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C‘HI( AN 8U TE UNUEF’QITYI IIIRAHIEL.

IIIIIII IIIIIIII‘IIIIIIIII III II II IIIII

31293 00571513

 

 

 

 

 

 

 

 

um
MICHIGAN STATE WWV
EAST LANSING, MICH. W10“

This is to certify that the

dissertation entitled

Thermogravitational Thermal Diffusion

of Electrolyte Solutions

presented by

Yuan Xu

has been accepted towards fulfillment
of the requirements for

PhD degree in Chemistry

 

 

(7/ Jl/IWAWM

Major professor

 

Date September 19, 1988

 

MS U is an Affirmative Acn'on/ Equal Opportunity Institution 0 - 12771

THERMOGRAVITATIONAL THERMAL
DIFFUSION OF ELECTROLYTE SOLUTIONS

By

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY

Department of Chemistry

May, 1988

A B S T R.A C T

THERMOGRAVITATIONAL THERMAL DIFFUSION

OF ELECTROLYTE SOLUTIONS

BY

YUAN XU

Thermogravitational Thermal Diffusion (TGTD) is used to separate the
components of fluid mixtures. The development of a.sem:<xf macrosc0pic
partial differential equations from nonequilibrium thermodynamics and
hydrodynamics Which describes the TGTD of electrolyflxn; is presented

here.

The goal of the research is the solution of the set of partial
differential equations for TGTD columns with or without reservoirs
at the ends. The solutions are obtained analytically under a variety of
'boundary and initial conditions. Through perturbation approaches, we
obtain a complete set of temperature,velocity, and concerMnnation

distribution functions for binary electrolyte solutions in the TGTD

column forlxniithe time-dependent regime and the steady state. The
space derivatives of the steady state concentration function confiums
previous results. However, the approach here is new and the results are
more complete. The average steady state concentration distribution
function along the TGTD column can be used to calculate the thermal

diffusion factor of binary electrolyte solutions.

The time-dependent solution of the concentration equation allows us
to estimate the relaxation time required to reach steady state, and the

solution can be used as a guide for designing the TGTD column.

The reservoir theory of the TGTD column based on an isothermal
diffushnlnmdel is also reported here. Due to the complexity of the
theory, we obtain only a rough but simple formuhatx>accountjfinrthe
reservoir effect on the TGTD column early in the experiment. The formula
can be applied to calculate the thermal diffusion factor of electrolyte
solutions while the concentration-time-dependence is still linear if the
TGTD coltmuiis connected to reservoirs at the ends. We may also use the
formula to guide the proper design of the reservoirs and to explain

recent strange experimental results.

mwmimxmwfimxa =

fi¥ffi 9E5;
3.153%)“: id:

$308333? EEB‘QEQEZQ‘iE‘EE -

56.: 3 i (ii?)

{ah—Ii-‘NU kit-13525 aerawmnxgflnume.
* fl 3 i i fil. I H $5.

3.

To my parents, Mr. Ziwei Xu, and Mrs. Shensu Xie,

who believed in education and in me.

and

a special gift for the 60’s birthday of my father

Acknowledgments

I wish to thank the Department of Chemistry for financial support in
the form of teaching assistantships during my years at Michigan State
University.

I am especially grateful to my advisor, Dr. F. H. Horne, who
directed this research. He and I have worked together on the many
problems involved in the dissertation. Without his continuous
inspiration and encouragement, I probably could not have completed the
research and this dissertation. I also appreciate very much the personal
scholarship set up by him to support the research during the summer of
1987.

I thank my doctoral committee members, who have contributed to this
dissertation. I thank Dr. K. Hunt for her critical reading and
improvement of my dissertation, I also thank Dr. H. A” Kick, Dr. E.
Grulke and Dr. R.H. Schwendeman for their useful suggestions while
reading my manuscript. In addition, I thank Dr. R. I. Cukier for his
important suggestions and useful discussions concerning the diffusion
problem of chapter 8.

I thank Dr. Bruce Borey, Mr. D. Y. Yang, and Dr. R. H. Huang for
their helpfulness, good discussions and friendliness, during the years
at M.S.U. I thank my brother, Mr. Feng Hsu, for his assistance for the
computer work that has led to the final dissertation.

Finally, I thank my family: my parents, Mr. Ziwei Xu and Mrs. Shensu
Xie, who started me on the long road of education many years ago, and

did not stop educating me even during the ten years of " the Great

III

Culture Revolution" tragedy. They never doubted that I could eventually’
reach the target long hoped for; I thank my wife, Liling Shen, who has
been encouraging me spiritually through the wonderful Pipa and Piano

music slipping from her finger tips whenever I need it.

TABLE OF CONTENTS

Chapter

LIST OF TABLES .

LIST OF FIGURES.

1. INTRODUCTION.
A. Thermal Diffusion .
B. Objectives of the Research.

C. Plan of the Dissertation.

2. FUNDAMENTAL EQUATIONS OF TGTD .
A. Introduction.
B. Basic Assumptions
C. Mass Balance.
D. Momentum Balance.
E. Energy Balance.
F. Onsager Equations .
G. Practical Transport Equations
H. Simplifying Assumptions .
I. Velocity Equation .
J. The Temperature Equation.

K. Summary .

Page

VIII

ll
12
13

15

25
26

27

Chapter

3. BOUNDARY AND INITIAL CONDITIONS.

General Remarks .

Boundary and Initial conditions
for Temperature .

Boundary and Initial conditions
for Velocity.

Boundary and Initial conditions

for Concentration .

TEMPERATURE DISTRIBUTION.

perturbation Scheme .

Steady State Temperature Equation .
Time dependent Equation .
Asymptotic Solution .

Discussion of the Solution.

5. VELOCITY DISTRIBUTION .

Perturbation and Other Assumptions.

Solution of the Zeroth Order Equation .

Asymptotic Form of the solution
for Large Argument.
Steady State Velocity Profile

and Discussion.

6. STEADY STATE CONCENTRATION DISTRIBUTION.

A.

Solution of the Equation.

Page

31

31

32

33

34

37

37

40

A3

56

59

64

64

67

72

7h

78

78

Chapter

B.

Discussion of the solution.

7. TIME DEPENDENT CONCENTRATION DISTRIBUTION

IN THE COLUMN WITH TWO ENDS ARE CLOSED.

A.

Introduction.
Time dependent Differential Equation.
Discussions of the Time Dependent Solution.

Working Equations .

8. THEORY OF RESERVOIRS.

A.

B.

General Remarks .

Differential Equation of Diffusion.

Solution of Differential Equations.

Discussion of The Source Function .

Concentration Distribution in The Reservoir With
Exponential Decay and Constant Source Functions .
Concentration Distribution in The Reservoir With
Linear Source Function.

Summary And Discussion.

9. SUGGESTIONS FOR FUTURE WORK .

APPENDIX A - Solution of Differential Equation in Chapter 8.

APPENDIX B - Solution of Partial Differential Equation(8-6).

REFERENCES .

Page

85

88
88
89
102

105

120
120
123
125

128

129

134

144

149

153

161

167

LIST OF TABLES

Table

2-1 Numerical Values of Physical Properties of
NaCl and KCl at 25C°, 1 atm., and 0.5mol/dm3

7-1 Numerical Comparison Between the
Infinite Sum and Its Asymptotic Form .

8-1 The Approximation of Eq.(8-42) as a Function
of Reservoir Dimensions. . . . . . . .

8-2 The first roots of Eq.(8-18 & 46).

VIII

Page

28

119

139

141

LIST OF FIGURES

Figure

3 - 1 The TGTD Column Profile.
4 - 1 Temperature Distributions.
5 - 1 Steady State Velocity Distribution .

7 - 1 Concentration Distribution as a Function
of x at a Given Time and z

7 — 2 Concentration Distribution as a Function
of 2 at a Given Time and x .

7 - 3 Concentration Distribution as a Function
of time at given x and z

7 - 4 Average Concentration Distribution
as a Function of Time for Given L.

Page

36
63

77

112

114

116

118

C H A P T E R 1

INTRODUCTION

A. THERMAL DIFFUSION

Application of a temperature gradient to an electrolyte solution.or'
to any multicomponent liquid or gas mixture causes redistribution of the
components. The motion of the components leads to the establishment of a
concentration gradient which ultimately achieves a constant value that
depends on the thermodynamic and transport properties of the system. The
final concentration distribution is not uniform.

Thermal diffusion in salt solutions was first demonstrated by Ludwig
[1856] and was re-discovered by Soret [1879], who more thoroughly
investigated the phenomenon. Thermal diffusion of aqueous salt solutions
is often called the Soret Effect. Very few binary non-electrolyte
solution (Wereide [1914]), aqueous electrolyte and nonJelectrolyte
solution.systems (Eilert [1914]), were studied before World War II. The
Soret Effect has since been studied in liquid alloys (Winter and

Drickamer [1955]), in mixtures of molten salts (Hirota, Ma:sunaga, and

Tunaka [1943]) , and in solutions of macromolecules and polymers (Debye
and Bueche [1954], Gaeta and Cursio [1969]). Several studies have been
made of mixtures of organic liquids (Prigogine [1950], Rutherford,
Dougherty, and Drickamer [1954], Horne and Bearman [1962-68] , Turner,
Butler, and Story [1967], Turner and Story [1969], Johson and Beyerlein
[1978], Ma and Beyerlein [1983]). Thermal diffusion in gases is well
known and has been extensively investigated both experimentally and
theoretically (Furry, Jones, and Onsager [1939], Bardeen [1940], Jones
and Furry [1946], Crew and Ibbs [1952], Greene, Hoglund, and Halle
[1966], Rutherford [1973], Santamaria, Saviron, and Yarza [1976],
Navarro, Madariaga, and Saviron [1983]).

There are two major experimental thermal diffusion methods,
thermogravitational thermal diffusion (TGTD) and pure thermal diffusion
(PTD). PTD is characterized by a vertical temperature gradient directed
so that there is no density induced convection (for most mixtures this
requires that the system is heated from above). PTD is theoretically
simpler since the steady state concentration gradient is proportional to
the temperature gradient. The operational theory based on Onsager
thermodynamics has been developed by deGroot [1947] , Rutherford [1954] ,
Bierlein [1955], Agar [1960], Horne and Anderson [1970], and Navarro g;
11. [1983] for both electrolyte and non-electrolyte solutions. A good
summary of early work was given by Tyrrell [1961].

TGTD is experimentally quite different from PTD. In TGTD, the
mixture is contained between two vertical plates or two cylindrical
columns. The outer and inner surfaces are kept at different
temperatures. Thermal diffusion takes place horizontally. In solutions
of electrolytes, the solute usually moves towards t cold region and

solvent to the warm region. Because of the density gradient produced in

the horizontal direction by thermal expansion under the temperature
gradient, natural convection deve10ps due to gravity. The solute
enriched fluid nearer the cold wall descends to the bottom of the
column, and the less concentrated solution near the hot wall ascends to
the top of the column. Of the two vector components of the steady state
concentration gradient, the vertical component is independent of the
magnitude of the horizontal temperature gradient. The horizontal
component of the steady state concentration gradient is smaller than
what would be caused by PTD because convection reduces the concentration
difference. Clusius and Dickel [1938] invented this technique and
applied it to separate gaseous isotopes. The theory of TGTD for
separation of gaseous isotopic materials was developed by Furry;
Jones,and Onsager [1939] , and reformulated by Furry and Jones [1946] .
Uranium isotope separation by TGTD was of considerable interest in both
Germany and the United States during World War II. Bardeen [1940]
studied the time dependent theory of TGTD for gases.

The operational theory of TGTD for liquid mixtures was outlined by
Debye [1939], Hiby and Wirtz [1940], deGroot [1945] and Prigogine
[1950]. The theory was similar to that of gases. Horne and Bearman
[1962, 1966, 1968] developed the detailed operational theory of TGTD for
liquids at steady state. The phenomenon is a very complicated function
of the geometric parameters of the column and the physical properties of

the solution.

B. OBJECTIVES OF THE RESEARCH

Although the working theory of TGTD has been treated extensively, it

has not previously been approached using a full, rigorous nonequilibrium

thermodynamic analysis including time as a variable. All previous
approaches followed the general pattern of Furry, Jones and Onsager
[1939] , which was developed for gases. Only a few experimental TGTD
studies of electrolyte solutions have been reported (Hiorta, Matsanaga,
and Tanaka [1942, 1943, 1944, 1950], Gillespie [1941,1949], Alexander
[1954], Longsworth [1957], Gaeta, Cursio, Perna, Scala, and Belluccl
[1969, 1982], and Naokata and Kimie [1984]).

Thermal diffusion in electrolyte solutions has attracted a good
deal of attention because the Soret coefficient and the heat of
transport are important characteristics of ion-ion and ion-solvent
interactions for nonequilibrium situations. Interest in electrolyte
solutions has accelerated in recent years due to three developments: (1)
increased, efficient use of TGTD as a means of separating liquid
solution components (Naokata and Kimie [1984]); (2) improved approaches
to the long sought but so far elusive goal of an explicit usable
molecular theory of coupled mass and heat flows in mixtures (Wolynes
[1980] , Kahana and Lin [1981], Mauzerall and Ballard [1982], Calef and
Deutsch [1983], Fries and Patey [1984], Petit, Hwang, and Lin [1986],
and Kincaid, Cohen, and Lopez de Haro [1987]); and (3) the published
reports of Gaeta, Perna, Scala, and Bellucci [1982], whose TGTD
experiments appear to imply phase transition behavior in dilute sodium
chloride and potassium chloride solutions. Petit, Renner, and Lin
[1984] , using a pure thermal diffusion technique, and Naokata and Kimie
[1984], using a TGTD technique, did not find the behavior suggested by
Gaeta, et a1. Since Gaeta, et a1. and other TGTD experimentalists used
for [their experimental calculations only the very approximate equations
developed long ago for gas mixtures (Furry, Jones, and Onsager [1939])

and since the Gaeta, et a1 results are so intriguing, it is appropriate

to obtain accurate time-dependent equations for TGTD in electrolyte
solutions. The results may be readily adapted to nonelectrolyte liquid
mixtures and to gas mixtures.

The principal objective of the research reported here was to describe
TGTD of electrolyte solutions by equations based on the thermodynamics
of irreversible process and hydrodynamics. We formulate rigorously a
set of partial differential equations and upon applying certain
assumptions, we solve these differential equations analytically, . where
possible, to obtain the temperature, velocity, and concentration
distributions in a cylindrical TGTD column.

The results presented here should lead to a greater understanding of
TGTD in general. More specifically, it is hoped that these results lead
to clarification of the recent contradictory experimental results and
that the working equation derived from our theoretical results can be
used to calculate Soret coefficients at both the steady state and at

early time in a TGTD experiment in electrolytes.

C. PLAN OF THE DISSERTATION

Chapter 2 begins with some basic assumptions for the nonequilibrium
thermodynamic and hydrodynamic equations of TGTD . On the basis of these
fundamental assumptions, we formulate a set of TGTD transport equations,
and we discuss, in detail, the physical significance of these transport
equations. Because of the importance of boundary and initial conditions
the entire chapter 3 is devoted to them. In chapter 4, we obtain the
equation for the temperature distribution. The temperature equation is

solved analytically by a perturbation scheme. We give, for the first

time, a complete time and space dependent temperature distribution
functiLHI. Chapter 5 deals with the velocity distribution in the column.
Chapter 6 describes the steady state concentration distribution. The
concentration derivative with respect to the vertical variable agrees
with previous results (Horne and Bearman [1967]). The result presented
here is more complete than theirs.

we devote chapter 7 to the time dependent solution of the
concentration equation of a column without reservoirs. It is found that
the steady state result is independent of whether or not there are
reservoirs, and the result agrees with the steady state solution
derived in the previous chapter. Chapter 8 deals with the concentration
distribution in top and bottom reservoirs. The partial differential
equation for diffusion is solved, and it is seen that the
concentration distribution in the two reservoirs is a very complicated
function of reservoir dimensions and time. We show again that at steady'
state, the average concentrations in the reservoirs are the same as they
would be at the two ends of a TGTD column without reservoirs. For the
rest of chapter 8, we derive a working equation from which the Soret
coefficients can be determined, if the average concentration change with
time in the two reservoirs can.be measured. We also discuss the
contradictory TGTD experimental results.

Finally in the last chapter, we discuss the need for some numerical
calculations to obtain a better working equation as well as some of the
mathematical difficulties for deriving a limiting form for the sum of

the infinite series at small times.

C H A P T E R 2

FUNDAMENTAL EQUATIONS OI" TGTD

A. INTRODUCTION

In this chapter, we use the set of basic hydrodynamic and
thermodynamic equations to develop the theory of TGTD for liquids.

These equations can be found in the literature of thermodynamics of
irreversible processes and of fluid mechanics (de Groot and Mazur,
[1962], Fitts, [1962], Horne [1966]).

After presenting the fundamental equations of nonequilibrium
thermodynamics and of hydrodynamics, we transform the set of coupled
partial differential equations to the Hittorf reference frame, the frame
most suitable for electrolyte solutions. The transformed equations are
then solved under experimental initial and boundary conditions
appropriate to TGTD. In order to facilitate the solution, a number of

carefully specified assumptions and simplifications are made.

B. BASIC ASSUMPTIONS

The “thermodynamics of irreversible processes" could also be called
the "thermodynamic-phenomenological theory of irreversible processes",
for it consists of both a thermodynamic and a phenomenological part. The
thermodynamic part of the theory follows the terminology of classical
thermodynamics extended to the nonequilibrium regime. The
phenomenological part of the theory introduces a postulate new to
macroscopic theory, the "phenomenological equations" or the "Onsager

equations", which are mathematically expressed as

m
Ji-Eaijxj; (1-1,2,3-o-)
j-l

These homogeneous linear relations are the phenomenological equations,

where J1 is the ith generalized flux and XJ. is the jth generalized

driving force. The quantities a are called phenomenological

iJ'
coefficients or Onsager coefficients. Thus the assumption is the linear
dependence of the generalized fluxes on the generalized forces. The

generalized fluxes and the generalized forces all individually vanish at

equilibrium. The "Onsager reciprocal relations" aij - aji are motivated

by the molecular theoretical foundation ( especially by the notion of
microscopic reversibility ), and have been proved correct in all
experimental tests in near-equilibrium systems.

In general, matter in a gravitational, centrifugal, or
electromagnetic field constitutes a continuous system in which
properties such as concentration, density, pressure, temperature and the
chemical potentials depend, even in equilibrium, on the space

coordinates in a continuous way if we exclude the phase boundaries and

do not consider discontinuous fields. We restrict our derivations to the
case of time-invariant (stationary) conservative force fields, as
represented by the earth's gravitational field, the centrifugal field at
constant angular speed, and the electrostatic field. We assume isotropic

media and exclude the polarization of matter.

In a continuous system, intensive properties such as density,
pressure, temperature, and concentration depend on the space coordinates
in.a continuous manner.fflnun those quantities are, in general,
functions of time and position for irreversible processes. Only in the
case of a steady state are intensive state functions constant in time,
although they still may depend on the position coordinates. In summary,

the general assumptions are:

(1) The system is isotropic.

(2) External force fields are constant in time.

(3) Electric and magnetic polarization of the material do not appear.

(4) For electrical phenomena, the Lorentz force which acts on moving

charges in magnetic fields, can be neglected.
(5) The irreversible processes take place near equilibrium.

(6) Electrical fields can be neglected for TGTD of electrolyte

solutions because of bulk electroneutrality.

C. MASS BALANCE

For a continuous, isotropic, nonreacting binary mixture, the

equations of conservation of mass are

10

ac

-—1 + V-(ci v

at - 0 , 1 a 1, 2 . ( 2 - 1 )

i)
where Cl is the molar density of component i and V1 is its local vector
velocity. The operator a/at denotes the derivative with respect to time

at fixed position, so the equation is a local balance equation. The

barycentric velocity v is the mass-fraction sum of the component

 

velocities.
‘v - wlvi + w2v§ , ( 2 - 2 )
with
xiMi
wi - M - ciMi/p , i - 1, 2 ( 2 - 3 )
where w

i is mass fraction, xi mole fraction, Mi is molar mass of

component i, M is mean molar mass,

and p is density,

where V is the molar volume of the solution. The equation for

conservation of'total.mass is obtained by summing Eq. ( 2 - 1,2, and 3

11

), with the familar result

3% + V-(pv) - O .

(2-6)

Eq. (2-6) is the local total mass balance equation, which is called

the equation of continuity of matter in hydrodynamics. If we introduce

the diffusion current density or diffusion flux

B .
ji - ci( vi - v ) , 1 - 1, 2

with v defined by Eq. (2-2), then Eq. (2-1) becomes

— + V-(civ)+V-(j§)-0.

D. MOMENTUM BALANCE
A general form for momentum balance is:

8v
p—— + pV‘VV + v-n - E c.K = 0 ,
at 1 1

(2-7)
(2-8)
(2-9)

where v is the barycentric velocity, Ki is the molar external force

acting on species i , and II is the pressure tensor, which for viscous

fluids is (Fitts, [1962]),

12

n-[(%q-¢)V-V+P]l-2nsym(VV). <2-10)

where n and o are, respectively, the shear and bulk viscosity, 1 is the

unit tensor, P is pressure, and

sym(Vv)-%(Vv+VvT), (2-11)

where (vv)T is the transpose of Vv. Eq.(2-9) becomes, with Eq.(2-10),

av

p—+pv0Vv+V-[(§‘n-¢)V-V+P]1-2VOnsym(VV)-§c.l(.-O , ( 2 - 12 )
at l 1

The term E ciKi is the resultant of the force density of the external

forces.

The equation of momentum balance for a Newtonian fluid subject to no

external field except gravity leads to the equation of motion (Horne

[1966]).

8v 2
p-—+VP+pg&Vo[(-n-¢)V-v]+pvav-2V-nsyva=0 , ( 2 - 13 )
at 3

where g represents the gravitational field.

E. ENERGY BALANCE

13

The most useful form of the equation of energy transport for

experimental purposes is (Horne [1966]),

Jo:

£1
at

<llu03

“FVbVT-av'VP-agg-(n3Pl):Vv+V-q

+J?-V[fi.-<M1/M2>fi2]-o <2 - 14>

where Cpis the molar constant pressure heat capacity, T is the

temperature, a is the thermal expansivity,

(aV/a'l')?’x1 (c9/J/c'3T)p,x1

a__.,__ _-— (2-15)

V p

The thermal and mass flux terms in Eq.(2-4) contain the heat flux q and
the molar diffusion flux j? relative to the barycentric velocity, which

is given by Eq.(2-7). Note by Eq.(2-2,3,and 7),
B B
M1.11 + M2.12 ' 0- ( 2 ' 16 )

The last term on the left hand.side of Eq (2-14) contains the partial
molar enthalpies H1 and H2; this term is proportional to the heat of

mixing (Ingle and Horne [1973]; Rowley and Horne [1980]).

F. ONSAGER EQUATIONS

14

The Onsager equations that relate the heat and matter fluxes to the

partial derivatives of temperature and chemical potential are:

B

T
B
' jg - Ozlval + 022VT/J2 + 020VIDT, ( 2 " 17 )
with
VTpi-Vpi+§iw, (2-18)

where ”i is the chemical potential of component i and §i is its partial

molar entropy.

The Onsager coefficients Oi]. are not all independent (Bartelt and

Horne [1969]),

1 1 1 1

2 2

E MiMj oij - o - E MiMjOij ( 2 - 19 )
1-1 j=1

In the independent Onsager coefficients 002,(h¢, and.000, Eqs.(2-l7)

become

15

'12--[VTflz‘(“z/M1)vf#1]012(Mi/M2)+020V1nT - ( 2 - 20 )
Now (Horne [1966]),

vffll ‘ V1VP ‘ (x2/x1)p22vx2 ‘ M18

Vfflz ' V2VP ' (x2/x1)p22Vx2 ‘ M28 ( 2 ’ 21 )
where
ng-(ap22/8x2)T P=(RT/x2)[1+(61nf2/61nx2)T P] ( 2 - 22 )

and f2 is the mole fraction based activity coefficient of the solute. In

the experimentally measurable properties mole fraction, pressure, and

temperature, Eqs.(2-20) become:

pi#22 V2 V1 V1
x1M1 M2 M1 ( 2 _ 23 )

B fiflzz v2 V1
'j2-'01 2mvx2 ‘01 2M1[M—2-'M—1']VP+02 OVIHT .

G PRACTICAL TRANSPORT EQUATIONS

Although Eqs.(2-1,l3,14, and 23) suffice as the differential

l6

equations for TGTD in a binary fluid system, they are not those used in
practice. In this section we first convert to more common transport
parameters such as the mutual diffusivity D, the Soret coefficient a,
and the thermal conductivity x. In order to identify the Onsager

coefficients of Eqs.(2-23) with conventionally tabulated parameters, it
is necessary to define precisely the experimental conditidns that

underlie the various definitions. A particularly important result of

this section is the identification of the Soret coefficient 0*
determined in TGTD experiments on electrolytes.

The thermal diffusion factor a2 is defined(Horne and Bearman [1962])

by the experimental equation for the steady state of one-dimensional

pure thermal diffusion experiment in the absence of a pressure gradient,

dx2 dlnT
dz ' “2x1“ dz ( 2 - 24 )
By Eqs.(2-27 and 28)
M2
- 0 0 =———— . 2 - 25
a2 ( 20/ 12)[ Mx2p22 ] ( )

Note that (11 - -a2 when 021 is defined by the equation symmetric to

Eq.(2-24). The Soret coefficient is simply (deGroot [1945], Haase

[1969])

a-az/T <2-26)

The sedimentation coefficient 52 is defined by the experimental

l7

equation for the isothermal equilibrium one-dimensional composition

gradient due to a pressure gradient,

 

dxz dP
3;“ - -szx1x2E; ( 2 - 27 )
whence
[M1M2 ][ \7, V1] (2 28
s - ~ "“ - - . -
2 Mx2/‘22 M2 M1 ' )

Again, 31 - -52 if 51 is defined symmetrically. Note that $2 is not a

transport property since sedimentation is an equilibrium phenomenon.

'To obtain the relationship between 012 and the mutual diffusion
coefficient D defined by Fick's Law, consider the Fickian flux j?

defined relative to the volume velocity vy,

j: - ci(‘vi - vV ), vV - clVl'vl + c2V2v2 . ( 2 — 29 )
where ciVi - ( xiVi/V ) is the volume fraction of component i. Fick's
First Law is the experimental equation for the relationship between the
one-dimensional Fickian diffusion flux j: and the concentration gradient

in a binary, isothermal, isobaric system,

11; - D( dci/dz) (2 - 30)

18

By Eq.(2-33),

~ F ~ F
V111 + V232 ‘ O ( 2 - 31 )

the definition of D in Eq.(2-30) is consisternzxvith Eq-(2-31) and the

general Gibbs-Duhem result for uniform temperature and pressure

V1dc, + v2dc, - o . ( 2 - 32 )

The relationship between the Fickian diffusion fhncjg and the

barycentric molar diffusion flux jgis

jg - p( V,/M, )jg . ( 2 - 33 )

where we have used Eqs.(2-7,l6, and 29).

To complete the relationship between D and 012, we need the
relationship between dxz and dc2, which we obtain from x2 - CQV auui the

chain rule for dV

av - ( V, - V, )dx2 + a VdT -3 VdP , ( 2 - 34 )‘

where 6 is the isothermal compressibility. Then

Vac, --( v,/v )dx, - x2e dT + x28 dP . ( 2 - 3s )

19

For the one dimensional, isothermal, isobaric Fick's experiment, from

Eq.(2-34,36 and 38)

- 35 - [Ml/(pV>ID[V1/V21<dx./dz> . < 2 - 36 )
or

- J? - [Ml/(fiv>10<dx2/dz> .

and by Eqs.(2-25 and 23),

Vii2fl22
] ( 2 - 37 )

D - - 012[ x1M1M2

The heat of transport Q* is experimentally obtained, in principle,
by determining the heat flux due to matter flux under isothermal

conditions. Thus (Bearman, Kirkwood, and Fixman [1958], Rowley and Horne

[1980]), for VlnT - o,
*
q - Qiji ( 2 - 38 )

and by Eq.(2-24),

 

 

* M2 Q02 2 39
Q" '[!h ][Q.J ' ‘ ' ’

By Eqs.(2-16 and 42),

20

Q’f--(M1/M2>Q’§. (2-40)

With Eqs.(2-25 and 39) Onsager reciprocity implies

* fixzflzz
Q2"[—]02- (2-41)

Two thermal conductivity parameters must be distinguished, in
principle, in thermal diffusion experiments. At the beginning of the
experiment, when no chemical potential gradient has developed, Fourier's

First Law is

- q - nOVT ( 2 - 42 )

and, by the first of Eqs.(2-24).

K0 - Goo/T ( 2 ‘ 43 )

At the steady state of a thermal diffusion experiment, the diffusion

flux vanishes and Fourier's Law in the form

-q-rcmVT, (2-44)

combined with Eq.(2-20) yields

 

002020 002020 ]

,1
"m ‘ [ 000 + (Mz/M1)[ 012 ]]T = "o + (Mz/M1)[ "ffi:;

21

*
- no + x1x2[ Vi )anD . ( 2 - 45 )

With "practical" transport parameters replacing Onsager

coefficients, the flux equations are

M
B 1
- 12 - [ z: ]D[ Vx2 - XlxzaVI + x1x2529E ] ( 2- 46 )

Following is a summary of theerelationships between the Onsager

coefficients and the practical transport parameters:

M1M, MIT
012 ' ' x1x2[ ;;;;;§fi2]0 . 020 ' ' x1x2[ -§fi_ ]UD

M? *

Although the volume frame of reference is the basis of Fick's Law
and is the reference frame of choice for concentrated electrolyte
solutions and for non-electrolyte mixtures, the Hittorf frame, with the
solvent velocity as reference velocity, is the better choice for dilute
solutions. Moreover, the composition variable usually chosen for dilute

solutions is the molar concentration c2 rather than mole fraction x2.

With Eq.(2-34), the first of Eq.(2-45) becomes:

MV
B 1 ~ ~
-j2-[vlfi]D[Vc2-(clczvla-c2a)VT+(c1c2V1s2-c23)VP]. ( 2 - 48 )

 

22

The Hittorf diffusion flux is defined by
H
32 ' C2( v2 ' v1 )- ( 2 ' 49 )

To obtain the relationship between the Hittorf flux and the molar
‘barycentric fluxq it is useful to add and subtract V'in Eq.(2-49), and

to use Eq.(2-l6), where

3’2 - c2[v2-v-(v1-v)]-jE-(c2/c1)j?

C2M2

B .13 ~ .
- j: + c m 32 == [M/(x1M1)J§ . (2 - 50)
1 1 .

 

Note, for later use, that

v-w1v1+w2vz--w1(v2-v1)+v2--(w1/c2)j}2{+v2 ( 2 - 51 )
or
x1M1 CIMI
C2V2-C2W R jz-C2v+ p j2 . ( 2 ' 52 )

By Eqs.(2-48 and 50),

H * * *
-j2-D (Vtz-cza VT+c252VP) , ( 2 - 53 )

where

* ~
D ' D/( Clvl )

23

Note that c1171 is the volume fraction of solvent and is nearly equal to
unity. Even for 0.5 M sodium or potassium chloride solutions, however,

the difference, 1 - clV1 - c2V2 is about 0.02, and for 1”()li solutions

is about 0.04. It is thus not prudent to replace of], with 1 if 1% or
better accuracy is desirable.

'The equation of mass concentration for the solute is, from Eq.(2-

1).

302

5? + V-(czvz) - 0 . ( 2 ' 55 )

With Eqs.(2-52 and 53), this becomes

302 DM1 * *
'a—t’ -v.[v1—p [VCz'a C2VT+S C2VP]‘C2V]=O . ( 2 - S6 )

Eq.(2-56) describes the concentration distribution in space and in time.
Obviously, this equation cannot be solved alone because it is coupled

with equations for temperature, pressure, and fluid velocity.

H. SIMPLIFYING ASSUMPTIONS

24

A typical TGTD apparatus is shown in Fig. 1. For the thermal steady

state the constant temperature TH of the inner cylinder, radius r1, is
maintained hotter than the constant temperature TC of the outer
cylinder, radius r2 (a.cylindrical jacket surrounds the apparatus). To

begin the experiment the apparatus is filled with a solution.of

concentration c3 and is allowed to come to isothermal equilibrium, which

is also the sedimentation equilibrium of Eq.(2-27). By Eq.(2-13)
aP/az - - pg ( 2 - 57 )

at mechanical equilibrium (Bartelt and Horne [1970]),and the pressure is

constant in both of the other two directions. By Eqs.(2-53 and 57)
( 2 - 58 )

] z -0.7 X 10“ m3kg—1 and sjpg z - 3.0 X 10"5m-1

(see table 2-1). Thus, c2 varies by only 0.003% per meter. This is

undetectably small in most thermal diffusion experiments. Similarly, for

RIF-<1

] z —0.7 x 10-‘ m3kg-1 and sjpg z -2.0 x 10-5m-1. Thus,

for practical purposes, (1) the composition is uniform at the beginning
of the thermal diffusion experiment amml (2) pressure gradient
contributions to the composition gradient are always negligible..An

immediate consequence is that Eqs.(2-53 and 56) become

25

ac2 DMI *
.2 [-—-[w]]

I. VELOCITY EQUATION

The equation for the convective velocity in a gravitational field is
Eq.(2-l3). This equation, as it stands, cannot be solved exactly because
it is nonlinear. Moreover, the quantities n and ¢ are functions of
pressure and temperature and in general '7 and 46 are not constant
throughout the fluid. In most cases, however, the viscosity coefficients
'vary only slightly in a fluid which does not contain large temperature,
composition, or pressure gradients, and they can then usually taken to
be constants. We do this here.

The next simplification comes from the so-called incompressibility
assumption. This is based on Eq.(2-6) and the chain rule equation for
the pressure dependence of the density of a pure isothermal substance.

At steady state,

Vp - pfl VP. ( 2 ' 60 )

The conservation Eq.(2-6) can be rewritten as

erv + v-Vp - 0 , ( 2 - 61 )

which, when combined with Eq (2-64), yields

26

V-v-i-fiv-VP-O. (2-62)

Thus, if 8 - 0, then the fluid is incompressible and V-v - 0. It is

customarily assumed that the total density is constant throughout the
system. This would be ludicrous for TGTD because the chief driving force
is the thermal expansivity of the fluid. We do assume that the
divergence of the velocity vanishes in the steady state and also for the

time dependent state. Eq.(2-13) becomes

8v
p-—+pv-Vv-2V°nsym(Vv)+pg+V-P=O ( 2 - 63 )
at

For subsequent use, note that V-v= 0 implies that the vertical
component of v is independent of the vertical direction i_f the other

components of'v vanish.

J. THE TEMPERATURE EQUATION

The temperature equation has been discussed in great detail for both
TGTD and PTD (Horne and Bearman [1967], Horne and Anderson [1970]). The
chief simplifying result is that all terms but the first and V-q are
‘very small in Eq.(2-14). The initial heat flow is caused by the
temperature gradient. Since the steady temperature distribution is
established very quickly, the flow of heat is dominated by thermal
conduction rather than heat of transport or heat 5 mixing. Under these

assumptions, Eq.(2-14) yields

27

5 3T

TSP _- o u -

v at ”“1 0, <2 64)
or

‘6 6T

=p — - O - -

v at v (NW) 0, (2 65)

where n - - Goo/T. That is, we neglect both the contribution of the heat
of transport term and the difference between no and mm. Eq.(2-65) is the

well-known heat conduction equation of Fourier.

K. SUMMARY

In this chapter, we have obtained the three basic partial

differential equations describing TGTD for binary electrolyte solutions.

They are
‘ép 6T
V at - V-(nVT) - 0
av
p- + pv(-Vv) - 2V-qsym( Vv )1 - pg + V-Pl = 0
at
6c2 DM1 *

These three equations will be solved under experimental boundary and

initial conditions appropriate for TGTD.

28

Table 2-1

Approximate values of some thermodynamic and transport properties

for'().5M binary aqueous NaCl and KCl solutions at 25°C. Solute is

component 2, solvent (H20) is component 1

 

 

Property NaCl soln. KCl soln. References
-2 -1
M1/10 kg.mol 1.80 1.80
-2 -1
M2/10 kg.mol 5.84 7.46
\7,/1o'5m3mo1’1 1.8 1.8 a
V2/10'5m3mo1'1 1.81 1.97 a
3 -3
p/lO kg m 1.02 1.02 b
p-1(ap/6c)T P /1o'5m3mol'1 3.9 4.5 c
a/1o"‘1<'1 -2.29 -2.86 d
p/lo'lopa'1 4 92 4.48 d
EP/103J.kgilx'1 4.03 4.1 e
n/1o‘lJ.s'1K'1m'l 6.04 5.99 f
x'1(an/aT)C/1o'3x’1 2.45 2.47 f

5 3

n'1(an/ac)T/1o’ m mo1'1 -9.2 -2.25 f

29

Table 2-1 ( continued)

n/10'3kg.s‘1m‘l 0.93 0.9 g
-1 -1

n (an/8T)c/K -0.02 -0.02 g

n'1(an/aC)T/10'6m3mo1‘l 0.9 3.5 g

0/10'9m25'1 1.47 1.85 d,h

0'1(80/ac)T/10‘51:13mo1‘1 0.5 3.7 d,h
-1 -1 .

D (aD/aT)C/K 0.02 0.02 1

M1, M2, V1, V2 are, respectively, the molar masses of water,

salts, and partial molar volumes of water and salts, p the density , a

thermal expansivity, B isothermal compressibility, C

P heat capacity, n

thermal conductivity, :7 shear viscosity and D diffusion coefficient of

the salt solutions.

Millero F.J., J. Phys. Chem. 74, 356(1970)

Timmermans , "Phys. Chem. Constants of Binary Systems", Interscience

Publisher Inc., New York (1960)

Batuecas T.,Iunh Real Acad. Cienc. Exactas, Fis. Natur. Madrid.,

61(3), 563(1967).

d)

e)

f)

8)

h)

i)

30

Harned H.S., "Phys. Chem. of Electrolyte Solutions", Reinhold

Publishing Co., New York, 88(1958).
Simard M.A., and Fortier J.L., Can. J. Chem. 59, 3208(1981).

Out, D.J.P., and Los J M., J. Solution Chem. 9(1), 19(1980).

Kestin j., Sokolov M., and Wakeham W. A., J. Phys. Chem. Ref. Data,

7(3), 941(1978)

Rard J.A., and Miller D.G., J. Solution Chem. 8(10), 701(1979).

Estimated on assumption that the product nD is independent of

temperature

CHAPTER 3

BOUNDARY AND INITIAL CONDITIONS

A. GENERAL REMARKS

The set of partial differential equations that describe general TGTD
cannot be solved without experimental boundary and initial conditions.
The expression "experimental boundary and initial conditions" is
intended to imply that the conditions may vary under different TGTD
column designs and specific experimental operations. (Tyrell [1961],
Horne and Bearman [1962], Gaeta et a1 [1982], Naokata and Kimie [1984]).

The apparatus in question here (Fig.3-1) consists of two vertical,
concentric cylinders closed at both ends, and it contains an electrolyte
solution. At the beginning of the experiment the system in the TGTD
column is effectively homogeneous, which means that the temperature and
concentration are uniform throughout the annulus and the fluid is
static, and the convection velocity is zero. In fact, there is an
initial concentration gradient in the column due to the gravitational

force but this concentration gradient is experimentally undetectable, as

31

32

already discussed in chapter 2. we assume that the apparatus is
cylindrically symmetric and that all physical properties are independent

of the azimuthal coordinate.

B. BOUNDARY AND INITIAL CONDITIONS FOR TEMPERATURE

The experiment starts when a horizontal temperature difference is
imposed by suddenly increasing the temperature of the inner wall
relative to the outer wall. For a brief interval, the temperature of the
fluid in the column remains uniform due to the time required for thermal
conductionLthrough the walls. This phenomenon is called the "warming-up
effect" (Horne and Anderson, [1970]. There is also a slight lag because
it is not possible experimentally to change the wall temperatures
instantaneously.

It is, nevertheless, possible to determine empirically the time
required for both the inner wall and the outer wall to reach their
steady state temperatures. This time depends for both walls on the
column material and thickness as well as the means of maintaining the
temperatures of the inner and outer walls. A useful way to take account

of the warming-up effect is (Horne and Anderson [1970])

l -t/‘rH
TH(r1,t) - TM + §AT( l - e )

1 -t/rC ( 3 - l )
TC(r2’t) - TM - 2AT< l - e ) ,

where AT is (TH-TC), the applied temperature difference, TM is the

33

arithmetic mean temperature, and TH and r are, respectively, the

C

relaxation times at the hot and cold walls, best obtained
experimentally. Usually the walls reach their steady state temperature
distribution much sooner than the over-all system attains its steady

temperature distribution. The initial condition for temperature is

T(r,t-0) - T ( 3 ' 2 )

M .
Eqs.(3-l and 2) are the boundary and initial conditions for the
temperature equation given by Eq.(2-69).
C. BOUNDARY AND INITIAL CONDITIONS FOR VELOCITY

The initial condition for velocity stems from the requirement that

at zero time, the system in the TGTD column is uniform, and there is rm)

convection. Then the vertical and radial components of'v are initially

vz(r,z,0) - 0 - vr(r,z,0) . ( 3 - 3 )

Because the fluid is contained within the column, all velocity

components vanish at the cylinder boundaries:

Vr(r1,z,t) = 0 . Vr(r2,z,t),
vr(r,0,t) - 0 - vr(r,L,t),

Vz(r1,z,t) - 0 - vz(r2’zvt)v

34

vz(r,0,t) - 0 - vz(r,L,t). ( 3 - 4 )

D. BOUNDARY AND INITIAL CONDITIONS FOR CONCENTRATION

“For concentration, the experimental initial condition is that at the
beginning the concentration is uniform when we ignore the sedimentation

equilibrium concentration distribution.

c2(r,z,0) - cg . (3 - 5)

We can say nothing a priori about the concentration at the boundary
at any time later than zero because the concentration varies at every
point of the boundary. This causes no difficulty, however, because the
theoretical boundary condition is that the diffusion flux perpendicular'
to the wall vanishes at the wall for all times. This is because neither
the solvent nor the solute leaves the column. Thus the boundary

conditions for concentration are

j§r(r1,2,t)- O - jgr(r2vzst)9 (3 " 6 )

where jgr is the radial component of the flux jg. The vertical component

of the flux vanishes at the top and bottom of the apparatus. The
presence or absence of reservoirs determines the form of the

corresponding equations. We deal with this in chapters 7 and 8.

35

Figure (3-1)

Schematic profile of TGTD apparatus (not to scale). Radius r1 is

maintained at higher constant temperature T the outer cylinder radius

R)
r2 is maintained at lower constant temperature TC' 26r=a is the
annular spacing and h is the height of the reservoirs. (We assume that

the two reservoirs are identical).

36

 

 

 

 

K 2 ’3 2'
7r ‘ ’
h J
L W

L TH TC

 

 

 

 

 

 

TOP
RESERVOIR

Tc

BOTTOM
RESERVOIR

(3H1AIPT'E11 4

TEMPEMKUHULDESTRIEUFHNI

A. PERTURBATION SCHEME

In general the thermodynamic and transport parameters are not
constants, but instead, they depend on composition and temperature. To
take into account the temperature and composition dependences of
coefficients, we formally use the perturbation scheme of Horne and
Anderson [1970], which is based on the fact that the thermodynamic and
transport properties vary only slightly with composition and

temperature. For any coefficient L, we write

L-i+e[(T-TM)LT+(c,-cg)ic]‘

+£2[%(T-TM)2£TT+(T-Tk)(C2’C3)£Tc+%(CQ-Cg)2£CC]

+O(e3), ( 4 - l )

where

37

38

L - L( TM,cg ),

- 21 ' - QL

LT a [ 6T ] TM,cg ’ LC 3 [ 6c2]TM,c2 ’

- ELL - _L_2 L

LTT ' [ 6T2 ] TM,cg ’ LTC ' [ aTac2 ] TM,cg ' ( 4 ‘ 2 )

with TM the mean temperature, and c3 the initial uniform concentration

A

of solute. When 5 = 1, L - L. Except for the ordering parameter c,
Eq.(4-1) is simply a Taylor’s series expansion of a property L about the

mean temperature and initial concentration. For the variables T and C?

the perturbation expansions are

T-TM+0, 6-00+€61+€292+€363+... (4-3)
c2 - cg + 1, 7 - 70 + £11 + 6272 + 6373 + . . .
Substitution of Eqs.(4-3) into Eq.(4-1) yields
L-L+e o ' +7 1 +52 192' +7 0 ' +l721 +9 ’ +7 1
011 °c 2°LTT °°ch2°cc 1L1 10
+0(€3). ( 4 - 4 )

The partial differential equation for temperature, Eq.(2-69)

becomes, in cylindrical coordinates,

39

ET _ 2L T
at r ar[ n ari] ’ ( 4 ' 5 )
where

Since mass diffusion is very slow compared with thermal conductirnl, the
concentration terms in Eq.(4-4) have no effect on the temperature
distribution at the outset. Moreover, neither v nor n is sufficiently

dependent on concentration that the small concentration.gradient.at
steady state has any discernible effect on the steady state temperature
distribution. Substitution of Eqs.(4-3 and 4) into Eq.(4-5) yields, with

neglect of concentration terms,

as, 80, ac,

- l
at +Eat +5 at +O(e ) [u+evT00+c (uT01+2uTT60)]

xlé‘ E+ex 0 +52(n 0 +ln 62) X raga-Her331+e'"rifl2
rar T ° T 1 2 TT 0 8r 6r 8r

+0(€3) . ( 4 - 6 )

The zeroth-order equation is

80, as
_ _ "ll. r_°
at wcr ar[ 6r ’ ( 4 ' 7 )

with boundary and initial conditions

00(r1,t)-%AT[1-e-t/T]; 00(r2,t)=%AT[e-t/T-l]

00(1‘. '0)-0 . (4 -8 )

40

where we assume that the warming-up relaxation time T - Th ‘Tc is the

same at the inner wall as it is at the outer.‘This assumption is
experimentally testable, and may be removed, if necessary, vdthout

appreciable increase in complexity.

The first order equation is

as, 89 a 80 aa
_ _--_a_ r—1 -_°L r_° - 12. _° _
6t V~r6r[ 8r ]+VTKr 6r 6r +V'cTrc’ir raoar ’ ( 4 9 )

with boundary and initial conditions

o,(r,,c)-0-o,(r,,c); a,(r,,0)=0 . ( 4- 10 )

The second order equation is very similar, and like Eq.(4-9)
contains terms involving lower order solutions. We shall demonstrate

that the maximum contribution of 01 to the temperature distribution is
negligible and shall then neglect 01 and all higher order contrdlnitions

to the temperature distribution.

B. STEADY STATE TEMPERATURE DISTRIBUTION

We first solve the steady state problem for 00 and 01. At steady

state,

d 80,
a;[ r5; ] = O , ( 4 - 11 )

41

with

a, ( r, ) - 1AT;00( r, ) - -%AT . ( 4 - 12 )

The steady state problem for 01 is

d0 n d0
L r—1 —.—I d— —o - -
dr[ dr ] + n dr[ ro°dr 0 ’ ( 4 13 )

with boundary condition Eq.(4-10).

The solution of the zeroth order steady state equation is

60 - - [ I:T%f7f:7 ][ 1n( r/./r2r1 ) ]. ( 4 - l4 )

For 91 the solution is

o,--%(nT/k) IEIEI7EIT] [1n( r/r2)]ln(r/r1) ( 4 - 15 )

The maximum contribution of 01 to T occurs at i» - /r2r1, the geometric

mean annular radius, and is

o.< i ) - %< nT/k >< AT )2 ( 4 - l6 )

For ( nT/n ) - ( alum/6T )TMz 0.002/K and AT z 10K, ( 01 )max z 0.05K.

42

This is negligible compared with T z 300K.

M
For thermal diffusion, the temperature gradient is more important
than the temperature itself. It is therefore necessary to compare the

derivative r( dao/dr ) with r( dal/dr ). By Eq.(4-14)

80,
r; - -AT/1n( rz/r1 ) - ( 4 ' 17 )

By Eq.(4-15)

do, _ AT 2 _
r5?” - - ( KT/ n )[ I;7;;7;:; ] 1n( r / r ) . ( 4 - 18 )

Thus, in the steady state,

rig-rE%--[AT/ln(r2/rl)][1+(nT/E)[AT/1n(r2/r1)]ln(r/r)].( 4 - 19 >

The maximum contribution of the first order term occurs at the walls,

where the bracketed term in Eq.(4-l9) becomes [ 1 i %( nT/ 2 )AT ].

For ( nT/ R ) - 0.002I(—1 and AT - 10K, the maximum contribution of 61,

to the gradient is less than 1%.
Further insight into the steady state temperature gradient is gained
by converting the logarithmic radial dependence to a linear form by

using a transformation similar to that of Horne and Bearman [1962].

r - res , s - 1n( r/r ); i = frzr1 , s(rl) = -6

43

s(r2) - 6 , 5(2) = O , (rz/rl) = e26 , ( 4 - 20 )
-5-8'-. -

rz-rl-r(e -e )=2r31nh6z2r6-a ,
then

a. - -s< 9% ); a. - %< 4T / k >< 52 - s2 )< ޤ >2 < 4 - 21 >
and

T-TM-s(%§)+%<nT/k><62-s2)(%%>2

-TM-%(AT)(s/6)+%(nT/k)(AT)2[1-(s/6)2] ( 4 - 22 )
Further,

35--<%§)[1+%<~T/k><41><s/6>]- ( 4 - 23 )

[NIH
2:12;

C. TIME DEPENDENT TEMPERATURE EQUATION

The zeroth order time dependent temperature problem with initial
and boundary conditions is displayed in Eqs.(4-7 and 8). Since the first
order contributions are negligible in the steady state, we henceforth
retain only the zeroth order term. We solve the zeroth order time-

dependent problem by Laplace transform. Multiplication of Eqs.(4-7 and

8) by e-pt and integration from zero to infinity yields

(0

80 80
-pt_° _, -pt 1 L ]._°
e atdt I e Qr ar[ 6r dt ’

0%8

0

Q m
I 00(r1,t)e-ptdt - I %AT[ e-t/T - l ]e-ptdt , ( 4 - 24 )
o o

44

Q

J00(r2,t)e-ptdt "
0

%AT[ 1 - e't/' ]e'ptdt ,

0"—08

where p is a complex number, Q-wc, and we drop the subscript for 0.
Integrating the left hand side of the first equation of Eqs. (4-24) by

parts, and defining
Q
0 - I o e'Ptdt , ( 4 - 25 )
o

and then performing the integrations for the two boundary conditions, we

obtain the transformed differential equation and boundary conditions.

 

_fld2 1.51% A

dr2 + r dr ' on - 0 ’

A A1 1

9(r1:P)"2[p(p7+1)], (4'26)

ID
H

 

A l
0(r2:P)" 2[p(p1’+1)]

The transformed partial differential equation is an ordinary
differential equation with two constant boundary conditions, which can
‘be solved easily. The solution of Eqs.(4-26) is a linear combination of
modified zeroth order Bessel functions of the first and second kinds

(Watson [1958], Abramowitz and Stegun [1970]),

2 (r,p) - AIO(Ar) + BKO(Ar) ,

45

A2 - p/Q . ( 4 - 27 )

The two constants are obtained from the required boundary conditions,

with the result

 

3(1. )- AT x
'9 2p<pr+1>[K0<Ar2>Io<Ar1>‘Ko<*r1>lo(*r2)1

{K0(Ar)[Io(Ar1)+Io(Ar2)]-I0(Ar)[K0(Ar,)+Ko(Ar2)]} . ( a - 28 )

For greater simplicity of notation, we write

AT

”Wm

f(r,p) .

f(r,p) K0(Ar)[10(Ar1)+Io(Ar2)]-10(Ar)[K0(Ar1)+Ko(Ar2)]}G(r,p)-'1

G(r,p)-K0(Ar2)Io(Ar1)-K0(Ar1)Io(Ar2) . ( a - 29 )

'To obtain the solution for 0 (r,t), we must find the inverse transform

of Eq.(4-29). That is, we must evaluate the inverse transform integral

a (r,t) - 5%?19 e pto (r,p)dp . ( 4 - 30 )

The integral is performed along any simple closed contour 6 around po

described in the positive sense, such that the integral is analytic on

the contour 6 and interior to it except at at the point po itself, where
p0 is a singular point in Slde the contour.. The straightforward way to

evaluate Eq.(h-BO) is by Cauchy's residue theorem,

46

o (r,t) - 5%; 0 e+pt00(r,p)dp - E pn<r.t>. ( a - 31 >

n

where pn(r,t) is the nth residue of the integrand, at the rnfli isolated.

singular point of the integrand. From Eq.(4-29), this integrand is

ATept

2p(pr + 1) f(r,p) . ( 4 - 32 )

0 (r,p)ept -

The singular points for 9 (r,p)ept are those that make the denominator
vanish. These singular points are those at p - O, p - - l/r as well as

those such that

f(r,p) - m

or C(r.P) ‘ K0(Ar2)Io(Ar1) ' K0(Ar1)lo(xr2) ' 0 ( 4 ' 33 )

The next step is to evaluate the residues (Spiegel [1964], Churchill,
Brown, and Verhey [1974]). Because the integrand has a simple pole at

p-O, the residue there is

Zim pATept
p + 0 2p(pr + 1)

ii

“ t
p<r.c.0> - p T 0 pa <p.r>ep - f(r,p)

£im ATept
- p 4 0 2(pT + 1) f(r.P) ( 4 ' 34 )
or

m K0(Ar)[10(Ar1)+Io(Ar2)]-Io(xr)[K0(Ar1)+Ko(Ar2)]

‘

2
p +0 K0(Ar2)IO(Ar1)-K0(Ar1)Io(Ar2)

 

r

( 4 - 35 )

47

From Eq.(4-27), A.approaches zero as p approaches zero, and we can use

the limiting forms for Bessel functions of small arguments.

R

Iu(z) (z/2)”/r(u + 1), v ¢ - 1, - 2, - 3, . . .

Ko(z) z - lnz , ( 4 - 36 )

where F(u + 1) is the Gamma Function of order v. Of course, 10(2)»1 as

 

2+0. Then
p(r c 0) - .1 21m ' 21“(*r) ' [ ' 1n(xr‘) '1n(xr2) ]
. . 2 p 4 o [ - 1n(Ar2) + 1n(Ar1) ]
1n r r /r2 » '
AT [ 1 2 1 AT _lairzrl_ . ( 4 - 37 )

 

' ‘5 1n( r,/r2 ) ‘ 1n(r1/r2)

This is the residue at p - 0.
The pole at p - -l/r is also a simple pole, so

p<c.-1/r> - gif(_1/,)( p + 1/r >0 (r.p>ept

41 -t/7 21m i2_:_lALi
2 ‘ p »<-1/r> p<pr + 1) f‘r’p)
- - AI .-t/' 11m f(r,p) . < a - 38>

2 p *(-l/r)

Now as p+(-1/r), then A+i/J(7Q) by Eq.(4-27).
Let

X - 1/(JTQ) . < a - 39 >

48

Then A-vii as p»(-l/r). Some of the useful Bessel Function identities

that permit conversion between real and complex arguments are

10(2) Jo(iz) )

J0(2) ‘ Jo(‘z) ,

H;(z) Ju(z) + iYV(z) ,

( 4 - 4o )

2
Hv(z) JV(z) - iYV(z) ,

5. «vi/2 1 .
KV(z) 21e Hy(1z)

1 mni sin(1-m)yn.1 sinmun 2
Hu(ze ) - sinyw dv(z) - sinvn Hu(;) ’

 

where JV and Yu are, respectively, Bessel functions of the first and

1 2
second kind of order v, and Hy(z) and Hv(z) are, respectively, Hankel
functions of the first and second kind of order v.

Of specific use for evaluating p(t,-1/r) from Eqs.(4-38 and 29) are
- - - 2 -
Io(iAr) - J0(Ar) , Ko(iAr) - -(n/2)iHo(Ar) . ( 4 - 41 )

These yield, for G(-1/r)

G(-1/r) - Ko(iir,)10(iir,) - Ko(iir,)10(iir2)

- n/2[ Jo(i r,)Yo(ir,) - Jo(ir,)Yo(ir2) ] . ( 4 - 42 )

Similarly, the numerator of f(r,-1/r) is, from Eq.(4-29),

49

Ko(iir)[10(1Xr,)+10(iir2)]-Io(iir)[Ko(iir,)+xo(iir,)] -

«/2{Jo(ir)[Yo(ir,)+yo(ir,)]-Yo(ir )[Jo(ir,)+Jo(ir2)]}. ( 4 - 43 )
Then the residue at ~1/r is

p<r.c.-1/r) - - fil e't/' f(r.-1/r>, < 4 - 44 >

v ‘i —

_{J0(Sr)[Yo(ir,)+yo(ir,)]-Yo(ir )[Jo(ir,)+Jo(ir,)]}
f(r.-1/r) ~ _ _ _ - .
[J0(A r2)Yo(Ar1)-Jo(Ar1)Yo(Ar2)]

The last set of singular points contains those that make G(r,p)
vanish. We must evaluate the residues at these singular points as p
approaches any one of the roots of G(r,p). In general, the procedure for
evaluating residues at these roots is algebraically very tedious, and is
not of great interest. In fact, we work out the residues here because of
the absence of published results in the literature; the method, however
is available (Bateman [1954] , Roberts and Kaufman [1966]). We assume
that there are no duplicate roots, that all roots are real, that all
roots are isolated, and that the derivative of G(r,p) exists as the

argument approaches any one root. Rewrite Eq.(4-29),

 

A pt _ ATept a Ngr,t,pz
o (r:P)¢ 2P(PT + 1) f(r:P) D(r,p) v ( a ' 45 ).

where the numerator is

N(r,t,p) - [K0(Ar)[ 10(Ar1) + 10(Ar2) ]

50

- Io(Ar)[ K0(Ar1) + K0(Ar2) “(Hept , ( 4 - 46 )

and the denominator is

D(r,p) - 29(pr+1)G(r.p)
With these definitions, the residues can be evaluated

flimt N(r.t,Q)

, 4 - 47
pe -q§ dD(r.p>/dp ( )

pn(t.r) -

where -q; is the nth root of G(r,p), and pn‘t,r) denotes the residue at
the nth root of G(r,p).

We evaluate dD(r,p)/dp first.

dD/dp - 2p( pr + l )[ r2K5(Ar2)lo(Ar1) + r1K0(Ar2)Ié(Ar1)

‘ r1K6(Ar1)Io(Ar2)~r2Ko(Ar1)16(Ar2) ](dA/dp)| 2
P ' -qn

- (Qp51/2 p< pr+1>[r2K5<Ar2>Io<Ar1>+r1Ko<Ar2>Ia<Ar.)

- r1K6(Ar1)Io(Xr2)-r2Ko(Ar1)15(Ar2)] ( 4 - 48 )

-_2
p qn

Here K5 and 16 denote respectively the derivatives of K0 and Io with
respect to Ar. In deriving Eq.(4-47), we have used the fact that D(r,p)

vanishes for p = -q; and Eq.(4-33). Now define

51

E - r/JQ ( 4 - 49 )

and rewrite Eq.(4-48) with the help of Eq.(4-27)

dD/dp - Jp<pr+1>[E2[K5(JpE2>Io<Jp?1>-K.</pE.>Ia</pE.>]

+¥1[Ko</p?2)Ia</pr1>-Ka</p?.>Io<JpE2)]]p_,q2 .( 4 - 50 >
n

As p+-q: , one has to consider two cases for Jp.
Case 1: Jp 4 iqn.
Case 2: /p 4 ~iqn.

For case 1 Eq.(4-47) is

dD/dpl g-iq<-rq§+1)[E2[Ka<ian2>Io<ian1>-Ko<ian.>Ia<ian2>]

p+-q

+E,[Ko(1an,)15(ian,)-K5(ian,)Io(ian,)]] . ( 4 - 51 )
Using the relations among Bessel functions

16(2) - I1(Z) , K6(Z) ' ”K1(Z) , ( 4 ' 52 )

then

dD/dpl

pi_q;-iqn(-rq;+1)[E1[Ko(iqn?2)11(iqn?1>+K1(iqn?1)Io(iqn?2)]-

E,[K,(ian,)Io(ian,)+Ko(ian,)I,(ian,)]] . ( 4 - 53 )

52

To rewrite Eq.(4-50) in terms of the Bessel functions of'tflua first and

second kinds, we use

IV(z) - e'(”"i)/2JV(12), -« < arg(z) s w/Z
Jy(zem"i) - em"”iJu(z), m - i 1, i 2, . . - ( 4 - 54 )

mni -mxui
) - e

Yu(ze YV(z) + 2isin(mun)cos(vn)JV(z)

Making use of these relations and Eqs.(4-37), then

Kl(ian1) - “/2[ J1(an1) + iY1(an1) ] ,
K,(ian,) - «/2[ J,(an,) + 1Y,(an,) ] , ( 4 _ 55 )
Ko(iqn?,) - in/2[ Jo(an,) + 1Yo(an,) ] ,

Ko(ian,) - in/2[ Jo(an,) + iYo(an,) ]

Thus we have converted, for case 1, the modified complex Bessel
functions of the first and second kinds into the Bessel functions of the
first and second kinds, which involve no complex arguments. The same

work must be done for case 2, namely, for /p = -iqn. Substituting Jp - -

iqn into Eq.(4-47) we derive

dD/dp--iqn(l—q:r)[E2[K5(-iqnf2)IO(-iqnfl)-Ko(-ian1)Ia(-iqn?2)]+

El[Ko('iqn;2)16('ian1)'K6(‘ian1)Io(‘iqn;2)]]p__q:

53

--iqn<1-qgr>[E2[-K.<-iqn?2>Io<-iqn?1>-Ko<—ian.>I.(-iqn?2>]+

f1[Ko(-iqn§2)Il(-iqnfl)+K1(-ian1)Io(-ian2)]]p__q: . ( 4 - 56 )

For case 2, we need to convert K1, K0, 11, I0 into J1, J0, Y1, Y0. These

relations can be worked out, but we omit the details and only list these

relations

Io(-ian) - Jo<qn?) , I.<-ian> - -J1<an> .
Ko(-ian) - in/2[ Jo(an ) + iYo(an ) ] ,

K,(1an) - iw/2[ J,(an ) + iY,(an ) ] . ( 4 - 57 )

2
To obtain dD/dp, as p -+ -qn, in terms of J0, J1, Y0, Y1 , we substitute

Eqs.(4-52) into Eq.(4-50) and Eqs.(4-S4) into Eq.(4-53). With lengthy

algebraic operations and great care, the results are:

for Im(qn) > 0 ,

(dD/dp) 1; ~ ~ ~ ~ ~
[iqn(1-q;1)] p4-q:' 2{r1[Jo(r2qn)Y1(r1qn)-J1(r1qn)Yo(r2qn)]+

f2[J1(rzqn)Yo(E1qn)-Jo(r1qn)Y1(rzqn)]} ; ( 4 - 58 )

for Im(qn) < O

54

(dD/dp) «1{~
- - 2 4- 2
[ iqn(1 an)] p qn 2

— r1[JO(E2qn)Y1(E1qn)‘J1(Elqn)Yo(E2qn)]+

E,[J,(E,qn)yo(f,qn)-JO(E,qn)Y,(E2qn)]} . ( 4 - 59 )

Because Eq.(4-58) is identical with Eq.(4-59), we conclude that as p v -
q;, the function dD/dp is single valued even though two choices, Jb - i
iqn, are made. In order to obtain the residues of Eq.(4-44), we evaluate
the function N(r,p) for the two cases /p - i iqn.
For Im(qn) > 0, from Eq.(4-43)

-q2t

1’1

N(r,p)-(«/2)ATe {J0(Eqn)[Y0(E1qn)+Yo(r2qn)]

-Y,(Eqn)[JO(E,qn)+JO(E,qn)]} . ( 4 - 60 )
For Im(qn) < 0, from Eq.(4-43), then

-q2t ~ ~ ~
N(r,p)-(w/2)ATe n {Jo(rqn)[Yo(r1qn)+Yo(r2qn)]

-Yo(Eqn)[JO(E,qn)+Jo(E,qn)]} . ( 4 - 61 )

where E-r/JQ. Because Eqs.(4-6O and 61) are the same, we consider only

Im( ) > O, for evaluation of the residues at sin ular oints 2 for n -
qn 8 P qn

1, 2, 3, o o 0 from Eq.(4-47). Combining Eqs.(4-S8,60) and Eq.(4-47), to

give the residue at the nth singular point qn ,

55

_ 2
qnt

pn(rst)- (Q27'1)q X{90(Eqn)[Y0(E1qn)+Yo(;2qn)]
II n

-Yo<Eqn>[ J0<E1qn> + Jo(22qn> ]}+
‘{E1[30(E2qn)Y1(Elqn)'J1(Elqn)Yo(E29n)]

+E,[J,(E,qn)Yo(E,qn)-Jo(§,qn)y,(E2qn)]} , ( 4 - 62)

- qflt
-<AT)e (Tn/Bn>/[qn<qgr-1)1

where
Tn'{Jo(§qn)[Y0(E1qn)+Yo(Ezqn)]'Y0(Eqn)[ J0(E1qn) + J0(E2qn) ]}
Bn'Ei[Jo(;2qn)Y1(Ean)'Jl(Elqn)Yo(E2qn)] +E2[J1(E2qn)Yo(E1qn)-
J0(E1qn)Y1(E2qn)] ( 4 ‘ 63 )
In this equation E-r/JQ, Q-rcV/Cp, and p --q;"l is the nth root of Eq.(4-

33). In general there are infinitely many roots. The sum of these

residues is

Q

p(r,t) - E Pn(r,t)

n-l

( 4 - 64 )

The complete inverse Laplace transform of Eq.(4-31) is

m

+ r,t
p=-1/T E pn< )|P=-q§

n=1

0(r.t)=p(r.t) ( 4 - 65 )

 

 

p=0+p(rv t)

S6

 

 

2 w -q2t
-- (AT)AT lnLrlrz/r ] +lA(AT) e -t/T+§ TnATe n
2 1n( rl/rg ) 2 (qgt-1)ann’
n=1

 

-{Jo(ir)[Yo(ir,)+yo(ir,)]-Yo(ir) Jo(ir,)+Jo(ir,)]}
A - , 4 - . ( 4 - 66)

[Jo(i r,)Yo(ir,)-Jo(ir,)Yo(ir,)]

D. ASYMPTOTIC SOLUTION

For electrolyte solutions, n=0.6J.s.m-1K—1, V=l8xlO-°m3mol-1,

Cp-4X103J.kg-1K-1, and hence Q-1.5xlO-7m2s—1. For metal walls, 7260 sec
(Anderson and Horne [1970]), and X-l//(1Q)=333m—1. For r2-1.1cm, and

r1-1.0cm, irz3.3, and we may use the asymptotic relations

1 1

)3 (cosz-fin), Yo(z)=[ fl: ]; (SinZ-ifl) ( 4 ' 67 )

J°(z)-[ a:

 

 

to simplify Eq.(4-66),with the result,

1 sin{i(r-r1)}//r1- sin{i(r2-r)}//r2
A--(r2r1/r)2

 

251n{i(t,-r,)) ' ( 4 ' 68 )

where i-1//(Qr). Note that i(r2-r,)eo.33. Since the

argument of sin is small, we may simplify Eq.(4-66) by taking

sin(z)zz whence, after some algebra,

S7

(r-r)
A'/r(/r2-/r1) ' ( 4 ' 69 )

To simplify the infinite sum term requires the roots of Eq.(4-33). To

obtain these roots, we use

Io(iZ)-Jo(Z). Ko(iZ)=(iW/2)[Jo(Z)+iYo(Z)]. ( 4 - 70 )

so that Eq.(4-33) becomes

G(r,p)'Jo(xn)Yo(Pxn)'Jo(#xn)Yo(xn) , ( 4 ' 71 )

where xn-anI and p-(Ia, /r1 ). The nth root of Eq.(4-71) is given by

(Abramowitz and Stegun [1970])

 

 

n1rr1 r2‘r1 r2-r1 3
x- - ..[[ 1]. (.42.
n rz-r1 8n1rr2 nu

Retaining only the leading term of Eq.(4-72) since the rest of terms are

much smaller, we have

 

n1rr1
x - , and q - 9519. . ( 4 - 73 )

Since ;_;££§_’ the arguments in Tn and Bn of Eq.(4-63) are about 10nn
2' 1

for r2-r1-0.1. We may confidently use asymptotic expansions of Eqs. (4-

66) for all the Bessel functions, along with

58

 

 

Y1(z)--Jo(z), J1(z)—Yo(z). ( 4 - 74 )
Then
2(r,-r,) n
Tn--[—;;3;7;—][(-1) /r1+/r2]sin[nn(r-r1)/(r2-r1)]
Bn-2(-1)“(r,-r,)2/(n«2r) , ( 4 - 75 )
and
pn<r.t)--n,?T§;}}:.e:n:t:;;[<-1>“/r.+/r2]sin[[r:_:1]<r-r.)] .
'( 4 - 76 )
where
r'-(r2-r.>2/<«2Q> . < 4 - 77 >
For r,-r,-o.1cm and Q=1.5x10'7m25‘1, r'- 0.6755, and pm is better

expressed as

pn(r.t)-

(I)

n -n7t/r'
-§ AT(-1) e (r2-r1)2[(-1)n/rl+/r2]sin[[ n ” ](r-r1)]

[ n31'37/(I‘Q) ] rz-r1

 

 

n-l

( 4 - 78 )

where we ne lect 1' com ared with n27. The com lete as totic solution
8 P P YmP

is a combination of Eqs.(4-62,6S,69 and 78).

114T11n(r/r) (r-r) -t/r

0(r,t)z- 1n(r1/r2) +2/r(/r2-/r1)e

59

w
n énzt/r'

+7 .662)... [][[11

n-l

 

 

( 4 - 79 )

This solution satisfies the boundary and initial conditions. It is
reduced considerably if we (1) neglect all but the first term of the
infinite series, an excellent approximation for t>l sec., and (2)
convert from the radial variable r to the linear variable 5 by using

Eqs.(4-20); then

 

0(r.t)-AT[[ 2: ](e-t/T-1)+%,sin[%fl]«ft/1'] < 4 - so >
and
T(r,t)-TM+0(r,t) ( 4 - 81 )

Eq.(4-80) is of the form (Carslaw and Jaeger [1959], Horne and Anderson

[1970]), usually found for one-dimensional cartesian system.

E. DISCUSSION OF THE SOLUTION

The general solution of 9(r,t) consists of two parts, the steady
state and time dependent part. The time required to reach the steady
state temperature distribution is controlled by two relaxation times. 1
is the relaxation time which characterizes the time interval required
for a column wall to reach its steady state temperature from the instant
it is brought into contact with a reservoir of the desired temperature.

1 is nearly the same for both the inner and the outer walls of the

60

column. Another relaxation time 1' hathat of the fluid, which
characterizes the time interval required for the fluid to achieve a
steady state temperature gradient, after the temperature gradient is
established at the walls. The relaxation time 1 depends mainly on the
thickness, heat capacity, and thermal conductivity of the column
material. The relaxation time of the fluid depends on the thermal
conductivity, heat capacity, density, and the square of the annular gap
width a. The relaxation time ratio r'/r affects the time dependent
temperature distribution significantly only for large values of a. After
about at most 47 of starting the experiment, the temperature
distribution in a TGTD column is the steady state distribution. The
steady state temperature gradient is established long before the
concentration begins to change detectably. This has been taken as an
assumption by many authors (Jones and Furry [1946], Tyrell [1961],
Navarro et al [1983]). It is important to note that we assume here that
the relaxation time of the walls is much larger than that of the fluid
in the annulus. If, however, the column is made of non-metal the thermal
conductivity of the material will be very small. This leads to a much
larger relaxation time, and if it is much larger than one minute, then
Eq.(4-66) cannot be simplified. Under such a situation one has a very
complicated form for the temperature distribution and mass diffusion
will develop before the steady state temperature is reached. This will
in turn lead to a much more complicated situation for the velocity
equation as well as for the concentration distribution.

In this chapter we solved both the steady state and the time
dependent temperature equations. Our zeroth order solution for steady
state agrees with previous results. The time dependent temperature

distribution has not been obtained for the TGTD column before due to its

61

complicated form. The temperature distribution as a function of r and
time is given in Fig.4.l. The equation used for the plot is Eq.(4-79).

Numerically, except at short time, say t<O.Sr, Eq.(4-79) is in good

agreement with Eq.(4-80).

62

Figure(4-l)

Temperature distribution in a TGTD column, for r -l.0cm, rz-l.lcm,
AT-lOK, TM-298K. The lines from top to bottom represent individually

the temperature distributions at t-w, 100 sec., 60 sec., 30 sec., 10

sec., and 0 sec. Eq.(4-79) is used for the plot.

63

30

28

N
O)

V

Temperature (°C)
N
.p.
l

20 ' I v I I I . I I
1.00 1.02 1.04 1.06 1.08 1.10

Column Width (cm)

 

 

C H A.P T E R 5

VELOCITY DISTRIBUTION

A. PERTURBATION AND OTHER ASSUMPTIONS

If we neglect all vr terms (Horne and Bearman [l962,l967]) and

assume that the fluid is incompressible, then by Eq.(2-67)

2:12

RIC
o>m
el

22
az+g-o, (5-1)

2’]?
+
‘OIH

with v-vz, w-(n/p)-constant, and homogeneous boundary and initial

conditions

v(r1,t)-0-v(r2,t),

v(r,0)-0 . ( 5 ' 2 )

Following the general perturbation expansions of the temperature

and concentration given by Eqs.(4-l to 4), the perturbation assumptions

64

65

here are

p - 5+[5T00+;C70]+
1 - - 12- - - 2
e zongT+0010pTC+270pCC+91pT+71pC +O(6 ) ,

v - vo + ev1+ €2V2 + . - - . ( 5 - 3 )

where in order to accommodate both (1) the essential equivalence«of
ap/az and —pg and (2) the physical requirement that convection in a

temperature field is due to temperature-induced density differences, we

have assumed 5T and 5C to be zeroth order. Thus,

ap/az--5g. <5-4)

The zeroth order velocity equation is then

av, w Q_ av, _ _ _ _
E';arra—£+8(PT90/P+PC7O/p)’0’ (5‘5)
with
”1: [[aT]P,c,]TM,cg 8", ”c [[ac,]T,P]Tm,eg ' ( 5 6 )

where a is the thermal expansivity evaluated at (TM,cg).

Since p - clM1 + c2M2,

a2 6C1
[302]T,P - [602]T,PM1 '1' M2 = -(V2M1/V1)+ M2

66

M1M2
- -§:—[( Vl/M1)-(V2/M2 )1 . ( 5 - 7 )

and Eq.(S-S) becomes

avo ai- 3V0 MIMZ
3; -rarr3; -ga00+g-:§:[(VI/M1)'(V2/M2)170:0 - ( 5 - 8 )

Since this equation contains the concentration term 70, the velocity
cannot be found unless 70 is known. On the other hand, by Eq.(2-63), '10
cannot be found unless V0 is known.

The composition dependence of the density has been the object of much

concern in thermal diffusion studies. Its effect on TGTD was called

"l’effet oublie" - the forgotten effect-by deGroot [1945]. Horne and
Bearman [1968] showed that the steady state effect on thermal diffusion
is about 1% for liquid mixtures of carbon tetrachloride and cyclohexane,
the system in which the effect should be maximal because of the great
difference in densities of the two pure components. The forgotten effect
should be considerably less important for electrolyte solutions.

In order to solve Eq.(S-B), we suppress the concentration term 10 in
Eq.(5-8) and later evaluate its importance after determining 10. With

this suppression Eq.(S-B) reduces to

6v,
5: - r 6r r 5; - g as, - 0 . ( S - 9 )

 

We showed in chapter 4 that the steady state temperature

67

distribution is established within about 2 minutes after the beginning
of the experiment. Convection starts as soon as tflua'temperature
difference is imposed but becomes established only after the temperature
gradient is established. It is satisfactory for our purpose (determining

7) to use the steady state result for the temperature distribution. We

shall see that the velocity distribution becomes steady very rapidly.
Since higher order perturbation contributions to the velocity depend
on very small terms [see Eq.(5-3)], we obtain only the zeroth order

result.

B. SOLUTION OF THE ZEROTH ORDER EQUATION

The partial differential equation for the convective velocity vo-v

with boundary and initial conditions is

5%-? grrgf-geoo-o, v(t,r,)=o, v(t,r,),

v(t-0,r)-0 . ( 5 - 10)
Let

v (r,t) - ((r) + {(r,t) ( 5 - 11 )

The f(r) term represents the steady state velocity. Furthermore, to
satisfy the boundary and initial conditions, we require that E(r,t)
vanish as time goes to infinity. Both £(r,t)enul§(r) vanish at the

walls. With Eq.(S-ll), Eq.(5-10) yields

- gee, = o , ( 5 - 12 )

filS

CLIC-
H
3%

68

{(r1) - 0 - ((rz) .

and

2:12:
'1 l8
er»
:12
I
O

. ( 5 ' 13 )
€(r15t) ' 0 - €(r2,t)

503,0) - -§(r) 9

where the t-O condition for 6 takes the specified form because

v(r,0)-0.

From Eq.(4-14),

1n r r
00(r) - - AT[ ——‘-—4—)—1n(r2/r1)] , (5 - 14)

where r - J(r1r2). Thus,

d g§__gaAT Ingrzrz -
dr r dr w r[ 1n(r2/r1) ] ° ( 5 15 )

Successive integration of Eq.(5-15) and imposition of the boundary

conditions yield

_ 1
_ 2 Meg _r_- , L 1-—
{(r) 4q1n(r1/r2)[r2[1n r 1] r§[ln r2][2 1n(r1/r2)]

l
_r__1_____
-r§[ln r1] 2-ln(r2/r1)]] I ( 5 - 16 )

69

To solve Eq.(5-13), we assume a solution of form

€(r.t) - W(t)X(r) . ( 5 - l7 )
Then
dW/dt + AZW - 0 , ( 5 - 18 )
and
2 2
X(r1) -O-X(r2) v

where A2 is the separation constant. The solution for W(t) is

W(t) - Ke-AQC ( 5 - 20 )

where K is a constant of integration.

To solve Eq.(5-l9), we make the independent variable transformation

2 - Ar/JE ; ( 5 - 21 )
then Eq.(5-24) becomes

es; 162””,

dz2 + 2 dz ’ ( 5 ' 22 )

X(zl) - O - X(22)

Eq.(5-22) is Bessel's differential equation of order zero, whose
solution is a linear combination of zeroth-order Bessel functions of the

first and second kinds.

 

70

X(z) - AJo(z) + BYo(z) . ( 5 - 23 )

To satisfy the boundary conditions, we must have

A - -BYo(zl)/Jo(zl),

One of the solutions is

X(z) - B[ Jo(z,)Yo(z ) -Jo(z )Yo(zl) ] . ( 5 - 25 )

The general solution is

m -A§t
f(r.t) - E Ban(Anr/Jw)e . ( 5 - 26 )

n-l

where An is proportional to the nth root of Eq.(5-24). Moreover,

0, n # m

r
I §Xm(z)xn(z)dr - 2 Jg(z,) ( 5 - 27 )
‘1 «213/6[Jg(z,) ]' n ' m

 

For simplicity of notation, we abbreviate by z, 21, and 22 what are

actually z(n), 21(n) and 22(n), with

z(n)-Anr//B . ( 5 - 28 )

71

The initial condition is, from Eq.(5-l3),

m

e<r.t-0> - E anxn<xnr/Jw> = - ((r)

n-l
Using Eq.(5-27), we obtain from Eq.(5-29)

r2
-J r§(r)Xn(Anr//w)dr

r1
J2(z )
2 [ o 1 - 1 ]

nzkg/w Jg(z2)

 

Bn(An) -

 

( 5 - 29 )

( 5 - 30 )

The solution of Eq.(5-13) is Eq.(5-26), with the constants Bn(An)

obtained from Eq.(5-30). The difficulty is evaluation of the integral

r2
I r§(r)Xn(Anr/Jw)dr .
r1H

with ((r) given by Eq.(5-l6). The calculation

is extremely complicated

due to the combinations of Bessel functions and rg’(r). With the help of

Tranter [1968], after considerable work, we find that the integral has

the very simple form

r2

Then

I
I§g(r)xn(,\nr//e)dr --L:r%122{[Yo(zl)/Yo(zz)]+l} .
n

( 5 - 31 )

72

BaAan{[Y°(zl)/Yo(zz)] + 1}

9 (5-32)
20): {[Jo(z1)/Jo(22)] - I}

Bn(An) -

and

[Y (2 )/Y (z )]+1
o 1 ° 2 %[Jo(zl)Yo(z )-J0(z )Yo(z,)]e‘*§t

 

vz(r,t)-§ -a «w

n-l 2"*§ [J0(zi)/Jo(22)]-l

 

B ATag r —E— 1 l
+4flln(r1/r2)[r [1n r 1] r1[1n r2][2 1n(r1/r2)]

l
_r__1._________
-r§[ln r1][2-1n(r2/r1)]] ’ ( 5 - 33 )

with.An proportional to the nth root of Eq.(5-24) and z(n) given by

Eq.(5-28).

C. ASYMPTOTIC FORM OF THE SOLUTION FOR LARGE ARGUMENT

This form of the velocity is very complicated. It simplifies
quickly once we determine that the roots of Eq.(5-25) are large and
therefore that the argument 2 is large enough to express the Bessel
functions asymptotically.

By Abramowitz and Stegun [1970] the roots qn of Eq.(5-25) can be

written as a series expansion (this approach was also used in Chapter 4,

Eq.(4-72),

qn/JE - fin+ p/fln+ (g-p2)/flg + ° ' . . ( 5 - 34 )

fin ‘ nn/[(r2/r1 '1)] .

73

P ' -l/[8(r2/r1)] .

‘

 

25[ (rz/rl)3 - 1
g - 6(4r2/r1)3(r2/r1-l)

 

Thus for r2/r1 - 1.1,

ql/JE - 31.4 -(o.12/31.4)+(o.13/31.42)+--.e31.4

and
qn/JS - 10nfl . ( 5 - 35 )

Now z(n) -rAn/J;, with

An-nnjz/(r2-r1) , ( 5 - 36 )

and therefore

z(n)-rnx/(rz-rl) . ( 5 - 37 )

Since zlenn, asymptotic Bessel function formulas can be used to
simplify the expression for the time dependent part of the velocity.

Repeated use of Eq.(4-65) yields

2 AT m [(‘1)n/r2fjr1] . nn(r-r1)
”Fa/r «3 $1“ a
n—l

e-n2(n2w/a2)t

 

€(r9t)- ( 5 ' 38 )

The viscous relaxation time («zw/a) is very large since w-n/p

le‘°mzs“1. Then (1r1'c.3/a2)=1r"’s-1 for a=0.lcm, and the convection steady

state is attained about 0.4 second (4/n2) after the establishment of

74

the steady state temperature gradient. We may thus ignore the time-
dependent part of the velocity equation.

Use of the steady state convection velocity function for the
treatment of a TGTD column has been taken as an assumption by previous
authors (Jones and Furry [1946], Tyrrell [1961] , and Navarro et a1.
[1983]). We here have established a solid foundation for the assumption

by solving the time dependent velocity equation.

D. STEADY STATE VELOCITY PROFILE AND DISCUSSION

As discussed above, the time-independent part of the convective
velocity suffices for solving the concentration equation. It is
convection that brings about a measurable concentration gradient along
the column. The steady state velocity distribution as a function of r is

displayed in Figure(5 -). Clearly, {(r) vanishes at r1 and r2 and

effectively at r. The vertical velocity is positive (upward) for the
warmer portion of the annulus because the density there is smaller and
the material rises against gravity; similarly, the velocity is negative
in the cooler portion of the annulus.

Because the algebraic form of ((r) is 1n(r) dependent and it is not
easy to work with the logarithmic form in solving the concentration
equation, we use the linear transformations given by Eqs.(4-20).
Applying these transforms, and neglecting the time-dependent terms,

Eq.(5-32) becomes

v(s) --m§%’6fl‘m[(l-s)(cosh26-e25)+(s-62)6-sinh26] . ( 5 - 39 )

75

If we expand the hyperbolic functions and the exponential functions and

truncate after the first order in 6, we find

v(s)z-2%%§ga2[l-(s/6)2]{(s/6)-6[l+(s/6)2]+0(62)} . ( 5 - 4o )

Ignoring the second and higher terms in the curly-bracketed part of this
equation introduces as much as 10% error since 0.0526, but yields a

very simple form for the steady state vertical velocity,

v(s) z -é%§%Ia2[l-(s/6)2](s/6) . ( 5 - 41 )

This equation has previously been derived by Jones and Furry [1946] and
Horne and Bearman [1962]. For careful work, the second term in the curly
brackets of Eq.(5-40) should be retained.

The vertical velocity is directly proportional to the gravitational
constant, the thermal expansivity, the temperature difference, and the

square of gap width, and is inversely proportional to the kinematic

viscosity (n/fi).

76

Figure(5-l)

Steady state convection velocity profile. r and AT are the same as in
Fig.(4-1). Eq.(5-l6) is used for the plot. If Eq.(5-4l) is plotted

against s/6, the diagram will be symmetric at s=0.

77

2.5-

 

 

1.54
A
0
G)
(O
E
o 0.5-
v
N
O
x —.5-
6?."
O
.2
0 - .
> 154
:9
2
L1.
-2.54
-:5.5

 

1.00 1.62 1.64 1.66 1.68 1.10
Column Width (cm)

C H A P T E R 6

STEADY STATE CONCENTRATION DISTRIBUTION

A. SOLUTION OF THE CONCENTRATION EQUATION

In this chapter, we establish the steady state concentration
distribution by using a perturbation scheme based on the smallness of
the Soret coefficient. The result here is of considerable use in finding
the zeroth order time-dependent solution of the concentration equation
in the next chapter.

To obtain the steady state radial concentration distribution, we use

Eq.(2-76) for steady state,

DM1 *
V-[ V__ [ VC2 - a c2VT ] - C2V’] - 0 . ( 2 - 76 )
1P .

For reasonably small temperature gradients, the properties represented

~

by p, V1, and D are constants to be evaluated at the mean temperature

78

79

and initial composition. In that case,

* *

(6-1)
where,
0*211- 6 2
”17.7 ('>
Since Vtv, V2T, and vr are all zero, Eq.(6-l) becomes
ac 82c 6c
*1 Q_ 2 * 6T *___2 __2 -
D r ar[r5; -c20 r5;]+D 622 -vzaz O ’ ( 6 3 )

*
where we have also taken a

to be constant. The wall boundary conditions
are from Eq.(3-6),

6c2

-— -c 0*g1 =0
6r 2 dr r1,r2

(6-4)

Additional boundary conditions are required to obtain c2 as a

function of both r and 2.

These, too,

stem from conservation
requirements.

By symmetry, the average value of c2 at the vertical

center plane must be the initial concentration. Thus,

L

( 6 - 5 )

with

 

80

r,
J rfdr

r1 ‘
<r>-—-—. (6-6)

r r2
rdr
r1

In the steady state, Gauss' theorem requires that

I (V-c2v§)dV- (c2v2)-ds , ( 6 - 7 )
V s

or, by Eqs.(6-3 and 4),

1

In the linear variable 5 defined by Eq.(4-20), Eqs.(6-3 and 4)

become

a 8C2 *AT i2 25 6C2 _2 25 82C2
85(83 +c20 26].D*e v5; -r e 622 ’

ac, *AT

[5; “=20 ELI-0 ' ‘ 6 ’ 9 ’

where we have used the steady state temperature result of Eq.(4-21),

£1141. <6-10>

Further simplification is obtained by defining

81

x-(s/6), x(-6)--l, x(6)-l. ( 6 - ll )

Then, with Eq.(5-4l) for the velocity,

8 3C2 6C2 1 6202

- -— _- - 2 - - 2 __ __ 2 ___

ax[ax +ec2] 9(1 x )[x 6(1 x )]82 4a (1+26x)322 ,

8c,

[5; +ec2]i1-O . ( 6 - 12 )

where
* _ 62agvla‘AT
e-a AT/2, a=26r, e=__I92;M:D— . ( 6 - l3 )

If‘we neglect the terms of order 6 and neglect the second z-derivative

of c2, then

a ac2 6c2 6c2
3;[5; +ec2]--6x(l-x2)3; , [5; +£C2]i1=0 . ( 6 - 14 )
Moreover, Eqs.(6-6 and 8) become
l l
2] c2(x,L/2)dx-cg ,
-l
‘1 1 6C2
I [2325; +9x(l-x2)c2]dx-0 . ( 6 - 15 )

-1

Now assume that c2 is separable according to

82

cz-e'Kz[cg+U(x)]+¢(z)+R(x) , ( 6 - l6 )

where K is a constant and c3 is the initial concentration. Substitution

of Eq.(6-l6) into Eqs.(6-l2) yields

d dU
E;[a;+e(c3+u)]-k6(c3+U><1'X2)X '

[§E+e(cg+U)]i1-O , ( 6 - l7 )
and
d2

E;[gfi+eR]--ex(l-x2)§§ ,

[§§+ea]i1-o . -( 6 - 18 )

In order to satisfy Eq.(6-18), (dQ/dz) must be constant, or
¢-A+Bz, ( 6 - 19 )

where A and B are both constants.
Eqs.(6-l8 and 19) are satisfied if
93+eR-[-Bex2(2-x2)2]/4 . ( 6 - 20 )

dx

This yields, through first order in e,

R- E%—BG[(15x-10x3+3x5)-% 6(15X2-5x4+x6)]+C(l-ex) , ( 6 - 21 )

83

where B and C are to be determined by using Eq.(6-15). The w and R parts

of the second of Eqs.(6-15) are

 

1 2
I [$35.4 56 B92x(l-x2)(15x-10x3+3x5)-eer2(l-X2)]dx'0, < 6 - 22 >
-1

'where only'even terms appear because odd terms vanish upon integration.

This yields
1 315 2 -l
B -%5£c[1+3232] . ( 6 - 23 )

Since (a/9)-192nM1D/(pagV1a3AT)z0.01 for aalmm and AT-lOK, we safely
neglect the second term in the denominator of Eq.(6-23), and
21. -
B- 496 . ( 6 24 )

* - 4 -
Note that (e/G)-96a nMID/(pagV1a4)z0.05m 1 for 0 =10 3/K.

To solve Eqs.(6-l7), we suppose that

U-euo+52u1+0(€3),
K9-£k°+€2k1+9(€3) - o o . ( 6 - 25 )

Then

1.10

d —+cg]- -koc3x(1-x2) ,

dx dx

uo

84

l
I x(l-x2)uodx-0 , ( 6 - 26 )
-l

where the last of Eqs.(6-26) is from the second of Eqs.(6-15) with

neglect of tenms of order (a/e)2 compared to the retained term. The
results are
0%

uo-- §6[25x-70x3+21x5] . ( 6 - 27 )

ko-Zl/‘l .

Thus, through terms of order 5,

 

c - c°-:E§(25x-70x3+21x5) ex -2162 +zl£C(z-L/2)+C
2 2 80 P 49 4e

ecg

-§6—x(25-70x2+21x‘) ( 6 - 28 )

By the first of Eqs.(6-15),

21 L
C-c3-c3exp[-‘Zé a] , ( 6- 29 )

and then

[1—11— )1—[[— 11112-8

 

ecg
21 z 21 L
-§6—x(25-70x2+21x‘)[8XP[’ 4; ]-exp['-Z§ 5]]

ecg
-§6—x(25-7Ox2+21x4) . ( 6 - 3O )

85

B. DISCUSSION OF THE SOLUTION

Eq.(6-30) is the steady state concentration distribution function.

However, for practical application of Eq.(6-30) and because

e/9z0.OSm—1, we expand all exponential terms to first order in 5. Then

0. 0
C2 °2+ 49c2 48 2 2 z

2.. 1— 111- I

ec°
-§Bz{x(25-70x2+21x4))[1+Z%é[%-z]] , ( 6 - 31 )

01‘ 2 . 2112.1.
-c2 -§6x(25-70x +21x ) 1+ 49 2-2

and

acg *
— __.A_I£ 0 _42 2 21 ‘ 21.6 L-
8x 32°2[[1 5 x *5 x 1 4e 2
T * 42 21
z-—§%cg[l-§—x2+§—x‘] , ( 6 - 32 )

acg Zl£ e
_-- 0 - _ 2 4
62 49c2[l 36x(25 70x +21x )]

*
50 4nDMla

z- 1 co
angvla‘ 2

 

( 6 - 33 )

Eqs.(6-32 and.33), the derivatives of our zeroth order solution of
the steady state concentration Eq.(6-31), agree with the previous
results (Horne and Bearman [1967, 1968]). The present result is more
accurate at higher order, and for the first time we obtain explicitly
the steady state concentration distribution itself rather than the first
derivative. The previous results for the first order derivative cannot

Ina integrated to obtain our results because the integration constant is

86

usually a function of z. This function is important also for the time
dependent solution.

In chapter 5 , we assumed that the forgotten effect is not important
for electrolyte solutions. We then neglected the composition dependence
of the density in solving the velocity equation. This assumption can be
verified if one knows the steady state concentration function. Since the
concentration difference for TGTD reaches its maximum in the steady'
state, the forgotten effect should be maximal then.The effect is now

easily estimated with the help of Eq.(6-31). The 70 in Eq.(5-8) is, from

Eq.(6-31)

.21; o _Zl£ L L_
7° 49°? 1 49 2 2 z ’

ecg
-§6—{x(25-7Ox2+21x‘))[1+;%é[%-z]] , ( 6 - 34 )

10 has its maximum at z=0 and x=1 and 70z(21/8)ch(e/6). Using the data

3

given in table 1, for cg-0.5 mol dm- , L-0.1 m, and e/9z0.0S m'l, then

70:6.3 mol m'3. The fourth and fifth terms of Eq.(5-8) are, then,

respectively, for AT-lOK (and suppression of g in both terms),

00 oz 1 . 5x10_3 ,

M1M2
-:§:[(V1/Ml)-(V2/M2)]1oz 0.28x10-3.

Thus, the fifth term is at most about 18% of the fourth term for KCl

solutions, and at most about 12% for NaCl solutions. For higher

87

concentrations, longer tubes or smaller temperature differences, neglect

of the concentration term in Eq.(S-8) must be re-examined.

CHAPTER7

TIME DEPENDENT CONCENTRATION DISTRIBUTION

INTHECOLUHNWITHTUOENDSCLOSED

A . INTRODUCTION

Although the time dependent solution of the concentration equation
for liquid mixtures is very important for both theoretical and practical
purposes, an accurate time dependent solution has not previously been
achieved. The TGTD experiments for binary liquid solutions involve a
very long waiting period, usually several hours, to achieve the steady
state concentration distribution. The time dependent equation usually
used to calculate liquid thermal diffusion coefficients from non-steady
state experimental data is based on the approximate theory of Furry and
Jones [1946], derived for gaseous mixtures. An assumption in that theory
is that the convection velocity profile is a step function. In this
chapter we present the derivations of the time dependent concentration
distribution in the annulus and of the working equations for both steady
state and time dependent evaluation of Soret coefficients. The steady

state working equation applies to a column with or without reservoirs,

88

89

but the time dependent working equation is applicable only to a column

without reservoirs.

B. TIME DEPENDENT DIFFERENTIAL EQUATION

Our starting equation is Eq.(2-60)

3C2 DM1
v.[

*
5E - VIp [Vbz-a c2VT]-c2v]=0 . ( 7 - l )

In cylindrical coordinates, the equation is

ac 62c Be Be
* 2 :k 2 2 2
r r 6r ar

azz'VzE'EE (7'2)

As before, we assume that the temperature and convection velocity are
both time independent. With the independent variable transformation

relations Eqs.(h-ZO),

a 3C2 6C2 1 62C2 £02302
-' — _ 2 _ _ 2 — — 2 — _— _ _
8x[6x +ec2]+9(l x )[x 6(1 X )]az +4a (1+26X)azz D*3t ( 7 3 )

where we have made use of Eqs.(6-10,ll, and 5-41). The initial condition
is Eq.(3-3), but the boundary condition depends upon the design of the

column. For a column without reservoirs, i.e. both ends closed, the

boundary condition is

9O

J§(x.z.t)-0. x-il . j§(X.z,t)-O, z=0,L . ( 7- a )

To solve Eq.(7-3), we assume that the solution consists of two

parts, Y and R:

c2-Y(x,z,t)+R(x,t) . ( 7 - 5 )

We call R(x,t) the pure thermal diffusion effect in the TGTD column.

Eq.(7-3) becomes, with Eq.(7-5)

BY BY 82Y wzaY 62R 6R wzaR
— — _ 2 — 2——_— — — —_— _- -
ax[ax+‘Y]+9(1 x )xaz+“ 622 D*at+ax2+‘ax D*ac 0 ' ( 7 6 )
w2-32/4, ( 7 ' 7 )

where we ignore terms of order 6. We require that both Y and R must

satisfy the following two equations as well as boundary and initial

conditions:
aY aY 62Y w26Y
5x[5;+eY]+6(1-x2)x5;+w23;3-;*52-0 , ( 7 - 8 )
BY
[5;+€Y]x-il-O; Y(x,t-O)-O ,
82R 6R wzaR 6R
5x3+€5;-;*5f‘0 , [3;+6R]x=i1-O , ( 7 - 9 )

R(x,t—0)-cg

9l

Eq.(7-9) is easy to solve, but Eq.(7-8) is solvable only by use of a

perturbation approach.

C. SOLUTIONS OF DIFFERENTIAL EQUATIONS.

The solution of Eq.(7-9) is

co

 

 

 

266° n e
R(x,t)- 2e"X-aecge"x (“”)2[1‘(‘1) ‘ 1x ( 7 - 10 )
ee-e-e n-l [62+(nfl)2]2
6 I‘Qfl!2+§2ln*t
[cos[(x+l)n«/2]- nu sin[(x+l)nx/2]]exp[- a2 ]

We call R(x,t) the pure thermal diffusion effect in the TGTD column
because R(x,t) is analogous to the time dependent pure thermal diffusion

results of Horne and Anderson [1970]. Note that R + cg+e(e)

as C”.

To solve Eq.(7-8) we try
Y-eK(L/2'z)[cg+u(x,z,t)] , ( 7 - ll )

then

azu au Bu
5;;+eg;-9Kx(l-x2)[cg+u]+ex(l-x2)5;

azu au au *
+w2[ Egg-2K5;+K2[cg+u]]- b252=0 , b2=w2/D , ( 7 - 12 )

92

au
[5;+e(cg+u)]x_i1-O , u(t-0)--cg

We take u(x,z,t) as a perturbation term because u(x,z,t) is much smaller

than initial concentration cg except at the boundaries, where it takes
its extreme values. Thus, Eq.(7-12) becomes
azu au

ax2+63;'

au
6Kx(l-x2)[cg+Au]+9x(l-x2)5; ,

azu an au

+w2[ 3;;-2K52+K2[c3+xu]]-b23E=O , b2=w2/D* , ( 7 - 13 )

au
[3;+e(cg+u)]x_i1-O , u(t-O)=-cg , and

u(x,z,t)-E Anun(x,z,t) . ( 7 - l4 )

n-O

Combining Eqs.(7-l3 and 1A) and noting that the summation variable is a

duemmy variable, we obtain

azuo an, auo azuo auo
3;;+e5; +9x(1-x2)5; +w25;; -2Kw25; +(Kw)2c3-9Kx(l-x2)c3
an,
- bzgz -O . ( 7 - 15 )

6‘10
[5; +e(cg+uo)]x_il-O; uo(t=0)=-cg

For n20 the general form of the perturbation equations is

93

azun Bun aun 82un aun
___ __ _ 2 __ 2——— - 2__ 2 - - 2
6x2 +£8x +6x(l x )62 +w 822 2Kw 62 +(Kw) un_1 9Kx(l x )un_1
au
-b?En-O , < 7 - 16 >

Bun
[5; +euan-il-O , un(t-O)-O .

To solve Eq.(7-15), we note that one of the terms, that duee to

convection, is a function of x only. We assume

uo(x,z,t)-Wo(x,t)+¢o(x,z,t) . ( 7 - l7 )
Then
62Wo 6W0 6W0
___ __ _ _ 2 o, 2__ - -
8x2 +€6x 9Kx(l x )c2 b at 0 , ( 7 18 )
6WD

[5; +‘(°3+W°)]x-:1'O; Wo(t-O)=-c3 ,

and
62¢o 6¢o 2 8¢o 262¢o 26¢o 2 o 26¢o
ax? +eax +ex(1-x )az +w 622 -2Kw az +(Kw) c2-b at =0 ,
a¢o
[ax +c¢o]x_i1-O , ¢o(t-0)-O . ( 7 - 19 )

Since the boundary condition for Eq.(7-18) requires tflmat 6Wo/at and

awo/ax are order of e, we omit terms of second order in e and find

32w, awO
___ _ _ 2 o_ 2__ = ,
6x2 er(1 x )c2 b at o ( 7 20 )
with
aw0
-— +ecg-O at x--1,l, Wo(x,O)--cg . ( 7 - 21 )

at

94

Eq.(7-20) is a second order linear inhomogenous partial differential

equation whose solution (Boyce and DiPrima [1977]) is

CD

 

 

- - n - 2 _ 2
wo(x,t)-cg 1(éflig [462 (nu/2b) t_2i:§2[ _ (::)2][1_e (nu/2b) t]
n-l
xcos[n«(x+1)/2]+ech-cg . ( 7 - 22 )
As time goes to infinity, Eq.(7-22) becomes
m n
Wo(x,t)--cg [1-(2:;)l329K [1-'(::)2]cos[nw(x+l)/2]
n-l
+ec3x-c3 . ( 7 - 23 )
Now
_:_ In]
c3} [1 (:«)‘326K[1_ z%%32]cos[nn(x+l)/2]
n-l
-c2K9(x3/6-x5/20-x/4) . ( 7 - 24 )

This identity may be verified by expanding the right hand side of Eq.(7-
24) in terms of cos[n«(x+1)/2] for x from -1 to 1. Applying this

identity we rewrite Eq.(7-23) as

W0(x)--ch9(x3/6-x5/2O-x/4)+ech-cg . ( 7 - 25 )

If, as we expect, from chapter 6, K9=215/4, then

95

CS
Wo(x)--§66[25x-7Ox3+21x5]-cg . ( 7 - 26 )

Now'we turn to Eq.(7-19). First we ignore the (wK)2 term in this

equation. Then

62¢o a6, 2 a6, 262¢o 26¢0 26¢0
6x2 +eax +9x(l-x )az +w 622 -2Kw 5; -b 5: =0 ,
a6,
[5; +e¢o]x_fl=0. ¢o<c=0>=o . ( 7 - 27 >

j§(x,z,c)-o, z=O,L .

If we first integrate Eq.(7-27) for x from ‘1 to 1 and then use the

boundary condition for x, Eq.(7-27) becomes

1 3260 1 a6, 1 a6, 1 660
w2 5;; dx-J 2Kw25; dx +J f(x)5; dx-sz 5; dx-O , ( 7 - 28 )

with f(x)-6x(1-x2) and

1 82¢0
x-ii'I F(")axazd" '

 

l 6¢o 8¢0
I f(x)5; dx-E; F(x)

 

F(x)- f(x)dx . ( 7 - 29 )

32¢o
To derive an expression.for'axaz, we integrate indefinitely the first

 

equation of Eqs.(7-27) with respect to x then differentiate with respect

to 2. Thus

96

 

32¢o 3¢o 2 33¢o 232¢odx 62¢o 2 32¢o
5;5;--eaz -w 62 de+JZKw dx-ZJ.f(x)a2 dx+b I azatdx

Now substituting this expression into Eqs.(7-29) and rearranging to

3¢o

give Hf(x) zdx, we use this integral to

1 3¢o
eliminate I f(x)5; dx in Eqs.(7-28). This leads to
-1

2 ___ , 2__ __
w 622 dx 2Kw 62 dx+az F(x)

1 82¢o 1 63¢0
+J: F(x)[IF(x)::: odx]dx-2Kw2I F(x)[j‘5;; dx]dx+w2I F(x) 5;; dedx

1 62¢o 1 5¢o 6¢0 l 6¢0
I x=il+ +6! F(x)——“dx

 

 

l a¢o
-bzl:F(x)[1:22::dx]dx-sz 5; dx- 0 . ( 7 - 3O )

In order to simplify Eq.(7-30), we assume that for the zeroth order

approximation, ¢o is independent of x. This is effectively true for TGTD

of the liquid mixture because the concentration gradient along the z
direction due to fluid convection along the same direction is much
larger than the concentration gradient due to the temperature gradient
along the x direction. Making use of this assumption, we have the

following very simple equations

2 32¢o 2 6¢o 26¢o
[2w +E]az2 +[-4Kw +H]az -2b at -0 , ( 7 - 31 )

and

97

1 l
HerF(x)dx , E-IF2(x)dx . ( 7 - 32 )
-1 . -1

To derive Eq.(ffifl), we have applied the odd and even function
'properties of'f(x) and F(x) respectively. The two constants H and E are

easily evaluated:

F(x)-9(x2/2-x4/4-1/4); H--859/30; E-1692/315 . ( 7 - 33 )
By Eqs.(6-13), GzO.lm, €25x10-3, ale-3m, and K-Zle/(46)-O.25m-1. Then
EszlO'amz and fizl.3x10'4m. Thus E>>2w2—a2/2=5x10'7m2 and

fi>>4Kw2-a2Kz2.Sx lO-7m. Neglecting Zen2 and Asz, Eq.(7-3l) becomes

62¢o a¢0 a¢0

5;; -(H/E)5; -(2b2/E)3; =0 . fi--H . ( 7 - 34 )

Now fi/E-Zle/(he). From Eqs.(6-25 and 27) 215/(49)-K. Thus fi/E-K. We have
thus obtained from the time-dependent equation the K factor, which is
very important in TGTD (Tyrrell [1961], Horne and Bearman [1967]).
Although the general approaches are quite different, the factor K
appears independently in both the steady state and the time-dependent
solutions for TGTD. This result supports the validity of the assumptions
made earlier in this section.

To solve Eq.(7-34), the boundary condition in the z direction must
be known. For a column closed at both ends the flux along the column at
both top and bottom must vanish, and from this condition we must be able
to derive a proper boundary condition for the 2 component. The flux

along the z direction for TGTD can be written as (ch. 2)

98

6C2

jz-vz(x)c2-D*5; . ( 7 - 35 )

By the second of Eqs.(7-4), jz vanishes at the boundary, so

vz(x)c2-D 5; -O , at z-O,L . ( 7 - 36 )

Since c2 is the sum of R(x,t) and Y(x,z,t) by Eq.(7-S), we have

gE-v(x)Y/D*+v(x)R(x,t)/D* , at z-0,L, ( 7 - 37 )

where we drop the subscript. Keeping in mind that starting from Eq.(7-
37) all the following mathematical manipulations are true only at z-O
and L, we then combine Eq.(7-ll) with Eq.(7-37) to obtain

K(z-L/2)z

fig K(cg+u)-v(x)(cg+u)/D*+v(x)R(x,t)e /D* . ( 7 - 38 )

62'

At this point, we introduece a perturbation device. With the help of

Eq.(7-14), in orders of An Eq.(7-38) becomes

au
EEO-x<c2+uo)-v<x>(c3+uo>/D*+v<x>R(x.t>eK(z'L/2)/D*. < 7 — 39 >
32 -[v(x)/D +K]un , n21 . ( 7 - 40 )

The zeroth order equation can be rewritten in terms of Wo(x,t) and

¢o(X.2.t)

99

3¢

* 0 *
D 5; -[v(x)+D K][cg+Wo(x,t)+¢o(x,z,t)]

+eK(z'L/2)R(x,c)v(x), ( 7- 41 )

where we used Eqs.(7-l7). Eq.(7-41) cannot be used as it stands, because

both W0 and R are time dependent. However the relaxation times of W0 and

* -
R are typically of order (a2/n2D*). Taking az0.1cm, D le 5cmz/sec. for

n~1,the relaxation time is about 100 seconds. Thus the exponential

*
- 2 2
factor 9 (nu) D t/a

is very small after about 8 minutes. The relaxation
time along the column height is a few hours or longer (Naokata and Kimie
[1984]). By comparing these two relaxation times, we see that the steady
state concentration in the x direction is reached when the vertical
concentration gradient is still insignificant. On the other hand, the

horizontal concentration gradient is very small compared to the vertical

gradient and we therefore take only steady state parts for Wo(x,t) and

R(x,t). This introdueces no significant error but simplifies our

vertical boundary condition tremendously. Hence Eq.(7-41) becomes

*3¢o

D 5; -[v(x)+D*K][Wo(x)+¢o(x,z,t)]+e

K(z‘L/Z’RS<><)v<x>. < 7- 42 >

where Wo(x)-Wos+cg, and both WOs and RS are steady state concentrations.
The following treatment for boundary conditions is the same as before

for differential equation ¢0. We first integrate Eq.(7-42) for x from -1

to l and note that v(x)/D*-4f(x)/a2, f(x)=ex(l-x2). After some

computations we end up with

100

a¢ 1 1
E;;°-fi¢o+I f(x)wo(x)dx+eK(z'L/2)J f(x)RS(x)dx , ( 7 - 43 )

where H and E are defined by Eqs.(7-33). The two integrals are evaluated

easily.

1 l l .
I f(x)Wo(x)dx-I f(x)[Wo§x)+c3]=26c36/8OI (x-x3)[25x-70x3+21x5]dx-0

 

l Zecg 1 ex heecg
.______ _ 2 ' z- -
I f(x)dex e -5 9I x(l x ) e dx 15 . ( 7 44 )
_1 e -e _1

To evaluate these two integrals, we used Eqs.(7-lO and 26), expanded

'EX
e

and neglected terms of order 62 in deriving the second integral.

 

Finally,
8¢o heecg
__ _ _ R(z-L/Z) _
az K¢o 15E e , z O,L . ( 7 - 45 )

Because by Eqs.(7-34) ee=30fi/8, then

a¢o _
5; -K[¢o-cgeK(z L/2)

] ! 2-0,L 0 ( 7 ' 46)
Eq.(7-46) is the boundary condition subject to differential equation (7-
34). Having Eq.(7-46) in hand, we can solve Eqs.(7-34) without

difficulties. The equations to be solved are

101

32¢o - 3¢o 2 a¢o
5;; -(H/E)5; -(2b /E)at =0 .

6¢
EEO-K[¢o-cgeK(Z-L/2)] , z-0,L , ( 7 ' 47 )

¢o(z,t-O)-0 .

K(z-L/2) -

We first let ¢o(z,t)-$ (z,t)-chze . Then in terms of ¢

628 a; a8

_-__2_=

822 K62 2b /Eat O ’

63

5; -K$ , z-0,L , ( 7 ' 48 )

a (z,t=O)-chgeK (2-1/2) ,

where we have neglected terms of order K2 in the first of Eqs.(7-48).
The method of solving Eqs.(7-48) can be found in any partial

differential equations text book. The solution is

 

a-Bo$o(z)+§ Bn$nrn(c), Bo-[1-E§KL-1]cg ,
n-l

‘LZK BE 2 1_(_1)neKL/2] -E
B“ L [L] [(K/2)2+(nfl/L)Y]2 , Tn(t)=exP[E[(K/2)2+(n’r/L)2]t] ,

$o-€K(Z-L/2)

 

KL

$n-[cos(nwz/L)+(2nfl)sin(nnz/L)]eK(z-L/2)/2

 

( 7 - 49 )

Thus the solution for Eq.(7-47) is

102

m

¢o(z,t)-Bo$o(z)+§ BnanTn(t)-chgeK(z'L/2) . ( 7 - 50 )

'n=1

C. DISCUSSIONS OF THE TIME DEPENDENT SOLUTION

In this section we discuss some of the results derived in this
chapteru IIt is clear that we have solved the problem of the
concentration distribution as a function of space and time to zeroth
order. This has not been done before. Now it is possible to predict the
concentration at any point any time in the column while the experiment
is in progress.

For convenience in discussing our solution, we combine Eqs. (7-5,ll

and 17) and write

c2(x,z,t)-eK(L/2-z)[cg+Wo(x,t)+¢o(z,t)]+R(x,t),

 

 

m - - n - 2 - 2
wo(x.c)-c3 175% [4“ (mt/2b) c-3i:._1)<2[ _ 7.1152] (1-, (mt/a) c”
n-l
xcos[nw(x+l)/2]+ech-Cg .
m n KL/2
( t)-0 KL 1 1((2-1/2) + o} -_2_K[M]2 [1'('1) ' ]
¢o 2, c2 _KL' e C2 L L [(K/2)2+(n«/L)2]2

l-e n-l

 

x[cos(nnz/L)+(§n:)sin(nnz/L)]eK(z—L/2)/2

xexp[-f[(KL/2)2+(n«)2]]-ch3eK(z-L/2) ,

103

 

 

Zecg m 2 n e

R(x,t).._,-ex-4,co,-ex§ (n1r) [1-(-1> e 1

‘e‘e-e [62+(n1r)2]2

e Ignxzz+gzlp*c
x[cos[(x+1)nn/2]- nu sin[(x+l)nn/2]]exp[- a2 ] ,
362880D* L 2

_ 2 2 ___—___. __n___ -
1 2b L /E a5 [ATpoag] . ( 7 51 )

Where K, e and 0* all have been defined before.

The solution is the sum of three terms. ¢o(z,t) is a function of
column height and time only, while R(x,t) is a function of column width
and time only. We call ¢o(z,t) the pure convection contribution to TGTD.
¢o(z,t) does not directly depend on the temperature gradient. Instead,

the temperature gradient affects only the progress towards steady state
concentration distribution along the column height because the
relaxation time is inversely proportional to (AT)2. A higher temperature
gradient leads to faster convection and a higher velocity redueces the

time required to reach steady state. When sttady state is attained ¢o is

independent of'AI. In.general, from the definition of 1, large AT,
thermal expansivity a, density p, gravitational force g (if the

experiment is performed on a planet with large g) and small viscosity 0
*
will lead to a small relaxation time. Since a, n, D and p0 are almost'

constant and do not change much duering the course of the experiment for
small AT, the dimensions a and L, the annular gap width and column
height, are very important in setting 1. Usually the column height can

vary from a few centimeters up to about half an meter. Experimentally

104

Naokata and Kimie [1984] have verified the strong dependence of r on L.
A longer column requires a longer time to reach steady state, but will
Lead to a higher concentration difference along the column. When steady
state is established one has the largest separation of solute from
solvent along the column. This can be seen from our numerical
calculations (Figs.7-l, 2, and 3)). Up to now in this discussion we have
been using Eqs.(7-Sl) for the relaxation time. The real relaxation time
is r/[(KL/2)2+(nx)2]. (KL)2 is usually very small compared with «2
'unless L.is over 10 meters, which is unlikely. We therefore ignore this
term and use only r'-r/x2 hereafter.

The most important factor which affects the r' is a“, the annular
spacing of the column. A small change in a will change 1' very
significantly. To obtain a higher concentration gradient, one prefers a
narrower annular spacing, but the time required to reach steady state
increases dramatically with smaller a. It is interesting to note that
when the annular spacing approaches zero there is no thermogravitational
thermal diffusion because convection of the fluid will not occur.

To finish the discussion, we compare the steady state with that

obtained in chapter 6. At steady state the functions 450(2), Wo(z,x) and

R(x,t) take the following forms

 

26c°
KL 2 -ex
¢°(z)-[1_e-KL -1-Kz]c3 . R(X)=ee-e'6 e
‘C3 K(L/2- )
Wo(z,x)--‘§6—[x(25-70x2+21x3)]e z . ( 7 - 52 )
With c2(x,z) -$o(z)+wo(z,x)+R(x) , ( 7 - s3 )

L 603 L
c2(x,z)-c3+ch[‘2'-z]- 8O [x(25-70x2+21x3)]{1+K(§-z)}

 

105

e
-cg[l-‘—86x(25-70x2+21x3)]{1+K(I§'—z)}. ( 7 - 54 )

Here we have expanded all exponential terms up to first order fort!( and
c. This result is the same as Eq.(6-31). Although the general approaches
of chapter 6 and.7 are very different, the steady state results are the
same for the zeroth order solution. This validates the assumptions made
in dealing with the time dependent solution.

Figures(7-l, 2, and 3) display the concentration distribution. These

curves are calculated from Eq.(7-Sl).

D . WORKING EQUATIONS

The measurements of solute concentration at the two ends of the TGTD
column can be made at either steady state or transient state (Gaeta,
Perna, Scala, and Bellucci [1982], and Naokata and Kimie [1984]). The
advantage of steady state measurements is that the concentration
gradient has reached its extreme values at both ends, so it is easy to
measure with a relatively small error. The disadvantage is that it is
very time consuming to reach steady state. Because of this, one also
measures the concentration change at early times. In order to use our
theory to account for the experimental results at early times, some
further work is needed.

Because experimentally one monitors the average concentration'change
at the column ends, it is necessary to convert our concentration
distribution equation by averaging along the annular dimension.

Mathematically, we evaluate the integral

106

r2
I rc2(r,z,t)dr
r

 

<c2(z,t)>-* 1 . ( 7 - 55 )

r2
rdr
r1

With x as variable, Eq.(7-55) is

l
I rc2(x,z,t)e26xdx

1
l
I e26x dx
-1

 

<c2(z,t)>- ( 7 - 56 )

where we have used Eqs.(4-20). The next step is to substitute c2(x,z,t)

given by Eqs.(7-Sl) into Eq.(7-S6) to evaluate the integral. We omit the

lengthy details of calculations and simply write down the result.

(c (z, t)>-A -A2t m -A2t
2 i B w(z) n ]+C+§ Ene n J +[-ELjEi-1-Kz]

 

 

-1, 3, s-- n-l 1“
+eK(L/2-z)/2 E Gn(z)e-t/r'(n) , ( 7 _ 57 )
n-l
with
46

1-(-l)ne
B (2)-6466 [ ][1(n«)2[(nx)211]1eK(L/2-z)/2’ A-26/(e46-l),
“ (n«)2[(46)2+(n«)2] .

8KL(2n«)2[( 1)“e KL/2-1 KL
G (2)-7 V, 1rZ—n-fl—[cos(n1rz/L)+ sin(nnz/L)] ,
“ [(KL)2+(2mr)2 ]

 

107

 

 

2[e45-26_1 15

c V, - , 362880D* 9L 2 -2

-[ '26] ’ Tn a6 ATp ag (In) ’

l-e (26-c) °
46- e
1-(-1)n e ]( )“e -1]
2

A3—[19§%—]D*, En--3266«2 62 2 ~ . ( 7 - 58 )

[;2+(n«)2] [(:6- 1:)2+(n1r)2]

Now we define a new function

A<c2>-c2(0,t)-c2(L,t) ( 7 - 59 )

Thus A<c§> represents the concentration difference between the bottom

and top of the column. Substituting Eqs.(7—52) back into Eq.(7-57), then

letting z-O and L respectively, Eq.(7-59) becomes

 

A<c2> 8 e-t/rfi
cg -KL[l-;2§ (2n+1)2]
n-O
+-—§—(1+e5)sinh(KL/2)[1-e‘*3t], ( 7 - 6O )

*
where A3 is the first term of A3—(2n+1)2«2D /a2. The reason for just

-A2t
taking the n-O term is that e n is nearly zero for n21 when t260

seconds. To derive Eq.(7-60) we have used relation

 

'- . ( 7 - 6l )

m 42 12
1-(n«)2[(nn)2-1] l
«4(2n+l)‘ 960

n-l

 

108

We also have ignored terms such as (KL)2, (46)2, in comparison with

(mt)2 term, and have expanded terms like e26 and 846. Eq.(7-60) redueces

to zero at time zero.

As time goes to infinity, we have the maximum for A<c2>, which is

A<c2>
'23——-KL+Z%%£(1+e45)sinh(KL/2)
2

5 4 D AT
~_Q_2_fl__L |__Q -

Eq.(7-62) is the working equation for steady state evaluation of the
Soret coefficient. A<co> is er'nerimentally measured. By solving Eq. (7-
62) for 0* one obtains the Soret coefficient.

Eq.(7-60) is the working equation before steady state, and can be
used at any time duering the experiment. To avoid the series, it is

desirable to derive a simple equation which can be applied to a certain

a)

-t/r'
time period. To do this we must evaluate Em. We redefine this
1 n=0
term as
” -p(2n+l)2 * .
f(fl)‘ _£_____; ’ #_ 2é2%§%2_ __nL__ 2 -1t, ( 7 _ 63 )
(2n+l) a « ATpoag

n-O

then for small p

109

- 2 - 2
dgifll" e p(2n+l) z-Jme p(2n+l) dn ( 7 - 64 )
n-l

The integral is tabulated (Gradshteyn and Ryzhik [1980]), and Eq.(7-64)

becomes

 

gfifiui-(w/fl)1/2 [l-ezfljufl ( 7 _ 65 )

or f(p)-(«p)l/Zerf(/p)+l/(2e‘#)-(xp)l/2/2+C; where erf stands for

error function and the constant is evaluated at p-O. Thus

 

1/2 -p 1/2 .
f(p)_11g12erf(1u)+% _ 13%) +§2_ ( 7 _ 66 )

NIH

By expanding Eq.(7-66) in powers of p and retaining only the first two

terms, we derive

m

/2 -p(2n+l)2
f(p)==1r2/8-‘(§£1 z} -%§;:I;;—— ( 7 - 67 )
n—O

The accuracy of Eq.(7-67) depends on how small p is. Table 7-1 gives a
comparison between the function f(p) and the infinite summation. As we
can see from the table, for about 1% error, p. can be as big as 0.7.

Taking p be 0.5 , Eq.(7-67) will be a very good approximation. Because

t1r2a°(ATpoag)2
#- * . (7-68)
362880D (’71.)2

 

110

if pz0.5, z3 to 7 hours, depending on the values of L, a and.AT. With

the constraints, the time dependent working equation for a column

without reservoirs is

A<c2> AKLJp 26¢
c° ' 3/2+ 15
2 x

 

sinh<KL/2)[1-e‘*3t] . ( 7 - 69 )

We also give two additional working equations, which can be used for

measurements at either the top or the bottom,

 

 

<c2(0,t)>-cg 2KLJp -KL _ 2
+54 . 22][1-.Aot] ,

cg ' ”3/2 15 3

 

 

cg " 3/2 15 3 ( 7 ' 7° )

1f

<c2(0,t)>-c3 ZKL/p -KL _ 2
-66[ e 2W2][1-€ AOC]

lll

Figure(7-1)

Concentration distribution as a function of column width x at a

given time t and column height 2. For cg-O.5mol./dm3,

AT-lOK, column height L-lOcm, From top to bottom, line 1 represents the
concentration distribution at t-l hour, z-O (bottom of the column); line
2, t-O.2 hour, z-O; line 3, c-o, ‘2; line a, c-o.2 hour, z-lOcm (top of
the column); line 5, t=l hour, z-lOcm. Note that the TGTD steady state
concentration distribution as a function of column width for any 2 is
not linear because of convection along the column. At t-l.0 hour, the
distribution is almost steady state for L-lOcm. However, for PTD, the
staedy state concentration distribution is linear (Bierlein [1955],

Horne and Anderson [1970]).

112

523 5:28

 

 

 

wd .vd 0.0 .v.l 0.! N._.l,
..—-L.—..p.rpp. bbF—Lb- 0*..0
\\\.‘\ lmV.O
s T l road

a
\X\ fl

(I/‘lOW)°ONOO

113

Figure(7-2)

Concentration distributions as functions of column height at a given
x and t. Parameters are as in Fig.(7-l). Upper group curves ,from top to
bottom, represent the concentration distribution at x-l (at cold wall)
and t-w, 0.5 and 0.25 hours respectively. Lower group curves from top to
bottom, are at x--1 (at hot wall) and t=oo, 0.5, and 0.25 hours

respectively.

114

Z.

r—o:

PIOEI 223.50
N. m n

b . _ . _

 

 

(I/‘lOW)'ONOO

115

Figure(7-3)

Concentration distributions as function of time t at a given x and
2. Parameters are as in Fig.(7-l and 2). Upper group curves:(at Um;
bottom of the column, z-O), from top to bottom, x-l, 0 and -1. Lower

group curves:(at the top of the column, z=10cm), from top to bottom,

x-l, O, and -l.

iCONC.(MOL/L)

0.54-

0.53~

0.52-

0.51- J

 

0.50
0.49d ‘
0.48-

0.47-

116

l

.L

 

 

 

 

 

 

0.46
0.0

T

130 ' 210 5 3:0
TGTD TIME (hour)

117

Figure(7-4)

Average concentration distributions at the top and bottom of the column
as function of time t for a given column height L. Here, parameters as
in Fig.(7-1) and Eq. (7-57) are used for the plots. Upper group curves
are from top to bottom, at L—50cm, 30cm, and 10cm, and at z-O. Lower

group curves, from top to bottom, L=10cm, 30cm, and 500m and at z-L.

Average Conc. (Mol/L)

0.65-

0.60-

0.55-

 

0.50

0.45-q

0.40-

 

0.35

118

 

0

r

2 .

l'l'l

. , . . .fi
4 6 8 1O 12 14
TGTD TIME (hour)

119

Table 7-1

Numerical comparision between the infinite summation and its.asymptotic

form for a given p.

 

 

 

” _ 2
e p(2n+l) £2- « 1/2
“ (2n+1)2 8 2
n=0
0.00 «2/8 «2/8
.01 1.14507785 1.14507758
.1 .953450989 .953451000
.5 .6078 .60704

 

We have summed up to 500 terms in evaluating this infinite summation.

C H A P T E R 8

THEORY OF RESERVOIRS

A. GENERAL REMARKS

In chapter 7, we discussed the time-dependent theory of a column with
both ends closed. However, for the experimental purpose of evaluating
Soret coefficients, the column with both ends closed is not the most
useful. This is because the annular gap is usually very small axui it is
not easy to measure the concentration difference between the two ends,
except at steady state. Many of the experimental studies of Soret
coefficients have used a TGTD column with reservoirs connected to both
ends of the column. The volumes of the reservoirs vary from about 15 Lu)
to 500cm3, and the two reservoirs may have equal or different volumes
(de Groot [1945], Prigogine, de Brouckere, and Amand [1950] , Horne and
Bearman [1962], Beyerlein and Bearman [1961+], Gaeta, Perna, Scala, &
Belluccl [1982], Naokata and Kimie [1984]).

In general the temperature gradient is applied only to the column,

120

121

not to the reservoirs. Initially, the column and the reservoirs are
filled with solution, with the concentration distribution uniform for
the whole system. After the temperature gradient is applied to the
column, solute will, in general, migrate downward and solvent upward.
After a while, the lower reservoir becomes more concentrated than the
solution in the column, and inside the reservoir, isothermal diffusion
begins. Similarly, isothermal diffusion occurs inside the upper
reservoir because the solute concentration is smaller at the entrance to
the upper reservoir than it is within the reservoir. At steady state,
the reservoir concentrations are the same as the concentrations at the
entrances to the reservoirs. Thus the difference in reservoir
concentration at steady state is the same as the difference in
concentration between the two ends of the column. Because the volume of
the reservoirs is much larger than that of the column, it is easier to
monitor the concentration change in the reservoirs than in the column.
That is why most experiments have utilized reservoirs.

‘Pwo kinds of apparatus have been used. One is the cylindrical type
which we deal with in this work, while the other is a rectangular
thermogravitational cell, combining two flat vertical plates, one
heated and the other cooled (de Groot [1946], Tyrell [1961]).
Traditionally, the theory of a time-dependent TGTD cohum1vflth two
reservoirs has been based on a theory developed for gaseous mixtures by
Furry, Jones, and.0nsager (1939), and Furry and Jones (1946). An
approximate time dependent TGTD theory for dilute binary solutions was
developed by de Groot (1946). The main assumption made in. all previous
approaches is that material reaching the upper or lower reservoir is
almost instantaneously distributed uniformly throughout the reservoir.

This assumption was first presented by Furry, Jones, and Onsager for the

122

treatment of TGTD in gases. It is nearly true for gases, because gas
molecules diffuse much faster than liquid molecules. The diffusion

constant is of order 0.1cm2/sec for gases, but for liquid mixtures is of

order 10-5cm2/sec. Thus, for gases, molecules reaching the reservoirs

can diffuse rapidly into them and they are rapidly distributed uniformly
throughout the reservoirs. For liquid mixtures, the diffusion process is
very slow. The working equation (Tyrell [1961]) for the concentration
distribution in a TGTD apparatus with two equal reservoirs at short time

based on the above assumption is

c -
B 0 ba3 AT 2
{—cT-1]-L3——-L)—36Onv st, . ( 8 - 1 )

where s is the experimental Soret coefficient, V the volume of
reservoir, b is the width of the plate, and the other coefficients have
their usual meanings. Thus if one could measure the solute

concentrations cB and cT in the bottom and top reservoirs at a given

time and given initial concentration, one will be able to evaluate 5

through Eq.(8-1). As we can see from Eq.(8-l) if cB/cT>1, (for TM>4°C),
then s is positive and if cB/cT<1, then s is negative. Clearly, if

solute indeed concentrates in the lower reserwoir then the Soret

coefficient is positive. Soret coefficients determined by pure thermal

diffusion experiments are positive and of order of 10-3K. Recently,
Gaeta and coworkers (1982) reported from their TGTD experiments that in

certain concentration regions the ratio cB/cT<1, and then s becomes

negative for NaCl and KCl binary aqueous solutions. That is, in those

concentration regions, the solute is enriched in the upper reservoir

123

rather than in the lower one. To explain this unusual experimental
result,t1mw’suggest order-disorder transitions between solvent and
solute which involve sharp changes in solvation. Later, however, Naokata
and coworkers [1984] could not reproduce the unusual concentration
dependence of Soret coefficient in their TGTD column. Gaeta et al. used
a rectangular TGTD cell. Earlier, Prigogine et a1. [1950] reported

similar results from a rectangular TGTD cell.

In this work, the reservoir geometry is cylindrical, not rectangular.
Experimentally, it is much easier to attain precise geometry and precise

temperature control in a cylindrical apparatus.

B. DIFFERENTIAL EQUATION OF DIFFUSION

The fundamental differential equation of diffusion in an isotropic

isothermal medium is, by Eq.(2-76),

3C2

*
at D v c2 ( 8 2 )

*
where c2 is the solute concentration and D the modified diffusion

coefficient. We neglect the convection velocity term v-V'c2 in writing

Eq. (8-2) because the reservoirs are assumed isothermal and convection

will not occur. In cylindrical coordinates, Eq.(8-2) is

acgD :c218202 82 c2
252w }

at r +; 533 +r

124

Assuming 8c2/6¢-0, we have

at r ar( ar )+razz ( 8 - 4 )

To solve Eq. (8-4), boundary and initial conditions must be known. The

initial condition is that at the time of starting the experiment, c2 iJI
the reservoirs is cg, duainitial concentration. One end of the

reservoir is closed and the other end is connected to the column. The
boundary conditions for the closed end and for the outer wall reflect
the fact that nothing will diffuse through the reservoir walls. The
situation is very complicated for the end connected to the column
because the concentration at the junction between column and reservoir
is a function of space and time. If, however, the annular spacing of the
column is much much less than that of the reservoir, and the

concentration variation along r from r1 to g is very small in the
column, we can replace c2(r,t) by its average concentrationi<c2(t)>. we

write for the boundary conditions at the lower junction

c2(r2,z-0, t)-<c2(t)>-g(t), rlerrQ, ( 8 - 5 )

where g(t) is the average concentration at the bottom end of the column.
For the diffusion process in the bottom reservoir, the differential

equation and boundary conditions are

r2<r<r3, h<z<0, t>0,

at -r r- :2)+r32

aczD 62c2
”{3-< z

125

8C2 6c2

5?

 

 

02(r2,t,2-0)=g(t), =0, t>0,

r3: 5; h

C2(t-O)-Cg ( 8 ' 6 )

Eqs.(8-6) tell us that there is a point concentration source at the
upper entrance to the bottom reservoir. Note that if the source term

g(t) is just.c92, the solution of Eqs.(8-6) will be just cz-cg. If
g(t)>cg, then solute will diffuse into the reservoir, while if g(t)<c2,

solute will diffuse out of the reservoir.

C. SOLUTION OF DIFFERENTIAL EQUATIONS
We first let

Then

6U D* Q_ 8U 62U
__ _. { (r—— )+r——— }, r2<r<r3, h<z<0 t>0,

 

 

at r ar ar 822
- Q! Q! =
U (r2,t,z-0)-g(t), ar ran 62 h 0, t>0,
U (t-0>-0. é<t>-g<c>-c3 < 8 - 8 >

Eqs.(8-8) can solved if the solution of the following partial

differential equation can be obtained

To derive Eqs.(8-12 and 13),

126

82W
at -r D*{arfl 6r )+r5;; }, r2<r<r3, t>0,
a_w 6.11 =
W (r2,z 0)-l, 6r r3 62 h 0, t>0,

 

 

W(t-0)-0,

then (Carslaw and Jaeger [1959])

U(r, z ,c)- -I: g(A)% :(t A)dA .

If we suppose a form of solution for W(r,z,t)

W(r,z,t)=R(r,t)V(z,t),

Eqs.(8-9) become

62V

 

a—V — a fl=
at 322' V(0,t) l, az'h o, t>0,
V(t-0,z)-0,
3R
__DL
at r ar(r5?) r2<r<r3, t>0,
R (r2)-1, %% raao, c>o,
R(t=0)-0.

we have assumed

(8-9)
( 8 — 10 )
( 8 -1l )
( 8 ~12 )
( 8 - 13)

that R(r=r2,t)=V(z-0,t)=1,

127

t>0, which means that at the origin, both functions equal unity, the

same as unit concentration c3. This assumption is reliable if we take a

look at the second equation of Eqs.(8-9). Physically, it tells us that

at the origin, or at r=r2, z=0, there is a constant concentration source

of unity for t>0, and at t=0 the concentration in the reservoir is

initially zero. Solute does not flow out at r3 and z-h. At the origin,

the solute concentration will be unity along any direction, or the
diffusion from the z direction is independent of the r direction, Thus
the solution for W can be written as the form of the products of V and

R.

The solution for V has two forms. Using a Laplace transformation, we

have

 

“ m 2h(m+1)-z ” m 2hm-z
V(z,t)-E (-l) erfc[—*]+§ (—l) erf[ * ], (8 - 14 )
2/(D c) m=0 2/(D )

m=0

2hm-z

 

where erf represents the error function with argument [ ] and the

*

2/(0 )

complementary error function is defined by erfc(x)=-1-erf(x). An
alternative form can be derived by the method of separation of

variables,

.-

(2m+1) «z *
2h exp[_(2m+l)2n2D c]

2
(2m + I)2 “h

 

 

 

4 sin
V(z,t)-l-;§ ( 8 - 15 )

m==0

The solution for R(r,t) is

128

cu *

 

 

«221) t J¥(a r3)
R-1+« a(a )¢ (a r)e n a(a )- n ( 8 - 16 )
n 0 n , n Jg(a r2)'J¥(a r3),
n n
n-0
with
¢o(anr)-Jo(anr)Yo(anr2)-Jo(anr2)Yo(anr) ( 8 - 17 )
a satisfies
n
¢1(ar3)-J1(ar3)Yo(ar2)-Jo(ar2)Y1(ar3)=O. ( 8 - 18 )
For the details of solving these equations see Appendix A.
The solution for W is, by Eqs.(8-11,15 and 16),now
” -a:Dt 4 ” -d20*t
W - l+x§ a(an)¢o(anr)e ‘1-; E Om(z)e , ( 8 - 19 )
n-O m-O
with
”i” 1 I .
51“[21. "Z_ _ n (z) 2m+1 2n2 _ d, ( 8 - 20 )
2 m ’ 4h2 m '

(2m + 1)

The solution of Eqs.(8-6) is given by combining Eqs.(8-7,10 and 19)

02 t_ a m ~a:D*(t-A)
;§-1+ g(A)3; 1+«E a(an)¢o(anr)e
0

n-O

co *
-d"’D (t-A) .
x[1-$ E Om(z)e m ]}dx . ( 3 - 21 )

m-O

D. DISCUSSION OF THE SOURCE FUNCTION

129

To obtain an exact solution for Eq. (8-21), an actual form for the
source function g(A) must be known. From chapter 7, the average
concentration distribution function as a function of time at the two

ends of the column can be written approximately as

 

<c2>T-cg-7(1-e‘92D*t)cg , <c2>B-cg+7(1-e-ezD*t)cg , ( 8 - 22 )
where
2 asnz ATpoag 2
y-KL/Z' e - 362880(D*)2[ "L ] ( 8 - 23 )

Experimentally, it has been observed (Naokata and Kimie, [1984]) that
the average concentration change in time at the column ends (for a
column with reservoirs) has the same form as Eq.(8-22), but 7 and 9 are
essentially adjustable parameters to the experimentalists.

Now we make the following assumptions for source functions at the two

column ends.

*
92D t

92D*t
), ( 8 - 24 )

83(t)-CS+163(1-e' ). gT(t)-cgoycg(I-e'

and for short times the above equations reduce to

* *
gB(t)zcg(l+762D c), gT(t)zcg(1-792D c) . ( 3 - 25 )

E. CONCENTRATION DISTRIBUTION IN THE RESERVOIR WITH EXPONENTIAL AND

CONSTANT SOURCE FUNCTIONS

130

Substituting Eqs.(8-24) into Eq.(8-21) and making use of the last
equation of Eqs.(8-8), the general solution is found for the
concentration in the bottom reservoir. The detailed mathematical

manipulations are presented in Appendix B.

 

B m 2 _ 2 * m 2
33-1+ ." ana(an)¢o(anr) e anD t_e-G2Dt +3 dm 0m62)X
cg 7 a: - e2 n d; - e2

n-O m=0

 

-d7D*t * m ” a(a )¢ (a r)fl (z)E 2 -E 29*: *
[ m -92D t] E E n 0 n m mn[ mn -92D t]
e -e +4 e -e

E 2 - e2
m-On-O mn
” m -E 20*: a ” -d;D*c

+4 E E a(an)¢o(anr)0&z)[l-e mn ]+; E 0m(z)[l-e ]

m-On-O m=0

w ~a§D*t
-« E a(an)¢o(anr)[l-e ] , ( 8 - 26 )

n-O

where E 2 -a2+d2.
mn n m

Putting a minus sign in front of 7 for Eq.(8-26), we obtain the
concentration in the top reservoir. Note that at t=O, Eq.(8-26) is

simply unity and as t approaches infinity, we get

c2 w w m
Eg - 1+7[ 4 E E 3(an)¢°(anr)0mz) + % E flm(Z)
m-On-O m=0
- N E a(an)¢o(anr)]. ( 8 - 27 )
n=0

Using the relations (Appendix B)

131

n E a(an)¢o(anr)--1, % E Om(z)-l , ( 8 - 28 )
n-O m=0

we get from Eq.(8-27)

E? - 1 + 7 . ( 8 - 29 )
2

Therefore, when a steady state is established, the concentrations in the
reservoirs will be equal to the concentrations at the two ends of the
column. If we solve Eqs.(8-6) for steady state, we end up with the same
result. This is also the previous result since upon averaging Eq.(7-54)
and taking z-O, it is just Eq.(8-29).

Now we need to evaluate the average concentration change with time in

the reservoirs. The average concentration is

B 1 2w r3 B
<C2(t)>-_V— I I J rc2(r,z,t)dwdrdz, ( 8 - 30 )
0

‘where VR is the volume of reservoir. Making use of Eq.(8-26) as well as

the integrals in the Appendices A and B we have

 

*
<c2(t)> 2“ Q a(an) -a§D t HGQD t r§- -r§ w . 1
cg A1+ vR 2h E a2 - e2 [‘ ‘ ] h E d2-92 x
m
n~0 m-O

_ co co 2 - 2 *
dm D* t HGZD t 4 a(anflimn -82D*t EmnD t
' +h a2d2( E 2 e?) '9
m-On- 0 n m( mn

132

 

 

4 a(a ) -E 20*: rg-rg ” -d2D c
+- e m“ -1 + -—— l-e
h déa: h d;
m-On¥0 m=0
m a(an) -a;D*t
+2h E a2 [l-e ] . ( 8 - 31 )
n
n-O

At this stage, we have solved the reservoir problem. We have found the
concentration distribution in the reservoir as a function of space and
time as well as the average concentration change in time in the
reservoir. However, Eq.(8-31) is too complicated to apply for practical.
purposes. We want particularly to know what will happen during the
experimental time interval shortly after the beginning of the

experiment.

Before answering that question, it is interesting to examine Eq. (8-

31) for a special case. If 62 is much smaller than d; and a3, then
aa-ezza: and dg-Gszg. In other words, if the rate of flow into the

reservoir determines the rate of the process and the solute diffusing
into the reservoir will spread throughtout the whole reservoir

immediately, then Eq.(8-3l) reduces to

 

 

<c2(t)>-1 a m ” a(an ) rg-rg ” 1 ” a(an)
—c—3 lfi+h§§d2ag'h EEE-ZhEOS x
m-On -0 m-O n=-O
920*:
[e' -1] . ( 8 - 32 )

Using the equations in Appendix B

133

(rE-rg) . ( 8 - 33 )

( 8 - 34 )

This is just the same as Eq.(8-22). If 9240, then the concentration is
just the initial concentration in the reservoir. This case corresponds

to the zero time situation.

On the other hand we rewrite Eq.(8-3l) as

 

 

 

<c2(t)> 2' h 4 m ” a(an ) e2 -Em;D*t
o '1+-_1 (rg-r2 )' E E 2 2 2 2
c2 VR 2 h a nmd (EIn -9 )e
m-On-On
rg-rg m 2 -d2D*t m 92a(a ) -02D*t
+ ————e e m +2h —“ e n
h d2(d2-62) a2(a2-62)
m m n n
m-O n=0
Q w 2 2- 2 CD
+9 __fffan_a__ 920*t_:£_53 1 -e2D*t
h a2d2(Em:- -e2) h dg-e2‘
m-On-O n m m-O
Q
3(0 ) 2 *
n -6 D t
-2h E Eg-j—ag—e ] . ( 8 - 35 )
n
n-O

134

'To write down.Eq.(8-35), we have rearranged Eq.(8-33) and used Eqs.(8-

33). As 924w, Eq.(8-35) becomes,

 

 

 

<c2(t)>14 m m a(an ) -Em 2D* t rg-rg m l -d;D*t

o %(r§ r§)+— 2 2 e ' E 29
c2 h a nmd h dm

co *

a(an) -a:D t

-2h E a2 e . ( 8 - 36 )
n
n-O

TUnis case corresponds to the constant source concentration since Eq.(8-

36) could have been derived if we replace g(A) in Eq(8-21) by the
constant 15 and it tells us that if the source function is a constant,

the rate of diffusion is only dependent on the reservoir's dimensions.

F. CONCENTRATION DISTRIBUTION IN THE RESERVOIR WITH A LINEAR SOURCE

FUNCTION

If the source function is linear in time, we substitute Eqs.(8-25)

into Eq.(8-21) and make use of the last equation of Eqs.(8-8) to derive

 

 

 

c2 2 * a(a nn)¢o(a r) -a:D*t 4 ” 0m(z) -d;D*t
E§-1+79 D t+n no: [1 -e ]-W E d; [l-e ]
“‘0 m=0
” ” a(a )¢o(a r)0 (z) -E 2D*c ~
- 4 E E n E 2 m [l-e m“ J]. ( 8 - 37 )
m-On=0 mn

135

Applying Eq.(8-30), we have for the average concentration change in time

in the bottom reservoir

 

B - 2* 2- 2 a) - 2*
{52> 1 2 * Anhm a(a ) anD t 2n(r3 r2) 1 de t
o - +76 D t-‘V— ‘———:n‘ l- e -___hV-——_ d‘ l-e
c2 R an R m
n-O m=O
a(an ) -E 2D*t
mn
E M 41-. ]. (a-..)
VRm-On-O ma n Emn)

Obviously, there is no steady state solution for a source function
linear in time. However, since we are more interested in early time, we
assume that the time is short enough that we can expand all exponential
times in Eq.(8-38). On the other hand we notice that all summations in
Eq. (8-38) converge very fast. At early time, the average concentration

change will be almost linear. This is because the terms

* * *
-En;D t -a:D t -d;D t
l-e , l-e , and l-e will all be linear for small time

intervals and these infinite sums converge after only a few terms.

Although the main feature of Eq.(8-38) is that <c2> is almost linear in

tfinw, it is not easy to make a satisfactory simplification for this

a(a )

equation because the term E -;:—— cannot be written in a closed form
n-O n
(we can show that
k 2:10]:n ) '
ifa for k-2, 3, ---, the sum i -————— has a closed form only for k=2,
ak
n-O n

and for k>2 there are no closed mathematical expressions). Here a(an)

and an are given by Eqs.(8-l6 and 17).). If 92 is known, Eq.(8-38)

136

should be used to evaluate the Soret coefficient. We still want a more
simplified.equation, and as an approximation, we take only one term for
all sums and expand the exponential terms. (The exact way of simplifing;
Eq.(8-38) is to work out the asymptotic forms for terms such as
m a(a )

E ___EE- and

a
n-O n

on *
a(an) -a;D t
E --E——¢ . This approach is fraught with mathematical

a
n-O n

difficulties.) Thus we have

” a(a ) -a2n*t * m 1 -d2D*t *
E __ZTB_[1‘9 n ]za(ao)(D t/ag), E a4[1'e m )zD t/dg
n
n-O m=0

” “ a(an) -Eng*t a(ao)D*t
E E (d [l-e Jz-—-————— ( 8 - 39 )

2 2 2
m-on-O manEmn) aodo

substituting these equations into Eq.(8-38) and rearranging, we obtain a

very approximate linear equation

B 4a(a )(«2-8)
<Lc§92-1+[1-[§ + o ]] 792D*t.
2

«2 a3«2(r§-r§)

 

 

8 4a(ao) 2 *
_1+[1-;2][l- a3(r§-r§)]7e D t, ( 8 - 40 )

Where we used the relation VR=nh(r§-r§).

The average concentration change in the upper reservoir is given by

putting a minus sign in front of 7 for Eq.(8-40), which is

137

 

4a(a )(«2- 8)
-;:-(()£2-1 [1- [— O ]]792D*t.

+
«2 ao«2(r§- -r2)

 

8 48(00) 2 *
-1-[1-«,][1- ag(r§-rg)]79 D t ( 8 - 41 )

The error caused by Eq.(8-40 and 41) will be dependent on the relaxation
time; the smaller the time, the better the Eqs.(8-40 and 41). This can

be seen by the following arguments. First we write

m 1 ‘dzD*t _16h‘ B(2 1)2
m m+
m-O m-O '
where
B—[EIE]2D*t. ‘ ( 8 - 43 )

Then, we put

E <2m+1).<1-e'3(2m+1’2>z8. < 8 - 44 >

m-O

Table(8-l) shows the numerical comparison between B and Eq. (8-42). For
BSO.2, the relative error due to Eq.(8-44) is about 3% and the error is

about 9% for 350.3. We assume 350.2, then

0.8h2
ts * . ( 8 - as )
«2D

 

This tells us that if Eq.(8-44) is used to replace the infinite sum, and

the error due to this approximation is expected to be less than 3%, then

138

t must satisfy Eq.(8-45). For h-Zcm, Eq.(S-AS) gives 1159 hours.

139

 

Table 8-1
B 0 0 0.1 0 2 0 3
w
_ 2
E 753i577(1-¢ 3(2m+1) ) 0.0 0.1046 0.194 0.271

m-O

 

140

 

-azD t
The situation for E [l-e ] is, however, much more

n-O

complicated, for only the smallest (the first) roots are tabulated for

r3/r2>l (Bogert [1951]). We are unable to give a comparison like Table
3, but we know that a:n increases very rapidly as n increases. The

asymptotic forms of an and a(an) are given by a8 and a(a$)

 

a.~_(_20:1.1«_ 8(0.)~ rzn‘smzafirsn ( 8 _ 46 )
n~2(r3-r2) ’ n ~r3[1+sin(2ar'lr2)]-r2[1+sin(2alf1r3)]

This relation is valid only for large n. In table 4 we give a comparison
‘between the first roots calculated.by Eq.(8-18) and Eqs.(8-46) for a

given ratio of r3/r2. a(a5) and a(a6) are calculated by the second
equation of Eqs.(8-46).

The table suggests that an increases as n increases. We therefore

 

assume
Q a(a ) -a2D*t a(ao) -agD*t m a(a') -(a')2D*t
n n n n
-——7—— l-e ~ ‘ l-e + , ‘ l-e
Earl Ja.[ 1§<an>[ ]
n-0 n-1
*
za(ao)D t/ag . ( 8 - 47 )
*

agntsA (8-48)

For error less than 3% for above equation, A50.2.

141

 

 

 

 

 

 

 

Table 8-2

r3/r2 1.01 1.10 1.20 1.50 2.00 3.00 5.00
a ,156‘8 15,41 7,52 2,90 1,36 0,63 9,28

0 r2 r2 r2 r2 r2 r2 r2
a' 121,1 15,71 2,85 3,14 1.52 0,29 9,39

0 r2 r2 r2 r2 r2 r2 r2
a(ao) ---- 16.08 5.82 1.42 1.54 0.68 0.44
a(a5) 100 10.0 5.00 2.00 1.00 0.33 0.207
a' 421,2 42,1 33,6 9,42 4.71 2,36 1,18

0 r2 r2 r2 r2 r2 r2 r2

 

142

If we require that the relaxation time along the z direction is the
same as along r, then from Eq.(8-48) and Eq.(8-43) (taking the equal

Siegn), the height of reservoir is related to r by

___«38
h2 4013(3). ( 8 - 49 )

Thus Eq.(8-49) must be used for proper design of the reservoirs, and the

constants A and B are determined by Eq.(8-47) and Eq.(8-44). If an is
given by Eq.(8-46) and r3-r2-h, then by Eq.(8-48 and 43), A-B and

Eq.(8-49) is an identity.

Now we combine Eqs.(8-40 and 41) to derive

- *
<c2> 270 62t
-1_4——-—————- . ( 8 - 50 )
(cg) 1-1D Sgt

 

 

 

4a(a )(«2-8) 4a(a )
-4--*—1 ° 1111-22111- ° 14

«2 a3«2(r§-r§) a3(r§-r§)

If §D*92t is much less than 1, (true for t55 hours) this leads to

-*
<c2> 21D 92c _ *
T-l— _* 427002: (8-51)
<c2> 1-7D 82t

 

Eq.(8-51) is our working equation for calculating the Soret coefficient
from a TGTD experiment at an early time period. The time length is

controlled by the dimensions of the reservoir. Eq.(8-49) gives the

143

relation between h and r and A and B are determined properly from the

accuracy requirement of the approximation of Eqs.(8-44 and 47).

Now we replace 92 by Eqs.(8-23) to give a practical form of the

working equation

8

 

 

 

<C2> [ [:1112 poag 8 48(00)(fl2-8) *
< T '1' 6! [ flL ][1'[«2+ agflz<r§_rg)]]0 t ( 8 - 52 )
c2>

Eq.(8-52) is to be compared with Eq.(8-l). Using Gaeta and coworker's

data (1982), VR-15cm3, a-0.045, b-8cm, LP4.8cm and AT-16°C, we calculate
the numerical coefficients for these two equations. We take VR-nh(r23-
r§)-15cm3 and assume rg/rg-Z, then use Eq.(8-49) to evaluate h (we have
taken AFB'O.2). We get h-l.155cm, r2-1.174cm, r3-2.34 and a(a°) as well
as ea is from table 4. Substituting these values into Eq.(8-52) and

Eq.(8-1) we find

<c2>

<c$>

 

 

9008 , '
]a*t, a*-c1V15-a ( 8 - 53 )

-1zs.4x10'5[

<c2>

<c§>

 

 

_5 P008
-1z3.5x10 [ 0 )st, ( 8 - 54 )

* -
Because 3 and a are of the order of 10 3, we expect Eqs.(8-53 and 54)
are also the same order; then Eq.(8-52) and Eq.(8-1) are qualitatively

equal. A better result is derived if we remember

144

#hxmuxu. . .; < 8 - 55 )
then a better approximation for Eq.(8-50) is

<c2> 250*92t _ * _ *
T -1- _ * ~21D 62t(1+7D 92:), ( 8 - 56 )
<c2> 1-1D 92t

 

 

where the 110*82c is given by Eqs.(8-23). Eq.(8-56) is a second order

algebraic equation for 0*. By solving it we will have two values for 0*,
and only a meaningful root will be applicable to evaluate s from the
last equation of Eqs.(8-53).

As we pointed out the working equation Eq.(8-52) is only an
approximation, but we do think it will be applicable at least
qualitatively under the requisite experimental conditions. Furthermore,

justification of usage of 62 from Eq.(8-23) must be done experimentally.

G. SUMMARY AND DISCUSSION

In this chapter we developed the theory of TGTD column with two equal
volume reservoirs. The theory is based on diffusion. The practical
differential equation for the diffusion process in the reservoirs is
established using this model. The equation is solved to obtain the
concentration distribution in the reservoirs as a function of space and
time. The solution is dependent on the choices of the boundary
concentration distribution, i.e the source function. Several special

cases were discussed, and corresponding equations were developed.

145

We were particularly interested in deriving a working equation
applicable at an early stage of the experiment and from which the
thermal diffusion coefficients or Soret coefficients could be estimated.'
The result is given in section F. For certain restrictions of reservoir
dimensions as well as time, we do obtain a working equation to estimate
the Soret coefficients if the average concentrations in both top and
bottom reservoirs are measured. Because at present we do not know the

asymptotic expansions such as

 

a(a ) -a2D t a(a ) -E :D t
1—4 4: 1 11...... 14 1
n-O n m-On-O m n mn

the accuracy of the working equation given in section F is uncertain. If
possible Eq.(8-38) should be used. However, at present, only the

smallest roots are given for different ratios of r3/r2. The difficulty

of computing the roots of Eq.(8-18) hinders usage of Eq.(8-38). We are
unable to find the asymptotic roots for Eq.(8-18) because the arguments
cannot be made large enough to do so. We hope this difficulty will be
solved later.

Another important aspect of TGTD with reservoirs is that although the
general form of exponential decay type source function is confirmed
experimentally (Naokata and Kimie [1984]) and used in our problem, the
actual form of relaxation time for such exponential decay is not yet
established. The source function relaxation time used to derive our
working equation was borrowed from the theory of TGTD without
reservoirs. Our linear source function came from the direct expansion of

the exponential term as time t is small. This "short-time" scale depends

146

upon the dimensions of the column, and the temperature gradient as well
as the physical properties of the solution, as can be seen from Eqs.(8-
23). Usually, this "short-time" is about a few hours for a typical
column and.AT. Time dependent TGTD is a very sophisticated problem even
without reservoirs. For TGTD with reservoirs, we used our diffusion
model so that the problem can be attacked, and find a very approximate

working equation to estimate Soret coefficients.

There is a marked discrepancy between the working equation derived by
us and the one used before. However we see from our numerical
calculation that the two working equations are of the same order, which
nmmns that a cylindrical type TGTD column gives about the same
separation of solute from solvent with a rectangular cell type TGTD
column. However, the biggest difference between our equation and the one
used before is that Eq.(8-52) is made of two terms with opposite signs.

This can be seen by rewriting Eq.(8-52)

 

 

<cB> p ag 4a(a )
2 2 0 O
-1_L—Ll)—la A («2-8) 1-——2 2 2 Jr. ( 8 - 58 )

Because all terms outside the square bracket are positive, the sign
change depends on the two terms in the bracket. If the second term in

<e2>

 

the bracket is larger than 1, then is less than 1, the

<c2>

concentration in the bottom reservoir is less than that in the top
reservoir, thus instead of migrating to the bottom reservoir, solute

moves against the temperature gradient up to top reservoir. This is true

147

if the reservoir is very small. Using table 4 we found that if ra/r2<1.2

then the square bracket term is negative, and <c§> is less than <c§>.
But if r3/r2>1.2, then solute concentrated in the bottom reservoir as it

usually does (We remind the readers here that we did find by calculating
the concentration distributions in the column without reservoirs at very
early time period that when column length is over 20 cm, the solution is
a little bit more concentrated at upper section of the column.).
Therefore, in order to ensure that the solution is more concentrated in
the bottom reservoir, one has to design one's reservoir carefully, and
our equation provides a useful qualitative criteria for that purpose.

<c2>

<c§>

 

The disadvantage of the old working equation is that if is less

than 1, one obtains a negative Soret coefficient or thermal
diffusion coefficient from the old equation. Then to explain such an
unusual situation of electrolyte solutions at low concentrations( about

3to 1.3x10-1mol) and an average temperature of around 30°C, the

5x10"
authors (Gaeta et a1. [1982]) claimed that there must be a phase
transition under the conditions described above. But from our working
equation, it is apparent that the possibility that a negative sign

occurs for Eq.(8-58) is due to the improper choices of the dimensions

of the reservoirs such that rg-rg is too small. In other words, for

small reservoirs, it is possible to make a conversion of direction of
regular TGTD during the early time of experiment. Because we did not
work out the TGTD theory for a rectangular cell, we are unable to apply
our working equation to recalculate Gaeta et al.'s experimental results.

If larger reservoirs had been used in their experiments, negative Soret

148

coefficients would probably not have occurred. Our conclusion implies
that it is unlikely that there is any kind of phase transition in dilute
electrolyte solution. The conversion of TGTD is more likely due to
improper design of the apparatus. We also mention here that Naokata and
Kimie [1984] were unable to reproduce the results of Gaeta et a1. From
our point of view this is because they used relatively larger and
cylindrical geometry reservoirs. Moreover the pure thermal diffusion
experiments of Petit, Renner, and Lin (1984) yielded only positive Soret
coefficients.

Since the Japanese workers reported explicitly the detailed time

course of their results (through curves), it is should be possible to

*
obtain a from their paper as long as the dimensions of reservoirs are

given. It is n_o_§ possible to recalculate the Italian results to obtain

*
reliable 0 since the Italians report only their calculated results, not

their experimental results.

CHAPTER 9

SUGGESTIONS FOR FUTURE WORK

We have already seen from previous chapters that the complexity of
time-dependent TGTD prevents us from solving the problem exactly. Only
the zeroth or for some cases at most the first order solutions are
obtained, and nothing can be done for the solutions with order higher
than 1. However, we still have had a clear and deep look at the time-
dependent TGTD problem and have established a solid foundation for any
further research on the problem. Whenever possible, a numerical solution
should be done to check the accuracy of the perturbation solutions. For
the steady state, we have found an accurate result, but more research is
required for the transient state, particularly for the transient state
with two reservoirs.

The situation for the transient state with reservoirs is extremely
complicated due to the uncertain boundary conditions at the column ends.
We thus hope to reformulate a proper mathematical form for matter flux
at the interfaces between the column and reservoirs either empirically
or theoretically, so that one can derive the corresponding boundary

conditions in the transient state. At the interfaces where the

149

150

temperature gradient vanishes, the formulation of matter flux at
interfaces is not easy, and even if it could be done, the boundary
conditions would be too complicated to hold out much hope of solving the

concentration equation.

In chapter 8, we found the concentration1distribution in.the
reservoirs based on diffusion models, but we still do not know the
concentration distribution in the column. The average matter flux has

following form:

-A

2
j: - Ko30* ( L - 22 )( 1 - e t ) at z - 0, L , ( 9 - 1 )

*
where chD ( L - 2z ) is the steady state flux of the column without

*
reservoirs. Note that j: (z=0)=-jz(L), which is to say that the flux is

antisymmetric at the two ends. When a final steady state is reached, the
concentrations in the top or bottom reservoirs are the concentrations
at the respective ends of the column. The only disturbance in the
reservoir comes from the interfaces where concentration gradients are
built up due to TGTD in the column. We expect that the factor A2 will be
a very complicated quantity dependent on the dimensions of reservoirs as
well on the properties of electrolyte solutions. One possibility is

that

QQ(AT)2D* *
12 - C(L,a) " 0 v ( 9 - 2 )

R 3

where C(L,a) is a constant which depends on the dimensions of the column

151

and VR is the volume of the reservoirs,(Tyrrell [1962]), Naokata and

Kimie [1984]). On the other hand, the zeroth order average flux in the

column is from chapter 7,

2* -KL - a¢o _
Jz - 8a”: ( H¢°' E 5; ' ( 9 - 3 )

On the boundaries, or at the immerfaces, these two fluxes must be the

same, thus at z - O, L,

* -KL 8%,

-——8:2e 1118,- r 5; -jz. < 9 - 4)

A2t

j:-Keg(L-2z)(1-e' )

An alternative way of looking at this problem is that the net flux in

the column is

jN-JZ-jz’ (9'5)

where the additional term is due to the reservoir effect. After

subtracting this effect we have the modified flux jN’ whic is still zero

Combining Eq. (7-48) with (9-4), we have the following differential

equation with proper boundary and initial conditions.

152

 

 

 

 

a¢o 82 o ‘2

- eA DV t LK
—- K¢o- ch(l - e T )e _
63° K8 a2 K 0(1 -S2DvTc) LK
_ - 0+ C2 " e e = ,
Bz 8D*E z L

*

- _ 2
80- xzcg eKL, at r-o, A2-C(L,a)2flé§1—2 , ( 9 - 6 )

where VT and VB are respectively the volume of top and bottom

reservoirs. The above equation reduces to Eq. (7-48) as V approaches to

R
zero. Solution of Eqs.(9-6) will be not easy because of time dependent
boundary conditions and the actual form of A2 must be given. It is hoped
that this can be done to get the concentration distribution in the TGTD
column with two reservoirs.

In chapter 8, the concentration distribution in the reservoir is
given, and a simple transient working equation is derived to estimate
the Soret coefficients. Because of the difficulty of evaluating the
higher roots of Eq. (8-18), the first term is used to accomplish the
transient working equation. Numerical work is needed to give some higher
order roots for Eq. (8-18), thus a more accurate working equation could
be given by counting more terms of the infinite summations. From the
point of view of mathematics, the best way is to work out the asymptotic

forms of following sums such as

an 2 Q
E 31321 -a Dt } 35331 -o20c
e n , e n
2 2- 2
n—l an n=1 (a e )

 

 

 

A P P E N D I X A

SOLUTION OF PARTIAL DIFFERENTIAL EQUATION ( 8 - 13 )

 

 

Qflj a_ a_R _ _ «it: _
6t-r 8r[rar] ’ R(r-r2)—l, 8r r=r3—O

R(t=O)=O . . ( A - 1 )
If

R(r,t)=<1>(r.t)+\/(r), ( A - 2 )
then Eqs.(A-l) becomes

@1112 6_ a_d_> _ __ as _

6t_r 6r[rar] ’ ¢(r—r2)—O, 8r r=r3—O

<I>(t=0)=-V(r), ( A - 3 )
and

Q. 9! _ _ QM _ -

dr[rdr]-—0, V(r2)—1, dr rB—O . ( A 4 )
The solution of Eqs.(A-4) is

V(r)=1 ( A - 5 )

153

154

We solve Eqs.(A-l) by the method of separation of variables. Writing

¢(r.t)-T(t)¢(r). ( A - 6 )
we have

Q. d 22

dr[ dr]+a2¢-ov ¢(r2)=1, dr 1:330, ( A ' 7 )
and

%%--a2DT, ( A ' 8 )

with a2 an arbitrary separation constant. The solution of Eq.(A-8) is

T(r)-ae'°‘29t , ( A - 9 )

where a is an integration constant. To solve Eqs.(A-7), we make the

independent variable transform
r-fix. ( A - 10 )
This gives, by Eqs.(A-7 and 10),

If afl-l, then

r-x/a ( A - 12 )

155

Eq.(A-ll) is a standard form of Bessel's differential equation of the

zeroth order if Eq.(A-12) is satisfied. One of the the solution is

Jo(ar2)
¢o(r)-AJo(ar)+BYo(ar), B--§;?;;;7A ,
¢5(ar3)-J5(ar3)Yo(ar2)-Jo(ar2)Y5(ar3)=0, ( A - l3 )
or _
¢1(ar3)-J1(ar3)Yo(ar2)-Jo(ar2)Y1(ar3)-O, ( A - 14 )

where we have made use of Eqs.(A-7), and J0, J1, Y0, Y1, J5, Y5 are

respectively the zeroth and first order Bessel's function of the first
and second kind and their derivatives. Furthermore, the constant a must
be the root of the second of Eq.(A-l3) or of Eq.(A-14). Because there
are infinitely many positive nondegenerate roots (Bogert,[1951]), we

rewrite the first of Eqs.(A-13)

¢o(anr)-A:[Jo(anr)Yo(anrz)-Jo(anr2)Yo(anr)]

n-l, 2, 3, . . ., A*-A/Yo(onr,)-An , ( A - 15 )

where an satisfies Eq.(A-14). The general solution of Eqs.(A-7) is the

infinite sum of Eqs.(A-lS)

Q

¢- E An¢o(anr) . ( A --16 )

n-l

Now, combining Eqs.(A-3,4,5,6,9, and 16), we have

156

Q

_ath
¢(r,t)- E An¢o(anr)e
n-l
¢o(anr)-Jo(anr)Yo(anr2)-Jo(anr2)Yo(anr) . ( A - l7 )
¢(r,t-0)- E An¢o(anr)--l, ( A - 18 )
n-l

where we have redefined ¢o(anr) and An is a new constant.

For Eq.(A-18) to be true, we must expand the constant 1 in terms of

¢o(anr), and An must be the nth coefficient of the expansion. To this

end we must compute the required integrals since they do not appear in

the literature. Assuming the two differential equations

d2¢o(anr) d¢o(a r)
2

drz +. dr“ +agr2¢o(anr>-0. ( A - 19 )

 

r

2d2¢o(amr) d¢o(amr)

+r-——————-+a;r2¢o(amr)-O, ( A ' 20 )

r dr2 dr

n-1,2,---, m-1,2,---,

where an and am are any two roots of Eq.(A-lA), we multiply Eq.(A-l9) by
¢o(amr), and Eq.(A-ZO) by ¢o(anr), then subtract Eq.(A-ZO) from Eq.(A-

19), and finally integrate the result from r2 to r3 to obtain

r3
(cg-a;)Ir r¢o(anr)¢o(amr)dr
2

r3
—r[am¢o(anr)¢6(amr)-an¢o(amr)¢6(aar)]r . ( A - 21 )
2

157

Here, ¢5 is the derivative with respect to the argument anr or amr, not

just r. For min, Eq.(A-Zl) reduces to

r,
(cg-a;)Ir r¢o(anr)¢o(amr)dr
2

-r3 [am¢o(anr3)¢6(amr3)]-r2[-an¢o(amr2)¢6(al'1r2):l, ( A - 22 )

where we have applied Eqs.(A-l3,14 and 17). Because ¢o(amr2) and

¢6(amr3) are also zero by Eqs.(A-l3,14 and 17), we have

r3
I r¢o(anr)¢o(amr)dr=0 , for am# an. ( A - 23 )
r2

For am- an, the situation is complicated and we give only an outline

of the proof. We rewrite Eq.(A-21) as

r3
I r¢o(anr)¢o(amr)dr
r2
r r3
-___[am¢o(anr)¢6(amr)-an¢o(amr)¢6(ar'lr)] . ( A - 24 )
ag-a; r2

When anéam the right hand side of Eq.(A-24) requires application of

L'Hospital's rule, which yields after some manipulations,

158
r
I 3r¢3(anr)dr
r2
2 2
.;3[[¢6(anr3)]2+¢g(anr3)]-;fl[[¢6(anr2)]2+¢g(anr2)]. ( A - 25 )

To obtain Eq.(A-25), we have used Eqs.(A-l3,14 and 17 and 19), the

Wronskian (Abramowits and Stegun [1970])

 

WIJV(2>.YV<2)1-JV+1<2)YV(2)-Jy<z>YV+1<z)- "22 . < A - 26 >

and the identities

Jy+1(z)-§Jy(z)-J;(z) , Yy+1(z)-:Yu(z)-Y;(z). ( A - 27 )

Then

2

1rar
n2

 

¢6(anr3)-’¢1(anr3)'o v ¢6<anr2)"¢1(anr2)" ( A ' 28 )

Using these relations, we finally derive from Eqs.(A-25 and 23)

2 ' [J3(anr2) ]
r -———— ——————— - , n=m
I 3r¢o(°nr)¢o(amr)dr (mm)2 JRant?)

r
2 0, nfim

( A - 29 )

Eq.(A-29) is very important, and we will repeatedly apply it.

Having obtained Eq.(A-29), we now expand Eq.(A-l8). Because r¢o(anr)
forms a complete set of orthogonal functions with different an, we

multiply Eq.(A-18) by r¢o(amr) and then integrate from r2 to r3 to give

159

rs
I r¢o(anr)dr
r2 (wan)2 J3(anr2)

-1 r3

An-- r3 = 2 [J§(a r2)-1] I r¢o(anr)dr. ( A - 3o )
n r
I r¢§(a r)dr , 2

 

In order to evaluate the integral in Eq.(A-30), we derive a general form

for this type of integral.

1
ap+2
n

 

Irp+1¢o(anr)dr- Jzu+l¢o(z)dz, z=anr ( A - 31 )

with p a constant. The following two relations are needed for evaluating

Eq.(A-3l) (Tranter, [1968]),

Iz”+1Jy(z)dz-z“+lJV+1(z)+(p-u)z“JV(z)-(y2-u2)Iz“+1Jv(z)dz
sz+1YV(z)dz-zp+1Yu+1(z)+(p-u)szV(z)

~(p2-v2)Izp+lYV(z)dz ( A ~ 32 )
We substitute Eqs.(A-32) into Eq.(A-31) and use Eqs.(A-l7) to derive

Irfl+1¢o(anr)dr=a;("+2)[zp+l¢l(2)+pzfl+l¢o(2)

-p2Iz”'1¢o(z)dz] . ( A - 33 )

We note that Eq.(A-33) is exactly the same as Eqs.(A-32) in form with

v-O, and ¢u is replaced by either JV or YV. We immediately derive from

160
Eq.(A-33) for p-O

r3 2
J r¢o(anr)dr--;;; , ( A - 34 )
r n

where Eqs.(A-14 and 26) are used. The integral in Eq.(A-30) now is

eliminated by Eq.(A-34), and An is then,

J¥(anr3)
An/fl- Jg(anr2)-J§(anr3) ' ( A - 35 )

 

The solution of Eqs.(A-l) is, by Eqs.(A-2,5,9,l7 and 34),

” -a20c J§(anr,)

n
R-1+w§ a(an)¢o(anr)e , 8(an)-J3(anr2)-J¥(anr3)'

 

( A - 36 )

n-l

161

A P P E N D I X B

SOLUTION OF PARTIAL DIFFERENTIAL EQUATION (8-6)

The solution of Eqs.(8-6) is obtained by differentiating the square

bracket product in Eq.(8-21),

c2 ” t a:D(A-t)
Eg-l-«D E a:a(an)¢o(anr) e g(A)dA
n-l 0
” ” t Em;D(A-t)
+4D E E a(an)¢o(anr)0&z)Em: e g(A)dA
m-ln-l
” t d2D(A-t)
+50 dzfl (z) e m (A)dA E 2-a2+d2
n m m ‘ g ’ nm n m'
m-l O
-92DA
g(A)-7(1-e )

For the g(A) given, Eq.(B-l) becomes

( B - 1 )

 

162

 

” 2 _ 2 m 2
co 1 ana(an)¢o(anr) anDt -92Dt 5 dm amaz)
n-l .n n-l m

E 2 _ 92

_ 2 w m 2 _ 2
e e +4 e -e
mn

°° m -Em§Dt 4 °° 4;th
+4 E E a(an)¢o(anr)0&z)[l-e ]+; E 0m(z)[1-e ]
m-ln-l m=l
m -a;Dt
-« E a(an)¢o(anr)[l-e ] . _( B - 2 )
n-l

Eq.(B-2) contains many infinite sums, and is difficult to apply in

practice. We are able to perform some of the sums. Consider first the

summation
@
d2 0 82)
Y(z)- E 5%—795; . ( B - 3 )
m
n-l

Using Eqs.(8-20) to eliminate Om(z) and d2m , we derive

m
(2m+1) i [(2 1) I __ 82(2h22
Y(z)' E (2m+:)? _ g: x v X'gfig {23 «2 ° ( B ' A )
n-O

To sum Eqs.(B-h), we use the two relations (Gradshteyn and Ryzhik,

[1980])

163

i ki_ll_§inikzl _Eéifll£12m£;§ll

k2 - :2 25in(§w) ’ ( B ‘ 5 )
n-l
kgigik§1_ «sin 2m+1 n -x _
E k2-§2 25in(§n) ' ( B 6 )
n-l

If we subtract Eq.(B-6) from Eq.(B-S), then

i____l_l_ii____l_l= ________
E 2k:%k:1?2 ?k;; X =Asin(§W ){sin[ (2m+1)n§- -x§]- -sin[§(2m«- x)]},

O<x<«, m-O, i1, :2, ~--. ( B - 7 )

Taking m-O, Eqs.(B-7) reduce to

m

E 12k:%fi:in£(Bk;l)x1_asi:(gw){sin[(n-x)§]-sin(§x)}, ( B - 8 )

n-0

and comparing Eq.(B-S) with Eqs.(B-B and 4), we obtain

man z)- zsin[(2h- z)91- -sin 92 ( B _ 9 )
d2 - 92- sin(2h6) '
n-l

Integrating both sides of Eq.(B-9) from O to h for 2, we obtain

E di%954 = -3- t8(h9) ( B ' 10 )
m

164

It is also possible to perform the infinite sum

° a2a<a >¢o<a r)
X(r)-x E n a2n- 92 n
n-l n

 

To evaluate this summation, we define

Do(9r)-Jo(6r)Yo(er3)-J0(er3)Yo(6r)
¢o(er)-Jo(9r)Yo(er2)-Jo(er2)Yo(er)

¢1(9r)-J1(9r)Y0(9r2)-Jo(er2)Y1(er).

The following relationships are satisfied,

D1(9r)--D6(6r), Do(9r2)--¢(9r3), ¢1(Gr)=-¢6(8r).

( B - 11 )
( B -12 )
( B - 13 )

The trick here is to expand ¢o(er) in terms of orthogonal functions

¢o(anr). We write

¢o(6r)-§ Cn¢o(anr).

Then

r,
I r¢o(anr)¢o(9r)dr

r2

 

C-

n r3
I r¢3(anr)dr
1'2

and because ( Luke, [1962])

( B - 14 )

( B - 15 )

165

 

r3 r39¢o(anr3)¢1(9r3)
I r¢o(anr)¢o(9r)dr-- 02 _ 92 , ( B - 16 )
r2 n

we obtain

 

C r39¢o(anr3)¢l(9r3)

n a2 - 92
n

(flan)2a(an). ( B - 17 )

where we have used Eq.(A-29). By Eq.(B-lh),

 

 

e¢o(anr3)¢1(anr) 2 2¢0(9r)
E a; - 62 (flan) a(an)-.r39¢1(0r3) ( B - 18 )
The next step is to expand Do(9r) in terms of ¢o(anr),
Do(9r)-§ Bn¢o(anr). ( B - l9 )

As before, we obtain for Bn

(1mn )2a(an
Bn-- 2n(a2- 92)[r39¢0(anr3)D1(er3)+r2an¢1(anr2)Do(er)].( B - 20 )

 

Combining Eqs.(B-20,19 and 18), we finally obtain for Eq(B-ll)

 

 

S ana(an )¢o(an r) ¢0(9r)D1(9r3)-D0(9r)¢1(9r3)

a2 - 92 ’ Do<er2)¢1<er3> '
n-l

r2<r<r3. ( B - 21 )

166

As 640, we can show that the limiting form of Eq.(B-21) is
a} a(an)¢o(anr)--l, ( B - 22 )

and Eq.(B-22) is just Eq.(A-l8), the initial condition of the partial

differential equation(A-B).

If we multiply both sides of Eq.(B-22) by r, then integrate from r2 to

r3, the result is

___4___
E a(an) r2 ¢1(6r3)D1(9r2)- n2r2r392

 

 

a: - e2 "29 Do(er2)¢,(er3) ' ( B ' 23 )
n-l

where we have used Eqs.(A-26,3l,32, 33, and 34). At G-O, Eq.(B-23)

reduces to a very simple form. To show this, we rewrite Eq.(B-23)

 

 

 

___‘_+___.
m a(a ) r2 ¢1(6r3)D1(9r2)62- n’rar2
“ ---; ( B - 24 )
a2 - 92 e 200(6r2)¢1(er3)
n-l n
Then
” a(a ) r39¢o(9r3)D1(9r2)+9r2¢1(9r3)Do(9r2)
E a2 '538 'r2 aeno(er2)¢,(er3)
n-l n
-(rg-rg)/a . ( B - 25 >

To get the last line of Eq.(B-25), we have successively employed
L'Hospital's rule. A simpler way to show this is to multiply Eq.(B-22)

by r, then integrate from r2 to r3 to lead to Eq.(B-25).

l67

BIBLIOGRAPHY

Abramowitz, M., and Stegun, I. A., "Handbook of Mathematical Functions"
Dover Publications Inc., N. Y. (1970).

Agar, J.N., Trans. Faraday Soc. 56, 776(1960).

Alexander, H.F., Z. Physik. Chem. 203, 212(1954).

Bardeen, J., Phys. Rev. 57, 35(1940).

Bartelt, J.L., and Horne, F.H., J. Chem. Phys. 51, 210(1969)

Bartelt, J.L., and Horne, F.H., Pure and Applied Chem. 22, 349(1970).

Bateman, T., "Tables of Integral Transforms" Mc Graw-Hill Book Company
Inc. (1954).

Batuecas, T., Rev. Real Acad. Cienc. Exactas, Fis. Natur. Madrid 61(3),
563(1967).

Bearman, R.J., Kirkwood, J.G., and Fixman, M., Advances in Chem. Phys.
1, 1(1958), Intersciences Publishers, New York.

Bierlein, J. A., J. Chem. Phys. 23, 10(1955).
Bogert, B.P., J. of Mathematics and Physics 30, 102(1951).

Boyce, W. E., and DiPrima, R. C., "Elementary Differential Equations",
John Wiley and Sons, Inc. (1977).

Calef, D. F., and Deutch, J. M., Annu. Rev. Phys. Chem. 34, 493(1983).
Carr, R., J. Chem. Phys., 12 349(1944).

Carslaw, H. S., and Jaeger, J. C., "Conduction of Heat in Solids",
Oxford University Press, London, England (1959).

Churchill, R. V., Brown, J. W., and Verhey, F. R., "Complex Variables
and Applications, McCraw-Hill, Inc., New York (1974).

Clusius, R., and Dickel, C., Nalurwissenschaften 26, 546(1938).
Clusius, R., and Dickel, C., Nalurwissenschaften 27, 148(1939).
Debye, P., Ann. Physik. [5] 36, 248(1939).

IDebye, P., and Bueche, A.M., "Collected Paper of P. W. J. Debye"
Interscience, New York and London, 443(1954).

de Groot, S.R., "L Effet Soret", North-Holland, Amsterdam (1945).

168

de Groot, S.R., and Mazur, R., "Non-equilibrium Thermodynamics", North:
Holland, Amsterdam (1962).

Eilert, A.Z., Anorg. Chem. 88, 1(1914).

Pitts, D.D., "Nonequilibrium Thermodynamics" McGraw Hill Book Company
Inc., New York(l962)

Fries, P. H., and Patey, G. N., J. Chem. Phys. 80, 6253(1984).
Furry, W.H., Jones, R.C., and Onsager, L., Phys. Rev. 55, 1083(1939).
Gaeta, F.S., and Cursio, N.M., J. Poly. Sci. 7, 1697(1969).

Gaeta, F.S., Perna, C., Scala, G. and Belluccl, F. J. Phys. Chem. 86,
2967(1982).

Gillespie, L.J., and Breck, S., J. Chem. Phys. 9, 370(1941).

Gradshteyn, I. S., and Ryzhik, I. M., "Tables of Integrals, Series, and
Products", Academic Press (1980).

Greene, E., Hoglund, R. L., and Halle, E. V., UNion Carbide Corp. Report
K, 1469(1966).

Grew, K.E., and Ibbs, T.L., "Thermal Diffusion in Gsaes", Cambridge
University Press (1952).

Guthrie, C., Wilson, J.N., and Schomaker, V., J. Chem. Phys. 17,
310(1949).

Haase, R., "Thermodynamics of Irreversible Process", Addison-Wesley
Publishing Company Inc. (1969).

Harned, H.S., "Phys. Chem. of Electrolyte Solutions", Reinhold
Publishing Co., New York, 88(1958).

Hiby, J.W., and Wirtz, K., Phys. Zeits. 41, 77(1940).
Hirota, ., J. Chem. Soc. Japan 62, 480(1941).
Hirota, ., J. Chem. Soc. Japan 63, 999(1942).
Hirota, ., Bull. Chem. Soc. Japan 16, 475(1941).

Hirota,
811(1943

Matsunaga I. and Tanaka Y., J. Chem. Soc. Japan 64,

K
K
Hirota, K., Bull. Chem. Soc. Japan 16, 232(1941).
K
K
).

Hirota, R., J. Chem. Phys. 18, 396(1950).

Horne, F.H., Ph.D Thesis, The University of Kansas (1962),(available
from University of Michigan, Microfilms, Ann Arbor, MI)

Horne, F.H:, and Bearman, R.J., J. Chem. Phys. 37, 2842(1962).

169

Horne, F.H., and Bearman, R.J., J. Chem. Phys. 45, 3069(1966).
Horne, F.H., and Bearman, R.J., J. Chem. Phys. 46, 4128(1967).
Horne, F.H., and Bearman, R.J., J. Chem. Phys. 49, 2457(1968).
Horne, F.H., and Anderson, T.G., J. Chem. Phys. 53, 2321(1970).
Ingle, S.E., and Horne, F.H., J. Chem. Phys. 59, 5882(1973).

Ivory, C. F., Gobie, W. A., Beckwith, J. B., Hergenrother R.,
and Malec, N., Science 238, 58(1987)

Johson, C., and Beyerlein, A. L., J. Phys. Chem. 82, 1430(1978).
Jones, R.C., and Furry, W.H., Rev. Modern Phys. 18, 151(1946).
Kahana, P. Y., and Lin, J. L., J. Chem. Phys. 74, 2995(1981).

Kestin, J., Sokolov, M., and Wakeham, W.A., J. Phys. Chem. Ref. Data
7(3), 941(1978).

Kincaid, J.M., Cohen, E.G.D., and Lopez de Haro, M., J. Chem. Phys. 86,
963(1987).

Korsching, R., and Wirtz, K., Naturwiss 27, 367(1939).
Ludwig, C., Math. Naturwiss k1.20, 539(1856).

Luke, "Integrals of Bessel Functions", Mc Graw-Book Company Inc.
(1962).

Ma, R., Stanford, D., and Beyerlein, A., J. Phys. Chem. 87, 5461(1983).
Mauzerall, D., and Ballard, S. G., Annu. Rev. Phys. Chem. 33, 377(1982).
Mclaughlin, B., Chem. Rev. 64, 389(1964).

Millero, F.J., J. Phys. Chem. 74, 356(1970),and articles cited.

Naokata, T., and Kimie, N., Bull. Chem. Soc. Japan, 57, 349(1984).
Nagasaka, Y., Okada, M., Phys. Chem. 87, 859(1983).

Navarro,.J.In, Madariaga, J. A., and Saviron, J. M., J. Phys. Soc.
Japn. 52, 478(1983).

Onsager, L., Phys. Rev. 37, 405(1931).
Onsager, L., Phys. Rev. 38, 2265(1931).
Out, D.J.P., and Los, J.M., J. Solution Chem. 9(1), 19(1980).

Petit, C. J., Renner, K.E., and Lin, J.L., J. Phys. Chem. 88,
2435(1984).

Petit, C. J., Hwang, M. B., and Lin, J. L., Int. J. Thermophys. 7,

170

687(1986).
Prigogine, I., de Brouckere, L., and Amand, R., Physica 166, 577(1950).
Rard, J.A., and Miller, D.G., J. Solution Chem. 8(10), 701(1979).

Roberts, (I. B., and Kaufman” H., "Table of Laplace Transforms",
Saunders,Phi1adelphia (1966).

Rowley, R.L., and Horne, F.H., J. Chem. Phys. 72, 131(1980).

Rutherford, W.M., Dougherty, E. L., and Drickamer, H.G., J. Chem. Phys.
22, 1289(1954).

Rutherford, W. M., J. Chem. Phys. 59, 6061(1973).
Rutherford, W. M., and Drickamer, H. G., J. Chem. Phys. 22, 1157(1954).

Santamaria, C. M., Saviron, J. M., and Yarza, J. C., J. Chem. Phys.
3, 1095(1976).

Simard, M.A., and Fortier, J.L., Can. J. Chem. 59, 3208(1981).
Soret, Arch. Sci. Phys. Nat. Geneve 2, 48(1879).

Spiegel, M. R., "Complex Variables", McGraw-Hill Book Company Inc.
N.Y. (1964).

Story, M. J., and Turner, J. C. R., Tans. Faraday Soc. 65, 1523(1969).
Tanner, J.E., and Lamb, F.W., J. Solution Chem. 7(4), 303(1978).

Timmermans, A., "Phys. Chem. Constants of Binary Systems", Interscience
Publisher Inc., New York (1960).

Tranter, C. J., "Bessel Functions with Some Physical Applications",
New York, Hart Pub. Co. (1968).

Turner, J. C. R., Butler, B. D., and Story, M. J., Trans. Faraday Soc.
63, 1906(1967).

Tyrell, H.J.V., "Diffusion and Heat Flow in Liquids", London,
Butterworths(l961).

Watson, G. N., "A Treatise on the Theory of Bessel Functions", 2nd. ed.
Cambrige University Press, Cambridge, England, (1958).

Wereide, T., Ann. Physique 56 67(1914).
Winter, F.R., and Drickamer, H.G., J. Phys. Chem. 59, 1229(1955).

WOlf, A.V., Brown, M.G., and Pentiss, P.G., "Handbook of Chem. and
Phys.", CRC Press 53RD, D-l8l-210(l973).

Wolynes, P. C., Annu. Rev. Phys. Chem. 31, 345(1980).

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