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,