
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UF00097815/00001
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:
 1968
 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 )
nonfiction ( 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 highspeed 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 nonequilibrium 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 24 contain a detailed study of the free
volume theory of viscosity and selfdiffusion 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:
 ThesisUniversity of Florida, 1968.
 Bibliography:
 Bibliography: leaves 155163.
 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 nonprofit 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 highspeed 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 nonequilibrium 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 24 contain a detailed study of the free
volume theory of viscosity and selfdiffusion 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 ThreeDimensional Oscillator..... 151
LITERATURE CITED...................... ...... .... ........ ......... 155
BIOGRAPHICAL SKETCH .................... ..... ................... 164
LIST OF TABLES
Table Page
1.1 Test of RiceAllnatt Theory for Shear Viscosity
of Argon................................. ............. 6
1.2 Test of RiceAllnatt 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 SelfDiffusivity Data Sources and Range of Conditions 48
4.4 Parameters for SelfDiffusivity 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 SelfDiffusivity
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 SelfDiffusivity
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 SquareWell Model............... 9
1.2 Theoretical and Experimental Saturated Liquid Self
Diffusion Coefficient of Argon SquareWell 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 SelfDiffusivity ............................. 53
5.1 Smoothing LennardJones [6,12] Parameters, vs. cu.. 68
kT
c
P 1/3
5.2 Smoothing LennardJones [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 OneDimensional 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
SelfDiffusivity 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 ThreeDimensional 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
= preexponential factor of free volume equation
for viscosity
= A/T1/2
= arbitrary constants
= constants
= constants
= coefficients of viscosity equation
= molecular diameter
= preexponential factor of free volume equation for
selfdiffusivity
= 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 = selfdiffusion coefficient
DAB = binary diffusion coefficient
'. 1/2 1/2
D" = Dm l2/GI = reduced selfdiffusivity by simple
correspondence principle
'1/2 1/2
D = Dm /co = reduced selfdiffusivity 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 selfdiffusivity 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. = ycomponent 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/n6)(n/6)6/(n6)
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 AB
cAB
3
P = Pa /c = reduced pressure by simple correspondence
principle
= Pa /ce = reduced pressure by improved correspondence
principle
p = momentum
ix = momentum in xdirection 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 AB
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 xdirection
= (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. = xcomponent 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 = xcomponent 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 selfdiffusivity
of a dilute gas
p" = reduced density
a = characteristic distance parameter in [6,n.] potential
Gl ,2 = characteristic distance parameter in squarewell
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. T1 and In(D/T /2) vs. T1 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 selfdiffusion coefficient behavior
of liquids composed of simple, nonpolar molecules that may be expected
to approximately obey a LennardJones [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
selfdiffusion 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 selfdiffusion 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 fluctuationdissipation 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 selfdiffusion of Brownian particle (2). The following
expressions for transport coefficients are obtained.
0O
Selfdiffusion 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 xdirection 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
(46). 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 FokkerPlanck type. The main disadvantage in this theory is
that a large number of approximations have to be made in obtaining the
(1) (2)
FokkerPlanck 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
Selfdiffusion 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 RiceAllnatt 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 FokkerPlanck equation. These authors therefore treated the
rate of change of the distribution function f due to hard core colli
sions by means of an Enskogtype 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 RiceAllnatt 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 = 0.73x103 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 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 RiceAllnatt 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 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
cl, which was obtained from equation of state data.
TABLE 1.1
TEST OF RICEALLNATT THEORY FOR
SHEAR VISCOSITY OF ARGON
33
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 RiceAllnatt 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 selfdiffusion 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 RICEALLNATT 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 RiceAllnatt
theory by using the squarewell potential energy function. Because of
the simple form of this function the RiceAllnatt 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 squarewell 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 RiceAllnatt expressions. They
then calculated the viscosity coefficient and selfdiffusion 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 selfdiffusion
coefficient is up to 34%. The experimental selfdiffusivity data
used by Davis et al. as shown in Figure 1.2 are apparently extrapolated
values obtained from the selfdiffusion coefficient data of Naghizadeh
and Rice (18). Luks, Miller and Davis (22) have also made use of the
modified RiceAllnatt 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 14
U
F4
(asTod) O c : x
c..J
0
u
r c
**
E
4
r0
Ea
3
,1
i1
. J
c
c
o 11
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.
Selfdiffusion 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
centeredcubic 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 selfdiffusion 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 (3032). 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 selfdiffusion 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 facecentered
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
LennardJones 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 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 = Ca + Ba/T),
but cannot account for nonArrhenius liquid behavior. In order to
overcome this deficiency Doolittle (4042) proposed empirical expres
sions which related liquid transport properties to the free volume,
defined by
vf = v Vo (2.5)
where v is the hardcore 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 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.
They obtained the following expressions for the viscosity and self
diffusion coefficient:
r = 1 2 exp (2.6)
3ra 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 selfdiffusion 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 selfdiffusion 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 preexponential 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 preexponential 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 preexponential 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 VRT"
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 preexponential 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 selfdiffusivity 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
oo
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 (K1)
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/(n6)
1 n6 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 twoparticle 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,
2i
U 1i
,. ,
r..)
u
c\
C4
z
D 0 0 0
3 0 0 0 0
( aiom'lwo)Az
0
44
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 iC4H10 (54)
g CS2 (54)
v nC6H14 (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 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 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 selfdiffusion 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, 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 1215%.
The only high pressure selfdiffusivity 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(K1)
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 selfdiffusivity 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 selfdiffusion 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
selfdiffusion 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 selfdiffusion
coefficient data of Naghizadeh and Rice (18), very few high pressure
viscosity and selfdiffusivity data are available. Thus most of the
viscosity and selfdiffusivity 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 selfdiffusion coefficient of the molecules
studied.
The average percent deviation between theory and experiment
is about 6% or less for viscosity. For selfdiffusion 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 selfdiffusion 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
O1
O N'
(U*z ^ v
0 r C4 0 11o c0 Lt
<\ 11CO ~
>~ 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 44
( a) o co
p Pc; ID It
0
oD
CO
c
SJ
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
01
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)
E4
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
Oi
o o w Lr
,4 C :
S C O0 0I O0 CO
O
0
O uC)
UH 0
0 4 0
U) L
." ( ) O 00 00 r4 < <
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 iI CM F '4 >4
CCC L0 0 CD C0 \ CO
0l
Cl
E4
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
i4
co ON czt r Cm
cm c uii m c) r
C ,
0 0 0 0 0
11
0 C N Co 0 0
F ,,
iii
E I
1o co
> 0 o 0 L L
a1 (N cC N m c
z O
ii
E N ( N (N rc 0
EI Cl) '" Co 'i '0 oC1 Co
14 4i
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 K1)
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
(asod) 01 x
to V = 49.5 cc/mole). Figure 4.4 compares theory and experiment for
the selfdiffusion coefficient of saturated liquid methane.
The expressions for the preexponential factors of the
viscosity and selfdiffusion 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 preexponential 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. T1 and In(D/T1/2) vs. T1 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 selfdiffusivity behavior of liquids composed of
r
0
CC
*r1
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
( DoasuI ) o0 Y
z Z
simple, nonpolar molecules that may be expected to approximately obey
a LennardJones [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 selfdiffusivity 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 24 accurately
predict viscosities and selfdiffusion 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 selfdiffusion 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 selfdiffusion coefficient is
given below as an example of the procedure.
The selfdiffusion 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.. dN
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 selfdiffusion 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 (7981). 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 CiniCastagnoli,
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).
CiniCastagnoli, 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.
Selfdiffusion 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
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
critical parameters were used. However, in practice the group of
molecules do not all accurately obey Assumption 4, and the potential
parameters are forcefitted to some semiempirical equation, in this
case the LennardJones (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 twopara
meter LennardJones 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 selfdiffusion 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
I3
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 II 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
4J
*4
>
'41
CO 4c
U4
c*
41
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
ln1 *2 a3T
In' 1 = al + a22T + aT2 + a T*3 + a5T 4 (5.42)
and InX' = b1 + b2T1 + b3T*2 + b T*3 + b 5T4 (5.43)
at constant reduced pressure or along the saturation curve. The
reduced selfdiffusion 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 highspeed 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 nonequilibriian 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 24 contain a detailed study of the free volxome theory of viscosity and selfdiffusion 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 ThreeDimensional Oscillator 151 LITERATURE CITED 155 BIOGRAPHICAL SKETCH 164 VI
PAGE 8
Table LIST OF TABLES Page 1.1 Test of RiceAllnatt Theory for Shear Viscosity of Argon 5 1.2 Test of RiceAllnatt 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 SelfDiffusivity Data Sources and Range of Conditions 48 4.4 Parameters for SelfDiffusivity 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 SelfDiffusivity 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 SelfDiffusivity 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 SquareWell Model 9 1.2 Theoretical and Experimental Saturated Liquid SelfDiffusion Coefficient of Argon SquareWell 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 SelfDif fusivity 53 5.1 Smoothing LennardJones [6,12] Parameters,^Â— Â— vs , O). , 68 /^c \^/^ 5.2 Smoothing LennardJones [6,12] Parametersfrr Â— Iff vs . CO .'V. .c .J. 69 5.3 Correspondence Principle for Saturated Liquid Viscosity Monatomic Molecules 70 IX
PAGE 11
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 OneDimensional 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 SelfDif 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
PAGE 12
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
PAGE 13
LIST OF SYMBOLS = preexponential 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 = preexponential factor of free volume equation for selfdiffusivity 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 cÂ„ = (CtO/v ) = constant J 1 m c, = e/kT = constant 4 m Xll
PAGE 14
D = selfdiffusion coefficient D,Â„ = binary diffusion coefficient AB D = Dm / Oc = reduced selfdif fusivity by simple correspondence principle 'Wf 1/2 1/2 D = Dm /coe = reduced selfdif fusivity by proposed correspondence principle D = reduced mutual diffusion coefficient defined by equation (5.33) d = distance of closest approach of 2 molecules d, ,dÂ„ = coefficients of selfdif 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. = ycomponent 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
PAGE 15
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/n6)(n/6)^/^'^^^ k = Boltzmann constant k^ ^ = defined by equation (6.8) kj^2 ~ defined by equation (6.9) kÂ„Â„ = 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
PAGE 16
P = pressure P = critical pressure c P Â„ = critical pressure of a mixture AB 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 xdirection of molecule i IX p = probability of a molecule having activation energy ^ e. and free volume v 1 P * 1/2 p = p/(mÂ£) = 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 AB cAB T = melting temperature m T* = kT/e = reduced temperature by simple correspondence principle XV
PAGE 17
'V', T" = kT/cÂ£ = reduced temperature by improved correspondence principle Oj T = kT/[Â£(s/c), ] = reduced temperature used by Hermsen and Prausnitz (156) t = t ime * 1/2,1/2^.., t = tÂ£ / CT m = dimensionless time U(r) k N ) u(,r..; = pair potential i
PAGE 18
= 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. = xcomponent 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 Â£ = 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(mÂ£) = reduced shear viscosity by improved correspondence principle T) = defined by equation (5.36) = angle vector (in terms of Eulerian angles d,
PAGE 19
e X /c K. K ov TT a xcomponent of an angular rotation = bulk viscosity * J,, vl/2 /CO /CmÂ£; = reduced bulk viscosity by simple correspondence principle * _ Â„ 2, , ,111 K.a /c(inÂ£) = 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 /ckÂ£ = 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 selfdif fusivity of a dilute gas P' = reduced density characteristic distance parameter in C6,n] potential Â°^1'^2 ~ characteristic distance parameter in squarewell 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
PAGE 20
^ = angle associated with superimposed angular velocity tp ,i1j^ = the radial functions in the spherical harmonic expansion of the nonequilibrium distribution function g(r) Â„(1,1)^ r^(2,2)* ^. . ^ ,, . . . ii , it ' ' = dimensionless collision integrals Oi = Pitzer factor CO = angular velocity CO, = angular velocity in a harmonic motion XIX
PAGE 21
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 selfdiffusion 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 selfdiffusion coefficient prediction are melting and critical parameters, together with one experimental value of viscosity (or selfdif fusivity) . XX
PAGE 22
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 selfdiffusion 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
PAGE 23
It is also shown that the proposed model can account in a reasonable way for thermodynamic properties of polyatomic fluids. XXI 1
PAGE 24
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 fluctuationdissipation 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
PAGE 25
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 selfdiffusion of Brownian particle (2). The following expressions for transport coefficients are obtained. CO Selfdiffusion 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 xdirection 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)
PAGE 26
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 (46) . 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
PAGE 27
Liouville equation for the distribution function into an equation of the FokkerPlanck type. The main disadvantage in this theory is that a large number of approximations have to be made in obtaining the FokkerPlanck 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 , Â„ oo T] =^^^ r' ^ Z^(r)^,(r)ar (1.8) ^ 15kT ^ ar o z Bulk viscosity r ^ Â°Â° Thermal conductivity Â„ . 5k niT nVTTkT Â°Â° A ~ O 00 dr (^ St Â•) r^dr (1.10) Sg^(r) 2 oT kT Selfdiffusion coefficient D = Â— r (1.11) where Â°Â° 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
PAGE 28
momentum change is small. In the RiceAllnatt 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 FokkerPlanck equation. These authors therefore treated the rate of change of the distribution function f due to hard core collisions by means of an Enskogtype 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 RiceAllnatt 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,
PAGE 29
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 RiceAllnatt 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 RICEALLNATT 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 RiceAllnatt 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 selfdiffusion 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 RICEALLNATT THEORY FOR THERMAL CONDUCTIVITY OF ARGON XxlO^, cal.cmT'secT'^c" _3 Density, gem.
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In general a , aÂ„ and e for the squarewell 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 RiceAllnatt expressions. They then calculated the viscosity coefficient and selfdiffusion 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 selfdiffusion coefficient is up to 34%. The experimental selfdif fusivity data used by Davis et al. as shown in Figure 1.2 are apparently extrapolated values obtained from the selfdiffusion coefficient data of Naghizadeh and Rice (18). Luks, Miller and Davis (22) have also made use of the modified RiceAllnatt 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
PAGE 32
o
PAGE 33
10 c o Â•rl n3
PAGE 34
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 = Â— 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 Selfdiffusion 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 =Â— X The value of y may be obtained from the equation of state by Â— = 1 + y RT ^ (1.18) Enskog suggested as an empirical modification of the hard sphere
PAGE 35
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 " Â° 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] . . Â° ^ ''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 0Â°C and 75Â°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 facecenteredcubic lattice structure for the liquid and that transfer of thermal energy down the temperature gradient was due to two causes:
PAGE 38
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. 15 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
PAGE 41
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 40Â°C that is 36% too low (29). Similar empirical expressions are also available for liquid thermal conductivity and selfdiffusion 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 (3032). 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 riri^= 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
PAGE 42
19
PAGE 43
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 .
PAGE 45
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 selfdiffusion coefficient are: ^ = 4^f "in) (2TltnkT)^/2e'Â° (2.3) 2X, 1 . ., Â„ V. 1/2 e^/kT D =Â— 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 facecenteredcubic 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 (4042) 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 hardcore 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. Â° o They obtained the following expressions for the viscosity and selfdiffusion coefficient: Tne term "free volume" in these theories has a different
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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 selfdiffusion 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 selfdiffusion coefficient are 7V T] = Aexp f Â—V ) exp f^ ) (2.8) V V ; '^ VRT o 7V N. / E \ D = Bexp ( Â—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 preexponential 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 preexponential 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 preexponential 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 preexponential 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 sclfdif 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 Â•10.0 39.15 Â•10.5 J L 2.0 4.0 6.0 3.0 10.0 1/T X 10^ (Â°k"S 12.0 14.0 Figure 3.1. Viscosity of Argon.
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30 Â•8.0 Â•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 = Â£K w^s^e / \ X \ 6/(n6) 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
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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^oparticle system to the potential energy of the system at the distance of closest approach, d: 3kT = Â£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 Â• 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 Â•oo Â•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 Â•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_Â».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 Â— 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
PAGE 60
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 ZÂ£ r / \ n c, = v/r: a_ K ZÂ£ 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 ^ ' /^Â— ^ Uxp(^^^ (3.13) TT^= A exp f T^^^ Â° ^V V (T) and / V (T) \ / E (V) Â° = B exp (T7Vt^ ) exp r ^Â— ] (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 selfdiffusion 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 12157o. The only high pressure selfdif 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
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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 ^IMn/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 (Â°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 Â•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^(Â°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 Â° nitrogen may be used to form the basis of corresponding states correlations of these parameters as described in Chapter 3, from which viscosity and selfdif 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 selfdiffusion 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 selfdiffusion 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 selfdiffusion coefficient data of Naghizadeh and Rice (18), very few high pressure viscosity and selfdif fusivity data are available. Thus most of the viscosity and selfdif 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 selfdiffusion coefficient of the molecules studied. The average percent deviation bettvreen theory and experiment is about 67o or less for viscosity. For selfdiffusion 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 selfdiffusion 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
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49 tu
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Â•7.0 50 Free Volume Theory o Experiment (64,65) Â•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^(Â°k"^) 6.0 Figure 4.2. Test of Free Volume Theory for Liquid Xenon Viscosity. 7.0
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51
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52 to V = 49.5 cc/mole) . Figure 4.4 compares theory and experiment for the selfdiffusion coefficient of saturated liquid methane. The expressions for the preexponential factors of the viscosity and selfdiffusion 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 preexponential 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 selfdif fusivity behavior of liquids composed of
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53 Â•a Â•el 3 u
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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 selfdif 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 24 accurately predict viscosities and selfdiffusion 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 Â£ = 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 Â•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 selfdiffusion 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 selfdiffusion coefficient is given below as an example of the procedure. The selfdiffusion 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 / Â• Â•/"\(0)\(t)e^%^. . .dÂ£^dr^. . .dr^ = ^^.^^ (5.7) ' ' /SH e dÂ£^...dÂ£^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
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59 and if Assumptions 3 and 4 are satisfied, we may write U = N Â£ ) 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 ' Â£ ' ) (5.13) volume v" = v/cr (5.14) then foo ,v Â£ 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 selfdiffusion 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" = Â° = 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 (7981). 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 CiniCastagnoli, 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). CiniCastagnoli, et_ al . applied the principle to the viscosity and thermal conductivity of saturated liquid Ar, CO, NÂ„ , CH and 0Â„ . 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 , HÂ» , 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
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closely, but even relatively simple polyatomic molecules such as , C'd/ and CD, were found to deviate markedly. Selfdiffusion 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^,Â„ (5.34) AB ^(l,l)>' They plotted their results as D" against P^ at constant T^ where ^ ^ AB R R P T PÂ„ = Â— and TÂ„ = Â— . 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 Â° 16 TT n^^'^^" and 2 ri''=?i=. (5.37)
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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
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66 critical parameters were used. However, in practice the group of molecules do not all accurately obey Assumption 4, and the potential parameters are forcefitted to some semiempirical 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 Â£ , 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 twoparameter 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
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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 ( Â—2Â— ^ and :; for twenty different molecules are V kT y Â° 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 selfdiffusion coefficient as
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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 Â•w to 3 41 CO w o 141 a Â•11 o c Â•H ^ Â• en dJ OJ U 1H C 3 0) U t3 01 C .H o o CTj a o U rl u B o o en m (U ii 3 M Â•H O CM i I I I O o o o I I L. o u1 o CN o
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71 o 1 c^ sj 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
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72 u u Oi oi < i2, X ^ P Q. ^ 00 VO J I I u. 00 o I CO D 141 141 Â•rt cr Â•ii a> jj CO !j D iJ ca en Sl o c
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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
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74 o 60 < Â•rl O cr Â•n O) 41 CO 4J O Â•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 Inrj'' = a^ + a^T""^ + a^T*"^ + a^f''"^ + a^T*"^ (5.42) ,* "'1 , K t'''2 , L rp"V3 , rp*^ and InX = ^^ + b2T""^ + b^T "^ + b^T ^ + b^T""^ (5.43) at constant reduced pressure or along the saturation curve. The reduced selfdiffusion 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 ui o C3 (1) a e u 0) (0 c o Â•tl o c D w Â• 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 1Â—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 1Â°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 selfdiffusion coefficient data of different workers are especially large. Thus the seifdif fusivity of xenon reported by two v7orkers differ by as much as 100% (18,57). Equation (5.44) for selfdif 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 ll 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 15 o s it O ^ M oS M O < O n o < (^ o M o M o 00 o nj si ^1 SI > 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 selfdiffusion 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 reexamination 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\^70parameter potential energy functions, such as the LennardJones [6,12] equation, do not accurately account for properties of polyatomic fluids (92). In fact threeparameter 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 rootmeansquare deviations of 7.47o and 5.4% respectively. The correlation for selfdiffusion coefficient was comparatively poorer, showing a rootmeansquare deviation of 157c. Their treatment seems open to criticism on several counts. Firstly, by forcefitting 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 gasphase 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
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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 Â— I r< IÂ— I cH 1Â— I rsl K 00pf^^s>K<) 00 *. \ \ \ \ \ \ \ \ CO 0) CO to > < ^ \ \ s^ i. ^ \ \ \ ^ \ \ ^s^\\\ \ < \ V ^' o \<:v^^^ k \ ^ \ \ ^^\ N V V Â« oN \ \ c ^. Â«\. v\ 9s aj 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 ii IB ;j 3 Â•U CO C/5 o Sl P. (U CO c o O 0) CO O
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92 Â•H/kT , X X V^.(0)V^.(t)e ^dÂ£^ Â•Â•dV^e Â•Â•Â•d0^ 1 = ^ X X ^ ../ e dÂ£^ Â•Â•Â•dr^dÂ£g ..d^^ (^l) 1 where H(p ...r ..pn Â— Q) = KÂ„ (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 onedimensional 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 onedimensional motion is assumed. However, it can be readily extended "o the threedimensional case by assuming an isotropic system in which the motion along any of the 3 axes is the same. Figure 6.3. A OneDimensional Oscillator Model. Consider a molecule oscillating linearly along the xaxis 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 zaxis (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) + â€¢ (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 pÂ„, 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 Â— the new angular coordinates p ,, , so that =q) Â— = v^(0)V^(t)e^^/^^ d^ dr^dp dO^ ^.Te^/^^dp^ dr^dÂ£^ 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"? . Â„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} (69) 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 12 1 ^ K = mx^ + ^ lY (6.12) 2 2 X ' _ dx * dY where x ^ and ^J/^ = Â— , and the Lagrangian of the system becomes L = K(x,Y ) U(x,Y ) 1 .2 Â— 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^ + kÂ„.^ + 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 â€¢ = 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 + !<,,Â•? =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'cosCDjt + B"sinJa t Â•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 Â„ _ 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 Â„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 timeaverage 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 xdirection, 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^ = Sf J... I e dÂ£^...dÂ£^.dr^...dr^ 00 o 3N 2 Zcp. ^ + cU(r^...V i=l Now define the reduced temperature T as (6.40) T" = Â— (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 threedimensional 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 selfdif 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
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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, selfdif 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 IÂ— 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 ^2Â«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 longchain 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 selfdiffusion 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<4Â»o^Â» V> 00 ffi u 01 c cd X o 00 o CM ^ en i< 0) o !i Ph O 0) rl > CO O O ^1 O a, en B Â•'' 1^ > vO u o M ' ' o o o o CM ' J L_ LTl o o CM O ?P
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Ill 0) X u < CO o w n to ii c 0) CNJ O <]Â•:=; r^ ii 2 2; 00 o CM CO
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112 o CO
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ii: 0) u 03 to V4 O Ul u o o c o a. i o o 'O > o u a. B >> >. ij Ul ri 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 Â« S 1Â—1 P o O M ^g < o Pi o o < PS H < CO fi^ o + rII h O ij
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121 o u
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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 rmers (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
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125 00 u CO o u ^ ej u > a < 25 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 rÂ— , CO. \ r^ + cU^r r.J (657) and therefore CM Z ^Â™ 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 KÂ» foo y ik Since 1=1 2mkT e clÂ£i''''^Â£Â»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 Â• o,/> / / ' y / / / / f / / ^ ^ ' ^ / y ' >' 0^ O CO o cy !i D CO CO 0) PL, o aÂ•rl o r^ Â•H !i Pi O o s r; O O. CO <1> Vi 51 O o Cu S Â•H cy D r( L ' ' O O CM O o o o CM O Â— i o o o
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131 u 150^ '^ ^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 selfdiffusion 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 selfdiffusion 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 selfdif 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 (169171). 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 Â— 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^^ Â£_ ^ 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 = Â— (a + fie . + Av ) 1 p' e ^ Y"! ri (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 ^ (Â«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=Â— (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) =expfJ (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 selfdif fusivity and fluidity are directly proportional to this probability, one may write for the expressions of viscosity and selfdif 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 preexponential factor is obtained.
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APPENDIX 2 FURTHER TEST OF PROPOSED FREE VOLUME THEORY
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145 Â•S.O In Â•8.5 9.0 n2 9.5 10.0 Free Volume Theory o Experiment (49) 32.6 cm."'raole Â•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 ^
PAGE 170
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^=Â° (A3.1) and 2 d ^ dc z ZZ X IZ (A3. 2) In terms of differential operators one may write and (mD + k )x + k.Â„^ = 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^^^^ = Â° (A3. 6) Equation (A3. 6) may be solved to give roots of D as n (k22in+k^,^I)+y(k22m+k^^I)^ 4lm(k^^k22k^,) 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^^k22k^2) 1/2 2Im (A3. 9) Therefore ico t +ico t iCJÂ„t ico^t X = A.e + AÂ„e + AÂ„e ' + A, e 12 3 4 (A3. 10) Similarly, eliminating x from equations (A3. 3) and (A3. 4) gives (mD^ + k^)(ID^ + kÂ„)^ k^.Y = li 2.1 yi 1/ X (A3. 11) Equation (A3. 11) is expanded to give [iraD^ r (k^2Â™ + k^^I)D^ + k^,^k22 k^2^^ " Â° (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 + BÂ„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)^ Â• This is true if (k22m+k^^I)^ 4lm(k^^k22k^2^ = Â° (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 >Â• 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 TIIREEDIMENSIONAL OSCILLATOR X Figure A4.1. A ThreeDimensional Oscillator Model Consider a molecule oscillating in an isotropic harmonic manner in a 3dimensicnal 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 CJÂ° (_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,_Â§) = a).(x)e + CD (y)e + O^Jz)e + a)Â°(^)e X X y z ^ J J. + a3Â°(0)e +6DÂ°(9)e (A4.4) y Â— 2 ^ ~3 The angular velocities at the cell center can be expressed in terms of Eulerian angles as (173) CDÂ°(6;) = 0COS0 +ijj sin0sin0 (A4.5) X COÂ°(e) = 0sin0 Â•^sin9sin0 (A4.&) GOÂ°(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 xaxis, one along the yaxis and one along the zaxis. These oscillators may be considered separately. The oscillation associated with the xaxis will be as shown in Figure 6.3. We define the angles associated with co as Â° 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^ Â• (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 xdirection 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 = Â— 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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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

