• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of symbols
 Abstract
 Introduction
 Thermodynamic theory
 Group contributions
 Formulation of models for direct...
 Model parameterization
 Use of the group contribution model...
 Discussion
 Appendices
 References
 Biographical sketch














Title: Thermodynamic properties of multicomponent mixtures from the solution of groups approach to direct correlation function solution theory
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Title: Thermodynamic properties of multicomponent mixtures from the solution of groups approach to direct correlation function solution theory
Physical Description: xi, 167 leaves : ill. ; 28 cm.
Language: English
Creator: Telotte, John Charles, 1955-
Copyright Date: 1985
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Subject: Thermodynamics -- Tables   ( lcsh )
Thermochemistry   ( lcsh )
Statistical mechanics   ( lcsh )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
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 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1985.
Bibliography: Bibliography: leaves 163-166.
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by John Charles Telotte.
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Volume ID: VID00001
Source Institution: University of Florida
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Table of Contents
    Title Page
        Page i
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
    List of symbols
        Page vi
        Page vii
        Page viii
        Page ix
    Abstract
        Page x
        Page xi
    Introduction
        Page 1
        Page 2
        Page 3
    Thermodynamic theory
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
    Group contributions
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
    Formulation of models for direct correlation functions
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
    Model parameterization
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
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        Page 47
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        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
    Use of the group contribution model for pure fluids
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
    Discussion
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
    Appendices
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
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        Page 157
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        Page 159
        Page 160
        Page 161
        Page 162
    References
        Page 163
        Page 164
        Page 165
        Page 166
    Biographical sketch
        Page 167
        Page 168
        Page 169
        Page 170
Full Text













THERMODYNAMIC PROPERTIES OF MULTICOMPONENT MIXTURES FROM
THE SOLUTION OF GROUPS APPROACH TO DIRECT
CORRELATION FUNCTION SOLUTION THEORY



By

JCILH CHARLES TELOTTE


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


UNIVERSITY OF FLORIDA

1985

























To Bonnie, for her support and encouragement

and patience and love















ACKNOWLEDGEMENTS


The author would like to thank Dr. John P. O'Connell

for his guidance and understanding throughout the years.

Thanks also go to Dr. Randy Perry for many helpful discus-

sions and the members of the supervisory committee.
















TABLE OF CONTENTS


PAGE


ACKNOWLEDGEMENTS................................ iii

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

ABSTRACT........................................ x

CHAPTER

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

2 THERMODYNAMIC THEORY................ ...... .. 4

Thermodynamic Properties of Interest....... 5
Calculation of Liquid Volumes.............. 6
Chemical Potentials and Fugacities......... 7
Fluctuation Solution Theory................ 9
Summary .................................... 13

3 GROUP CONTRIBUTIONS ........ ............... 14

"Reactive" System Theory ................... 18
RISM Theory. ................................ 19
Differences between RISM Theory and
"Reactive" Solution Theory............... 20
Thermodynamic Properties from Group
Functions. ................................ 21
Summary................... ................... 24

4 FORMULATION OF MODELS FOR DIRECT
CORRELATION FUNCTIONS ...................... 25

Thermodynamic Consistency .................. 25
Basis for Model Development................ 28
Analysis of the Model....................... 30
Property Changes Using the Proposed Model.. 32

5 MODEL PARAMETERIZATION..................... 34

Corresponding States Theory................ 35
Choice of a Reference Component............ 39
Method of Data Analysis..................... 40










Analysis of Argon Data..................... 44
Analysis of Methane Data................... 49
Use of the Correlations to Calculate
Pressure Changes. .................. ...... 53

6 USE OF THE GROUP CONTRIBUTION MODEL FOR
PURE FLUIDS ................................ 59

Extension of the Model to Multigroup
Systems.................................. 59
Designation of Groups ...................... 61
Pressure Change Calculations............... 61
Summary .................................... 68

7 DISCUSSION ................................. 73

Fluctuation Solution Theory................ 74
Approximations ............................. 74
Comparison Calculation.......... ............. 82
Summary .................................... 83

8 CONCLUSIONS ................................ 84

APPENDICES

1 FLUCTUATION DERIVATIVES IN TERMS OF DIRECT
CORRELATION FUNCTION INTEGRALS ............. 86

2 PERTURBATION THEORY FOR DIRECT CORRELATION
FUNCTION INTEGRALS USING THE RISM THEORY... 92

3 THREE BODY DIRECTION CORRELATION FUNCTION
INTEGRALS .................................... 96

4 NONSPHERICITY EFFECTS...................... 98

5 HARD SPHERE PROPERTIES..................... 110

6 COMPUTER PROGRAMS........ .................. 119

7 COMPRESSIBILITY THEOREM FROM RISM THEORY... 156

REFERENCES...................................... 163

BIOGRAPHICAL SKETCH ............................. 167
















LIST OF SYMBOLS


an,aB nth order expansion coefficient in perturbation
function for pair aB

C matrix of direct correlation function integrals

CO matrix of short-ranged group direct correlation
function integrals

C matrix of purely intermolecular group direct
correlation function integrals

C.. molecular direct correlation function integrals

C05 group direct correlation function integral

c.. molecular direct correlation function

cas group direct correlation function

Cijk three body direct correlation function integral

c.. group direct correlation function

f temperature dependent function

g temperature dependent function

gij pair correlation function

H matrix of total correlation function integrals

h supermatrix of group total correlation functions

h.. total correlation function
1J
h group total correlation function
ij3
HE excess enthalpy

I, I identity matrix and supermatrix

i molecule










K matrix of differences between direct correlation
function and its angle average

k wave vector

k Boltzmann's constant

M general property

M general partial molar property

N total number of moles, number of components

N. number of moles of species i
1
O lowest order

P pressure

R gas constant

Position vector

r distance

r position vector

T temperature

V, V volume, molar volume

V. partial molar volume
1
W matrix of intramolecular correlation functions

W intramolecular correlation function integral

X matrix of mole fractions

X independent set of mole fractions

X. mole fraction of molecule i
1
Z compressibility factor

a generalized hard sphere parameter

= 1/k T, dimensionless inverse temperature

Yi activity coefficient

5(.) Dirac delta function


vii










6.. Kroniker delta

E energy

n packing fraction

i. chemical potential

v matrix of stoichiometric coefficients

5X th order packing fraction

p total density

p vector of densities

P matrix of densities

pipa density of species i or group a

0 supermatrix of intramolecular correlation functions

o. diameter
1
pa8 perturbation function

2 orientation normalization factor

Q matrix of correlation functions

2 correlation function integral

_,] matrix or supermatrix of intramolecular
correlation functions


Subscripts


i,j,... species quantity

n nth order term in series expansion

a,B,... group quantity


Superscripts


E excess property

HS hard sphere property

viii










mix mixture property

o reference state property

PYC, PYV using Percus-Yevick compressibility or virial
forms

r residual property

ref reference

T transpose of a matrix or vector


Special Symbols


vector

matrix

supermatrix

partial molar property

< > ensemble average or angle average

fourier transform

















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of
Doctor of Philosophy


THERMODYNAMIC PROPERTIES OF MULTICOMPONENT MIXTURES
FROM THE SOLUTION OF GROUPS APPROACH TO DIRECT
CORRELATION FUNCTION SOLUTION THEORY

By

John Charles Telotte

May 1985

Chairman: John P. O'Connell
Major Department: Chemical Engineering

A solution of groups technique was developed for use

with fluctuation solution theory. The general expressions

for calculation of pressure and chemical potential changes

from some fixed reference states have been shown. A new

corresponding states theory correlation for direct correla-

tion function integrals was proposed and used with the

group contribution technique for calculation of pressure

changes during compression for several n-alkanes and

methanol.

This work gives a detailed analysis of the RISM theory

of liquids. Shown are new results for perturbation theory

and a generalized compressibility theorem for RISM fluids.

The use of the RISM theory for calculation of thermodynamic

properties of real fluids also is given.










The use of hard sphere reference fluids for development

of equations of state has been explored. A generalized

hard sphere equation of state was developed. It was shown

that the most accurate hard sphere equation of state is

not the best reference system for construction of a liquid

phase equation of state of the van der Waals form.
















CHAPTER 1
INTRODUCTION


The rational design of chemical process equipment

requires knowledge of the thermodynamic properties of the

substances involved. More specifically, volumetric

properties are needed to size a piece of equipment,

enthalpies are needed to determine heat duties, and fugaci-

ties are used to decide feasibility of reactions and

separations.

The general problem in physical property correlation

and estimation is then to determine a procedure for calcula-

tion of volumes, enthalpies, and fugacities, for multi-

component, multiphase mixtures. This has been accomplished

for some pure components using an equation of state with

a large number of parameters. Here, a more modest problem

is addressed. A formalism is developed that allows the

volumetric properties and chemical potentials for liquid

mixtures to be expressed in terms of a single group of

parameterized functions.

The approach used here tries to recognize the essential

differences between liquid and vapor phase properties.

A thermodynamically consistent formulation for the volumetric

properties and chemical potentials is developed using an

equation of state. Here, unlike many other schemes, the

1










equation of state uses a liquid reference state. This

is all made possible by utilization of the Kirkwood-Buff

(1951) or fluctuation solution theory.

In an effort to allow for extrapolation of present

results to state conditions not examined, the correlations

are put into a corresponding states form. This can be

justified from analysis of the microscopic theory that

is the basis for this macroscopic approach.

Finally, to allow for actual physical property estima-

tion, the theory is cast in a "group-contribution" form.

This is easily accomplished formally because the fluctuation

solution theory relates molecular physical properties to

microscopic correlations. These quantities are well defined

for groups, or sites, as they will be often termed.

The next chapter in this work goes into detail in

discussing the thermodynamic aspects of the problem and

the possible approaches that are available to express the

required physical properties. Then, the flucutation solution

theory is outlined.

Chapter 3 is devoted to the discussion of group con-

tribution modeling. Two possible means of linking the

group contribution ideas to the fluctuation theory are

presented and their differences expounded upon. This chapter

concludes with the formulation of the physical property

relationships from group correlation function integrals.

Approaches to modeling correlation functions are the

emphasis of Chapter 4. General forms, as suggested by










microscopic perturbation theories, are outlined and

discussed. The thermodynamic consistency requirements

for model formulation are given along with the final expres-.

sions for the thermodynamic property changes from the chosen

form.

Because the correlation function integrals contain

unknown functions, expressed in corresponding states form,

some experimental data are required for the model parameteri-

zation. In Chapter 5 possible data for this parameterization

are discussed critically, and two sets of general correla-

tions are developed. One of these correlations is then

decided upon based on the problem of interest and accuracy.

Chapter 6 then presents calculations for several

n-alkane molecules' compressions. Both correlations and

predictions are shown. Chapter 7 is a general discussion

of all of the previously reported work, and Chapter 8 offers

several conclusions and recommendations for future work.

There are also several appendices which give greater

detail on developments presented in the body and listings

of several useful computer programs.
















CHAPTER 2
THERMODYNAMIC THEORY


Thermodynamics is a science that has its basis on

certain empirical observations that have become known as

laws. The simplest of these observations, and the ones

most often used in practice, are that mass and energy are

conserved quantities. These statements allow for the con-

struction of useful mass and energy balance equations.

These balances, when augmented with constitutive relations

such as the conditions of phase or reaction equilibria,

can be used for design of-chemical process units. However,

the power of these balance expressions cannot be fully

utilized unless the physical properties that appear in

the expressions are available.

In the following discussion one other empirical obser-

vation is required. This is not known as a law but is

often listed as a postulate of thermodynamics (Modell and

Reid, 1974). This result may be stated in many ways, but

the simplest formulation is as follows: For a simple,

homogeneous system of N components, with only P-V work

and thermal interactions with its surroundings, that N+l

independent, intensive variables are required to specify

the state of the system. The simplest use of this observa-

tion is the existence of equations of state, such as











P = f(T,V,X)


Thermodynamic Properties of Interest


As stated previously, this work is concerned with

the volumetric properties and chemical potentials of the

components in liquid mixtures. The chemical potentials

are of interest for phase or chemical equilibrium calcula-

tions because of the constraints that are imposed. For

equilibrium between phases a and B the constraints are



vi = 1. V. in both a and B (2-2)
1 1 1


and the chemical equilibrium constraints are



I v. v. = 0 V reaction j (2-3)
iJ 1


These equilibrium conditions are used to determine both

feasible operating conditions for chemical processes and

minimum work requirements for some processes.

To size a piece of process equipment one must have

knowledge of volumetric properties of the fluids involved.

The molar volume of a mixture is found from knowledge of

the component partial molar volumes by



V = V x. V. (2-4)
-- i 1


(2-1)









An important aspect that must be considered in con-

struction of thermodynamic models is the relationship between

chemical potentials and partial molar volumes



( ) = (2-5)
P T,X


When both the chemical potentials and partial molar volumes

are determined from the same equation of state, then equation

(2-5) will always be satisfied. However, if different

correlations are used for calculating these two different

properties, then a thermodynamic inconsistency exists and

can cause computational difficulties.


Calculation of Liquid Volumes


Reid, Prausnitz, and Sherwood (1977) give an extensive

review of techniques for correlating liquid molar volumes.

Most effort has been placed on calculation of saturated

liquid volumes because of the insensitivity of liquid volumes

to pressure. Most of the correlations are in corresponding

states form.

Calculations of compressed (subcooled) liquid volumes

often are based on a knowledge of the saturated liquid

volume at the system temperature. In general, the volumes

are found from expressions of the form


V = V(T,P)


(2-6a)









Vmix = V(T,P,X) (2-6b)



These are calculationally convenient forms. If a complete

equation of state is used, the volumes are found from solu-

tion of the implicit relation



P = P(V,T,X)



and care is often necessary to ensure that the proper volume

is calculated. Equation 2-1 can be satisfied for several

values of the volume at a fixed temperature and composition.

Partial molar volumes are found from the definition



V = (2-7)
1 T,P,Njfi


For most correlations this is evaluated by using one of

the pure component correlations with a set of mixing rules

applied to the parameters.


Chemical Potentials and Fugacities


Chemical potentials are used for solution of phase

or reaction equilibria problems. However, the chemical

potentials themselves are not often used because the

equilibrium expressions can be written in terms of the

fugacities, defined by



S= 1 + RT Zn (fi/f ) (2-8)
ii










where the superscript o refers to a given reference state

of a specified pressure, composition, and phase at the

system temperature, T. The fugacity can be found from

two general approaches based on different reference states.

For the ideal gas reference state a complete equation of

state is required, one that can reasonably predict the

system volume. Another, more common, approach is to use

a liquid reference state and an expression for the excess

Gibbs free energy. In this formulation the fugacity is

written as



f. = X. Y f (2-9)
1 1 1 1



The Y. term, known as the activity coefficient, is found
1
from the free energy model and is used to correct for com-

position nonideality in the liquid solution.

Mathias and O'Connell (1981) have proposed a slight

variant on this liquid reference state scheme. Their

approach is based on using the temperature and the component

densities, 0i = N./V as the independent variables to describe

the state of the system. Then, at constant temperature

the following relation holds:


3 Zn f.
d Zn f = ) dp (2-10)
1 3 j
J T 'Pk/j


Fugacity ratios can be calculated based on any reference

state if the partial derivatives in equation 2-10 are known.









Expressions for these derivatives in terms of integrals

of microscopic correlation functions are presented in the

next section.


Fluctuation Solution Theory


Fluctuation solution theory (Kirkwood and Buff, 1951;

O'Connell, 1971, 1982) is a bridge that connects the thermo-

dynamic derivatives to statistical mechanical correlation

function integrals. The basic relation of fluctuation

theory is

3
I-L = (-
j T,V,P j 1 1 (2-11)



where the brackets denote an average over an equilibrium

grand canonical ensemble and B = 1/k T. These averages

are related to correlation function integrals by



6.. = (i ) f dld2gij(1,2) (2-12)
1 3 13 1 Q2 ]


where fdl is an integration over all phase space coordinates

required for molecule 1 and Q is the normalization constant

for the orientation dependence. The function g.j(1,2)

is known as a pair correlation function and is directly

related to a two molecule conditional probability density.

It is often more convenient to work with the total correla-

tion function, h. (1,2), defined by









h..(1,2) = gi. (1,2) 1 (2-13)


Combination of equations 2-12 and 2-13 leads to


3
---! = .. + H. (2-14)
STV,,kj 13 1 1 3 13


where we have defined


H.i = --1 I dld2 hi (1,2) (2-15)


Now because of the translational invariance of an equilibrium

ensemble not subject to external fields


hij(l,2) h (R1,R2, ) = h ((R-R2, 2) (2-16)


and thus we can also write


1 dRd dh (, (2-17)
ij V I dRd2 2 1 2


To simplify the further analysis we rewrite equation 2-14

using matrix notation as



=1 ) = [N + NHN]i (2-18)
j T,V,Ik j

where the elements of the N and H matrices are


(N) = .ij (2-19a)











(H)i = H. (2-19b)



To calculate changes in chemical potentials, one is

interested in the inverse of equation 2-18


j) = {[N + NHNI- j (2-20)
j T,V, 13
kpj

This is most easily expressed in terms of integrals

of the direct correlation functions, c.i(1,2), introduced

by Ornstein and Zernike (1914). These direct correlation

functions are defined by


h. (1,2) = c. (1,2) + V- J d3cik(l,3)hkk(3,2) (2-21)
3 k


Because this integral is not of full convolution form,

it may seem that it is not possible to relate the fluctuation

derivatives to integrals of the direct correlation functions

but that is incorrect. Appendix 1 gives the details of

the relationship required. Using equations 2-20 and Al-23

we find


3Bu. -1
( i) = [Nh /v]j (2-22)
3 T,V,
k j
and if the independent variables used are the component

densities the result is










3ui -1
p ) = [e C]i (2-23)



If equation 2-23 is combined with equation 2-10 one

finds that

Snf
) -Cij (2-24)
TPkfj


These relations are most useful because they are

required to derive the differential equation of state.

If the Gibbs-Duhem equation (at constant T) is written

as



dPS = I pi [- d p. (2-25)
i j pj T,pkj


then it is shown that the equation of state can be found

through knowledge of the direct correlation function

integrals. Combining equations 2-25 and 2-23 yields



dPB = [ [1 C p. C..]d p (2-26)



A knowledge of the C.. then allows for the calculation

of both pressure changes and fugacity ratios relative to

any chosen reference state. These results can also be

used to determine partial molar volumes of all components

in a mixture.







13

Summary


This chapter has dealt with some of the properties

of interest for process design. The relation, required

by thermodynamic consistency, between the partial molar

volumes and the chemical potentials has been emphasized.

Mention was made of the common forms of correlations for

these quantities, asserting that often liquid reference

state approaches are used for both. The final section

presented the fluctuation solution theory that allows for

calculation of the partial molar volumes and chemical poten-

tials, based on any reference state, to be expressed simply

in terms of ope set of functions.
















CHAPTER 3
GROUP CONTRIBUTIONS


One of the more powerful tools developed for physical

property estimation has been the group contribution concept.

The term group normally refers to the organic and inorganic

radicals but can be more specific. A molecule of interest

can be described by the number of the different types of

groups of which it is composed.

There are two general methods for using the group

contribution concept. In the first approach some molecular

property is written in terms of the state variables and

a set of parameters, 6,



M = M(T,P; 8) (3-1)



and the parameters for a given set of substances are found

as sums of group contribution



i a 9 ia (3-2)
all
groups a

The most common forms of this type of formulations have

been for predicting critical properties (Lydersen, 1955)

and ideal gas specific heats (Verma and Doraiswamy, 1965).










"In some cases this technique can be justified on molecular

grounds.

The second use of group contribution ideas has been

to assume that the groups actually possess thermodynamic

properties and that the molecular properties are then a

sum of these group properties,


M. = M
1 a l a (3-3)
groups
a in i

This idea is the basis for two popular activity coefficient

correlations, ASOG (Derr and Deal, 1968) and UNIFAC

(Fredendslund et al., 1975). A model for the group proper-

ties must be developed for this technique to be useful.

The true utility of the group contribution approach

stems from its predictive ability. Because all of the

molecules in a homologous series are formed of the same

groups, only in different proportions, the data for several

of the elements of the series can be used to establish

the group property correlations. These can then be used

to predict the properties of the other series members.

This has even greater scope for mixtures.

Consider for example mixtures of n-alkanes and

n-alkanols. They can be considered to be made of only

three groups, -OH, -CH2, and -CH3. Thus, any mixture of

the alkanes and alkanols can be described by the concentra-

tion of these three groups. If a viable theory exists

for some property in terms of the group functions, then









the properties of all mixtures of these groups are set.

Figure 3-1 shows an application of these ideas for

calculation of the excess enthalpy of alkane-alkanol mix-

tures. The figure shows the surface of the excess enthalpy

for all mixtures of the hydroxyl, methyl, and methylene

groups calculated using the UNIFAC equation (Skjold-

Jirgensen et al., 1979). Any compound made of the three

groups-is represented by a point in the base plane. For

example, the point (XCH = 1/2, XCH = 0, XOH = 1/2) is

that for methanol. The possible group compositions for

any mixture are found along the line connecting two molecular

points. In the figure these lines are drawn in for methanol-

pentane, ethanol-pentane, and pentanol-pentane. The predic-

tion of the excess enthalpy is then found by the intersection

of a vertical plane along the composition path and the

property surface. To obtain the enthalpy prediction of

the molecular system, the ideal solution value must be

subtracted from the group estimate. The ideal solution

line simply connects the property surface at the points

of the pair molecules. The enthalpy prediction for a

molecular system of methanol and pentane and at (XME =
M
2/3, XE = 1/3) is shown as the value Ho in the figure.

The power here is that this one diagram can be used to

find excess enthalpies for all alkane-alkanol mixtures.









17



















\ LE-X- ", -












-- L
jOLrz ;O



-00














Z--












--- --- -----------

















systems at 298 K calculated using UNIFAC
theory.










"Reactive" System Theory


Equation 3-3 can also be written as



Mi = v ui Ma (3-4)

where via the stoichiometric coefficient, represents the

number of groups of type a in molecule i. This expression

is completely analogous to that found for a system of groups

"reacting" to form the molecule



I Via" i (3-5)



With this idea the group contribution expressions have

a physical interpretation and the thermodynamics of

"reactive" systems (Perry et al., 1981) can be applied

to obtain many results.

The motivation here has been to use the fluctuation

solution theory in terms of the direct correlation function

integrals. It is then desirable to determine the relation-

ship between the molecular and group direct correlation

functions. This has been accomplished in a very general

fashion by Perry (1980). The development is lengthy, but

the salient features are presented below.

The most important aspect of this approach is that

even if the groups do not have a thermodynamics, there

are still well-defined correlations between groups. This

allows for the development of the Kirkwood-Buff theory







19

in terms of group fluctuation, with the constraints offered

by equation 3-5. The result is


i T -1
j ~ = [MT (e' c' -]6)
jp T kj 1 (3-6)



where p' is the matrix of group densities and C' is a matrix

of group direct correlation function integrals.

While this theory is formally exact for the "reactive"

system, it can be difficult to apply. The correlation

function integrals contain both inter- and intramolecular

contributions. For the total correlation functions these

effects can be separated, but no known analogous result

exists for the direct correlation functions (Lowden and

Chandler, 1979). This is a problem that becomes of great

importance in attempting to model the correlation function

integrals.


RISM Theory


Chandler and Andersen (1973) have formulated a molecular

theory for hard sphere molecules known as RISM. The mole-

cules are assumed to be composed of overlapping hard spheres

or groups. They show how the molecular Ornstein-Zernike

equation 2-18 can be reduced to a group form with explicit

separation of the intermolecular and intramolecular correla-

tions. This allows them to define group direct correlation

function integrals that have only intermolecular







20

contributions. The development is detailed but the essential

result is



cij(1,2) = i Vje C B,(ri r (3-7)



Note here that the molecular correlation function has orien-

tation dependence even though the group functions,
(-a 6
caB(ri,rj), are written for spherically symmetric inter-

actions. Chandler and Andersen discuss many attributes

of these group functions but put no emphasis on the thermo-

dynamic ramifications of these findings.


Differences between RISM Theory and
"Reactive" Solution Theory


The major difference between the RISM theory and the

"reactive" solution theory is the nature of the direct

correlation functions. In Perry's formulation we have


4-
c B(r = direct correlation function (3-8)
between group a and group B



and the RISM theory uses



c "(ri,r) = direct correlation function (3-9)
Between group a on molecule i
and group B on molecule j



This shows that the RISM functions contain less information

but may be easier to model because they seer. analogous







21

to molecular functions for which models have been developed.

Chandler and Andersen have also presented a variational

theory which can be used to obtain the direct correlation

functions for hard body systems.

Figure 3-2 shows the results obtained using data from

Lowden and Chandler (1973) for a system of hard diatomic

molecules. The diatomic molecules are treated as overlapping

spheres of diameter o separated by a distance L. The figure

shows that the calculations agree to a reasonable degree

with the Monte Carlo calculations for this type of system.

Another important aspect of the Chandler-Andersen

theory is that they discuss how the c.B functions could

be written for real molecules. In general, they show that

perturbation theories that would be valid for molecular

systems would also apply to RISM group system. The RISM

formulation will be employed in the present work.


Thermodynamic Properties from Group Functions


Equations 2-23 and 2-26 can be combined with equation

3-7 to express the molecular thermodynamic differentials

in terms of group correlation function integrals. The

first quantity required is



C.. = i(y ) dld2 cij(1,2) (3-10)
1: s 1


Using equation 3-7 this is

























45








30









15








I I
0.0 0.3 0.6 0.9
3


Figure 3.2. Comparison of RISM (---) theory prediction and Monte
Carlo (e* ) for a hard, homonuclear diatomic molecule
with separation to diameter ratio of 0.6.









c. f dld2 c Cr., r.)
C'3 = r r Vi vjs ( 12 2) I dld2 ce6 (+a :


(3-11)


To evaluate this quantity, a coordinate transformation

is required.

dld2 = dR1dR2dSi, d 2 drl, dr2d~id 2 (3-12)



Here, i., represents the set of angles needed to specify

the orientation of molecule i with respect to a fixed

coordinate system. The above transformation is canonical,

and the angular integration can be performed to yield


C.. = I v, ( ) dr dr c (?, r.)
ii] ]B i 2 i j aO 1 ]


(3-13)


Now, the systems under

that


consideration are homogeneous so


c (ri?, rt) = c a(ri ) c t(r)
06 i ] j B 16


Then


Cj = .i vj6 Ce


where



CaB = ( d cB()


This leads to


(3-14)


(3-15)


(3-16)










(3BiJ_5 6.
i Ti n Uia B. 3 C 3 (3-17a)
j T,Pk7j i a B


or in differential form


do.
d. i v (j C dp ) (3-17b)
i p aB


and the corresponding expression for the pressure variation

is



dPB = dP D CaF dpg (3-18)
a b

Summary


Group contribution approaches are valuable for predict-

ing thermodynamic properties. In this chapter two general

approaches for incorporating group contributions into fluc-

tuation theory have been presented and analyzed. Finally,

the thermodynamic property differentials for molecular

systems have been expressed in terms of group direct correla-

tion function integrals. Actual property changes can be

formulated once models are expressed for the integrals.

This is the subject of the next chapter.
















CHEAPER 4
FORMULATION OF MODELS FOR
DIRECT CORRELATION FUNCTIONS


In previous chapters it has been shown that changes

in thermodynamic properties of molecular systems can be

expressed in terms of integrals of site direct correlation

function. Expressions for these integrals are required

before the calculations can be performed.

The purpose of this chapter is two-fold. First, thermo-

dynamic requirements on models for the direct correlation

functions will be presented and examined. With these

restrictions on model form established, a feasible model

for the correlation function integrals will be presented.


Thermodynamic Consistency


One aspect that has been emphasized throughout this

work is the necessity of the formalism to meet thermodynamic

consistency requirements. Because our proposed calculational

scheme employs direct correlation function integrals, the

models that are developed for these quantities are subject

to consistency tests. The easiest check that can be employed

is one of equality of cross partial derivatives of the

dimensionlesss) chemical potentials.









2 2
( ) = ( )a ] V i,j,k (4-1)
aPjSpk p kj


This can be expressed in terms of the direct correlation

functions integrals because


) = C. (4-2)
j T,Pkj


Combination of equations 4-1 and 4-2 leads to

aC. aC
DC-3 ik) (4-3)
r___ = (aplk
Pk T,pZ/k j Tr,/j



It must be noted that this is equivalent to the equality

of three-body direct correlation function integrals (Brelvi,

1973)



C.. = C. (4-4)
ijk = Cikj (4-4


where the subscripts can take on all values associated

with the species in the mixture. The case of interest

here is that in which the direct correlation function

integrals for the molecular species are written in terms

of group quantities



Cij = ( I vi vj CB (4-5)
a 8


Then equation 4-3 takes the form











[ k B B T, ik
nTT,p p



S a [K r Vio VkB CB ) (4-6)
j a B T,p j


This requirement can be expressed in terms of group prop-

erties if the chain rule is used for the derivatives


_P i YaP (4-7)
yV 1 P y y


Combination of equations 4-6 and 4-7 leads to





C
ac

aBy iy kB y Bp nny
i- Y UkVji (a --] (4-8)


This relation will only be satisifed in general if

a3C 3C
0y) T6,p (4-9)
T' -Y P Z-


Thus, the consistency requirements for group direct correla-

tion function integrals are equivalent to those of the

molecular quantities. This constraint will be employed

in formulation of the working models for the group direct

correlation function integrals.









Basis for Model Development


It has already been shown that the thermodynamic con-

sistency tests for a group formulation and a molecular

formulation are equivalent when written in terms of direct

correlation function integrals based on the RISM theory.

This should not be surprising considering that stability

conditions for fluids were shown to be equivalent for the

two approaches by Perry (1980). Thus, in developing models

for group direct correlation function integrals, the same

considerations should apply as those used by Mathias (1979)

in developing molecular models.

The general philosophy that will be employed is to

use as much theoretical information as possible to develop

these models. This requires use of some concepts of statis-

tical mechanical perturbation theory to obtain approximate

forms for the correlation function integrals. The analysis

of Chandler and Andersen (1972) has shown that for RISM

theory direct correlation function results of molecular

perturbation theory are easily extended to the group func-

tions. Appendix 2 contains a complete development of an

exact perturbation theory for the direct correlation func-

tions based on the RISM theory. In this chapter we shall

only deal with the important ramifications of these results.

It is always possible to write



B ref + a ref (4-10)
C =a6 + (Cc a (4-10)









where the superscript ref refers to some reference system.

The purpose of perturbation theory is to determine a refer-

ence system so that an approximate form for the perturbation

(second on the right-hand side) term can be made that yields

useful results. Because this work is concerned with dense

fluids, we require a reference system that can adequately

represent the behavior at high densities. In liquids the

optimal choice of a reference system seems to be one with

only repulsive forces (Weeks, Chandler, and Andersen, 1971).

Values of Ca3 are not available for this type of model

system. However, the system with purely repulsive forces

can be well represented by a system of hard spheres if

the hard sphere diameters are chosen as functions of tempera-

ture (Barker and Henderson, 1967). This approach will

be followed in this work. Even with this choice of reference

system the perturbation term cannot be exactly identified.

A further approximation used is that the zero density limit

of this term is adequate. Thus, the model employed here

is


-HS lim HS
C CB + lim (CaB C) (4-11)



Appendix 2 shows the evaluation of the required limit which

gives the working relation



Ca C H + (a) /Tn (4-12)
n=o aB







30

The constants in equation 4-12 are dependent on the inter-

molecular potential (written as a sum of group of interac-

tions). No explicit calculation of these terms is attempted

here because we choose to determine the expressions on

the basis of experimental data. This model form is com-

pletely analogous to the model used by Mathias (1979).


Analysis of the Model


Several interesting features of the direct correlation

function model seem to merit mention. Equation 4-12 is

highly similar to the RISM form of the mean spherical

approximation (Lebowitz and Percus, 1966; Chandler and

Andersen, 1972) and is essentially equivalent if the hard

sphere diameters are chosen as functions of temperature

only. The assumptions involved in obtaining the present

approximant and the mean spherical form are different,

but it seems that for our purposes this difference is

immaterial.

Of greater interest here is the relationship suggested

by the present model for the three-body direct correlation

function integrals. These can be found from


3C.
C = (--) (4-13)
ijk 8k T,p /k


The form for the C.. proposed in equation 4-12 then suggests

that









HS
C CHS (4-14)
ijk ijk


This is probably not a very accurate approximation, but

the flexibility in the model form inherent due to the

determination of the parameters from experimental data

may make this adequate. A better interpretation of this

analysis is that the density dependence of the three-body

direct correlation function is assumed to be approximated

by that of the hard sphere quantity. This analysis also

suggests an extension of the RISM theory to three-body

correlation functions, a derivation of which can be found

in Appendix 3.



Cjk i ji jky CaBy (4-15)


Finally, thermodynamic consistency of the proposed model

must be considered. If equation 4-12 is rewritten in

explicit form


HS
C (pT) = C (p,T) + B (T) (4-16)



then the consistency requirement is met if

ECHS aCHS
( "B ( y)
py Tp T,pp (4-17)


HS
In this work, the Cae are determined from the generalized

hard sphere equation of state, which is consistent, then









this must be true. Note that if terms of higher order

in density had been included in the perturbation term,

they also would have to have been chosen to be thermodynami-

cally consistent.

Also of interest here is the functional dependence

of the hard sphere diameters. If they are chosen to be

functions of density, as is normally done in the Week-

Chandler-Andersen perturbation scheme, then this dependence

would have to be such that equation 4-17 was satisfied.

This is an aspect not always appreciated.


Property Changes Using the Proposed Model


This chapter has presented and analyzed a feasible

model for group direct correlation function integrals.

The utility of this model can only be tested by its use

to calculate changes in molecular thermodynamic properties.

The required formulas are developed below.


Chemical Potential


Equation 3-17b expresses the differential of a molecular

chemical potential in terms of the group direct correlation

function integrals. This is conveniently rewritten in

terms of the residual chemical potential as



dBf = i vj CaB dPg (4-18)



Insertion of equation 4-16 and integration leads to









Ari = I )i [A HS + r A Ap] (4-19)
aa a


where ',HS is the residual chemical potential calculated

from the hard sphere equations for group a in a solution

of groups.


Pressure Change


To calculate pressure changes, it is again easiest

to work with residual properties. The required differential

form is



dP = p CB dp (4-20)
ca a


Insertion of the model and integration leads to



Apr = APr,HS + (1) Y A(p p ) (4-21)
6 5 2 a a aS


where A r,HS is the change in the residual pressure (divided

by k T) calculated from the hard sphere equation for the

solution of groups.

It should be noted that the ideal gas state used for

the residual property changes on either side of equation

4-21 are different. This should be expected for the group

equation of state is written in terms of the group densities

which sum to a larger value than the molecular densities.

However, because this form is not used as a complete equa-

tion of state, this should not present a problem.















CHAPTER 5
MODEL PARAMETERIZATION


This work has been concerned with formulating the

thermodynamic properties of molecular systems in terms

of integrals of group direct correlation functions. In

the last chapter a form for these terms was developed



C = CRE + (5-1)



and further, the reference state was identified as that

of a hard sphere system. The full dependence on state

variables is written as



C (p,T) = CHS (P,T; ) + 0a (1/T) (5-2)



where U is the vector of hard sphere diameters for the

groups in the system. The purpose of this chapter is to

develop empirical forms for the functions U'aS and to estab-

lish the dependence of the hard sphere diameters on state

parameters (Weeks,Chandler, and Andersen, 1971).










Corresponding States Theory


The utility of the form proposed in equation 5-2 can

be enhanced if the perturbation functions and hard sphere

diameters could be written in corresponding states form.

This is a two-step process:

1. Determine if a corresponding states principle

exists for the direct correlation function

integrals for liquids.

2. Determine if the functional form proposed in

equation 5-2 can represent the observed

correspondence.

To answer these questions simple molecules will be

considered first. The molecules considered here can be

considered to consist of only one group. The relation

to thermodynamics used in the analysis is



( ) = 1/(l-pC) (5-3)
T



Figure 5-1 shows a corresponding states correlation for

the quantity on the left-hand side of equation 5-3 for

simple molecules, using only one parameter. Further, Brelvi

and O'Connell (1972) showed that this behavior can be found

for polyatomic molecules also, if the density is large

enough (p > 2Pc). These results suggest that for large

densities a one-parameter corresponding states formulation

may be valid for the direct correlation function integrals.































-2









in o
CCl4
-3 4
0 CH

m CO2


-N






0.7 0.8 0.9
3
p3

Figure 5-1. Bulk modulus versus reduced ensity for simple
fluids. The line is argon data and o is
Lennard-Jones potential parameter.










Note that the temperature dependence seems to have little

effect. Inclusion of temperature dependence in these cor-

relations should be able to enhance the accuracy.

Figure 5-2 shows another test of this correspondence.

This is in an integrated form of equation 5-3.



PB = PBef + S (1-pC) dP (5-4)
Pref


Here, a characteristic volume for each molecule, V*, and

a characteristic temperature, T*, have been found to yield

the correspondence as



P = Pref' f(, T; ref) (5-5)



where

P = PV*/RT

= pV*

T = T/T*



The function, f, in equation 5-5 is a corresponding

states correlation for the density integral of the direct

correlation function. This is what is desired. The data

presented seem to suggest that a corresponding states formu-

lation can be found for pure component direct correlation

function integrals for dense fluids. A detailed discussion















100



argon

50 methanol

V acetic acid

2 n-Cl6H34
n-C16 H 34

A 25 38


20





10
T = T/T*
P = PV*

P =-+C
RT








2


T=0.92 T=0.60 T=0.50

T=0.80


Figure 5-2. Corresponding states correlation for liquids.


Figure 5-2. Corresponding states correlation for liquids.







39

of the origins of this correlation is presented in Appendix

4.

Mathias (1979) has shown that these concepts can be

extended to mixture-direct correlation function integrals

with the proper choice of mixing rules. He used a form

similar to equation 5-2, but the group contribution concept

was not employed.

The applicability of the two-parameter corresponding

states principle seems valid for the direct correlation

function integrals. The question of the use of the func-

tional form must now be addressed. This is again most

easily accomplished by analysis of a set of experimental

data for a simple molecule. After the noble gases are

used for this type of testing, however, the major interest

here is for hydrocarbons so that a hydrocarbon reference

system may prove to be more useful.


Choice of a Reference Component


Mathias (1979) had great success in developing a

corresponding states correlation for direct correlation

function integrals, using a form similar to equation 5-2,

for molecular systems, using argon as a reference component

to determine model parameters. This work is concerned

with hydrocarbon properties so that it seems reasonable

to examine the difference in direct correlation function

integrals between argon and a simple hydrocarbon, methane.










The comparison is performed here using the dimensionless

quantity 1-pC. Table 5-1 shows this comparison for two

isotherms, in the dense fluid region.

The behavior is similar on both isotherms. It is

seen that the fluids act quite similarly at lower densities

but behave differently as the density becomes larger.

Methane is increasingly more compressible as the density

increases. This can be explained by the polyatomicity

of methane. Both molecules are essentially spherical,

but at the higher densities the methane seems to allow

for interlocking of the hydrogens.

These effects can be important for the development

of the group contribution correlations. If the groups

were chosen as monatomic, then argon would probably be

the better choice of reference substance due to physical

similarity. However, if the groups are chosen as the common

organic radicals, then methane may be the better choice

of reference substance.

The actual choice of a reference substance will be

made after it has been determined whether equation 5-2

is a viable functional form for the direct correlation

function integrals.


Method of Data Analysis


This section addresses the question of whether the

form proposed in equation 5-2 can be used to fit the direct
















COMPARISON


TABLE 5-1

OF DIRECT CORRELATION FUNCTION INTEGRALS
FOR ARGON AND METHANE


T/T = 0.76 T/TC = 2.0


pVc (1-pC)argon ( 1-pC)methane pVc (1-pC)argon (1-pC)methane


2.26 6.44 6.35 2.1 7.23 7.14

2.31 7.97 7.82 2.2 8.50 8.35

2.37 9.68 9.44 2.3 9.96 9.73

2.43 11.51 11.25 2.4 11.65 11.29

2.48 13.57 13.24 2.5 13.58 13.07

2.54 15.83 15.43 2.6 15.79 15.06

2.59 18.30 17.83 2.7 18.29 17.30

2.65 21.05 20.46 2.8 21.13 19.91

2.70 23.88 23.33 2.9 24.33 22.59

2.76 27.10 26.45 3.0 27.94 25.68







42

correlation function integrals of real fluids. The analysis

will be performed in terms of the dimensionless quantity,

C, definedby


PBS
C pC = 1 ( (5-6)



The thermodynamic derivative is evaluated from correlated

experimental data. From equation (5-2) this is



C CHS (p, T; o) + p6 (5-7)



For this work the hard sphere diameter is treated as a

function only of temperature. Thus, along an isotherm,

if the proposed functional form is proper, for the proper

choice of hard sphere diameter, the value of 6 should be

constant given by



S= (CHS C)/p (5-8)



for any value of p. The suitability of the proposed model

is determined by how well this is obeyed. The hard sphere

properties are determined using the generalized hard sphere

expressions as presented in Appendix 5. It should be noted

that all of the hard sphere equations are generated by

the same microscopic form of the direct correlation

function. Here the hard sphere direct correlation function







43

integrals are found using equation 5-6 with the corresponding

form of the hard sphere equation of state. For these pure

component cases we have


-HS (1+a)n4 4(1+a)n3 + 2 8
C 4 (5-9)
(1-n)


where n = V/6.po



and in corresponding states form



n = [T/6 o3/V*][pV*] (5-10a)



= X pV* = Xp (5-10b)



Here, the characteristic volume, V*, has been introduced

as a reducing parameter for both the hard sphere volume

and the density. On each isotherm there are actually two

parameters that can be used to vary the model, a and X.

The analysis here has been done sequentially, a value of

a is chosen and then the optimum X has been found for that

value of a. The characteristic volume has also been used

to reduce the perturbation term 0 as



u = (t,/V*)V* = V*f(T) (5-11)


The model can then be written as












C(p,T) = CHS(n; T,X,a) + pf (5-12)



Figure 5-3 gives a simplified flowchart of how the calcula-

tions are performed for each isotherm for a fixed value

of a. The subscript i denotes the number of data points

on the isotherm, NP. The optimization routine, RQUADD,

can be found in Appendix 6.


Analysis of Argon Data

3P8
The values of ( needed for this analysis were generated


from the equation of state of Twu et al. (1980). These

authors claim to be able to reproduce dense fluid pressures

to within the experimental accuracy. Eleven temperatures,

evenly spaced between the triple and critical points, were

chosen to generate liquid phase data. Each isotherm con-

sisted of 16 density-correlation function integral pairs

evenly spaced between the vaporizing and freezing densities.

Supercritical data were also used. The isotherms were

for values of T/T from 1.1 to 3.0. The densities used

were in 0.1 increments of P/P from 1.5 to 3.0, or the

freezing density, whichever was lower. A listing of the

program used to generate the data is given in Appendix

6.




















































Figure 5-3. Flowchart for calculation of X at a fixed a.










For the initial analysis values of a of 0, -1, -2,

-3, -4, and -5 were used. Table 5-2 shows the average

absolute deviation in the direct correlation function

integrals found for the optimum value of x at each isotherm.

A noticeable minimum exists in the range of a = -4. Note

that the Carnoban-Starling equation (a = -1) does not yield

the best predictions, even though it is the "best" hard

sphere equation of state.

Several isotherms were chosen to examine the sensitivity

of the results to the value of a, on a finer scale. These

results are given in Table 5-3. For the liquid phase iso-

therms a value of a = -4.3 seems to be optimal. The super-

critical data are best fit with a of -4.0 or -4.1. It

was decided to use a constant value of a = -4.2 to fit

all of the argon data.

With a set at -4.2 the value of X and f could then

be determined. The optimum values of X found were correlated

as a polynomial in inverse temperature. For this analysis

the characteristic parameters were chosen to be equal to

the critical parameters. The form that fit X best was



X = 0.12342 + 0.15069 _0.18545 + 0.15947
T 2 3
T T



0.070261 0.012111 (5
+ (5-13)
4 -5
T T










TABLE 5-2

REGRESSION ANALYSIS OF ARGON DATA


T Average Absolute Deviation in C for = a


0.5966
0.6297
0.6629
0.6960
0.7292
0.7623
0.7954
0.8286
0.8949
0.9280
0.8617
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.2
2.4
2.6
2.8
3.0


0.0101
0.0392
0.0785
0.1225
0.1651
0.2062
0.2445
0.2758
0.3104
0.3053
0.2994
0.2840
0.2413
0.2175
0.1990
0.1837
0.1707
0.1611
0.1530
0.1498
0.1403
0.1310
0.1242
0.1192
0.1152
0.1126


0.0102
0.0395
0.0789
0.1228
0.1651
0.2059
0.2436
0.2736
0.3056
0.2990
0.2963
0.2764
0.2344
0.2088
0.1899
0.1740
0.1611
0.1504
0.1411
0.1344
0.1286
0.1196
0.1131
0.1084
0.1047
0.1023


0.0094
0.0361
0.0714
0.1100
0.1461
0.1798
0.2094
0.2304
0.2408
0.2374
0.2430
0.2142
0.1835
0.1580
0.1383
0.1227
0.1101
0.1003
0.0920
0.0854
0.0798
0.0713
0.0651
0.0610
0.0582
0.0566


0.0058
0.0219
0.0427
0.0642
0.0830
0.0983
0.1085
0.1104
0.1122
0.1280
0.1071


0.1260
0.1058
0.0912
0.0801
0.0713
0.0643


0.0536
0.0495
0.0425
0.0371
0.0324
0.0290
0.0261


0.7266
1.3274
1.7500
2.0472
2.2372


2.3746
2.3629
2.1992
2.0497
2.2885
1.4495
1.1049
0.8986


0.6364
0.5516
0.4845


0.3864
0.3505
0.2948
0.2536
0.2214
0.1960
0.1749







48



TABLE 5-3

EFFECT OF HS EQUATION ON FITTING ARGON DATA



Average Absolute Deviation in C for = a


-3.8 -3.9 -4.0 -4.1 -4.2 -4.2 -4.4


.7954 .1435 .1277 .1085 .0848 .0547 .0214 .0401

.8286 .1516 .1329 .1104 .0831 .0535 .0263 .0595

.8617 .1507 .1297 .1071 .0829 .0592 .0399 .0866

.8949 .1477 .1294 .1122 .0916 .0732 .0667 .1264

.9280 .1581 .1434 .1280 .1144 .1049 .1152 .1851

1.1 .1484 .1368 .1309 .1235 .1258

1.2 .1344 .1303 .1260 .1263 .1364

1.3 .1126 .1092 .1058 .1064 .1065


0590 .0585 .0618 .0698


1.8 .0607










The maximum error in calculated x values from equation

5-13 was 0.105% with an average error 0.022%. With x values

from equation 5-13 and a set at -4.2 the optimal values

of f were then found. These values were also fit to a

polynomial as


f = -14.70 23.237 +25.221 11.636
f = -1.4098 + +
2 ~3 ~4
T T T T



2.0463
+ 2 (5-14)
T-5



This polynomial form reproduced the f values with an average

error of 0.15% for the isotherms analyzed. The temperature

dependence of X and f is shown in Figure 5-4. Table 5-4

gives a summary of the values of X and f for argon along

with the average error in the prediction of the direct

correlation function integrals. In general, the correlation

function integrals were correlated to within the experimental

accuracy. The only range of temperatures for which the

fit was not excellent was near the critical point, which

is to be expected.


Analysis of Methane Data


There is less high quality P-V-T data available for

dense methane than for argon. In this work the equation





























9.0







l/V*







5.0


Figure 5-4.


.0.20






/I 3
- /V*
6



0.18












0.16


1.0 2.0
T/T*



Dimensionless hard sphere diameter and
perturbation function based on argon
reference.










TABLE 5-4

CORRELATION OF ARGON DCFI USING a = -4.2


Average Error
T X f V inC
c


0.5966
0.6297
0.6629
0.6960
0.7292
0.7623
0.7954
0.8286
0.8617
0.8949
0.9280
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1. 9
2.0
2.2
2.4
2.6
2.8
3.0


0.21159
0.20916
0.20692
0.20482
0.20283
0.20094
0.19915
0.19745
0.19583
0.19430
0.19286
0.18649
0.18348
0.18086
0.17852
0.17643
0.17453
0.17280
0.17120
0.16971
0.16833
0.16582
0.16359
0.16159
0.15978
0.15814


890.79
820.09
759.79
707.47
661.13
620.05
583.39
550.40
520.61
493.31
467.98
373.74
335.43
306.28
282.27
262.19
245.04
230.17
217.10
205.48
195.05
177.04
161.94
149.05
137.88
128.10


.0035
.0128
.0525
.0372
.0466
.0532
.0562
.0547
.0596
.0732
.1076
.1264
.1377
.1166
.1037
.0936
.0844
.0764
.0700
.0648
.0605
.0527
.0458
.0393
.0343
0308








52

of state of Mollerup (1980) was used to generate nine sub-

critical isotherms and one supercritical isotherm, as was

done for argon. These supercritical isotherms measured

by Robertson and Babb (1969) were also analyzed. The pro-

cedure used for the analysis of the methane data was the

same as for argon, except some positive values of a were

also investigated.

For the liquid phase isotherms a value of a = -4.2

was clearly the best. For the supercritical data values

of a from -3.0 to -4.3 could produce almost equivalent

results, but a = -4.0 was the optimum. It should be noted

that for the supercritical isotherm values of a = 9 could

also offer a reasonable representation of the data, but

not as good as -4.0. Thus, a value of -4.2 was chosen

to represent the methane data.

Proceeding as was done before for the argon analysis,

the values of X and f were fit to polynomials in inverse

temperature. The resultant forms were


0.026446 0.080054 0.089994
X = 0.13832 + + -2 -3
T T T

S0.039085 0.0060984
-4 ~5 (5-15a)
T T

23.21 50.827 40.662 15.818
f = 5.0851 -+ -2 3 +
T T T T

2.3173
-5 (5-15b)
T










where the critical properties were used as the reducing

parameters. These functions are plotted in Figure 5-5.

Table 5-5 gives the summary of results for the methane

correlation. The proposed model was able to reproduce

the data to within the experimental uncertainty.


Use of the Correlations to Calculate Pressure Changes


The development of the DCFI models for the pure com-

ponents was first tested by performing equation of state

calculations for the base substances, argon and methane.

The method of calculation was to choose the lowest density

point on each isotherm as a reference and then to calculate

pressure changes for all other points on the isotherm from

that reference. There are thus two criteria for goodness

of fit, the average percent error in pressure changes,

defined by


ND LPcalc Aexp
aP = [ ] / ND (5-16)
i=1 APexp


and the sum squared percent error, defined by


ND APcalc Aexp 2
SSEAP = [ ] (5-17)
i=l APexp


where in both cases ND is the total number of data points

for a compound. When parameters were found to best "fit"

a set of data, the quantity SSEAP was minimized using a











TABLE 5-5

CORRELATION OF METHANE DCFI USING a = -4.2


Average Error
T X f V in C
c


.4986 .22182 1545.26 .0015
.5511 .21596 1309.43 .0155
.6036 .21095 1131.07 .0188
.6561 .20671 996.35 .0157
.7085 .20302 890.93 .0140
.7610 .19974 805.32 .0335
.8135 .19675 733.53 .0612
.8660 .19400 671.78 .0946
.9185 .19145 617.18 .1413
1.1 .18381 473.60 .1484
1.6173 .16916 260.80 .2262
1.9585 .16316 204.86 .1437
2.4833 .15703 168.74 .0637




















13.0













9.0



)/V*







5.0


3.0


1.0 2.0
T/T*


Figure 5.5.


Dimensionless hard sphere diameter and
perturbation functions based on methane
reference.


1.20









113
-3/V*


. 18












. 16








56

nonlinear regression routine and the program GROUPFIT listed

in Appendix 6. However, the data comparisons will normally

be made in terms of the quantity AP. This procedure is

common and is used to eliminate bias from the error mini-

mization.

The first test performed was to calculate AP for argon

using the characteristic parameters as the critical

parameters and the functions given by equations 5-13 and

5-14. The average error was found to be 1.42%. This is

larger than expected, but if only the liquid phase data

reconsidered, the error is just 0.629%, and if the isotherm

that is within 10% of the critical temperature is neglected,

the error for the liquid isotherms is only 0.34%. This

is within the accuracy of the equation of state for this

region of the phase diagram. On the whole the largest

errors are found in the low density data near the critical

temperature. The equation is able to reproduce the highest

temperature isotherms (T/Tc = 2.8 and 3.0) with an average

error of 0.47%. So the fit of the argon data seems satis-

factory for the desired usage.

In an attempt to improve the argon representation,

a regression was performed to find the set of characteristic

parameters that minimized the sum square error in the pres-

sure changes as per equation 5-17. The parameters found

were T* = 144.45 K and V* = 74.31 cm3/gmole as compared

to T = 150.86 K and V = 74.57 cm /gmole. The average
c c








57

error for all of the data increased to 1.53%. This worsening

of the percentage error occurred because the fit was made

more even; that is, SSEAP was decreased from 4.549 x 103

to 3.034 x 10 but on the average the error was larger.

This is due to the larger localization of error in the

critical region.

A calculation was also performed on the methane data

using the argon functions and methane's critical parameters

as the characteristic parameters; the error was 3.13%.

And when a regression was performed on the methane data,

the characteristic parameters were found to be T* =

190.54 K and V* = 98.40 cm3/gmole (compare T = 190.53

K and V = 98.52 cm /gmole) with an error of 2.98%. This

shows that the trends found for the methane DCFI as compared

to argon carries over to the pressure change calculations.

Similar calculations to those described above were

performed using the hard sphere diameter and perturbation

term functions as given by equations 5-15. The average

error in pressure changes for all of the methane data was

0.57% with the largest error from the isotherm at 175 K

(within 9% of the critical temperature). Even at this

temperature the high density data are well described.

In the reduced temperature range of 0.5 to 0.89 the average

error was only 0.18%, well within the accuracy of the data.

As for argon, when the GROUPFIT program was employed to

find optimal characteristic parameters for methane, the









2 2
SSEAP was reduced from 4.13 x 10 to 2.44 x 10 but the

average error was increased to 0.727%. Again .the localiza-

tion of the error causes this phenomenon.

A calculation of the pressure changes for the argon

data using the methane functions gave an error of 4.65%,

and when optimized, the argon characteristic temperature

only changed 1.3% and the error reduction was only to 4.56%.

This again shows that there are noticeable differences

in the compressive behavior of argon and methane at high

densities.

In summary, the temperature dependence of the hard

sphere diameter and perturbation term in the DCFI model

are seen to adequately represent the data from which they

were developed. The difference in DCFI noticed for the

atomic species argon and the polyatomic methane are notice-

able in calculations of compressions.

The rest of this work will deal with calculations

of hydrocarbon compressions using the group contribution

formulation. The groups of interest are methyl (-CH3)

and methylene (-CH2). These are physically more similar

to methane than to argon. For this reason all of the group

contribution calculations will be performed using the methane

reference functions.
















CHAPTER 6
USE OF THE GROUP CONTRIBUTION MODEL FOR
PURE FLUIDS


The development thus far has been limited to the formal

expressions for calculating property changes of mixtures

based on the group contribution model of direct correlation

function integrals and the parameterization of the model

using data for argon and methane. In both cases the mole-

cules were assumed to consist of only one type of group.

This chapter presents the calculations of pressure changes

for pure fluids composed of different types of groups.

The first section discusses the extension of the cor-

relations required for systems of several groups. The

remainder of the chapter presents and discusses the results

of the calculations performed for several n-alkanes and

one n-alkanol.


Extension of the Model to Multigroup Systems


The expressions developed in Chapter 4 for calculation

of pressure changes involve two contributions.



APB APBS + APPERTURBATION (6-1)







60

The hard sphere contribution is calculated for the solution

of groups using the results shown in Appendix 5. To cal-

culate the perturbation contributions for multigroup systems,

the form of the 'P functions must be established for the

terms with a 3 B. In Chapter 5 a corresponding states

expression was developed



a = V* f (T/T ) (6-2)



where f is a universal function of the reduced temperature.

This result is extended to the unlike terms as



S= V* f (T/T* ) (6-3)



and the characteristic parameters are found from

1 1/3 1/3)
V = (V* )1/3 + (V )3) (6-4a)
aB 8 at B6


T* = (T* T )1/2 (-k) (6-4b)



where k is an empirically determined binary interaction

parameter for groups a and 3. It is important to note

that this binary parameter can still be determined from

analysis of pure component volumetric data.










Designation of Groups


Before the group contribution model can be used for

calculation of property changes, the decision as to which

collections of atoms in a molecule comprise a group must

be made. The simplest choice would be to consider each

atom as a group, and thus the properties of all molecules

could be described with minimal information. However,

that is an overly ambitious approach. Even for the molecules

considered here that contain only carbon, hydrogen, and

oxygen atoms, this is unworkable. Typically, the common

organic radicals are designated as the groups.

For this work it must be decided if the n-alkanes

are composed of only one type of group, or two. If they

are considered as being comprised of only one type of group,

then the model could never predict excess volumes for

n-alkane mixtures. This is reasonable at low pressures,

but for highly compressed systems, noticeable excess volumes

do exist (Snyder, Benson, Huang, and Winnick, 1974). Thus,

the n-alkanes will be considered to be composed of two

groups, methyl (CH3) and methylene (CH2). To analyze

methanol the hydroxyl moiety (OH) will also be considered

as a separate group.


Pressure Change Calculations


All of the molecules analyzed in this work will be

considered to be composed of only two types of groups.









The equations required for the calculation of the pressure

changes are


P6 (P)ref = PHS (pHS)ref
PB -(PB) PB (PB


1 p2_ 2 ref) 2 (6-5)
2 111+2 v212222) (6-5)


where here, v. is the number of groups of type i in the

molecule. The individual terms are found using



PBHS 1 1 3- 11 + 3623
1 (1-53) (1-63)


4.2 2 3 } (6-6a)
(1-3)

with



1 = P[vlo + 2c2 ] (6-6b)


and the oi terms are found using


3
o /V* = f(T/Tf) (6-6c)
6 i l 1


where the function of reduced temperature required in equation

6-6c is given in Chapter 5. The perturbation contributions

are found using equations 6-3 and 6-4 with the required

function of reduced temperature again given in Chapter 5.










These functional forms are dependent only on the reference

component chosen for their development, either argon or

methane. For the two group molecules there are then only

five independent parameters, V*, V*, T*, T*, and kl2.


Analysis of n-Alkane Compressions


In this section we consider the ability of the proposed

formulation to calculate changes in pressure during compres-

sion of four n-alkane molecules: ethane, propane,

n-pentadecane, and n-octadecane. Calculation of pressure

changes using the equations given above were performed

for both the argon- and methane-based reference functions

as derived in Chapter 5.

The initial plan of attack was to determine the methyl

parameters using ethane data and ethylene and cross parame-

ters from the propane data. The model capabilities would

then be examined by using these same parameters for calcula-

tions on the long chain molecules. The analysis was per-

formed both with and without a binary interaction coeffi-

cient. The characteristic parameters for the methyl and

methylene groups as determined by regression analysis are

shown in Table 6-1 along with the errors in calculated

pressure changes. Several conclusions can be drawn from

these results that will be of future interest. First,

the best fit of all of the data was obtained using the

methane-based correlations with the inclusion of a binary























or 0

-4


4.


.a
N^ N CN N 421
H0


m
-4:

Z










Z
4

















z
LO
<









C)













a z
mz
04






< o
204














u a
U E
e n














uen
men

EH M



0u


I-4


04
a0
2-
UH


3 C
4O CC
0 ., 13


< 0
0 a C
C

np t
4 0
0 C



c


o u
S44










Ln rn
-H (
41 0








0 0 PO
905


Oo



U
Ou








-H
4-)
4-3









4C1
0 4




a 4-
uCO

.0



p e
10 Co



0)444
Co Co

U o
re0
440


er
N*
32
U 9r













N







m *

U mC
N


o0 o0 4 .0
I-l M P *=


U

u

ri
0





rI









Q
-)















U) 0 0
o

























0 0



D m a
04
Co













c




S en
m




O
-4-
































H .0 0
0 71 0


0 H H
an

0



10 l




C 42 42,


Ln
N



C
o


oo
a-










interaction coefficient. This binary constant was much

larger than usual and thus gave a large value of the T*
12

parameter. This results from the fact that the model is

much more sensitive to the values of the characteristic

volumes than the characteristic temperatures so that large

adjustments to the binary constant are required to signifi-

cantly affect the calculations.

It was also seen that 2V* = 0.9 Ve and
CH C,ethane
2V* + V*C = 0.9 V This suggests an approach
CH 3 CH2 c,propane
for estimating characteristic volumes of the groups from

critical properties. The characteristic temperatures might

also be obtainable using critical values with some sort

of mixing rule applied.

While the results are not as good as the molecular

fit, they could be satisfactory. A more important question

is their ability to predict compression data for other

hydrocarbons. For n-pentadecane and n-octadecane results

were rather poor, the best agreement being an average error

in the pressure changes of 16.6% using parameter set 1.

This is discouraging but not unexpected. The hard sphere

contribution does not behave properly at high densities

as explained in Appendix 5. In addition, the improper

ideal gas limit of the equation of state affects the results.

In an effort to improve the model performance, the

data for the two long chain hydrocarbons were used to deter-

mine the methylene and interaction parameters.










The results of interest are shown in Table 6-2. The

methyl parameters were those found from analysis of the

ethane data. The results are rather encouraging in that

either the four- or five-parameter models could give an

adequate representation of data for both short- and long-

chain hydrocarbons. However, it should be noted that none

of the parameter sets could give a reasonable representation

of the propane pressure changes.

These results do yield two interesting conclusions.

First, that when the larger hydrocarbons are considered

for the parameter estimation that the binary parameter

becomes very important. Noticeable improvement in the

pressure change calculations occurs when the binary parameter

is included. This suggests that the methyl and methylene

groups should be considered as distinct entities. However,

the magnitude of the binary parameter is much greater than

would be expected. It can also be seen that the methane-

based correlations are more apropos for modeling the alkane

compounds because the groups are physically more akin to

methane than to argon.

After studying the aforementioned results, it was

decided to establish a final set of model parameters for

the methyl and methylene groups from an analysis of all

of the n-alkane data simultaneously. The methane-based

reference functions were used, and the calculations were

performed both with and without a binary parameter. These




















(N
0 rn
El


mn n

enNm N


mL o ef N 0 n

H *





0 N N V
ul


E



MW
z

a4








U
Hg




















O i
H Z
0







o
WE-E






<
Z 0
IEi Z




z WC I


z

H a

OZ
HOC

M0:





HZ




HU

U


01 N N
00 C C













n* *
Lfl m (n e


en en rN Lf





Lf3 (0 '1 L





(D c 00 .


\0

0 0

o a


0 u0
000o




040
0o00






u0
-H o

.m E

(0 m

0 1





-H> -







11 O C
o >




41






1 4J

40
00) N












N -


0U 0
0 -H

0 0


0 0


O
U o






0 0
r 0
Q) a)
S4



0

o
1 4J

(a E
1 r










results are shown in Table 6-3. It can be seen that use

of a binary parameter does improve the results and that

the order of magnitude of this parameter is reasonable.

Using only five adjustable parameters, the average absolute

error in pressure changes calculated for all four hydro-

carbons was 5.9%.


Analysis of Methanol Compressions


A set of compression measurements for methanol were

analyzed to determine if the present model could adequately

calculate pressure changes for molecules other than

n-alkanes. Methanol was considered to be composed of one

methyl group and one hydroxyl (OH) group. The methyl group

characteristic parameters used are those listed under

parameter set 11 in Table 6-3. The regression results

are shown in Table 6-4. The characteristic volume for

the hydroxyl group appears to be quite reasonable because

V* + VH 0.9 Vmethan which is the same pattern
CH OH c,methanol
seen for the alkanes. The characteristic temperature and

binary parameter found are surprising, but these values

were necessary to obtain a fit of the data about as good

as that for the alkanes.


Summary


While the calculations presented here are based on

recognized assumptions that are believed to be sound, there






















O



0
o






m;

LO
0U




I


z1
aZ


ME
W









Z

Z
0H


0
I 0



















MO
Urn
0 <










42
0















H2



















0


0U

in cn

0r


in




N *
0 <



O *
o ^
2


O-,
0)






0 (U




0 -J
Mi








0) 0
012--















40
100












OJ-







n 0
0 0
0 M
-H


















.-I a



40
O
04 -



4 C
Co
C)


004a
















Cow,

CO
12<


N CN








Co.
CDo














m *

m H






i- i i-


6iT
0 o

o 0






















OE
z
)-
0




H





0u

>4
=; 4
1 0

P E





z






04



EM
a
O o

CL4
H
<

0


U M

a-



O 0
OH
0
00




2-1


2-l


0 C)0
U) -4 tL -


aC 0







I 0 4-)
-4


OI 0 4




> c
0 4-1



H

4- )
4--



X oE O
000 E
X H 4E 0










00 >
U

4 4





K 0
O



4
-Hc) -
C)a


N 0













S C
o

o o


H N

P~ N
N- C
N- i0









N Cl
--4 .-










are certain disconcerting aspects. In particular, the

equation of state developed does not properly approach

the ideal gas limit. Even though the calculations are

always performed for pressure differences from a liquid,

or dense fluid, reference state this may be important.

Also, the percentage errors in the pressure change

calculations appear to be rather large. In most process

design calculations the temperature and pressure are known

and one wishes to calculate the volume. It would then

be of interest to know how the present model would perform

in calculating volume changes for given pressure changes.

A simple error analysis shows how this can be estimated.

At constant temperature errors in pressure calculations

and volume calculations are related by


3V
AV = (- ) AP (6-7)
T


Then average absolute errors in volumes and pressures must

be related by



/ / = PT /P/ (6-8)



where 8T is the isothermal compressibility. For liquids

the product PBT is almost always less than 0.2 and even

then is only that large at very high pressures. This sug-

gests that if the present correlation were used to calculate







72

a volume change for a given pressure change that the average

absolute errors would be less than 20% of the errors reported

here for the pressure changes. This shows that the present

model compares reasonably with existing correlations.

One reassuring result of regression analysis is the

magnitude of the group characteristic volumes that were

calculated. Ratios of the group characteristic volumes

found here are very nearly equal to the ratios of the van

der Waals volumes for these groups. This may allow for

estimation of these group parameters. This type of regu-

larity was not seen for the group characteristic tempera-

tures. Also, whereas the characteristic volumes for the

different groups did not vary much when determined from

different data sets, the characteristic temperatures did.

As a result, the binary constants did not follow a pattern.

The binary constant was necessary in all cases to obtain

an optimal representation of the data, but the improvement

obtained by introducing this additional parameter was

minimal.

No calculations have been reported for chemical poten-

tial changes or for pressure changes for multicomponent

systems. The necessary formulae have been presented in

Chapter 4 and Appendix 5. Pressure change calculations

for multicomponent systems are no more difficult than those

reported here because the molecules considered contained

more than one group.
















CHAPTER 7
DISCUSSION


This work has centered on the development of a group

contribution liquid phase equation of state. Fluctuation

solution theory was used to develop the exact relationships

between thermodynamic derivatives and direct correlation

function integrals. Two approximations were made to actually

construct the equation of state. The first approximation

made was the use of the interaction site formalism, or

RISM theory. The -last stage in the model formulation was

the choice of an approximate form for the group direct

correlation function integrals. After that development,

the general expressions for calculation of pressure and

chemical potential changes were written and used for

several compounds.

In this chapter, the important aspects of each stage

in the development described above will be examined in

detail. Great emphasis will be placed on the approximations

embodied in the model development as they pertain to the

numerical results.










Fluctuation Solution Theory


The relationship between integrals of molecular correla-

tion functions and thermodynamic derivatives have come

to be known as fluctuation solution theory (O'Connell,

1981). Kirkwood and Buff (1951) originally reported these

relationships though they have not been extensively used.

The original results, using total correlation function

integrals, are valid for both spherical and molecular sys-

tems. With the introduction of the direct correlation

functions, through the Ornstein-Zernike equation, the analy-

sis is not as simple. Appendix 1 details a procedure which

surmounts all difficulties encountered. This result is

not new, but the rigorous proof appears to be. This result

could also be derived through the use of the operator tech-

nique of Adelman and Deutch (1975).


Approximations


While equations 2-24 and 2-25 are exact results for

fluctuation derivatives, one must have values for the direct

correlation function integrals for these to be useful.

Because exact values of the required integrals are not

generally available, some approximations must be used to

obtain these values. The approximations used will determine,

to a large degree, the success of any thermodynamic calcula-

tions. In this work, two levels of approximation are used

and they will be discussed separately below.










Group Contributions


One of the major goals of this research was the develop-

ment of a group contribution formulation of fluctuation

solution theory. This appears to be a realistic and rational

goal. This is true because fluctuation solution theory

requires knowledge of correlation functions and correlations

among the groups are well defined. The major problem

encountered was the ability to separate the inter- and

intramolecular correlations. The RISM theory (Chandler

and Andersen, 1972) purports to affect this separation

and offer an approximate relationship between group direct

correlation functions and molecular correlations. The

calculations reported here are based on this original version

of the RISM theory.

Recently, the direct correlation functions calculated

using the RISM theory have become the objects of considerable

attention (Cummings and Stell, 1981, 1982b; Sullivan and

Gray, 1981). It has been shown that in many situations

for neutral molecules that the site-site direct correlation

functions are necessarily long-ranged. This means that

these functions are nonintegrable and thus no compressibility

relation involving these functions may exist. Cummings

and Stell (1982a) have shown how a general compressibility

relation can be derived in terms of a limiting operation,

but this is not a practical solution. However, for the

cases that have been considered in the literature it seem










this problem can be overcome. If the compressibility rela-

tion is viewed as the k o limit of a result in Fourier

space, where all of the correlation function transforms

exist, then all of the mathematical manipulations required

may be performed. Then as the long wavelength limit is

taken, we find (Appendix 7) the following relationship:




~PX = l-p XiXji iviB C, (o) (7-1)
DP T,X ij ab


While the individual functions C a(r) may be long-ranged,

it appears that the collection of terms wcy is well defined

in the k o limit. It is known (Cummings and Stell, 1982a)

that for diatomic molecules this is true, and thus a simple

compressibility theorem applies. Also for triatomics,

as examined by Cummings and Stell (1981), we find that

this summation of terms is nondivergent. As has now become

known, the possible divergencies in the C a(r) functions

are due to intramolecular effects that are not shown

explicitly. It appears that the projection by the v matrix

removes the divergent terms. This is analogous to the

ionic solution case (Perry and O'Connell, 1984) where the

charge neutrality constraint removes the divergent part

of the direct correlations.

Chandler et al. (1982) formulated the proper integral

equation theory for site-site correlations to offer an

exact formulation involving nondivergent CB(r) functions









that here are labeled C (r). These new functions are

related to the RISM C B(r) functions and are made nondiver-

gent by removal of intramolecular correlations involving

only sites a or 6. This requires use of auxiliary functions

SaB(r) that have a complicated density dependence. These

functions do possess the interesting property



eaB(o) = 0 Va,B (7-2)



It is then possible to derive a compressibility relation

involving only C (r) function integrals and 0



(1P 1 X.X. v. j
ST,X ij a s


J dr C C (r) (7-3)



In this case, the integrals always exist. However, we

have no knowledge of the functional dependence of 0 on

temperature or density.

One further aspect of the group contribution approach

used here needs to be mentioned. This formulation works

on the Ansatz that



cij(l,2) = I jia Ij c Cs B) (7-41)
jS a










which is only to be considered an approximate relation.

However, the compressibility relation obtained from the

above starting point and that found using the proper integral

equation formalism (as shown in Appendix 7) are the same.

The only difference is that all forms of the RISM theory

assume that each group, as opposed to each type of group,

have separate correlation, i.e., that the set of functions

cij r) should be considered, not just caB(r). This is

definitely true at the level of the correlation functions

themselves. However, it is always possible to define a
-)* aS -*
set of c C(r) functions that are averages of the c (r)

functions (Adelman and Deutch, 1975) and retain the same

compressibility relations.


Modeling of Direct Correlation Function Integrals (DCFI)


As the DCFI are functions of temperature and density,

it would be possible to model them using polynomial expan-

sions for the pure fluids and a solution theory for the

mixture quantities. In this work, we attempted to use

fluctuational forms that have a theoretical basis. This

approach seems to have both merits and demerits. For mole-

cules, Mathias (1979) had success in modeling DCFI using

an approach suggested by perturbation theory. In the present

case this is not of the greatest value because analytic

forms are available for group DCFI in only some limited

cases (Morris and Perram,1980; Cummings and Stell, 1982a),







79

and even then, the solutions are not in closed form. Thus,

some reasonable assumptions must be made to proceed with

the modeling.

It is known that the RISM correlation functions can

be related to Percus-Yevick correlation functions for

molecular species in some limiting cases (Chandler, 1976).

This led to the use of the form for the DCFI presented

in Chapter 5. The important aspect of this approach is

that the groups are dealt with as if they were all indepen-

dent. This is not physically the situation. The dependence

is caused by the intramolecular correlations, and they

must be properly accounted for to yield an accurate model.

Originally it was believed that the RISM theory took account

of this behavior, but as was mentioned in the preceding

section, this is not true. Even the proper integral equation

direct correlation functions, equation 7-3, while short

ranged, still involve intramolecular correlations due to

third body effects. Cummings and Sullivan (1982a) explicitly

show that while the h (r) functions are purely intermolecu-

lar quantities that the c (r) are not.

The exact resummation of the cluster expansion for

h o(r) shown by Chandler et al. (1982) suggests a way around

the aforementioned difficulties. It should be possible

to define a set of direct correlation functions, labeled

here as ca (r), that include only intermolecular effects.

Chandler's (1976) analysis can be examined to see that










these diagrams can be isolated. And then, in the same

fashion as was used to eliminate the long-ranged behavior

of the original RISM direct correlation functions, it would

be possible to write an exact proper integral equation

of the form



S+ Ph = [I Pc ] (7-5)



where the elements of the W matrix would contain all infor-

mation about intramolecular effects. It is then easy to

show the relationship between the different classes of

direct correlation functions, for example


y 1 -1 -1
= c + -( ) (7-6)



The elements of the W matrix would be functions of

temperature and density, but the exact functionality could

not be determined in general. It is possible to identify

some of the properties of the new functions, such as



olim W = (7-7)



and



klm = W V (7-8)
0 ( a3 a,8










Using equations 7-5 and 7-8, in conjunction with the

techniques of Appendix 7, a compressibility relation can

be derived in terms of these new group direct correlation

functions



(3P 1 p .
ST,X ij a i


J dr C (r) (7-9)



All of the above analyses show that there is no simple

solution to the modeling problem. One may either work

with a simple compressibility relation written in terms

of DCFI whose behavior is not certain, or in terms of well

behaved DCFI but have another unknown quantity present.

The former approach was taken for this work and is discussed

below.


DCFI Model


The rationale behind the development of the DCFI model

has been presented in Chapter 5. It is of interest to

note that the proposed form is a group variation of a van

der Waals, or mean field, model. Appendix 2 details how

higher order correction terms could be appended to the

proposed form.

Because the DCFI model was formulated as a correspond-

ing states correlation, the constants in the temperature










dependence of the hard sphere diameters and perturbation

terms had to be determined using experimental data. Volu-

metric data for both argon and methane were used to accom-

plish this task. The results of the compression calculations

show that the methane-based correlations were definitely

superior for representation of the properties of the larger

molecules.


Comparison Calculation


Chapter 6 presents the results of the group contribution

compression calculations for the n-alkane and methanol.

The results seem to indicate that the theoretical basis

is feasible but that the accuracy is not that desired.

It is an accomplishment to be able to perform the compression

calculations for all of the n-alkanes using only five

parameters, but the overall accuracy is not high. One

would hope to be able to calculate pressure changes with

an accuracy of about 1% (this allows for density change

calculations accurate to -0.2%), and this cannot be assured.

The theoretical basis behind the equation of state

development is now on firm footing. Our compressibility

relation is exact; one must simply have proper models for

the functions involved. Here, this appears to not be the

case. Because we use the RISM approximation, the c a(r)

functions must contain intramolecular effects that have

no analogy at the molecular level. Thus, our model that







83

is based on an analogy to a previously successful molecular

approach appears inadequate. Note, however, that we seem

to have little recourse. Because if we were to introduce

easily modelable cB8(r)'s, then the aspect of determining

the W function must be addressed, and this is still an

unsolved problem.


Summary


This study has revealed many interesting aspects about

site-site correlation functions and their relationship

to thermodynamic derivatives. A group contribution liquid

reference state equation of state has been formulated and

tested. The results are encouraging enough to warrant

further investigation but not accurate enough for practical

density calculations. It appears that the major problem

associated with the present approach is the inability to

model the required functions, and suggestions have been

made as to how one can alleviate that problem.
















CHAPTER 8
CONCLUSIONS


The goal of this work was the development of a group

contribution technique for calculation of liquid phase

properties of mixtures. The general results are available

but have only been tested on a few pure components. The

calculated pressure changes for four n-alkanes and methanol

are in general in error by less than 7%. This is reasonable

but not of high enough accuracy for process engineering

calculations. While the results of this study are not

outstanding, there are several interesting conclusions

that can be drawn.

It has been shown that it is possible to derive thermo-

dynamic property derivatives from a RISM theory. A general-

ized compressibility theorem was proven, and this was used

along with a model, based on a rigorous perturbation theory,

to develop a van der Waals equation of state.

The liquid phase equation of state was applied to

the representation of compression measurements of argon

and methane. It was shown that proper choice of the

repulsive contribution to the equation of state is

important. Also, that an optimum repulsive contribution










could be determined from a new hard sphere equation of

state.

The use of the group contribution model was limited.

Pressure changes could be calculated for both long- and

short-chain alkanes using a five-parameter model. The

results were of reasonable accuracy for these cases and

when the model was applied to methanol. While the volumetric

parameters in the model appeared to correlate well with

previous molecular results (Mathias, 1979), the temperature

parameters followed no pattern.

An analysis of the present work also shows several

areas that require future work. First, the present model

should be applied for calculation of chemical potential

changes for n-alkane mixtures. This, along with mixture

volumetric calculations, would be a strong test of the

model's ability. Secondly, some effort to identify the

intramolecular correlation functions of the form shown

in Chapter 7 may be required to improve the present model.

This appears to be the weakest aspect of the present work.















APPENDIX 1
FLUCTUATION DERIVATIVES IN TERMS OF DIRECT CORRELATION
FUNCTION INTEGRALS


To eliminate the difficulties associated with the

form of the general O-Z equation we define an angle averaged

total correlation function, by



= d2 d2 h i(1,2) (Al-l)



and then we implicitly define a new set of direct correlation

functions, the by an O-Z type equation



=



+ j k dR3 (Al-2)



Now, because the full h.. (1,2) is translationally invariant,

the average function must also be, and thus the
13
function must also possess this property. Then if we

integrate equation Al-2 over one coordinate we have



=
l] + < Jk ds (Al-3)
k ik
And if this is then integrated once more we find

86











= +
k


where we define


= 1 dR


= 1 1 dR
1j V i]


(Al-4)


(Al-5a)



(Al-5b)


and when these terms are collected in matrix form we find


= +


(Al-6)


Here, the terminology is the same as that used in Chapter

2 for the definition of the matrix elements. This relation

becomes most useful when it is realized that


= H


(Al-7)


as can be shown by direct substitution. And now rearrange-

ment leads to


[N + NHN]- = N-1 ]


(Al-8)


which shows that the fluctuation derivatives can be written

in terms of the integrals of the functions. The
1]








relation between cij and is not obvious but can be

derived. By definition


=


dR k dR3 Cik R1R3>kj 3,R2


and, if the O-Z equation is angle averaged, we find


= di d .2 cij(l,2)


+ -k f d3 ddd2 c i(1,3)h k(3,2) (Al-9)
kv-7 1 2 ik kj
k VQ

Now, when these two relations are combined, the link between

the angle averaged c.. and the functions are given

as


= ds d C (1,2)


+ [ 3 dihld cik(l,3)hkj(3,2)
k v3 2


J dR3 (Al-10)


And now insert the definition of to find









= d d c (1,2)


+ 3 I d3d~ld 2 cik(1,3)hkj(3,2)
k v1 2


N2 dd2hkj(3,2) (Al-11)
k V'2 2 ik 3 kj


and the terms can be grouped as


d d 2 ci (1,2)


1
<= N d3d2hkj (3,2) {1 dn c (1,3)- k VQ

(Al-12)


Now, use the fact that


= J d2 (Al-13)


to rewrite equation Al-12 as


11
1 d {

= --- + d3d(2 hkj(3,2) {- I dl cik(1,3)
k VQ

} (Al-14)


It is now advantageous to define




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