Citation
Corresponding states relationships for transport properties of pure dense fluids

Material Information

Title:
Corresponding states relationships for transport properties of pure dense fluids
Creator:
Tham, Min Jack, 1935- ( Dissertant )
Gubbins, K. E. ( Thesis advisor )
Reed, T. M. ( Reviewer )
Walker, R. D. ( Reviewer )
Blake, R. G. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1968
Language:
English
Physical Description:
xxii, 164 leaves. : illus. ; 28 cm.

Subjects

Subjects / Keywords:
Argon ( jstor )
Correlations ( jstor )
Correspondence principle ( jstor )
Fluids ( jstor )
Gases ( jstor )
Liquids ( jstor )
Molecules ( jstor )
Self diffusion ( jstor )
Transport phenomena ( jstor )
Viscosity ( jstor )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Fluid dynamics ( lcsh )
Fluids ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
An important consideration for the chemical engineer is his ability to predict reliable values for various properties of chemical substances that are needed for design equations. Because of the large number of combinations of substances, compositions, pressures, temperature, etc., that may be encountered, it is often neither feasible nor desirable to make experimental measurements of such properties. The need for reliable correlations has become more acute with the increasing use of high-speed electronic computers in designing chemical plants. Such design requires suitable mathematical expressions for properties as a function of operating conditions. Even when experimental data in tabulated form are available it is most conveniently introduced into the computer in the form of soundly based theoretical equations. In the long term the only satisfactory approach is one that is firmly based on molecular considerations. Theories for dilute fluids are well established. The kinetic theory of gases is capable of describing the thermodynamic and transport properties accurately. However, for dense fluids there is still no theory which can parallel the success of the dilute gas theory. Although there are several rigorous molecular theories for liquids, none have yet been developed to a stage that would yield numerical results. Theoretical study of transport properties is more difficult than that of thermodynamic properties. In thermodynamics, all equilibrium properties can be obtained once the partition function of the system is known. However, for transport processes each of the transport properties has to be formulated separately. This situation occurs because the thermodynamic equilibrium state is unique, whereas there are many types of non-equilibrium state. This dissertation considers corresponding states correlations of transport properties of fluids, particularly liquids. This principle has previously proved of great value to engineers in predicting thermodynamic properties. Chapter 1 provides a brief survey of the present status of theories of transport properties of dense fluids. Chapters 2-4 contain a detailed study of the free volume theory of viscosity and self-diffusion coefficient, with particular emphasis on developing corresponding states relations for the parameters involved. The next two chapters discuss a more direct corresponding states treatment, and a new theory is proposed for polyatomic molecules.
Thesis:
Thesis--University of Florida, 1968.
Bibliography:
Bibliography: leaves 155-163.
General Note:
Manuscript copy.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
030419653 ( AlephBibNum )
16996953 ( OCLC )
AER8377 ( NOTIS )

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Full Text












CORRESPONDING STATES RELATIONSHIPS
FOR TRANSPORT PROPERTIES OF
PURE DENSE FLUIDS










By
MIN JACK THAM













A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY










UNIVERSITY OF FLORIDA
1968










PREFACE


An important consideration for the chemical engineer is his

ability to predict reliable values for various properties of chemical

substances that are needed for design equations. Because of the

large number of combinations of substances, compositions, pressures,

temperature, etc.,that may be encountered, it is often neither

feasible nor desirable to make experimental measurements of such

properties. The need for reliable correlations has become more

acute with the increasing use of high-speed electronic computers in

designing chemical plants. Such design requires suitable mathe-

matical expressions for properties as a function of operating

conditions. Even when experimental data in tabulated form are

available it is most conveniently introduced into the computer in

the form of soundly based theoretical equations.

In the long term the only satisfactory approach is one that

is firmly based on molecular considerations. Theories for dilute

fluids are well established. The kinetic theory of gases is capable

of describing the thermodynamic and transport properties accurately.

However, for dense fluids there is still no theory which can parallel

the success of the dilute gas theory. Although there are several

rigorous molecular theories for liquids, none have yet been developed

to a stage that would yield numerical results.

Theoretical study of transport properties is more difficult

than that of thermodynamic properties. In thermodynamics, all

equilibrium properties can be obtained once the partition function










of the system is known. However, for transport processes each of

the transport properties has to be formulated separately. This

situation occurs because the thermodynamic equilibrium state is

unique, whereas there are many types of non-equilibrium state.

This dissertation considers corresponding states correla-

tions of transport properties of fluids, particularly liquids. This

principle has previously proved of great value to engineers in

predicting thermodynamic properties. Chapter 1 provides a brief

survey of the present status of theories of transport properties of

dense fluids. Chapters 2-4 contain a detailed study of the free

volume theory of viscosity and self-diffusion coefficient, with

particular emphasis on developing corresponding states relations

for the parameters involved. The next two chapters discuss a more

direct corresponding states treatment, and a new theory is proposed

for polyatomic molecules.

The author is greatly indebted to Dr. K. E. Gubbins, chairman

of his supervisory committee, for his interest, stimulation, advice

and encouragement during the course of this research. He is also

grateful to Dr. T. M. Reed for helpful discussions. Finally,he

expresses his sincere appreciation to Dr. T. M. Reed, Professor

R. D. Walker and Dr. R. G. Blake for serving on the committee.










TABLE OF CONTENTS


PREFACE.........................................................

LIST OF TABLES.................................................

LIST OF FIGURES............................................... ..

LIST OF SYMBOLS................................................

ABSTRACT........................................................

CHAPTERS:

1. INTRODUCTION.........................................

1.1 The Time Correlation Function Theory.............

1.2 The Kinetic Theory of Liquids...................

1.3 The Enskog Theory................................

1.4 Activation and Free Volume Theories..............

1.5 The Correspondence Principle.....................

1.6 Empirical Correlations ...........................

1.7 Summary..........................................

2. PREVIOUS WORK ON ACTIVATION AND FREE VOLUME THEORIES.

3. IMPROVED FREE VOLUME THEORY ..........................

3.1 Temperature Dependence of V ....................

3.2 Volume Dependence of E .........................

4. TEST OF IMPROVED FREE VOLUME THEORY ..................

4.1 Corresponding States Relationships..............

4.2 Test of Proposed Correlations ...................

4.3 Summary .........................................


Pa ge

ii

vii

ix

xii

xx



1

1

3

8

14

16

16

20

22

27

28

35

41

42

44

52










TABLE OF CONTENTS (Continued)


Page

5. PRINCIPLE OF CORRESPONDING STATES FOR MONATOMIC
FLUIDS............................................... 55

5.1 Molecular Basis of the Correspondence Principle. 55

5.2 Previous Work.................................... 63

5.3 Test of Correspondence Principle for Inert Gases 65

5.4 Summary......................................... 84

6. PRINCIPLE OF CORRESPONDING STATES FOR POLYATOMIC NON-
POLAR FLUIDS.. ....... ............................... 86

6.1 Problems in Polyatomic Fluids................... 86

6.2 Derivation of Corresponding States Principle for
Polyatomic Molecules.............................. 90

6.3 Test of Correspondence Principle for Polyatomic
Molecules.............................................. 103

6.4 Comparison with Theory for Thermodynamic Pro-
perties......................................... 124

7. CONCLUSIONS.......................................... 132

7.1 Free Volume Theory............................... 133

7.1.1 Extension to Mixtures.................... 133

7.1.2 Electrolyte Solutions..................... 133

7.2 Corresponding States Principle.................. 134

7.2.1 Mixtures................................. 134

7.2.2 Polar Substances......................... 134

7.2.3 Fused Salts.............................. 134

7.2.4 Thermodynamic Properties................. 134

7.2.5 Prediction of c.......................... 135









TABLE OF CONTENTS (Continued)


Page


APPENDICES..................................................... 136

1. Chung's Derivation of the Free Volume Equations...... 137

2. Further Test of Proposed Free Volume Theory........... 144

3. Solution of Equations of Motion for Oscillator....... 148

4. Hamiltonian for the Three-Dimensional Oscillator..... 151

LITERATURE CITED...................... ...... .... ........ ......... 155

BIOGRAPHICAL SKETCH .................... ..... ................... 164










LIST OF TABLES


Table Page
1.1 Test of Rice-Allnatt Theory for Shear Viscosity
of Argon................................. ............. 6

1.2 Test of Rice-Allnatt Theory for Thermal Conductivity
of Argon.............................................. 7

4.1 Viscosity Data Sources and Range of Conditions....... 46

4.2 Parameters for Viscosity Prediction.................. 47

4.3 Self-Diffusivity Data Sources and Range of Conditions 48

4.4 Parameters for Self-Diffusivity Prediction........... 49

5.1 List of Parameters of Monatomic Molecules............. 78

5.2 Coefficients of Saturated Liquid Viscosity Equation
for Monatomic Molecules .............................. 80

5.3 Coefficients of Saturated Liquid Thermal Conducti-
vity Equation for Monatomic Molecules ................ 81

5.4 Coefficients of Saturated Liquid Self-Diffusivity
Equation for Monatomic Molecules..................... 82

5.5 Coefficients of High Pressure Viscosity Equations
for Monatomic Molecules................................ 83

5.6 Coefficients of High Pressure Thermal Conductivity
Equations for Monatomic Molecules .................... 85

6.1 Parameters for Correspondence Correlation of Poly-
atomic Molecules. ................................... 105

6.2 Sources of Transport Property Data.................... 107

6.3 Coefficients of Saturated Liquid Viscosity Equation
for Polyatomic Molecules............................... 117

6.4 Coefficients of Saturated Liquid Thermal Conductivity
Equation for Polyatomic Molecules.................... 119

6.5 Coefficients of Saturated Liquid Self-Diffusivity
Equation for Polyatomic Molecules.................... 120

6.6 Coefficients of High Pressure Viscosity Equations
for Polyatomic Molecules............................. 122










LIST OF TABLES (Continued)


Table Page

6.7 Coefficients of Gas Thermal Conductivity Equation
for Polyatomic Molecules .............................. 125

6.8 Coefficients of High Pressure Thermal Conductivity
Equations for Polyatomic Molecules.................... 126

6.9 Comparison of c with Corresponding Parameter of
Hermsen and Prausnitz................................. 127


viii











LIST OF FIGURES


Figure Page
1.1 Theoretical and Experimental Saturated Liquid
Viscosity of Argon Square-Well Model............... 9

1.2 Theoretical and Experimental Saturated Liquid Self-
Diffusion Coefficient of Argon Square-Well Model... 10

1.3 Theoretical and Experimental Viscosity of Argon -
Enskog Theory........................................ 13

1.4 Theoretical and Experimental Viscosity of Argon -
Modified Enskog Theory ............................... 15

1.5 Theoretical and Experimental Thermal Conductivity -
Horrocks and McLaughlin Theory....................... 17

1.6 Viscosity Isotherms of Krypton as Functions of
Density............................................... 19

3.1 Viscosity of Argon.................................... 29

3.2 Viscosity of Nitrogen................................. 30

3.3 Variation of v* with T* .............................. 33

3.4 Activation Energy as a Function of Volume............ 36

3.5 Model of a Molecular Jump............................ 37

3.6 Corresponding States Correlation of Activation
Energy vs. Volume ....................................... 39

4.1 Test of Free Volume Theory for Liquid Argon Viscosity 43

4.2 Test of Free Volume Theory for Liquid Xenon Viscosity 50

4.3 Test of Free Volume Theory for Saturated Liquid
Methane Viscosity........................ ............... 51

4.4 Test of Free Volume Theory for Saturated Liquid
Methane Self-Diffusivity ............................. 53

5.1 Smoothing Lennard-Jones [6,12] Parameters,-- vs. cu.. 68
kT
c
P 1/3
5.2 Smoothing Lennard-Jones [6,12] Parameters,(--
vs. C ......................................Tc./ ..... 69

5.3 Correspondence Principle for Saturated Liquid Vis-
cosity Monatomic Molecules ......................... 70










LIST OF FIGURES (Continued)


Figure Page

5.4 Correspondence Principle for Saturated Liquid Thermal
Conductivity Monatomic Molecules ..................... 71

5.5 Correspondence Principle for Saturated Liquid Self-
Diffusivity Monatomic Molecules.................... 72

5.6 Reduced Bulk Viscosity of Saturated Liquid Argon........ 74

5.7 Reduced Viscosity Isobars as Functions of Reduced
Temperature Monatomic Molecules...................... 76

5.8 Reduced Thermal Conductivity'Isobars as Functions of
Reduced Temperature Monatomic Molecules............... 77

6.1 Simple Correspondence Principle Polyatomic Molecules. 89

6.2 Simple Corresponderm Principle for Saturated Liquid
Viscosity Polyatomic Molecules....................... 91

6.3 A One-Dimensional Oscillator Model..................... 93

6.4 Superimposed Rotational Motions of a Molecule.......... 94

6.5 Improved Correspondence Principle for Saturated Liquid
Viscosity Polyatomic Molecules.......................110

6.6 Improved Correspondence Principle for Saturated Liquid
Thermal Conductivity Polyatomic Molecules............ 111

6.7 Improved Correspondence Principle for Saturated Liquid
Self-Diffusivity Polyatomic Molecules ................112

6.8 Test of Improved Correspondence Principle for High
Pressure Viscosity......................................113

6.9 Test of Improved Correspondence Principle for Gas
Thermal Conductivity Polyatomic Molecules............ 115

6.10 Test of Improved Correspondence Principle for High
Pressure Thermal Conductivity..........................116

6.11 Simple Correspondence Principle Vapor Pressure....... 130

6.12 Improved Correspondence Principle Vapor Pressure.....131

A2.1 Test of Free Volume Theory for Liquid Nitrogen
Viscosity...............................................145

x









LIST OF FIGURES (Continued)


Figure Page

A2.2 Test of Free Volume Theory for Liquid Krypton
Viscosity............................................ 146

A2.3 Test of Free Volume Theory for Liquid Neon Viscosity. 147

A4.1 A Three-Dimensional Oscillator Model................. 151













A


A
o

AiA2,A3 ,A
A23


A' A"

al'a2'a3,a ,a5

a

B


B
o

o
B"
B1,B2 ,B3 ,B4

B' B"

B
a

b
o






C1
a

c



c
0

c 1

c2

c3

c4


LIST OF SYMBOLS

= pre-exponential factor of free volume equation
for viscosity

= A/T1/2

= arbitrary constants

= constants

= constants

= coefficients of viscosity equation

= molecular diameter

= pre-exponential factor of free volume equation for
self-diffusivity

= B/T1/2

1/2 1/2
= (B 12)/(ok )

= arbitrary constants

= constants

= arbitrary constant

= the rigid sphere second virial coefficient

= coefficients of thermal conductivity equation

= empirical constant

= arbitrary constant

= a characteristic factor defined by equation
(6.37)

= total concentration

3
= v/r = constant

= (r2/rl ) = constant

= (c 03/v-.) = constant
1 m
= 6/kT- = constant
m











D = self-diffusion coefficient

DAB = binary diffusion coefficient
'. 1/2 1/2
D" = Dm l2/GI = reduced self-diffusivity by simple
correspondence principle
'1/2 1/2
D = Dm /co- = reduced self-diffusivity by proposed
correspondence principle

DAB = reduced mutual diffusion coefficient defined by
equation (5.33)

d = distance of closest approach of 2 molecules

dl,d2 = coefficients of self-diffusivity equation

= average total energy of molecule i
1

E = activation energy at constant volume
v

E = E /RT = reduced activation energy
v v m

e = varying activation energy per molecule

= average activation energy per molecule

e. = energy level

e = height of potential barrier in equilibrium liquid

e = E /N
v v
F. = y-component of intermolecular force on molecule i
ly
() time smoothed singlet distribution function
f = time smoothed singlet distribution function

-(2)
f = time smoothed pair distribution function

f = a function of density given by equation (1.23)

g = universal function

go(r) = equilibrium radial distribution function

g = geometric factor

H = Hamiltonian


xiii










HCM = Hamiltonian of center of mass coordinates

h = Planck constant

h = h/(oam) = reduced Planck constant

I = moment of inertia

I = viscosity constant used in equation (1.22)

Jxx = defined by equation (1.6)

J = defined by equation (1.5)

K = kinetic energy

KCM = kinetic energy of center of mass coordinates

Krot = kinetic energy of rotational motion

K1 = (n/n-6)(n/6)6/(n-6)

k = Boltzmann constant

k11 = defined by equation (6.8)

k12 = defined by equation (6.9)

k22 = defined by equation (6.10)

L = Lagrangian

1 = jump distance

M = molecular weight

MA = molecular weight of component A

MB = molecular weight of component B

m = mass of molecule

N = number of molecules

N = Avogadro number

n = repulsive exponent in [6,n] potential law

n1 = singlet number density


xiv











P = pressure

P = critical pressure
c

P = critical pressure of a mixture A-B
cAB
3
P- = Pa /c = reduced pressure by simple correspondence
principle

= Pa /ce = reduced pressure by improved correspondence
principle

p = momentum

ix = momentum in x-direction of molecule i

(i)
p = probability of a molecule having activation energy
Se. and free volume v
1 p
1/2
p = p/(m) = reduced momentum

Q = partition function

Q = defined by equation (5.35)

q = thermodynamic free volume

r = position vector

r.. = r r.
--Lj -j --

r = r/C = reduced distance

(s/c) = a constant factor used by Hermsen and Prausnitz
k (166)

s = defined by equation (1.7)

T = temperature

T = critical temperature
c

T = critical temperature of a mixture A-B
cAB

T = melting temperature
m

T = kT/E = reduced temperature by simple correspondence
principle


xv













T



t

t


U(r)


U(O,O)


U(x,Yx)
x

U(x1....xN)

U(0)


V

V

Vf

V
x

V"


Vk


V






0
V

o
1
V
o
v1*
V
o


=kT/ce = reduced temperature by improved correspondence
principle

= kT/[e(s/c)k] = reduced temperature used by Hermsen

and Prausnitz (166)

= time

1/2 1/2
= tE /a m= dimensionless time

N

S u(r i) = pair potential

iij
i
= potential energy of.a molecule at the center of a
cell

= potential energy of a molecule in a cell

= potential energy of N molecules

= potential energy of N molecules at the centers of
their cells

= molal volume

= velocity vector

= molal free volume

= velocity in the x-direction

= (V/3 ) = reduced molal volume

= characteristic volume used by Hermsen and Prausnitz
(166)

= V/V
m

= V/V = reduced volume of Hermsen and Prausnitz
(166)

= minimum free volume per mole needed for molecular
flow to occur

= V /o = reduced minimum free volume
o

= minimum free volume when T* = 1.0

= V /V = reduced minimum free volume when T = 1.0
o m










v = V/N = average volume per molecule

vf = v v = free volume
f o
v = V /N
m m
v = V /N

v = varying free volume

v0 = (d/o)3

v =V /N = minimum free volume per molecule needed
o o
for molecule flow to occur

WIW2 = number of configurations for distribution of molecules

x. = x-component of position vector r of molecule i

Y = average percent error(see page 80)

y = b/V

Z = configurational partition function

z = number of nearest neighbors in a ring


a = Lagrange multiplier

= 1/kT = Lagrange multiplier

7 = free volume overlap factor

= characteristic energy parameter in [6,n] potential

= molecular friction constant

S= shear viscosity

S= dilute gas viscosity
*o

" = 2/(me)/2 = reduced shear viscosity by simple
correspondence principle
2 1/2
r = =r /c(me) reduced shear viscosity by improved
correspondence principle

o = defined by equation (5.36)

S= angle vector (in terms of Eulerian angles e,0,/)


xvii










e = x-component of an angular rotation
x

= bulk viscosity

K= Kc2/(m)/2 = reduced bulk viscosity by simple
correspondence principle
K2 1/2
C = IKc /c(rn) = reduced bulk viscosity by improved
correspondence principle

A = Lagrange multiplier

S= thermal conductivity

X = dilute gas thermal conductivity
o
2 1/2 1/2
S= XG m /kE = reduced thermal conductivity by
simple correspondence principle
2 1/2 1/2
S= X m /cke = reduced thermal conductivity by
improved correspondence principle

= B'/A' = constant

= constant

T = a constant which has a value of 22/7

p = density

(pD) = value of the product of density and self-diffusivity
of a dilute gas

p" = reduced density

a = characteristic distance parameter in [6,n.] potential

Gl ,2 = characteristic distance parameter in square-well
potential

0 = a universal potential energy function

0x = angle associated with constant angular velocity

= a universal function

X = a factor defined by equation (1.13)

Xh = probability of hole formation


xviii











x = angle associated with superimposed angular velocity
x

lo I,2 = the radial functions in the spherical harmonic
expansion of the nonequilibrium distribution
function g(r)

(1, 1)- (2,2)*
S (2 *= dimensionless collision integrals

o = Pitzer factor

O = angular velocity

O, = angular velocity in a harmonic motion


xix








Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy


CORRESPONDING STATES RELATIONSHIPS FOR TRANSPORT
PROPERTIES OF PURE DENSE FLUIDS

By

Min Jack Tham

June, 1968


Chairman: K. E. Gubbins

Major Department: Chemical Engineering


Two approaches have been considered for developing corres-

ponding states correlations for dense fluid transport properties.

The first is based on the free volume theory for viscosity and self-

diffusion coefficient. This theory has been improved to account

quantitatively for the nonlinearity of constant volume plots of

In(r/T /2) vs. T-1 and In(D/T /2) vs. T-1 for simple liquids, by the

introduction of the temperature dependence of V and the volume

dependence of E Corresponding states relations are presented for

these two quantities. The improved free volume theory accurately

describes the viscosity and self-diffusion coefficient behavior

of liquids composed of simple, nonpolar molecules that may be expected

to approximately obey a Lennard-Jones [6,121 potential law, for the

density range of p > 2p c. It is in this range that most other

theories fail. The only parameters required for the viscosity and

self-diffusion coefficient prediction are melting and critical para-

meters, together with one experimental value of viscosity (or self-

diffusivity).










Direct corresponding states correlations of transport pro-

perties have also been studied, and provide a more general if slightly

less accurate method of prediction. The simple corresponding states

principle derived by Helfand and Rice from the time correlation func-

tion expressions of statistical mechanics has been shown to apply

closely for the transport properties of monatomic molecules over the

entire range of temperature and density conditions for which measure-

ments are available. In this simple form, however, it was found not

to apply to polyatomic fluids in the dense phase. After careful re-

examination of the four assumptions on which the derivation of the

simple correspondence principle is based, it is shown that hindered

rotation at high density is the most probable cause for the deviation

of polyatomic fluids from the correspondence principle.

A correspondence principle for polyatomic molecules has been

derived, taking into account the effects of hindered rotation and

making use of a simple harmonic oscillator model. The proposed

equations have been tested exhaustively for the viscosity, thermal

conductivity and self-diffusion coefficient of a large number of

polyatomic fluids over a wide range of temperature and density condi-

tions. The molecules tested include saturated hydrocarbons from

methane to nonane, spherical molecules such as neopentane and carbon

tetrachloride, flat molecules such as benzene, and simple diatomic

molecules such as nitrogen. The improved correspondence principle

predicts the transport properties of saturated liquids within experi-

mental error. For high pressures the average percentage errors of

the predicted viscosity and thermal conductivity are slightly higher.


xxi










It is also shown that the proposed model can account in a reasonable

way for thermodynamic properties of polyatomic fluids.


xxii










CHAPTER 1

INTRODUCTION


In this chapter a brief survey and evaluation is presented of

the principal theories and empirical correlations that have been pro-

posed to explain dense fluid transport properties. For pure fluids

all transport properties are functions of temperature and density,

namely, ) = T)(T,p), K = c(T,p), D = D(T,p) and X = X(T,p). None of

the present theories provide a satisfactory means of predicting

transport properties over the entire range of temperature and density

conditions for even quite simple fluids. Also assumptions of dubious

validity must be made in most cases in order to be able to make any

comparison with experiment at all. In the following survey the more

rigorous approaches are first described, followed by simple model

theories and ending with the empirical correlations.

1.1 The Time Correlation Function Theory

In the last decade a new theory, called the time correlation

function theory (also known as the fluctuation-dissipation theory)

has been developed to treat transport processes. This theory treats

transport processes in terms of equilibrium ensembles whose properties

are known. In a sense the time correlation function plays a similar

role to the partition function in statistical thermodynamics. In

statistical thermodynamics, all thermodynamic properties of any parti-

cular system can be evaluated if the partition function is known. In

the same manner the transport coefficients can be obtained if the

appropriate time correlation functions can be calculated. However, in









one respect the analogy breaks down. In statistical thermodynamics a

single partition function determines all the thermodynamic properties,

whereas in transport processes different time correlation functions

are needed for different transport processes. A good review of the

time correlation function approach to transport processes is given by

Zwanzig (1).

The time correlation function expressions for the transport

coefficients may be derived by several methods. A simple derivation

involves starting with expressions analogous to the Einstein equation

for the self-diffusion of Brownian particle (2). The following

expressions for transport coefficients are obtained.
0O
Self-diffusion coefficient D = dt (1.1)
'0
SO
Shear viscosity 7 = Vk dt (1.2)
o


Bulk viscosity


C = dt
'o


Thermal conductivity X = -12 dt
VkT

where V (t) = velocity in the x-direction at time t
N
Jxy ~. Pixpiv 1
4 1 ly j

xx ix ix
jxx = + xF PV
i im i iX
i=l
N N
sx Pix ix 1>
2m i xx 2
i=l j=i


+ x..F.. 1
1] 1jX J


N
Pix _PV Pi
m Nm Pix
i=1


(1.3)



(1.4)





(1.5)



(1.6)


(ij xx


(1.7)









For dilute gases the time correlation functions may be evaluated and

the expressions for the transport coefficients can be reduced to the

forms obtained by solving the Boltzmann equation. However, for dense

fluids no satisfactory expressions have as yet been obtained, although

several approximate equations have been suggested (3). One approach

has been to assume that the time correlation function decays exponen-

tially. Alternatively one may perform molecular dynamics calculations

(4-6). In this method a very large amount of computation is involved;

the principal interest of such an approach is to provide data with

which to compare various theoretical expressions for the time correla-

tion functions.

The time correlation theory gives a formal description of

transport processes; however it is still not developed to a sufficiently

advanced stage to be of practical value in the calculation of trans-

port coefficients. To the chemical engineer, the most interesting

feature of the theory in its present state of development is that it

provides the basis for a rigorous development of the corresponding

states principle for dense fluid transport properties. This is dis-

cussed in a later section of this chapter.

1.2 The Kinetic Theory of Liquids

The kinetic theory of dilute gases is already very well

developed (7), and provides a method of accurately estimating their

transport properties. It is therefore reasonable to attempt a similar

development for dense gases and liquids. Such an attempt was first

made by Kirkwood (8,9),whose approach was based on casting the









Liouville equation for the distribution function into an equation of

the Fokker-Planck type. The main disadvantage in this theory is

that a large number of approximations have to be made in obtaining the
-(1) -(2)
Fokker-Planck equations for f and f the time smoothed singlet

and pair distribution functions respectively. In addition to these

deficiencies, Kirkwood's method has drawn much criticism for assuming

the mean momentum change for a collision to be small. The equations

derived by the method of Irving and Kirkwood (9) for transport coeffi-

cients of liquids are as follows (10)
Shear viscosity nkT +3 du
n mkT
S+ dr (r) )dr (1.8)
Bulk viscosity 2 0
o____ 3 du(r)
= 9kT r dr go(r)4o (r)dr (1.9)
9kT I dra0 0
Thermal conductivity 2 r
5k nTj n1 kT 1
C + C 3 u(r) u(r)) go(r)

dr ~oTgg() r dr + u(r) r u(r)


0go(r) 2
r dr (1.10)
kT
Self-diffusion coefficient D = (1.11)
where
2 2
= 3 p r 2g(r)7 u(r)dr (1.12)

and o,'2 = the radial functions in the spherical harmonic
expansion of the nonequilibrium distribution
function g(r)


A modification of the Kirkwood theory due to Rice and Allnatt

(11,12) attempts to avoid the assumption that the mean collisional











momentum change is small. In the Rice-Allnatt theory, a potential

energy function is assumed in which the molecule has a hard core

together with an outer shell which interacts with a soft attractive

potential. A hard core collision leads to a large momentum transfer

and tends to vitiate the Brownian motion approximation which is inherited

in the Fokker-Planck equation. These authors therefore treated the

rate of change of the distribution function f due to hard core colli-

sions by means of an Enskog-type collision term, and treated the rate

of change of the distribution function due to motion in the attractive

field of surrounding molecules between collision by Kirkwood's approach.

In order to calculate the transport coefficients from the

Kirkwood or Rice-Allnatt theory one must have accurate values for the

intermolecular potential, and also the equilibrium radial distribution

function has to be known quantitatively. The latter requirement is

especially difficult to meet since the experimental radial distribution

function has not been determined accurately.

Making use of the Kirkwood theory, Zwanzig, Kirkwood, Stripp

and Oppenheim (13) were able to calculate the shear viscosity and bulk

viscosity for liquid argon near the boiling point. Their calculated

value for shear viscosity was O = 0.73x10-3 poise, while the experimen-

tal value of shear viscosity for argon at the same temperature is
-3
71= 2.39x103 poise. Thus the calculated value is in error by roughly

a factor of three. For bulk viscosity, their computation yielded

S= 0.36x103 poise. However, according to Naugle's (14) measurement

of bulk viscosity for liquid argon at approximately the same temperature,








-3
K = 1.7x10 poise. The discrepancy between calculated and experimen-

tal values is more than a factor of four. Because of the non-trivial

mathematics involved, the Kirkwood theory has not been tested for the

temperature and density dependence even for very simple molecules such

as argon.

The Rice-Allnatt theory has been tested by Lowry, Rice and

Gray (15) for viscosity. They found quite good agreement between

calculated and experimental values of argon at densities slightly less

than the normal liquid density (Table 1.1). To compare theory and

experiment they assumed u(r) to be given by the Lennard-Jones [6,12]

potential. The radial distribution functions, g(r) for different

temperatures were those obtained theoretically by Kirkwood, Lewinson

and Alder (16). The radial distribution functions g(r) and the

derivative of u(r) were modified by introducing an empirical parameter

cl, which was obtained from equation of state data.


TABLE 1.1

TEST OF RICE-ALLNATT THEORY FOR
SHEAR VISCOSITY OF ARGON

3-3
T)xlO poise

Density, gcm 1.12 1.375

Temperature, OK 128 133.5 185.5 90

Pressure, atm. 50 100 500 1.3

c 0.9819 0.9827 0.9887 0.9705

r)(calc) 0.727 0.730 0.771 1.74

r(expt) 0.835 0.843 0.869 2.39

% error 13.0 13.4 11.3 27.2










Ikenberry and Rice (17) tested the Rice-Allnatt theory on

the thermal conductivity of argon for the same temperature and density

conditions as the viscosity values shown in Table 1.1. The calculated

values were not entirely theoretical since they made use of experi-

mental friction constants calculated from the self-diffusion coeffi-

cient data of Naghizadeh and Rice (18). Close agreement was found

between theory and experiment for the entire temperature and density

range considered as shown in Table 1.2.


-3
Density, gcm.

Temperature, OK

Pressure, atm.

X(calc)

X(expt)

% error


TABLE 1.2

TEST OF RICE-ALLNATT THEORY FOR
THERMAL CONDUCTIVITY OF ARGON

4 -1 -1
XxlO4, cal.cm. sec.l C

1.12

128 133.5 185.5

50 100 500

2.83 1.85 1.77

2.96 1.89 1.86

4.4 2.1 4.8


Davis, Rice and Sengers (19) further modified the Rice-Allnatt

theory by using the square-well potential energy function. Because of

the simple form of this function the Rice-Allnatt theory is greatly

simplified and the evaluation of the pair correlation function reduces

to the determination of g(cq) and g(2 ). In order to calculate the

transport coefficients from this theory, one must know the values of

the parameters ,' c2' g(01)' g( 2) and have equation of state data.


1.375

90

1.3

1.84

1.87

1.6










In general o1, a and G for the square-well potential are available

(20). In principle g(al) and g(2 ) can be calculated from equilibrium

statistical mechanics. However there is still no satisfactory method

of making such theoretical calculations accurately.

Davis and Luks (21) estimated the values of g(al) and g(2)

from equation of state and thermal conductivity data of argon by

solving simultaneously the modified Rice-Allnatt expressions. They

then calculated the viscosity coefficient and self-diffusion coeffi-

cient of argon at various temperatures and pressures. The results are

shown in Figures 1.1 and 1.2. The discrepancy between theory and

experiment for viscosity is as much as 19% and for the self-diffusion

coefficient is up to 34%. The experimental self-diffusivity data

used by Davis et al. as shown in Figure 1.2 are apparently extrapolated

values obtained from the self-diffusion coefficient data of Naghizadeh

and Rice (18). Luks, Miller and Davis (22) have also made use of the

modified Rice-Allnatt theory to calculate transport coefficients for

argon, krypton and xenon. Discrepancies between theory and experiment

become larger as the density increases.

1.3 The Enskog Theory (7)

Enskog extended the dilute gas kinetic theory to dense fluids

composed of hard spheres. As a gas is compressed the mean free path,

which for dilute gases is infinitely large with respect to the molecular

diameter, gradually decreases until at high density it is of the same

order of magnitude as the diameter of a molecule. During this process

the intermolecular collision frequency tends to get larger due to the

decrease in the mean free path; at the same time it decreases because





































































-4


C"








C)
-r







,--
rJ


-I
C)
I 1-4


U
F-4


(asTod) O c : x



c..J



0
u





r c
**





E-






4-
r0
E-a
3

,-1
i-1





-. J

c









c



o 1-1
0
1 )


1 I I I I I I I I I I I





10










I0


,-44




II


.1-





C)
C- c
Gu m



o / / 0



S/ a

u cu


//




0 O *t
S i H




















O N
//
/ / <
I I I-






/ / -"










I f / 0 *0
o -
So i
I I y. I o
















/ -0



0D 0 C) 0
(i"a/m) /O x H"











at close separations the molecules tend to shield one another from

collisions with more distant neighbors. The net change in collision

frequency was found to differ from that of a dilute gas by a factor

S, which is given by

b \b 02 /b 3
X = 1.0 + 0.625 ) b + 0.2869 2 + 0.115 ) (1.13)



where b 7 iNO3 = the rigid sphere second virial coefficient.
o 3

Assuming the collisional transfer of'momentum and energy between

colliding molecules to be instantaneous, Enskog modified the Boltzmann

equation for higher densities and solved it to obtain the following

equations for transport coefficients.


Self-diffusion coefficient



Shear viscosity

no

Bulk viscosity K =
7o


Thermal conductivity
(monatomic molecules)


where


The value of y may be


0
0


b
OD o
(pD) Vy



b
-- + 0.8 + 0.761y
V y

b
o
1.002 -- y
V



0 + 1.2 + 0.757y
V y


(1.14)



(1.15)



(1.16)


(1.17)


b
o
y X

obtained from the equation of state by


PV +
RT 1 + y
RT


Enskog suggested as an empirical modification of the hard sphere


(1.18)









theory the use of thermal pressure in place of external pressure.

Thus according to this modification y becomes

y = V( 1 (1.19)



Equations (1.15) and (1.17) predict that when ('rV) and Q~V) are

plotted as functions of y, the curves will go through minima at

y = 1.146 and y = 1.151 respectively, having


[V]min = 2.54570ob at y = 1.146 (1.20)


and [XV]. = 2.938X b at y = 1.151 (1.21)
mn o o

In the above transport coefficient expressions, b0 is the only unknown

parameter. It may be obtained from equations (1.20) and (1.21) by

using experimental values of [rV]min and [XV]min

The Enskog dense gas theory gives a useful approximate

description of transport coefficients above the critical temperature

and at densities less than the critical value. The theory fails at

temperature below the critical temperature and at densities higher than

the critical density. Under these conditions the rigid sphere approxi-

mation is in serious error.

The Enskog theory has been tested for thermal conductivity and

viscosity of argon at various temperatures and densities (23).

Figure 1.3 shows the comparison of calculated and experimental vis-

cosities of argon at 0C and 750C. The value of bo used in these

calculations was obtained from equation (1.20) using data at 0C. The

agreement between theory and experiment is moderately good at 0 C for






13













Co





O
C -

00



0 0r
ittS


oo




\ 8

0 o
\\ C

\O







\O

a o
00
0 0
0 u
Q-






\o I












co
co 0 < G

\ 0 0 (




(as\od) 0 x L.










densities up to about 0.8 g/cc. At higher densities the theory fails

to predict the correct density dependence. As can be seen in

Figure 1.3, the Enskog theory does not predict the correct temperature

dependence. Dymond and Alder (25) recently modified the Enskog

theory by using temperature dependent rigid sphere diameters obtained

from the Van der Waals equation of state. With this modification, the

theoretical and experimental values of viscosity are shown in

Figure 1.4. The predicted temperature dependence is found to improve

considerably; however the theory still fails to predict the density

dependence at densities above 0.8 g/cc.

1.4 Activation and Free Volume Theories

An activation theory for liquid transport properties was first

proposed by Eyring (26). This type of theory assumes that a molecule

spends a large fraction of its time oscillating about an equilibrium

position in a cell, and only occasionally does it leave one cell to

take up position in a neighboring vacant cell. Transport of mass and

momentum are assumed to occur during such molecular transitions.

Later modifications to Eyring's original theory have included the

introduction of the concept of the fluctuating free volume. These

theories are discussed in detail in Chapter 2. The free volume theories

apply only at densities above approximately twice the critical value.

Horrocks and McLaughlin (27) applied the activation and free

volume theory to the thermal conductivity. They assumed a face-

centered-cubic lattice structure for the liquid and that transfer of

thermal energy down the temperature gradient was due to two causes:







15















o







C---
C)l 60





0 O





o

N X

0 '- <
CO 0
0 C




0



Q-






\ CN -4 0
C















4 .0

\ \ 0 -e
0 O
0 Ca

1 0



\0d \O x
O O C
\' \ (











CC 0 0 C4
r(a c o 01 xL
\' \









the actual transit of a molecule from one lattice site to another,

and that due to the collisions of an oscillating molecule with its

neighbors. For liquids the contribution due to the former cause is

negligible when compared with that of the latter. The frequency of

oscillation is determined by the molecular mass and the intermolecular

force. Their theory has been tested for a number of simple liquids

and the agreement between theory and experiment is often within 20%.

Calculated and experimental thermal conductivities of a number of

liquids are compared in Figure 1.5.

1.5 The Correspondence Principle

The principle of corresponding states has been found very

useful in the calculation of equilibrium properties of dense gases and

liquids (28). However very few studies have been made of its applica-

tion to transport properties of dense gases and liquids. As the

principle of corresponding states will be discussed in great detail

in later chapters it will not be discussed further here.

1.6 Empirical Correlations

A very good review of the purely empirical methods of estimating

transport coefficients is given in a new book by Reid and Sherwood (29).

These methods have neither a theory nor a model to describe them. The

many expressions proposed usually contain one or more empirical para-

meters or constants which are said to characterize the structure or

properties of the molecules; in most cases these parameters have to be

supplied by the authors of the correlation. A typical example of these

correlations is Souder's method for estimating liquid viscosity, which

is one of the few empirical methods recommended by Reid and Sherwood.









































































0 -
wI

on


0, 0 0 0
~J ~ NJ


( I oas UI-,m 'I ) OT x y
1- 01- I- 7










Souder's equation is
I
log(log 10r) =- p 2.9 (1.22)


where T = liquid viscosity, centipoise
-3
p = liquid density, gcm.

I = viscosity constant calculated from atomic and structural
constants supplied by the author

M = molecular weight

The results yielded by such methods are usually not very satisfactory.

In spite of the author's claim that viscosity predictions are within

20%, errors are often greater than this. Thus the equation predicts a

viscosity for acetic acid at 400C that is 36% too low (29). Similar

empirical expressions are also available for liquid thermal conducti-

vity and self-diffusion coefficient; the discrepancies between these

equations and experimental data are of a similar order to that of the

viscosity correlations.

Among the empirical correlations, those based on residual-

viscosity concepts have received much attention (30-32). Figure 1.6

is a plot of viscosity isotherms for krypton as a function of density.

At the lower densities the curves are almost parallel, and this has

led a number of workers to correlate the residual viscosity r no with

density. The general form of this correlation is


S- = f1(p) (1.23)


where fl is a function of density only and oT is the dilute gas

viscosity. However, as can be seen in Figure 1.6,the isotherms

















































0-
C. -


























0-.


S L) C.)
000
0 0 0
L C) V 0C
CN r- un
,-<


(asTod) o0 x L


.CO






c~c
















N O
0






c4J







0










%-,

0
l1





o

cJn








O

O >
I-,
(-)



0
cJ^4-









intersect at higher densities, so that such correlations are not valid

over the whole density range. The intersection occurs at densities of

about twice the critical value. Above this density value the free

volume theory is found to work well.

1.7 Summary

Of the different approaches discussed above, the time correla-

tion function theory and the kinetic theory of liquids are the most

rigorous and correct descriptions of transport processes in dense

fluids. In their present stage of development, however, they offer

little immediate prospect of yielding methods of predicting transport

properties for fluids of interest to chemical engineers.

The Enskog dense gas theory provides good results for dense

gases, but fails at densities a little above the critical value. For

the lower densities quite good results may also be obtained from

empirical equations, such as those employing the residual viscosity

concept.

At present a particular need exists for satisfactory methods of

predicting transport properties for fluids at densities well above the

critical, that is in the normal liquid density region. Toward this

end two approaches are examined in detail in this dissertation. The

free volume theory is first studied, and examined as a framework for

developing predictive corresponding states relations for simple fluids.

Although such a model lacks the desirable rigor present in the formal

theories, it has the considerable advantage of being solvable, and

suggests ways in which expressions may be obtained for mixtures.

Chapter 2 briefly reviews previous work on free volume theories, and










Chapters 3 and 4 contain the new contributions to the theory.

The second approach consists of the development of corresponding

states relationships directly from the time correlation function theory.

The correspondence principle is particularly valuable when based on

the rigorous statistical mechanical approach, but no serious attempt to

apply the principle to dense fluid transport properties seems to have

been made previously. This approach is examined in detail in Chapters

5 and 6.










CHAPTER 2

PREVIOUS WORK ON ACTIVATION AND
FREE VOLUME THEORIES


Because of the present difficulties associated with developing

a rigorous kinetic theory of dense fluids analogous to that available

for dilute gases, considerable attention has been given to simplified

models of the liquid state. Assumptions are introduced in the initial

stages of the development, an attempt being made to obtain a model

which incorporates the essential features of real liquids, but which

is still solvable. Such attempts are exemplified by the activation and

free volume theories for liquid transport properties.

Eyring (26) may be considered a pioneer in the development of

activation and free volume theories for liquid transport properties.

In Eyring's original approach the equilibrium positions of molecules

in a liquid were considered to be on a regular lattice similar to the

crystal lattice in a solid. The transport of mass and momentum occurs

as a result of a molecule jumping from one equilibrium lattice site

onto a vacant neighboring site. In order to make such a jump a

molecule is required to have the necessary activation energy to over-

come the minimum energy barrier separating two adjacent sites, and at

the same time a vacant site must be available. Using this model Eyring

obtained the following expressions for liquid viscosity and self-

diffusivity by making use of the absolute reaction rate theory:

1 e /kT
S= 1/2 3(e m) (2.1)
pv









12kT -e /kT
D = 12 kT /e (2.2)
1/2 1/3 e
(27nkT) qf


More recently Weymann (35,36) used a statistical mechanical

approach, and arrived at expressions very similar to those of Eyring.

In his derivation of the equations for transport properties, Weymann

gave a clearer picture of the physical model used and of the signifi-

cance of the activation energy. Moreover his approach suggests a

possible extension to liquid mixtures. Weymann also derived an expres-

sion in which the probability of hole formation is related to the

volume and energy needed to form a hole. The equations obtained by

Weymann for viscosity and self-diffusion coefficient are:


l \ /2e /kT
3= 1 /3 (2kT) /2e (2.3)
4h ivl/3


2Xhl 2kT 1/2 -e /kT
D =-- -m e o (2.4)



where Xh is the probability of hole formation.

Eyring's activation theory was modified by McLaughlin (37)

by introducing the concept of the probability of hole formation, as

was done by Weymann. In his treatment, he assumed a face-centered-

cubic lattice for the liquid. The most important modification made by

McLaughlin was the attempt to relate the activation energy and energy

of hole formation to the intermolecular forces by means of the

Lennard-Jones and Devonshire cell theory of liquids. A similar

expression for liquid viscosity was also obtained by Majumdar (38)








using the tunnel model of Barker (39) in place of the Lennard-Jones

and Devonshire cell model.

The Eyring and Weymann theories yield good results for the

viscosity of Arrhenius liquids (ones whose viscosity varies with

temperature according to an equation of the type Inr = Ca + Ba/T),

but cannot account for non-Arrhenius liquid behavior. In order to

overcome this deficiency Doolittle (40-42) proposed empirical expres-

sions which related liquid transport properties to the free volume,

defined by

vf = v Vo (2.5)


where v is the hard-core volume of the molecule. His free volume
o
equations were placed on more solid theoretical grounds by Cohen and

Turnbull (43,44) who derived them by a statistical mechanical method.

They assumed that a molecule moves about in a cell in a gas-like

manner, while the free volume available to each molecule fluctuates

with time. A molecule is able to jump into a neighboring cell only if

a free volume of a size greater than some minimum value v is available.

They obtained the following expressions for the viscosity and self-

diffusion coefficient:


r = 1 2 exp (2.6)
3-ra f



D = gla )1/2 exp o (2.7)


The term "free volume" in these theories has a different








meaning from that implied in thermodynamic free volume theories. The

theories of Doolittle and Cohen and Turnbull assumed the molecules to

behave as hard spheres; the free volume referred to here is the space

in the fluid unoccupied by the hard spheres themselves. The average

free volume per molecule is defined as v vo, where v = V/N. The

free volume theory of these authors describes the viscosity behavior

at atmospheric pressure, but fails to predict the temperature depen-

dence of viscosity at constant volume.

Recently, Macedo and Litovitz (45) proposed expressions for

viscosity and self-diffusion coefficient in which the hole probability

in Weymann's equations was replaced by the free volume expression

obtained by Cohen and Turnbull. Molecular transport is assumed to

occur if a molecule has sufficient energy, e to overcome intermolecular

forces with its neighbors, and at the same time it has a free volume

greater than the minimum free volume v needed for a jump to occur.

The equations proposed for viscosity and self-diffusion coefficient

are


S= Aexp V v exp (2.8)



D Bexp exp (2.9)



By treating V and E as empirical constants Macedo and Litovitz
o V
have shown that equation (2.8) describes the viscosity behavior of a

number of liquids over a range of temperature. Chung (46) has pre-

sented an elegant statistical mechanical derivation of the equation of




26





Macedo and Litovitz. His derivation is presented in Appendix 1.

Macedo and Litovitz assume the pre-exponential factor A in

equation (2.8) to be proportional to temperature T although most other
1/2
workers predict a temperature dependence of T/2. Both types of

temperature dependence of the pre-exponential factor have been tested

in this work, and experimental results seem to give better agreement

with theory when a temperature dependence of T1/2 is used. Thus

throughout this work the pre-exponential factors A and B are assumed

to be proportional to T1/2. Thus equations (2.8) and (2.9) may be

rewritten as

V E
S= A exp ) exp (2.10)
T /2o V-RT"


D p 0o v (2.11)
B exp exp --
1/2 0 V V RT
T o

where A =
o 1/2
T

and B
o T1/2

Macedo and Litovitz original equations contain a constant

factor y which was introduced to account for the overlapping of free

volumes. In the early part of this work, the constant 7 was evaluated

for a number of molecules and was found to be close to unity. This

factor will therefore not be included in the equations of viscosity and

diffusivity.









CHAPTER 3

IMPROVED FREE VOLUME THEORY


The viscosity equation (2.8) has been tested by Macedo and

Litovitz (45) for a variety of liquids and by Kaelble (47) for poly-

meric substances. They treated the pre-exponential factor Ao, the

activation energy E and the minimum free volume V0 as adjustable

parameters characteristic of the substance considered. Equation

(2.8) was found to describe the viscosity behavior of the substances

tested by these workers moderately well. On the other hand, Naghizadeh

and Rice (18) tested the theory for the self-diffusivity of simple

fluids (such as the inert gases) and found that the agreement between

theory and experiment was poor, especially in the high density region.

If V is a constant, as assumed by Macedo and Litovitz,

differentiation of equation (2.10) with respect to 1/T at constant

volume (i.e. constant density) yields


1n(]/T1/2 E

L 3(1/T) R (3.1)

E represents the minimum energy required by a molecule to overcome

intermolecular forces in making a jump and was also assumed constant

by Macedo and Litovitz. Therefore equation (3.1) predicts that a

plot o" In('r/T1/2) versus reciprocal temperature at constant volume

should give a series of parallel straight lines. For a variety of

nonpolar and slightly polar liquids over a moderate density and

temperature range such a plot produces straight lines, but the slopes









of the lines vary with volume. Moreover, when the results are plotted

over a wide temperature range nonlinearity becomes apparent. Such

nonlinearity is particularly marked for fluids composed of simple

molecules. This is illustrated in Figures 3.1 and 3.2 for argon and

nitrogen, for which data are available at constant volume over wide

ranges of temperature and density.

From the above discussion it is apparent that the equations as

used by Macedo and Litovitz do not correctly predict qualitatively

the effect of temperature and volume on the viscosity, especially for

simple fluids. In addition, extensive experimental viscosity data are

needed for each fluid in order to fit the adjustable parameters Ao,

E and V ; when used in this way their equation is no more than an

empirical correlation.

In this chapter the physical significance of the parameters

E and Vo is examined in the light of the theory, and their dependence

on molecular type, temperature and volume is discussed. Interpretation

of these quantities on the molecular level suggests corresponding

states relationships which may be used to predict the parameters. The

improved theory also explains why the theory of Macedo and Litovitz

fails for simple fluids.

3.1 Temperature Dependence of Vo

The parameter Vo of equations (2.10) and (2.11) represents the

minimum free volume that must be available before a jump may occur. If

the molecules may be treated as rigid spheres, as in the smoothed

potential cell model of Prigogine (50), the free volume is independent
















-7.5

o- Experimental data (48)



-8.0

3 -1
29.14 cm.mole

o-o

-8.5 o 30.45


oz 0 32.75


-9.0 -
0vo 36.3




-9.5 0- 39.15

-P 0 V0VVV`'"/




-10.0





-1C.5 I I I l
2.0 4.0 6.0 8.0 10.0 12.0 14.0
1/T x 103 (K-1)

Figure 3.1. Viscosity of Argon.

















-o- Experimental data (49)

3 -1
32.6 cm.mole






35.1



o 37.2





44.5


S40.6

o oo

:^/^ /^ 56.0
o




/o


I I


, I


. I


6.0 8.0 10.0 12.0

1/T x 103 (K-)

Figure 3.2. Viscosity of Nitrogen.


I


14.0 16.0


-8.0


-8.5 H


-9.0




CM1

E -9.5


-10.0 L-


-10.5 -


-11.0


2.0


I


4.0


I I









of the temperature at constant volume, and V should therefore be a
0
constant for a particular molecule, independent of temperature and

density. Macedo and Litovitz (45) assumed this to be the case for all

molecules treated by them.

For more realistic potential models one would expect V to

decrease somewhat with increasing temperature, since as temperature

rises the average kinetic energy of the molecules increases, and

molecules are thus able to approach each other more closely. According

to such a viewpoint Vo should not be affected by the density at constant

temperature. The extent to which this parameter depends on temperature

will be determined largely by the repulsive portion of the intermolecu-

lar potential energy curve. Since this part of the curve rises less

steeply for simple molecules than for more complex polyatomic molecules,

one would expect the effect of varying V to be most evident for the

simpler molecules.

To obtain a general expression describing the temperature

dependence of Vo, a (6,n] pair potential energy function is assumed.

Thus


u= EK1 2 j (3.2)


where 6/(n-6)

1 n-6 6/


The parameter n indicates the steepness of the repulsive part of the

curve; for small molecules such as the inert gases n is close to 12,

whereas for more complex molecules the best values of n is 28 or









higher.

For a group of roughly spherical molecules V may be assumed
0

to be proportional to the cube of the molecular "diameter." An

estimate of the variation of the molecular diameter (and hence V ) with

temperature may be obtained by equating the average kinetic energy of

a two-particle system to the potential energy of the system at the

distance of closest approach, d:


3kT = EK ( 6]3.3)


or, in reduced form


3T = K1 [ 3 (--)2 (3.4)


where
T= kT
C
v a d7



Equation (3.4) may be solved to obtain v" as a function of T* for
a
various n values. The temperature dependence of v for several n

values is shown in Figure 3.3. Assuming that V0 is proportional to

d3, one may write


V = (v (3.5)
o a

where ( is constant for a particular molecule, V at any temperature

can be calculated from

V 1
V = V o (T (3.6)
o o v (T = 1)


























































2.0 4.0 6.0 8.0

kT/e
Figure 3.3. Variation of v with T .
Variaion a


1.0






0.9






0.8






0.7


0.6






0.5





0.4


10.0










1 1
where V is the value of V at T= 1. Thus a knowledge of V ,
o o o
together with n and E, suffices to calculate Vo at any temperature.

Some support for the above procedure is provided by values of the hard-

sphere diameter calculated at various temperatures from equation of

state data for inert gases by Dymond and Alder (25). These values are

included in Figure 3.3, and agree well with the curve for n = 12.

With n = c, no temperature dependence should be observed for

V Under these conditions, from equation (2.10)



[ ~1n(/T/2) (3.7)
L (1/T) J R

1/2 -1
When In(n/T /2) is plotted against T- at constant volume for such

nonpolar molecules as decahydronaphthalene, benzene, etc., approximately

linear behavior is observed; for these molecules n is large and the

variation of V0 with temperature is small. For simple molecules, such

as the inert gases, for which n 12, the temperature dependence is

more marked, and the model predicts noticeable nonlinearity on such a

plot. This is as observed experimentally, as shown in Figure 4.1 in

the next chapter.
1
In addition to the above comments, the parameter V should be

related in some way to molecular size. If V may be made dimension-

less with some suitable reducing parameter, the resulting reduced

minimum free volume may be quite constant for a series of similar

molecules. Since free volume theories apply best at high densities,

the molal volume at the melting point seems a more appropriate reducing

parameter than the critical volume. Thus











V1
o
V = v (3.8)
m


3.2 Volume Dependence of E

Macedo and Litovitz (45) treated the minimum activation energy

E as an empirical constant, independent of density and temperature.

However, as pointed out by Brummer (51), E may be expected to vary

with the average intermolecular distance, and thus with density. The

value of Ev becomes larger as the molal volume decreases because of

the increase in repulsive intermolecular force between molecules at

close separations.

From equations (2.10), (3.4) and (3.6) one may obtain


E 1/2
_v = Lim [ln(/T 1) (3.9)
R T-+0 d(1/T)

so that Ev may be calculated from experimental viscosity values. For

more complex molecules where n is large,a plot of In(7/T1/2) versus

1/T is found to be approximately linear, and the requirement T 0 in

equation (3.9) is less stringent. Figure 3.4 shows the volume depen-

dence of E for several fluids as calculated from equation (3.9).

The activation energy arises from the motion of the jumping

molecule from its initial equilibrium position through a region of

higher potential energy to its final position. A model of the situation

is shown in Figure 3.5 in which a molecule jumps from A to C, and

passes through a region B in which it has to squeeze through a ring of

z molecules. The activation energy may be written














































r..
U!
C-,





2-i
U 1-i










,. ,
r..)


u


c\
C4
z


D 0 0 0
3 0 0 0 0


( aiom'lwo)Az


0








-4-4






o







ii






o e
00 0
C.i

> >
c*


.,-4
C-)
o c
i- i
Q) >
i- &
0 !-
E










z Molecules


r0 1

0r


0
B


Figure 3.5. Model of a Molecular Jump.


I--











e = z[u(r2) u(r)



= KlZE 2


a n




- 6" (3.10)


c1 =


3
v/rl
1


c2 = (r2/r)3


c3 = C 1/Vm


c4 =
kT


Then

e = K zkT c 3 m 3 ) ( 3vm )2
v m4 c2V c2 (3.11)


c v n/3 C

The parameters cl, c2, c3, c4 and n, the repulsive eponent should be

approximately the same for a group of similar molecules and thus

equation (3.11) suggests a corresponding states relation of the form


E = E (V*)
V V


where

and


(3.12)


E
V
E -
v RT
m
V, V
V~ 7


Figure 3.6 shows the correlations of reduced activation energy as a

function of reduced molal volume for several nonpolar molecules. Good

agreement is obtained at high reduced volumes, but some scatter is


Define












o Ar (48)
o N2 (49)
CH4 (52)
a CO, (53)
0 i-C4H10 (54)
g CS2 (54)
v n-C6H14 (55)
. C6H5C1 (55)
.C C6H6 (55)
3 CC4 (55)


B






A
1



\

-\

v"




"\


oot-


1.0


V/V
mn


Figure 3.6.


Corresponding States Correlation of
Activation Energy vs. Volume.


7.0


6.0 1


5.0


4.0 ,-


-3.0




2.0




1.0




0.0


0.8


I










observed at lower values of V", where the curve rises steeply. This

may be attributed to differences in n, the repulsive exponent, for the

molecules within the group. Thus two curves A and B are plotted in

Figure 3.6. Curve A is for simple molecules obeying a Lennard-Jones

[6,12] potential. Curve B shows an approximate relationship for more

complex molecules which obey a potential law in which n is larger

than 12. Sources of viscosity data from which values of Ev are cal-

culated are given in Figure 3.6.

Taking into consideration the temperature dependence of V

and the volume dependence of Ev, the modified viscosity and self-

diffusion coefficient equations become:



-2 A exp -- ) exp E ) (3.13)
T' V V (T) RT
o

and

D/ V (T) / E (V)
2 -B exp ( ) exp ) (3.14)
1/2 V V (T) x RT
T o











CHAPTER 4

TEST OF IMPROVED FREE VOLUME THEORY


In order to perform a rigorous test of the theory, viscosity

and self-diffusion coefficient data over a wide range of temperature

and density conditions are most desirable. In this work only fluids

composed of simple molecules which approximately obey the Lennard-Jones

(6,12) potential law will be considered. These fluids include the

inert gases and most diatomic fluids-and perhaps methane, fluoro-

methane and tetradeuteromethane.

The only monatomic and diatomic fluids for which viscosity

data over a wide range of temperature and density conditions are avail-

able seem to be argon and nitrogen. Zhadanova (48,49) has reported

viscosity measurements for these fluids for experimental conditions

corresponding to pressures up to several thousand atmospheres. The

accuracy of the experimental argon data appears to be of the order 6%

over the entire range of conditions. However, the nitrogen data are

in poorer agreement with measurements of other workers, and the accuracy

appears to be of the order 12-15%.

The only high pressure self-diffusivity measurements for simple

liquids seem to be those of Naghizadeh and Rice (18), who reported

measurements for argon, krypton, xenon and methane over a reasonable

range of temperature and at pressures up to a more than one hundred

atmospheres. Naghizadeh and Rice claimed that the uncertainty of

their experimental data was less than 5%. However the disagreement

between their argon data and those of Corbett and Wang (56) is about










12%, while their xenon data differ from those of Yen and Norberg

(57) by as much as 86% and their reported methane data are about 11%

higher than those of Gaven, Waugh and Stockmayer (58).

4.1 Corresponding States Relationships

The following procedures were employed to test the theory for

viscosity and to establish the corresponding states relationships in

equations (3.8) and (3.12).

(1) The experimental data were plotted as ln(]/T1/2) against
-1
T at constant volume. Values of E were estimated at various volumes

from the lines extrapolated to low temperature, since from equation

(3.9)

v Lim ln('yn/T1/2) 14.
R T->0 (1/T)


(2) Values of the parameters A and V were obtained from
0 0
the best fit to the viscosity data using the computer, equations (3.4)

and (3.6) being used to obtain the temperature dependence of V .

Values of Ev for argon and nitrogen estimated by procedure (1)

are included in Figure 3.6. Theoretical and experimental liquid argon

viscosity values are shown in Figure 4.1. Agreement between theory and

experiment is found to be within 5%. Similar agreement between theory

and experiment is found for nitrogen, with V = 17.0 cc/mole and

A = 2.53x105 poise (K)-/2, and is shown in Appendix 2. The theory

provides a satisfactory fit to the data for densities above twice the

critical value. Thus as can be seen in Figure 4.1 the theory fails

for molal volumes of 39.15 cc/mole and above for argon. Similar

behavior is also observed for the nitrogen viscosity (see Figure A2.1


















3 -1
29.14 cm.mole


1, -,


30.45



34.4


36.3


^ 39.15 (theory)


S39.15 (expt.)
o-

Free Volume Theory
-- o--- 9hadanova (48)
o Lowry et al. (15)
r De Bock et al. (59)
0 Saturated Liquid
(60)


8.0 10.0

1/T x 103(K-1)


12.0


14.0


Figure 4.1. Test of Free Volume Theory for
Liquid Argon Viscosity.


-7.5


-8.0 -


-8.5





-9.0


-9.5


-10.0






-10.5






-11.0


2.0


4.0











in Appendix 2), where the theory fails for molal volumes of 56.0 cc/mole

and above. The density range in which the theory applies is sharply

defined and is similar to that in which similar models for thermo-

dynamic properties, such as the Prigogine smoothed potential model

(50), apply. The theory appears to work well for temperatures below

about 1.5 Tc, although the temperature range in which the theory

applies is less clearly defined.

The values of the parameters E and V found for argon and
v o

nitrogen may be used to form the basis of corresponding states cor-

relations of these parameters as described in Chapter 3, from which

viscosity and self-diffusivity values of other molecules may be pre-

dicted. The average value of the reduced minimum free volume for

argon and nitrogen is 0.554, so that

V1
V1 =_V = 0.554 (4.2)
o V
m

4.2 Test of Proposed Correlations

To use equations (2.10) and (2.11) to estimate the viscosity

and self-diffusion coefficient for simple molecules, values of V were
o
calculated from equations (3.4), (3.6) and (4.2), assuming n = 12 and

taking values of e/k determined from gas viscosity data (20). Values

of E were found using curve A of Figure 3.6. The best values of A
v 0

were obtained by substituting experimental viscosity data in equation

(2.10) for each molecule. In a similar manner, by fitting experimental

self-diffusion coefficient data of each molecule into equation (2.11)

the best values of Bo were obtained. Other than the viscosity data










for argon and nitrogen ofZhadanova (48, 49) and self-diffusion

coefficient data of Naghizadeh and Rice (18), very few high pressure

viscosity and self-diffusivity data are available. Thus most of the

viscosity and self-diffusivity data used are for saturated liquids.

Table 4.1 shows the viscosity data sources and ranges of temperature

and density for each of the molecules studied. The different para-

meters which are required for the estimation of viscosity with

equation (2.10) and the average percent deviation between theory and

experiment are shown in Table 4.2. Tables 4.3 and 4.4 provide

similar information for the self-diffusion coefficient of the molecules

studied.

The average percent deviation between theory and experiment

is about 6% or less for viscosity. For self-diffusion coefficient the

discrepancy between experiment and theory is found to be much larger,

being about 15% in most cases. The discrepancies between theory and

experiment for both viscosity and self-diffusion coefficient seem to

be of the same order as the accuracy of the experimental data for the

liquids studied. The viscosity data of Zhdanova for nitrogen at

high pressures seem to be in poor agreement with those of other workers

(66), and saturated liquid viscosity data (66) were used in obtaining

che A value given in Table 4.2 for this fluid.

Experimental and predicted viscosities for xenon and methane

are compared in Figures 4.2 and 4.3 (similar plots for Ne, Kr and N2

are shown in Appendix 2). The predicted values become less reliable

as density falls to values approaching 2pc (for methane this corresponds
























O-1


O N'
(U*z ^ v


0 r C4 0 11o c0 Lt
<-\ 1-1CO ~


>~ a o cr>
U0 0
co CNJ L 0n 'I cO r
- Q) LC ) I' 1.0 10 I' D c



> Pc


d)

ci c

C '.
cn 0 )
r- 4-4
( a) o co
p Pc; ID It


0
oD
CO



c
S-J




3 0






Q)
olc




HO




CO







?- s
0)
fr.


I--( 0N

I I
,d 04


0 'D '.0













uLt -d r. i ir cO cO

I I I I I I I
-"1 1* rC CNJ I D Z
cc u f m fc


0 Ln t ) "n 00 cc in i
0 r- C 0 ,t4 Ci I
CV C 11-- )- co 1-
I I ' I I I I i I
co0 o C4 0 o c r- cn
1 01% r- cc- I' O


0-1


Sz
0
z


4 C CM C4 0 C
SXz ; u u u


4' .


Nt Z





cj!







H


cj3
0
U
cj
I-


''








47





mo
















OO
0 B~2
0--













> o
)o ' Ci CO ,






o
Si cI t- c ^ cC-


H

C-)
E-4




M -


C)

c 0c
0 > Co


0
o-^-







vN-
c/L/








0 00


0 0 J. CNI CO I OC C Ni CP) Ci CO











O O









c) -7 r^ ui

Z < c in o 0 C- a-)
0-
A^-




























Q 0 0 D'- C CI <-l
O-i













o o w Lr
,4 C :
S C O0 0-I O0 CO
O










0
O uC)






UH 0
0 4 0
U) L




."- ( ) O 00 00 r-4 < <




c' Q <
0)


0 Cl




H o o a oo (,
3O 0 c -Z <-Z c< -Z LCi

EH Oc' oo C3 Cn C LCn LC
< > C'. c< <- ( ( <-Z


u0






CI3

CC
I Co

Q) C c 0


Q -4 D O CC -Z CrZ
E -4 i-I CM F- '-4 >-4

CCC L0 0 CD C0 \ CO


0l

Cl


E-4




Cl !-4 }H CL 0 C








49








M0C 00 0 0
Lu- r--. -. L"Th G' CM




Z



o > c GIN '-i









,i q 00 --1 0 0
0
i-4

co ON czt r- Cm

cm c uii m c) r-







C ,



0 0 0 0 0
1-1








0 C N Co 0 0
F- ,,


i-ii






E I
1o co









> 0 o 0 L L
a-1 (N c-C -N m c
-z O












i-i
-E N ( N (N rc- 0










EI Cl) '" Co 'i '0 oC1 Co
1-4 4-i


























0
E: V0 c0 'Z0 00 0













C C 0 (N f -

^^X U










-7.0

Free Volume Theory
o Experiment (64,65)

-7.5 -3
44.5 cm.mole



-8.0 -

50.


-8.5 5





I:
-9.0




-9.5




-10.0 I I I I
1.0 2.0 3.0 4.0 5.0 6.0 7.0
1/T x 103 K-1)

Figure 4.2. Test of Free Volume Theory for
Liquid Xenon Viscosity.























a U
J 0




oo
E /
o o C
0 D CC



-_J

-4
0 0O





0

o
L-

C
0 -~
o


0

O 0






4 0 >







"-4
U





a0 3
0 C








,- C- 0 0


(as-od) 01 x











to V = 49.5 cc/mole). Figure 4.4 compares theory and experiment for

the self-diffusion coefficient of saturated liquid methane.

The expressions for the pre-exponential factors of the

viscosity and self-diffusion coefficient equations of Eyring (26)
2/3 1/2
and Weymann (35) suggest that the groups (AoV //M ) and

(B M1/2/Vm /3) may be approximately the same for different molecules.

Values shown in Table 4.2 indicate that the quantity (AoVm2/3/M /2

is approximately constant for inert gas liquids, but for the other

molecules a range of values is found. Similar behavior is observed

for the group (B M1/2/Vm/3). Apparently the expressions given for

the pre-exponential factors by Eyring and Weymann are not generally

valid, and cannot form the basis of successful corresponding states

relationships for Ao and B In general the value A (or B ) can be

determined from a single experimental value of viscosity (or self-

diffusivity) for a given fluid. This can then be used together with

the proposed correlations for V and E to predict viscosity (or self-

diffusivity) values at other temperatures and densities.

4.3 Summary

By allowing for the variation of V with temperature it is

possible to account quantitatively for the nonlinearity of constant

volume plots of In(r/T1/2) vs. T-1 and In(D/T1/2) vs. T-1 for simple

liquids. The physical model predicts a density dependence of E and

thus explains the variation with density of the slope of constant

volume plots of In(T1/2) vs. T- and In(D/T1/2) vs. T1. With these

improvements the free volume theory is able to accurately describe

the viscosity and self-diffusivity behavior of liquids composed of























r-






0




CC
*r-1







o
0.

Cr)








E-' O-
C>
0 ao






c 0
\ 0
\ 0 -i *

















\ o C 0
o -o
0 r r-







\I 0 0



0 0 Lr 0


( Doas-uI ) o0 Y
z- Z











simple, nonpolar molecules that may be expected to approximately obey

a Lennard-Jones [6,12] potential law. The theory works well at

densities p > 2pc, where other correlations fail.

The fact that both V1 and E values vary in a corresponding
0 v

states way indicates that the free volume theory provides a reasonably

correct picture of viscosity and self-diffusivity behavior for these

liquids, and shows that the equations' success does not arise only

from the availability of three adjustable parameters. The simple

expressions given for A and B by Eyring and others, however, do

not seem to be generally applicable.











CHAPTER 5

PRINCIPLE OF CORRESPONDING STATES FOR
MONATOMIC FLUIDS


The free volume equations discussed in Chapters 2-4 accurately

predict viscosities and self-diffusion coefficients for nonpolar

fluids in the density range p > 2p. However, the theory cannot be

readily extended to include thermal conductivity, and is limited to

the above density range. In this chapter and the next a general and

direct application of the corresponding states principle is examined.

The treatment for monatomic molecules does not assume any simple

model, and should be applicable to all transport properties over the

entire range of densities and temperatures.

5.1 Molecular Basis of the Correspondence Principle

If correctly applied, the principle of corresponding states

provides an accurate and very useful method for predicting physical

properties of pure substances, under conditions for which no data are

available, from the measured properties of one or more substances.

The corresponding states principle has been widely applied to the

thermodynamic properties of pure substances; thus the generalized

charts of Hougen, Watson and Ragatz (28) and of Hirschfelder, Curtiss

and Bird (20) have proven of great practical use in engineering

calculations. The principle of corresponding states was first

suggested by Van der Waals in 1873 as a result of his equation of

state. It was not until 1939, however, that a rigorous and more

general theoretical derivation of the principle was attempted by

Pitzer (74) for thermodynamic properties using statistical mechanics.










Pitzer presented his derivation based on the following assumptions:

Assumption 1. Quantum effects are negligible.

Assumption 2. The intramolecular degrees of freedom
(rotational and vibrational) are indepen-
dent of density.

Assumption 3. The intermolecular potentials are pair-
wise additive.

Assumption 4. The potential energy for a pair of molecules
has the form



u = EO (5.1)


where r = the intermolecular distance

E = characteristic energy

S= characteristic distance

0 = a universal function


Using Assumptions 3 and 4 the configuration energy may be

written as
N
U C (5.2)

i
The configurational partition function may then be put in

reduced form (50)
V V N
Z exp dr .... dr
o o i



03N
Z = Z (T ,V*,N) (5.4)
N.











where N = number of molecules
-A- kT
T = reduced temperature

V
V = = reduced molal volume
a
Since all equilibrium configurational properties of interest

may be evaluated from the configurational partition function Z and

its temperature and volume derivatives, equation (5.4) represents a

general proof of the principle of corresponding states for thermo-

dynamic properties of pure substances, provided all the above

assumptions are observed. Thus the reduced equation of state is



SPc 03 T nZ
P - kTT
v )T,Ni


= P (T ,V ) (5.5)



where it is noted that intensive properties do not depend upon N.

The principle of corresponding states for thermodynamic properties may

also be derived by the method of dimensional analysis (75). However

in this method the conditions under which a molecule may obey the

corresponding states principle are not explicit.

One way of deriving the principle for transport properties

is by di.aensional analysis, with the reduction performed using either

the critical constants or appropriate combinations of molecular

parameters. A rigorous statistical mechanical derivation for trans-

port properties similar to that mentioned above for thermodynamic

properties, is possible,as has been shown by Helfand and Rice (76).











The transport coefficients may be expressed in terms of equilibrium

time correlation functions. The equations for the shear and bulk

viscosity, thermal conductivity and self-diffusion coefficients are

given in equations (1.1) to (1.4) of Chapter 1. Using the four

assumptions made in deriving the principle for thermodynamic pro-

perties, it is possible to put these equations in a reduced form.

The derivation of the principle for the self-diffusion coefficient is

given below as an example of the procedure.

The self-diffusion coefficient expressed in terms of the time

integrals of appropriate autocorrelation function has the form



D = dt (5.6)

0

Based on Assumptions 1 and 2 given above the time correlation function

may be written in terms of integrals over phase space



.V V

V (0)V (t)e- d. ...d Ndr ...drN

= 0 (5.7)

x V
e. ePHdf1. .d2Ndrl...drN
-co o



Assumption 2 is implicit, since the integrals are performed only over

center of mass coordinates. The Hamiltonian H for the center of mass

coordinates is given by

N 2
H = + U (5.8)
i=2m
i=l











and if Assumptions 3 and 4 are satisfied, we may write




iU
i(j


Now, if we define reduced quantities


distance r =


temperature T


time t =


momentum p =


volume v =


as

r
C-

kT


t 1/2/m 1/2
tc /m 0.


p/(ml/2 1/2)


v/o3


then N *2
SP (O)p (t")exp -
m -. 2T*
= -
S x v- N *2


T2T
-co o 1=1

N

1 Np -1\ L.. d-N

ij = -- g(T ,V* ,t )
1 ... (5.15)
-T T (r ) d 1...dE dr .d
i
where g is a universal function of T V and t. From equation (5.6)


1/2
D 1/---- g(T*,V )dt* (5.16)
m
o


(5.9)


(5.10)


(5.11)


(5.12)


(5.13)


(5.14)






60



or combining equations (5.15) and (5.16)


D = D^(T*,V*) (5.17)


where D* = D F (5.18)


Similar relationships may be derived for shear and bulk

viscosity and thermal conductivity,


]= = (T",V ) (5.19)


S= *(T",V*) (5.20)


= X'(T',V) (5.21)


where 2
7O (5.22)


K 02
= -= (5.23)



>1 = x (5.24)
k V

The corresponding states principle may be derived under less

restrictive conditions than those used above. Although it is then of

general application it loses the attractive simplicity of the two-

parameter laws expressed by equations (5.17), (5.19), (5.20) and

(5.21). Thus, Assumption 1 is readily removed by using quantum

mechanical expressions in place of integrals over phase space. This

results in the introduction of a reduced Planck constant in the

expressions for transport properties,












D D'(T ,V",h ) (5.25)


-' = r(T, V", h ) (5.26)


= ic'(T', V, h ) (5.27)


= (T V h ) (5.28)


where
h


Donth (77) showed that the corresponding states principle could be

proved for thermodynamic properties without assuming pairwise addi-

tivity of potentials (Assumption 3). Thus in place of Assumption 3

he assumed that the total potential energy of the system might be

expressed in the form


rr r
U = E a3, -3 -,... (5.29)


where 0 is a universal function of the reduced position coordinates.

Recently Hakala (75) derived a more general form of the corresponding

states principle for the thermodynamic properties by means of dimen-

sional analysis. The effect of his treatment is to remove Assumption

4. The corresponding states principle is shown to apply to substances

which do not obey a pair potential of the form given by equation (5.1),

provided the necessary additional reduced parameters are included in

the correlation.

It is of interest to note that the above corresponding states

relations for D and n may be derived from the free volume equations.










The assumptions implied are in this case less clearly defined. The

expression for self-diffusion coefficient derived from free volume

theory is given by equation (2.11) as


D p V 0
S/2= Boexp _v exp (- V Vo (5.30)


Equation (5.30) can be expressed in terms of reduced variables as


E" V
S = B exp exp (5.31)
o
or
D* = D (T*,V) (5.32)

where
E
S= E (V*) (see Chapter 3)
v NE v

V
3
V
V* = V(T) (see Chapter 3)
o 3 o
1/2
and it is assumed that B* o = constant.
o 1/2
ok
In this chapter we consider only fluids composed of monatomic

molecules (excluding He) so that Assumptions 1 and 2 may be regarded

as fully satisfied. The inert gas molecules have been found to obey

the principle of corresponding states for thermodynamic properties

(20,74) and transport properties at low density (78) within experi-

mental accuracy. Pairwise additivity of potentials seems to be a

good assumption up to reasonably high densities for the molecules

considered in this chapter (79-81). Ross and Alder (82) have pre-

sented experimental evidence that nonadditivity becomes a significant










factor for these fluids only at very high densities, of about 4 times

the critical value. Assumption 4 concerning the form of the pair

potential energy is only approximately satisfied by monatomic mole-

cules considered here, as discussed by Smith (83).

5.2 Previous Work

Previous work on the application of the principle to trans-

port coefficients of inert gases and other simple polyatomic molecules

in the dense gas and liquid states has been reported by Cini-Castagnoli,

Pizzella and Ricci (84), Kerrisk, Rogers and Hammel (85), Rogers and

Brickwedde (86), Boon and Thomaes (87,88), Boon, Legros and Thomaes

(64), Tee, Kuether, Robinson and Stewart (89), Trappeniers, Botzen,

Van Den Berg and Van Oosten (90), and Trappeniers, Botzen, Ten Seldam,

Van Den Berg and Van Oosten (91).

Cini-Castagnoli, et al. applied the principle to the viscosity and

thermal conductivity of saturated liquid Ar, CO, N2, CH4 and 02. Correla-

tions appeared to be poor, especially for the reduced viscosities.

Kerrisk et al. and Rogers and Brickwedde examined the principle of
3 4
corresponding states for the light molecules He He H2, D 2 T2

etc., where quantum effects are important. They tested the principle

for the saturated liquid viscosity and thermal conductivity of these

substances. In their correlation they introduced the reduced Planck
*
constant h as a third parameter as required by equations (5.26) and

(5.28). Boon et al. tested the validity of the corresponding states

principle for the saturated liquid viscosity of Ar, Kr, Xe, 02, N2,

CO2, CH and CD They found that the inert gases obey the principle









closely, but even relatively simple polyatomic molecules such as 02,

C':. and CD, wcre found to deviate markedly.

Self-diffusion coefficients and binary diffusion coefficients

of more than 14 simple molecules in the dilute gas, dense gas and

liquid states have been tested by Tee et al. They defined the reduced

diffusion coefficient D as
AB


c T cAB
DAB P cAB
oAB AB
T cAB
AB / *
AB M1 1
T cAB M M

where c here represents the total concentration, and obtained

the equation for the diffusion coefficient of a dilute gas the

T
T
D = 4.05 x 10- cAB
AB(1,1)*


(5.33)



from

relation



(5.34)


They plotted their results as D" against P at constant T where
AB R R
P T
P= P and TR However, the data they present seem to be too
c c
scattered to justify the test.

Trappeniers et al. applied the corresponding states principle

to the viscosity of inert gases up to high densities for temperatures

above critical. They defined the reduced viscosity as


Q = n


where


5 T
0 16 (2,2)*
16 n R


and


2
T ) -n


(5.35)


(5.36)


(5.37)











Q" was then related to p and T*.

5.3 Test of Correspondence Principle for Inert Gases

No systematic and consistent test of the principle for all

three transport properties of dense fluids over a wide range of tempera-

tures and densities seems to have been made. In particular the trans-

port properties of fluids at very high pressures (and thus high densi-

ties) have not been examined. The behavior at high densities is of

particular interest since it is under these conditions that Assumption

3 may become invalid. Also, errors arising from differences in the

repulsive portion of the intermolecular potential for the molecules

considered (Assumption 4) may become apparent at high density.

The essential simplifying factor in the case of monatomic (as

opposed to polyatomic) molecules is that Assumption 2 is fully

satisfied. A comparison of the correspondence behavior for monatomic

and polyatomic molecules may therefore throw light on the validity of

this assumption. Calculations are presented in this chapter to test

the correspondence principle for all three transport coefficients over

the full range of temperatures and pressures for which data are

available. The extension to polyatomic molecules is discussed in the

following chapter.

Reduction of the transport coefficients are performed using

Lennard-Jones [6,12.] potential parameters rather than critical constants

because the former have more theoretical significance. The use of

critical constants is justified when the group of substances accurately

obey the same potential function, Assumption 4 (50). In such a case

it would in principle be immaterial whether potential function or











critical parameters were used. However, in practice the group of

molecules do not all accurately obey Assumption 4, and the potential

parameters are force-fitted to some semi-empirical equation, in this

case the Lennard-Jones (6,12] potential function. Under such condi-

tions the use of potential parameters rather than critical constants

may improve the correlation obtained. In this work, reduction with

both critical constants and potential parameters (smoothed in the way

described below) was tried. The use of potential parameters was found

to give consistently better results.

The potential parameters reported in the literature by differ-

ent workers show substantial variations. Thus for krypton values of

C, the characteristic energy, reported by different authors vary by

as much as 14%. It is therefore desirable to find some means of

averaging these parameters. Moreover, Reed and McKinley (92) have

shown that polyatomic molecules do not all obey the simple two-para-

meter Lennard-Jones 16,12] potential energy function. Since Assumption

4 requires that all molecules should obey the same potential energy

function in order to arrive at the same corresponding states, an

attempt was made to force the polyatomic molecules to fit a two-

parameter potential energy function by adopting the method used by

Tee, Gotoh and Stewart (93). According to the Pitzer compressibility

correlation (94) the volumetric behavior of a fluid, and thus the

intermolecular forces present, are characterized by P T and the
c c

acentric factorO Thus the intermolecular force constants E and a

should depend on these three quantities, so that one obtains the











dimensionless relations



= f(() (5.38)
kT
c


P p 1/3
SkT ) a = f'() (5.39)
S c

where f and f' are universal functions. In Figures 5.1 and 5.2
P 1/3
values of (c and for twenty different molecules are
SkTC kT
c c
plotted against their corresponding Pitzer factor c. The best straight

lines were drawn through them by the method of least squares. Poten-

tial parameters used in these figures were values calculated from gas

viscosity data, and were taken from Hirschfelder, Curtiss and Bird

(20) and Tee, Gotoh and Stewart (93). Where several values for E and a

were reported for a given molecule the mean values are plotted in

Figures 5.1 and 5.2, and the maximum deviation is indicated. The

calculations presented in this chapter and the next make use of smoothed

potential parameters obtained from these two figures. Values of the

potential parameters used are shown in Tables 5.1 and 6.1. Values for

molecules other than those tabulated may be calculated from the

equations for the best lines shown in Figures 5.1 and 5.2 which are


= 0.7932 0.6723C (5.40)
kT
c

/ c \1/3 = 0.4583 + 0.1213C (5.41)
kT


Figures 5.3, 5.4 and 5.5 are reduced plots of saturated liquid

viscosity, thermal conductivity and self-diffusion coefficient as























'D \ CO
0 CM N =C O =
CM CM Q r- Ci \o CM M
NO U U U u





<> >
C)
c,
x
C)

0





' - r -- 'i














I,










,I-, -




~-----/-


/ -


0 0


0








3

0
o





ulU







F--



c



C)
0 u















0









0


o
Crm
0 \^









-o


I-3
0 0


0
':1


0
4-
C,
C)-


I

























o
,--1


N r Noo co> >









c ,c

jO H 0



0- p ^ C C O U V
o F te<^f Zo ( Q iU U tO


0 10 O0
2 o


u"8


C7


-J
C1)



04
r --'











SP









.C)
C

'0




C)-

C
O r(





O
O




o o


P,>


1



































































q6


\


o\


0 0 t 6
~~~Q

0MX


S I


I I I--I I I I -IIf-. I I I -


Ul)
0



















cn
0














00 C
O
*H

-l





0 0
-a














cC
O


;-I
o
uC

*- *-1


















LQl





0








':3


O









CC)
CO







o



-4

o
.-4


















0

C- -
o o


0 o










0
0 0




C)'







ou








00


03-





CO


I 1 I Ii I i i i i


I


,











































p.


oij~


C ri f) C4
0 0
0 0O
O O


4-J

*-4
>





'4-1
CO 4c















U-4
c*




4-1

















0
C/I
C)
0


















0
*rl
-11
0


C))

t 'ci








0o









Oi
*r-
i-











functions of temperature for the inert gases (argon, krypton, xenon

and neon). For pure saturated liquids there is only one degree of

freedom, and the reduced transport coefficients depend only on reduced

temperature. In the saturated liquid region, the inert gases appear

to obey the corresponding states principle for the above transport

properties reasonably well with the exception of the thermal conducti-

vity data of neon. The discrepancy observed in Figure 5.4 for neon

may be due to quantum effects, since neon is the second lightest

molecule among the inert gases. Experimental data for bulk viscosity

are scarce. The only bulk viscosity measurements made seem to be

those reported by a few workers for argon (14,95,96). Figure 5.6

is a reduced plot of bulk viscosity of saturated liquid argon

(extrapolated data) as a function of temperature. Great experimental

difficulties are involved in bulk viscosity measurements, and the data

are very scattered. Even among the data of the same worker the

average percent deviation is as much as 15%.

In the derivation of the correspondence principle for trans-

port properties shown above, reduced viscosity is found to be a

function of reduced temperature and reduced molal volume. Thus

according to equation (5.19) = (T"',V*). However it is more

convenient in practice to plot the transport properties as a function

of temperature and pressure. Since reduced molal volume may be shown

to be related to reduced pressure through the equation of state, one

may e::press as r"(T",P').

Reduced viscosities of inert gases plotted as a function of

reduced temperature at constant reduced pressures are shown in

































0






ClC












4J
0
0O



0






r--



M







0















SO0



O 0 0 0 0 0 0
N )











Figure 5.7. The corresponding states principle appears to apply

closely to viscosity of inert gases over the entire range of tempera-

ture and pressure for which experimental data are available. Data

plotted in Figure 5.7 cover densities from the dilute gas phase to

the dense gas and liquid phase, and correspond to pressures from zero

to over 2000 atmospheres. A similar plot for the thermal conductivity

of inert gases is shown in Figure 5.8. In this plot the correspondence

principle is found to apply well for argon, krypton and xenon over

the entire temperature and pressure range for which data are avail-

able. The neon data in Figure 5.8 obey the simple correspondence

principle at high temperatures, but not at low temperatures. This

behavior for neon suggests that quantum effects are important for

this fluid at low temperatures.

The reduced viscosity and reduced thermal conductivity data

are found to fit equations of the form

ln-1 *-2 a-3T
In' 1 = al + a22T + aT2 + a T*3 + a5T -4 (5.42)


and InX' = b1 + b2T-1 + b3T*-2 + b T*-3 + b 5T4 (5.43)


at constant reduced pressure or along the saturation curve. The

reduced self-diffusion coefficient, on the other hand, is found to

fit an equation of the form


InD = dl + d2 T (5.44)


The principal parameters used to test the correspondence

principle for monatomic molecules are shown in Table 5.1. The






































C)










0



0
C.-i








-e
u


















0
U1.








c,

0




O4
L)
N-

CO)

, ^

E- o (






G


N 0 0































C3
0
U






0




0
0






i-







C;
0
U

0














01
C)













C)
01




*r-


*l-
O-
U,




Full Text

PAGE 1

CORRESPONDING STATES RELATIONSHIPS FOR TRANSPORT PROPERTIES OF PURE DENSE FLUIDS By MIN JACK THAM A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1968

PAGE 2

ill"

PAGE 3

PREFACE An important consideration for the chemical engineer is his ability to predict reliable values for various properties of chemical substances that are needed for design equations. Because of the large number of combinations of substances, compositions, pressures, temperature, etc., that may be encountered, it is often neither feasible nor desirable to make experimental measurements of such properties. The need for reliable correlations has become more acute with the increasing use of high-speed electronic computers in designing chemical plants. Such design requires suitable mathematical expressions for properties as a function of operating conditions. Even when experimental data in tabulated form are available it is most conveniently introduced into the computer in the form of soundly based theoretical equations. In the long term the only satisfactory approach is one that is firmly based on molecular considerations. Theories for dilute fluids are well established. The kinetic theory of gases is capable of describing the thermodynamic and transport properties accurately. However, for dense fluids there is still no theory which can parallel the success of the dilute gas theory. Although there are several rigorous molecular theories for liquids, none have yet been developed to a stage that would yield numerical results. Theoretical study of transport properties is more difficult than that of thermodynamic properties. In thermodynamics, all equilibrium properties can be obtained once the partition function 11

PAGE 4

of the system is kno^^m. However, for transport processes each of the transport properties has to be formulated separately. This situation occurs because the thermodynamic equilibrium state is unique, whereas there are many types of non-equilibriian state. This dissertation considers corresponding states correlations of transport properties of fluids, particularly liquids. This principle has previously proved of great value to engineers in predicting thermodynamic properties. Chapter 1 provides a brief survey of the present status of theories of transport properties of dense fluids. Chapters 2-4 contain a detailed study of the free volxome theory of viscosity and self-diffusion coefficient, with particular emphasis on developing corresponding states relations for the parameters involved. The next two chapters discuss a more direct corresponding states treatment, and a new theory is proposed for polyatomic molecules. The author is greatly indebted to Dr. K. E. Gubbins, chairman of his supervisory committee, for his interest, stimulation, advice and encouragement during the course of this research. He is also grateful to Dr. T. M. Reed for helpful discussions. Finally, he expresses his sincere appreciation to Dr. T. M. Reed, Professor R. D. Walker and Dr. R. G. Blake for serving on the committee. Ill

PAGE 5

TABLE OF CONTENTS Pa^e PREFACE ii LIST OF TABLES vii LIST OF FIGURES ix LIST OF SYMBOLS xii ABSTRACT xx CHAPTERS : 1. INTRODUCTION 1 1.1 The Time Correlation Function Theory 1 1.2 The Kinetic Theory of Liquids 3 1 . 3 The Enskog Theory 8 1.4 Activation and Free Volume Theories 14 1 . 5 The Correspondence Principle 16 1 . 6 Empirical Correlations , 16 1 . 7 Summary 20 2. PREVIOUS WORK ON ACTIVATION AND FREE VOLUME THEORIES. 22 3 . IMPROVED FREE VOLUME THEORY 27 3 . 1 Temperature Dependence of V 28 3.2 Volume Dependence of E 35 V 4. TEST OF IMPROVED FREE VOLUME THEORY 41 4.1 Corresponding States Relationships 42 4.2 Test of Proposed Correlations 44 4 . 3 Summary 52 IV

PAGE 6

TABLE OF CONTENTS (Continued) 5. PRINCIPLE OF CORRESPONDING STATES FOR MONATOMIC FLUIDS 55 5.1 Molecular Basis of the Correspondence Principle. 55 5.2 Previous Work 63 5.3 Test of Correspondence Principle for Inert Gases 65 5.4 Summary 84 6. PRINCIPLE OF CORRESPONDING STATES FOR POLYATOMIC NONPOLAR FLUIDS . . .86 6.1 Problems in Polyatomic Fluids 86 6.2 Derivation of Corresponding States Principle for Polyatomic Molecules 90 6.3 Test of Correspondence Principle for Polyatomic Molecules 103 6.4 Comparison with Theory for Thermodynamic Properties 124 7. CONCLUSIONS 132 7.1 Free Volume Theory 133 7.1.1 Extension to Mixtures 133 7.1.2 Electrolyte Solutions 133 7.2 Corresponding States Principle 134 7.2.1 Mixtures 134 7.2.2 Polar Substances 134 7.2.3 Fused Salts 134 7.2.4 Thermodynamic Properties 134 7.2.5 Prediction of c 135

PAGE 7

TABLE OF CONTENTS (Continued) APPENDICES 136 1. Chung's Derivation of the Free Volume Equations 137 2. Further Test of Proposed Free Volume Theory 144 3. Solution of Equations of Motion for Oscillator 148 4. Hamiltonian for the Three-Dimensional Oscillator 151 LITERATURE CITED 155 BIOGRAPHICAL SKETCH 164 VI

PAGE 8

Table LIST OF TABLES Page 1.1 Test of Rice-Allnatt Theory for Shear Viscosity of Argon 5 1.2 Test of Rice-Allnatt Theory for Thermal Conductivity o f Argon 7 4.1 Viscosity Data Sources and Range of Conditions 46 4.2 Parameters for Viscosity Prediction 47 4.3 Self-Diffusivity Data Sources and Range of Conditions 48 4.4 Parameters for Self-Diffusivity Prediction 49 5.1 List of Parameters of Monatomic Molecules 78 5.2 Coefficients of Saturated Liquid Viscosity Equation for Monatomic Molecules 80 5.3 Coefficients of Saturated Liquid Thermal Conductivity Equation for Monatomic Molecules 81 5.4 Coefficients of Saturated Liquid Self-Diffusivity Equation for Monatomic Molecules 82 5.5 Coefficients of High Pressure Viscosity Equations for Monatomic Molecules 83 5.6 Coefficients of High Pressure Thermal Conductivity Equations for Monatomic Molecules 85 6.1 Parameters for Correspondence Correlation of Polyatomic Molecules 105 6.2 Sources of Transport Property Data 107 6.3 Coefficients of Saturated Liquid Viscosity Equation for Polyatomic Molecules 117 6.4 Coefficients of Saturated Liquid Thermal Conductivity Equation for Polyatomic Molecules 119 6.5 Coefficients of Saturated Liquid Self-Diffusivity Equation for Polyatomic Molecules 120 6.6 Coefficients of High Pressure Viscosity Equations for Polyatomic Molecules 122 VIL

PAGE 9

LIST OF TABLES (Continued) Table Page 6.7 Coefficients of Gas Thermal Conductivity Equation for Polyatomic Molecules ^^^ 6.8 Coefficients of High Pressure Thermal Conductivity Equations for Polyatomic Molecules 125 6.9 Comparison of c with Corresponding Parameter of Hermsen and Prausnitz ^^' Vlll

PAGE 10

Fisrure LIST OF FIGURES Page 1.1 Theoretical and Experimental Saturated Liquid Viscosity of Argon Square-Well Model 9 1.2 Theoretical and Experimental Saturated Liquid SelfDiffusion Coefficient of Argon Square-Well Model... 10 1.3 Theoretical and Experimental Viscosity of Argon Enskog Theory 13 1.4 Theoretical and Experimental Viscosity of Argon Modified Enskog Theory 15 1.5 Theoretical and Experimental Thermal Conductivity Horrocks and McLaughlin Theory 17 1.6 Viscosity Isotherms of Krypton as Functions of Dens ity 19 3.1 Viscosity of Argon 29 3.2 Viscosity of Nitrogen 30 3.3 Variation of v''' with T^' 33 3.4 Activation Energy as a Function of Volume 35 3.5 Model of a Molecular Jump 37 3.6 Corresponding States Correlation of Activation Energy vs. Volume 39 4.1 Test of Free Volume Theory for Liquid Argon Viscosity 43 4.2 Test of Free Volume Theory for Liquid Xenon Viscosity 50 4.3 Test of Free Volume Theory for Saturated Liquid Methane Viscosity 51 4.4 Test of Free Volume Theory for Saturated Liquid Methane Self-Dif fusivity 53 5.1 Smoothing LennardJones [6,12] Parameters,^— — vs , O). , 68 /^c \^/^ 5.2 Smoothing LennardJones [6,12] Parametersf-rr — Iff vs . CO .'V. .c .J. 69 5.3 Correspondence Principle for Saturated Liquid Viscosity Monatomic Molecules 70 IX

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LIST OF FIGURES (Continued) Figure Page 5.4 Correspondence Principle for Saturated Liquid Thermal Conductivity Monatomic Molecules 71 5.5 Correspondence Principle for Saturated Liquid SelfDif fusivity Monatomic Molecules 72 5.6 Reduced Bulk Viscosity of Saturated Liquid Argon 74 5.7 Reduced Viscosity Isobars as Functions of Reduced Temperature Monatomic Molecules 76 5.8 Reduced Thermal Conductivity Isobars as Functions of Reduced Temperature Monatomic Molecules 77 6.1 Simple Correspondence Principle Polyatomic Molecules. 89 6.2 Simple Corresponderre Principle for Saturated Liquid Viscosity Polyatomic Molecules 91 6.3 A One-Dimensional Oscillator Model 93 6.4 Superimposed Rotational Motions of a Molecule 94 6.5 Improved Correspondence Principle for Saturated Liquid Viscosity Polyatomic Molecules 110 5.5 Improved Correspondence Principle for Saturated Liquid Thermal Conductivity Polyatomic Molecules Ill 6.7 Improved Correspondence Principle for Saturated Liquid Self-Dif fusivity Polyatomic Molecules 112 6.8 Test of Improved Correspondence Principle for High Pressure Viscosity 113 6.9 Test of Improved Correspondence Principle for Gas Thermal Conductivity Polyatomic Molecules 115 6.10 Test of Improved Correspondence Principle for High Pressure Thermal Conductivity 116 6.11 Simple Correspondence Principle Vapor Pressure 130 6.12 Improved Correspondence Principle Vapor Pressure 131 A2 , 1 Test of Free Volume Theory for Liquid Nitrogen Viscosity 145 X

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LIST OF FIGURES (Continued) Figure Page A2.2 Test of Free Volume Theory for Liquid Krypton Viscosity 146 A2.3 Test of Free Volume Theory for Liquid Neon Viscosity. 147 A4.1 A ThreeDimensional Oscillator Model 151 xi

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LIST OF SYMBOLS = pre-exponential factor of free volume equation for viscosity o A.,A ,A ,A = arbitrary constants A' ,A' = constants A' ,A" = constants a ,a ,a ,a ,a = coefficients of viscosity equation a = molecular diameter B = pre-exponential factor of free volume equation for self-diffusivity B = B/T^^^ o b" = (B m^/^)/(Ok^''^) o o B ,B ,B ,E, = arbitrary constants B',B" = constants B = arbitrary constant a b = the rigid sphere second virial coefficient o b ,b ,b ,b, ,b = coefficients of thermal conductivity equation C^ = empirical constant C = arbitrary constant c = a characteristic factor defined by equation (6.37) c = total concentration o c 1 1 c / 3 = v/r^ = constant 3 2 = {xjr ) = constant c„ = (CtO/v ) = constant J 1 m c, = e/kT = constant 4 m Xll

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D = self-diffusion coefficient D,„ = binary diffusion coefficient AB D = Dm / Oc = reduced self-dif fusivity by simple correspondence principle 'Wf 1/2 1/2 D = Dm /coe = reduced self-dif fusivity by proposed correspondence principle D = reduced mutual diffusion coefficient defined by equation (5.33) d = distance of closest approach of 2 molecules d, ,d„ = coefficients of self-dif fusivity equation = average total energy of molecule i E = activation energy at constant volume V E'" = E /RT = reduced activation energy v V m e = varying activation energy per molecule = average activation energy per molecule e. = energy level J e = height of potential barrier in equilibrium liquid e = E /N V V F. = y-component of intermolecular force on molecule i f = time smoothed singlet distribution function -(2) f = time smoothed pair distribution function f^ = a function of density given by equation (1.23) g = universal function g (r) = equilibrium radial distribution function g = geometric factor H = Hamiltonian Xlll

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H_, = Hamiltonian of center of mass coordinates CM h = Planck constant h" = h/ (ay/ me) = reduced Planck constant I = moment of inertia I= viscosity constant used in equation (1.22) XX J = defined by equation (1.6) xy J = defined by equation (1.5) K = kinetic energy K = kinetic energy of center of mass coordinates K = kinetic energy of rotational motion K^ = (n/n-6)(n/6)^/^'^-^^ k = Boltzmann constant k^ ^ = defined by equation (6.8) kj^2 ~ defined by equation (6.9) k„„ = defined by equation (6.10) L = Lagrangian 1 = jump distance M = molecular weight M = molecular weight of component A Mg = molecular weight of component B m = mass of molecule N = number of molecules N = Avogadro number n = repulsive exponent in [6,n] potential law n^ = singlet number density XIV

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P = pressure P = critical pressure c P „ = critical pressure of a mixture A-B cAB * 3 P' = pa /t = reduced pressure by simple correspondence principle P' = Per Ice = reduced pressure by improved correspondence principle p = momentum p. = momentum in x-direction of molecule i IX p = probability of a molecule having activation energy ^ e. and free volume v 1 P * 1/2 p = p/(m£) = reduced momentum Q = partition function Q = defined by equation (5.35) q = thermodynamic free volume r^ = position vector r . = r . r . -ij -J -1 r = r/cr = reduced distance (s/c), = a constant factor used by Hermsen and Prausnitz "" (166) s = defined by equation (1.7) T = temperature T = critical temperature c T = critical temperature of a mixture A-B cAB T = melting temperature m T* = kT/e = reduced temperature by simple correspondence principle XV

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'V', T" = kT/c£ = reduced temperature by improved correspondence principle Oj T = kT/[£(s/c), ] = reduced temperature used by Hermsen and Prausnitz (156) t = t ime * 1/2,1/2^.., t = t£ / CT m = dimensionless time U(r) k N ) u(,r..; = pair potential i
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= V/N = average volume per molecule v^ = V V = free volume f o V = V /N * 'V. 'O '\y v" = v'Vn v = varying free volume vj = (d/a)^ v = V /N = minimum free volume per molecule needed o o for molecule flow to occur W ,W2 = number of configurations for distribution of molecules X. = x-component of position vector r of molecule i Y = average percent error (see page 80) y = b^x/v Z = conf igurational partition function 2 = number of nearest neighbors in a ring a = Lagrange multiplier ]S = 1/kT = Lagrange multiplier y = free volume overlap factor £ = characteristic energy parameter in [6,n] potential ^ = molecular friction constant T) = shear viscosity 7) = dilute gas viscosity 9 1/9 Tj'^ = 7)0 / (me) = reduced shear viscosity by simple correspondence principle "^j 2 1/2 1^ = T)CT /c(m£) = reduced shear viscosity by improved correspondence principle T) = defined by equation (5.36) = angle vector (in terms of Eulerian angles d,
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e X /c K. K ov TT a x-component of an angular rotation = bulk viscosity * J-,, vl/2 /CO /Cm£; = reduced bulk viscosity by simple correspondence principle * _ „ 2, , ,111 K.a /c(in£) = reduced bulk viscosity by improved correspondence principle A = Lagrange multiplier ^ = thermal conductivity '^Q dilute gas thermal conductivity y* ^^2 1/2,, 1/2 ^ AO m Ike = reduced thermal conductivity by simple correspondence principle >" _ A 2 1/2. , 1/2 A. AO m /ck£ = reduced thermal conductivity by improved correspondence principle ^ = B'/A' = constant s = constant = a constant which has a value of 22/7 P = density ^P'^\ = value of the product of density and self-dif fusivity of a dilute gas P' = reduced density characteristic distance parameter in C6,n] potential °^1'^2 ~ characteristic distance parameter in square-well potential ^ = a universal potential energy function '^x "" angle associated with constant angular velocity = a universal function X = a factor defined by equation (1.13) \ = probability of hole formation XVlll

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^ = angle associated with superimposed angular velocity tp ,i1j^ = the radial functions in the spherical harmonic expansion of the nonequilibrium distribution function g(r) „(1,1)-^ r^(2,2)* ^. . ^ ,, . . . ii , it ' ' = dimensionless collision integrals Oi = Pitzer factor CO = angular velocity CO, = angular velocity in a harmonic motion XIX

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Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CORRESPONDING STATES RELATIONSHIPS FOR TRANSPORT PROPERTIES OF PURE DENSE FLUIDS By Min Jack Tham June, 1968 Chairman: K. E. Gubbins Major Department: Chemical Engineering Two approaches have been considered for developing corresponding states correlations for dense fluid transport properties. The first is based on the free volume theory for viscosity and selfdiffusion coefficient. This theory has been improved to account quantitatively for the nonlinearity of constant volume plots of ln(T]/T ) vs. T~ and InCD/T''"' ) vs. t"''" for simple liquids, by the introduction of the temperature dependence of V and the volume dependence of E . Corresponding states relations are presented for these two quantities. The improved free volume theory accurately describes the viscosity and self-diffusion coefficient behavior of liquids composed of simple, nonpolar molecules that may be expected to approximately obey a LennardJones [6,12] potential law, for the density range of p > 2p . It is in this range that most other theories fail. The only parameters required for the viscosity and self-diffusion coefficient prediction are melting and critical parameters, together with one experimental value of viscosity (or selfdif fusivity) . XX

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Direct corresponding states correlations of transport properties have also been studied, and provide a more general if slightly less accurate method of prediction. The simple corresponding states principle derived by Helfand and R.ice from the time correlation function expressions of statistical mechanics has been shown to apply closely for the transport properties of monatomic molecules over the entire range of temperature and density conditions for which measurements are available. In this simple form, however, it was found not to apply to polyatomic fluids in the dense phase. After careful reexamination of the four assumptions on which the derivation of the simple correspondence principle is based, it is shown that hindered rotation at high density is the most probable cause for the deviation of polyatomic fluids from the correspondence principle. A correspondence principle for polyatomic molecules has been derived, taking into account the effects of hindered rotation and making use of a simple harmonic oscillator model. The proposed equations have been tested exhaustively for the viscosity, thermal conductivity and self-diffusion coefficient of a large number of polyatomic fluids over a wide range of temperature and density conditions. The molecules tested include saturated hydrocarbons from methane to nonane, spherical molecules such as neopentane and carbon tetrachloride, flat molecules such as benzene, and simple diatomic molecules such as nitrogen. The improved correspondence principle predicts the transport properties of saturated liquids within experimental error. For high pressures the average percentage errors of the predicted viscosity and thermal conductivity are slightly higher. XX i

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It is also shown that the proposed model can account in a reasonable way for thermodynamic properties of polyatomic fluids. XXI 1

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CHAPTER 1 INTRODUCTION In this chapter a brief survey and evaluation is presented of the principal theories and empirical correlations that have been proposed to explain dense fluid transport properties. For pure fluids all transport properties are functions of temperature and density, namely, r] = rj(T,p), /c = /c(T,p), D = D(T,p) and >, = \(T,p). None of the present theories provide a satisfactory means of predicting transport properties over the entire range of temperature and density conditions for even quite simple fluids. Also assumptions of dubious validity must be made in most cases in order to be able to make any comparison with experiment at all. In the following survey the more rigorous approaches are first described, followed by simple model theories and ending with the empirical correlations. 1 . 1 The Time Correlation Function Theory In the last decade a new theory, called the time correlation function theory (also known as the fluctuation-dissipation theory) has been developed to treat transport processes. This theory treats transport processes in terms of equilibrium ensembles whose properties are known. In a sense the time correlation function plays a similar role to the partition function in statistical thermodynamics. In statistical thermodynamics, all thermodynamic properties of any particular system can be evaluated if the partition function is known. In the same manner the transport coefficients can be obtained if the appropriate time correlation functions can be calculated. However, in

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one respect the analogy breaks down. In statistical thermodynamics a single partition function determines all the thermodynamic properties, whereas in transport processes different time correlation functions are needed for different transport processes. A good review of the time correlation function approach to transport processes is given by Zwanzig ( 1) . The time correlation function expressions for the transport coefficients may be derived by several methods. A simple derivation involves starting with expressions analogous to the Einstein equation for the self-diffusion of Brownian particle (2). The following expressions for transport coefficients are obtained. CO Self-diffusion coefficient Shear viscosity Bulk viscosity D= r r) = /c = dt / X X o 00 1 VkT VkT dt o 00 dt (1.1) (1.2) (1.3) Thermal conductivity X = ^<^o)sl:t)> dt VkT (1.4) where V (t) = velocity in the x-direction at time t "" N 4L pp"^ix ly N ,xx S^ = i=l N m PPIX IX m + x.F. 1 1 ly J + x.F. 1 IX i=l PV p. p. IX IX 2m 1 XX N 2 /_, L ij XX (1.5) (1.6) + X. .F. . N m Nm ^ ix i=l (1.7)

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For dilute gases the time correlation functions may be evaluated and the expressioiB for the transport coefficients can be reduced to the forms obtained by solving the Boltzmann equation. However, for dense fluids no satisfactory expressions have as yet been obtained, although several approximate equations have been suggested (3) . One approach has been to assume that the time correlation function decays exponentially. Alternatively one may perform molecular dynamics calculations (4-6) . In this method a very large amount of computation is involved; the principal interest of such an approach is to provide data with which to compare various theoretical expressions for the time correlation functions. The time correlation theory gives a formal description of transport processes; however it is still not developed to a sufficiently advanced stage to be of practical value in the calculation of transport coefficients. To the chemical engineer, the most interesting feature of the theory in its present state of development is that it provides the basis for a rigorous development of the corresponding states principle for dense fluid transport properties. This is discussed in a later section of this chapter. 1 . 2 The Kinetic Theory of Liquids The kinetic theory of dilute gases is already very well developed (7), and provides a method of accurately estimating their transport properties. It is therefore reasonable to attempt a similar development for dense gases and liquids. Such an attempt was first made by Kirkwood (8,9), whose approach was based on casting the

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Liouville equation for the distribution function into an equation of the Fokker-Planck type. The main disadvantage in this theory is that a large number of approximations have to be made in obtaining the Fokker-Planck equations for f and f , the time smoothed singlet and pair distribution functions respectively. In addition to these deficiencies, Kirkwood's method has drawn much criticism for assuming the mean momentum change for a collision to be small. The equations derived by the method of Irving and Kirkwood (9) for transport coefficients of liquids are as follows (10) Shear viscosity , „ oo T] =^-^^ r' ^ Z^(r)^,(r)ar (1.8) ^ 15kT ^ ar o z Bulk viscosity r ^ °° Thermal conductivity „ . 5k niT nVTTkT °° A ~ O 00 dr (^ St •) r^dr (1.10) Sg^(r) 2 oT kT Self-diffusion coefficient D = — r (1.11) where °° C =^p rr^g(r)v^u(r)dr (1.12) ^o and ^o'^2 ~ ^^^ radial functions in the spherical harmonic expansion of the nonequilibrium distribution function g(r) A modification of the Kirkwood theory due to Rice and Allnatt (11,12) attempts to avoid the assumption that the mean collisional

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momentum change is small. In the Rice-Allnatt theory, a potential energy function is assumed in which the molecule has a hard core together with an outer shell which interacts with a soft attractive potential. A hard core collision leads to a large momentum transfer and tends to vitiate the Brownian motion approximation which is inherited in the Fokker-Planck equation. These authors therefore treated the rate of change of the distribution function f due to hard core collisions by means of an Enskog-type collision term, and treated the rate of change of the distribution function due to motion in the attractive field of surrounding molecules betv^7een collision by Kirkwood's approach. In order to calculate the transport coefficients from the Kirkx^70od or Rice-Allnatt theory one must have accurate values for the intermolecular potential, and also the equilibrium radial distribution function has to be known quantitatively. The latter requirement is especially difficult to meet since the experimental radial distribution function has not been determined accurately. Making use of the Kirkwood theory, Zwanzig, Kirkwood, Stripp and Oppenheim (13) were able to calculate the shear viscosity and bulk viscosity for liquid argon near the boiling point. Their calculated value for shear viscosity was r] = 0.73x10 poise, while the experimental value of shear viscosity for argon at the same temperature is _3 T) = 2.39x10 poise. Thus the calculated value is in error by roughly a factor of three. For bulk viscosity, their computation yielded k; = 0.36x10 poise. However, according to Naugle's (14) measurement of bulk viscosity for liquid argon at approximately the same temperature,

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-3 K = 1.7x10 poise. The discrepancy between calculated and experimental values is more than a factor of four. Because of the nontrivial mathematics involved, the Kirkwood theory has not been tested for the temperature and density dependence even for very simple molecules such as argon. The Rice-Allnatt theory has been tested by Lowry, Rice and Gray (15) for viscosity. They found quite good agreement between calculated and experimental values of argon at densities slightly less than the normal liquid density (Table 1.1). To compare theory and experiment they assiomed u(r) to be given by the LennardJones [6,12] potential. The radial distribution functions, g(r) for different temperatures were those obtained theoretically by Kirkwood, Lewinson and Alder (16) . The radial distribution functions g(r) and the derivative of u(r) were modified by introducing an empirical parameter c^ , which was obtained from equation of state data. TABLE 1.1 TEST OF RICE-ALLNATT THEORY FOR SHEAR VISCOSITY OF ARGON -3 Density, gem Temperature, K Pressure, atm. ^ rj(calc) T](expt) 7o error r)xlO-

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Ikenberry and Rice (17) tested the Rice-Allnatt theory on the thermal conductivity of argon for the same temperature and density conditions as the viscosity values shown in Table 1.1. The calculated values were not entirely theoretical since they made use of experimental friction constants calculated from the self-diffusion coefficient data of Naghizadeh and Rice (18) . Close agreement was found bet\-;een theory and experiment for the entire temperature and density range considered as shown in Table 1.2. TABLE 1.2 TEST OF RICE-ALLNATT THEORY FOR THERMAL CONDUCTIVITY OF ARGON XxlO^, cal.cmT-'-secT-'-^c" _3 Density, gem.

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In general a , a„ and e for the square-well potential are available (20). In principle g(a ) and g(a ) can be calculated from equilibrivim statistical mechanics. However there is still no satisfactory method of making such theoretical calculations accurately. Davis and Luks (21) estimated the values of g(0'^) and g(0' ) from equation of state and thermal conductivity data of argon by solving simultaneously the modified Rice-Allnatt expressions. They then calculated the viscosity coefficient and self-diffusion coefficient of argon at various temperatures and pressures. The results are shown in Figures 1.1 and 1.2. The discrepancy between theory and experiment for viscosity is as much as 19% and for the self-diffusion coefficient is up to 34%. The experimental self-dif fusivity data used by Davis et al. as shown in Figure 1.2 are apparently extrapolated values obtained from the self-diffusion coefficient data of Naghizadeh and Rice (18). Luks, Miller and Davis (22) have also made use of the modified Rice-Allnatt theory to calculate transport coefficients for argon, krypton and xenon. Discrepancies between theory and experiment become larger as the density increases. 1.3 The Enskos Theory (7) Enskog extended the dilute gas kinetic theory to dense fluids composed of hard spheres. As a gas is compressed the mean free path, which for dilute gases is infinitely large with respect to the molecular diameter, gradually decreases until at high density it is of the same order of magnitude as the diameter of a molecule. During this process the intermolecular collision frequency tends to get larger due to the decrease in the mean free path; at the same time it decreases because

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o

PAGE 33

10 c o •r-l n3

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11 at close separations the molecules tend to shield one another from collisions with more distant neighbors. The net change in collision frequency was found to differ from that of a dilute gas by a factor ^, which is given by X = 1.0 + 0.625^-1)+ 0.2869 (^ ) +0.115i;:^ ) (1.13) + . . . where b = — ttNCt = the rigid sphere second virial coefficient. Assuming the collisional transfer of ' momentimi and energy between colliding molecules to be instantaneous, Enskog modified the Boltzmann equation for higher densities and solved it to obtain the following equations for transport coefficients. b Self-diffusion coefficient oD (pD)^ Vy (1.14) Shear viscosity _o V + 0.8 + 0.761y (1.15) Bulk viscosity 1.002 -^y (1.16) Thermal conductivity ^ b / . , , s A. o (monatomic molecules) r~ ~ o + 1.2 + 0.757y y (1.17) where y =— X The value of y may be obtained from the equation of state by — = 1 + y RT ^ (1.18) Enskog suggested as an empirical modification of the hard sphere

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12 theory the use of thermal pressure in place of external pressure. Thus according to this modification y becomes Equations (1.15) and (1.17) predict that when (r)V) and (XV) are plotced as functions of y, the curves will go through minima at y = 1.145 and y = 1.151 respectively, having [r]V] = 2.545r] b^ at y = 1.146 (1.20) min " ° and [XV] . = 2.938X b at y = 1.151 (1.21) mm o o In the above transport coefficient expressions, b is the only unknown parameter. It may be obtained from equations (1.20) and (1.21) by using experimental values of [rN] . and [XV] . . ° ^ '-'I Jmin -'mm The Enskog dense gas theory gives a useful approximate description of transport coefficients above the critical temperature and at densities less than the critical value. The theory fails at temperature below the critical temperature and at densities higher than the critical density. Under these conditions the rigid sphere approximation is in serious error. The Enskog theory has been tested for thermal conductivity and viscosity of argon at various temperatures and densities (23) . Figure 1.3 shows the comparison of calculated and experimental viscosities of argon at 0°C and 75°C. The value of b used in these calculations was obtained from equation (1.20) using data at C. The agreement between theory and experiment is moderately good at C for

PAGE 36

13

PAGE 37

14 densities up to about 0.8 g/cc. At hi^er densities the theory fails to predict the correct density dependence. As can be seen in Figure 1.3, the Enskog theory does not predict the correct temperature dependence. Dymond and Alder (25) recently modified the Enskog theory by using temperature dependent rigid sphere diameters obtained from the Van der Waals equation of state. With this modification, the theoretical and experimental values of viscosity are shown in Figure 1.4. The predicted temperature dependence is found to improve considerably; however the theory still fails to predict the density dependence at densities above 0.8 g/cc. 1 .4 Activation and Free Volume Theories An activation theory for liquid transport properties was first proposed by Eyring (26). This type of theory assumes that a molecule spends a large fraction of its time oscillating about an equilibrium position in a cell, and only occasionally does it leave one cell to take up position in a neighboring vacant cell. Transport of mass and momentum are assumed to occur during such molecular transitions. Later modifications to Eyring' s original theory have included the introduction of the concept of the fluctuating free volume. These theories are discussed in detail in Chapter 2. The free volume theories apply only at densities above approximately twice the critical value. Korrocks and McLaughlin (27) applied the activation and free volume theory to the thermal conductivity. They assumed a facecentered-cubic lattice structure for the liquid and that transfer of thermal energy down the temperature gradient was due to two causes:

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15

PAGE 39

16 the actual transit of a molecule from one lattice site to another, and that due to the collisions of an oscillating molecule with its neighbors. For liquids the contribution due to the former cause is negligible when compared with that of the latter. The frequency of oscillation is determined by the molecular mass and the intermolecular force. Their theory has been tested for a number of simple liquids and the agreement between theory and experiment is often within 20%. Calculated and experimental thermal conductivities of a number of liquids are compared in Figure 1.5. 1-5 The Correspondence Principle The principle of corresponding states has been found very useful in the calculation of equilibrium properties of dense gases and liquids (28). However very few studies have been made of its application to transport properties of dense gases and liquids. As the principle of corresponding states will be discussed in great detail in later chapters it will not be discussed further here. 1.6 Empirical Correlations A very good review of the purely empirical methods of estimating transport coefficients is given in a new book by Reid and Sherwood (29). These methods have neither a theory nor a model to describe them. The many expressions proposed usually contain one or more empirical parameters or constants which are said to characterize the structure or properties of the molecules; in most cases these parameters have to be supplied by the authors of the correlation. A typical example of these correlations is Souder's method for estimating liquid viscosity, which is one of the few empirical methods recommended by Reid and Sherwood.

PAGE 40

17 o CO

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18 Souder's equation is I. log(log IOt]) = ^ p 2.9 (1.22) where T] = liquid viscosity, centipoise -3 ,0 = liquid density, gem. I = viscosity constant calculated from atomic and structural constants supplied by the author M = molecular weight The results yielded by such methods are usually not very satisfactory. In spite of the author's claim that viscosity predictions are within 207o, errors are often greater than this. Thus the equation predicts a viscosity for acetic acid at 40°C that is 36% too low (29). Similar empirical expressions are also available for liquid thermal conductivity and self-diffusion coefficient; the discrepancies between these equations and experimental data are of a similar order to that of the viscosity correlations. Among the empirical correlations, those based on residualviscosity concepts have received much attention (30-32). Figure 1.6 is a plot of viscosity isotherms for krypton as a function of density. At the lower densities the curves are almost parallel, and this has led a nimber of workers to correlate the residual viscosity r] T] with density. The general form of this correlation is ri-ri^= f^(p) (1.23) where f^ is a function of density only and T) is the dilute gas viscosity. However, as can be seen in Figure 1.6, the isotherms

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19

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20 intersect at higher densities, so that such correlations are not valid over the Xv^hole density range. The intersection occurs at densities of about txcfice the critical value. Above this density value the free volume theory is found to work well. 1 .7 Summary Of the different approaches discussed above, the time correlation function theory and the kinetic theory of liquids are the most rigorous and correct descriptions of transport processes in dense fluids. In their present stage of development, however, they offer little immediate prospect of yielding methods of predicting transport properties for fluids of interest to chemical engineers. The Enskog dense gas theory provides good results for dense gases, but fails at densities a little above the critical value. For the lower densities quite good results may also be obtained from empirical equations, such as those employing the residual viscosity concept. At present a particular need exists for satisfactory methods of predicting transport properties for fluids at densities well above the critical, that is in the normal liquid density region. Toward this end t^
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21 Chapters 3 and 4 contain the new contributions to the theory. The second approach consists of the development of corresponding states relationships directly from the time correlation function theory. The correspondence principle is particularly valuable when based on the rigorous statistical mechanical approach, but no serious attempt to apply che principle to dense fluid transport properties seems to have been made previously. This approach is examined in detail in Chapters 5 and 5 .

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CHAPTER 2 PREVIOUS WORK ON ACTIVATION AND FREE VOLUME THEORIES Because of the present difficulties associated with developing a rigorous kinetic theory of dense fluids analogous to that available for dilute gases, considerable attention has been given to simplified models of the liquid state. Assumptions are introduced in the initial stages of the development, an attempt being made to obtain a model which incorporates the essential features of real liquids, but which is still solvable. Such attempts are exemplified by the activation and free volume theories for liquid transport properties. Eyring (26) may be considered a pioneer in the development of activation and free volume theories for liquid transport properties. In Eyring' s original approach the equilibrium positions of molecules in a liquid were considered to be on a regular lattice similar to the crystal lattice in a solid. The transport of mass and momentiim occurs as a result of a molecule jumping from one equilibriiom lattice site onto a vacant neighboring site. In order to make such a jxjmp a molecule is required to have the necessary activation energy to overcome the minimum energy barrier separating two adjacent sites, and at the same time a vacant site must be available. Using this model Eyring obtained the following expressions for liquid viscosity and selfdiffusivity by making use of the absolute reaction rate theory: 1 ,^ ,^xl/2 3 %^^^ T] = -fY3(271inkT)"' q^e " (2.1) pv 22

PAGE 46

23 2 -e^/kT ^ = 1/2 1/3 ^ ^2-^> (27nnkT)^^ q^ '^ More recently Weymann (35,36) used a statistical mechanical approach, and arrived at expressions very similar to those of Eyring. In his derivation of the equations for transport properties, Weymann gave a clearer picture of the physical model used and of the significance of the activation energy. Moreover his approach suggests a possible extension to liquid mixtures. Weymann also derived an expression in which the probability of hole formation is related to the volume and energy needed to form a hole. The equations obtained by Weymann for viscosity and self-diffusion coefficient are: ^ = 4^f "in) (2TltnkT)^/2e'° (2.3) 2X, 1 . ., „ V. 1/2 -e^/kT D =— ll2kT \ ^ o (2.4) 3 \ Tim / where Xy, is the probability of hole formation. Eyring' s activation theory was modified by McLaughlin (37) by introducing the concept of the probability of hole formation, as was done by Weymann. In his treatment, he assumed a face-centeredcubic lattice for the liquid. Tae most important modification made by McLaughlin was the attempt to relate the activation energy and energy of hole formation to the intermolecular forces by means of the LennardJones and Devonshire cell theory of liquids. A similar expression for liquid viscosity was also obtained by Majumdar (38)

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24 using the tunnel model of Barker (39) in place of the LennardJones and Devonshire cell model. The Eyring and Weymann theories yield good results for the viscosity of Arrhenius liquids (ones whose viscosity varies with temperature according to an equation of the type Inr) = C + B /T) , a a but cannot account for nonArrhenius liquid behavior. In order to overcome this deficiency Doolittle (40-42) proposed empirical expressions which related liquid transport properties to the free volume, defined by v^ = V V (2.5) f o where v is the hard-core volume of the molecule. His free volume o equations were placed on more solid theoretical grounds by Cohen and Turnbuli (43,44) who derived them by a statistical mechanical method. They assumed that a molecule moves about in a cell in a gaslike manner, while the free volume available to each molecule fluctuates with time. A molecule is able to jump into a neighboring cell only if a free volume of a size greater than some minimum value v is available. ° o They obtained the following expressions for the viscosity and selfdiffusion coefficient: Tne term "free volume" in these theories has a different

PAGE 48

25 meaning from that implied in thermodynamic free volxome theories. The theories of Doolittle and Cohen and Turnbull assumed the molecules to behave as hard spheres; the free volume referred to here is the space in the fluid unoccupied by the hard spheres themselves. The average free volume per molecule is defined as v v , where v = V/N. The free volume theory of these authors describes the viscosity behavior at atmospheric pressure, but fails to predict the temperature dependence of viscosity at constant volume. Recently, Macedo and Litovitz (45) proposed expressions for viscosity and self-diffusion coefficient in which the hole probability in Weymann's equations was replaced by the free volume expression obtained by Cohen and Turnbull. Molecular transport is assumed to occur if a molecule has sufficient energy, e to overcome intermolecular forces with its neighbors, and at the same time it has a free volume greater than the minimum free volume v needed for a jump to occur. The equations proposed for viscosity and self-diffusion coefficient are 7V T] = Aexp f -—V ) exp f^ ) (2.8) V V ; '^ VRT o 7V N. / E \ D = Bexp ( —V ] exp (^ ) (2.9) V V ; " \ KT J By treatirg V and E as empirical constants Macedo and Litovitz have shown that equation (2.8) describes the viscosity behavior of a number of liquids over a range of temperature. Chung (46) has presented an elegant statistical mechanical derivation of the equation of

PAGE 49

26 Macedo and Litovitz. His derivation is presented in Appendix 1. Macedo and Litovitz assume the pre-exponential factor A in equation (2.8) to be proportional to temperature T although most other 1/2 vjorkers predict a temperature dependence of T . Both types of temperature dependence of the pre-exponential factor have been tested in this work, and experimental results seem to give better agreement 1/2 with theory xvhen a temperature dependence of T is used. Thus throughout this work the pre-exponential factors A and B are assumed 1/2 to be proportional to T ' . Thus equations (2.8) and (2.9) may be rewritten as (2.10) where and V

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CHAPTER 3 IMPROVED FREE VOLIME THEORY The viscosity equation (2.8) has been tested by Macedo and Litovitz (45) for a variety of liquids and by Kaelble (47) for polymeric substances. They treated the pre-exponential factor A , the activation energy E and the minimum free volume V as adiustable "^ V o -^ parameters characteristic of the substance considered. Equation (2.8) was found to describe the viscosity behavior of the substances tested by these workers moderately well. On the other hand, Naghizadeh and Rice (18) tested the theory for the sclf-dif fusivity of simple fluids (such as the inert gases) and found that the agreement between theory and experiment was poor, especially in the high density region. If V is a constant, as assumed by Macedo and Litovitz, o ^ differentiation of equation (2.10) with respect to 1/T at constant volume (i.e. constant density) yields 5ln(Ti/T^^^ 1 =^ a(l/T) J^ R (3.1) E represents the minimum energy required by a molecule to overcome intermolecular forces in making a jump and was also assumed constant by Macedo and Litovitz. Therefore equation (3.1) predicts that a 1 / ? plot of ln('r)/T ' ) versus reciprocal temperature at constant volume should give a series of parallel straight lines. For a variety of nonpolar and slightly polar liquids over a moderate density and temperature range such a plot produces straight lines, but the slopes 27

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28 of the lines vary with volume. Moreover, when the results are plotted over a wide temperature range nonlinearity becomes apparent. Such nonlinearity is particularly marked for fluids composed of simple molecules. This is illustrated in Figures 3.1 and 3.2 for argon and nitrogen, for which data are available at constant volume over wide ranges of temperature and density. From the above discussion it is apparent that the equations as used by Macedo and Litovitz do not correctly predict qualitatively the effect of temperature and volume on the viscosity, especially for simple fluids. In addition, extensive experimental viscosity data are needed for each fluid in order to fit the adjustable parameters A , E and V : when used in this way their equation is no more than an empirical correlation. In this chapter the physical significance of the parameters E and V is examined in the light of the theory, and their dependence on molecular type, temperature and volume is discussed. Interpretation of these quantities on the molecular level suggests corresponding states relationships which may be used to predict the parameters. The improved theory also explains why the theory of Macedo and Litovitz fails for simple fluids. 3.1 Temperature Dependence of V The parameter V of equations (2.10) and (2.11) represents the minimum free volume that must be available before a jump may occur. If the molecules may be treated as rigid spheres, as in the smoothed potential cell model of Prigogine (50), the free volvime is independent

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-7,5 -8.0 29 Experimental data (48) i . D 9.0 9.5 3 -1 29.14 cm. mole •10.0 39.15 •10.5 J L 2.0 4.0 6.0 3.0 10.0 1/T X 10^ (°k"S 12.0 14.0 Figure 3.1. Viscosity of Argon.

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30 •8.0 •8.5 -9.0
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31 of the teniDerature at constant volume, and V should therefore be a o constant for a particular molecule, independent of temperature and density. Macedo and Litovitz (45) assumed this to be the case for all molecules treated by them. For more realistic potential models one would expect V to o decrease somewhat with increasing temperature, since as temperature rises the average kinetic energy of the molecules increases, and molecules are thus able to approach each other more closely. According to such a viewpoint V should not be affected by the density at constant temperature. The extent to which this parameter depends on temperature will be determined largely by the repulsive portion of the intermolecular potential energy curve. Since this part of the curve rises less steeply for simple molecules than for more complex polyatomic molecules, one would expect the effect of varying V to be most evident for the simpler molecules. To obtain a general expression describing the temperature dependence of V , a [6,n] pair potential energy function is assumed. Tnus a \ fa -I (3.2) u = £K w^s^e / \ X \ 6/(n-6) K = ^ n \ /n^ 1 The parameter n indicates the steepness of the repulsive part of the curve; for small molecules such as the inert gases n is close to 12, whereas for more complex molecules the best values of n is 28 or

PAGE 55

32 higher. For a group of roughly spherical molecules V may be assumed to be proportional to the cube of the molecular "diameter." An estimate of the variation of the molecular diameter (and hence V ) with ^ o' temperature may be obtained by equating the average kinetic energy of a t^v^o-particle system to the potential energy of the system at the distance of closest approach, d: 3kT = £K, (3.3) or, in reduced form 3T* = K, n/3 2 -1 (3.4) where T"" = kT V Equation (3.4) may be solved to obtain v ' as a function of T' for a various n values. The temperature dependence of v'! for several n values is shown in Figure 3.3. Assuming that V is proportional to d , one may write V = ^v o ^ a (3.5) where ^ is constant for a particular molecule, V at any temperature can be calculated from V V = V • -rr S: o o V" (T" = 1) a (3.6)

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33 1.0 t 0.9 0.8 0.7 0.6 0.5 •oo •c. Equation (3.4) > Equation of State Data (25) Ne Ar Kr Xe 0,4 L 0. 2.0 4.0 kT/e 6.0 8.0 Figure 3.3. Variation of v* with t'. 10.0

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34 where V is the value of V at T' = 1. Thus a knowledge of V , o o o together with n and e, suffices to calculate V at any temperature. Some support for the above procedure is provided by values of the hardsphere diameter calculated at various temperatures from equation of state data for inert gases by Dymond and Alder (25) . These values are included in Figure 3.3, and agree well with the curve for n = 12. With n = coj no temperature dependence should be observed for V . Under these conditions, from equation (2.10) ' 5ln(VT^^S l _ ^v ,^ ,. S(l/T) J^ -"i ^^-^^ 1/2 1 •When ln(rj/T ' ) is plotted against T at constant volume for such nonpolar molecules as decahydronaphthalene, benzene, etc., approximately linear behavior is observed; for these molecules n is large and the variation of V with temperature is small. For simple molecules, such as the inert gases, for which n % 12, the temperature dependence is more marked, and the model predicts noticeable nonlinearity on such a plot. This is as observed experimentally, as shown in Figure 4.1 in the next chapter. In addition to the above comments, the parameter V should be related in some way to molecular size. If V may be made dimensionless with some suitable reducing parameter, the resulting reduced minimum free volume may be quite constant for a series of similar molecules. Since free volume theories apply best at high densities, the molal volume at the melting point seems a more appropriate reducing parameter than the critical volume. Thus

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35 1'' ,,i" o ^o =F (3.8) m 3.2 Volume Dependence of E Macedo and Litovitz (45) treated the minimum activation energy E as an empirical constant, independent of density and temperature. However, as pointed out by Brummer (51) , E may be expected to vary with the average intermolecular distance, and thus with density. The value of E becomes larger as the molal volume decreases because of the increase in repulsive intermolecular force between molecules at close separations. From equations (2.10), (3.4) and (3.6) one may obtain E ,. V _ Lim R T_».0 ^inlnZl^l (3.9) a(l/T) J V so that E may be calculated from experimental viscosity values. For 1/2 more complex molecules where n is large, a plot of ln('n/T ) versus 1/T is found to be approximately linear, and the requirement T — in equation (3.9) is less stringent. Figure 3.4 shows the volume dependence of E for several fluids as calculated from equation (3.9). The activation energy arises from the motion of the jumping molecule from its initial equilibrium position through a region of higher potential energy to its final position. A model of the situation is shown in Figure 3.5 in which a molecule jumps from A to C , and passes through a region B in which it has to squeeze through a ring of 2 molecules. The activation energy may be written

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36 2 o

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37 z Molecules V \ y Figure 3,5. Model of a Molecular Jump.

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38 Define e^ = z [uCr^) u(r^) = K Z£ r / \ n c, = v/r: a_ K Z£ 0_ r. (3.10) c_ = c (r2/ri) C/. = kT m Then e = K.zkT c, V 1 m 4 I 3 n i c^v n/3 3 m (3.11) The parameters c^, z^, c,, c^ and n, the repulsive ej/ponent should be approximately the same for a group of similar molecules and thus equation (3.11) suggests a corresponding states relation of the form E" = E"(V'') V V "^ (3.12) where and V RT m r^, V_ V'" = V m Figure 3.6 shows the correlations of reduced activation energy as a function of reduced molal volume for several nonpolar molecules. Good agreement is obtained at high reduced volumes, but some scatter is

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7.0 39 6.0 5.0 A.O b 3.0 > 2,0 1.0 o

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40 observed at lower values of v", where the curve rises steeply. This may be attributed to differences in n, the repulsive exponent, for the molecules within the group. Thus t\-Jo curves A and B are plotted in Figure 3.6. Curve A is for simple molecules obeying a LennardJones [6,12] potential. Curve B shows an approximate relationship for more complex molecules which obey a potential law in which n is larger than 12. Sources of viscosity data from which values of E are calculated are given in Figure 3.6. Taking into consideration the temperature dependence of V^ and the volume dependence of E , the modified viscosity and selfdiffusion coefficient equations become: . V (T) X . E (V) X ^ -' -/^— ^ Uxp(^^^ (3.13) TT^= A exp f T^^^ ° ^V V (T) and / V (T) \ / E (V) ° = B exp (T7-Vt^ ) exp r ^— ] (3.14) ^1/2 o""^ \^ V V^(T) J '-^'^ \ RT

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CHAPTER 4 TEST OF IMPROVED FREE VOLUME THEORY In order to perform a rigorous test of the theory, viscosity and self-diffusion coefficient data over a wide range of temperature and density conditions are most desirable. In this work only fluids composed of simple molecules which approximately obey the LennardJones (6,12) potential law will be considered. These fluids include the inert gases and most diatomic fluidsand perhaps methane, fluoromethane and te trade uterome thane. The only monatomic and diatomic fluids for which viscosity data over a wide range of temperature and density conditions are available seem to be argon and nitrogen. Zhadanova (48,49) has reported viscosity measurements for these fluids for experimental conditions corresponding to pressures up to several thousand atmospheres. The accuracy of the experimental argon data appears to be of the order 67o over the entire range of conditions. However, the nitrogen data are in poorer agreement with measurements of other workers, and the accuracy appears to be of the order 12-157o. The only high pressure self-dif fusivity measurements for simple liquids seem to be those of Naghizadeh and Rice (18) , who reported measurements for argon, krypton, xenon and methane over a reasonable range of temperature and at pressures up to a more than one hundred atmospheres. Naghizadeh and Rice claimed that the uncertainty of their experimental data was less than 5%. However the disagreement between their argon data and those of Corbett and Wang (56) is about 41

PAGE 65

42 12?o, while their xenon data differ from those of Yen and Norberg (57) by as much as 86% and their reported methane data are about 11% higher than those of Gaven, Waugh and Stockmayer (58). 4.1 Corresponding States Relationships The following procedures were employed to test the theory for viscosity and to establish the corresponding states relationships in equations (3.8) and (3.12). (1) The experimental data were plotted as ln(r]/T^''^) against T at constant volim:ie. Values of E^ were estimated at various volumes from the lines extrapolated to low temperature, since from equation (3.9) !v ^ Lim r ^IM-n/T'^^) 1 R T->0 [_ 8(1/T) J ^^^^ (2) Values of the parameters A and V were obtained from the best fit to the viscosity data using the computer, equations (3.4) and (3.5) being usee to obtain the temperature deoendence of V ' o Values of E^ for argon and nitrogen estimated by procedure (1) are included in Figure 3.6. Tneoreticai and experimental liquid argon viscosity values are shown in Figure 4.1. Agreement between theory and experiment is found to be within 5%. Similar agreement between theory and experiment is found for nitrogen, with V""" = 17.0 cc/mole and A^ = 2.53x10 poise (°K) ~, and is shown in Appendix 2. The theory provides a satisfactory fit to the data for densities above twice the critical value. Thus as can be seen in Figure 4.1 the theory fails for molal volumes of 39.15 cc/mole and above for argon. Similar behavior is also observed for the nitrogen viscosity (see Figure A2.1

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43 •7.5 8.0 9.0 -9.5 c -10.0 10.5 11.0 2.0 3 -1 29.14 cm. mole 30.45 39.15 (theory) 39.15 (expt.) oo Free Volume Theory 2h a da nova (48) Lowry _et al . (15) De Bock _et al^ (59) Saturated Liquid (60) 4.0 6.0 8.0 10.0 12.0 14.0 1/T X 10^(°k'S Figure 4.1. Test of Free Volume Theory for Liquid Argon Viscosity.

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44 in Appendix 2), where the theory fails for molal volumes of 56.0 cc/mole and above. The density range in which the theory applies is sharply defined and is similar to that in which similar models for thermodynamic properties, such as the Prigogine smoothed potential model (50), apply. The theory appears to work well for temperatures below about 1.5 T , although the temperature range in which the theory applies is less clearly defined. The values of the parameters E and V found for argon and ^ . V o ° nitrogen may be used to form the basis of corresponding states correlations of these parameters as described in Chapter 3, from which viscosity and self-dif fusivity values of other molecules may be predicted. The average value of the reduced minimum free volume for argon and nitrogen is 0.554, so that vj" =^ = 0.554 (4.2) m L 9 Test of Proposed Correlations To use equations (2.10) and (2.11) to estimate the viscosity and self-diffusion coefficient for simple molecules, values of V were o calculated from equations (3.4), (3.6) and (4.2), assuming n = 12 and taking values of e/k determined from gas viscosity data (20) . Values of E were found using curve A of Figure 3.6. The best values of A were obtained by substituting experimental viscosity data in equation (2.10) for each molecule. In a similar manner, by fitting experimental self-diffusion coefficient data of each molecule into equation (2.11) the best values of B were obtained. Other than the viscosity data

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45 for argon and nitrogen of Zhadanova (48, 49) and self-diffusion coefficient data of Naghizadeh and Rice (18), very few high pressure viscosity and self-dif fusivity data are available. Thus most of the viscosity and self-dif fusivity data used are for saturated liquids. Table 4.1 shows the viscosity data sources and ranges of temperature and density for each of the molecules studied. The different parameters which are required for the estimation of viscosity with equation (2.10) and the average percent deviation between theory and experiment are shovm in Table 4.2. Tables 4.3 and 4.4 provide similar information for the self-diffusion coefficient of the molecules studied. The average percent deviation bettvreen theory and experiment is about 67o or less for viscosity. For self-diffusion coefficient the discrepancy between experiment and theory is found to be much larger, being about 157o in most cases. The discrepancies between theory and experiment for both viscosity and self-diffusion coefficient seem to be of the same order as the accuracy of che experimental data for the liquids studied. The viscosity data of Zhdanova for nitrogen at high pressures seem to be in poor agreement with those of other workers (66), and saturated liquid viscosity data (66) were used in obtaining the Aq value given in Table 4.2 for this fluid. Experimental and predicted viscosities for xenon and methane are compared in Figures 4.2 and 4.3 (similar plots for Ne, Kr and N are shovjn in Appendix 2) . Tne predicted values become less reliable as density falls to values approaching Ip (for methane this corresponds

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46 o C o O 00 CO o M H M P 7^ O o p4 o tJ o >^

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47 Qi

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48 m

PAGE 72

49 tu

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•7.0 50 Free Volume Theory o Experiment (64,65) •7.5 -8.0 jo. -8.5 9.0 // R 3^-1 44.5 cm, mole -9.5 10.0 1.0 2.0 3.0 4.0 5.0 1/T X 10^(°k"^) 6.0 Figure 4.2. Test of Free Volume Theory for Liquid Xenon Viscosity. 7.0

PAGE 74

51

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52 to V = 49.5 cc/mole) . Figure 4.4 compares theory and experiment for the self-diffusion coefficient of saturated liquid methane. The expressions for the pre-exponential factors of the viscosity and self-diffusion coefficient equations of Eyring (26) 2/3 ,,,1/2 and Weymann (35) suggest that the groups (A^V /M ) and (B M /V ) may be approximately the same for different molecules. 2/3 1/2 Values shown in Table 4.2 indicate that the quantity (A V /M ) is approximately constant for inert gas liquids, but for the other molecules a range of values is found. Similar behavior is observed 1/2 1/3 for the group (B M /V^ ) . Apparently the expressions given for the pre-exponential factors by Eyring and Weymann are not generally valid, and cannot form the basis of successful corresponding states relationships for A^ and B . In general the value A (or B ) can be determined from a single experimental value of viscosity (or selfdiffusivity) for a given fluid. This can then be used together with the proposed correlations for V and E to predict viscosity (or selfdiffusivity) values at other temperatures and densities. 4.3 Summary By allowing for the variation of V with temperature it is possible to account quantitatively for the nonlinearity of constant volume plots of ln(ryT-'"'^) vs. T"-*" and ln(D/T-'-''^) vs. T" "' for simple liquids. The physical model predicts a density dependence of E , and thus explains the variation with density of the slope of constant volume plots of ln(T]/T ' ) vs. t' and ln(D/T-'-/^) vs. t'''". With these improvements the free volume theory is able to accurately describe the viscosity and self-dif fusivity behavior of liquids composed of

PAGE 76

53 •a •e-l 3 u

PAGE 77

54 simple, nonpolar molecules that may be expected to approximately obey a LennardJones [6,12] potential law. The theory works well at densities p > 2^ , where other correlations fail. The fact that both V and E values vary in a corresponding states way indicates that the free volume theory provides a reasonably correct picture of viscosity and self-dif fusivity behavior for these liquids, and shows that the equations' success does not arise only from the availability of three adjustable parameters. The simple expressions given for A and B by Eyring and others, however, do not seem to be generally applicable.

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CHAPTER 5 PRINCIPLE OF CORRESPONDING STATES FOR MONATOMIC FLUIDS The free volume equations discussed in Chapters 2-4 accurately predict viscosities and self-diffusion coefficients for nonpolar fluids in the density range p > 2p . However, the theory cannot be readily extended to include thermal conductivity, and is limited to the above density range. In this chapter and the next a general and direct application of the corresponding states principle is examined. The treatment for monatomic molecules does not assume any simple model, and should be applicable to all transport properties over the entire range of densities and temperatures. 5 . 1 Molecular Basis of the Correspondence Principle If correctly applied, the principle of corresponding states provides an accurate and very useful method for predicting physical properties of pure substances, under conditions for which no data are available, from the measured properties of one or more substances. The corresponding states principle has been widely applied to the thermodynamic properties of pure substances; thus the generalized charts of Hougen, Watson and Ragatz (28) and of Hirschfelder , Curtiss and Bird (20) have proven of great practical use in engineering calculations. The principle of corresponding states was first suggested by Van der Waals in 1873 as a result of his equation of state. It was not until 1939, however, that a rigorous and more general theoretical derivation of the principle was attempted by Pitzer (74) for thermodynamic properties using statistical mechanics. 55

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56 Pitzer presented his derivation based on the following assimptions; Assumption 1. Quantiim effects are negligible. Assimiption 2. The intramolecular degrees of freedom (rotational and vibrational) are independent of density. Assumption 3, The intermolecular potentials are pairwise additive. Assumption 4. The potential energy for a pair of molecules has the form u = eg (5.1) where r = the intermolecular distance £ = characteristic energy a = characteristic distance = a universal function Using Assumptions 3 and 4 the configuration energy may be written as N U = a (5.2) i
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57 where N = number of molecules •AkT T = ~~ = reduced temperature V V" = ~~T = reduced molal volume a Since all equilibrium conf igurational properties of interest may be evaluated from the configurational partition function Z and its temperature and volume derivatives, equation (5.4) represents a general proof of the principle of corresponding states for thermodynamic properties of pure substances, provided all the above assumptions are observed. Thus the reduced equation of state is e e V ov , = P"(T",V") (5.5) where it is noted that intensive properties do not depend upon N. The principle of corresponding states for thermodynamic properties may also be derived by the method of dimensional analysis (75) . However in this method the conditions under which a molecule may obey the corresponding states principle are not explicit. One way of deriving the principle for transport properties is by dimensional analysis, with the reduction performed using either the critical constants or appropriate combinations of molecular parameters. A rigorous statistical mechanical derivation for transport properties similar to that mentioned above for thermodynamic properties, is possible, as has been shown by Helfand and Rice (76).

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58 The transport coefficients may be expressed in terms of equilibrium time correlation functions. The equations for the shear and bulk viscosity, thermal conductivity and self-diffusion coefficients are given in equations (1.1) to (1.4) of Chapter 1. Using the four assumptions made in deriving the principle for thermodynamic properties, it is possible to put these equations in a reduced form. The derivation of the principle for the self-diffusion coefficient is given below as an example of the procedure. The self-diffusion coefficient expressed in terms of the time integrals of appropriate autocorrelation function has the form D = / dt (5.6) J X X o Based on Assumptions 1 and 2 given above the time correlation function may be written in terms of integrals over phase space V V / • •/"\(0)\(t)e-^%^. . .d£^dr^. . .dr^ = ^^.^^ (5.7) ' ' -/SH e d£^...d£^dr^...dr^ -00 o Assumption 2 is implicit, since the integrals are performed only over center of mass coordinates. The Hamiltonian H for the center of mass coordinates is given by N 2 H = \ TT^ + U ^2^^' (5.8) 1=1

PAGE 82

59 and if Assumptions 3 and 4 are satisfied, we may write U = N £ ) L 11 a (5.9) Now, if we define reduced quantities as distance temperature time r T" = a kT e , 1/2, 1/2 = te /m a (5.10) (5.11) (5.12) momentum -:^ ,, 111 111. 3 = p/(ni ' £ ' ) (5.13) volume v" = v/cr (5.14) then -foo ,v £ m ^ p;(o)p:;(t")exp N ':<1 Pi = -00 o V i=l ir ^ N^ exp -00 o N -,v2 2T* i=l N i^ o (5.16)

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60 or combining equations (5.15) and (5.16) D" = D"(t",v''') (5.17) where D" = ^ f^ (5.18) Similar relationships may be derived for shear and bulk viscosity and thermal conductivity, r{' = Ti"(T",v'') (5.19) /c* = k:"(T",v'') (5.20) X'' = xa*X') (5.21) where 2 n* =^ (5.22) v me /c" =^ (5.23) /me X'^.^yY (3.24) The corresponding states principle may be derived under less restrictive conditions than those used above. Although it is then of general application it loses the attractive simplicity of the tijoparameter laws expressed by equations (5.17), (5.19), (5.20) and (5.21). Thus, Assumption 1 is readily removed by using quantum mechanical expressions in place of integrals over phase space. This results in the introduction of a reduced Planck constant in the expressions for transport properties.

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61 where D" = D"(T",V",h') (5.25) T)* = r]"(T", V", h") (5.26) K* = k:"(T", V", h'") (5.27) X*= X"(t", v", h") (5.28) u h" = Donth (77) showed that the corresponding states principle could be proved for thermodynamic properties without assuming pairwise additivity of potentials (Assumption 3) . Thus in place of Assumption 3 he assumed that the total potential energy of the system might be expressed in the form U=e0(% %....=!) (5.29) where is a universal function of the reduced position coordinates. Recently Hakala (75) derived a more general form of the corresponding states principle for the thermodynamic properties by means of dimensional analysis. The effect of his treatment is to remove Assumption 4. The corresponding states principle is shown to apply to substances which do not obey a pair potential of the form given by equation (5.1), provided the necessary additional reduced parameters are included in the correlation. It is of interest to note that the above corresponding states relations for D and rj may be derived from the free volume equations.

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62 The assumptions implied are in this case less clearly defined. The expression for self-diffusion coefficient derived from free volimie theory is given by equation (2.11) as P ^ /^ ^v \ / ^o \ ^1/2 ^o^^P V ^ J ^""^ ( " ^^^ ) ^^-^^^ Equation (5.30) can be expressed in terms of reduced variables as or where u _-;<:/ V \ / o T \ T \ V V o d'' = D"(T",V") (5.32) E ^v = ^ = eJ'(V*) (see Chapter 3) V Vq = "3 = V^'(T") (see Chapter 3) B m and it is asstimed that B" = ° = constant. Ok ' in this chapter we consider only fluids composed of monatomic molecules (excluding He) so that Assumptions 1 and 2 may be regarded as fully satisfied. The inert gas molecules have been found to obey the principle of corresponding states for thermodynamic properties (20,74) and transport properties at low density (78) within experimental accuracy. Pairx^ise additivity of potentials seems to be a good assumption up to reasonably high densities for the molecules considered in this chapter (79-81). Ross and Alder (82) have presented experimental evidence that nonadditivity becomes a significant

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63 factor for these fluids only at very high densities, of about 4 times the critical value. Assumption 4 concerning the form of the pair potential energy is only approximately satisfied by monatomic molecules considered here, as discussed by Smith (83). 5 .2 Previous Work Previous work on the application of the principle to transport coefficients of inert gases and other simple polyatomic molecules in the dense gas and liquid states has been reported by Cini-Castagnoli, Pizzella and Ricci (84), Kerrisk, Rogers and Hammel (85), Rogers and Brickwedde (86), Boon and Thomaes (87,88), Boon, Legros and Thomaes (64), Tee, Kuether, Robinson and Stewart (89), Trappeniers, Botzen, Van Den Berg and Van Oosten (90), and Trappeniers, Botzen, Ten Seldam, Van Den Berg and Van Oosten (91). Cini-Castagnoli, et_ al . applied the principle to the viscosity and thermal conductivity of saturated liquid Ar, CO, N„ , CH and 0„ . Correlations appeared to be poor, especially for the reduced viscosities. Kerrisk et_ al. and Rogers and Brickwedde examined the principle of 3 4 corresponding states for the light molecules He , He , H» , D , T , etc., where quantum effects are important. They tested the principle for the saturated liquid viscosity and thermal conductivity of these substances. In their correlation they introduced the reduced Planck constant h as a third parameter as required by equations (5.26) and (5.28). Boon et al . tested the validity of the corresponding states principle for the saturated liquid viscosity of Ar, Kr, Xe, 0_ , N , C0_, CH, and CD,. They found that the inert gases obey the principle

PAGE 87

closely, but even relatively simple polyatomic molecules such as , C'd/ and CD, were found to deviate markedly. Self-diffusion coefficients and binary diffusion coefficients of more than 14 simple molecules in the dilute gas, dense gas and liquid states have been tested by Tee _et al. They defined the reduced diffusion coefficient d" as AB T n ' CAB c D, "o AB V P ,^ AB I ^ ^ ^ N (5.33) "^cAB V m" "*" M~ j \ A B where c here represents the total concentration, and obtained from the equation for the diffusion coefficient of a dilute gas the relation D"" = 4.05 X 10" "" ^f^,„ (5.34) AB ^(l,l)->' They plotted their results as D" against P^ at constant T^ where ^ ^ AB R R P T P„ = — and T„ = — . However, the data they present seem to be too c c scattered to justify the test. Trappeniers e_t al. applied the corresponding states principle to the viscosity of inert gases up to high densities for temperatures above critical. They defined the reduced viscosity as Q* = 2(5.35) o where ° 16 TT n^^'^^" and 2 ri''=?i=. (5.37)

PAGE 88

65 Q" was then related to p" and T" . 5 3 Test of Correspondence Principle for Inert Gases No systematic and consistent test of the principle for all three transport properties of dense fluids over a wide range of temperatures and densities seems to have been made. In particular the transport properties of fluids at very high pressures (and thus high densities) have not been examined. The behavior at high densities is of particular interest since it is under these conditions that Assumption 3 may become invalid. Also, errors arising from differences in the repulsive portion of the intermolecular potential for the molecules considered (Assumption 4) may become apparent at high density. The essential simplifying factor in the case of monatomic (as opposed to polyatomic) molecules is that Assumption 2 is fully satisfied. A comparison of the correspondence behavior for monatomic and polyatomic molecules may therefore throw light on the validity of this assumption. Calculations are presented in this chapter to test the correspondence principle for all three transport coefficients over the full range of temperatures and pressures for which data are available. The extension to polyatomic molecules is discussed in the following chapter. Reduction of the transport coefficients are performed using LennardJones [6,12] potential parameters rather than critical constants because the former have more theoretical significance. The use of critical constants is justified when the group of substances accurately obey the same potential function, Assumption 4 (50), In such a case it would in principle be immaterial whether potential function or

PAGE 89

66 critical parameters were used. However, in practice the group of molecules do not all accurately obey Assumption 4, and the potential parameters are force-fitted to some semi-empirical equation, in this case the LennardJones [6,12] potential function. Under such conditions the use of potential parameters rather than critical constants may improve the correlation obtained. In this worl<, reduction with both critical constants and potential parameters (smoothed in the way described below) was tried. The use of potential parameters was found to give consistently better results. The potential parameters reported in the literature by different workers show substantial variations. Thus for krypton values of £ , the characteristic energy, reported by different authors vary by as much as 14%. It is therefore desirable to find some means of averaging these parameters. Moreover, Reed and McKinley (92) have shown that polyatomic molecules do not all obey the simple two-parameter LennardJones [6,12] potential energy function. Since Assumption 4 requires that all molecules should obey the same potential energy function in order to arrive at the same corresponding states, an attempt was made to force the polyatomic molecules to fit a twoparameter potential energy function by adopting the method used by Tee, Gotoh and Stewart (93). According to the Pitzer compressibility correlation (94) the volumetric behavior of a fluid, and thus the intermolecular forces present, are characterized by P , T and the c c acentric factorco . Thus the intermolecular force constants e and a should depend on these three quantities, so that one obtains the

PAGE 90

67 dimensionless relations ^ = fCcu) (5.38) c P N 1/3 ^ ) a = f (CO) (5.39) c ^ where f and f are universal functions. In Figures 5.1 and 5.2 values of ( —2— ^ and :; for twenty different molecules are V kT y ° kT ^ c / c , plotted against their corresponding Pitzer factor co. The best straight lines X'/ere drawn through them by the method of least squares. Potential parameters used in these figures were values calculated from gas viscosity data, and were taken from Hirschfelder , Curtiss and Bird (20) and Tee, Gotoh and Stewart (93). Where several values fore and a were reported for a given molecule the mean values are plotted in Figures 5.1 and 5.2, and the maximum deviation is indicated. The calculations presented in this chapter and the next make use of smoothed potential parameters obtained from these two figures. Values of the potential parameters used are shown in Tables 5.1 and 6.1. Values for molecules other than those tabulated may be calculated from the equations for the best lines shown in Figures 5.1 and 5.2 which are -^ = 0.7932 0.672300 (5.40) kT c ^ Y/2^= 0.4583 + 0.1213a) (5.41) kT ] c ^ Figures 5.3, 5.4 and 5.5 are reduced plots of saturated liquid viscosity, thermal conductivity and self-diffusion coefficient as

PAGE 91

68 o

PAGE 92

69 O

PAGE 93

70 o CM CO c o CO o O CQ U U i) v <: 1^ X 2 o ff U (^ VO (N 00 o 'V 3 XT (U •w to 3 4-1 CO w o 14-1 a •1-1 o c •H ^ • en dJ OJ U 1-H C 3 0) U t3 01 C .-H o o CTj a o U -r-l u B o o en m (U i-i 3 M •H O CM i I I I O o o o -I I L. o u-1 o CN o

PAGE 94

71 o 1 c^ s-j V( o a < M X S o p "a if 00 CX5 o TO E (U 3 a" , O jj c > O -rW CL J-" cn u OJ 3 ti 73 ;^ G O O 0) 3 o o o o o CM X o c CM

PAGE 95

72 u u Oi oi < i2, X ^ P -Q. ^ 00 VO J I I u. 00 o I CO D 14-1 14-1 •r-t cr •i-i a> jj CO !j D iJ ca en S-l o c

PAGE 96

73 functions of temperature for the inert gases (argon, krypton, xenon and neon) . For pure saturated liquids there is only one degree of freedom, and the reduced transport coefficients depend only on reduced temperature. In the saturated liquid region, the inert gases appear to obey the corresponding states principle for the above transport properties reasonably well with the exception of the thermal conductivity data of neon. The discrepancy observed in Figure 5.4 for neon may be due to quantum effects, since neon is the second lightest molecule among the inert gases. Experimental data for bulk viscosity are scarce. The only bulk viscosity measurements made seem to be those reported by a few workers for argon (14,95,96). Figure 5.6 is a reduced plot of bulk viscosity of saturated liquid argon (extrapolated data) as a function of temperature. Great experimental difficulties are involved in bulk viscosity measurements, and the data are very scattered. Even among the data of the same worker the average percent deviation is as much as 157<,. In the derivation of the correspondence principle for transport properties shown above, reduced viscosity is found to be a function of reduced temperature and reduced molal vol;ame. Thus according to equation (5.19) T]' = r)^'(T'%V"). -However it is more convenient in practice to plot the transport properties as a function of temperature and pressure. Since reduced molal volume may be shown to be related to reduced pressure through the equation of state, one may express t}" as r)"(T",P"). Reduced viscosities of inert gases plotted as a function of reduced temperature at constant reduced pressures are shown in

PAGE 97

74 o 60 < •r-l O cr •n O) 4-1 CO 4-J O •H CO O u w > M 0) O D dJ Pi

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75 Figure 5.7. The corresponding states principle appears to apply closely to viscosity of inert gases over the entire range of temperature and pressure for which experimental data are available. Data plotted in Figure 5.7 cover densities from the dilute gas phase to the dense gas and liquid phase, and correspond to pressures from zero to over 2000 atmospheres. A similar plot for the thermal conductivity of inert gases is shown in Figure 5.8. In this plot the correspondence principle is found to apply well for argon, krypton and xenon over the entire temperature and pressure range for which data are available. The neon data in Figure 5.8 obey the simple correspondence principle at high temperatures, but not at low temperatures. This behavior for neon suggests that quantum effects are important for this fluid at low temperatures. The reduced viscosity and reduced thermal conductivity data are found to fit equations of the form In-rj'' = a^ + a^T""^ + a^T*"^ + a^f''"^ + a^T*"^ (5.42) ,* "'-1 , K t'''-2 , -L rp"V-3 , rp*-^ and InX = ^^ + b2T""-^ + b^T "^ + b^T -^ + b^T""^ (5.43) at constant reduced pressure or along the saturation curve. The reduced self-diffusion coefficient, on the other hand, is found to fit an equation of the form InD* = d + d t""^ (5.44) The principal parameters used to test the correspondence principle for monatomic molecules are shown in Table 5.1. The

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76 J L <; ^ X z o o o u-i o C3 (1) a e u 0) (0 c o •t-l o c D w • CO (U u U2 CO E c o o D (1) o * u 3

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77 u 3 "a o w o w

PAGE 101

78 o o o o o o o o CO 1—1 o CM ON CO o en CO CO e

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79 potential parameters included in this table are obtained from Figures 5.1 and 5,2, The coefficients of equation (5,42) evaluated by computer for saturated liquid viscosity of monatomic molecules (Ar, Kr, Xe, Ne) together with the average error between the equation and experiment are shown in Table 5,2, The average percentage errors betivfeen prediction and experiment for the inert gases are within kX, and this is of about the same order as the experimental accuracy. Tables 5,3 and 5.4 exhibit the coefficients of equations (5.43) and (5.44) respectively for monatomic fluids. The maximum disagreement between reported thermal conductivity data of different workers for monatomic molecules is estimated to be approximately 1°L. Equation (5,43) X'/hen used with the values of coefficients shov/n in Table 5,3 gives an average percentage error well within 47o (thermal conductivity data of neon were not used to fit the coefficients) , Disagreements between self-diffusion coefficient data of different workers are especially large. Thus the seif-dif fusivity of xenon reported by two v7orkers differ by as much as 100% (18,57). Equation (5.44) for self-dif fusivity of monatomic molecules using the values of coefficients in Table 5.4 yields an average percentage error of about 5%. Only the data of Naghizadeh and Rice (18) were used in obtaining these coefficients. Table 5.5 lists the coefficients of the viscosity equation together v;ith percentage errors for the viscosity isobars shown in Figure 5.7. The average errors are largest at the highest reduced pressures, v/hich correspond to pressures up to approximately 2000 atmospheres. These larger errors may be due to a failure of

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80 Z O M H < P C M ryj O U CO zn M W :=> Q O M w CO u M O H < o H < CO Pi t^ O O fjH CO Z, M M U l-l o ai

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0) u to o

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82 o M cy w X H M > H 00 M W Q 1-5 o s i-t O ^ M o-S M O < O n o < (^ o M o M o 00 o nj s-i ^1 S-I > S o CO CM d

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83 o to u ^ o CM 00

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84 Assijmption 3 at these high pressures, or to experimental errors. The coefficients of the ther.:ial conductivity equations for the isobars shCT^7n in Figure 5.8 and the average percentage error between experiment and the equations are given in Table 5.6. The errors for all the equations are within experimental error. Sources of viscosity, thermal conductivity and self-diffusion coefficient data are also included in Tables 5.2 5.6. 5 .4 Summary The good agreement of the experimental data with the principle of corresponding states leads one to conclude that the 4 assumptions on which the molecular derivation is based are good approximations for monatomic fluids. An exception is neon, for which quantum effects must be taken into account at low temperatures. It is of particular interest that Assumption 3 concerning pairvise additivity of the intermolecular potential does not lead to significant errors even at the highest densities for which measurements are available. Similarly, Assumption 4 does not give rise to significant errors for these fluids in spite of known differences in the potential energy form (S3).

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85

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CHAPTER 6 PRINCIPLE OF CORRESPONDING STATES FOR POLYATOMIC NONPOLAR FLUIDS It has long been realized that the simple correspondence principle does not apply accurately to polyatomic fluids in the condensed phase. In the previous chapter it has been shown that monatomic molecules obey the principle of corresponding states over the entire range of temperature and pressure conditions. In this chapter a careful re-examination of the four basic assumptions used in deriving the principle is first made. A simple model of the molecular behavior of a polyatomic liquid is proposed, and provides the basis for a rederivation of the correspondence principle. 6 . 1 Problems in Polyatomic Fluids Polyatomic fluids should obey Assumption 1 closely since their molecular weights are usually large and thus quantiom effects are negligible. Assumption 3 concerning pairwise additivity of potentials does not seem to be a major source of error for either monatomic or polyatomic fluids (81) . Therefore it appears likely that the deviation of polyatomic fluids from the correspondence principle is due to the fact that they do not satisfy either or both . Assumptions 2 and 4. T\^70-parameter potential energy functions, such as the LennardJones [6,12] equation, do not accurately account for properties of polyatomic fluids (92). In fact three-parameter energy functions, such as the LennardJones [6,n] or the Kihara potential give better results for these molecules. Thus polyatom.ic fluids in general do not satisfy Assumption 4 closely. Failure of polyatomic fluids to satisfy 86

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Assimiption 4 may be corrected by introducing a suitable third parameter. Alternatively a tt-.'oparameter potential may be used, taking suitable potential parameters in the way suggested by Tee, Gotoh and Stewart (93). Recently Preston, Chapman and Prausnitz (112) examined the corresponding states principle for the transport coefficients of nine monatomic and polyatomic fluids in the saturated liquid state. They proposed that potential parameters of substances in the liquid state may take different values from those in the gaseous state. Liquid phase potential parameters x^7ere obtained from the saturated liquid viscosity and thermal conductivity data by assimiing that the simple correspondence principl.^ applied. They then used these parameters to test the correspondence principle for the saturated liquid viscosity, thermal conductivity and self -diffusion coefficient of the nine molecules. This procedure gave reasonably good correlations for liquid viscosity and thermal conductivity, with root-meansquare deviations of 7.47o and 5.4% respectively. The correlation for self-diffusion coefficient was comparatively poorer, showing a root-meansquare deviation of 157c. Their treatment seems open to criticism on several counts. Firstly, by force-fitting the parameters using liquid phase data the validity of their test is doubtful. In particular, certain facets of the behavior of polyatomic molecules may be hidden in this way; thus hindered rotation in the liquid state may be important, as discussed below. Further, because separate sets of potential parameters are iiecessary for the gas and liquid phases, problems will arise in choosing the most suitable parameters for the intermediate densities,

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88 and it will not be possible to predict transport properties when no experimental liquid phase data are available. Little attention seenis to have been given previously to the possible failure for liquids of Assumption 2 concerning the aensity independence of intramolecular degrees of freedom. Because vibrational quanta and bond energies are large one would not expect intermolecular forces to have much effect on the vibrational energy of a polyatomic molecule. However, rotational energies are smaller and it seems quite likely that free rotation of polyatomic molecules may not be possible at high densities. Figure 6.1 shows a plot of r)" at P" = for a variety of monatomic and polyatomic molecules in both the gas and liquid states. Reduction of the dilute gas and liquid viscosity coefficients is performed using smoothed molecular parameters determined from gas viscosity data (see Figures 5.1 and 5.2). In the dilute gas phase both monatomic and polyatomic molecules obey the correspondence principle very well. In the liquid phase, however, only the monatomic molecules obey the principle. Deviations for the polyatomic molecules increase as the density increases. Were these deviations due only to incorrect potential parameters, as suggested by Preston et al. (112), one would expect that discrepancies might occur for monatomic molecules in the liquid state also. The fact that gas-phase parameters work well for the latter molecules in the liquid states, together with the excellent agreement found for polyatomic fluids at low density, strongly suggests a failure of Assumption 2 in the liquid phase. It

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89 a

PAGE 113

90 seems likely, therefore, that hindered rotation has a significant effect on the transport coefficients of polyatomic fluids at high densities, and that this effect causes increasingly large deviations as density rises. Davis and Matheson (113) have presented evidence to show that rotational degrees of freedom are restricted at high densities. Figure 6.2 shows reduced viscosity for 20 saturated liquids. The same trends as in Figure 6.1 are seen; deviations from the curve for monatomic liquids are greatest for molecules having larger Pitzer factors . 6.2 Derivation of Corresponding States Principle for Polyatomic Molecules In the gas phase molecular separations are usually large and intermolecular forces are small. A molecule can therefore rotate freely about its body axes. However, as the density increases intermolecular forces become more important, and if these forces are noncentral the molecules exert a torque on one another. Polyatomic molecules thus rotate in a potential energy field which is a function of the relative orientations and center of mass positions of the molecules; in the case of very high densities the rotational motion might be reduced to that of angular vibrations about the body axes. For dense fluids the rotational degrees of freedom thus become dependent on the center of mass coordinates, and cannot be factored out from the phase integral as in the case of dilute gases. The time correlation function instead should be written as

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91 c cN — I r< I— I c-H 1— I rsl K 00pf^^s>K<) 00 *. \ \ \ \ \ \ \ \ CO 0) CO to > < ^ \ \ s^ i. ^ \ \ \ ^ \ \ ^s^\\\ \ < \ V ^' o \<:v^^^ k \ ^ \ \ ^^\ N V V « oN \ \ -c ^. «\. v\ 9s a-j I L f I ! I I o o o o U1 o CN4 ITl O 00 O o CM O (0 o o CO 3 cr H hJ t3 G i-i IB ;j 3 •U CO C/5 o S-l P-. (U CO c o O 0) CO O
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92 •H/kT , X X V^.(0)V^.(t)e ^d£^ ••dV^e •••d0^ 1 = ^ X X ^ ../ e d£^ •••dr^d£g .-.d^^ (^-l) 1 where H(p ...r ..pn — Q) = K„ (p ...) +K (p^ ...) r^ -N ^2. ~^ CM^-'=^r, '^ rot^-^& '^ + U(r^...0^...) The phase integral in this form will be difficult to handle, since we have little knowledge as to the form of the potential energy term. Although the kinetic energies of translation (Kpv) ^^'^ rotation (K ) are separable, the potential energy term is not. Evaluation of the integrals therefore requires that a suitable expression be obtained for U(_r,_0) . The part of the Hamiltonian arising from intramolecular vibrations is here assumed to be separable and independent of density for the reasons discussed above. To account in an approximate way for the coupling between translational and rotational degrees of freedom, a highly simplified one-dimensional model is proposed. Consider a harmonic oscillator cell model similar to the type used by Prigogine (50) . In this model each molecule is assijmed to spend a large fraction of its time oscillating in a cell or cage. Since the oscillator is assumed harmonic, the restoring force acting on a ir.olecule at any time is proportional to the displacement of the molecule from its equilibrium position along each of its orthogonal coordinates. The nearest neighbor molecules are assumed in an average situation to be on their lattice sites, and

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93 are treated as smeared over a sphere passing through their lattice positions. For the present a one-dimensional motion is assumed. However, it can be readily extended "o the three-dimensional case by assuming an isotropic system in which the motion along any of the 3 axes is the same. Figure 6.3. A One-Dimensional Oscillator Model. Consider a molecule oscillating linearly along the x-axis in a cell (Figure 6.3). For convenience the molecule is represented as a rigid diatomic one, but similar considerations will apply in general. Because of the linear oscillation, the molecule rotates in an oscillating potential field, and this gives rise to hindered rotation. The rotational motion of the molecule about the z-axis (perpendicular to the plane of the paper) may be considered to consist of tvjo parts: a constant velocity rotation plus a harmonic angular oscillation superimposed on it. The harmonic angular oscillation is assumed to be

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94 in phase with the linear oscillator. The constant velocity rotation is the value that would be observed if no linear oscillation occurred, that is when the molecule remained at 0. The angular coordinates can be written as e (x,t) = (D (t) + • (x) (6.2) XXX ^e^'^^ = P(i) "^ P^ (^^ (6.3) X XX where (t) and prefer to the constant velocity rotational motion X at 0, while ^ (x) and p (x) account for the superimposed oscillation X or angular perturbation. Pg =P0 + P^ X X ^ X rotating reference axis Figure 6.4. Superimposed Rotational Motions of a Molecule. The coordinates are shown in Figure 6.4 for an instant in time when the molecular axis lags behind the rotating reference axis. The reference axis rotates with the constant velocity C> , and viewed x

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95 relative to this axis the molecule performs angular vibrations. Because the t^^ro oscillations are in phase the quantities ^ and p„, X Y X will depend only on the linear displacement x. Using equations (6.2) and (6.3) it is possible to transform from the phase space coordinates p , used in equation (6.1) to a — the new angular coordinates p ,, , so that -=-q) — = |v^(0)V^(t)e-^^/^^ d^ dr^dp dO^ ^.Te-^/^^dp^ dr^d£^ dO^ ^ 11 (6.4) where H = H(p^, 1, V^^ and it is noted that for coordinate transformation in phase space the Jacobian is unity (114) . In practice the above transformation requires knowledge of the functions Y (x) and p^ (x) . Expressions for these X quantities are obtained below. For small oscillations the total potential energy can be expressed in terms of a Taylor's series expansion as \ y x=0 \ X / x=0 \ dx x=0 Y =0 ^ =0 Y =0 XX X \ ^^^^x y x=o ^ ^\b^^ J ^ =0 ^ x=0 ^ w =0 X

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96 Since U(x,Y ) is a minimum at x = and ^ = X X f^^ 8u V ox / ^ V o"? . „x=0 V X / x= X ^ X = Denoting (6.6) Neglecting terms higher than second order, \-je have ^ ox / x=0 \ X / x=0 ^ 2 ( ~ ) \ (6.7) Vox ^^^^Q ^2= (Ur} (6-9) x=0 ^22 = ( 7~2 ) (6.10) x=0 equation (6.7) can be written as U(x,^if^) = U(0,0) + I k^^x^ + k^2^Y^ + I k22T^ (6.11) For small oscillations the total kinetic energy may be written as 1-2 1 -^ K = mx^ + -^ lY (6.12) 2 2 X ' _ dx * dY where x -^ and ^J/^ = — , and the Lagrangian of the system becomes L = K(x,Y ) U(x,Y ) 1 .2 — mx + ^ I^^ [U(0,0) +1 k^^x^ + k^2^Y^ (6.13) + I ^22^]

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97 The Xa'-i^rangian equation of motion with respect to x is given by dt \ ox / ox So that from equation (6.13) mii + k^ 'X + k.-^ = (6.15) Similarly the Lagrangian equation of motion with respect to ^ is dt U\^ V a\^ (6.16) or Let I^ + k„.^ + k..x = (6.17) x 22 X 12 X = A'cosOO t + A"sinXi,t (6.18) ^"^^ ^ = B'cosco.t + B"siilO,t ^^-^^^ X 1 1 be the solutions to equations (5.15) and (6.17). A complete solution of equations (6.15) and (6.17) is shown in Appendix 3. Differentiating equations (6.18) and (6.19) twice with respect to time yields X* = -CO, [A'cosoo^t + A"sina),t] 1 i 1 = -CD^x (6.20) and 2 • = -CUUB'cosoo^t + B"sim),t] X 1 1 1 = -a)?7 (6.21) 1 X Substituting equations (6.20) and (6.21) into equations (6.15) and

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98 (6.17) ?mx + k,.x + !<,,•? =0 (6.22) -cc^r? + k-.Y +k.-x = (6.23) i X 22 X 12 2 Eliniinating CD and rearranging we have 9 IT^ 2 mx c The potential energy U(x,T ) now becomes U(x,-^) = U(0,0) +1 [k^^x^ 4k^2-\] +i ^^22^^ + k^2^^?J mx (6.25) Let us consider at any time t = t x = and Y 0, then = A'cosa>[_t^ + A"sina>[_t^ (6.26) ^ = -cotCO^t (6.27) A 1 o and = B'cosoo. t + B"sinait (6.28) or therefore 1 o p= -cotco^t^ (6.29) A^ ^ b;^ (6.30) A' B' Combining equations (6.18) and (6.19) we have B'cosCD-jt + B"sinJa t •a = X X A'cosco,t + A"sina) t

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99 B" B'[cosco t +-7 sincu t] i B 1 A'Lcosco^t + YT Sim), t] 1 A 1 ^ (6.31) where „ _ B' "" " A^ (6.32) Equation (6.25) can be rewritten as U(x,^ ) = U(0,0) + [ ^=^ + 1] -i [k^.x^ + k .x^ ] (6.33) >m 2 11 12 X Differentiating equation (6.31) with respect to time we get ?^ = Sx (6.34) The total kinetic energy is given by K =^ mi^ +-^ I^^ (6.35) 2 2 ^x and making use of equation (6.32) we have ^ 1 .2 _^ 1 ^r^.T2 K = -T mx + -r ILcxJ „2 = 1 mx^[ ^^-^ 1] (6.36) Now we define c = [i^ + 1] (6.37) m

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100 The moment of inertia appearing in the above equations is that for the molecule rotating about an axis fixed in space, rather than about a body axis. It may be regarded as a time-average of the principal m.oments of inertia for the molecule. In the limit where hindered rotation is absent, as for dilute gases and monatomic molecules at all densities, B' and B" are zero and so is S, so that c reduces to unity. The Hamiltonian for the system of N molecules in one dimension can be written as ^ cp2 ^ 4 i = ^_lx^eU(x^...^)+ ^ _^ ^U(O) i=l i=i (6.38) where N N U(x^...x^) =\ V i\A^\2-^\^-\ -\ fkii +-^2^ X 4 i=l i=l and U(0) is the potential energy when all molecules are at their cell \~' 2 centers. In the above equation > p. /2m represents the kinetic energy due to oscillations in the x-direction, and U(x . . -x ) is the potential energy associated with the linear oscillations. The term c thus accounts for energies associated with the angular vibrations. The last two terms in equation (6.38) corresponding to the cj) coordinate are now separable, and the center of mass Hamiltonian for the one dimensional case is N 2 \ = l_ 1^ +CU(X^...X^) ^g_3g) i=l

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101 Comparison of this equation with equation (5.8) for the center of mass Hamiltonian when there is free rotation shows that the two expressions are of the same form. In fact, as far as the equations of classical mechanics are concerned, we may treat the hindered rotation case in the same way as free rotation, by replacing the usual kinetic and potential energy terms by those in equation (6.39), that is by N ) cp /2m and cU respectively. The new kinetic energy term implies A ix 1^ 1/2 that momentum p should be replaced by c p. Assuming the oscillation is isotropic, so that the motions of the molecule along all the 3 orthogonal coordinates are equivalent, the time correlation function can now be written as f. . . rcV^(0)V^(t)e"^CM/'-T ^^^ _ _ -dp^dr^. . .dr^ = S-f J... I e d£^...d£^.dr^...dr^ -00 o 3N 2 Zcp. ^ + cU(r^...V i=l Now define the reduced temperature T as (6.40) T" = — (6.41) ce where c is given by equation (6.37). Writing equation (6.40) in terms of reduced variables, yields ^The derivation for the three-dimensional case is considered in more detail in Appendix 4.

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102
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103 conductivity are given by 'n = ~^ZZ (6.49) /me 2 k: = (6.50) c /me '^ For dilute gases and monatomic molecules c reduces to unity and therefore equations (6.45), (6.49), (6.50) and (6.51) all reduce to the forms obtained by the simple correspondence principle derivation in equations (5.18), (5.22), (5.23) and (5.24). Thus the above is a general derivation of the correspondence principle which should apply to dilute and dense phases, and to both monatomic and polyatomic nonpolar fluids. 6 . 3 Test of Correspondence Principle for Polyatomic Molecules Several workers (87,88) have pointed out that the corresponding states principle in its simple form does not apply to transport properties of even relatively simple polyatomic molecules, such as nitrogen and carbon monoxide, in the dense phase. Some investigators have adopted the pessimistic viewpoint that the principle is inapplicable for such fluids. In this section the modified corresponding states principle derived above is to be tested for the viscosity, thermal conductivity and self-dif fusivity of polyatomic molecules over a wide range of temperature and density conditions. To make a test of the proposed equations, reasonably accurate values of the characteristic factor c of different fluids must be

PAGE 127

104 determined. This factor can be evaluated from one experimental value for any one of the transport coefficients. Saturated liquid viscosities were used to determine c, as they seem to be the most accurately measured of the transport coefficients, and are most readily available in the literature. The factor c of 25 monatomic and polyatomic molecules thus evaluated is shown in Table 6.1. A complete list of the smoothed potential parameters evaluated from Figures 5.1 and 5.2 is also included in Table 6.1. Table 6.2 gives the sources of transport property data used here to test the correspondence principle. The viscosity data for the low boiling molecules are estimated to have a maximum disagreement bet^.een workers of approximately 81; an exception is oxygen for which the maximum disagreement is 20%. Saturated liquid viscosity data for the hydrocarbons are primarily obtained from a single source (69) and the accuracy of these data are not made known. The high pressure viscosity data are less accurate. Thus for nitrogen at a pressure of 250 atmospheres the maximum disagreement between workers is as much as 14%, and for methane and ethane discrepancies are over 10%. Saturated liquid thermal conductivities appear to be of quite good accuracy for most heavy fluids, being of the order 2%. However, for inert gases, for which extensive measurements have been made, discrepancies are found to be as much as 7%. Of all the three transport coefficients, self-dif fusivity measurements are the least accurate. Thus the methane data reported by different workers show discrepancies as large as 18%, and for ethane differences are 20%.

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105 I— I O o CO o o c o o -< o LO

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106 CM

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107 TABLE 6.2 SOURCES OF TRANSPORT PROPERTY DATA Molecule CH, CD, Ne CO ^2«6 Reference 52,64,68,69, 115,116,117, 118,119 64 61,62,90, 107 64,67,117, 120 49,61,64, 67,68,98, 108,116,121, 122 69,119,123, 124 59,64,67,68, 70,104,108, 120,121,125 66 X Reference 17,27,66,138, 139,140,141 66,103,111 27,66,138 27,66,100, 101,102,142, 143 , 144 140,145 66,146 Reference 18,58,72 71 149,150 Ar

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108 TABLE 6.2 (Continued)

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109 Figure 6.5 is a reduced plot of saturated liqiid viscosity as a function of temperature using the improved correspondence principle, where the viscosity and temperature are now reduced according to equations (o.49) and (6.41) respectively. For clarity, only representative molecules of different molecular structures and weights are included in this figure. However, similar agreement is found for the other molecules listed in Table 6.1, The molecules used to test the modified correspondence principle include long-chain molecules such as nonane, spherical molecules such as neopentane, flat molecules such as benzene, very light molecules such as methane and neon, heavy molecules such as carbon tetrachloride and isotopic molecules such as tetradeuterome thane . Figure 6.6 is a similar plot for reduced saturated liquid thermal conductivity. The thermal conductivity is reduced according to equation (6.51). Good agreement is found except for the data of carbon tetrachloride, benzene and neon. The deviation for neon may be due to quantum effects as has been pointed out in the previous chapter. The poor correlation for carbon tetrachloride and benzene may be due to the Eucken effect. This effect appears to be more important in the gaseous state than for liquids, and is discussed in more detail later. Figure 6,7 is a plot of reduced self-diffusion coefficient for saturated liquids. The correlation is seen to be poorer than for the other two transport coefficients. This may be due to experimental inaccuracies. Figure 6.8 shows a test of the principle for viscosity over a very wide range of temperatures and pressures. Pressures up to almost

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110 o vo re X u o<4»-o-^» V> 00 ffi u 01 c cd X o 00 o CM ^ en i-< 0) o !-i Ph O 0) -r-l > CO O O ^-1 O a, en B •'-' 1-^ > vO u o M ' ' o o o o CM ' J L_ LTl o o CM O ?P

PAGE 134

Ill 0) X u < CO o w n to i-i c 0) CNJ O <]•:=; r-^ i-i 2 2; 00 o CM CO

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112 o CO

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ii: 0) u 03 to V4 O U-l u o o c o a. -i o o 'O > o u a. B >-> >. ij U-l -r-i O CO O iJ O CO CO QJ -H H > 00
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114 1000 atmospheres are included. With the modified principle good ~vV 1 correlation is observed for the gas and liquid viscosities at P =0. However, the correlation is poorer at higher reduced pressures. This may be due to the fact that c is slightly density dependent. Within the normal liquid density range the factor c is found to be approximately constant. The simple correspondence principle does not apply to the thermal conductivity of polyatomic fluids in the gas phase, because of the Eucken effect. The Eucken effect can be largely corrected for by plotting the gas thermal conductivity as logW?^" vs. T"" , where o X " is arbitrarily chosen as the value of X" at T" = 0.1. Figure 6.9 o shows such a plot for a nuiriier of monatomic and polyatomic molecules. Using this method of presentation, corresponding states correlations are found to be greatly improved. It seems that the Eucken effect is more important in the gas than in the liquid phase, for good correlation is observed in the liquid phase (with the exception of benzene and carbon tetrachloride) without correction for the effect. A test of the modified principle for the thermal conductivity of polyatomic molecules at P = 0.0 (liquid phase) and ?" = 1.0 is shown in Figure 6.10. Tables 6.3, 6.4 and 6.5 give the coefficients of the equations (5.42), (5.43) and (5.44) fitted for the curves shown in Figures 6.5, 6.6 and 6.7 respectively. Estimated average percent errors for these equations are also g iven in the tables. Table 6.6 gives the coeffi1 ~v.The reduced pressure P used for polyatomic molecules differs from that used in Chapter 5, and is defined by equation (6.62). It involves the term c. This definition arises for the equation of state, and is discussed in the next section.

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115 r^ c u to o o

PAGE 139

116 c CO ^H

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117 a

PAGE 141

118 o u

PAGE 142

119 OJ u

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120 O M H W ix H M > M CO M W O [d I hJ W W CO hJ O « S 1—1 P o O M ^g < o Pi o o < PS H < CO fi^ o + rII h O ij

PAGE 144

121 o u

PAGE 145

122 0)

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123 o u

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124 cients of equations for the curves in Figure 6.8 and Tables 5.7 and 6.8 give coefficients of the thermal conductivity equations for the curves in Figures 6.9 and 6.10 respectively. A list of the values of X" for molecules considered in Figure 6.9 is also included in Table 6.7. 6 .4 Comparison with Theory for Thermodynamic Properties Recently, Hermsen and Prausnitz (166) proposed a corresponding states treatment for the thermodynamic properties of liquid hydrocarbons. Their approach was based on Prigogine's corresponding states theory for r-mers (50) . Th^ defined the reduced temperature as T = , )\ (6.52) and the reduced molal volume as V = ^ (6.53) k where V" is an arbitrary characteristic volume. Potential parameters k determined from second virial coefficient data as given by Hirschfelder, Curtiss and Bird (20) were used. By fitting experimental molal volume data using a high speed computer they obtained (s/c) and V" for a k k large number of hydrocarbons. It is of interest to compare their values of (s/c) obtained from molal volume data with the values of c k obtained here from viscosity data. These quantities, together with the potential parameters used in each case, are compared in Table 6.9. Hermsen and Prausnitz arbitrarily assumed the value of (s/c) for methane co be unity. After multiplying their values of (s/c)^ by a

PAGE 148

125 00 u CO o u ^ ej u > a < 2-5 O in

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126 CU !-i

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127 1

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128 constnnt factor chosen such that the value of (s/lO^^ for methane agrees with the value of c for methane used in this work, it is seen from Table 6.9 that the txvro parameters are approximately the same for most hydrocarbons. This suggests that the correspondence principle should apply quite well to both transport and thermodynamic properties in the liquid phase using a single set of c values. However, it may be necessary to use different sets of potential parameters in the t\Jo cases. It is of interest to examine the modified correspondence principle for thermodynamic properties of polyatomic fluids. From statistical thermodynamics, the pressure is given in terms of the canonical partition function as where the partition function may be separated into a densityindependent Dart and a part Q which depends on density, ^ ^i ^ CM Q=i7QQ (6.55) where 400 V -co o 3N 2 r— , CO. \ -r^ + cU^r r.J (6-57) and therefore CM Z ^™ -i 1^ i=l U(r,...r^.)-. V *(^) (6.58)

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129 P = kT V V ^(In /. . . / e ^ dr^. . .dv^) bv (6.59) N cpf -K» -foo y ik Since 1=1 2mkT e cl£i''''^£»T is independent of density. -00 -00 Equation (6.59) may be written in terms of reduced variables to give ce ~* V V S(ln exp o o L T 0(rt.) i
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130 (0 o to ca u o CO MSOOOOUU o<>t:i'<>-5? >::;ti • -o,/> / / -' y / / / / f / / ^ ^ ' ^ / y -' ->-' 0^ O CO o cy !-i D CO CO 0) PL, o a•r-l o r^ •H !-i P-i O o s r; O O. CO <1> Vi 5-1 O o Cu S •H cy D r-( -L ' ' O O CM O o o o CM O — i o o o

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131 u 15-0^ '^ ^H O 00 o -I I 1o c

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CHAPTER 7 CONCLUSIONS The free volxur.e equation provides a useful basis for predicting viscosities and self-diffusion coefficients of liquids at high densities, where most other correlations are unsuccessful. The fact that both V and E values vary in a corresponding states way indicates that the equation's success does not arise only from the availability of three adjustable parameters. The simple expressions given for A by Eyring and others, however, do not seem to be generally applicable. The free volume theory seems to provide somewhat more accurate values of transport properties than does the direct corresponding states approach, but has several limitations. In particular it applies only ac high density (p> 2p ), and cannot be used to obtain thermal conductivities. The direct corresponding states approach is of more general application, and in the form presented in Chapter 6 it may be used for all transport properties and over a wide range of densities. An attractive feature of the principle presented for polyatomic fluids is that, although a third parameter is introduced, the transport coefficients are expressed as functions of only two independent variables (T and P ) . In view of the simplicity of the proposed model, it may at first sight seem somewhat surprising that the correlation is so successful. It should be remembered, however, that the model is only used to obtain the form of the reduced relationships, and not to directly calculate transport coefficients. In this 132

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133 forra the correspondence principle applies to a wide variety of substances, including fluids composed of relatively complex molecules X'/hich are of interest to engineers. It is easy to apply, the only parameters required for prediction being T , P , M, CD and c. The first four parameters are available in the literature, and the latter may be obtained from one experimental measurement of any one of the transport properties. I^Jhen no such measurements are available it may be evaluated using thermodynamic data (e.g., vapor pressure). In the light of the promising results of this investigation, it is felt that the free volume theory and the corresponding states principle are very useful methods for the prediction of physical properties of dense and condensed fluids. For future work the following studies are recommended. 7 .1 Free Volume Theory 7.1.1 Extension to Mixtures In the course of this work the free volume theory has been tested for the viscosity of binary mixtures of simple molecules (167) . For the mixtures tested it is found that the predicted viscosity is within the experimental accuracy over the entire range of compositions. In the prediction of the mixture viscosity no mixture data are required. For future studies, the free volume theory may be extended to binary diffusion coefficients and viscosity of multicomponent systems. 7.1.2 Electrolyte Solutions Podolsky (168) has utilized the Eyring cell theory to predict the self-diffusion coefficient of electrolyte solutions.

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134 However, no attempt has yet been made to apply the free volume theory to the transport properties of electrolytes. Based on the promising results demonstrated by the free volume theory for the viscosity and self-dif fusivity of pure fluids, it is believed that this theory may serve as an improved basis for predicting transport properties of electrolytes. 7.2 Corresponding States Principle 7.2.1 Mixtures The corresponding states principle may be derived in the same manner as for pure substances, by starting with the time correlation function equations for the transport properties of mixtures (169-171). 7.2.2 Polar Substances In this work only nor.polar substances are considered. It will be useful to extend the correspondence principle to polar substances. This can be done by introducing suitable reduced dipole or quadrupole moments , 7.2.3 Fused Salts Transport properties of fused salts are very important for a number of practical applications. As measurements of transport properties of fused salts are difficult, it will be of interest to extend the corresponding states principle to these substances. Such liquids are composed of ions which may often be regarded as roughly spherical charged particles. 7.2.4 Thermodynamic Properties A further extension of the proposed principle for polyatomic

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135 molecules will be a detailed study of its application to thermodynamic properties in the der.Sj i/iu condensed phase. A preliminary study shows that the proposed correspondence principle greatly iu.proves the correlation for vapor pressure. With suitable reduced expressions, it seems likely that the correspondence principle may apply to all other thermodynamic properties. 7.2.5 Prediction of c Although an approximate expression has been obtained for the factor c from the derivation of the correspondence principle for polyatomic molecules, no extensive study of the behavior of this parameter has been made. The parameter c is a function of the moment of inertia, mass and potential parameters of the molecule, as well as the density and temperature. A useful extension of this work will be the study of the properties of the parameter c and methods of predicting it. Any more detailed examination should include a study of the density dependence of this parameter.

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APPENDICES

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APPENDIX 1 CHUNG'S DERIVATION OF THE FREE VOLUXE EQUATIONS Chung (46) considered a aiolecule to spend a large fraction of the time oscillating about an equilibrium position in a cell. The m.olecule can leave the cell to jump into a neighboring cell only when the following b
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138 as \1^ ^(i), (Al.l) i=l Similarly, if W is the number of ways of assigning N molecules in V , N "^ in V ... etc., where i = 1 to j , it follows that 12 2 ^ ^ , N^^^: 1 r N^J^' p=l p p=l p 1=1 j k ^ri), (A1.2) i=l p=l P The total number of configuration becomes W = WW iN(^>: n: i=i TT N^ ' ! TT 7T N^ ' I i=l i=l p=l P n: " j k^,(i), (A1.3) i=l p=l P Making use of Stirling's approximation we may write j k InW = NlnN ) V N^^^^lnN^^^ (A1.4) i=l p=l The statistical mechanical formulation of entropy S is given by S = klnW . , = k[NlnN V Vw^^-'lnN^^'*] (A1.5) _ -^ P P i=l p=l

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139 The system is isolated and is considered to be at thermodynamic equilibriiim. Thus the entropy is a maximum, and dS = j k ^ j k = k) V — 7TT I NlnN 1=1 p=l p m=l n=l N^^^nN^^^ n n dN (i) J t^ = k ?^ i=l p=l P P P dN .(i) J k -k yy [InN^^) -M]d4^^ £_ ^ p i=l p=l (A1.6) or j k y y [inN^^^ + l]dN<^^^ = / / D D z_ i=l p=l (A1.7) The system is subject to the restraints of constant number of molecules, constant total free volume and constant energy, so that j k or P i=l p=l 1 1 ^A'' = ^ i=l p=l

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140 Using the Lagrange undetermined multiplier procedure we may multiply equations (Al. 11), (A1.12) and (A1.13) by (a1), /3 and A respectively, and add the resulting equations to equation (A1.7), to give J k ^ > [a + InN^^-* + /Se. + Av ] dN ^^^ = L. i. p ^ 1 p' p i=l p=l Choosing suitable values for a, jS and A, one may write (A1.14) or a + InN^^-* + fie. -rAv =0 (A1.15) pip. -.^x -(a + pe. + Av ) Np = e ^ P for all p,i (A1.1&) The probability distribution of free volume and energy may be written as (l) D Pp = -— -(a + fie . + Av ) 1 p' e ^ Y"! r-i (a + /3e , + Av ) ^ ) e 1 P (A1.17) i=l p"^l From equation (A1.16), on summing over p and i, we have X X (a + /5e . + Av ) N = V V e ^ ? i=l p=l Solving for a gives (Al.lS) -Q. N e ^ V -<^^i-^%) e (A1.19) 1=1 p=i Making use of equations (ALIO), (A1.16) and (A1.19), and allowing the values of free volume v^^ to tend to the continuum limit, we have

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141 X ^ -(«e. +Av^) 1 D P N ) e ) V e '^ _ i=l D=2 = j ^ k , (A1.20) I "^ I e P thus i='l i=l V J ve dv f N r=^ (A1.21) a'^ -Av ~ e dv where v = fluctuating free voliome or A=— (A1.22) ^f Similarly combining equations (A1.9), (A1.15) and (A1.19) gives ^ -Be. ^ 1 V e.e 1 ^ X (A1.23) -Be. e "i=l :t is obvious from equation (A1.23) that /3 has its usual significance. that is '^ " kT ^=h V exp(-v /v )exp(-e./kT) Z_; 4_ P ^ ^ (A1.25) 1=1 p=l

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142 Since the values of v are assumed to be continuous, the sicnmation P can be replaced by an integral and the subscript p dropped, so that p(v) =-expf-J (A1.26) To get an explicit expression for the energy distribution, Chung assumed the molecules to be harmonic oscillators. In the classical limit (172) ?(e) = 1 \ /'e_ \kT 2kT exp e kT (A1.27) The probability that a molecule simultaneously has an energy greater than some value e and a free volume greater than some value v is: P(v > v ,e > e ) = O V p(v)p(e)dedv v e o V , , e \ 2 1 / V 2 \. kT + V kT + 1 exp r /e V \ kT v^ (A1.28) Based on the assijmption that self-dif fusivity and fluidity are directly proportional to this probability, one may write for the expressions of viscosity and self-dif fusivity E V r 1 (A1.29) and D = Bexp / E V f _Z ^ 2_ V RT V V (A1.30) where

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143 V . = V V t o V = Nv V = Nv o o E = Ne V V and A and B are assumed roughly constant. The form of the preexponential factors is not made clear in the derivation. The preexponential term appearing in equation (A1.2S) depends on the assumed form of the molecular energy. Thus if the molecules are assumed to be in a potential well, rather than behaving as oscillators, a different pre-exponential factor is obtained.

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APPENDIX 2 FURTHER TEST OF PROPOSED FREE VOLUME THEORY

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145 •S.O In •8.5 9.0 n2 9.5 10.0 Free Vol-ume Theory o Experiment (49) 32.6 cm."'raole •10.5 56.0 (Theory) 56.0 (Expt.) -11.0 4.0 6.0 8.0 10.0 ^x lO^V^ 12.0 14.0 16.0 Figure A2.1. Test of Free Volume Theory for Liquid Nitrogen Viscosity.

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146 -^

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147 o CO en O O

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APPENDIX 3 SOLUTION OF EQUATIONS OF MOTION FOR OSCILLATOR From equations (6.15) and (6.17) we have m d^x dt 2-^^l^ + ^2^=° (A3.1) and 2 d -^ d-c z ZZ X IZ (A3. 2) In terms of differential operators one may write and (mD + k )x + k.„^ = 11 12 X (ID + k.,)? + k.^x = ZZ X 12 (A3. 3) (A3. 4) Eliminating Y.from equations (A3. 3) and (A3. 4) gives (ID^ + k22)(mD^ + k^^x k^^^ = ^ (A3. 5) Expanding equation (A3. 5) we have [imD^ + (k^^m + k^^I)D^ + k^^k^, k^^^^ = ° (A3. 6) Equation (A3. 6) may be solved to give roots of D as n (k22in+k^,^I)+y(k22m+k^^I)^ 4lm(k^^k22-k^,) 1/2 (A3. 7) Let 148

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149 ^1 = (k^.,ni+k^^T') -\/ (k.^./>ilk^, l)"^ 4Im(k^^k^2"'^i9) " 1/2 2Im (A3. 8) 0), = (k2om+k^^I) +/(k22m+k^^I)^ 4lm(k^^k22-k^2) 1/2 2Im (A3. 9) Therefore -ico t +ico t -iCJ„t ico^t X = A.e + A„e + A„e ' + A, e 12 3 4 (A3. 10) Similarly, eliminating x from equations (A3. 3) and (A3. 4) gives (mD^ + k-^)(ID^ + k„)^ k^.Y = li 2.1 yi 1/ X (A3. 11) Equation (A3. 11) is expanded to give [iraD^ -r (k^2™ + k^^I)D^ + k^,^k22 k^2^^ " ° (A3. 12) Solving for the roots of D we have D = ' i / ~ 2/ 1/ j(k22m+k^I) + ^/(k2^m+k^I) 4Im(k k^^-k^) -, 2Im (A3. 13) Making use of equations (A3. 13), (A3. 8) and (A3. 9), -iaj t ico, t -iCD^t i6D2t H' = E,e + B.e + B„e + B, e X 1 2 3 4 (A3. 14) Since v/e have assumed that the translational and angular oscillations represented by equations (A3.1) and (A3. 2) are harmonic and that the oscillations are in phase, then 60 , which represents the angular

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150 velocity of a harmonic motion, must be equal to o)^ • This is true if (k22m+k^^I)^ 4lm(k^^k22-k^2^ = ° (A3. 15) Equation (A3. 10) now reduces to X = A'e ^ + A^e""'^!*' (A3. 16) -icd t Since e = cosOO t isimo t (A3. 17) i(X>1 and e = cosCD c + isinO). t (A3. 18) 1 ->• Equation (A3. 16) may be written as X = A'cosO), t + A"sinJO, t (A3. 19) 1 i. where A' = A| + A' A" = i(A' Ap Similarly equation (A3. 14) may be written as ^ = B'cosco.t + B"sinw,t (A3. 20) X 1 i

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APPENDIX 4 HAMILTONIAN FOR THE TIIREE-DIMENSIONAL OSCILLATOR X Figure A4.1. A Three-Dimensional Oscillator Model Consider a molecule oscillating in an isotropic harmonic manner in a 3-dimensicnal cell. At any particular position _r of the molecule, it has a linear velocity V(_r) and an angular velocity co(_r,_0) , where represents the Eulerian angles. One may resolve _r and V(_r) along the x, y and z axes as r = xe + ye + ze (A4.1) V(r) = V^(x)e^ + V (y)e2 + V^(z)e3 (A4.2) Let to(_r,_9) be made up of two parts; a constant angular velocity CJ° (_r = O,_0) together with a harmonic angular oscillator which is center of mass coordinates dependent, a)(_r) , superimposed on it. In other v7ords we assume (^(r,e) = a.(r) + (a ( r = 0,0) (A4.3) 151

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152 Resolving this along the 3 Cartesian coordinates we have a^(r,_§) = a).(x)e + CD (y)e + O^Jz)e + a)°(^)e X X y z ^ J J. + a3°(0)e +6D°(9)e (A4.4) y — 2 ^ ~3 The angular velocities at the cell center can be expressed in terms of Eulerian angles as (173) CD°(6;) = 0COS0 +ijj sin0sin0 (A4.5) X CO°(e) = 0sin0 •^sin9sin0 (A4.&) GO°(0) = + 7//cose (A4.7) z ~" Suppose that the niolecule is oscillating in a particular direction with instantaneous velocity V at any position r, and at the same time it has a harmonic angular oscillation which is in phase with the linear one. We now consider the components of V and Cd(r,6) . We pair CV CO ) , (V ,03 ) and (V ,co ) , and consider them as the linear and X z y X z y angular velocities of 3 oscillators, one along the x-axis, one along the y-axis and one along the z-axis. These oscillators may be considered separately. The oscillation associated with the x-axis will be as shown in Figure 6.3. We define the angles associated with co as ° X t e (t) = r CO (x,0)dt ^J ^ (A4.8) t o

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153 ^ (t) = / co,(e)dt (A4.9) t o t = r o) (x^ ^ (x) = / O) (x)dt (A4.10) X ,' X t o The potential and kinetic energies are expressed as before as 1 .2 1 '2 K = ^ mx + ^ I^ • (A4.12) X 2 2 X The equations of motion and their solution are nov/ identical to those given in section 6.2. The final equations for the potential and kinetic energies associated with the x-direction are = I c[k^^ + Hk^^'^^ (A4.13) K = ^ cxax(A4.14) where H and c are defined by equations (6.32) and (6.37). Similarly for the y and z axes we have ^/y.\) = ^ c[k^^ + Hk^2^y2 (A4.15) U^(.,Y^) 4 c[k^^ + Hk^2^z2 (A4.16) K = -J cmy^ (A4.17) y 2 ^

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154 K = — cmz z 2 (A4.18) The Hamiltonian for the center of mass coordinates becomes H CM N ) K. + K. + K. + U. + U. + U. + U(0,0) ^_, IX ly 12 IX ly iz v j^-' i=l N i=l ~ -^ = Vfc 2,2,2 i=l zm (p. + p. + p. ) + cU(r r ,) (A4.19)

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LITERATURE CITED 1. Zv7an2ig, R, , Ann. Rev. Phys . Chem., J_6, 67 (1965). 2. Helfand, E., Phys. Rev., _119 , 1 (1960). 3. Rice, S. A. and Gray, P., "The Statistical Mechanics of Simple Liquids," Interscience Publ., New York, 1965. 4. Rahman, A., Phys. Rev., _136, A405 (1964). 5. Rahman, A., J. Chem. Phys., 45, 2585 (1966). 6. Berne, B. J. and Harp, G., "Linear and Angular Momentum Correlations in Liquids and Gases," Paper presented at New York A.I.Ch.E. meeting (November, 1967). 7. Chapman, S. and Cowling, T. G. , "The Mathematical Theory of NonUniform Gases," Cambridge University Press, New York (1952). 8. Kirkwood, J. G., J. Chem. Phys., _14, 180 (1946). 9. Irving, J. H. and Kirlcwood, J. G., J. Chem, Phys., 18, 817 (1950). — 10. Mazo, R. M., "Statistical Mechanical Theories of Transport Processes," Pergamon Press, Inc., London (1967). 11. Rice, S. A. and Allnatt, A. R. , J. Chem. Phys., M, 2144 (1961). 12. Allnatt, A. R., and Rice, S. A., J. Chem. Phys,, 3U, 2156 (1961). 13. Zv/anzig, R. W., Kirkwood, J. G., Stripp, K. F. and Oppenheim, I., J. Chem. Phys.,n> 2050 (1953). 14. Naugle, D, G., J. Chem. Phys., 44, 741 (1966). 15. Lowry, B. A., Rice, S. A. and Gray, P., J. Chem. Phys., 40, 3673 (1964). 16. Kirkwood, J. G., Lewinson, V. A. and Alder, B. J., J. Chem. Phys., 20, 929 (1952). 17. Ikenberry, L. D. and Rice, S. A., J. Chem. Phys., 39, 1561 (1963). ~ 18. Naghizadeh, J. and Rice, S. A., J. Chem. Phys., 36, 2710 (1962). 155

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156 19. Davis, H. T., Rice, S. A. and Sengers, J. V., J. Chem. Phys . , 35, 2210 (1961). 20. Hirschfelder, J. 0., Curtiss, C. F, and Bird, R. B., "Molecular Theory of Gases and Liquids," John Wiley and Sons, New York (1954). 21. Davis, H. T. and Luks , K. D., J. Phys. Chem., 69_, 869 (1965). 22. Luks, K. D., Miller, M. A. and Davis, H. T., A.I.Ch.E.J., 12_, 1079 (1966). 23. Reed, T. M.,III and Gubhins, K. E., "The Molecular Approach to Physical Property Correlations," Chapter 12, to be published. 24. Michels, A., Botzen, A. and Schuurman, W. , Physica, 2G, 1141 (1954). 25. Dymond, J. H. and Alder, B. J., J. Chem. Phys., 45, 2061 (1966). 26. Glasstone, S., Laidler, K. J. and Eyring, H. , "The Theory of Rate Processes," McGraw-Kill Book Co., Inc., New York (1941). 27. Horrocks , J. K. and McLaughlin, E., Trans. Faraday Soc, 56 , 206 (1960). 28. Hougen, 0. A., Watson, K. M. and Ragatz, R. A., "Chemical Process Principles," 2nd ed., pt. 2, Thermodynamics, John Wiley & Sons, Inc., New York (1959). 29. Reid, R. C. and Sherwood, T. K. , "The Properties of Gases and Liquids," 2nd ed. , McGraw-Hill Book Co., New York (1966). 30. Eakin, B. E. and Ellington, R. T., J. Petrol. Technol., 15, 210 (1963). 31. Jossi, J. A., Stiel, L. I. and Thodos, G., A.I.Ch.E.J., 8_, 59 (1962). 32. Stiel, L. I. and Tnodos, G. , A.I.Ch.E.J., 10_, 275 (1964). 33. Trappcniers, N. J., Botzen, A., Van Gostcn, J. and Van Den Berg, H. R., Physica, 31_, 945 (1965). 34. Reynes, E. G. , Ph.D. Dissertation, Northwestern University, Evanston, Illinois (1964). •5 5. Weymann, H. , Kolloide Zeit. , 138, 41 (1954) 36. Weymann, K. , Kolloide Zeit., 181, 131 (1962)

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157 37. McLaughlin, E., Trans. Faraday Soc, _55, 28 (1959). 3S. >:ajutr.dar, D. K. , J. Phys. Chem., 62, 1374 (1963). 39. Barker, J. A., "Lattice Theories of the Liquid State," Pergamon Press, Oxford (1963). 40. Doolittle, A. K. , J. Appl, Pays., 22, 1471 (1951). 41. Doolittle, A. K., J. Appl. Phys., 22_, 1031 (1951). 42. Doolittle, A. K. , J. Appl. Phys., 23, 236 (1952). 43. Cohen, M. H. and Turnbull, D. J., Chem. Phys,, 31, 1164 (1959) 44. Turnbull, D. and Cohen, M. H., J. Chem, Phys., 34, 120 (1961). 45. Macedo, P. B. and Litovitz, T. A., J. Chem. Phys., 42, 245 (1965), "" 46, 47, 48, Chung, H, S,, J. Chem, Phys., 44, 1362 (1966), Kaelble, D, H., in "Rheology IV," ed. by F. R. Eirich, Academic Press, Kew York, to be published. Zhadanova, N. F., Soviet Physics J.E.T.P., 4, 749 (1957). 49. Zhadanova, N. F., Soviet Physics J.E.T.P,, 4, 19 (1957). 50. Prigogine, I., Bellemans, A., andMathot, V., "The Molecular Theory of Solutions," North-Holland Publ. Co., Amsterdam (1957), 51. Brummer, S. B., J. Chem. Phys., 42, 4317 (1965). 52. Haung, E. T. S., Swift, G. W. and Kurata, F., A.I.Ch.E.J., 12, 932 (1966). 53. "International Critical Tables," Vol, _5, P. 10, McGraw-Hill Book Co,, New York (1929), 54. "International Critical Tables," Vol. 1_, P. 222, McGraw-Hill Book Co., New York (1930), 55. Jobling, A. and Lawrence, A, S. C, Proc. Roy. Soc, A2C6 , 257 (1951). Corbett, J. W. and Wang, J. H., J. Chem. Phys,, 2_5, 422 (1956). 56, 57. Yen, W. M. and Norberg, R. E,, Phys, Rev,, l^l, 269 (1963)

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158 53. Gaven, J, V., Waugh, J. S, and Stockmayer, W, H. , J. Chem. Phys., 28, 287 (1963). 59. D<3 Bock, A., Grevendonk, W. and Awouters, H. , Physica, 34 , 49 (1967). 60. Cook, G. A., ed., "Argon, Helium and the Rare Gases," In jer science Publ., New York (1961). 61. Forster, S., Cryogenics, _3> 176 (1963). 62. Kuth, ?., Cryogenics, 2, 368 (1962). 63. Van Itterbeek, A., Verbeke, 0. and Staes, K. , Physica, 29, 742 (1963). 64. Boon, J. P., Legros, J. C. and Thomaes, G., Physica, 33 , 547 (1967). 65. Reynes, E. G. and Thodos, G,, Physica, 30' 1^29 (1964). 66. Johnson, V. J., Wadd Technical Report 60-56, Wright-Patterson Air Force Base, Ohio (1960). 67. Rudenko, N. S., and Schubnikov, L. W. , Phys. Z. Sowjetunion, 6, 470 (1934). 68. Rudenko, N. S., Zh. Eksp. Teo . Fiz., 9, 1078 (1939). 69. Rossini, F. D., at al . , "Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds," Carnegie Press, Pittsburgh (1953). 70. Van Itterbeek, A. and Van Paemel, 0., Physica, _8, 133 (1941). 71. Cini Castagnoli, G., Physica, 30, 953 (1964). 72. Rugheimer, J. H. and Hubbard, P. S., J. Chem, Phys., 3_9, 552 (1963). 73. Reed, T. M. , III, "Fluorine Chemistry," Vol. 5, Chapter 2, Academic Press, Inc., New York (1963). 74. Pitzer, K. S., J. Chem. Phys., 7_, 583 (1939). 75. Hakala, R. W. , J. Phys. Chem., 71, 1880 (1967). 76. Helfand, E. and Rice, S. A., J. Chem. Phys., 32., 1642 (1960). 77. Donth, E., Physica, 32, 913 (1966).

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159 78. Danon, F, and Rossi, J. C, J. Phys. Chem. , 70, 942 (1966). 79. Danon, F. and Pitzer, K. S., J. Am. Chem. Soc, 66, 583 (1962), 80. Keeler, R. N., Van Thiel, M. and Alder, B. J., Physica, 31, 1437 (1965). 81. Crivelli, I. and Danon, F., J. Phys. Chem., 7_1, 2650 (1967). 82. Ross, M. and Alder, B., J. Chem. Phys., 46, 4203 (1967). 83. Smith, E. B., Chem. Soc. Annual Reports, 63, 13 (1966). 84. Cini Castagnoli, G., Pizzella, G. and Ricci, F. P., Nuovo Cim., U,, 466 (1959). 85. Kerrisk, J, F,, Rogers, J, D, and Hammel, E, F,, Advances in Cryogenic Eng,, _9, 188 (1964). 86. Rogers, J, D. and Brickwedde, F. G., Physica, 32, 1001(1966). 87. Boon, J. P. and Thomaes, G,, Physica, 28, 1074 (1962), 88. Boon, J. P. and Thomaes, G., Physica, 22, 208 (1963). 89. Tee, L. S., Kuether, G. F., Robinson, R. C. and Stewart, W. E., Paper presented at the 31st Mid-year Meeting of the Am. Petrol. Inst. Division of Refining in Houston, Texas (1966). 90. Trappeniers, N. J., Botzen, A., Van Den Berg, H. R. and Van Oosten, J., Physica, 30, 985 (1964). 91. Trappeniers, N. J. Botzen, A., Ten Seldam, C, A., Van Den Berg, H. R. and Van Oosten, J,, Physica, 31. 1681 (1965). 92. Reed, T. M., Ill and McKinley, M. D., J. Chem. & Eng. Data, 9, 533 (1964). 93. Tee, L. S., Gotoh, S. and Stewart, W, E., I.&E.C. Fundamentals, 5, 356 (1966). 94. Pitzer, K. S., J. Am. Chem. Soc, 77^, 3427 (1955); ibid. , 77, 3433 (1955), 95. Madigosky, W. M. , J. Chem. Phys., 46, 4441 (1967). 96. Naugle, D. G., Lunsford, J. H. and Singer, J. R., J. Chem. Phys,, 45, 4669 (1966). 97. Rudenko, N. S., Zh. Tekh. Fiz,, _18, 1123 (1948),

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160 98. Van Itterbeek, A., Zink, K. and Van Paemel, 0., Cryogenics, 2, 210 (1962). 99. Saji, Y. and Kobayaski, S., Cryogenics, 4, 13c '1964). 100. Keyes, F. G., Trans, Am. Soc. Mech. Eng . , 11_, 13S5 (1955). 101. Uhlir, A., J. Chem. Phys,, 20, 463 (1952). 102. Zieland, H. and Burton, J. T. A., Brit. J. Appl. Phys., 9_, 52 (195S). 103. Lochtermann, E., Cryogenics, _3, 44 (1963). 104. Kiyama, R. and Makita, T., 3.ev. Phys. Chem. (Japan), 22 , 49 (1952). 105. Michels, A., Botzen, A. and Schuurman, W., Physica, 2£, 1141 (1954). 106. Kestin, J, and Leidenfrost, W. , Physica, _25, 1033 (1959). 107. Flynn, G, P. Hanks, R, V., Lemaire, N. A. and Ross, J., J. Chem. Phys., 38, 154 (1963). 108. Van Itterbeek, A,, Hellemans, J., Zink, H. and Van Cauteren, M. , Physica, yi, 2171 (1966). 109. Michels, A., Sengers, J. V. and Van De Klundert, L. J. M. , Physica, 29, 149 (1963). 110. Rosenbaum, B. M, , Oshen, S. and Thodos, G., J. Chem. Phys., 44, 2831 (1966). 111. Sengers, J. V., Eolk, W, T. and Stiger, C. J,, Physica, _30. 1018 (1964). 112. Preston, G. T., Chapman, T. W. and Prausnitz, J. M. Cryogenics, 7, 274 (1967). 113. Bavis, D. B. and Matheson, A. J., J, Chem, Phys., 45, 1000 (1966) 114. Wilson, A. H., "Thermodynamics and Statistical Mechanics," Cambridge University Press (1957). 115. Gerf, S. F. and Galkov, G, I., Zh. Tekh. Fiz., _11, 801 (1941). 116. Ross, J. F. and Brown, G, M. , Ind. Eng, Chem,, 49, 2026 (1957). 117. Barua, A, K. , Aizal, M,, Flynn, G. P. and Ross, J., J. Chem, I'hys,, 41, 374 (1964). T31.

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161 lis. Giddings, J. G., Koa, J. T. F. and Kobayashi, R., J. Chem. Phys., 45, 578 (1966). 119. Baron, J. D. , Roof, J. G. and Wolls, F. W. , J. Chem. Eng . Data, 4, 2S3 (1959). 120. Galkov, G. I. and Gerf, S. F., Zh. Tekh. Fiz., ^I, 613 (1941). 121. Van Itterbeek, A., Zink, H. and Hellemans, J., Physica, 32 , 489 (1966). 122. Michel, A. and Gibson, R. 0., Proc. Roy. Soc, A134, 288 (1932). 123. Carmichael, L. T. and Sage, B. H. , J. Chem. Eng. Data, 8, 94 (1963). 124. Smith, A. S. and '£>vo\
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162 136, Handbook of Chemistry and Physics, 47th ed,, Chemical Rubber Publishing Co. (1966). 137, Bleakney, W. M., Physica, 3, 123 (1932), 138, Johnston, H. L, and Grilly, E, R,, J. Chem, Phys,, ]A, 233 (1945). 139, Lenoir, J, M, and Comings, E. W,, Chem. Eng , Prog,, 47, 223 (1951), 140, Lenoir, J, M. , Junk, W, A. and Comings, E, W. , Chem, Eng, Prog,, ^, 539 (1953), 141, Carmichael, L. T., Reamer, H, H, and Sage, B, H., J, Chem, Eng, Data, _11, 52 (1966), 142, Michels, A. and Botzen, A., Physica, 19, 585 (1953), 143, Johannin, P,, Proc. Conf. Thermo. S: Transport Props. Fluids, London (1957). 144, Johannin, P. and Vodar, B,, Ind, Eng. Chem,, 49, 2040 (1957), 145, Carmichael, L. T., Berry, V. and Sage, B. H., J, Chem, Eng, Data, 8, 281 (1963), 146, International Critical Tables, Vol, _5, P, 214, McGraw-Hill Book Co . , New York ( 1929 ) , 147, Lambert, J, D,, Staines, E. N, and Woods, S, D,, Proc, Roy, Soc, (London), A20Q , 262 (1949), 148, Carmichael, L. T. and Sage, B. H., J. Chem. Eng. Data, 9, 511 (1964). 149, Gaven, J. V., Stockmayer, W, H, and Waugh, J, S., J. Chem, Phys,, 37, 1188 (1952). 150, VJade, C. G, and Waugh, J. S., J. Chem. Phys,, 43, 3555 (1965), 151, Timmerhaus, K. D. and Drickamer, H. G., J. Chem. Phys., 20, 981 (1952). 152, Robb, W, L, and Drickamer, H, G., J, Chem, Phys,, _19, 1504 (1951). 153, Robinson, R. C, and Stewart, W, E., "SelfDiffusion in Liquid Carbon Dioxide and Propane," Private Communication.

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163 154. Fishman, E., J. Phys. Chem. , 59, 469 (1955). 155. Douglass, D. C. and McCall, D. W. , J. Phys. Chem., 62, 1102 (1958). 156. McCall, D. W, , Douglass, D. C. and Anderson, E. W. , J. Chem. Phys., 31, 1555 (1959). 157. Douglass, D. C, McCall, D. W. and Anderson, E. W. , J. Chem. Phys., 34, 152 (1961). 158. Hiraoka, H., Osugi, J. and Jono, W. , Rev, Phys. Chem. Japan, 28, 52 (1958), 159. Grauoner, K, and Winter, E. R. S., J. Chem, Soc, 1952 . 1145 (1952). 160. Johnson, P. A. and Babb, A. L, , J. Phys. Chem., 60, 14 (1956). 161. Birkett, J. D. and Lyons, P. A,, J. Phys. Chem., 69, 2732 (1965) 162. Rathbun, R. E, and Babb, A, L., J. Phys, Chem,, 65, 1072 (1961), 163. Kulkarni, M. V., Allen, G. F. and Lyons, P. A,, J. Phys. Chem,, 69, 2491 (1965). 164. Watts, H., Alder, B. J. and Hilderbrand, J. H., J. Chem. Phys., 23, 559 (1955). 165. Carman, P. C. and Miller, L. , Trans. Far. Soc, 55, 1833 (1959). 166. Hermsen, R. W. and Prausnitz, J. M. , Chem. Eng . Sci., 2A, 791 (1966). 167. Gubbins, K. E. and Tham, M, J., "Free Volume Theory for Viocosity of Simple Nonpolar Liquids. II Mixtures," to be published in A.I.Ch.E.J, 168. Podolsky, R. J., J. /jn. Chem. Soc, 80, 4442 (1958). 169. Kawasaki, K. , Phys. Rev., ^50, 291 (1966). 170. Mori, H., Phys. Rev., n2. 1S29 (1958). 171. McLennan, J. A., Jr., Phys, Fluids, 3, 493 (1960), 172. Rushbrooke, G. S., "Introduction to Statistical Mechanics," Chap. 5, Oxford University Press, London (1949). 173. Eyring, H., Henderson, D. , Stover, B. J. and Eyring, E. M., "Statistical Mechanics and Dynamics," John Wiley & Sons, Inc., New York (1964).

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BIOGRAPHICAL SKETCH ViT . Min Jack Tham was born on April 14, 1935 in Shanghai, China. He received all his early education in Rangoon, Burma. In May, 1953, he entered the University of Rangoon and passed the Intermediate Science Examinations in March, 1955. He was adiriitted to the College of Engineering, University of Rangoon in the same year, and received a Bachelor of Science degree in Chemical Engineering in March, 1959. During his six years in the University of Rangoon, he passed each year's examinations (except the 3rd Year Engineering Examinations) with Distinction. From 1959 to 1963 he served at the Ngwe Zin Yan Oil and Flour Mills, Rangoon, as chief engineer. In September, 1963, he came to the United States of America for graduate study in chemical engineering. He entered the University of Florida and received a M.S.E. degree in Chemical Engineering in December, 1964. 164

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This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Engineering and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Docror of Philosophy. June, 1968 eajr. College of Engineering Supervisory Committee: Chairmi^n Dean, Graduate School