyu% nu.
..zbbfiooooo .10
Izle/luooooo
’ .u /. 12
nu
\ \ \\oooo.a 4
3.2.1....
r P _ . _ _
m. m m. w m. m m
o o o ... ... ...

z/a

Figure 14 .

156

mixing are conducive to easier measurement of the Dufour
effect, and that the Dufour effect becomes very small as
the initial concentration difference decreases.

Using our results we can analyze the work of
Rastogi gt_gl. (1965, 1969, 1970). First, they neglected
heat of mixing. We find this a very poor assumption in
that the temperature at all points of the adiabatic cell
depends strongly on the heat of mixing. However, our re-
sults do show that, if the points between which the tem-
perature difference is measured are symmetric about 2 = 0,
the heat of mixing does not appreciably affect the tem-
perature difference measured. The assumption by Rastogi
gt_§1. that coefficients are constant is not correct but
does not introduce large errors. Their cell design is
not really comparable with our idealized cell in that
their initially sharp concentration boundary was achieved
by a stop cock whose diameter was only half that of the
cell. Thisconstriction was not taken into account in

their results. Their cell was 23 cm long and measurement

was made at z/a i 0.24. This falls in the region which
does not attain a steady temperature difference as dis-

cussed in Section 7. The maximum temperature difference
between these two points was reached at very low reduced
time; and the concentration difference between the points

was assumed to be the initial concentration difference.

From Figures 3 and 4 we observe that the maximum

157

temperature difference varies with the z/a used while the
assumption that the concentration difference is the initial
one is still reasonable. Rastogi and Yadava (1970) recog-
nized this and took the variations into account with their
experimentally determined parameter 6. Unfortunately their
results are incomprehensible because one given distance
between their thermocouple and stop cock is greater than
the given half length of the cell. As we have shown, it
is the value z/a, not 2, which is important to the tempera-
ture distribution.

We hope that the results of our work will provide
a useful guide to the experimentalist. The solutions given
are such that, if all diffusion coefficients, heat capaci-
ties, 232;! are known, they can be substituted into the
temperature equation, and the heat of tranSport can be
adjusted to fit the measured temperature variations. Our
results also indicate that the Dufour effect causes no
complications in diffusion experiments with very small

concentration differences.

BIBLIOGRAPHY

BIBLIOGRAPHY
Bartelt, J. L., Ph.D. Thesis, Michigan State University,
1968.
and F. H. Horne, Pure Appl. Chem., 22, 349 (1970).

Bearman, R. J. and J. G. Kirkwood, J. Chem. Phys., 33, 136
(1958).

Carslaw, H. S. and J. C. Jaeger, Conduction of Heat in
Solids (Oxford U. P., London, 1959), 2nd ed.

 

Clusius, K. and L. Waldmann, Naturwiss., 32, 711 (1942).

Coleman, B. D. and V. J. Mizel, Arch. Rational Mech. Anal.,
13, 245 (1963).

and v. J. Mizel, J. Chem. Phys., 4_o, 1116 (1964).

and W. Noll, Arch. Rational Mech. Anal., 12, 167
(1963).

Doria, M. L., Arch. Rational Mech. Anal., 3, 343 (1969).
Dufour, L., Poggend. Ann. Physik, 148, 490 (1873).

Fitts, D. D., Nonequilibrium Thermodynamics (McGraw-Hill,
New York, 1962).

 

Grew, K. E. and T. L. Ibbs, Thermal Diffusion in Gases
(University Press, Cambridge, 1952).

 

de Groot, S. R., L'Effet Soret (North Holland Pub. Co.,
Amsterdam, 1945).

 

, Thermodynamics of Irreversible Processes (North
Holland Pub. Co., Amsterdam, 1952).

 

and P. Mazur, Nonequilibrium Thermodynamics
(North Holland Pub. Co., Amsterdam, 1962).

 

Gyarmati, I., Nonequilibrium Thermodynamics (Springer-
Verlag, New Yofk, N.Y., 1970).

 

158

159

 

Haase, R., Thermod namics of Irreversible Processes (Addison-
WesIey P55. Co., Reading, Mass., 1969).

Horne, F. H. and T. G. Anderson, J. Chem. Phys., 53, 2332
(1970). 7"

and R. J. Bearman, J. Chem. Phys., 46, 4128 (1967).

 

Kestin, J., A Course in Thermodynamics, Vol. I. (Blaisdell
Pub. Co., Waltham, Mass., 1966).

 

Kirkwood, J. G. and B. Crawford, J., J. Phys. Chem., 56,
1048 (1952).

Lewis, G. N., M. Randall, K. S. Pitzer, and L. Brewer,
Thermodynamics (McGraw-Hill, New York, N.Y., 1961)
2nd ed.

 

McLaughlin, E., Chem. Rev., 34, 389 (1964).
Meixner, J., Ann. Phys., 22, 244 (1943).

and H. G. Reik, Handbuch der Physik, Vol. III/2,
S. Flugge, Ed. (Springer-Verlag, Berlin, 1960).

 

 

Miller, L., z. Naturforsch, 43, 262 (1949).
Mfiller, 1., Arch. Rational Mech. Anal., 36, 118 (1967).

, Arch. Rational Mech. Anal., _2_§_, 1 (1968).

 

Olson, J. D., Ph.D. Thesis, Michigan State University,

Onsager, L., Phys. Rev., 31, 405; 38, 2265 (1931).

Prigogine, I., Etude Thermodynamique des Phenomenes
Irreversible (Editions Desoer, Liege, 1947).

, Introduction to Thermodynamics of Irreversible
Processes (Interscience Pub., New York, N.Y., 1955).

Rastogi, R. P. and G. L. Madan, J. Chem. Phys., 43, 4179
(1965).

and B. L. S. Yadava, J. Chem. Phys., 51, 2826
(1969).

 

and B. L. S. Yadava, J. Chem. Phys., 52, 2791
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160

Rock, P. A., Chemical Thermodynamics (MacMillan Co.,
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Tree, D. R. and W. Leidenfrost, "Thermal Conductivity
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Truesdell, C., Rend. Lincei (8) 22, 33 (1957).

and W. Noll, Handbuch der Physik, Vol. III/3,
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and R. A. Toupin, Handbuch der Ph sik, Vol. III/1,
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, Z. Naturforsch, 1, 59 (1946).

 

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, Z. Naturforsch., 43, 105 (1949).

 

APPENDICES

APPENDIX A

SOME EQUATIONS OF THERMOSTATICS

AI = EI - TS (A.1)
dEI = TdS - Pd(1/p) + (01 - u2)dwl (A.2)
HI = EI + VP (A.3)
- dP + wldml - p2) + SdT = 0 (A.4)
pHI = {paid (A.5)
Ha = 1.1a + TS“ (A.6)
s = - aAI/BT (A.7)
Ewan“ = AI + P/p (A.8)
11a = AI(l - wa) (aAI/aw1)p,'r+92(3AI/3°)w1,'r (A.9)
pa = AI + p(8AI/8pa)pB’T (11.10)
(“1"“2/9)‘=(3AI/3°1)p2,m"(aAI/392)pl,T
.= 1/°(3AI/3w1)m,p (A.ll)
P = 991(3A1/391)p2,m + 992(3AI/892)91,T
= pz‘aAI/3°)w1,m (A.12)

161

APPENDIX B

A RATIONAL MECHANICAL DEVELOPMENT

OF THE TIP THEORY

The heuristic but very successful TIP results of
nonequilibrium thermodynamics have often been criticized
for their lack of mathematical rigor by the prOponents of
rational mechanics. It is, however, quite possible to de-
rive the results of TIP with the formalism of rational
mechanics by redefinition of thermodynamic process for a
mixture in such a way that there is no difference between
the specific partial stress tensors and there are no in-‘
ternal forces among components. The thermodynamic process
is then defined by the Bv + 5 functions which satisfy the
balance of mass for each component, the balance of linear
momentum for the mixture as a whole, and the balance of
energy. Now we can further assume that the external forces,
bf, may be solved for using the linear momentum balance
equation. There are no interaction forces. With this
limited definition of a mixture, the TIP results are imme-
diately derivable from a linear rational mechanical for-

malism.

Our balance of energy equation is

162

163

p(dE/dt) = (avi/axj)Tij-(3q;/3xj) + pr

+ (0102/0)ui(bi - b?

l). (3.1)

This can be placed in the entropy balance equation using

the thermodynamic identity
A = E — TS. (B.2)
The entropy balance equation which results is

pT(dS/dt) = - p(dA/dt) - pS(dT/dt) + (avi/ij)Tij

- (aqg/axj) + (9192/9)ui(bi - bi).(B.3)

Until this point we have not needed to distinguish among
the independent variables and the functionals determined
by constitutive equations. Studying our definition of
thermodynamic process and those terms which appear in the
energy balance equation we let our independent variables

be
3p1 3p2 8T 3V1

91' 92' T' 3x.’ 8x.’ ax.' 3x.’
3 J J J

 

 

(E.4)

and our constitutive relations in terms of these variables

will define

ui, Tij, qg, fi, A, s. (3.5)

Following the principle of equipresence in the

linear limit, we obtain constitutive relations of the form

164

A = A(pl, 02: T) (8.6)
Tij = - Paij + ¢(8vk/axk)sij + 2n(3vj/axi) (B.7)
* = _ n _ n _ n

qj qu(3p1/3xj) Cq2(892/8xj) CqT(3T/3xj)
(8.8)

kj = - Cfll(3p1/3xj) - C£2(392/3xj) - C£T(3T/3xj)
(8.9)

uj = - C31(apl/3xj) - C32(apZ/8xj) - C3T(8T/8xj),
(3.10)

where k3. is defined by

q?
k. = f - -l (3.11)

The derivatives dA/dt and Bkj/axj can be expanded in terms
of the independent variables and dpa/dt can then be re-
placed by the balance equations:
(do /dt> = - (92/9)u (39 /3x ) - (92/9)u (89 /ax )
l 2 j 1 j l j 2 j

- 9192(8uj/3xj) - 991(3vj/3xj)

2 2
(dpz/dt) (92/9)uj(391/3xj) + (pl/p)uj(392/3xj)

+ Bu. 3x. — 3v 3x. . 8.12
9192( J/ J) 992( j/ J) ( )
An immediate consequence of this formalism is

kj = 0. (B.13)

The total entropy production is readily rearranged to the

form

165

9T9 = (“1 - 02)[3(9192/9)uj/3xj]
*
+ (Tij + P)(8vi/3xj) + qj(8T/3xj)

+ (0102/0)ui(b% - bi). (9.14)

The coefficient P is identified as the thermostatic

pressure,
P = ptpl(aA/ap1) + 02(8A/302)] (3.15)

and the usual thermostatic definition of chemical potential

has been used. The result
8 = - aA/aT (8.16)

is a consequence of the Clausius-Duhem inequality. Re—
arrangement of the entrOpy balance equation (8.3), then

leads to the TIP result

9T¢ = 3[(ui - 95)9192uj/p]/3xj
(9192/9)uj[3(ui - u§)/3xj] + q§(8T/3xj)

+ (Tij + Pdij)(3vi/8xj). (8.17)

.-- 3

In: ml.‘-—'.al—-._t.vdn -.' I.
h
:L

 

 

 

APPENDIX C

THE TEMPERATURE SOLUTION, T FOR THERMAL

ll
CONDUCTIVITY OF A PURE FLUID

IN A FLAT PLATE CELL

The solution of equation (III.3.13) for n = 1,

subject to the conditions Y = Yu = YL and Tod-TL:=2TM, is

T1(z,t) = (4/fl32(AT)2££0[(KT/Ko)(2£+1)-l

(2£+1)-2{1 - exp(-(21+1)2t/T]}

2x[(22+1)2x-11-1{exp(-t/y)

exPI-(22+l)2t/T]}

+ x[(22+1)2x-2]-1{exp(-2t/Y) - epr-(2£+l)2t/r]}
+ 2 -+ Z](-l)2cos[(2£+l)nz/a] (c.1)
l 2
‘where
x = Y/T. (0.2)

2 = Z (4n2x - 1)'1{2(30(22+1)‘l
l 1v=l
- [2(KT/Ko) - 301(22+1)[2n+22+11’1[2n-(22+1)1‘1

- [(pCé)T/(QC§)O - (KT/Ko)]

166

 

 

 

+

X

167

16n(22+1)[2n+22+1]‘2[2n-(22+1)1’2}

X[(2£+1)2X - l]_l{exp(-t/Y) - exp]-(22+l)2t/r]}
[(21+1)2-4n21'1{exp[-4n2t/r] - exp[-(2£+1)2t/T]}
x[(22+1)2x - 21'1{exp(-2t/y) - exp[-(22+1)2t/T]}

x{((22+1)2—4n21x - 1}-l{exp[-4n2(t/T)-(t/Y)]

exp [-(22+1)2t/T]} (
2 (4n2x - l)-l(4m2x - l)-l

m=l
(480(2n-(2£+l)]-l[2n+(2£+l)]_14-{[(KT/Ko)-Bo]4(2£+l)
[4n2+4m2+(2£+l)2]

[(pEé)T/(pfié)o - (KT/Ko)]32(2£+l)m2 }
{[2(n+m)+2£+1][2(n+m)-(2£+l)][2(n-m)+(2£+1)]
[2(n-m)—(22+1)1}"l

(x{[(22+1)2-4m21x - 1}‘1{exp[-4m2(t/r)+4m2(t/y)1
exp[-2(2£+1)2t/T]}
[(22+1)2-4(n2+m2)1‘1{epr-4(n2+m2)t/rl
exp(-(22+1)2t/rl}

xt(22+1)2x-21‘1{exp[-2t/y1 - exPI-(2£+l)2t/T]}
x{[(2£+1)2-4n2]x - 1}‘1{epr-4n2<t/r)~(t/y>1

exp[-(22+l)2t/T]}

C.3)

(C. 4)

 

APPENDIX D

THE TEMPERATURE SOLUTIONS, T66' FOR THE

DUFOUR EFFECT IN ADIABATIC CELLS AND

CELLS WITH DIATHERMIC ENDS

Adiabatic

In the adiabatic cell, the solution T86 from Eqs.

(IV.5.18) and (IV.4.17) has the form

2

_ oz -2 -1 ” -
T86 - 2ml n Ko 2: [(22+1) B1(£)

o
+ Z (-l)mBz(£,m)cos(2mnx/a)]
m=l

where

31(2) = c1(2) + c2(1)

(v.1)

B2(£,m) = C3(£,m) + C4(£,m) + C5(£,m) + C6(2,m) + C7(£,m),

with

1/0 = y

C1(2) = 0.5{(95§)W(DD03)0[(95') K (l‘Y)]-

p o o

(9.2)

1

-1 —- - —| -1
+ (£np)w(pDQi)o[Ko(1-Y)] - (9D)O(Hl-H2)w[(ocp)onol }

x {exp [-2(22+1)2(t/e)] - 1}

168

rd “Ali—m‘l' ... ' ”I.
.n
.- I
t
....

 

169

_ _ - - _ -l
C2(£) - { (90%)w(pDQi)o [(pCp)o(1 y)]

- (2np)w<p'ép)o (pnopo [Kou-m'l

x {exp [—(22+1)2(r'1+e’1)t1 - 1} (1-y)‘1,

C3(£.m) = {(pD)o(Hi-Hé)w - (pEé)o(pDQi)000(2np)w[Ko(l-Y)1'1

+ [2(Kw/Ko)(pDQi)o(1—y)'1-2(pnog)w]m(22+1-2m)‘1

41-1‘h V

_ (pEP)WDo(pDQi)O(2£+1)[Ko(1-y)(22.+1-2m)]-1

_ Dwmnoi)08m3[4D0(2£+1—2m)2(251+1)l"1

.TA‘EfmeT-t" 0

[iii—1

+ (£np)w(pDQi)o4m2[(22+l-2m)(22+1)]-1}
X {expl-(22+l-2m)2(t/O) - (22+1)2(t/0)] - epr-4m2(t/T)]}

x {4m2 - y[(2£+1-2m)2 + (22+1)2]}'1.

c4(2.m) = {(po)o(fii-fié)w

- (p6?)O(poog)ono(2np)w[Ko(1—y)1‘1

- [2KwK;1(pDQi%§l-Y)-l-2(pDQi)w]m(2£+l+2m)-l

— (pE§)wDo(pnoi)o(2z+1)[Ko(1-Y)(22+1+2m)1'1

3 , 2 -1
+ Dw(pDQ{)08m [400(22+1+2m) (21+1)]

+ (lnp)w(pDQi)o4m2[(2£+l)(22+l+2m)]-1}

x {eXpl—(2£+l+2m)2(t/0) - (2£+1)2(t/0)] - exp[-4m2(t/T)]}
x {4m2 — y[(2£+1+2m)2 + (22+1)2]}’1.

c5(2,m) = {(95?)oDo(lnp)w(pDQi)o[Ko(1-Y)1‘1

- 2(9DQ§)wm[(1-Y)(22+1-2m)]'1

170

+ <pE§)w(pnoi)o(2z+1)((pE§)o(1-y)(22+1-2m)1’1}

x {epr-(2£+l-2m)2(t/0) - (22+l)2(t/r)] - exp(-4m2(t/T)]}
x {4m2-(22+1)2 - y(21+1-2m)2}'1,

c6(2.m)==i(p6?)090(2np)w(pnoi)olxo(1-y)1‘1

+ 2(onog)wm[(1-y)(22+1+2m)1'1

+ (p5§)w(pDQi)o(2£+l)[(OCé)o(l-Y)(21+1+2m)]-1}

{exp(-(22+1+2m)2(t/e) - (22+l)2(t/T)] - exp(-4m2(t/T)]}

X

2 2 2 -1
{4m - (22+1) - y(21+1+2m) } ,

X

C7(£,m) = {(Dw/Do)(000i)02m[(22+1)2-4m2]-2

+ 2(2np)w(pDQi)o[(2£+l)2-4m2]-1}

x {exp(-4m2(t/0)] - exp(44m2(t/t)1}(1-y)’1.

Diathermic

 

In the cell with diathermic ends and adiabatic

walls, the solution Ts from (IV.5.14) and (IV.6.2) has

6
the form

w m £+n
r£0[n£0 9.=Z-0(-1) E1(r,£,n)

o2 -3

_ -1 -
T66 — 4m1 n Ko (pCp)o

(D

co m — -1
+ mil ££0(-1) (pDQi)o(pCp)o 32(r,2,m)1eos[(2r+1)nx/a1
(v.3)

where

r...

Ann—‘1
‘. \
1 .‘ .

.*J

171

El(r,£,n) = F1(r,£,n) + F2(r,£,n) + F3(r,£,n)
E2(r,£,m) = F4(r,£,m) + F5(r,£,m) (D.4)
with

Fl(r,2,n) =((pD)o(Hi-Hé)w(pC§);l{2(2r+l)[(2r+1)2 - (22-2n)2]’

2(2r+1)[(2r+l)2-(2£+2n+2)2]-1}
+ (poof)w(p5§);1(2n+1)'1{2(2n—22)(2r+1)_

I(22+1yf-(2n-22)31‘1-2(2n+22+2)(2r+1)

X

[(2r+1)2 - (2n+22+2)2]'1}

X

+ 2(pDQi)oK;l(2£+1)(2n-2r)[(2n-2r)2-(2£+1)2]-l

[(2n-2r)2-y(22+1)21'1

X

1

X

_ _ -1 2 _
[(pCP)w(pCP)o Do(2£+1) (2n+l) + Do(2np)w(2n-2r)
1

+ Kw(pC§)-l(2n-2r)(2r+1)(2n+l)— ]

O

+ 2(pDQi)oK;l(2£+1)(2n+2r+2)[(2n+2r+2)2 - (222.1)21"1
x [(2r+2n+2)2 - y(22+1)2]'1
x [-(pCé)w(pCé);lDo(2£+1)2(2n+1)-1 - Do(2np)w(2n+2r+2)

+ Kw(pC§);1(2r+1)(2n+2r+2)(2n+1)-1])
x(exp{-l(22+1)2 + (2n+1)2](t/e)} - exP{-(2r+l)2(t/T)}

x {(2r+l)2 - y[(29.+1)2 + (2n+1)2]}-lr

F2(r,£,m) = 2(pDQi)o(22+l)(2n-2r)2

x {Ko[(2n-2r)2-(22+1)2][(Zn-Zr)2 - y(22+1)2]}'l

1

..nm

humun‘n‘mfllrkfi‘ ‘ "i‘
. , :
f

[L

172

1

X

_7 _ 2 -
{-Do(£np)w-(on)wKo(2n-2r)[(pcp)o(2n+l)]
1}

(2r+l)Kw[(2n+l)(DEé)o]-

{exp[-(2n+l)2(t/G) - (Zn-2r)2(t/T)] - expte(2r+1)2(t/r)1}

X

{(2r+1)2 - Y[(2n+1)2 + (2n-2r)21}‘1,

X

F3(r,2,n) = 2(pDQi)o(2£+l)(2n+2r+2)2{Ko[(2n+2r+2)2 - (22+1)2]

[(2n+2r+2)2 - y(2.Q.+l)2]}-'1

X

1

X

{Do(£np)w + (pCé)wKo(2n+2r+2)[(pCfi):(2n+l)]-
1}

(2r+l)l<W [(2n+1) (95p)o]-

{exp(-(2n+l)2(t/0) - (2n+2r+2)2(t/T)]

X

exP [-(2r+1)2(t/T)]}

x {(2r+1)2 - y[(2n+1)2 + (2n+2r+2)2]}-1r

F4(r,£,m) = {2(2np)w - 2mpw[Do(22+1-2m)]‘1} (2r+1)4m2
x {(22+1)[(2r+1)2-4m2] (22+1-2m)}-1

x {exp[-(22+l)2(t/0) - (22+1—2m)2(t/e)1

exp[-(2r+l)2t/0)]}

{(2r+l)2 - Y[(22.+1)2 + (23?.+l—2m)2]}-1

X

{exp[-4m2(t/G)] - expl-(2r+l)2(t/0)]}

X [(2r+1)2 - y4m21-1 ,

F5(r,2,m) = {2(2np)w + 2mnw[no(22+1+2m)]'1} (2r+1)4m2

x {(22+l) [(2r+1)2 - 4m2](2!l,+l+2m)}-1

;;:3

. .
‘.
’ ‘

 

 

 

X

X

X

173

{exp(-(22+1)2(t/e) - (22+1+2m)2(t/0)
exp [-(2r+1)2(t/T)]}

2 2 2 '-
{(2r+l) — y[(2£+l) + (22+1+2m) ]}
{exp[-4m2(t/G) - exPI-(2r+l)2(t/T)]}

[(2r+1)2 - y4m21'1),

]

1

«3

lI.‘h-.‘. -. A
i
1

~14.

APPENDIX E

EQUATIONS FOR Te AND THEIR SOLUTION

66
IN AN ADIABATIC CELL

The right hand side of Eq. (IV.5.6) for T€66 is

C.‘
II

555 "[(Dap)w/(OEP)olwo(3T55/3t) - [(pEp)w/(p-ép)o]w6(BTG/at)

[(pCp)T/(pEp) 0111S (aTG/at)

+

2(pfié);1[(pn)o(fii-fié)w + (DDQi)w](3w5/3Z)(3wo/3Z)

+

(p6§);1[(pn)o(fii-fi:)T+-(poog)T1(aTS/az)(awo/az)
+ Kw(pC§);1mo(32T56/822) + Kw(pC§)o(8m6/Bz)(3T6/Bz)

- -l 2 — -l 2 2
+ K,I,(pCp)o MTG/32) + Kw(pCp)o m6(8 TG/Bz )

_. -1 2 . 2 ._ -l 2 2
+ KTme)o T5(3 T‘s/3z )+ (pDQi)o(pCp)o (3 “566/32 )

+

(pnog)w(p6§)glwo(azw5/822) + (poog)w(pfié)ow5(asz/azz)

-l
o

+

(oDQi)T(95§) T6(azwo/azz) - v€<aras/az) - V€5(3T6/32).

(11.1)

After application of (IV.5.8), we have the greatly

simplified form,

U€66 - [(90§)T/(OC§)OJT6(3T5/3t)

+

(ODQi)T(pCé);lT6(32w0/322)
+ (DEP)61[(OD)0(El-E2)T + (pnog)T1(325/az)(awo/az)

174

 

175
— -1 2 2 — —1 2
+ KT(pCp)o T6(8 TG/Bz ) + KT(pCp)o (BTG/az)
- v€6(8T5/az). (E.2)

The equation for order 662 corresponding to (IV.5.18)
is
a

3g€56m(t)/3t + (mz/T)g€55m(t) = (2/a) é 0865cos(mwx/a) (E.3)

and solution of this equation under adiabatic conditions

gives

_ 02 2 _ -l w
T656 — 2m1 (pDQi)o[w Ko(1 y)] [2&0 Gl(£)

co

+ (DDQI)0K;1 2 z G2(£,m)(-1)mcos(2mnx/a)], (E.4)
m=l 2—0

where

61(2) = H1(2) + H2(2)

G2 = H3(2.,m) + H4(2I,m) + H5(£Im) + H6(2'rm) + H7(R’Im) I (E.5)

with

Y = T/G

— .— -1
Hl(£) = {(pCp)T(pDQi)°[(pCp)oKo(l-y)l
— <— 9— -1

- (pD)o(Hl-H2)T[Do(pcp)o] }

x {1 - exp[-2(2£+1)2(t/0)]}[2(2£+1)2]-lr

—» -— —1
H2(£) = {(pCP)T(pDQi)o[(pCp)oKo(l-Y)]

+ (DD)o(fii-H2)T[Ko(1-Y)]_1}

176

1 2

x {1 - exp[-(2£+1)2(T- +e'1)t]}(22+1)' ,

H3(2.m) = (o‘c'p)T<pDont(pEp)oKo(1-y)1'1

x {1 — exp[-2(2£+1)2(t/T)]} [2(22+1)21'1.

H4(£,m) =‘Do(pCp)T(2£+l)[n2K0(l-Y)(2£+1—2m)]-l

+ 2m(22+1-2m)‘1{(pnog T<onoi);1 - KTIKO(1-y)1'1}

— .— -1 — -1
(pD)o(HI-H2)T<pnog)o + (nnp)TDo(pcp)o[Ko(1-v)1 )

X

({expt-(2£+1)2(t/0)-(2£+1-2m)2(t/0)l - expl-4m2(t/r)]}

{4m2-y[(22+1)2+(22+1-2m)2]}"l

X

{exp[-(2£+l)2(t/0)-(2£+l-2m)2(t/T)] - exp[-4m2(t/T)]}

X

{4m2-(2£+1-2m)2-y(22+1)2}-1)r

115(1),...) ='(Do(pCP)T(25L+l) (KO-(Ly) (22+1+2m) 1‘1

+ 2m(2z+1+2m)‘1{KT[Ko(1-y)1*L-(pnqi)T(ani);1}

- (pD)o(Hi-Hé)T(pDQi);1 + (£np)TDo(pC§)o[Ko(l-Y)1‘1)
x({expl-(22+1)2(t/0)-(2£+l+2m)2(t/0)] - exp [-4m2(t/T)]}

x {4m2-Y[(22+1)2+(2£+l+2m)2]}-1

- {expl-(2£+l)2(t/0)-(2£+1+2m)2(t/T)] - exp[-4m2(t/T)]}

x {4m2-(22+1+2m)2--\((2)L+1)2}-1 .

H6(n,m) = {-(2np)T(1-y)’1 - (22+1)(pC§)T[(2£+1-2m)(pCb)o(l-y)]-l
+ 2mKT[(22+1-2m)Ko(1-y)1‘1}

x {eXPI-(22+l)2(t/r)-(2£+1-2m)2(t/0)] - exp[-4m2(t/T)]}

177

x {4m2-(22+1)2-y(2£+1-2m)2}-1

- {exPI-(2£+l)2(t/T)-(2£+1-2m)2(t/T)] - exp[-4m2(t/T)]}
x {4m2-(2£+1)2-(2£+1-2m)2}-1 ,

H7(z.m) = {-(fin9)T(1-Y)-l-(22+1)(pC§)T[(22+1+2m)(pC§)o(l—Y)]-l

- 2mKT[(22+1+2m)Ko(1-y)1‘1}

X

{exp[-(22+l)2(t/T)-(22+1+2m)2(t/0)] - exp[-4m2(t/t)]}

{4m2-(22+1)2-y(22+l+2m)2}-1

X

- {exp[-(2£+1)2(t/1)-(2£+1+2m)2(t/I)] - exp[-4m2(t/T)]}

{4m2-(2£+1)2-(2£+l+2m)2}-l .

X

(E.6)

APPENDIX F

THE RELATIONSHIP OF HEAT OF MIXING

T0 3(Hl - H2)/3wl

The measured heat of mixing of a binary mixture
can easily be related to the derivative of the difference
of the partial specific enthalpies with respect to the
mass fraction if the mixture is assumed to be regular.
This is a satisfactory assumption in our case since we
need only an estimate of 3(Hi - Hé)/8wl. This derivative

can be rewritten using the Euler relation
H + w H = 0 (F.l)
so that
8(H1 - H2)/3wl = (1/w2)(3Hi/awl). (F.2)

The mass fraction derivative is related to the mole frac-

tion derivative by
(a/awl) = (wlwz/xlszB/Bxl) = (Mlmz/fizna/axl).
(F.3)

where M is the average molecular weight, so with (F.3) we

have
aml - g2)/aw1 =(M1M2/M2w2) (ail/3x1) . (13.4).

178

179

We can replace the partial specific enthalpy by a partial
molar enthalpy,

— —~M
Hl = H1 /M1. (9.5)

and thus

3(fi

1 - fi2)/awl (M2/M2w2)(3HiM/3xl)

(l/Mx2)(3HiM/3xl). (9.5)

For regular solutions (Rock, 1969; Lewis, Randall,

Pitzer, and Brewer, 1961),

afiiM/axl = - 2 x [w - T(3w/8T)] (F.7)

2
where w is a constant independent of x1. Thus,

3(1'1'1 - fi2)/awl = - 2[w'- T(8w/3T)]/M. (F.8)

Now, by regular solution theory

M — _ O
AHm — xllew T(8w/8T)], (F.9)
therefore,
_ _ _ _ M _
3(H1 - H2)/8w1 — (2/x1x2)AHm /M, (F.10)

where Ang is the molar heat of mixing. Thus, for an equi-

molar mixture

3(H1 - H2)/3w1 = - 8AH

ml (F.11)

where Afih is the specific heat of mixing of the equimolar'

mixture.

    
  

RS

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