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THESlS

This is to certify that the

thesis entitled

A Study of the Simplified Local Density Model

presented by

Hena Pyada

has been accepted towards fulfillment
of the requirements for

Master's degreein Chemical Engr.

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A STUDY OF THE SIMPLIFIED LOCAL DENSITY MODEL

By

Hena Pyada

THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Chemical Engineering

1994

ABSTRACT

A STUDY OF THE SIMPLIFIED LOCAL DENSITY MODEL

By

Hena Pyada

The simplified local density (SLD) model of adsorption [Rangarajan B.,PhD
Thesis, Michigan State University, 1992] is modified by using the Peng -
Robinson equation (PRSLD) to describe fluid structure. The adsorption
isotherms predicted by the two SLD models were compared with
experimental data for different combinations of fluids and system
conditions. For the same solid - fluid interaction parameter efw, the PRSLD
predicted higher surface excess values. The predicted and experimental
isotherms showed qualitative agreement. The quantitative agreement
depended on the value of £fw parameter. For ethylene on graphon the
PRSLD predicts isotherms in good agreement with experiment for the same
value of cfw over a wide range of temperatures (263 - 313 K).

In another approach, called the IE approach the SLD density profile is used
as an initial guess to solve an integral equation iteratively. The resulting
density profile is used to calculate surface excess values. The SLD and IE
adsorption isotherms were compared to determine conditions when the
SLD is a good approximation. The SLD performs well when there is a

strong fluid - solid attraction and when pressure is low (Pr < 0.6).

List of Tables
List of Figures

Chapter 1

Chapter 2

TABLE OF CONTENTS

Introduction

Theories Of Adsorption

Adsorption

Kinetic Theories
Langmuir theory
BET theory

Potential Theories
Polanyi theory
Dubnin-Radhuskevich equation
FHH theory

Equation Of State Approach
Hill-deBoer theory
Model of Barrer and Robins
Simplified local density model

Chemical Physical Models
Lattice gas model
Distribution function approach
Integral equation theories
Density functional theory

Summary

Comm-QQCJ'IUIAA

HHHHHHHHHI—I
CDQODAfirhODNI—‘O

Chapter 3

Chapter 4
Chapter 5
Appendix A

Appendix B

Appendix C

Bibliography

The Simplified Local Density Model
Fluid-Solid Potential
Fluid-Fluid Potential
Reduced distance units : ETA
and BETA
Modified SLDzPRSLD
The Integral Equation
Summary and Conclusion
Details of solving Integral Equation
Program code for calculating
Surface excess using PRSLD.
Program code for calculating

Surface excess using density profile

calculated from vdWSLD.

II

22
22
23

27
27

38
61

70

105

Table 3 - 1:

Table 4 - 1:

Table4-2:

Table 4 - 3:

Table 4 - 4:

Table 4 - 5:

Table 4 - 6:

LIST OF TABLES

A comparison of the experimental values of
bulk saturation liquid densities for ethylene
(Tc = 282.4 K, Pc = 50.4 bar) with values
predicted by the vdW and PR equations of
state.

"X" values for Water (Tc = 647.3 K, Pc = 22.34
bar) and Ethylene (Tc = 282.4 K, Pc = 51.07
bar) at different reduced temperatures.

The Ofi‘, 23* values for ethylene, ethane and
carbon dioxide.

Different reduced temperatures (in K)
selected for ethylene, ethane and carbon

dioxide.

8 values for ethylene, ethane and carbon
dioxide.

The % error values for Carbon dioxide at Tr =
0.95 for different values of the ratio ewl 83.

The % error values for Ethylene at Tr = 0.95
for different values of the ratio aw/ Eff.

III

30

42

45

45

47

Table4 — 7:

Table 4 - 8:

Table 4 - 9:

Table 4 - 10 :

Table 4 - 11:

Table 4 - 12 :

Table 4 - 13 :

The % error values for Ethane at Tr = 0.95 for
different values of the ratio sw/ Efi‘.

The % error values for Carbon dioxide at Tr =
1.05 for different values of the ratio sw/ eff.

The % error values for Ethylene at Tr = 1.05
for different values of the ratio ew/ Eff.

The % error values for Ethane at Tr = 1.05 for
different values of the ratio 8w / Efi‘.

The % error values for Carbon dioxide at Tr =
1.10 for different values of the ratio sw/ Efi‘.

The % error values for Ethylene at Tr = 1.10
for different values of the ratio ew/ £3.

The % error values for Ethane at Tr = 1.10 for
different values of the ratio ew/ Eff.

IV

48

49

49

50

50

51

51

Figure2-1:

Figure 3-1 :

Figure 3-2 :

Figure 3-3 :

Figure 3-4 :

Figure 3-5 :

LIST OF FIGURES

A brief classification of adsorption models.

Adsorption isotherms predicted using the
vdWSLD & PRSLD models for Carbon dioxide
(Tc=304.1 K, Pc=73.8 bar) for the ratio cw / SFF

= 0.25 .

Adsorption isotherms predicted using the
vdWSLD & PRSLD models for Ethane (Tc =
305.4 K,Pc = 48.8 bar) for the ratios cw / EFF =

0.25.

Adsorption isotherms predicted using the
vdWSLD & PRSLD models for Ethylene (Tc =
282.4 K, Pc = 50.4 bar) for the ratios 8W / EFF =

0.25 .

Comparison of the density profiles generated
by vdWSLD & PRSLD for Ethylene (Tc=282.4

K, Pc= 50.4 bar) on a solid at Tr=0.95 and the
ratio cw / EFF = 2.0 .

A comparison of adsorption isotherms
predicted using the vdWSLD & PRSLD models
for Ethylene (Tc=282.4 K, Pc=50.4 bar) versus
the experimental data at 263K.

31

32

Figure 3-6 :

Figure 3-7 :

Figure 4-1 :

Figure4-2:

Figure 4-3 :

Figure 4-4 :

Figure 4-5 :

A comparison of adsorption isotherms
predicted using the vdWSLD & PRSLD models
for Ethylene (Tc=282.4 K, Pc=50.4 bar) versus
the experimental data at 293K

A comparison of adsorption isotherms
predicted using the vdWSLD & PRSLD models
for Ethylene (Tc=282.4 K, Pc=50.4 bar) versus
the experimental data at 313K.

Example of a typical density profile generated
by the SLD model for the case of vapor liquid
interface

Example of a typical density profile generated
by solving the IE for the case of a vapor liquid
interface.

Comparison of the density profiles generated
by SLD & IE for Ethylene (Tc=282.4 K, Pc=50 .4
bar) on a solid at Tr=0.95 and the ratio 8W /
EFF = 0.1.

Adsorption isotherms predicted using the IE
& vdWSLD models for Ethylene (Tc=282.4 K Pc
= 50.4 bar) for the ratios aw / EFF =2.0 .

Adsorption isotherms predicted using the IE
& vdWSLD models for Ethane (Tc=305.4 K,
Pc=48.8 bar) for the ratios cw / EFF =2.0 .

VI

36

37

40

41

52

53

Figure 4-6 :

Figure 4-7 :

Figure 4-8 :

Figure 4-9 :

Figure 4-10 :

Figure 4-11 :

Figure 4-12 :

Adsorption isotherms predicted using the IE
& vdWSLD models for Carbon
dioxide(Tc=304.1 K, Pc= 73.8 bar) for the ratios
8W / EFF = 2.0 .

Effect of temperature on % error for the ratio
8W / EFF = 1.0 , for Ethylene (Tc=282.4 K, Pc=

50.4 bar).

Effect of temperature on % error for the ratio
cw / EFF = 1.0 , for Ethane (Tc=305.4 K, Pc=48.4

bar) .

Effect of temperature on % error for the ratio
cw / EFF = 1.0 , for Carbon dioxide (Tc=304.1 K,

Pc= 73.8 bar).

Effect of the ratio cw / EFF on % error at Tr
=1.05 for Ethylene (Tc=282.4 K, Pc = 50.4 bar).

Effect of the ratio 2w / EFF on % error at Tr
=1.05 for Ethane (Tc=305.4 K, Pc = 48.4 bar).

Effect of the ratio aw / app on % error at Tr

=1.05 for Carbon dioxide (Tc=304.1 K, Pc =73.8
bar).

VII

54

55

56

57

58

59

CMEB 1

INTRODUCTION

Adsorption at high pressures is important in supercritical

extraction from solids, in pressure swing adsorption of non-ideal gases and
for regeneration of adsorbents with high pressure gases in the area of
waste treatment, fermentation and food processing industries. There is a
need for a simple theory of adsorption which gives good predictions of
adsorption behavior for engineering calculations. The theory of adsorption
by Langmuir (1918) is simple to use but it predicts only monolayer coverage.
Realistic theories must predict multilayer coverage. The BET equation of
Brunauer et al., (1938) is a well known multilayer model but it is applicable
only to ideal gases and can not be used above the critical temperature of the
gas. Another multilayer theory based on the Polanyi equation developed by
Dubinin and coworkers (1960) is derived using empirical assumptions.

The engineering models based on principles of chemical physics give
promising results. The models of Sullivan (1979), Teletzke (1982) and
Tarazona and Evans (1984) which are density functional theories predict
multilayer coverage but involve the solution of integral equations, which
can be tedious and time consuming.

Somewhere between the empirical theories and the rigorous
chemical physical models lie the adsorption theories based on equation of

state approach. This approach simplifies the description of the fluid

structure by assuming it to obey a particular equation of state, e.g. the van

der Waals' (vdW), the Redlich-Kwong (RK) or the Peng-Robinson (PR). The
model of Hill-deBoer [deBoer (1953) and Ross and Oliver (1964)] assumes
that the adsorbed fluid is a two-dimensional van der Waals' fluid.

Barrer and Robins (1951), developed an equation of state model for
multilayer adsorption which assumes that the chemical potential of a fluid
molecule at a distance '2' from the adsorbent surface is made up of the
following two terms:

(1) the chemical potential of the usual van der Waals' bulk fluid,
appropriate to the temperature T and local density p(z) (the local
density approximation);

(2) the fluid-solid interaction potential, which is assumed to be the
attractive part (2'3) of the integrated 9-3 Lennard-Jones (LJ)
potential.

The Simplified Local Density (SLD) model of Rangarajan (1992), is
based on the same principles as the model of Barrer and Robins, but uses a
better form of the fluid-solid potential (Lennard-Jones partially integrated
10-4 potential) and the attractive van der Waals' parameter varies with
distance from adsorption surface.

Actually, the van der Waals' contribution to the local chemical
potential does not depend on the local density [Hill (1952,1951)], but on the
entire density profile p(z). To calculate the actual van der Waals'
contribution an integral equation has to be solved. The rigorous solution of
the integral equation is very complicated. An iteration scheme for solving

the integral equation was developed which uses the density profile obtained

from the SLD as a first guess. This approach shall be referred to as the
Integral Equation (IE) approach. Though the IE presents a more accurate
picture of the forces involved in the adsorption process, it is still an
approximation. The objectives of this thesis are:

(1) To modify the SLD by using the Peng-Robinson equation of state to
calculate the fluid properties, this modified SLD will be referred to
as the Peng-Robinson SLD (PRSLD) and the original SLD will be
the van der Waals' SLD (vdWSLD).

(2) Compare the density profiles and surface excess curves developed
by the PRSLD with the vdWSLD.

(3) Investigate whether the SLD density profiles are good first
approximations for solving the IE.

(4) Study and compare the density profiles and adsorption isotherms

predicted by the SLD and the IE approaches.

A brief review of different theories of adsorption is presented in
Chapter 2. The SLD is discussed and modified using the Peng-Robinson
equation in Chapter 3. SLD and IE approaches are discused and compared
in Chapter 4. Chapter 5 summarizes the conclusions reached based on

these studies.

CHAPTER 2

THEORIES OF ABSORPTION

A s i n
Sorption is the general term used to define the phenomena in
multi-component, multi-phase systems when at least one component has a
higher concentration at the interface (between the two different phases)
than in any bulk phase. Adsorption is the special case when the sorbed
molecules are unable to penetrate into the second bulk phase. Adsorption
processes are divided into two areas, physical adsorption and
chemisorption. Physical adsorption or physisorption results from
intermolecular forces, where as chemisorption results from chemical
bonding between fluid (adsorbate) and solid (adsorbent).
Physical adsorption has been the focus of several published studies
and reviews. Some of the well known reviews are by Brunauer (1945),
Young and Crowell (1962), Ross and Oliver (1964), Defay and Prigogine
(1966), Flood (1967) and Nicholson and Parsonage (1982).
Adsorption may be defined by the surface excess I‘ex = nex/Ae = excess

moles per unit area,

rex =J' (p(z)-p(b)) dz 2 - 1
20

where, p(z) is the density of the fluid at a distance 2 from the wall and p(b) is
the bulk density of the fluid, 20 is the plane above the surface of the solid
where the gas-solid potential is zero. The surface excess is defined as the
number of excess moles per unit area of adsorbent.

To test the accuracy of different adsorption models, reliable
experimental data are needed. Among the best data available at high
pressure are those collected by Findenegg and co-workers (1983) for gases
like krypton, argon, methane, ethylene and propane on graphon.

The following is a summary of some important adsorption models.
All the models presented are for adsorption of a pure fluid (mainly gases)
adsorbing on a solid adsorbent. Some of the different theories discussed

here and a rough guide to the way they are related is shown in Figure 2 - 1.

KINETIC THEORIES:
The Langmuir theory and the BET theory were derived using a kinetic
approach. The BET equation is often used for surface area and pore size

distribution measurements of adsorbents.

Langmuir Thegu;
The Langmuir theory [Langmuir, ( 1918)] is one of the first theories.
It has a two parameter equation
O=W/Wm=kP/(1+kP) 2-2
where, W is the mass adsorbed, Wm is the amount adsorbed in a mono-
layer and k is an empirical constant. The Langmuir theory predicts mono-

layer coverage and adsorption isotherms exhibited by microporous

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Models of
Adsorption
1. Chemical 2. Equation 3. Kinetic 4. Potential
Physical of State theories theories
theories approach
1. Chemical
Physical
theories
Lattice Gas Distribution Density Integral
theory function functional Equation
approach approach theories
Theories Born-Green-
based on the Yvon
closure of the equation
Ornstein-
2. Equation 239ml“
of State equation
approach
I 3. Kinetic
J theories
Two Three
dimensional dimensional
J ] Lagieggnuu BET theory
Hill deBoer Barrer and Simplified ry
theory Robins local density
theory model
4. Potential
theories
1 7
Polanyi Slab theory Dubinin-
theory Radhuskevich
equation

 

 

 

Figure 2 - 1 : A brief classification of adsorption models.

 

 

 

adsorbents and thus is more valid for chemisorption applications. It
assumes that molecules are adsorbed at specific sites on the surface and
vibrate around these sites. Adsorbed molecules do not move laterally, i.e.

adsorption is localized.

W

This theory was proposed by Brunauer, Emmett and Teller
(Brunauer et al., 1938) and it extends the Langmuir isotherm to multi-
layers.

W/Wm = N/Nm = C(P/Po) / [(l-P/Po)(l-P/PO+C(P/Po)] 2 - 3
where, Po is the vapor pressure of a pure fluid, N is the number of moles
adsorbed and C is a constant related to the fraction of surface that is
uncovered.

The theory assumes that molecules adsorb in stacks or layers and
that the upper most molecules in the adsorbed stacks are in equilibrium
with the vapor. The first adsorbed layer is treated separately and all the
layers above the first layer are condensed and treated as a liquid. Other
assumptions are that the surface is homogeneous, there are no lateral
interactions between adsorbed molecules i.e. localized adsorption and the
fluid is an ideal gas. The BET equation is quite successful in the range 0.05
< P/Po < 0.35 and sub-critical temperatures of the fluid.

POTENTIAL THEORIES:

The potential theories assume that a potential field exists on the surface of

the solid which attracts the fluid particle. The oldest of these is the Polanyi

theory which states that the cumulative volume of the adsorbed space is a
function of the potential energy of the surface. This function is called the
characteristic curve. Dubinin postulated different forms of the
characteristic curve for different types of adsorbents. The Slab theory or the
Frenkel-Halsey-Hill theory (FHH) gives an approximation of the form of the
adsorption potential. This form is valid at large thicknesses of the adsorbed

layer.

Polanyi Theory;
The Polanyi equation (Polanyi, 1914) can be written as follows :
ln(P/Po)=ei/kT 2-4

where, 8i is the potential energy of an equipotential surface enclosing a
volume Vi which is being filled at a relative pressure ( P/Po ) and k is the
Boltzmann constant. The cumulative volume of the adsorbed space is a
function of ti. This function is characteristic of the particular gas-solid
system and hence is referred to as the characteristic curve. The
characteristic curve is independent of temperature since the adsorption
potential expresses the work of temperature independent dispersion forces.
If the characteristic curve is determined at one temperature it is possible to
predict adsorption isotherms at other temperatures for the same gas-solid
system. One of the drawbacks of this theory is that 8i is a free energy and

not a potential energy and therefore it is a function of temperature.

Duhinin-Radhgskevich Equation

The approach of Dubinin and his co-workers ( 1960), is empirical and
based on the Polanyi equation. They postulated the functional form of the
characteristic curve using semi-empirical functions for two types of
adsorbents. For microporous adsorbents like activated carbon the following
form, often called the Dubinin-Radushkevich equation is used:

W=Wo exp(-ke2/BZ) 2-5
where, W0 is the limiting volume of the adsorbed space, which equals the
micropore volume, B is the affinity coefficient characterizing the
polarizability of the adsorbate and k is the Boltzmann constant. The affinity
coefficient, 6, is intended to be a shifting factor to bring the characteristic
curves of all gases on the same adsorbent into a single curve. For
adsorbents with large pores like carbon black, the following characteristic

curve is used:

W=Woexp(-ke/B) 2—6

FHH Thgzg
The Slab theory [Frenkel (1946), Halsey (1948, 1952) and Hill

(1949)]also called the Frenkel-Halsey-Hill (FHH) theory allows the
adsorption potential to vary as the inverse cube of the distance from the
surface. The simplest form of the FHH isotherm is:
ln(P/Po)=-A81/n3kT 2-7
where, n is the layer number, -Ael is the net interaction energy of the
first layer and k is the Boltzmann constant. Layers are assumed to develop

at discrete distances from the surface, thus the isotherm predicts that

10

discrete steps exist, a fact that can be found in only a very few experimental
cases.

In order to predict realistic multilayer isotherms the solid surface
has to be heterogeneous, which means that the adsorption potential can no
longer be a function of distance from the surface. One has to add the
tangential variation of the potential. This changes the n3 dependence to a 91‘
dependence where 6 is the surface coverage and r is a measure of the rate
of decay of the gas-solid interaction energy. The value of r is slightly less
than 3.

The heterogenity of the adsorbate can be handled by a perturbation
technique. When the solid surface is homogeneous the 91’ dependence
becomes ()3 i.e. r = 3. Singelton and Halsey (1954) included a perturbation of
the adsorbate by introducing two parameters:

1. A lateral interaction parameter, w.
2. A factor 'g' which measures the compatibility of the adsorbed film with
the bulk adsorbate when perturbed by the adsorbent.

They proposed the following isotherm:

ln(P/Po)=-Ae,/03kT+w(1-g)/kT 2-8
where 0 is the surface coverage and k is the Boltzmann constant. The

empirical nature of parameters w and g is a major limitation of this theory.

EQUATION OF STATE APPROACH:
The theories based on the equation of state approach simplify the
description of the fluid structure. The Hill-deBoer model assumes that the

adsorbed layer can be represented by a two-dimensional van der Waals'

11

fluid. The model of Barrer and Robins and the simplified local density
model (SLD) assume that the fluid obeys the vdW equation of state and the
fluid-wall potential used is some form of the integrated Lennard-Jones
potential. Equation of state-based isotherms have been successfully used to

model supercritical adsorbers [Akman and Sunol ( 1991)].

Hill-deBoer Theory
The Hill-deBoer theory is discussed in detail in the books by deBoer

(1953) and Ross and Oliver (1964). In this model, the adsorbed film is
considered to be a two dimensional gas. Unlike the Langmuir theory, the
Hill-deBoer theory allows the adsorbed molecules to move laterally within
the 2-D phase and desorption can take place from anywhere on the surface.
Hence in this case the film is mobile.
The two dimensional gas is represented by a two dimensional van der
Waals equation of state
(n+a/02)(o-B)=kT 2-9
where, n is the two dimensional spreading pressure, 0 is area per
molecule, a and B are the 2-D analogs of a and b in the three dimensional
van der Waals' equation of state and k is the Boltzmann constant.
The isotherm equation is :
p=K0/(1-6)exp[0/(1-B)-2a6/(kTB)] 2-10
where, p is the pressure, K is a constant and 6 is the fraction of surface

covered by adsorbed molecules.

Th M l f B rr r n in :

Barrer and Robins (1951), assumed that the adsorbate fluid obeyed
van der Waals' equation of state and the interaction energy between the
adsorbent and the fluid could be approximated by the integrated 9-3
Lennard Jones potential (the repulsive part was neglected). At equilibrium,
the chemical potential of the bulk gas lib is equal to the chemical potential of

the adsorbed gas it, i.e.,

l1 = Pb 2 - 11
0/(1-0) eXp( 0/(1—0) -2a0/bRT - CN/r3RT) = Ob/(l-Bb) eXp(0b/(l-9b) -2a0b/bRT)
2 - 12

where a and b are van der Waals constants, R is the universal gas constant,
T is the temperature, N is total number of molecules in the system, 11 is the
total number of moles in the system, 0 is bn/V; the fraction of the total
volume V occupied by closely packed molecules, 9b is bn/Vb; the fraction of
the total volume Vb occupied by closely packed molecules in the bulk of the
fluid, r is the distance of the fluid molecule from the surface and C is an
empirical constant.

Let v equal the molar volume of the fluid in contact with the
adsorbent and let vb equal the molar volume of the bulk fluid, then equation
2 - 12 may be rewritten in the more conventional form as follows :

1/(v-b) expl b/(v-b) - 2a/vRT - CN/13RT] = l/(vb-b) exp [ b/(vb-b) - 2a/vaT] 2 -13

This model predicts isotherms that give qualitative agreement with
experimental results but in order to get quantitative agreement one needs to
manipulate the value of C. One possible reason for this quantitative

disagreement is the form of the fluid-wall interaction potential. Steele (1969)

13

and Lee (1990) show that the 9-3 Lennard-Jones potential under predicts the
interaction energy and suggest the use of the partially integrated 10-4
potential. Hill (1951) suggests further improvements for this model. Hill
(1952), replaces the repulsive part of the van der Waals equation by Tonk's
expression for hard spheres. Also the chemical potential at a distance r'
from the surface is not determined from the local density contribution (as in
the case of Barrer and Robins model) but from the entire density
distribution. The disadvantage of this treatment is the fact that a nonlinear

integral equation must be solved.

Th im lifi al Den i M 1'

This approach was developed by Rangarajan( 1992). It is similar to
the model of Barrer and Robins. The basic assumptions of this model are :
1) The fluid obeys the van der Waals' equation of state.

2) The fluid-wall potential is in the form of the 10-4 integrated Lennard-
Jones potential, both the attractive and repulsive parts are considered in
calculations.

3) The attractive van der Waals' parameter is forced to vary with distance
from the adsorbing surface.

The model gives good qualitative predictions of adsorption isotherms,
density profiles of the adsorbed fluid and phase transition near the surface
of the adsorbent. To get quantitative agreement, the fluid - solid interaction
parameter has to be adjusted. One reason could be that the van der Waals'
equation of state is not good at predicting properties of liquids. More details
of this theory will be discussed in Chapters 3 and 4.

14

CHEMICAL PHYSICAL MODELS :
These models are based on the principles of statistical mechanics. The
Lattice Gas model predicts only monolayer coverage. The distribution
function approach of Ross and Steele was one of the first multilayer
theories. As the theory of liquid states advanced; a number of approaches
based on solving the integral equations for the radial distribution function
were developed. An entirely different approach is the various density
functional theories. Vanderlick et a1. (1989), have compared different
density functional calculations with the BGY method. Evans et al. (1983)
lists advantages and disadvantages of different density functional and

integral equation theories.

LaiiicafiaaMudeh

In the case of localized adsorption, the surface of the solid has sites
on which fluid molecules are adsorbed. The adsorbed molecules vibrate
around the sites. These sites are considered to be lattice points. This is the
Lattice Gas Model which has been studied by Pandit et al. (1982). With
simplifying assumptions this model gives the Langmuir isotherm and the

BET isotherm.

is ri i n f n i n r
The distribution function approach was developed by Steele and Ross
(1960). They treated a one-component fluid with two-body forces in an
external field (the external field is generated by interaction between wall

and fluid molecules) which is a function of position only. The total potential

energy of the system of N atoms is

U (r1,...,rN) = Z uslri) +2u(rij) 2-14
where, us(ri) is the energy of the ith atom at position ri due to the external
field and 11(l'ij) is the mutual interaction energy of atoms i and j in the fluid
a distance l'jj apart.

For Steele's approach the singlet probability distribution function
p1(r1) and the pair probability distribution function p2(r1,r2) are considered.
The potential energy of the ith molecule at its position ri is given by

U(ri) = p1(ri)us(ri) + I p2(ri,rj) u2(ri,rj) drj 2 - 15

The relationship between the distribution function and the local
pressure tensor Pxx(ri), Pyy(ri) and Pzz(ri) are derived. The component
Pzz(ri) is normal to the plane of the surface of the fluid and is equal to the
ordinary pressure exerted by a homogeneous phase in equilibrium with the
adsorbed phase.

At equilibrium Pzz(ri) is independent of 21, therefore the partial
differential of Pzz(ri) w.r.t. 21 is zero, which results in an integral equation
relating p1(ri) to p2(ri,rj) in terms of the intermolecular forces. For given
intermolecular potential functions p1(ri) and p2(ri,rj) can be computed and
hence the number of molecules adsorbed can be calculated. The
calculations are exact for fluids with pairwise interactions. For low
densities the distribution functions can be expanded in terms of the activity,
which yields cluster integrals and virial expansions of equations. For
higher adsorbate densities (in the case of multi-layer adsorption) the theory
becomes more complicated. [Pierotti and Thomas (1971), Hill and

Greenschlag (1961), Hill and Saito (1961)].

16

Integral Equation Theories:

The fluid near the wall behaves like a liquid, as the fluid moves away
from the wall it behaves more like a gas. Therefore the equations used to
calculate the fluid local density should represent the behavior of non-ideal
fluids. One of the first attempts at calculating the local density was by using
a virial expansion of the local density in terms of the bulk density [Pierotti
and Thomas (1971)]. With advances in the theory of the liquid state, other
approaches were developed.

The central idea in most modern theories of the liquid state is the

radial distribution function, g(r). It is defined as [McQuarrie (1976)] :

V2 N ! I .J e-BUN dr3...drN
g(r) = 2 - 16
N2 (N-2)! ZN

where, there is a system of N particles in a volume V and at temperature T,

 

ZN is the configurational integral, r,,r2,r3... are coordinates in space and r
is the intermolecular distance between two coordinates. If the total potential
energy of an N-body system can be assumed to pair-wise additive, then all
the thermodynamic properties of the system can be derived in terms of the
radial distribution function.

A number of approximate equations have been derived for g(r). These
different equations are collectively known as Integral Equations. Two
approaches have been the most important; the hierarchy approach used to
derive the Kirkwood integral equation and the Born-Green-Yvon (BGY)
equation, and the direct correlation approach which includes the Ornstein-

Ziernike equation, which can be used to derive the Percus-Yevick (PY)

17

equation and the hypernetted chain (HNC) equation. Details of deriving
these equations and comparisons of their predictions are discussed by
McQuarrie ( 1976).

The use of the Ornstein-Ziernike equations for interface calculations
has limited success. For the case of hard spheres in contact with a hard
wall, the PY and the HNC closures yield density profiles which are in
considerable error close to the wall. In this approach the the wall must be
plane and cannot be allowed to show any atomic structure therefore this
approach can never be used to model realistic fluid - solid situations.

Fischer and Methfessel (1980), developed the Born-Green-Yvon (BGY)
approach to local densities of a fluid at interfaces. The local density of the
non-uniform fluid was calculated from the first equation of the BGY
hierarchy by modelling the pair correlation function. The mean force term
was divided into a repulsive part and an attractive part. The repulsive part
was approximated by a hard-sphere interaction and the pair correlation
function was taken locally. The particles were assumed to be uncorrelated
in the attractive part. They studied three cases : (a) the free liquid surface,
(b) gas adsorption on a wall at low temperatures, and (c) a liquid in contact
with a wall. In all three cases the results were in good agreement with

computer simulations.

The key concept in the density functional theory is that there exists a

functional 9 of the local number density p(r), such that the equilibrium p(r)

minimizes $2. This concept has been successfully applied to liquid helium

18

and the electron gas, and an exact form of the functional can be calculated
for classical fluids [Ebner et a1. (1976); Ebner and Saam (197 7) and Saam
and Ebner ( 1977)].

In the density functional theory, a Helmholtz free energy functional
F is minimized at equilibrium.

F = F (f(r), p(r), C(r1.r2) ) 2 - 17
where, f(r) is the local free energy, p(r) is the local density and c(r1,r2) is the
direct correlation function. At equilibrium F has a minimum value. This
condition gives an integral equation for local density p(r) in terms of the
fluid-fluid and fluid-wall potentials and the direct correlation function.

Sullivan (1979), developed the van der Waals' model for adsorption,
which can be considered to be an approximate form of the density
functional theory. Teletzke et al. (1983), generalized Sullivan's model and
applied it to predict wetting transitions for thin films. Tarazona and Evans
( 1984), developed a simple free energy functional which incorporated both
'local' thermodynamics and short ranged correlations.

Kierlik and Rosinberg (1990, 1991) proposed a simplified version of the
free-energy density functional for the inhomogeneous hard-sphere fluid
mixture, which requires four distinct weight functions and generates a
triplet direct correlation function for the one component fluid. This theory
gives results in good agreement with Monte Carlo simulation results and
performs well in predicting the density profile of a hard-sphere fluid in

contact with hard and soft walls.

SUMMARY

For the conceptual design of an adsorbent system, the knowledge of
adsorption isotherms is required for adsorbent selection, adsorption
configuration, selection of regenerant and in selection of pretreatment
methods. Of the considerable number of theories to predict adsorption
isotherms none are known to make satisfactory predictions in all
conditions. For engineering design purposes a model needs to give an
accurate representation of the equilibrium between the fluid phase and the
solid surface, the accuracy of the molecular interactions is not always
important.

This is why the early theories of adsorption such as the Langmuir
theory, the BET theory and the Polanyi theory are used in calculations, even
though their assumptions have limited validity. Unfortunately, these
theories have limited applicability which is mainly due to their invalid
assumptions. Another disadvantage is that in order to use any of these
models one has to perform experiments to generate data to evaluate the
parameters for the model. This can be a time consuming process which
can not always be justified for conceptual design of the system.

It would be desirable to have a general theory of adsorption which
can be applicable to all types of fluid-solid systems, this would require the
theory to describe the molecular interactions with more precision. The
interactions in an adsorption system is complex, it deals with matter
density ranges from gas to dense liquids and even solids, also the surface of
the adsorbent is heterogeneous and there is no uniformity of the potential.

Advances in the field of chemical physics has lead to several molecular

theories of adsorption which discuss all these complex molecular
interactions. Of the molecular theories, the different density functional
theories seem to hold the most promise. However due to the lengthy
calculations involved and the difficulty in obtaining parameters for
different substances required for successful applications of these theories,
their use in designing/modeling adsorption equipment has been avoided.
The equation of state based models are somewhere between the early
empirical models and the advanced molecular theories based on chemical
physics. These models recognize the idea that molecular interactions
between the adsorbent and the adsorbate are important to get an accurate
prediction of equilibrium conditions. However they simplify the description
of the fluid structure by choosing an equation of state to represent fluid
properties. This approach thus compromises accuracy of molecular
interactions with simplicity. For design purposes one is primarily
interested in the equilibrium characteristics and most popular equations of
states can give excellent representation of equilibrium properties inspite of
their inaccuracy on the molecular level. Therefore the model of Barrer and
Robins or the SLD can be used to predict adsorption isotherms which would
satisfy the needs for conceptual design of adsorption systems. One of the
parameters in Barrer and Robins' model requires experimental data to
determine it's value. The SLD does not require experimental data for it's
parameters: it uses Lennard-Jones parameters for the adsorbent and
adsorbate which are readily available or can be easily calculated using
different correlations and the critical properties of the fluid which are also

available in tabulated form.

21

Therefore the SLD model is a desirable candidate for predicting
adsorption isotherms for use in design of adsorption systems. All the
parameters used are easily available in literature so there is no need to
perform adsorption experiments to obtain parameters. Due to the
assumptions it makes on the molecular level it is applicable over a wide
range of pressures and temperatures and is not limited to ideal gases. This
thesis will study the application of the SLD and determine the range of

pressures and temperatures where it is most applicable.

CHAPTER 3

THE SIMPLIFIED LOCAL DENSITY MODEL

The simplified local density model was proposed by Rangarajan (1992). A
fluid molecule at any position from the solid interacts with the solid adsorbent
and with the bulk fluid molecules. At equilibrium, these interactions are
balanced. Therefore the chemical potential is uniform for the entire system.
According to Rangarajan ( 1992), the chemical potential of the fluid at a position
2 from the solid wall can be written as:

M2) = mazH mm = mo 3 -1
where, p(z) is the chemical potential, umz) is chemical potential due to fluid-
fluid interactions, ufw(z) is the chemical potential due to fluid-solid wall

interactions and lib is the chemical potential of the bulk fluid.

Th Flui - 1i P ni mg)

The fluid-solid potential is a function of the distance of the fluid molecule
from the solid, but for a given position it is independent of the number of
molecules at or around that position. The potential is also independent of the
temperature of the system. The fluid-solid potential chosen is the partially
integrated Lennard-Jones potential recommended by Lee (1990).

ufw(z) = 4 it pa wa 0fw6 { obs/5mm - 1/2*1/Zzi4 } 3 - 2
where, pa=density of solid, Efw and Ofw are the Lennard-Jones parameters for

the fluid-solid system, 2 is the distance from the solid and zi is the distance from

the ith layer of the solid.

The Lennard-Jones parameters for the fluid-solid system are determined
as follows:
0fw=(0a"+0w)/2 . 3-3
€fw=V/(8fi‘£w) 3-4
where, the subscripts ff and w represent pure fluid and pure solid parameters,
w is used instead of s to represent the fact that in the case of adsorption one is
interested in the surface properties of the solid adsorbent and often the solid

surface is referred to as a "wall".

The fluid-fluid putentigl ufl(_z_)
As the distance from the solid increases, the effect of ufw decreases.

When 2 is very large, the chemical potential p(z) equals the chemical potential

of the bulk fluid lib:

NZ) = m2) = lib 3 - 5
For a fluid the chemical potential can be given by:
lib = RTlnfb 3 - 6
where, fb is the fugacity of the bulk fluid given by
P P
ln(f/P) = I (v/RT-l/P) dP = f (Z-1)/P dP 3 - 7
O 0

where, v is the molar volume, T is temperature, P is the pressure, R is the
universal gas constant and Z = Pv/RT.
The integral can be determined for a known equation of state. Let the
fluid properties be represented by the van der Waals' equation of state, then
ln fb = ln(RT/(vb-b)) +b/(vb-b) -2ab/(vaT) 3 - 8
where, b is the close packed volume and ab is related to the attractive potential

between fluid molecules. The subscript b represents bulk properties. According

to the pressure equation [McQuarrie ( 1976)] the constant "a" can be written as:

a = (fie/6*] x(du/dx) g(x) 410:2 dx , x = r/o 3 - 9
where, u is the fluid-fluid interaction potential. In the van der Waals' equation
this is the Sutherland potential, which can be expressed as follows :

u(r) = °° r < o 3 -10 (a)

u(r)=s(o/r)6 0<r<°° 3-10(b)
where, r is the intermolecular distance and g(r) is the radial distribution
function. Substituting in equation 3 - 9 we get :

a=(4/3)rteo3 3-11(a)

If instead of the pressure equation approach, one were to use the partition
function approach, then we would have :

W=Hrep+llatt 3-11(b)
where limp is the contribution of repulsive forces and platt is the contribution of
the attractive forces. Rangarajan (1992) uses this approach to calculate m. The
difference between the pressure equation approach and the partition function
approach appears in the attractive term 'a'

a=(2/3)ateo3 3-11(c)

Another complication is the value of 8 and O. For the fluid - solid potential
Lennard - Jones parameters are used, but the vdW EOS assumes that the fluid
molecules interact according to Sutherland potential so the 8 and O values for
calculating 'a' are different. There is also some disagreement on the actual
value of these parameters for pure substances. To avoid these problems we
chose to calculate a and b from critical properties of the fluid as follows :

b=0.125RTc/Pc 3-12

a=27l64(RTc)2/Pc 3-13

The integral of equation 3 - 9, can be applied to any substance. van der
Waals' assumed that the integral was independent of the number density p and
the temperature T. The actual radial distribution function is a function of p and
T, but it's value oscillates around unity and therefore when integrated it shows
a weak density dependence.

For a homogeneous fluid the van der Waals' equation of state may be
derived from the pressure equation of statistical mechanics [McQuarrie (1976)].

P/ka = 1 - p/sk'rf r(du/dr) g(r) 41:13 dr 3 - 14

or P/pkt = 1 - (1039/6ka x(du/dx) g(x) m2 dx x=rlo 3 - 15
where p is the number density and k is the Boltzmann constant.

When the fluid is near the solid it is no longer homogeneous, therefore
the number density should be put back in the integral. The two body potential is
short ranged (~ r45) and the product of the density and the partial derivative of
the potential with respect to x appears in the integral. The major contribution to
the integral will be from the values of the potential and p near x. Therefore it
might be possible to use the local density at x and pull p out of the integral. This
is the local density approximation. This works because at distances far from x
the two body interaction potential is negligible so there is no significant
contribution to the integral.

Therefore at a distance 2 from the wall the chemical potential due to

fluid-fluid interactions will be:

ufl(z) - u, = RTln(f(z)/fo) 3 -16
ln(f(z)) = ln(RT/(v(z)-b)) + b/(v(z)-b) - 2a(z)/v(z)/R/T 3 -17
ln(f(z» =1n(fb) - NAMfw 3 -18

where, “0 and fo are respectively the chemical potential and the fugacity of the

fluid at a fixed reference state.

The details of the calculation of a(z), v(z) and the computer program for
adsorption calculations using this model are described by Rangarajan (1992).
The attractive term a(z) is calculated as follows depending on the distance of the

fluid molecule from the adsorbent.

AtO.50$le.50

a(z) = abulk [ 5/16 + (6/16) (z/0)] 3 - 19
At 1.5 0 S 2 S 00
a(z) = abulk [1 - (1/8){z/0-1/2}3] 3 - 20

where 0 is the Lennard-Jones parameter "0" which can be either 03‘ or Ofw.

The value of v(z), is calculated from equation 3-17, using numerical
techniques. However, below the critical point there maybe three values of v that
satisfy the equation and the stable value of v is the one which gives the highest
pressure. Rangarajan ( 1992) suggested the following methods to solve for the
roots:

(1) Rewrite equation 3 - 17 as

v(z) = b+b/[ln f(z)(v(z)-b)/RT) + 2a(z)/(v(z)RT)] 3 - 21

Using successive substitution this equation will provide a decreasing
value of v(z). The initial guess is taken to be v(z)=1.1*b. This method will
converge to a liquid root.

(2) Rewrite equation 3 - 17 as

v(z) = b + {RT/f(z)}exp[ b/(v(z)—b) - 2a(z)/(v(z)R’I‘)] 3 - 22

Using successive substitution this converges to give the vapor root. The
initial guess for v(z) used is v(z) = RT/f(z). Once the roots are obtained they are
arranged in increasing order of magnitude and the one that gives the highest
value for pressure is selected. Stability test that dP/dv < 0 is made to ensure that
the root selected is stable. Once the density profile is generated the surface

excess is calculated using the following equation:

00

Fex =f [p(z) - Pbulk] dz 3 - 23
20

Reduced distanee units : ETA and BETA
For the sake of convenience it was decided to use reduced distance units
throughout the program code. To calculate fluid-wall interaction potential i.e.
mm), the reduced unit used was ETA which is calculated as follows :
E'I‘A-—-z/ofw+0.50w/0fw 3-24
But for a(z) calculations one requires that the reduced distance units be in
terms of the fluid-fluid parameters 03. So for a(z) calculations the reduced unit
BETA was used.
BETA=Z/Ofi'. 3-25
The two reduced units can be interrelated as follows:

BETA={ETA-O.5ow/0fw}*0fw/0fi~ 3-26

The meuified SLD ; PRSLD

To represent the fluid properties in the SLD model any equation of state
may be used. The advantage of using the van der Waals' equation [Rangarajan
( 1992)], is that it is the simplest and most convenient equation of state that has a
theoretical basis. The drawbacks are that it is not an accurate equation of state
and gives poor predictions of vapor pressure and molar volume. The vdW
equation does a good job in qualitative description of fluid behavior but it
performs poorly in the field of quantitative predictions.

One of the objectives of this thesis was to improve the original SLD model

by using a better equation of state to represent the fluid properties. As shown in

equation 3 - 13, van der Waals assumed 'a' to be independent of temperature.
Most improved equations of state propose semi-empirical corrections for 'a',
and have the form a = a(T). Of these equations the Peng-Robinson (1976),
equation of state gives the best representation of fluid properties such as

saturation pressure and liquid densities. The Peng-Robinson equation is:

P = RT/(v-b) - a/(v2+2bv-b2) 3 - 27
where:

b=O.O778TcR/Pc 3-28

a= 0.45724 (RTc)2 a / Pc 3 - 29

a = [1-i-(0.37464+1.54226m-0.26992(02)(1-(T,-)"O.5)]2 3 - 30

w = -1 ~10810(P83t/Pc) l Tr=0.7 3 - 31

where, w is the acentric factor, P83It is the saturation pressure or the vapor
pressure of the fluid, T, is the reduced temperature, Tr=T/Tc .

This equation can be rewritten as

Z3 - (1-B)Z2+ (A—3B-2B)Z - (AB-BZ-B3)=O 3 -32
where:

A = aP/R2T2 3 - 33

B = bP/RT 3 - 34

Z = vP/RT 3 - 35

The fugacity of a Peng-Robinson fluid may be calculated from the
following equation:

ln(f/P) = Z-1-ln(Z-B)-A/(2.824B)*ln((Z+2.414B)/(Z-0.414B)) 3 — 36

The attractive term a(z) is calculated using equations 3 - 19 and 3 - 20. The
value of abulk in these equations is calculated using equations 3 - 29 thru 3 - 31.

The same type of algorithms are used to calculate v(z) the corresponding

equations are as follows:

For method (1) the equation equivalent to equation 3 - 21 is :

v(z)=b+b/DENO 3-37
DENO = {lnflz) + ln(v(z)-b)/RT +a(Z)V(Z)/(RT(V(Z)2+2bV(Z)-b2))
+a(z)/(2.824RTb)* ln[(v(z)+2.414b)/(v(z)-0.414b)]} 3 - 38
For method (2) the corresponding equation for a Peng - Robinson fluid is:
v(z) = b+RT/f(z) exp FUNC 3 - 39
FUNC = {b/(v(z)-b) - a(z)V(z)/[RT(v(z)2~l-2bv(z)-b2)l + a(z)/[2.824RTb]
*ln[(v(z)-O.414b)/(v(z)+2.414b)]} 3 - 40

The computer program used to calculate surface excess using the PRSLD is
described in Appendix B.

The PR equation of state is better at predicting vapor pressure and molar
volume than the vdW equation, it was expected that by using the PR equation in
the SLD would improve the quantitative aspect of the predicted adsorption
isotherms. This is because the PR equation predicts more realistic values of
fluid densities near the solid - fluid interface and also predicts more accurate
values of the vapor pressure for subcritical temperatures. Figures 3 - 1, 3 - 2 and
3 - 3 compare the adsorption isotherms predicted for carbon dioxide, ethane and
ethylene, using vdW and PR equations of state.

The PRSLD model predicts higher adsorption than the vdWSLD for the
same value at sfw at same T and P. The reason is that the PRSLD predicts a
higher value of density at each position from the wall, p(z), than the vdWSLD.
Thus the term (p(z)-pbu1k) is higher at each position from the wall for the
PRSLD model. As seen in Figure 3 - 4 the PRSLD predicts higher density values
near the solid wall than the vdWSLD. This is because the vdW equation under
predicts liquid densities. A comparison between experimental saturation

densities for ethylene (Angus et. al., 1974) and values predicted using the vdW

and PR equations of state are shown in Table 3 - 1.

Table 3 — 1 : A comparison of the experimental values of bulk saturation liquid
densities for ethylene (Tc = 282.4 K, Pc = 50.4 bar) with values predicted by the
vdW and PR equations of state.

 

T (K) Bulk saturation liquid density (mol/cu.m)
Expt. vdW PR
269.000 12937 .9496 723186069 121899189
255.000 14640.6454 7929.71104 14573.9878
241.000 159078617 848716317 16372.6949
225.000 170779609 904200009 180305907
211.000 179652552 948514626 19247.7591

 

Comparison of the vdWSLD, the PRSLD and experimental isotherms is
complicated by the differences in predicted vapor pressure values given by the
vdW and PR equations respectively and the experimentally obtained value of the
vapor pressure. The predicted adsorption isotherms depend on the efw value
chosen. Since the relationship between abulk and eff is uncertain it was decided
to let efw be an adjustable parameter. Figures 3 - 5, 3 - 6 and 3 - 7 compare
predicted isotherms of vdWSLD and PRSLD models against experimental
values of Findenegg ( 1983) for ethylene on Graphon. The efw value for the best
fit was the same for each model for all temperatures, 350 for vdWSLD and 150
for PRSLD.

60

50

b
0

Surface Excess(micromoles/sq.m)
N 03
O O

10

31

 

 

 

 

Illllllll

IIIIIIIII

 

— PRSLD Tr=0.95

PRSLD Tl' = 1.05

. PRSLD Tr=1.1
, vdWSLD Tr=0.95

vdWSLD Tr=1.05
vdWSLD Tr=1.1

 

 

 

 

 

Figure 3-1 : Adsorption isotherms predicted using the vdWSLD &

80 100 120 140
Pressure (bar)

PRSLD models for Carbon dioxide (T c=304.1 K, Pc=73.8 bar) for the

32

 

 

 

 

7o _
ll
: . . . PRSLD Tr=0.95
6° 7 i ......... PRSLD Tr=1.05
: — PRSLD Tr=1.1
: — vdWSLD Tr=0.95
5° ” : — — vdWSLD Tr=1.05
E : vdWSLD Tr=1.1
3 :
2
o ._ I
E 40 .
9 .
.2
E I
(I)
(I) I
8 .
x 30 _ .
UJ
Q) l
<:> I
E
:3 I
‘0 :
20 _— -
f
l
l
l
.'

   
 
 

10

 
 

I"

~~

Illlr

 

l l l l

O 20 40 60 80 1 00
Pressure (bar)

- _ u
_ ll"IIIIIHIIHHIMIHIIIIHHIIMIIII

l l l

120 140 160

 

 

Figure 3-2 : Adsorption isotherms predicted using the vdWSLD &
PRSLD models for Ethane (T c = 305.4 K,Pc = 48.8 bar) for the ratios

8w lsFF = 0.25.

Surface Excess (micromoles/sq.m)

33

40

 

 

— PRSLD Tr=0.95
35 +— ........ PRSLDTr=1.05
. . . PRSLD Tr=1.1

- - . vdWSLD Tr=0.95
3° ” ........ vdWSLD Tr=1.05
— vdWSLD Tr=1.1

 

 

 

N
01
l

 

  
  

N
O

 

_.L
0"

 

 

 

0 20 40 60 80 100 120 140 160

Pressure (bar)

Figure 3-3 : Adsorption isotherms predicted using the vdWSLD &
PRSLD models for Ethylene (T c = 282.4 K, Pc = 50.4 bar) for the

ratios 8w I eFF = 0.25 .

Density (gmoleslcum)

30000

25000

20000

15000

1 0000

5000

34

 

 

 

 

 

 

 

 

 

 

 

 

—— PRSLD
........ vdWSLD
l l 1 1
5 10 1 5 20 25

Reduced distance ETA

Figure 3-4 : Comparision of the density profiles generated by vdWSLD
& PRSLD for Ethylene (T c=282.4 K, Pc= 50.4 bar) on a solid at Tr=0.95

and the ratio 8w I s FF = 2.0 .

35

 

 

 

45

.0- i
+ eXpI.
- - , vdW

35 —
Ollllllll PR

 

 

 

 

N [0 OJ
0 U1 0

Surface Excess (micromoleslsqm)
01

10

 

 

 

o 1 ML 1 l l l l

0 5 1 0 15 20 25 3O 35 40
Pressure (bar)

 

Figure 3-5 : A comparison of adsorption isotherms predicted using
the vdWSLD & PRSLD models for Ethylene (T c=282.4 K, Pc=50.4
bar) versus the experimental data at 263K.

36

 

 

 

 

 

 

 

 

20
+ expt.
18 ~
- - . vdW
“w, IHHIII' PR
\‘ '—
16 7 3‘ '2
.3 ' ‘:
SI ‘ E
:I '
5‘, I
14 ~— ‘ I
A ‘
E. I E
8 . =
q I
h E
ii 12 L ,os‘ ' a
'6 03‘ ' E
E x: I a
9 0' O: ‘ :'
.2 ,o . t.
' c
g 10 P ’0 :5 . 3’
(D : | 64.
(D s‘ ’
a) I, 3 \ I'o,
o O s \ l’o
X '0 e ’I,
“J I s ‘ I",
a) 8 r— ’ :é ‘ I
U I s‘ a "I,
(U I 3 "h
t ' s \‘ ’I,
3 I 5 s
(D : s
I 5 ‘s
6 *1 5 x
5 \
r : x
4 —
2 _
0 l l l l l
O 20 40 60 80 100 120

Pressure (bar)

Figure 3-6 : A comparison of adsorption isotherms predicted using
the vdWSLD & PRSLD models for Ethylene (T c=282.4 K, Pc=50.4
bar) versus the experimental data at 293K.

37

 

Surface Excess (micromoleslsqm)

 

 

 

 

 

 

 

+ expt.
- - . vdW
“‘ 3"1.‘
"nun: PR \‘
- ’ "’
' ‘ "‘
Q ’1
\ 6
j.) \ ’I”
4" s ‘2
s I.
c‘ \ I’O
I
“5“ ‘ '3
_ 's‘ ‘ ’o
S
0'3. \ ’9”
O s: ‘ .9
I S ’9
"~¢Q \ v’.’
\ I
I ‘Q “ ’I”
' :‘ ’I
0' c. ‘ I,”
— s‘
’ s
. ‘
O 3. \
I s \
' s ‘
I s \
I o‘ s
l 3 s
3 s
L._ '5 S‘
l 5 ‘
l 5 ~ \
5 ~
'5
M L l J l
20 4O 60 80 100 120

Pressure (bar)

Figure 3-7 : A comparison of adsorption isotherms predicted using
the vdWSLD & PRSLD models for Ethylene (T c:282.4 K, Pc=50.4
bar) versus the experimental data at 313K.

CHAPE4

THE INTEGRAL EQUATION

The simplified local density model of Rangarajan ( 1992) and the
model of Barrer and Robins (1951) assume that the fluid chemical potential
can be calculated with the local density of the fluid. Actually the entire
density profile is required to find the true fluid-fluid potential. This
approach requires the solution of an integral equation and will be referred
to as the Integral Equation (IE) approach. The details of the equations used
in this approach are described in appendix A and the computer program is
in appendix C.

If a molecule is at a distance 2' from the wall or in terms of reduced
units 1] it is at a reduced distance of n' from the wall, where n = z/o, 11' = 270
and 'o' is the molecular diameter of the fluid molecule; the chemical

potential of that molecule is :

For 0.5 S 11' S 1.5

 

n'+1 °°
lnf(n')=ln(RT )+b - 3_3bulk|,r p(nidq+f amen I 4-1
v(n')-b v(n')-b 4 RT 1/2 n+1 ("-104
When n'21.5
n'+1 n'+1 °°
lnfln') = ln(BT_)_+L_- 3am If pin)_¢n+l p(n)dn +1 p(nLdnl
V(1]')-b v(n')-b 4 RT 1/2 ("-7154 n'-1 n'+1 ("-1134

4-2

To solve for n' we first need to solve the density profile integrals. This
can be done numerically. A first guess of the density profile is required.
This can be obtained from the SLD.

Instead of integrating the function p(n) / (ii—11')4 to 00 we need a finite
limit. In order to obtain that finite limit one has to determine to what
distance the value of the integral has a significant contribution. As the
distance (n-n') increases the contribution to the integral decreases. In order
to obtain maximum distance to which the function p(n) / (11-1134 has to be
integrated the case of a vapor-liquid interface was solved. For this case the

following density profile equation applies :

n~1 n'+1 0°

1 = If elm—dn + f p(nldn if nlnLdnl 4-3
.00 ("'n')4 ”£1 “0 +1 (1] _ 1")4

lnfm') = ln(RT ) +b - 3.2ka I 44

 

V(n')-b v(n')—b 4 RT

The initial profile used is shown in the Figure 4-1 which is the type
generated by the SLD model. On solving the IE we get a profile of the type
shown in Figure 4-2. For this case p(-°°) = Pliquid and p(+°°) = pvapor.
Assume that at a certain distance n'-x, p(n'-x) ~ p(im) = Pvapor. Then
instead of integrating from(-°°) to +00 we need to integrate from the finite
distances of (if-x) to (n'+x). To find these finite distances we substituted
different values of 'x' to solve the integral. If 'x' was such that p(n'-x) <

Pliquid and p(n'+x) > Pvapor. So we continued to increase the value of 'x' until

20000

1 8000

1 6000

1 4000

mi

.12000

Density (gmoles/cu
Si
O
O

8000

6000

4000

2000

40

 

 

l

I

l

 

 

 

l l l l l

 

 

0 5 10 15 20 25

Reduced Distance ETA

Figure 4-1 : Example of a typical density profile generated by
the SLD model for the case of vapor liquid interface

30

20000

1 8000

1 6000

1 4000

m

’fizooo

1 0000

8000

Density (gmoleslcu

6000

4000

2000

41

 

 

 

l l l l l

 

 

5 10 15 20 25
Reduced distance ETA

Figure 4 - 2 : Example of a typical density profile generated by
solving the IE for the case of a vapor liquid interface.

30

p(rf—x) ~ pfiqujd and p(n'+x) ~ Pvapor within a i 0.001% error. This was done
for water and ethylene for the range of reduced temperatures of 0.3 S Tr S
0.9. The results are tabulated in Table 4 - 1.

The relation between Tr and X can not be generalized and needs to be
determined for each fluid one is interested in. Based on these calculations it
was decided to use the limit of n'-l-17 for the case of adsorption of ethylene,
ethane and carbon dioxide on a graphon wall (lower limit for adsorption
case is 1/2 ). Figure 4 - 3 compares density profiles generated by vdWSLD
and [E for the adsorption of a fluid (Tc = 282.2 K, Pc = 51.07 bar, 03: 3.79 A)
on a solid ow = 3.4 A with a solid - fluid attraction of efw = 75.21 at 1 bar

pressure.

Table 4 - 1 : "X" values for Water (Tc = 647.3 K, Pc = 22.34 bar)
and Ethylene (Tc = 282.4 K, Pc = 51.07 bar) at different reduced
temperatures.

 

 

Ethylene Water
Tr X Tr X
0.3 7 0.3 6
0.4 7 0.4 6
0.5 7 0.5 6
0.6 9 0.6 6
0.7 6 0.7 5
0.8 16 0.8 4
0.9 17 0.9 4

 

Density(gmoles/cum)

43

 

 

 

 

 

 

 

 

 

100000
———— lE
llllllfli SLD
10000 ~;
1000 — 2%
100 —
10 —
1 l l l J l l
o 2 4 6 a 10 12 14

Reduced distance BETA

Figure 4-3 : Comparison of the density profiles generated by SLD 8. IE
for Ethylene (Tc=282.4 K, Pc=50.4 bar) on a solid at Tr=0.95 and the

ratio 8w / EFF: 0.1.

The other point to consider is the number of iterations required to
solve for the density profile. Once the integrals are solved then then the
chemical potential is solved numerically for obtaining the new densities.
Let the original be represented by p0(n') and the new density be represented
by pn(r|') then if, 90(11') - pn(n') / po(n') < 0.001, the new profile is accepted if
not the iteration continues. Thus the iteration is considered complete only
when the two profiles differ at each position by less than i 0.1%.

One of the objectives of this thesis is to determine, if the density
profile generated by the SLD is a good first guess for solving the IE. To
measure how good a first guess the SLD was it was decided to use the

% error which is defined as :

%error= ESLDmrlE * 100 4-5
FSLD
where, FSLD= Surface excess obtained from the SLD model

F IE = Surface excess obtained from the IE model.

Low values of % error indicate a first good guess. For the case of
adsorption of fluid on a wall three fluids were chosen : Ethylene, Ethane
and Carbon dioxide. Three different reduced temperatures selected were
Tr=0.95, T, = 1.05 & Tr=1.10. For each fluid at each reduced temperature
three solids were selected such that the relative solid - fluid attractions
(measured by Ew/Sfi‘) were 0.25 (low solid - fluid attraction), 1.0 (solid - fluid
attraction in the same range of fluid - fluid attraction) and 2.0 (solid - fluid
attraction much higher than fluid - fluid attraction). The hard core
diameter of the solid was taken to be : ow = 3.4A. These different parameters
are summarized in Tables 4 - 2, 4 - 3 and 4 - 4 [ Reid et a1. ( 1987)].

Table 4 - 2 : The 05, t-Ifl‘ values for ethylene, ethane and carbon dioxide.

 

 

Fluid T..(K) P..(bar) redo moi)
Ethylene 282.4 50.4 224.7 4.163
Ethane 305.4 48.8 215.7 4.443
Carbon 304.1 73.8 195.2 3.941
dioxide

 

Table, 4 - 3 : Different reduced temperatures (in K) selected for ethylene,
ethane and carbon dioxide.

 

 

 

Ethylene Mane Carbon dioxide
Tr T T T

0.95 268.28 290.13 288.895

1.05 296.52 320.775 319.305

1.1. 310.64 335.94 334.51

 

Table 4 - 4 : a values for ethylene, ethane and carbon dioxide

 

 

 

Ethylene Ethane Carbon dioxide
8w/fif'f 8w 8fw 3w 8fw 8w 8fw
0.25 56.175 112.35 53.925 107.85 48.8 97.6
1.0 224.7 224.7 215.7 215.7 195.2 195.2

2.0 449.4 317.773 431.4 305.05 390.4 276.05

 

The range of the % error for different reduced pressures (or P/Psat
for subcritical temperatures) for these different conditions are tabulated in
Tables 4 - 5 thru 4 - 13. These values are approximate and are meant as a
guide for developing generalizations only. Figures 4 - 4 thru 4 - 12 show the
effect of temperature and raw/cg on the % error for different fluids. For the
SLD to be a good first guess for solving the IE, the absolute value of the
% error should below 20%. For the conditions chosen the SLD is a good first
guess in certain regions for each fluid-wall system.
Willem:

( 1) The % error is a function of the fluid — solid system chosen (represented
by the ratio sw/EFF) and the temperature and pressure of the system.

(2) When the temperature of a chosen fluid - solid system increases from
near critical to super critical the absolute value of the % error is lowered.
(3) The SLD is a good first guess for super critical fluids when the reduced
pressure(P/Pc) is between 0.1 and 0.6. The % error increases drastically
when Pr > 0.7. For subcritical fluids the SLD is a good first guess (i.e. the
absolute value of the % error is below 20%) when the ratio of P/Psat is less
than 0.7.

(4) On increasing the ratio of fiw/EFF for the same system temperature and
pressure, the absolute value of the % error lowers, i.e. SLD is a better first

guess when the fluid-wall potential is higher than the fluid-fluid potential

The IE for the PR equation is difficult to formulate. In the case of the
vdW equation one can equate the attractive term with the attractive term in

the pressure equation to get :

47

-a/v2 = -2 at Eff 03NA2/ (3 v2) 4 - 6
Thus,a=2rtsffo3NA2/3 4-7
The corresponding activity results in the equation

- apR/(V2+2prV-pr2) = -2 it Efl' 03NA2 / (3v2) 4 - 8
where the subscript PR stands for Peng - Robinson. The problem with the
relation in equation 4 - 8 is that it is difficult to obtain a straightforward
relation between apR and Eff and 0 which is independent of the molar
volume v and the repulsive term pr. Also, the assumptions of vdW are
known and can be rewritten as an integral, the assumptions of the PR
equation are not known well enough to be rewritten in the form of an

integral.

Table 4 - 5 : The % error values for Carbon dioxide at Tr = 0.95 for different
values of the ratio raw/£3.

 

P/Psat 8w/8fl==0.25 8w/eff=1.0 €w/8ff=2.0
<01 42 ~ 20 8.0 ~ -1.0 3.0 ~ 4.0
0.1-0.2 2.0 ~ 9.0 -1.0 ~ -6.0 -4.0 ~ -7.0
0.2-0.3 9.0 ~ 1.0 -6.0 ~ -9.0 -7.0 ~ -10.0
0.3-0.4 1.0 ~ -3.0 -9.0 ~ -11.0 -10.0 ~ -12.0
0.4-0.5 -3.0 ~ -9.0 -11.0 ~ -14.0 -12.0 ~ -13.0
0.5-0.6 -9.0 ~ 420 440 ~ -15.0 430 ~ 450
0.60.7 -12.0 ~ -15.0 -15.0 ~ -17.0 -15.0 ~ -17.0

> 0.7

-15.0 ~ 500

-17.0 ~ 430

-17.0 ~ -44.0

 

Table 4 - 6 : The % error values for Ethylene at Tr = 0.95 for different values
of the ratio (aw/8g.

 

P/Psat 8w/8ff=0.25 8w/eff=1.0 8w/81r=2.0
<01 23 ~ 9.0 4.0 ~ -4.0 0.3 ~ -6.0
0.1-0.2 9.0 ~ 1.0 -40 ~ -7.0 -6.0 ~ -8.0
0.2-0.3 1.0 ~ -5.0 -7.0 ~ -10.0 .80 ~ .100
0.3-0.4 -5.0 ~ -9.0 .100 ~ .120 -100 ~ -120
0.4-0.5 -9.0 ~ -130 -120 ~ 15.0 .120 ~ -140
0.5-0.6 430 ~ -16.0 150 ~ -16.0 -140 ~ -16.0
0.6-0.7 -16.0 ~ 20.0 -16.0 ~ 20.0 -16.0 ~ -18.0
>07 20.0 ~ 610 20.0 ~ -500 -18.0 ~ 47.0

 

Table 4 - 7 : The % error values for Ethane at Tr = 0.95 for different values of
the ratio raw/cg.

 

P/Psat ew/etr=0.25 ew/Efr=1.0 8w/8tr=2.0
<0.1 26.0 ~ 11.0 4.0 ~ -3.0 0.3 ~ -5.0
0.1-0.2 11.0 ~ 2.0 -30 ~ -8.0 -5.0 ~ .90
0.20.3 2.0 ~ 4.0 -8.0 ~ .110 -9.0 ~ -120
0.3-0.4 -4.0 ~ .100 -110 ~ -130 -120 ~ -130
0.405 100 ~ -150 130 ~ -16.0 130 ~ -16.0
0.5-0.6 -150 ~ .190 -16.0 ~ -190 -16.0 ~ -190
0.30.7 -190 ~ 23.0 -190 ~ 23.0 -190 ~ 22.0
>07 23.0 ~ -67.0 23.0 ~ -61.0 22.0 ~ -570

 

Table 4 - 8 : The % error values for Carbon dioxide at Tr = 1.05 for different

values of the ratio avg/8g.

 

Pr €w/8ff=0.25 8w/err=1.0 ew/err=2.0
<0.1 42.0 ~ 18.0 9.0 ~ 1.0 3.0 ~ .4.0
0.1-0.2 18.0 ~ 8.0 1.0 ~ -5.0 -40 ~ -7.0
0.2-0.3 8.0 ~ 0.5 -50 ~ 9.0 -7.0 ~10.0
0.3-0.4 0.5 ~ 4.0 9.0 ~11.0 10.0 ~ 12.0
0.405 -40 ~ 9.0 11.0 ~ 13.0 12.0 ~ 13.0
0.5-0.6 9.0 ~13.0 13.0 ~ 15.0 13.0 ~ 15.0
0.30.7 13.0 ~ 17.0 15.0 ~ -18.0 15.0 ~ 17.0
>07 17.0 ~ -500 -18.0 ~ 39.0 17.0 ~ -36.0

 

Table 4 - 9 : The % error values for Ethylene at Tr = 1.05 for different values
of the ratio Sw/Efi‘.

 

Pr Sw/Eff=0.25 8w/efi”=1.0 €w/8ff=2.0
<0.1 23.0 ~ 10.0 4.0 ~ -3.0 -0.3 ~ -5.0
0.1-0.2 10.0 ~ 1.0 -3.0 ~ -8.0 5.0 ~ -8.0
0.2-0.3 1.0 ~ -4.0 -8.0 ~ -10.0 -8.0 ~ -10.0
0.3-0.4 -4.0 ~ -10.0 -10.0 ~ -12.0 -10.0 ~ -12.0
0.40.5 -10.0 ~ -13.0 -12.0 ~ -14.0 -12.0 ~ -14.0
0.5-0.6 -13.0 ~ -16.0 -14.0 ~ -16.0 -14.0 ~ 150
0.60.7 -16.0 ~ -19.0 -16.0 ~ -18.0 -15.0 ~ -17.0
>07 -19.0 ~ -52.0 -18.0 ~ 420 -17.0 ~ 359

 

Table 4 - 10 : The % error values for Ethane at Tr = 1.05 for different values
of the ratio Sw/Efi‘.

 

Pr 8w/Etr=0.25 Ew/Eff=l.0 8w/81r=2.0
<0.1 25.0 ~ 11.0 5.0 ~ -30 0.6 ~ 5.0
0.102 11.0 ~ 0.1 3.0 ~ -8.0 -5.0 ~ -9.0
0.2-0.3 0.1 ~ -5.0 -8.0 ~11.0 -9.0 ~-11.0
0.3-0.4 -5.0 ~ 10.0 11.0 ~ 14.0 11.0 ~ 13.0
0.4-0.5 10.0 ~ 15.0 14.0 ~ 16.0 13.0 ~ 15.0
0.5-0.6 15.0 ~ 18.0 16.0 ~ 19.0 15.0 ~ 18.0
0.60.7 180 ~ 22.0 19.0 ~ 21.0 -18.0 ~ 20.0
>07 21.0 ~ 51.0 20.0 ~ 45.0

220 ~ -61.0

 

Table 4 - 11 : The % error values for Carbon dioxide at Tr = 1.10 for different

values of the ratio Ew/Efi‘.

 

Pr 8w/81r=0.25 8w/8ff=1.0 8w/eff=2.0
<0.1 42.0 ~ 20.0 9.0 ~ 1.0 3.0 ~ -3.0
0102 20.0 ~ 11.0 1.00 ~ -4.0 -3.0 ~ -6.0
0.20.3 11.0 ~ 4.0 -4.0 ~ -7.0 -6.0 ~ —8.0
0.30.4 4.0 ~ 2.0 -7.0 ~ 10.0 -8.0 ~ 10.0
0.405 2.0 ~ -6.0 10.0 ~ 12.0 10.0 ~ 12.0
0.5-0.6 -6.0 ~ 10.0 12.0 ~ 13.0 12.0 ~ 13.0
0.60.7 10.0 ~ 13.0 13.0 ~ 16.0 13.0 ~ 15.0
>07 13.0 ~ 49.0 16.0 ~ 360 15.0 ~ -310

 

51

Table 4 - 12 : The % error values for Ethylene at Tr = 1.10 for different values
of the ratio Sw/Efi‘.

 

PT 8w/8ff=0.25 8w/Eff=1.0 Ew/Eff=2.0
<0.1 22.0 ~ 9.0 3.0 ~ -2.0 -0.7 ~ -5.0
0.1-0.2 9.0 ~ 2.0 -2.0 ~ -6.0 -5.0 ~ -7.0
0.2-0.3 2.0 ~ -3.0 -6.0 ~ -9.0 -7.0 ~ -10.0
0.3-0.4 -3.0 ~ -7.0 -9.0 ~ -11.0 ~10.0 ~ -11.0
0.4-0.5 -7.0 ~ -10.0 -11.0 ~ -13.0 -11.0 ~ -12.0
0.5-0.6 -10.0 ~ -13.0 -13.0 ~ -14.0 -12.0 ~ -14.0
0.6-0.7 -13.0 ~ -17.0 -14.0 ~ —16.0 -14.0 ~ -16.0
>07 -17.0 ~ 500 16.0 ~ 360 -16.0 ~ -33.0

 

Table 4 - 13 : The % error values for Ethane at Tr = 1.10 for different values
of the ratio Ew/Efl‘.

 

Pr 8w/81r=0.25 8w/8ff=1.0 8w/8ff=2.0
<01 23 ~ 10.0 4.0 ~ 2.0 0.4 ~ 5.0
0.102 10.0 ~ 4.0 . 2.0 ~ -6.0 -5.0 ~ -8.0
0.20.3 4.0 ~ 2.0 -6.0 ~ 10.0 -8.0 ~ 10.0
0.3-0.4 2.0 ~ -7.0 10.0 ~ 12.0 10.0 ~ 12.0
0.40.5 -7.0 ~ 12.0 120 ~ 14.0 12.0 ~ 14.0
0.5-0.6 12.0 ~ 15.0 14.0 ~ 16.0 14.0 ~ 15.0
0.60.7 150 ~ 20.0 16.0 ~ 19.0 150 ~ -18.0
>07 20.0 ~ 58.0 19.0 ~ 44.0 -18.0 ~ -390

 

52

 

 

 

 

 

 

 

4O
- - vdWSLD Tr=0.95
--------- vdWSLD Tr=1 .05
35 ’ — VdWSLD Tr=1.1
, nuuui 'E Tr=105
30 — 1'
, I I I IE Tr=1.1
I
A I
E f
' I
S 25 - .
e .
g l
O l :‘t
a 2 T.
._ a E .2
g 20 — . s a
(D ' .5 :2
o ' -' ‘
“>3 I f '3: s
I ~‘ 0 :
8 15 ,1 ,5 O E .
‘g b ’ c‘. '0 é ‘
3 / ‘é‘ .0 e.‘ "a .
CD es '08... ‘2 \
.Q“ 0....9‘.. 2 .
0. 0.0.. 2 .
.0. 0.... $5, .
10 - a, ’
0.275.. " ’5’ §
:? 00. ’lllhh’
g e "I" ~ ~ . .
. I
: “e.“ . -
5 ...... ' I
O l l l l l l l

 

 

 

0 20 4O 60 80 100 120 140 160
Pressure (bar)

Figure 4-4 : Adsorption isotherms predicted using the IE & vdWSLD

models for Ethylene (T c=282.4 KPc = 50.4 bar) for the ratios

53

 

4O

 

- - vdWSLD Tr=0.95
35 e -------- vdWSLD Tr=1.05
—— vdWSLD Tr=1.1
— lE Tr=0.95

IE Tr=1.05

. . . IE Tr=1.1

30——

 

 

 

N
01
l

 

Surface Excess (micromoles/sqm)
a '3
r

10

 

 

 

0 l l L l l J l 4

0 20 40 60 80 100 120 140 160 180
Pressure (bar)

 

Figure 4-5 : Adsorption isotherms predicted using the IE & vdWSLD
models for Ethane (T c=305.4 K, Pc=48.8 bar) for the ratios

8W IEFF=2.0 .

Surface Excess (micromoles/sqm)

54

 

 

 

 

 

 

50
- - vdWSLD Tr=0.95
45 —
-------- vdWSLD Tr=1.05
— VdWSLDTr=1.1O
40 r _ IE Tr=0.95
Illlllll- IE Tr=1.05
35 — : - . IETr=1.10
30 :

M
01

N
O

15

 

 

 

 

O l l l I l
0 20 40 60 80 100 120 140 160

Pressure (bar)

Figure 4-6 : Adsorption isotherms predicted using the IE & vdWSLD
models for Carbon dioxide(Tc=304.1 K, Pc= 73.8 bar) for the ratios

8w/8FF=2.0.

% Error in Surface Excess

1O

55

 

 

 

 

 

 

 

l J l l L

 

 

0 0.5 1 1.5 2 2.5

Reduced Pressure : Pr

Figure 4-7 : Effect of temperature on % error for the ratio
8w / sFF = 1.0 , for Ethylene (T c=282.4 K, Pc= 50.4 bar).

3.5

% Error in Surface Excess

1O

-60

56

 

 

 

 

 

 

 

l l l l l l

 

 

0.5 1 1.5 2 2.5 3
Reduced pressure : Pr

Figure 4-8 : Effect of temperature on % error for the ratio
8w I aFF = 1.0 , for Ethane (T c=305.4 K, Pc=48.4 bar) .

3.5

"/0 Error in Surface Excess

10

—40

57

 

 

. —— Tr=1.05
.\ - - Tr=1.10

 

 

 

 

 

l J l l l l l l l

 

 

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Reduced Pressure : Pr

Figure 4-9 : Effect of temperature on °/o error for the ratio
aw IaFF = 1.0 , for Carbon dioxide (T c=304.1 K, P0: 73.8 bar).

% Error in Sudace Excess

3O

20

1O

58

 

 

 

,,,,,,,,,,,,, 8w leFF

8w ”EFF

\ - - - 8W [EFF

 

= 2.0

=1.0

= 0.25

 

 

 

 

O 0.5 1

1.5

Reduced Pressure : Pr

Figure 4-10 : Effect of the ratio 9w [EFF on % error at Tr =1 .05 for

Ethylene (T c=282.4 K, Pc = 50.4 bar)

 

 

% Error in Surface Excess

59

 

 

 

30
20 r‘. ............ 8w lsFF=20
“ 8w [EFF = 1.0
10 L ‘
‘l _ _ _ 8w lcFF = 0.25

 

 

 

do
0

-40

 

 

 

 

O 0.5 1 1.5 2
Reduced Pressure : Pr

Figure 4-11 : Effect of the ratio 8w leFF on % error at Tr =1.05 for

Ethane (T c=305.4 K, Pc = 48.4 bar).

2.5

°/o Error in Surface Excess

60

40

20

60

 

 

l

l

 

 

 

 

 

 

l l l l

 

 

0.2

0.4

0.6 0.8 1 1.2
Reduced Pressure : Pr

1.4 1.6 1.8

Figure 4-12 : Effect of the ratio 8W IsFF on % error atTr =1 .05 for
Carbon dioxide (T c=304.1 K, Pc =73.8 bar).

CHAPTER 5
SUMMARY AND CONCLUSION

The objectives of this thesis as stated in Chapter 1, were to modify the
simplified local density (SLD) model by using the Peng—Robinson equation of
state to calculate the fluid properties and to investigate whether the SLD
density profiles are good first approximations for solving the the integral
equation (IE).

In chapter 2 the different adsorption models were described and in
the end it was emphasized that for the conceptual design of adsorption
systems the desirable characteristics of an adsorption model are:

(1) dependable prediction of equilibrium conditions at a wide range of
temperatures and pressures for different kinds of fluid - solid systems.

(2) easy availability of the different parameters required to calculate
adsorption isotherms.

(3) quick calculations and readily understandable results.

The SLD was modified in chapter 3 by introducing the Peng -
Robinson equation of state to represent the fluid properties. The original
SLD model used the van der Waals' equation and was thus named the
vdWSLD, while the modified form was called the PRSLD. The adsorption
isotherms predicted by these two models were compared for different fluids

at different temperatures. For the same solid - fluid interaction parameter

Efw, the PRSLD predicted higher P values than the vdWSLD. Adsorption

61

isotherms predicted by the PRSLD and the vdWSLD were also compared

with experimental data. The efw parameter was adjusted for both SLD
models to get the best fit isotherm. Over a difference of 50 C the efw value for
the best fit isotherm was the same. The PRSLD gave better quantitaive
agreement.

In chapter 4 a different approach was taken. The density profile
generated by the SLD was used as a first guess to solve an integral equation
to obtain a more accurate density profile. This approach was called the IE
approach. It is similar to the mean - field approach developed in density
functional theories. Here the aim was to find out in which ranges of
temperature and pressure the SLD will be a good first guess for solving the
IE. The adsorption isotherms generated using the SLD and the IE were
compared for ethylene, ethane and carbon dioxide on three different types of
adsorbents : low solid — fluid interaction potential, comparable potential and
high potential. Three different temperatures were selected for each fluid,
one to represent subcritical range, one to represent near critical range and
one to represent supercritical range. Based on these comparisons it was
concluded that the SLD is a good first guess when:

( 1) for super critical fluids when the reduced pressure(P/Pc) is between 0.1
and 0.6 and for subcritical fluids when the ratio of P/Psat is less than 0.7.
(2) when the fluid-wall potential is higher than the fluid-fluid potential.
Also the SLD performs better when the temperature is in the supercritical
region.

In the above mentioned comparisons between the SLD and IE only

the vdWSLD could be used. It is difficult to formulate an IE solution

corresponding to the PRSLD. This is because it is difficult to obtain a
relation between the attractive term, apR and eff and a which is
independent of the molar volume v and the repulsive term pr.

In conclusion the SLD developed by Rangarajan ( 1992) predicts
adsorption isotherms in qualitative agreement with experimental data.
Quantitative agreement is improved by modifying it with the Peng -
Robinson equation of state. Another approach to improving it's predictions
is by using it as a first guess to solve an integral equation i.e. the IE

approach.

Work;

More work needs to be done in the following areas :

(1) The SLD should be extended to the case of a mixture of fluids contacting
a solid adsorbent. When comparing predictions with experimental data, the
best fit value of the efw parameter should be obtained and used.

(2) A relation between am; and Eff and a needs to be developed which is
independent of v and pr in order that the corresponding IE can be formed
for the PRSLD. If a relation can not be formed then some method of solving

the existing relation needs to be formed so that the IE can be formulated.

APPENDIX A

APPENDIX A

Details of solving the Integral Equation

Rangarajan (1992), solved the fluid-fluid interaction potential for a

non-homogeneous fluid by considering the following two cases :

(1) A fluid molecule in the vicinity of the wall 0.5 0 5 z s 1.5 a

(2) A fluid molecule far form the wall 1.5 a s z 5 0°
where 0 is the molecular diameter

2 is the distance fmm the wall.

While performing the integration the simplified local density (SLD)
assumption was used p(x) = p(z) ( where “x" is the axial distance coordinate
in the cylindrical system with "r" as the radial distance coordinate). For the
integral equation approach the entire density profile is required not just the
local density. In this approach the fluid configurational integral will be
solved in a slightly different manner. For a van der Waals' fluid the
fugacity can be written as

lnf = ln(RT/ (v-b) ) + bl (v-b) - Za/VRT A - 1
where R = universal gas constant
T = temperature

v = molar volume

f = fugacity
b=0.125RTc/Pc A'Z
a = 21(RTc/P92 A - 3

64

Or a = ()3 E/oj x d4; g(x)41tx2dx A - 4
dx
where x = r / u

a = Hard core diameter

5 = Fluid-fluid interaction potential

u(r) = 0° r< a A - 5
u(r)=z:(o/r)6 o<r<°° A-6

The fluid-fluid interaction contribution to the fugacity equation can also be

writtenas:
-_a =-.1_1‘lA I-e(o/rd)6ld_v A-7
VRT kT V V(z)
-3 = -mA f-e(o/rd)6dvf m A-8
VRT ZkT V V v(z)

where rd is the distance from the center of molecule. So the fluid - fluid
interaction contribution consists of an integration of the Sutherland

potential and an integration of the density profile.

Consider the integral I £416 dv A - 9
v rd6
Let y = distance from the center of the fluid molecule.

a = diameter of the fluid molecule.
8 = maximum energy of attraction between pair of fluid
molecules.
This integral is solved in cylindrical coordinates where r is the radial
distance and x is the axial distance. Hence we have:

I I Mdrdx A-lO
0 0 (x2+r2)3

This integral will be solved in two parts 0 S y S 0 and 0 S y 5 0°

For 0 S y S 0
y 00
2am6 I I -r d; dx
0 Wrz—x2) (x2+r2)3

y
21w06 j r d; I
0 4(r2-+x2)2 «OZ-x2)
znmzy A- 11
2

ForaSySoo

y 00
:“803+ 2“... I $255M
2 o 0 (r2+x2)3

oo

= $159341! f, “.316m I
2 2 0 (r2+x2)2 0
y y

=mfls+nms f -151; =Jt803+1L€Q6(1/x3) l
2 2 O x4 6 0

= -2nm3+nm6
3 6 y3
= -22r1-:Q3(1- 03/4y3) A - 12
3

So we have I—s (o/rd)6 dv
V

%ng (0350) A-13
2

=-2u£03(1- 03/ 4y3) (USySOO) A - 14
3

But for a van der Waals' fluid we know that when r< o the potential is 0°. So

the integral of the Sutherland potential becomes for y = 0°.

238932;? = -4m;o3 A - 15
3 3
Now we have f-e (o/rd)6 Ldy A - 16
v v(z)

Consider a molecule at an axial distance 2' from the wall. It will be
assumed that the potential is most influential at distance 0 the molecule at

2'. Here a is equal to the diameter of the molecule. The Sutherland potential

becomes

~II£QZU=-II£_Q'3 A-17
2 2

Now our molecule of interest is located at a distance 2' from the wall. To

integrate the density profile p(z) or 1/v(z) around 2' where z is the distance

of any other molecule from the wall, we need to split the integral. When 2

is very near the wall i.e. 0/2 < z' < 1.5 o, where o is diameter of the

molecule, the density profile may be integrated as follows :

2 +0 00
I p(z)dz + f pm A _ 18
0/2 z'+0 (2-234

When 2' is far from the wall i.e. z' > 1.50, the density profile is integrated as

z'-o z'+0 0°
I p(z) dz + f p(z)dz + I p12) dz A— 19
“/2 (z-z')4 2'0 2+0 (z-z')4

Using reduced units of distance = n, where n = z/o, 11': 270 we get :

When 0.5 < n' < 1.5

n'+1 °°
I 901) dn + J. p(nLdn A - 20
1/2 n'+l ("4'54

And when n' > 1.5
1121 n'+1

I plnl_dn +f phndn + I pluLfin A-

1/2 ("-1194 n'-1 '1'“ (n-n')4

The vdW a; z 2 term can be replaced as follows:
v(z)RT

When 0.5 < n'< 1.5

 

n'-1 °°
-1NA | _1[£Q—3 f p(nldn _11£g3 J. p(n)_dn IA-
2 kT 2 1/2 2 ""1 (n-n')4
When n' > 1.5
'a'-l 11'+1 °°
ANAL m3! plnl_dn_nm3,l p(n) dn _1I£Q—3 I plum A-
2 kT 2 ”2 (1]—1]')4 2 ""1 2 11'4”1 (114194
For a homogeneous fluid the density is uniform so
fibulk.= -1* NA *1 I -8(o/rd)6 dv A -
VRT 2* WV V
abulk = _1*_1\IA*_1_ * 4&93 A-
vRT 2* kT'“v 3
am" = fibulk“ 3 * it. A-
2 R 4 N A
JE£03 = 3* abulk“: A-
2 4 R NA
Using this relation we substitute for m3
2
For 0.5 .<_ 11' _<. 1.5
n'+1 °°
3.3:ka | I 901) dn + f :11an I A -

8 RT 1’2 '1'“ (n-n')4

21

22

23

24

25

26

27

28

 

For 11' 2 1.5
n'-1 n'+1 °°
3_am1k l I pull—(111 + I p(n)dn + f plnuhi | A-29
8 RT 1’2 (ii-1194 ""1 '1'“ (1)-1154
lnf(1f)= ln( RT )+- b - gan') A - 3O
v(n')-b V(n')-b V(1]')RT

For 0.5 S n' S 1.5 we have,

 

 

n'+1 °°
lnfln') = ln(RT )+b - fiabuik If p(n)dn+.l~ p(nLdn I A-31
v(n')-b v(n') - b 4 RT 1/2 n'+1 (“-1154
And 11' 2 1.5
n'+1 n'+1 °°
1nf<n'> = ln(RT )+b - 3_asu1k If @an p<n>dn +I aim—dill
v(n')-b v(n') - b 4 RT 1/2 (“-1134 n'-l n'+1 (”4‘94

A-32

APPENDIX B

APPENDIX B

Program code for calculating surface excess using PRSLD

C-k'k'k'k'k'k'k'k'k***************************************************

C234567

C Program to calculate surface excess or adsorption isotherms

C
C The program uses Peng Robinson equation of state, with a
C modified a to account for exclusion. The strength of the
C adsorption potential and its dependence on position
C is given by Psi.
C The fluid-fluid potential uff = u total - u ads.
C From this potential the various fluid properties
C are calculated.
C
C
C The variables used are defined below.
C All units used are SI unless otherwise stated.
C
COMMON R,TC,PC,OMEGA
C
C Define Psi as a statement function
C Use a 10-4 potential where PSI is the negative of the

70

O (3 O F) (1 O

71

intermolecular potential.

SIGFW = sigma fluid-wall, in Angstroms.

EPSFW is the fluid-wall potential in K

ALPHA is the ratio of the spacing of the graphite
basal planes (3.35 A) to SIGFW.

The number density of C atoms in graphite is
0.382 atoms/A‘Z.

The equation used is that suggested by Lee (1.5).

PSI(ETA)=4.0*3.1415926*0.382*SIGFW*SIGFW*EPSFW*(‘0.2
l/ETA**10+0.5/ETA**4+O.S/(ETAiALPHA)**4+0.5/(ETA+2.0*

*ALPHA)**4+0.5/(ETA+3.0*ALPHA)**4+O.5/(ETA+4.0*ALPHA)

***4)

OPEN (UNIT-=3 , FILE: 'ADSORP . DAT' , STATUS: ' UNKNOWN' )
OPEN (UNITE; , FILE: 'DENPRO . DAT‘ , STATUS= ' UNKNOWN ' )
OPEN (UNIT=8 , FILE= ' EXCESS . DAT' ,STATUS= ' UNKNOWN' )
OPEN (UNIT=4 , FILE= ' VlCAL . DAT' , STATUS= ' UNKNOWN' )

OPEN(UNIT=2,FILE='P.DAT’,STATUS='UNKNOWN')

R = Gas constant J/K/mol
Tc = Critical Temperature K

Pc = Critical Pressure N/m2

O 000 000 O 00 O

OMEGA.= Acentric Factor
SIGMA = Sigma gas-solid in m.
T = Temperature K

PLIM = Bulk pressure limit.

EPSFW = Gas-solid potential well depth in K

SIGFW = Sigma gas-solid in Angstroms
ALPHA.= Spacing of graphite planes / Siggs
SIGFF = Sigma fluid-fluid in Angstroms

SIGWW = Sigma wall-wall in Angstroms

AB

Bulk Peng-Robinson 'a'

B8 = Bulk Peng-RObinson 'b'

R=8.3l4

TC=304.1

PC=73.8E5
OMEGA.= 0.239
FOMEGA.= 0.37464+l.54226*OMEGA-0.26992*(OMEGA**2)
T=334.Sl

=T/TC

PLIM=l3OE5

EPSFW=97.6

SIGFF = 4.163

SIGWW

3.4

SIGFW = (SIGWW+SIGFF)/2

SIGMA SIGFW*lE-10

ALPHA:3.35/SIGFW

Calculate a bulk and b
AB=O.45724*(R*TC*(1+FOMEG*(l-SQRT(TR))))**2/PC
BB=0.07780*R*TC/PC

Write parameters to files

PCB=PC/l.OES

WRITE(7,*) TC

WRITE(7,*) PC

WRITE(7,*) SIGMA

WRITE (7 , *) SIGWW

WRITE(7,*) SIGFW

WRITE(7,*) SIGFF

WRITE(7,*) AB

WRITE(7,*) BB

WRITE(7,*) T
CHECK WHETHER CONDITIONS ARE SUBCRITICAL OR
SUPERCRITICAL

IF(TR.GE.1) GOTO 2
IF SUBCRITICAL FIND SATURATION PRESSURE, FUGACITY
AND VAPOR AND LIQUID DENSITIES.
CALL FSAT(AB,BB,T,FUGS,PS,VV,VL)
DELP=PLIM/40.
LOOP FOR BULK PRESSURE

PB=0.0E5

74

J = 0
FOR EACH BULK PRESSURE VALUE
PB=PB+DELP
B=BB
A;AB
FIRST CALCULATE BULK DENSITY (DENB) and BULK

FUGACITY (FB)

CALL BVCAL(A,B,T,PB,PS,V,FB)

CALL PV(A,B,T,V,P)

DENB=l.O/V

BP=P

WRITE(7,*) P
WRITE(7,*) DENB

WRITE(2,*)P/1E5, DENB, ALOG(FB)

DELETA=O.1

ETA=.9

EXCESS=0.0
K=0

ETA=ETA+DELETA

Calculate local fugacity
Use the following formula to get the local

fugacity (F), for a given position ETA.

000 0

F=FB*EXP(PSI(ETA)/T)
ALNFB=ALOG(FB)
ALNF=ALNFB+PSI (ETA) /T
PSIV:PSI(ETA)

C Calculate local a
CALL ACALC(ETAqAB,A,SIGWW,SIGFW,SIGFF)

C Using local parameters calculate V and local density DENL

CALL VCALC(A4B,T,ALNF,BP,PS,V)

DENL=l.0/V

EXCESS=SIGMA*(DENL-DENB)*DELETA*1.E6+EXCESS
C DENLG is the density in gmoles/cc

DENLG=DENL/l.0E6
WRITE (7,102)ETA,DENL,ALNF

K = K+l

IF(K.LT.200) GOTO 6

BP=P/1.0E5

DENBG=DENB/1.E6

WRITE(3,*)BP,EXCESS

WRITE(8,*) BP

76

J = J+l
IFlJ.LT.40) GOTO 5
102 FORMAT(1X,F8.2,2X,F12.S,2X,F10.5)

END

C************************************************************

C234567
C SUBROUTINES

C************************************************************

C THIS SUBROUTINE CALCULATES BULK FUGACITY & DENSITY

SUBROUTINE BVCAL(A,B,T,P,PS,V,FB)

COMMON R,TC,PC,OMEGA

PMX = o
PMN = o
ASTAR = A*P/(R*T)**2
BSTAR = B*P/R/T

A2 = BS'I‘AR-l

A1 = ASTAR-BSTAR*(2+3*BSTAR)

.A0 = BSTAR*(BSTAR**2+BSTAR-ASTAR)

CALL CUBIC(A2,A1,AO,R1,R2,R3,Cl,C2,C3,IFLAG)

WRITE(2,*)'IFLAG',IFLAG

IF (IFLAG.EQ.l) THEN

ZV=Rl

v: ZV*R*T/P
ALNF=A/2.82/R/T/B*ALOG((V-O.4l4*B)/(V+2.4l4*B))
ALNF=ALNF+ALOG(R*T/(V-B))+B/(V-B)
ALNF=ALNF-AfiV/R/T/(V*V+2*B*V-B*B)

FB = EXP(ALNF)
ELSE IF(IFLAG.EQ.2) THEN

ZL = R1

IF(ZL.GT.R2) ZL=R2

ZV= Rl

IF(ZV.LT.R2) ZV=R2
ELSE
CALL ARANGE(R1,R2,R3)
ZL = R3
zv = R1

ENDIF

IF (IFLAG.EQ.I) GOTO 14

VMAX=ZV*R*T/P
VMIN=ZL*R*T/P
VB=l.l*B

WRITE(2, *) VMAX,VMIN

CALL PV(A,B,T,VMAX,PMX)

CALL PV(A,B,T,VMIN,PMN)

v = VMIN

CALL DPDV(A,B,T,VMAX,IDPDV1)
IF(VMIN.EQ.O) GOTO 11

CALL DPDV(A4B,T,VMIN,IDPDV3)

11 IF (P.LE.PS) v=VMAX
IF (VMIN.EQ.O) =VMAX
FUGFl=(A/2.8284)*ALOG((V-O.4l4*B)/(V+2.4l4*B))
ALNFMX=ALOG(R*T/(V~B))+B/(V-B)+FUGFl/B/R/T

ALNFMX=ALNFMX-AfiV/R/T/(V**2+2*B*V-B*B)

FB = EXP(ALNFMX)

l4 WRITE(6,*)l/V}FB
CONTINUE
RETURN

END

C************************************************************

C THIS SUBROUTINE CALCULATES THE LOCAL VOLUME (WHICH

c IS USED TO CALCULATE THE LOCAL DENSITY)

SUBROUTINE VCALC(A,B,T,ALNF,BP,PS,V)

COMMON R,TC,PC,OMEGA

IV=l
IV3=1
DPDVl=l
DPDV3=1
Pl=0
P3=O
V=O
CALL VlCALC(AqB,T,ALNF,Vl,IVl)
IF(IV1.EQ.1) GOTO 10
CALL DPDVLA,B,T,V1,IDPDV1)
IFlIDPDVl.EQ.l) GOTO 10

10 CONTINUE

CALL V3CALC(A,B,T,ALNF,V3,IV3)
IF (IV3.EQ.1) GOTO 14
CALL DPDV (A, B, T, v3 , IDPDV3)
IF(IDPDV3.EQ.0) THEN

CALL PV(A,B,T,V3,P3)

ELSE
CONTINUE

ENDIF

l4 CONTINUE
IF (Pl.GT.P3) THEN
V=Vl
ELSE
V=V3
ENDIF
CONTINUE
GOTO 15
15 RETURN

END

C************************************************************

C THIS SUBROUTINE CALCULATES SATURATION FUGACITY
C FOR SUBCRITICAL CONDITIONS
SUBROUTINE FSATlA,B,T,FUGS,PS,VMX,VMN)
COMMON R , TC , PC , OMEGA

TR = T/TC

SLOPE=-(l+OMEGA)/O.4286
FPR=(SLOPE/TR)-SLOPE
PR=10**FPR

P=PR*PC
20 CONTINUE

K = 0

81

BSTAR=B*P/R/T
ASTAR=A*P/(R*T)**2
A2=BSTAR-1
Al=ASTAR-BSTAR*(2+3*BSTAR)
A0=BSTAR*(BSTAR**2+BSTAR-ASTAR)
CALL CUBIC(A2,Ai,Ao,R1,R2,R3,c1,c2,c3,IFLAG)
IF (IFLAG.EQ.1) GOTO 32
COTINUE
IF(IFLAG.EQ.2) THEN
ZL = R1
IF(Rl.GT.R2) ZL=R2
zv = R1
IF(ZV.LT.R2) zv=2
ELSE
CALL ARANGE(R1,R2,R3)
ZL = R3
zv = R1
ENDIF

VL=ZL*R*T/P

=ZV*R*T/P
RLT=ALOG((2*ZV+BSTAR*(2.+8.**O.5))/(2*ZL+BSTAR*(2.+
18.**0.5)))

IF (BSTAR.GE.ZV) THEN

STOP
ENDIF
RLT2=ALOG(zv-BSTAR)
RLNPHI=(ZV-l)-RLT2-ASTAR*RLT/BSTAR/(8.**0.5)
FUGV=EXP(RLNPHI)*P
RLT=ALOG((2*ZL+BSTAR*(2+8**0.5))/(2*ZL+BSTAR*(2-
l8**0.5)))
IF(BSTAR.GE.ZL) THEN
STOP
ENDIF
RLT2=ALOG(ZL-BSTAR)
RLNPHI=(ZL-l)-RLT2-ASTAR*RLT/BSTAR/(8**0.5)
FUGL=EXP(RLNPHI)*P
K = K+l
IF(K.GT.200) THEN
GOTO 31
ENDIF
TEST=ABS(l-FUGL/FUGV)

IF(TEST.LT.0.001) GOTO 30

P=P*FUGL/FUGV
GOTO 20

31 WRITE(6,*) 'NOT ITERATED'
STOP

32 Z = R1

vv: Z*R*T/P

VL=VV
RLT=ALOG((Z+2.4l4*BSTAR)/(Z-0.4l4*BSTAR)
RLT2=ALOG(Z-BSTAR)
RLNPHI=Z~1-RLT2-RLTtASTAR/2.sz/BSTAR
FUGS=EXP(RLNPHI)*P

3o FUGS=FUGL
vmx=
VMN=VL
PS=P
WRITE(6,*)ALOG(FUGL),ALOG(FUGV),VL,VV
RETURN

END

C************************************************************

SUBROUTINE CONV(AWB,T,ALNF,VI,V3,V,ERROR)

COMMON R

This subroutine iterates between values VLOW = V1
and VHIGH = V3 for a V which is satisfactory.

This subroutine uses a linear interpolation scheme

GOOD

(Secant or Regula-Falsi) method and can be very slow.

ALNFlC=B/(Vl-B)‘AfiVl/R/T/(Vl*Vl+2*B*Vl-B*B)
ALNF1C=ALNF1C+ALOG(R*T/(VI-B))

ALNFlC=ALNFlC+A/2.824*ALOG((Vl-O.4l4*B)/(Vl+2.4l4*B))/

17

4O

*B/R/T
ALNFl=ALNFlC-ALNF

Fl=EXP(ALNFl)

ALNF3C=B/(V3-B)-A*V3/R/T/(V3*V3+2*B*V3-B*B)
ALNF3C=ALNF3C+ALOG(R*T/(V3-B))
ALNFC3=ALNFC3+A/2.824*ALOG((V3-O.4l4*B)/(V3+2.4l4*B))/
*B/R/T
ALNF3=ALNF3C'ALNF

F3=EXP(ALNF3)

v=(V3-Vl)/(F3-Fl)+(Vl*F3-V3*Fl)/(F3-Fl)
ALNFC=B/(V-B)-2.0*A/R/T/V
ALNFC=ALNFC+ALOG(R*T/(V-B))
ERROR=EXP(ALNFC-ALNF)-l.0
IF (ABS(ERROR).LE.0.001) GOTO 40
IF (ERROR.LE.0) THEN

V3=V

F3=EXP(ALNFC-ALNF)
ELSE

v1=v

Fl=EXP(ALNFC-ALNF)
ENDIF
GOTO 17

CONTINUE

RETURN

END

C "k***************************************************

C234567
SUBROUTINE DPDV(A,B,T,V,IDPDV)

COMMON R

Input A,B,T,V and Output = IDPDV
THIS SUBROUTINE EVALUATES THE DERIVATIVE dP/dv

AND RETURNS A.VALUE OF 1 FOR IDPDV IF THE SLOPE

O (3 O ()

IS POSITIVE, i.e. THE ROOT IS UNSTABLE.

IDPDv=0
DP=-R*T/(V-B)/(V-B)+2.0*A*(V+B)/((V*V+2*B*V-B*B)**2)
IF (DP.GT.0) IDPDV:1

RETURN

END

C'k'k'k*‘k'k'k‘k'k'k'k'k'k'k‘k*********************************************

SUBROUTINE VlCALC(A,B,T,ALNF,V1,IV1)

C This subroutine uses a successive substituion method
C to get the small root Vl.
C Inputs A,B,T,ALNF and Output V1, IVl

COMMON R,TC,PC,OMEGA

IVl=O

C

20

30

Starting guess for calculating v
Vl=l.l*B

Iterate 60 times

DO 20 I = l,60
ARG=(Vl-B)/R/T
IF((Vl.LE.B).OR.(ARG.LE.0.0)) THEN
IV1=1
GOTO 30
ELSE
DENO=ALNF+ALOG(ARG)+A*Vl/R/T/(Vl**2+2*B*Vl-B**2)
DENO=DENO+A/2.82/R/T/B*ALOG((V1+2.4l4*B)/(Vl-O.4l4*B))
Vl=B+B/DENO
ENDIF
CONTINUE
ALNFC=B/(V1-B)-Arv1/R/T/(V1*V1-2*B*V1*B*B)

ALNFC=ALNFC+ALOG(R*T/(VI-B))

ALNFC=ALNFC+A/2.82/R/T/B*ALOG((VlO.4l4*B)/(Vl+2.4l4*B))
ERROR=EXP(ALNFc-ALNF)-1.o

IF (ABS(ERROR).GE.0.001) IV1=1

CONTINUE

RETURN

END

C************************************************************

C234567

SUBROUTINE V3CALC(A,B,T,ALNF,V3,IV3)

COMMON R,TC,PC,OMEGA
IV3=O
C STARTING GUESS FOR V3,: RT/F
ALNV3=ALOGlR*T)-ALNF
V3=EXP(ALNV3)
C ITERATE TILL DONE OR 40 TIMES
DO 20 I=l,40
IF (V3.LE.B) THEN
IV3=l
GOTO 30
ENDIF
ARG=B/(V3'B)'AfiV3/R/T/(V3**2+2*B*V3-B**2)
IFlARG.GT.40.) THEN
C D3=l./V3
IV3=l
GOTO 30
ELSE
DENO=A/2.824/R/T/B*ALOG((V3'0.4l4*B)/(V3+2.4l4*B))

ALNVBB=ALOG(R*T)'ALNF+ARG+DENO

V3=B+EXP (ALNV3B)
ENDIF
2 o CONTINUE

ALNFC=B/(V3-B)-A*V3/R/T/(V3*V3+2*V3*B-B*B)

ALNFC=ALNFC+ALOG(R*T/(VB-B))
ALNFC=ALNFC+A/2.824*ALOG((V3-O.4l4*B)/(V3+2.4l4*B)/
*B/R/T
ERROR=EXP(ALNFC-ALNF)'l.0
IF(ABS(ERROR).GE.0.00I) IV3=1
3O CONTINUE
RETURN

END

C'k'k'k'k'k'k'k'k'k***************************************************

C234567

SUBROUTINE PV(A,B,T,V,P)

COMMON R
C This subroutine returns the value of P given a,b,T,v
C using the Peng-RObinson equation of state.

P=R*T/(V-B)-A/(V8V+2*B*V'B*B)
RETURN

END

C********************************************************

C234567
SUBROUTINE ARANGE(RI,R2,R3)

C PROGRAM TO PUT 3 NUMBERS IN DESCENDING ORDER
DO 20 J=l,3

IF(R2.GT.R1) THEN

TEMP=R1
R1=R2
R2=TEMP

ENDIF

IF(R3.GT.R2) THEN
TEMP=R2
R2=R3
R3=TEMP

ENDIF

20 CONTINUE
RETURN

END

C*************************************************

C234567
SUBROUTINE ACALC(ETA,AB,AqSIGWW,SIGFW,SIGFF)

C This subroutine calculates the 'a' term

000 000000 000000 000

after taking into account the effect of exclusion.
AB = Value of a in the bulk
ETA = reduced distance from the center of wall
The main program sends a reduced distance ETA based on
the distance from the center of the wall molecule to
the center of the first fluid molecule(SIGFW), which
is the basis for the integrated 9-3 potential.
However the integrations for configurational energy
have been done from the edge of the wall molecule.
Therefore it is necessary to translate the
coordinate. BETA is the distance from the edge of the
wall in reduced units
If the wall molecule and the fluid molecule were of
the same size
BETA.= ETA - 0.5
However, the wall molecules and the fluid molecules
are of different sizes, hence the conversion is

slightly complicated.

BETA = (ETA - (O.5*SIGWW/SIGFW))*(SIGFW/SIGFF)
IF (BETAtLE.l.5) THEN
AFAB*(5.O+6.O*BETA)/16.O
ELSE
A;AB*(l.0-l.0/8.0/(BETA-O.5)**3)

ENDIF

91

RETURN

END

C**********************************************************

**********************************************************

C***********************************************************

C

SUBROUTINE CUBIC

C***********************************************************

C* THIS SUBROUTINE FINDS THE ROOTS OF A.CUBIC EQUATION OF *
C* THE FORM X**3 +.A2*X**2 +.Al*X + A0 = 0 ANALYTICALLY. *

C************************************************************

C* VARIABLES *
Q************************************************************
C* AO~ ------ THE ZEROETH ORDER TERM OF THE NORMALIZED CUBIC *
C* EQUATION *
C*.Al ------ THE FIRST ORDER TERM OF THE NORMALIZED CUBIC *
C* EQUATION *
C* A2 ------ THE SECOND ORDER TERM OF THE NORMALIZED CUBIC *
C* EQUATION *
C* Cl ------ THE COMPLEX ARGUMENT OF ROOT #1 OF THE EQUATION*
C* C2 ------ THE COMPLEX ARGUMENT OF ROOT #2 OF THE EQUATION*
C* C3 ------ THE COMPLEX ARGUMENT OF ROOT #3 OF THE EQUATION*
C* CCHECK -- THE SAME AS "CHECK" BUT CONVERTED TO COMPLEX *
C* NUMBER FORMAT *
C* CHECK ‘-- Q**3 + R**2, USED TO CHECK FOR THE CASE OF THE *
C* SOLUTION AND IN FINDING THE ROOTS OF THE *
C* EQUATION, DOUBLE PRECISION *
C* DAD ----- “A0" CONVERTED TO DOUBLE PRECSION *
C* DAl ----- "Al" CONVERTED TO DOUBLE PRECSION *

C*

DA2 ----- "A2" CONVERTED TO DOUBLE PRECSION
ESl ----- AN INTERMEDIATE CALCULATION TO USED IN THE
CALCULATION OF "SI"
ES2 ----- AN INTERMEDIATE CALCULATION TO USED IN THE
CALCULATION OF "S2"
IFLAG ~-- A.FLAG TO INDICATE THE CASE OF THE SOLUTION OF
THE EQUATION:=1 ONE REAL + TWO COMPLEX ROOTS,
=2 REAL ROOTS,AT LEAST TWO THE SAME
=3 THREE DISTINCT REAL ROOTS
Pl ----- AN INTERMEDIATE SUM USED IN THE CALCULATION OF
"$31"
P2 ----- AN INTERMEDIATE SUM USED IN THE CALCULATION OF
"SS2"
Q ------ AN INTERMEDIATE SUM USED IN CALCULATING "CHECK"
R ------ AN INTERMEDIATE SUM USED IN CALCULATING "CHECK"
Rl ----- THE REAL ARGUMENT OF ROOT #1 OF THE EQUATION
R2 ----- THE REAL ARGUMENT OF ROOT #2 OF THE EQUATION
R3 ----- THE REAL ARGUMENT OF ROOT #3 OF THE EQUATION
RECK --- THE SAME AS "CHECK", BUT SINGLE PRECISION REAL
Sl ----- AN INTERMEDIATE VALUE USED TO FIND THE ROOTS OF
THE EQUATION, COMPLEX NUMBER
82 ----- AN INTERMEDIATE VALUE USED TO FIND THE ROOTS OF
THE EQUATION, COMPLEX NUMBER
881 ---' THE SAME.AS Sl BUT DOUBLE PRECSION REAL
SS2 ---- THE SAME AS 82 BUT DOUBLE PRECSION REAL
Z1 ----- ROOT #1 OF THE EQUATION, COMPLEX NUMBER
22 ----- ROOT #2 OF THE EQUATION, COMPLEX NUMBER
Z3 ----- ROOT #3 OF THE EQUATION, COMPLEX NUMBER

*

*

C************************************************************

SUBROUTINE CUBIC(A2,Al,AD,Rl,R2,R3,Cl,C2,C3,IFLAG1

DOUBLE PRECISION CHECK,DAO,DA1,DA2,Pl,P2,Q,R,SSl,SS2

COMPLEX ESl,E82,Sl,SZ,Zl,ZZ,Z3,CCHECK

DAo = DBLE(AO)
DAl = DBLE(A1)
DA2 = DBLE(A2)

Q = DAl/3.DOO - DA2*DA2/9.DOO

R

(DAl*DA2 - 3.DOO*DAO)/6.DOO - (DA2/3.DOO)**3

CHECK = Q**3 + R*R

IF (CHECK.GT.0.0) THEN

IFLAG = 1

P1 = R + DSQRT(CHECK)

P2 = R - DSQRT(CHECK)

IF (Pl.LT.
SSl
ELSE
SSl
ENDIF
IF (P2.LT.
SSZ
ELSE
SSZ

ENDIF

O

O

.0) THEN

-DEXP((DLOG(-l.DOO*Pl))/3.DOO)

DEXP((DLOG(Pl))/3.DOO)

.0) THEN

-DEXP((DLOG(-l.DOO*P2))/3.DOO)

DEXP((DLOG(P2))/3.DOO)

R1 = $81 + SS2 - DA2/3.DOO

R2 = -(SSl + SSZ) - DA2/3.DOO

R3 = R2

Cl 0.0

c2 = (SQRT(3.))*(SSl - SSZ)/2.DOO
c3 = -C2
ELSE IF (CHECK.LT.o.o) THEN
IFLAG = 3
RR = l.*R

RECK = l.*CHECK

CCHECK = CMPLX(RECK,0.0)

ESl = CLOG(RR + CSQRT(CCHECK))/3.
Esz = CLOG(RR - CSQRT(CCHECK))/3.
s1 = CEXP(ESl)
$2 = CEXP(ESZ)
22 = -(S1 + sz)/2 - A2/3 + (CMPLX(0.0,3**.5))*(Sl -
*s2)/2
23 = -(s1 + s2)/2 - A2/3 - (CMPLX(0.0,3**.5))*(Sl -
*S2)/2
R1 = REAL(Z1)
R2 = REAL(zz)
R3 = REAL(Z3)
C1 = 0.0
c2 = c1
c3 = c1
ELSE

C************************************************************

C * IF THE ROOTS OF THE EQUATION ARE VERY, VERY SMALL AND *

C * VERY, VERY CLOSE TOGETHER, THIS SUBROUTINE MAY

C * ERRONEOUSLY REPORT THAT THE EQUATION HAS ONLY ONE *

C * ROOT NEAR ZERO

C************************************************************

IFLAG = 2

IF (R.LT.0.0) THEN

SSZ

R1

R2

R3

C1

C2

C3

ENDIF

RETURN

881 = -DEXP((DLOG(-1.DOO*R))/3.D00)

ELSE IF (R.EQ.0.0) THEN

$81 = 0.0

ELSE

SS1 = DEXP((DLOG(R))/3.DOO)

ENDIF

$81
$81 + SS2 ‘ DA2/3.D00
-(SSl + SSZ)/2 - DA2/3.DOO
R2

0.0

C1

C2

APPENDIX C

APPENDIX C

Program code for calculating the surface excess using the density
profile calculated from the vdWSLD as a first guess to solve the

integral equation

C ADPRO.F

C This program calculates density profile and surface excess.
C Some of the constants used are R=8.314 J/K*mol, ETA is the
C dimensionless distance, DEN is the density, ALNF is the

C natural log of fugacity. BETA is corrected reduced

C distance, BULD is bulk density.

REAL*8 ETA(261),DEN(261),A1261),ANSl,ANSZ,ANSB,ER,ANS,TC
REAL*8 Vl(26l),V2,E,Y(151),X(151),T,DENl9261).EXCESS
REAL*8 PC,AB,BB,P,DENB,BETA(261),ALNF(261),DEN2(261)
INTEGER I,K2,K3,K4,K,J,L,N,N2,IFAIL,N1,M

EXTERNAL DOIGAF

C DOlGAF IS A.NAG SUBROUTINE WHICH PERFORMS
C NUMERICAL INTEGRATION.

OPEN(UNIT=3,FILE='DENPRO.DAT',STATUS='UNKNOWN')

93

C DENPRO.DAT is the data file generated by
C the VdWSLD program.
OPEN(UNIT=4,FILE='EXCES.DAT',STATUS='UNKNOWN')

C This file stores surface excess data.

OPEN(8,STATUS='UNKNOWN', FILE='IE3. DAT')

OPEN(2,STATUS='UNKNOWN', FILE='D3. DAT')

TC=Critical temperature, K
PC=Critical pressure. N/mfi*2
SIGMA:Sigma fluid - wall, m
SIGWW=Sigma wall - wall,Angstroms

SIGFW=Sigma fluid - wall, Angstorms

0 00 0 00

SIGFF= Sigma fluid - fluid, Angstorms

READ(3,*)TC
READ(3,*)PC
READ(3,*)SIGMA
READ(3,*)SIGWW
READ(3,*)SIGFW
READ(3.*)SIGFF
READ(3,*)AB
READ(3.*)BB
READ(3,*)T

IER=0.0

C T=temperature,K

C P=pressure, N/m**2

C BULD=bulk density,gmol/mfi*3
10 READ(3,*,END=ll)P

READ(3,*)DENB

DO 25 I=l,200
READ(3,102,END=12)ETA(I),DEN(I),ALNF(I)
12 BETA(I)= (ETA(I) -o . 5* (SIGWW/SIGFW) )* (SIGFW/SIGFF)
c 12 BETA(I)=ETA(I)-o.5
DEN1 (I)=DEN(I)
25 CONTINUE
c 'c' IS A LOOP COUNTER

c=1

C NOW THE FUNCTION VALUES ARE CALCULATED
5 DO 45 I=l,l40
K2=I'10
K3=I+10

K4=I+6O

IF (BETA(I).GT. 1.5) GOTO 66

DO 55 J=l,K3
A(J)=DEN(J)
JJ=J~l+l
X(JJ)=A(J)

55 CONTINUE

C NOW THE INTEGRAL IS CALCULATED USING DOlGAF
N:K3-l+l
IFAIL.=O

CALL DOlGIF (X,Y,N,ANSl,ER,IFAIL)

DO 65 K=K3 , x4

A(K)=DEN(K)/( (BETA(I) )**4)
KK=K-K3+l

X(KK)=BETA(K)

YlKKl=A(K)

65 CONTINUE
N1=K4~K3+1
IFAIL=O

CALL DOlGIF (X,Y.ANSZ,ER,IFAIL)

ANS=ANSl+ANSZ
GOTO 35
C DOlGAF CAN'T BE USED IF THERE ARE LESS THAN FOUR DATA

C POINTS FOR THIS CASE USE TRAPEZOIDAL RULE

66

73

67

76

100

IF (K2-l.LT.4) GOTO 67

DO 75 L=l,K2

A(L)=DEN(L)/( (BETA(L)-BETA(I) )**4 )

LL=L—1+1
X(LL)=BETA(L)
Y(LL)=A(L)

CONTINUE
N=K2-1+1
IFAIL=0

CALL DOlGAF(X,Y,N,ANSl,ER,IFAIL)

SUM=0.0
DO 76 L=1,K2
A(L)=DEN(L)/( ( EETA(L)-BETA(I)
SUM=SUM£A(L)
CONTINUE
H= (BETA(K2)-BETA(1) )/(K2-1+1)
SUM1=SUM- (A.(1) +A(K2) )/2
ANSl=H*SUMl
DO 85 M=K2,K3
A(M)=DEN(M)
MM:M-K2+1
X(MM)=BETA(M)

Y (MM) =A(M)

)**4 )

101

85 CONTINUE
IFAIL=0
N1=K3'K2+l

CALL D01GAF(X,Y,N1,ANSZ,ER,IFAIL)

DO 95 N=K3,K4

A(N)=DEN(N) / ( (BETA(N)-BETA(I) )**4 )

NN=N-K3+l

Y(NN)=A(N)

X(NN)=BETA(N)
95 CONTINUE

IFAIL=O
N2=K4-K3+1

CALL D01GAF(X,Y,N2,ANS3,ER,IFAIL)

ANS=ANSl+ANSZ+ANS3

C THE NEW DENSITIES ARE CALCULATED
35 F=ALNF(I)-LOG( (8.314*T) )

E=(ANS*.7S) / (T*8.3l4)*AB

v1 (I)= 1/DEN(1)
400 =LOG(V1 (1)-BB )+F+E-BB/ (Vl(I)-BB)

DFUN=Vl(I)/((V1(I)-BB)**2)

102

68 V2=Vl (I)-FUN/DFUN
IF(ABS(V1 (1)-V2 ).LT.lE-6 ) GOTO 77
'VI (I)=V2
GOTO 400
77 DEN2 (I)=l/Vl(I)
45 CONTINUE

CONDITION FOR ENDING IS THE NUMBER OF LOOPS SHOULD NOT
EXCEED 100 AND THE DIFFERENCE BETWEEN VALUES IN OLD

AND NEW PROFILE SHOULD NOT EXCEED 0.1%

c=c+1

IF(C.GT.lOl) GOTO 19

DO 7 KM=1,140
DIFF=(ABS(DEN(KM)-DEN2(KM) )/DEN(KM)

DIFFP = DIFF*100

IF (DIFFP.GT.0.1) GOTO 8
7 CONTINUE
GOTO 190
8 CONTINUE

DO 18 KJ=1,140
DEN(KJ)=DEN2(KJ)
18 CONTINUE

GOTO 5

103

Do 17 KL 141,200
DEN(KL) = DEN(140)
17 CONTINUE
GOTO 5
IF THE LOOP DOES NOT CONVERGE AFTER 100 ITERATIONS IER=1
19 IER=l.O
WRITE(6,*)'IER',IER
190 EXCESS=0.0
Exc1=o.o
DENERR=o.o
SUM:o.o
SUMl=0.0
Do 9 I=1,14o

DEN(I)=DEN2(I)

EXCESS=SIGMA*0 . 1* (DEN (I) -DENB) *1E6+EXCESS

EXC1=SIGMA*0 . 1* (DEN1 (I) *DENB) *1E6+EXC1

SUM=DEN(I) ‘DENB+SUM
SUMl=DENl (I) -DENB+SUM1
DENERR=1‘SUM/SUM1
WRITE(6,*) 'C' ,C
WRITE(2,*) ETA(I),DEN(I)
9 CONTINUE

EXERR=(1- (EXCl/EXCESS) )*100

104

WRITE(8,*)P,EXC1,EXCESS
PDENER=DENERR*100
WRITE (4,*)P,EXERR
GOTO 10
11 WRITE(6,*)'END OF PROGRAM

WRITE(6,*)IER

102 FORMAT(1X,F8.2,2X,F12.5,2X,F10.5)

103 FORMAT(1X,F8.2,2X,F12.5,2X,F12.5)

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