Reference

NATL INST. OF STAND & TECH

NBS

Publi-

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AlllDS TbflTfib

•"•CAU 0» " NBS TECHNICAL NOTE 1061

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

Phase Equilibria:

An Informal Symposium

—QC
NATIONAL BUREAU OF STANDARDS

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^
Phase Equilibria:

An Informal Symposium

NATIONAL BUHEAa. OF STANDAHOS

UBRAHT

WAR "7 TQB3

B. E. Eaton *t

J. F. Ely

H. J. M. Hanleyt

R. D. McCarty

J. C. Rainwater

Thermophysical Properties Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

* Department of Chennical Engineering, University of Colorado, Boulder, CO 80307

tin collaboration with J. Stecki and P. Wielopolski, Institute of Physical Chennistry, Polish

Acadenny of Sciences, Warsaw, Poland.

e

'^^ATESO^"^

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Annbler, Director

Issued January 1983
National Bureau of Standards Technical Note 1061

Nat. Bur. Stand. (U.S.), Tech Note 1061, 156 pages (Jan. 1983)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1983

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402

Price $6.50

(Add 25 percent for other than U.S. mailing)
PHASE EQUILIBRIA: AN INFOkHAL SYMPOoIUM

B. £. Eaton*"'', J. F. Ely, H. J. M. Hdnley''",

R. D. hlcCcirty and d. C. Rainwater

Therrnopiiysical Properties Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

PREFACE

Phase equilibria in fluid mixtures is a classical problem in the theory of

liquids v/fiich is not properly resolved, even today. The problem is of special

relevance because of current emphasis on synthetic fuels, the use of new feedstocks, and the need to conserve and increase productivity with existing

fluid technology. Hov^ever, the data base required for these new developments is

deficient and, in any case, the tasK of measuring all data that might be required

is prohibitive. One needs predictive procedures which can only be based on an understanding of fluid behavior backed up by the results of controlled

experiments on well-defined systems.

One of the tasks of the Boulder Fluid Properties Group, National Engineering

Laboratory, is to undertake such a program: namely, a study of mixtures via

theory, experiment and data correlation. We felt tnat some of our theoretical

ideas and approaches should be discussed and coordinated so an informal symposium

v/as held in October 1980 to do this. This Technical Note reproduces the main

presentations and is organized as follows: First, Ely reviews the state-of-theart of phase equilibria of nonelectrolyte systems, McCarty then reports on his

study of the procedure knov;n as extended corresponding states with special

emphasis on the nitrogen/methane system. This followed by a discussion by

Rainwater on an alternative attack based on the approach of Griffith and Wheeler

which addresses the prediction of VLE (vapor, liquid equilibrium) of mixtures

near the critical locus. Finally, Eaton and coworkers discuss the calculation of

the critical line itself and possible implications of the calculation to VLE in

general

* Department of Chemical Engineering, University of Colorado, boulder, CO

80307.

"*" In collaboration with J. Stecki and P. Wielopolski, Institute of Physical

Chemistry, Polish Acadeniy of Sciences, Warsaw, Poland.

m
Uo claim is made that the symposium covered other than a limited segment of

the total phase equilibria problem, but soi.ie themes and difficulties did emerge

which are general. Perhaps the most important of these is the well-known dilemma

between the strickly programmatic seiiii-empirical viewpoint, and the more

systematic, more academic counterpart. In the long run, of course, the latter

viev;point is preferable, but one has to be realistic and appreciate the need for

procedures which v/ork for the systems of interest today. It was felt that the

technique of extended corresponding states is an attractive coi.ipromise: it v/orks

well for nonpolar mixtures and is a systematic, soundly based, theory which can

still be developed; it applies to the v/hole of the liquid phase diagram without

parameter adjustment while most other techniques do not. However, the method is

based on the concept that a mixture can be replicated by a hypothetical pure

substance, and McCarty points out soi.ie subtle differences in mixture versus pure

behavior which need to bo discussed further. Clearly, too, the critical region

has to be included in a systematic manner, and we are studying combining the

extended corresponding states theory with that proposed by Rainwater. With

respect to this. Rainwater points out that the logical variables for a phase

equilibria study are the intensive set of pressure, temperature and chemical

potential. The engineer, of course, prefers, say, pressure, temperature and mole

fraction. It would bo interesting and important to try to resolve this

fundamental disagreement.

The probleiii of mixing rules and interaction parameters was discussed. The

latter difficulty is serious. As pointed out by Eaton, even interaction

parameters obtained by fitting the critical line, which is a sensitive task

involving second and third derivatives of the chemical potential, have no significance away from the critical region. As of today, the parameters tend to

obscure any unambiguous assessment of how well a theory can represent data.

This is one more reason why it is important to understand the assumptions which

go into a theory before the theory is developed and broadened.

Much of the work reported here was supported by the Office of Standard

Reference Data, and we thank Dr. Howard White, who attended the meeting, for his

interest and support. We also thank Mrs. Karen Bowie for typing and for other

help in preparing this Technical Note.

H. J. M. Hanley

IV
CONTENTS*

A REVIEW OF FLUID PHASE EQUILIBKIA PREDICTION METHODS

James F. Ely 1 THE EXTENDED CORRESPONDING STATES METHOD APPLIED TO THE

NITROGEN-METHANE SYSTEM

Robert D. McCarty 73

VAPOR-LIQUID EQUILIBRIUM OF BINARY MIXTURES NEAR THE CRITICAL

LOCUS

James C. Rainwater 83

PREDICTION OF THE CRITICAL LINE OF A BINARY MIXTURE: EVALUATION

OF THE INTERACTION PARAMETERS

B. E. Eaton, J. StecKi , P. Wielopolski and H. J. M. Hanley ... 125

Tne occasional use of non-S.I. units in tnis document arose because the

authors sought to compare their calculations directly with existing experimental

measurements.

A REVIEW OF FLUID PHASE EQUILIBRIA PREDICTION METHODS

James F. Ely

Thermophysical Properties Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

The accurate prediction of phase equilibria plays an important

role in the chemical process industries. A brief overview of fluid

phase equilibria predictive techniques is presented with special

emphasis on methods in current use in industry. Areas where better

fundamental understanding will lead to improved models are discussed

whenever possible.

Key words: activity coefficients; chemical potential; equations of

state; fugacity; group contribution models; phase equilibria.

1. Introduction

The prediction of thermophysical properties of mixtures presents

complications which are not encountered with pure fluids -- namely, that the

composition as well as the temperature and pressure dependence of the property

must be considered. This composition dependence introduces size and polarity

difference effects in the properties of single phase mixtures. From a predictive

point of view, hov/ever, the most difficult task is that of predicting the number

and compositions of coexisting phases at a known temperature, pressure and bulk

composition, e.g., the phase equilibria. Note that in predicting the properties

of single phase mixtures we are concerned with the properties of the fluid as a whole. However, in the case of the phase equilibrium prediction, we are interested in the partial properties of the individual components which

constitute the mixture.

Generally speaking, there is a vast base of experimental data for fluid

phase equilibria, especially when compared to the available experimental mixture

PVT, enthalpy and transport data. Partially due to this vast arnount of phase

equilibrium data, many simple, phenomenological models have been developed to

predict and correlate the observed phase behavior. Frequently, the simple

predictive models fail in a quantitative sense, especially when they are applied

to systems which contain species which differ substantially in size and polarity.

1
In order to make accurate predictions on these systems, statistical mechanical

models which can explicitly account for size and polarity effects must be used.

Unfortunately, the potential of the molecular models to predict complex fluid

phase equilibria has not been fully realized due to the mathematical complexity

of the problem and, to some degree, ignorance concerning the interactions of

chemically dissimilar molecules. For this reason, the engineering community is

forced to use simple models to make predictions, regardless of the accuracy

achieved.

The purpose of this chapter is to review the methods which are commonly used

to predict fluid phase equilibria for engineering applications. Areas where

fundamental research and further molecular understanding will improve our predictive models or perhaps lead to new models will be identified. Techniques

which have a more fundamental basis and those which have been developed to deal

with large size and polarity difference effects will be emphasized whenever

possible. The review will be limited to nonelectrolyte systems with the

exception being water-common inorganic (COp, HpS,..) systems. Solid-liquid

and sol id- vapor systems will also be excluded.

The structure of this article is as follows: section 2 examines some binary

mixture phase diagrams to illustrate some of the common types of fluid phase

equilibria. In section 3, the thermodynamic criteria for phase equilibrium and

mathematical methods for predicting the component equilibrium concentrations are discussed. Section 4 reviews mixture equation of state methods for predicting

chemical potentials and section 5 reviews the liquid phase activity-vapor phase

fugacity approach to phase equilibria, including the group solution methods.

2. Qualitative Phase Behavior in Mixtures

Phase diagrams for mixtures are considerably more difficult to visualize

than those for pure components due to the fact that the composition must be

considered. In addition to this added dimension, a casual inspection of

different phase diagrams shows great disparity in behavior from one binary system

to another. For example, some systems have azeotropes, isolated regions of

immiscibility and three phase lines. Von Konynenburg and Scott [1,2] have

proposed a convenient classification scheme which accounts for most of the

possible types of behavior. Their method is based on the existence or absence of

three phase lines, the number of critical lines and the manner in which the

critical lines connect with the pure component critical points and three phase

2
lines. Azeotropy gives rise to subclasses but does not change the qualitative

structure of the classification scheme.

In this system there are six basic types of phase diagrams which are illustrated as p-T projections of their corresponding three dimensional space

models in figure 1 [3]. In this diagram the dashed lines are critical loci and

solid lines are either pure component vapor pressure curves, three phase lines or azeotrope lines. Type I systems are the simplest of those encountered and

typically have a continuous critical locus which connects the critical points of

the two pure fluids, a common example of which is methane/propane. As is shown

in figure 1, the critical locus can be monotonically increasing or can have a maximum or minimum value. Figure 2 shows the three dimensional space model which

is typical of a type I system and an isoplethal (constant composition) of the

space model

Phase behavior in the region of a mixture critical point is usually more

complex than in a pure fluid because the two phase region can extend to pressures

and temperatures which are higher than the critical values. This type of

phenomenon is known as retrograde behavior and is very common in mixtures. It was first discovered and investigated by Kuenen in 1893 [4]. Figure 3 shows this

behavior more clearly. In this figure the highest temperature and pressure at

which the two phase exists (the maxcondentherm and maxcondenbar, respectively) do

not correspond with the critical point. Moving in the direction of increasing

temperature along the isobar AB in figure 3a, we see that we intersect the two

phase region at the point B, , where a less dense phase appears. Continuing along

this line more and more of this phae forms until we reach a point where it begins

to disappear and we emerge from the two phase region at B^ into a single,

dense liquid like phase. This process is called retrograde vaporization . Similar behavior is observed along the line isotherm oB in figure 3b which is

called retrograde condensation . If this behavior is not complicated enough, figure 4 shows a relatively

unknown type of behavior called double retrograde vaporization. This behavior

has been seldom discussed in the literature [5,6] even though it has been

experimentally observed in hydrogen/n-hexane mixtures [7]. In light of current

technological interest in hydrogen mixtures (e.g., coal liquefaction) further

experimental studies in this area seem warranted.

Figure 5 shows some isobaric cross sections of the type I space model

projected on the T-X plane. The upper curves are called the dew point

3
tei.iperature curve which gives the temperatures as a function of composition at

which liquid will condense at a fixed pressure. The lower curves are called

bubble point curves and give the temperatures at which the liquids begin to

vaporize. The sections at ^^

^"*^ P4 ^^ow the mixture critical points which Are

extrema on these projections. Note that in the cross section at p, there are tvw critical points. This corresponds to a critical locus which has an extremum

between tho critical points of the two pure fluids. Similar projections can be

made at constant temperature to obtain p-x curves as shown in figure 6.

In figures 5 and 6 the dew and bubble points were shown to be monotonic

functions of composition. In reality, this is frequently not the case. It is

SQr^ coi.imon to encounter chemical systems in which the pressure for an isothermal cross section or the temperature for an isobaric section attains a minimum or maximum value and the dew and bubble points become identical. These systems are called azeotropic mixtures and the different possibilities are shown in figure 7.

At the azeotropic point, the liquid and vapor phase compositions become identical

as is shown by the vertical, dashed line. From an industrial point of view this means that the components cannot be separated by simple distillation. There are occasions where the formation of an azeotrope is desired, e.g., azeotropic

distillation in a multi component mixture, so that an otherwise difficult

secondary separation may be made. It is also relatively common for the

azeotropic line in a binary mixture to intersect the critical locus thereby

giving rise to a critical azeotrope as is shown in figure 8.

Another variation in the behavior of mixtures is that of a miscibility gap . Usually this type of behavior is only observed for liquid or solid phases, but it can also occur in the high pressure vapor phase giving rise to the so-called

gas-gas or fluid-fluid equilibria. This type of behavior is shown in the

types II-VI exai.iples in figure 1. Usually the components are partially miscible

as is shown in figure 9. Within the closed loop shov/n in figure 9, two liquid

phases exist with compositions being given by the ends of the tie lines. The

temperatures and pressures labeled T , T. , P and P. are called the upper

and lower critical solution or consolute temperatures and pressures , respectively. Above or below these points the coiaponents are completely miscible

and form a single liquid phase.

Many systems only exhibit an upper or lov/er consolute point in which case the miscibility gap can intersect a two phase liquid-vapor region. This is shown
in figure 10. To further complicate matters azeotropic systems can also exhibit

partial miscibility as is shown in figure 11.

This brief description of qualitative fluid phase behavior has been included

to present a picture of the vast possibilities. For a more complete qualitative

discussion of different types of phase behavior see references [8-10].

3. Prediction of Phase Equilibria

The basic problem that we are faced with in mixture thermodynamics is to

predict the composition of the various phases which are in equilibrium. Once we have obtained the compositions of these phases v/e are then faced with the

secondary problem of predicting the thermophysical properties (e.g., entropy,

viscosity, enthalpy, etc.) of those phases. Normally the temperature and

pressure are specified along with the total, bulk composition of the system, but

frequently we must also calculate the dew or bubble point temperature or pressure

given only the overall composition and another variable. Certainly part of this

general problem is to be able to know how many and what types of phases coexist

at certain conditions. Even though this problem is quite complex, both

mathematically and conceptually, thermodynamics gives us some extremely powerful

tools for tackl ing it.

From thermodynamics we know that the condition for equilibrium in a closed

system is that at constant entropy and volume, for any infintesimal change in

state

dU^^, =0 (1)

Since we can always consider a multiphase system to be one large closed system we are thereby led to the conclusion that for equilibrium to exist v/e must have the

conditions

,(1) = p(2) ^ ^

p(ir)

(3)

yP^ = \l\^^ = . . . = u\'^ i = 1, N (4)

where T^^ , p^"^^ and u- ^ are the temperature, pressure and chemical potential

of the i component in the j phase. The super

"^" denotes a property of a
component in solution. Physically these equations mean that there is no heat

transfer between phases, no boundary expansion and no potential for mass transfer. The chemical potential is an intensive parameter v/hich, in general, a function of temperature, pressure and composition. From a predictive viewpoint

we can assume that the temperature and pressure of the coexisting phases are equal, so the prediction of phase equilibrium reduces to the problaa of

predicting the chemical potential at some specified temperature and pressure and

(to be determined) composition.

Most engineers do not, however, work directly with the chemical potential

but rather with a related quantity, the fugacity . The fugacity was defined by

G. N. Lewis [11] by the differential relation

du^. = RT d £n f^ (5)

and has the units of pressure. Mathematically the. fugacity is easier to work

with than the chemical potential for two reasons: (1) it approaches the pressure

as the fluid approaches the ideal gas state whereas the chemical potential

approaches minus infinity and (2) unlike the chemical potential it can be

calculated without a knowledge of thermal properties of the component, e.g.,

ideal gas heat capacities. The use of fugacity does not affect the criteria for

equilibrium -- eq (4) is merely transformed to be

fP = fj2^ = . . . = f^) i = 1, N (6)

e.g., the fugacity merely replaces the chemical potential.

Fugacity is frequently referred to as the escaping tendency and in order to

get a better feel for eq (6), it is appropriate to consider some qualitative

molecular aspects of phase equilibrium. In a system which is capable of existing

in two phases at some pressure and temperature, the molecules will have a tendency to "escape" from the phases in which they exist to another phase because

of their thermal energy. Unless this escaping tendency is exactly balanced by

the tendency of molecules to return to the given phase by escaping from another

coexisting phase [e.g., no potential for mass transfer], the phase in which the

escaping tendency is greater will disappear in favor of the phase of smaller

escaping tendency. The fugacity (and chemical potential) is a measure of this

escaping tendency and thus v/e can grasp the meaning of the criteria that
As was mentioned previously, the prediction of phase equilibrium reduces to

the prediction of fugacity. There are basically two ways of achieving this

goal: (1) equations of state v/hich incorporate a corresponding states principle

and (2) specific correlations or models for fugacity.

3.1 Mathematical Considerations for Calculating Phase Equilibria

Before proceeding to discuss specific means of calculating phase equilibria,

let's briefly consider the philosophy of how one actually performs the

calculations. From thermodynamics we know that at equilibrium the Gibbs energy

of the entire system is at a minimum, i.e., at constant temperature and pressure

dG^^p < (7)

Since for a composite system comprised of several phases in equilibrium we have

titG. = L n|^'^ G^J) (8)

' ' j=l '

and for every phase in equilibrium

G^J) = E np^ yjJ'^ (9)

i = l

^ ^

we can use a non-linear minimization routine to find the absolute minimum of the

Gibbs energy of the system, subject to the material balance constraints that

nx = E nV^ and n. = L n^^ (10)

' j=l ' '' j=l '

i.e., material balance. In these equations G^"^' is the molar Gibbs energy of

the i phase, jj| ^ is the chemical potential of the j component in the i phase and r\\'^' is the number of moles of the i component in the j phase and

the subscript "T" denotes the total, composite system.

This procedure, which is conceptually simple, is time consuming from a numerical point of view and is susceptible to the usual problems of non-linear

optimization, i.e., local minima. In addition, in order for the procedure to be

perfectly general it requires a "phase splitting" algorithm which is capable of

deciding when another phase will form and what its major components will be.

This is obviously a yery difficult task and only recently has there been an algorithm proposed which shows promise of being successful. This method was developed by Guatam and Seider L12,13J and has been included in the project ASPEN

7
simulation program. Several other Gibbs energy minimization algorithms have also

been proposed [14-16].

Because of the difficulties with direct minimization, most investigators use what is called a multiphase flash algorithm for phase equilibria calculations.

The basic limitation of this type of calculation is that it is not easily

expanded to more than three phases and can be unstable near a critical point,

nonetheless this method is applicable to a wide range of phase behavior and is

probably used in 99 percent of all phase equilibrium calculations.

A flash calculation is essentially a direct solution of the material balance

eq (10) for the coexisting phases. Its name is derived from what is called a flash separator or single stage distillation in chemical engineering. The

general scheme of a 2-phase flash calculation is as follows. If there are n. moles of a component in a mixture which potentially could be two-phased, we can write without loss of generality a material balance equation

xf + y^V = Z.F = n. i=l, N (11)

where L, V and F are the total number of moles in the liquid, vapor and bulk

mixture phases and x. , y. and Z. are the mole fractions of component i in the

liquid, vapor and feed phases respectively. By overall material balance for the

mixture

and we may write

and

F = L + V (12)

Z. = [(1 - R) K. + R] X. (13)

Z^ = L(l - R) + R/K^] y^ (14)

where R is the so-called liquid to feed distribution ratio, L/F, and k. is the

equilibrium K-value y./x.. The K-value is yery common in engineering distillation calculations and gives the equilibrium distribution of a component between

coexisting phases (in this case liquid and vapor). By rearranging these

equations and summing over all components we find that

E Zi L(l - R)K. + R]-' = E Z.K. [(1 - R)K. + R]-' = 1 (15)

i=l ^

^ i=l ^ ^ ^

Subtracting, v/e find that our material balance equation takes the form

8
Z^. (1 - K.)L(1 - R)K. + R]""" = (16)

The method of calculating phase compositions is then to make an initial guess at

the K-values (typically Raoult's law) for e'^ery component in the system and then

numerically solving the material balance equation for R. Then, new phase

compositions are calculated using the relations

X. = Z./[(l - R)K. + R] (17)

y^ = Z.k./[(1 - R)K. + R] . (18)

These new compositions are then used in the calculation of the component

fugacities. If the fugacities of the components do not match, new K values

are calculated from the expression K. = (i>\ ' /<^\ ' where (t)J /\/\ 111 111 '' = fj /x.p and

^y) = fvVy-p, i.e., the fugacity coefficients in each phase, and a new

and set of compositions are calculated. The iteration stops when two successive

values of R match and the fugacities of all components satisfy the criteria

del ineated earl ier.

There are also some variations of this procedure, namely dew and bubble

point calculations which are performed by forcing R to be either or 1

,

respectively, and searching for the temperature and pressure that make the

fugacities match. For extensions of this method to three phases see [17-20].

4. Mixture Equations of State

As mentioned previously, the first method of calculating fugacities for

phase equilibrium uses an equation of state. Given an equation of state and the

ideal gas heat capacities for the mixture, it is possible via straightforward

thermodynamics to calculate all of the thermophysical properties of interest,

e.g., enthalpy, entropy, energy, etc. In order to calculate the fugacity, one

uses the relationship

f.

RT £n

"" X.P

K X

3P or -^ - KT iln Z (19)

As of today, no investigator has reported a perfectly general equation of state

which can be applied to a wide spectrum of mixtures whose components differ

greatly in polarity and size. Instead, specialized equations have been developed

v/hich apply to limited ranges of niixtures, for example, natural gases and gas

liquids, and perhaps mixtures of aliphatic alcohols. The primary hinderance to
the development of what might be called a master equation of state is rooted in our level of understanding concerning the interactions of chemically dissimilar

molecules. Nevertheless equation of state methods for predicting phase

equilibria in nonpolar mixtures have developed rapidly during the last ten years.

Generally speaking these equations may be classified as being a member of one of

four possible families [21]: (1) van der Waals; (2) Benedict-Webb-Rubin;

(3) Reference Fluids Equations or (4) Augmented Rigid Body Equations.

4.1 van der Waals Family

Equations of state within this family are probably the most widely used for

engineering calculations. This popularity arises from the simplicity of the

equations which enables them to be expressed in terms of dimensionless two or three parameter corresponding states functions [21]. Generically these equations

have a cubic volume dependence and may be written as

RT

aCT^u))

P = V-b (1^,0))

- g(V,T^,a,) (20)

where g is a function of volume which typically depends on the b parameter and,

perhaps another parameter. The most important variants of this equation are summarized in table 1. These equations are typically written for pure fluids and

then extended to mixtures as discussed below. Before proceedings to that

discussion it should be noted that once the pure fluid critical point constraints

have been imposed

= (s), iV)

only the values of b{l,a)) a(l,a)) and g(V ,l,a)) are fixed. This means then that

the temperature dependence of these functions may be adjusted to obtain optimal

agreement which some selected properly. The members of this family which are the

most successful in predicting phase equilibria (Soave, Peng-Robinson and

Harmens-Knapp) adjust this temperature dependence so as to provide good agreement

with pure component vapor pressures. For example, with the Peng-Robinson

equation

R]L_ '(V*"^^

P v-b " V(V-b) - b(V+b)

the constant b is given by Q,b where ^, is a universal constant and be b

10
2

b^ = RT^/Pc» a(Tp,tD) = a^a(T^,a)) where a^ = ^g{RT^) /p^ and

a(T^,a)) = [1 + m((.)(l - 1^^'^)^

m((jo) = m-, + mM + mva)

The function a was chosen to fit vapor pressures for a series of compounds

including the normal paraffins in the temperature range 0.7 <. T < 1 and

generally gives good phase equilibria predictions for mixtures containing those

fluids.

Table 1. Summary of Common v

Equation Reference

van der Waals [96]

Redl ich-Kwong [97]

Soave [98]

Peng-Robinson [99]

Fuller [100]

Harmens [101]

Usdin [102]

g(V.T.a))

T^/^ V(V + b)

V(V + b)

V(V + b) - b(V - b)

V(V + cb)

V^ + Vcb - (c - l)b^

V(V + d)

As pointed out by Lei and [21] the selection of a(T ,0)) to fit pure

component vapor pressures results in poorer representation of other features of

the PVT surface, e.g., enthalpy. This arises because of theoretical flaws in the

Van der Waals family of equations, primarily in the repulsive, hard sphere

pressure term RT/(V-b) [23].

In order to apply these equations to mixtures, mixing rules must be

introduced. For the van der Waals family, the appropriate mixing rules can be

deduced on theoretical grounds by considering the conformal interactions between

different molecular species and the proportionality between the intermolecular

potential parameters and critical constants [24]. The end result is

a ,

= E E x.xj a.j (21)

". = ^ ^Vi "ij (22)

11
These are called the van der Waals one-fluid mixing rules. In order to apply

them to a mixture, combining rules which relate the binary parameters a. . and

b. . to the pure component parameters. For this family they are defined by

a,.j = (a^aj)l/2 (1 - k.^) (23)

,3 .. = l(b]/3^b]/3) (1.,..) (24a)

or

b.j = 1 (b. . bj)(l - £.j) (24b)

where k.. and £. . are called binary interaction coefficients which must be

determined from experimental data. Usually the volume interaction coefficient,

£.., is taken as being zero. Compilations of k.. values applicable to the

Van der Waals and other families of equations of state have been reported

[25,26]. In addition, several correlations for k. . have been given in the

literature [27-31].

As mentioned previously, recent members of the van der Waals family have

been very successful in predicting the phase behavior of hydrocarbon mixtures

which possibly include common inorganics such as CO2 , Np, etc. Many comparisons

of predicted and experimental phase equilibria have been given in the literature

for these equations [32-40]. For our purposes figure 12, which was taken from

the original Peng-Robinson (PRS) [32] paper suffices. This figure compares the

equilibrium K-values for a six-component mixture as a function of pressure. In

general the results are excellent with the only serious errors being in the

heaviest component at low pressures. The Soave equation (RKS) gives similar

results for this system. Both equations give results which are accurate to

within +5.8 percent which in many cases corresponds to the uncertainty in the

experimental data. The PRS equation has the advantage of being more accurate in

its predictions of the liquid phase density.

More recent activity related to this family of equations has dealt with

extending them to water/hydrocarbon systems. Application of these equations to

this type of system without modification tends to overpredict the solubility of

water in the hydrocarbon rich phase and underpredict the hydrocarbon solubility

in the water rich phase. This can be explained intuitively by observing that

these equations use the critical constants of the pure fluids as a measure of the

12
strength of intermolecular interactions and further, have no nieasure of induction

effects (e.g., induced dipoles in nj^drocarbons due to strong electric fields in

water) or of structure breaking effects on water. Thus, the critical constants

of water in a fiydrocarbon phase are too large since its hydrovjen bonded structure

has been broken (predicted solubility is too hign) and the hydrocarbon critical s

in a water phase are too small since they have induced dipoles which interact

strongly with the water (predicted solubility ii too Sinall).

There have been three basic approaches to this problem. The first raethod

which nas been proposed [41,42] is to use temperature dependent binary

interaction coefficients in the water rich phases, i.e..

a,, = {a,.a,)-(1-Ij(T))

and in a hydrocarbon rich phase

a, . = (a,a.)'/'^(l - k..)

This method although highly correlative in nature has been yery successful.

Figures 13 and 14 demonstrate some typical results which were reported by

Robinson [41].

The second approach concentrates primarily on the wter solubility in the

hydrocarbon phase and amounts dividing the attractive a parameter into a polar

and nonpolar contribution and allowing them to interact tensorial ly , e.g..

^ - s,)

^^^

where the superscripts "P" and "N" denote polar and nonpolar. B> allowing

p a. to nave an induced contribution from the water it is also possible to

obtain an improved representation of the hydrocarbon solubility in ttie polar

phase. This approach was proposed in some fonn by Chueh [43], Nakamura,

et al . [44] and Won [45] and has been moderately successful. Figures 15 and 16

show some typical results.

The third approach which has been proposed handle polar-nonpol ar systems

with cubic equations of state has been to use a binary interaction coefficient

with tne repulsive, b jjarai.ieter, as indicated in eq (24). This approach, which

has been proposed by Heidemann [46] and Erbar [47] is also reasonably successful.

Figures 17 and 18 demonstrate some of their results.

13
In spite of the relative success of these modifications in correlating

polar/nonpolar phase equilibria, they still leave much to be desired. For

example, it is not generally possible to a priori predict the required binary

interaction coefficients nor is it possible to determine the polar/nonpolar

separation of the attractive coefficients. In addition, this type of model is

merely correcting a model which is basically too simple and incorrect in detail

to make it work for more complicated systems.

Another approach to the correlation and prediction of phase equilibrium

based on a van der Waals family equation has been proposed by several

investigators. It is based on modifying the composition dependence of the mixing

rule, eq (21). This method enables one to correlate data for complex mixtures

but the equation of state parameters can no longer be identified with critical

constants. In addition, these models do not give the theoretically correct

composition dependence of the second virial coefficient. For details of this

method see references [48-51].

4.2 Benedict-Webb-Rubin Family

The second class of equations which are commonly used in engineering phase

equilibria are those of the Benedict-Webb-Rubin (BWR) family. The original

BWR [52] equation contains eight adjustable constants and is given by

P = pRT + (BqRT - Aq - Cq/T^)p^ + (bRT - a)p^ + aap^

(25)

2 + (cp^/t2)(1 + YP^)e-^P

In order to use this equation for mixtures, mixing rules such as those used for

the van der Waals equations are used. In general they are given by

^m (? > :"1

Values of r for each coefficient are given in table 2. It should be noted,

however, that unlike the van der Waals equation, these mixing rules have no theoretical basis. This type of equation offers vastly improved thermodynamic

property predictions but does not offer any better results for the phase

equilibria. It suffers from a second short coming in that it requires a large

number of coefficients for eyery component in the mixture and it tends to fail at

low reduced temperatures.

14
Table l. Benedict-Webb-Rubin Mixing Rules

n \ . J J /

Constant (a^^) £

«0 1 ^

2

^0 2

a 3

b 3

c 3

a 3

Y 2

Starling [53,54] addressed these latter two shortcomings and thereby greatly

expanded the use of this type of equation. The basic form of his equation (BURS)

is as follows

P = pRT + (BqRT - A^ - Cq/T^ + Dq/T^ - Eq/t'^)p^ + (bRT - a - d/T)p^

+ a(a + d/T)p^ + (cp'^/T^)(l + yp^) exp(- yp^)

(26)

which has eleven adjustable constants. In its original formulation, mixing rules

such as those for the BUR were used except geometric means were recommended for

the A^, C , D and E coefficients. These mixing rules are summarized in

o' ^ table 3. Later hov/ever Han and Starling L55] generalized the BWRS constants in

terms of the critical constants and Pitzer's acentric factor, for example

The appropriate equations and coefficients are summarized in table 4.

15
Table 3. Benedict-Webb-Rubin Starling Equation of State

Mixing Rules

^m = It "j^j

;

Constant

(^n,) r ^

1

a 3

b 3

c 3

d 3 a 3

Y 2

a„ = E E x,x. a|/^ a]/^ (I - .,.,"

1 J

Constant (a„)m

16
Table 4. Generalized Coefficients for the BURS Equation

p .B . = A, + B,(jo.

^Cl 01 111

p .A . ^Cl 01 ^ ^ •)

ci

p .C . Cl 01 A . D

^' ci

P^i^i = ^4 ^ ^4'^i

p2^.b. = A5.B3C..

P^i^

p^^.a. = A^ + B^a..

p -C • ^ Cl 1 „ ^ D

'^' ci

p .D . Cl 01 A J. D -1 = ^9 ^ S^i

RT^ . J y 1 ci

p2^ .d.

Cl 1 _ „ , R ;Z2— - ^0 ^ ^10^-

^' Ci

^^ = A^^ +B^^u3. exp(- 3.8..)

'^' ci

This equation has been extremely successful in predicting the properties of

natural gas mixtures and has been used to design a substantial number of pipepipelines and gas processing plants. Recently, Starling and coworkers [56j have

given this model a face-lift by essentially making the acentric factor an adjustable variable and by using molecular-size and energy parameters for reduction of

the density and temperature. In the generalization in terms of the critical

constants, the acentric factor has been eliminated in favor of a parai;ieter

labelled as y which is usually close to w. Generally speaking this equation

17
offers only marginal ii,iproveinent over the original BWRS. Other modifications of

the BWR equation have been proposed and the interested reader should consult

references [57-60].

The BWR family of equations has not been successfully extended to mixtures

containing highly polar components such as water. The primary problem in this

area lies in the large number of parameters used in this equation and the lack of

any theoretical guidelines. In fact, the BWRS has never been successfully fitted

to pure water so the parameters required for the mixing rules are not available.

This problem is under current investigation by Starling and his coworkers L61j.

4.3 Reference Fluid Equations of State

During the past 15 years large quantities of highly precise (and accurate)

PVT and thermodynamic property data have been measured by various laboratories.

With the advent of these data, complex equations of state have been developed to

represent these data without regard to mathematica-1 simplicity or eventual

generalization to other fluids. Notable examples of this class of equations are the 32 term BWR proposed by Stewart and Jacobsen L62] and the nonanalytic

equation developed by Goodwin [63].

In order to apply this type of equation of state to other pure fluids or mixtures, conformal solution theory must be used. This theory is based on the

assumption that within classes (e.g., homologous series) of fluids the

intermolecular potentials are given by

Uj (r) = ej f{r/o.)

for a pure fluid and

u^j (r) = e,j f(r/o.j)

for an unlike binary pair. This leads to the conclusion [64,65] for this class

of fluids that

and

Z^.(V,T) = ZQ(V/h., T/f.) (27a)

A^ =

-^i Ao(V/h.. T/f.) (27b)

18
D where Z is the compressibility factor, A is the residual Helmholtz free energy

and f. and h. are called equivalent substance reducing ratios which are defined

by

h. = v^/V^^ and f. = T^/T^^ .

The subscript "o" denotes the reference fluid. Lei and and coworkers [66,67]

further extended this two parameter corresponding states model to fluids having

assymetric, but not dipolar interactions by introducing shape factors in the

equivalent substance reducing ratios, viz.

h. = (V^/V^) *. (T^ , V^ , 0).) (28)

fi = (^/^o) S- ^V.'V.'^') (29)

1 1

The shape factors have been fit to a generalized mathematical form and are given

in reference [67]. Given eqs (27-29) it is possible to calculate all of the

thermodynamic properties of a pure fluid belonging to the same conformal class.

In order to extend this method to mixtures, mixing and combining rules must be

introduced for the parameters f and h. Usually they are chosen in accordance

with the van der Waals one-fluid theory [68]

a 3

and

Ot p

although other choices are possible. The combining rules used with this model are

f o = (^ ^o)^^^ (1 - k J (32) ag ^ a 3 a3

and

\e ' I ^^'l" ' ^^r^'^' - ^cb'

(33^)

or

19
Using eqs (30-33) in conjunction with (27-29) enables one to calculate any

mixture property of interest. For example, for the fugacity of a component in

solution

U*^ 9f 3h

X a X a

where Uq and Zq are the residual internal energy and coinpressibil ity factor of

the reference fluid. Several authors [69-72] have explored the predictions of

this method using a methane reference and figures 19 and 20 show some typical

results. In general, results obtained with this i.iethod using a methane reference

are yery accurate if the system doesn't contain components with molecular weights

greater than Cy or associating components [69]. Furthermore, the method suffers

from being mathematically coniplex which historically hinders industrial

acceptance.

A second reference fluid equation of state method has been proposed by

several authors [73-76]. It is based on the original Pitzer corresponding states

i.iodel [75], but uses more than one reference fluid. For a two-fluid model [74]

it takes the form

Z(P,,T^) Z ^(2) _ Jl)

LZ Z ]

where the superscripts (1) and (2) denote the reference fluid values. In order

to apply this method to i^iixtures, pseudocritical parameters must be defined via

mixing rules such as^n^cm = ^ ^

^i^j 'c. ^c,.

03 2-, X. CO. , etc,

111 . 1 1

most applications of this method have dealt with prediction of mixture density

and enthalpy and the results are very good [75]. Current work with this approach

deals with phase equilibria and critical lines [77j.

4.4 Augmented Rigid Body Equations of State

The final category of equations of state is similar in some respects to the

van der kJaals family, but is set apart because of the abandonment of the van der

20
Waals repulsion term RT/(V-b) . These equations start with theoretically based

rigid body equations of state and add terms to account for the effect of

molecular attraction. The rigid body terms are the Wertheim-Thiele [82],

Carnahan-Starl ing equation of state for hard spheres [78] and the equations of

Gibbons [79], Boublik [80] or Nezbeda and Lei and [81] for rigid nonspherical

bodies. Examples of this class of equations are the perturbed hard chain theory

[83,84], augmented van der Waals theory [85-87] and the Hlavaty equation of state

[88,89] among others [99-94].

Of particular interest in this class are the augmented van der Waals

equation developed by Kregleski and cov/orkers [85-87] and the perturbed hard

chain theory of Prausnitz, et al . [83,84]. These two models have been applied to

polar/nonpolar systems with moderate success. Recently [95] the Prausnitz model

has been successfully applied to water and water/alcohol /hydrocarbon containing

mixtures. Generally speaking, this family of equations of state is in a developmental stage and has not yet found widespread industrial use. They

appear, however, to offer the most economical route to phase equilibria in

polar/nonpolar systems.

4.5 Critical Loci From Cubic Equations of State

Latter portions of this report deal with the prediction of critical loci

using the reference fluid equation of state approach and the Leung-Griffiths

model [103-105]. It should be pointed out in passing that a considerable amount

of effort has been made in predicting these loci using members of the van der

Waals (cubic) family of equations of state. Scott and von Konynenburg [1,2] have

shown that with the appropriate choice of parameters in the original van der

Waals equation, all known types of critical lines in binary mixtures may be

qualitatively predicted. More recently, Peng and Robinson [106] have developed a numerical method for predicting critical lines in multicomponent mixtures. They

applied this method using their equation of state to both binary and

multicomponent liquid-vapor mixtures having up to 12 components. Their

comparisons showed prediction of the mixture critical temperature to within an average absolute error of 4 K (% 1 .3 percent) and pressures to within 173 kPa

(>. 2.3 percent). Predictions of the critical volumes were substantially worse

(x 12 percent error) which is not surprising since cubic equations of state are

not very accurate for density prediction.

21
The main advances in this area have been in the coiTiputational methods that

can be used with cubic equations of state. In particular the algorithm of

Heidemann and Khalil [107] which is based on the Helmholtz (rather than Gibbs)

free energy. This method used with a cubic equation of state requires only a few

milliseconds computational time regardless of the number of mixture

components l107].

5. Phase Equilibria From Liquid Phase Activity Methods

As was stated in section 3 there are two methods of predicting and/or

correlating phase equilibria -- equations of state and activity coefficient

methods. By and large, equation of state methods are primarily limited to

hydrocarbon systems with some current methods being directed tov/ards

polar/nonpolar systems such as water hydrocarbon systems. Obviously there is a wide spectrum of other iiiixtures which are routinely processed and for which phase

equilibria are predicted for engineering design purposes. The technique which is

used for these calculations is not new and is what is called classical solution

thermodynamics. It amounts to defining an idealized model of a mixture called an

ideal solution and describing deviations from this model in terns of excess functions.

The necessary and sufficient condition for phase equilibrium is that the

temperature and pressures of all phases are equal and that

f,P) = ff = . . . = f(") i = 1. N

Ideally it would be nice if a solution behaved like a group of individual pure

components weighted by some measure of their concentration. In other words, we would like for a liquid

where f. is the pure component fugacity and for a vapor

i.e., ideal gas. The combination of these leads to a definition of an ideal

solution, i.e..

22
y^.p = x.f.

Considering the case of a low temperature, low pressure liquid f? = p^(T),

e.g., the vapor pressure and we find

y^P = Pi = x.p^

which is Raoult's law. Given this definition one can define all of the

thermodynai.iic functions of interest in the ideal solution which are given in

table 5.

Table 5. Thermodynamic Properties of an Ideal Solution

Property Dependence on Mixing Contribution

Pure Component Properties

Gibbs Energy^•^ G = zn.u° + RT 2 n . £n x. RT 2n. iln x. 11 11 11

Enthalpy H = 2n.h°

Entropy S = ^n.s^ - Rzn. in x. - Rzn. £n x.

Volume V = J:n.v^

Heat Capacity C = zn.c .°

K Value K. = p*^(T)/p

Relative Volatility a.. = p?{T)/p^(T)

The superscript refers to a pure component value -- not an ideal gas. Also,

lower case symbols for extensive properties refer to molar values.

23
Just as no real gas behaves ideally no real solution behaves ideally. What

we do, therefore, is to define theri.iodynamic excess functions which are over and

above those of the ideal mixture, e.g.,

M^ = M - M^^

where M is any thermodynamic property. Obviously the excess functions satisfy

the same thermodynamic relations that the total functions satisfy. Next without

any loss of generality

and

where ({>. is called the vapor phase fugacity coefficient and y. is called the

liquid phase activity coefficient.

Normally one calculates (j). from a simple vapor phase equation of state such

as the PRS, RKS or virial equation of state via the integral relationship presented in our discussion of equations of state. Even though this leads to some inaccuracies (pressures lov^er than 10-20 atm) the quantities seldom deviate from

unity by more than 10 percent. On the other hand, liquid phase nonideal ities as reflected in activity coefficients can be and frequently are VQry large and in

fact can change by many orders of magnitude. For example y ^or a hydrocarbon in

9

water may be around 10 whereas for methane in ethane it is around 2.

Thus in this formulation the approach to phase equilibria is to develop

predictive and correlative methods for the liquid phase activity coefficient.

This, in effect, is to develop models for the excess Gibbs energy since from

thermodynamics

G^ = G - G^^

E /3g!\ U\ (^\

V'"l/T.p.n, V"lA.p,n, V"lA.p,n.

G^ = RT tn (f^/fj''^'^

24
or

G^ = RT in Y^

where the bar indicates a partial molar quantity. This partial molar excess Gibb's energy is simply the log of the activity coefficient. Since

G^ = S n.G^

we see that

G^ = RT E n. £n y.

In this overview consideration will be given to four of the most popular

models for excess Gibbs energy in liquids which are correlative and two

predictive models which are based on a group contribution concept. The first of

these is the Margules model. Before doing that, however, let us consider some

general points about activity coefficients.

Figure 21 illustrates the different kinds of deviations from ideality that are commonly encountered in vapor-liquid systems in terms of partial pressures,

activity coefficients, and y-x diagrams. There are two important features that

are shown in this figure. The first, and most important, is that the activity

coefficient of any component approaches a finite limiting value as the

concentration of that component approaches zero. This limiting value is of

upmost importance in activity coefficient correlations (or Gp correlations)

and is given the symt

activity coefficient

and is given the symbol of y^ ory.. It is called the infinite dilution

lim y. = y°

X. -^0

As will be shown, a knowledge of only the infinite dilution activity coefficients

enables us to calculate the activity coefficient over the entire composition

range. It will also be shown that infinite dilution activity coefficients follow

very systematic trends within a homologous series which enables one to predict

y.'s for many compounds based on experimental measurement of a few key members

of a homologous series.

25
The second point which is of interest with regard to figure 21 has to do

with azeotrope formation and miscibility gaps. In the case labeled "large

positive deviations" we see in the y-x diagram the vapor and liquid compositions

actually become identical at x = .85. This corresponds to azeotrope formation,

which as you can see by comparison is not possible in an ideal solution. In the

system with \/ery large positive deviations, the dotted lines indicate a miscibility gap, which also is not possible in an ideal mixture. Thus, it would

not be possible to do azeotropic distillation or liquid-liquid extractions if it were not for liquid phase non- idealities.

Returning to the expression for k-values in non-ideal systems, it is

convenient from a computational point of view to separate the pressure dependence

of the pure component fugacity and activity coefficient, formally, the greatest

contribution to the pure component liquid fugacity comes from the vapor pressure,

therefore it is convenient to rewrite this term to make that dependence explicit.

Since

\ ^P A.n " «T

the change in the fugacity in compressing the pure fluid from vapor pressure

p. to the system pressure p is

e, v°

ion f°(T,p)/f°(T,p^) = / rT ^P

•10 Pi

Since for the pure component at p?, f. = 4!:{p^,T)p^, we have

P V?

•/o

f?(T,p) = P^ l-^-lp^.T) exp r ^dp (35j

Pi

There is also pressure dependence in the activity coefficient. Most correlations

of excess Gibbs energy (i.e., activity coefficients) are for a standard or reference pressure, p . The pressure dependence is then separated in the same manner that we separated the pressure dependence of the pure component liquid

fugacity, only using the equation

26
3 £n f^. V.

3p RT

After performing all the appropriate integrations and Manipulations we find that

P V. - V.

Yi{p,T, \x.\) = Y^(p*,T, \x.\) exp T \j dp . (36)

As was mentioned earlier, one of the key quantities in VLE is the k value

y./x.. Using the solution thermodynamics formulation of fugacities we find

for the k value

^i <^.P

substituting in the results from eqs (35-36) we find

k

i e^p

where e^ is defined as

^(T^p.iyil) (

9^. = exp < - ^{T.p") '

and is called (the "vapor imperfection coefficient" by some authors. Nori.ial ly the

reference pressure, p*, is chosen as one atmosphere. Also, at low operating

pressures, the integral terms are small and may be neglected so that e. reduces

to 4»^/<t)^ which below 2 atm is also close to unity. Table 6 summarizes the

important thermodynamic relationships for non-ideal solution theriiiodynamics. The

expressions given in this table coupled with those in table 5 enable one to

calculate all the thermodynamic properties of mixtures. The important thing to

remember is that once one has the activity coefficients of all the mixture

coinponents as a function of temperature and pressure, one can calculate all of

the mixture thermodynamic properties.

27
Table 6, Relations Between the Activity Coefficient and Thermodynamic

Functions Which are Useful in Vapor-Liquid Equilibria

Function or Variable Relation to the Activity Coefficient

Excess Gibbs Energy, G G = RT z n . i2,n y^

Excess Enthalpy, H^ H^ = RT^ [3(G^/T)/3Tj ^

= - RT^ £n.0£nY/3T)p^^

Excess Entropy, S*" S^ = - (9G^/3T) ^

= - R Zn.L^ny^ - T(3£nY ./3T)p^^j

Excess Volume, V^ V^ = (3G^/3p)-p

^

= RTZn.{3£nY./3p)^^^

Activity a. = y,-x.

k-value K. = y.p^.{l)/<^.p

Relative Volatility a.^ = Y^.p°(T) ^yYjPj(T) 4.^.

^i (v° - V.) p V.

e. = [*./.°] exp / -^^^ dp - jr (^) dp

Pi

5.1 Azeotropes and Miscibility Gaps

Before going on to specific correlations for the excess Gibbs energy, we need to briefly consider the circumstances under which azeotropes and multiple

liquid phases are formed. For the sake of simplicity, let's only consider a binary mixture.

Azeotropes occur whenever the relative volatility becomes unity, i.e., the k

values of the two components are identical. There really is not anything magical

about this type of behavior, it just so happens that

28
o o

4>-|P '^2^

at some temperature, pressure and composition. If the pressure is low enough so that (f>. = 1 (say one atmosphere for most systems), this identity reduces to

a a YiPi = Y2P2

The term y.p? is frequently called the volatility and is like a corrected

vapor pressure of a component in a liquid mixture.

Given the infinite dilution activity coefficients and the assumption that

the vapor phase behaves ideally, it is possible to predict from the pure

component vapor pressures whether or not an azeotrope will form. Consider, for

example, an isothermal system which exhibits positive deviations from ideality.

In this case both activity coefficients are greater than unity. If we number the

components such that "1" indicates the more volatile component, the maximum value

the relative volatility a-.^ = K-i/Ko can attain is

(a^2) =

li"! (y-|P^/Y2P2) max x-, -^0

" /_0 ,_0\

= Yf(Pi/P2)

At the other end of the composition range (x-. = 1 )

, we find that the other

limiting value of the relative volatility is given by

O /_0\ / 00 (^12) . = (Pi/P2)/Y2

min

Thus, the total range of a, « is given by

a a h 1 Pi oc a 00 _< "12 < a ^1

h ^2 P2

If a,2 is unity (azeotrope formation), we find that by multiplying this

o ,_a inequality by p^/Pi that

29
— < (pp/p?) < Yr (positive deviations)

If this criterion is satisfied, there will be a minimum boiling azeotrope.

In a similar fashion one can show that for systems in which there are negative deviations from ideality, a maximum boiling azeotrope will form if

yT < (Pp/P-i) ^ I/Yo (negative deviations)

The main thrust of this discussion is to point out that there are no fundamental

behavioral differences between systems which are azeotropic and those which are

not. It just so happens that in the former case, the vapor pressures and

activity coefficients have magnitudes such that y-i P? = Yp P?*

One last point concerning azeotropes is that if the vapor phase is ideal,

Y^-x^.p'^/^^.y^p = 1 , or since x^ = y^. , y^- = (p/p^) • Thus, azeotropic data give

activity coefficients directly.

Now let us briefly turn our attention to immiscibility. At a fixed

temperature and pressure, a stable state is one in which the Gibbs energy is a minimum, i.e., for any infintesimal change in state, 5G _< 0. This means that a liquid mixture will only split into two distinct phases if upon doing so, it can lower its Gibbs energy. If we were to expand the Gibbs energy of mixing in a Taylor series, we would find that mathematically the criterion for immiscibility

is that

2

^-^ < (constant, T,p)

9x^

For a binary mixture, this amounts to the criterion that

9x^ \^1 ^ ^2/

where the second term on the left hand side comes from the ideal Gibbs energy of

mixing. Thus, if a mixture is to split into two liquid phases, it must

"overcome" the ideal mixing contribution. Since G is only a function of

temperature and the magnitudes of the activity coefficients, we see again that

immiscibility does not represent any abnormal behavior, but rather is a consequence of the non-ideality of the system.

30
Finally, the equilibrium criteria for phase equilibrium require that for a system which exhibits two liquid phases

or, using our definition of liquid fugacity in terms of activity coefficients

Yi X. = Yi X.

Thus, a knowledge of the liquid phase compositions (also called the solubility

limits) gives the ratio of the activity coefficients directly.

5.2 Excess Gibbs Energy Correlations

Thus far, statistical mechanics has not provided us with an adequate

theoretical basis on which we can develop prediction or correlation techniques

for liquid mixture properties containing chemically dissimilar species. Whenever

there is a lack of a definitive theory, there are always many seemingly different

correlations for the same property and the excess Gibbs energy is no exception.

5.2.1 Margules Equation

At a fixed temperature, the excess Gibbs energy of a mixture depends on the

composition of the mixture and, to a lesser extent, the pressure. If we consider

a binary mixture where the excess properties are taken with reference to an ideal

solution where the standard state is the pure component fugacity at p and T, the

molar excess Gibbs energy, g

= G /n must satisfy two boundary conditions:

g = when x-, or Xp =

The simplest non-trivial expression which obeys these conditions is

g = A x^Xg

where A is a function of temperature, but is independent of composition. Since

(9G /9n,-)p T D

~ '^^ ^"

'''i' ^^ ^^^^ ^^^^ differentiation

2

RT £n Y-i = A Xp

and

31
2

RT £n Y2 = A x^

These equations are called the two-suffix Margules equations [109] and are reasonable representations of simple (nearly ideal) liquid mixtures. Notice that

the predicted activity coefficients are symmetrical. Also, this correlation

implies that both infinite dilution activity coefficients are equal, i.e.,

Yf = Y2 = exp (A/RT)

The two suffix Margules equation is yery simple and requires only one piece

of data (y-i or Yo) ^^^ its application. A convenient extension of this

equation due to Redlich and Kister [110] is given by

E ^

g = x-jXp E a (x, - Xp)

' '^ n=0 " ' '^

This type of expansion leads to power series expansions for the activity

coefficients of the form

RT £n Yi = 4 ^

""n^

""

' "^ n=0 "

X,

RT iln Yo = x^ E a[^^ x^

"^

' n=0 " '

which are called M + 2 suffix margules equations. [A k-suffix Margules equation

gives £n y-i (or iln Yo) ^s a polynomial in Xp (or x-, ) of degree K.] Since, in

general, aj^ ' = aj^ ' , these relations do not predict symmetrical activity

coefficient curves. Most physical models for g have in them an implicit assumption concerning

the structure of the liquid phase, i.e., they imply that the local structure of

the liquid is determined solely by the interactions of binary pairs. This

certainly is not the case in reality, but it is a necessary simplifying

assumption. If we make this assumption, it is a trivial matter to extend the

two-suffix Margules equation to multicomponent systems. We find

32
where both sums extend over the number of components in the solution,

A.. = A.. =0, and the factor of 1/2 is included to avoid double counting.

Upon differentiation with respect to one of the mole numbers, we find that

RT Hn Y,, = E E (A,^ - -1 A.j) x.Xj

where all the A. . are determined from binary data.

5.2.2 van Laar Equation

One of the earliest attempts to form a rational physical model for liquid

phase mixtures is due to van Laar [111], van Laar considered a mixture of two

liquids and assumed that they mixed at constant temperature and pressure such

E E E E that V and S were identically zero, in which case G = .He then devised

a thermodynamic cycle for the mixing process and used the van der Waals equation

of state to calculate the energy changes during the cycle. The net result was the expression:

Sl. = /l2 ^21 ^1^2

RT (A^2^l ^ ^21^2^

Differentiating, we find for the activity coefficients

iln Y^ = A^2 ^^ ^ (A^2/^21^^^l/^2^^"^

and

^n Y2 = A2^ [1 + (A2^/A^2)(^2'^^l^^"^

These equations provide a direct relationship between the equation parameters

Ai2 and A21 and the infinite dilution activity coefficients, viz.

and

In Y^ = A^2

^nY2 = A21

The van Laar equation is extremely easy to work with and can adequately represent

moderately non-ideal systems. In general, however, it is not capable of

33
representing strongly non-ideal systems, especially those which exhibit

association or strong physical interactions.

5.2.3 Wilson Equation

For mixtures which have no excess enthalpy (athermal solutions) but whose

components do differ in size, Flory and Huggins [112] derived the following

expression for the excess Gibbs energy

^ = E X. £n (X./x.)

where X. is some ineasure of the size of the molecules, e.g., a volume fraction.

Wilson considered the case where the molecules not only differed in size but also

in the intermolecular interactions. These differences in intermolecular

interactions lead to microscopic deviations from the random mixing notion which

is inherent in many phase equilibrium models. Viewed microscopically a solution

is not homogeneous but has local domains which differ in composition. This is

illustrated in figure 22 which was taken from Prausnitz's review on phase

equilibrium [113]. There is no easy way to relate the local composition to the

bulk composition but Wilson proposed a Boltzmann factor type approach, i.e..

A/^ T AntyKZ -Ajj/RT

X„ -X„/RT

x-je

and

A T rt At"

-X^2^RT

Xrt^ — A^^/K

1

Xj,e

where Xp-i is the concentration of molecules of type "1" around a central "2,"

etc., and \.. is an unspecified parameter. Wilson then defined a local volume

fraction by

1 v,x,i +V2,

where v. is a measure of size and with an analogous equation for X . Wilson

then substituted his volume fraction into the Flory-Huggins expression for G . By defining

34
^ = !i exp .^]1_L^

^21 V2

^

RT

he found for a binary mixture

%f = - x-| £n (x, + A-ipXp) - Xp an (x^ + Ap-iX,)

The activity coefficients obtained from this expression are

A,« A,

£n Y^ = - £n(x^ ^ A-igX^) ^ X3[

^^ ^^^^^^^

- ^^^^^\ ^j

and

£n Yo = " ^"(Xo •" ^21^1^ ^ ^1

12 A,21

X-i » '^ 1 Q '^o 01 1 O

£n Yi = 1 - iln A12 - A^i

and

00 iln Yo - 1 - ^n Api - A-.^ .

The extension of Wilson's equation to multicomponent mixtures is very simple,

viz.

^ = - E X RT Y

i £n E X. A. .

and the corresponding activity coefficients are given by

An Yi. = - ^n

.? ''o^o

.1 - E

i

X, A.,/E Xj A..

The Wilson equation has proved to be an extremely valuable tool in correlatTng

highly non-ideal vapor-liquid equilibria data. As written, however, this

equation is not capable of predicting miscibility gaps. This problem has been

overcome, however, but we will not go into the details here. An important point

35
relating to this equation is that it forms the basis for a predictive method of

calculating vapor-liquid equilibria knov/n as ASOG (Analytical Solution of Groups)

which has been developed by E. L. Derr. This method along with another

predictive method for activity coefficients will be discussed in section 5.3.

5.2.4 NRTL Equation

In the discussion of the equation of state methods for predicting phase

equilibria we alluded to a one-fluid i,x)del which simply stated implies that the

properties of a mixture can be related to those of a hypothetical pure fluid.

There are also n-fluid theories of mixtures which state that the properties of a mixture can be related to those of an ideal mixture of several fluids of

different behavior. For example, a two fluid theory says that the residual Gibbs

energy of a mixture is given by

= X, G, + x^ G2

where the "R" refers to a residual value. The procedure then is to identify all

components of the mixture with either of the two fluids via a one-fluid

corresponding states principle and then mix them ideally.

Renon [114] combined the two-fluid approach with the local mole fraction

ideas of Wilson except he used the quasi-chemical approximation rather than the

Flory-Huggins term. He found that for the local mole fractions

X21 X2 exp(- a^^^g^^^/RJ)

T^ ~ x^ exp(- a^29ll/^^^

Proceeding as in Wilson's case he found that

g = x^x^

^21 ^21 ^ ^^12^2

X-i ' '^o'*oi o 110

where A. .= exp(-a .

.

t. .) , t.. = 3../RT and a.. = a... This equation has three

parameters per binary pair, C-io* Boi and a,^, unlike the Wilson equation

which has only two parameters per pair. This equation does, however, appear to

adequately represent strongly non-ideal systems, including those which exhibit

liquid-liquid immiscibil ity. The activity coefficients for the MRTL equation for

a binary mixture are

36
£n Y"! = x^ T^^

V^l "^21/ (X, .x,A,,)2j

and

2 r / ^12 \ ^

^21 ^21

1 p2\^X2 + x^A^2>/

(X, +x,A,

in Yp = X

At infinite dilution we find the relations

1 "2"21 A,i)'J

RT iln Y^ = ^2] "^ ^12 ®^P^' °'l2^12^'^^^

and

RT In Y2 = Bi2 + B21 exp(- a^2^21^'^"'^^ '

Notice that unless we arbitrarily assign some value to a, ^j knowledge of the

infinite dilution activity coefficients alone is insufficient for the

determination of the equation parameters.

This equation, like the Wilson equation, may be readily extended to

multicomponent mixtures, for which we have the following relations:

^

' = -E E^ii^i^ X RT . ^i E \ \i

k

and

£n Y^ = E x.Aj, ../S. . E (x/,j/Sj)[,,j - E x,A^- „,-/Sjj

where

^k = ^ ^j ^•k

•

5.2.5 UNIQUAC

Recently, another correlation for the excess Gibbs energy called UNIQUAC,

(UNIversal QUAsi-Chemical) , has been proposed by Prausnitz and his

37
co-workers [115]. This correlation is also based on semi-theoretical arguments

and divides the excess Gibbs energy into a combinatorial part and what Prausnitz

calls a residual part, i.e.,

E E E

g = 9 (combinatorial) + g (residual)

The combinatorial part comes from considering the fluid to be described by a statistical mechanical lattice model to which the quasi-chemical approximation

has been applied. The residual part was obtained by considering interaction

energies, much in the same way as Wilson and Renon.

This equation is quite complex, although it only uses two adjustable

parameters per binary pair. Preliminary indications from other investigators are

that it is capable of correlating highly non-ideal systems, including those which

are immiscible. For the sake of brevity, we will only give the multicomponent

form of this equation and the activity coefficients derived from it.

g^/RT = Ex.£n($^./x.) + 5 E e^.x^.£n(e./$^.) - L e^.x^.£n ( E e.A. . j

where $. = x.ct)./

^

x.(i). and e. = 9,-x./

^

^i^i*

"^^^ adjustable parameters in

this equation are the A.. = exp(-B. ./RT) . The terms 4). and e. are pure

component parameters which are measures of the molecular volumes and surface

areas, respectively.

The activity coefficients for this equation are given by

£n Y^- = ini^./x.) + 5£n(e^./$^) + £^. - ($^./x.)x.£.

- ^-^^ f

Vji" ^• - ^ fVij/? 'k\j

where

i. = 5(*. - e. )-(*.- 1)

5.3 Group Contribution Models

All of the activity coefficient expressions- that we have considered thus far are correlative in nature. They require specific data (at least the infinite

dilution activity coefficients) to determine the parameters of the correlation.

Derr and his coworkers [116-118] have developed an empirical method of predicting

molecular activity coefficients given only the molecular structure of the mixture

38
components which is known as the analytical solution of Groups or ASOG. The

general idea is that there are many less functional groups, e.g., CHo-j-OH.-NHp,

etc. than there are molecules. Thus if it is possible to correlate activity

coefficients in terms of groups it is possible to cross correlate or predict

activity coefficients in many molecular systems.

Initial efforts in predicting activity coefficients on the basis of

structural groups were performed by Pierotti, Deal and Derr [108] and consisted

of studying the infinite dilution activity coefficient of a homologous series of

solutes in some solvent. Some typical results are shown in figure 23. As one

can see from this figure there is a regular behavior as the number of (CH„)

groups increase in the polute. Pierotti, et al. performed this type of analysis

on many types of systems and found that the results could be correlated by the

equation

^"^1 = ^2 ^ -H^ ^ ij ^ °'"l -"2)^-4

where n is the number of -CH - groups. A-F are parameters and the subscript "1"

denotes the solute. The coefficients for various binary systems have been

sunmarized elsewhere [122,123] and will not be repeated here.

The study on infinite dilution activity coefficients led to an investigation

of whether the group method could be applied to the entire range of composition.

To do this we define group concentrations by the equation

where X. is the concentration of group k, n.. is the number of k groups in the

th "^ j molecular species, viz..

The group activity coefficient model has four basic assumptions [123]:

1) The molecular activity coefficient is made up of two contributions, one due

to size differences £n y and the other due to interactions of the groups.

in Y^.

c p

in Y,- = ^n Y,- + ^n y^

39
2) The size difference term may be obtained from an idealized model such as the

Flory-Huggins athermal solution or Gugenheim's lattice model [124]. For

example for the Flory-Huggins term

-v^ = ^"(tV])-ty]

+

where vj is some measure of the molecular size,

3) The contribution from the interactions of the groups is given by the term

where r^^- is the standard state activity coefficient of the k group in

the j molecular and r|^ is the activity coefficient of the k group in the

i.iixture. The standard state group coefficients must be included to ensure

that y. approaches unity when x.-^l.

J J 4) The final assumption is that the group activity coefficients may be obtained

from a correlation like the Wilson equation except that group concentrations

and parameters are used, e.g..

£n r. = 1 - £n

f h ^u

. E '^

"'''

k

] h ^i

There have been two very successful group contribution models -- ASOG [116-118]

(Analytical Solution of Groups) which uses the Flory-Huggins size term and the

Wilson equation, and UNIFAC [121] which uses the Gugenheim lattice model and the

UNIQUAC equation. Tables of group parameters are available for both of these

models which allow extensive predictions. Both of these models are accurate to

within 10-20 percent for systems wheref the group parameters are available. This

is usually sufficient for engineering screening design calculations.

5.4 Corarnents on Excess Gibbs Energy Correlations

A few coLiments are in order with regard to the activity coefficient

correlations which vie have presented. First of all, by design we have removed

the pressure dependence of y- ^nd placed it in the "vapor imperfection

coefficient," e . Although this is formally correct, we still need to know V.

and V. as a function of pressure to accurately calculate the liquid fugacity.

Normally, V. is known with sufficient accuracy, but V- is not. In fact.

40
experimental data for V. are scarcer than those for the activity coefficients

themselves. In this case, a common assumption is that V. = V., in which case we only have to worry about the pressure dependence of the latter term. At low

pressures, however, the activity coefficient is very insensitive to the pressure

and the errors caused by these approximations are negligible. At higher

pressures, however, especially in the critical region of a component, these

approximations can lead to serious errors. Secondly, and much more importantly, all of the excess Gibbs energy

correlations that we presented are for only one temperature, including those

which show an explicit temperature dependence. This is a particularly

troublesome problem, since we may require activity coefficients at temperatures

where we do not have experimental data. Since (9G /9T) = -S , the problem of

predicting the temperature dependence amounts to predicting or knowing the excess entropy as a function of temperature. Since this type of data is also seldom

available we are forced into a situation where we have to make some serious

approximations.

Two possible approximations are that the solution is athermal or that it is

regular. In the athermal case (i.e., H^ = 0), we are lead to the conclusion

that at constant composition, the activity coefficients are independent of

temperature. Generally this is a very poor approximation. The second choice is

to assume that we have a regular solution (i.e., S = 0) which leads us to

the conclusion that the activity coefficient varies as 1/T. Normally this

assumption is better than the first, but both are far from adequate.

The optimal situation (other than having data at the temperatures that we

want) is to have experimental data at two or more other temperatures which we can use to fit the Gibbs energy correlation coefficients or the activity coefficients

themselves, to some polynomial in temperature. For example

£n y^. = a + bT + c/t"

where a., b., c. and possibly even n, are determined by curve fitting

experimental data at known temperatures.

41
6. Conclusions

This completes our brief overview of phase equilibria prediction methods.

Although equation of state methods have developed rapidly during the past few

years, there is still a definite need for models which represent systems which

have large size and polarity differences. Activity coefficient methods which are predictive, e.g., ASOG, lack a sound theoretical basis upon which they may be

improved. Further work in this area is needed.

42
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183 (1976).

[91] Oellrich, L. R., Knapp, H. and Prausnitz, J. M., Fluid Phase Equil. ^,

1963 (1978).

[92] Carnahan, N. F. and Starling, K. E., A.I.Ch.E. J. 18, 1184 (1972).

[93] Haar, L. and Shenker, S. H., Proc. 5th Symp. Thermophysical Properties,

p 223, 1970.

[94] Bienkowski, P. R., Denenholz, H. S. and Chao, K. C, A.I.Ch.E. J. 19, 167

(1973).

[95] Whiting, W. B. and Prausnitz, J. M., "A New Equation of State for Fluid

Water Based on Hard-Sphere Perturbation Theory and Dimerization

Equilibria," 9th Int. Conf. on Properties of Steam, Munich, 1979.

[96] See, Threlfall, R. and Adair, J. F., "Physical Memoirs," Vol. 1, part 3,

Phys. Soc, London, 1890.

[97] Redlich, 0. and Kwong, J. N. S., Chem. Rev. 44, 233 (1949).

[98] Soave, G., Chem. Eng. Sci . 27, 1197 (1972).

[99] Peng, D.-Y. and Robinson, D. B., Ind. Eng. Chem. Fundam. 15, 59 (1976).

[100] Fuller, G. G., Ind. Eng. Chem. Fundam. 15, 254 (1976).

[101] Harmens, A. and Knapp, H., Ind. Eng. Chem. Fundam. 19, 291 (1980).

[102] IJsdin, E. and McAuliffe, J. C, Chem. Eng. Sci. H, 1077 (1976).

[103] Leung, S. S. and Griffiths, R. B., Phys. Rev. A8, 2670 (1973).

[104] d'Arrigo, L. M. and Tartaglia, P., Phys. Rev. A12, 2587 (1975).

[105] Moldover, M. R. and Gallagher, J. S., A.I.Ch.E. J. 24, 267 (1978).

[106] Peng, D.-Y. and Robinson, D. B., A.I.Ch.E. J. 23, 137 (1977).

[107] Michelson, M. L. and Heidemann, R. A., A.I.Ch.E. J. 27, 521 (1981).

47
[108] Pierotti, G. J. and Derr, E. L., Ind. Eng. Chem. 51, 95 (1959).

[109] Margules, M., Sitzber. Akad. Wiss. Math. Naturwiss. Klasse II 104 , 1243

(1895).

[110] Redlich, 0. and Kister, A. T., Ind. Eng. Chem. 40, 345 (1948).

[Ill] van Laar, J. J., Z. Phys. Chem. 72, 723 (1929).

[112] Flory, P. J., J. Chem. Phys. 9, 660 (1941); JO, 51 (1942); Huggins,

M. L., J. Chem. Phys. 9, 440 (1941).

[113] Prausnitz, J. M. "State-of-the-Art Review of Phase Equilibria," in "Phase

Equilibria and Fluid Properties in the Chemical Industry," Storvich, T. S. and Sandler, S. I., eds., ACS Symposium Series No. 60, American Chemical

Society, Washington, D.C., 1977.

[114] Renon, H. and Prausnitz, J. M., A.I.Ch.E. J. 24, 135 (1968).

[115] Abrams, D. S. and Prausnitz, J. M., A.I.Ch.E. J. H, 116 (1975).

[116] Derr, E. L. and Deal, C. H., I. Chem. E. Symp. Ser. 32, 40 (1969).

[117] Deal, C. H. and Derr, E. L., Ind. Eng. Chem. 60, 28 (1968).

[118] Derr, E. L. and Deal, C. H., Adv. Chem. Series 124, p 11 (1973).

[119] Wilson, G. M. and Deal, C. H., Ind. Eng. Chem. Fundam. J_, 20 (1962).

[120] Gugenheim, E. A., "Mixtures," Oxford University Press, Oxford, 1952.

[121] Fredenslund, A. A., Jones, R. L. and Prausnitz, J. M., A.I.Ch.E. J. 2J[,

1086 (1975).

[122] Chao, K. C. and Greenkorn, R. A., "Thermodynamics of Fluids," Marcel

Dekker, New York, New York (1975).

[123] Reid, R. C, Prausnitz, J. M. and Sherwood, T. K., "The Properties of

Gases and Liquids," McGraw-Hill, New York, New York (1977).

48
H->._o

1=1

$1

X

t

a

(U3

>>

00

0)

«/) V)

ITS U

4-»

(U

S-0)

o

CO

+-> o

Qi <-5 O

sQ.

OO

o

r— >

49
COMPOSITION B

u

3

COm

111 ae

a.

s # M ^Cr M L M.>^J ^L*^"P^

Sii^/^ ^

Hf^^ G prr

TEMPERATURE

Figure 2. Three dii.iensional space model for a type I system and corresponding

isoplethel cross section l8].

50
Pm A ^^ J^2 B

Pc - Liquip" "V^

p

/ Liquid-Vapor

/ / Vapor

m

Liquid

Vapor

J L

"'c "^m

Figure 3. Retrograde vaporization (a) and condensation (b)

51
Figure 4. Double retrograde vaporization.

52
P = Constant

C.P

c

C.P.

/

^

C.P.

^

(

Dew Point Curvgi^- ^

/''^LiqLjid-Vapor,^.-.-'''^ ^1

^Bubble Point Curve

X

Figure 5. T-x diagram for a type I system,

53
T = Constant

C.P.

T3

C.P.

V_\^

, Bubble Point Curve

^\V ^

^^^^

Dew Point Curve^ -—

X

Figure 6. p-x diagram for a type I system.

54
00

cL"^ »-" V

f

V\ o

o

1 / V

-i l\. / \

Q.N Ol

ll

(/) a;Q.

o

5-

+-> X O

o 0)M

1- fO o

Q. c

"r4-> o

u -Q

•r" x:X

a»

00G

O)

+-> en>»

CO

s_o"+-

CO

r:fO

s_CD

fO

*r* -aX

CL z

o a

c

1- (T3 co X

o 1 CL 1— s

o

o • r^

cu

S-CD 3UnS83tid 3tiniVtJ3dlfl31

55
T= Constant

Critical Azeotrope

Critical LocuSv^

Figure 8. p-x diagram for a system exhibiting a critical azeotrope,

56
COMPOSITION

Figure 9. T-x and p-x diagrams for a system which exhibits a miscibility

gap [8].

57
llJ <K

bi

Q.S

UJ

kJK

3

COCO

llJ a:

a.

COMPOSITION

Figure 10. p-x and T-x projections for a system havimj an upper critical solution

pressure, li^^ and lov/er critical solution temperature Kj|_ l8].

58
Ill oe

oeUJ

a.

111

uic

3

COm

u

QC

a.

COMPOSITION

Figure 11. Azeotropic systems which also have a lower critical solution

temperature Kjl or upper critical solution pressure K L8].

59
100

X

<

CO

CO

o

LU

10

1.0

0.1

0.01

I I I I I I ll

1 TT

Composition,

mol. Fract.

C^ = 0.8097

C2 = 0.0566

63 = 0.0306

nC5=0.0457

nC7. = 0.0330

nCi 0=0.0244 , •, • , L^I I I I I I III 1*1 I

I L_L

10 100 1000

PRESSURE, psia

Figure 12. Comparison of Peng-Robinson predicted K values with data for a six-component mixture [32],

60
LU

9

X

o

Q

z

o

DC

<

O

O

<

cc

Li.

Ill

-J

o

^isd '3anss3dd

s_cu t-

+-> o

2 3.

-> •a>

^

r—

>» o

+J r^

•1— r-^ r^5 •

a

^— A o u_

LO o

ro ^ o I—

1

i-3

^ 4

(XS +f 1 u.0) T3 ^ •r— •r— 3

i- cr

0) •f— :^ —I X

0)

• -o 1—

1

^ r—

fO

1 1 o

OJ u

-!-> +J

•a3 to u

M- a o

J

c 1+- o

o

t-> c <n • o zs LU

I/) o- O

*r— 0) r^

1- r>^ (O 00

a. 2i

Ci_ O

5

:_) O) A

j^ '_l_ • +-> ro r^

1—1 :d lO c I—

1

OJ •n- 3

co3 <

31

61
4000

.5

CO

a

LIl

cc

CO

CO

HI

oc

Q3000-

2000-

1000-

0.80 0.90 1.00

MOLE FRACTION HYDROGEN SULFIDE

Fiyure 14. Cohiparison uf the calculdted dnd experimentdl solubilities of H2S in

water usiny the PKS equation of state [41]. • 34U°F, o 28U°H, O 2:20°F,

A 160°F.

62
°o

^X

1 1 1 i2| CM

n — ^

/

21

/ i^CO o

/ ccoo

c — o

(0

•!3

O

O

CM

CQ

c

•r—

5

•r—> "

CO

CVJ / , O o- >:

iral

Eqi

f

State 1 O

CO

To

CM

LlT

4-O

+3c — y

^^'

2m o > o_

1 CO

to

53 — /

y

1

1 _ O

00

CO

LU 4-4-

3

l-> X

: / S: (0 ca C

llllllll

OC

0.

O

/

/

r

\ 1

Industri

Importi Regio

"""'''''''''"'"'

\^

O

>>

4-> •p—O

cuc

43

0)

rr O

O

o

1—

1

•^

1—

1

O 3 00

-a (U

1 1 ^ 4->

1 1 CM -a -

/-s a; •t— l/l CO t-> «M^

1

/ ,

• _ OO •r— 'o

4-O

/ / / / CM ^ 4- "^

t/> /

/ / / CO •r- 4-> c

o

m^mm ^

/

/ /

/ L

/

/

O

CO

JQ 4-

gj

4-> 3

o

CM / /

/

/ /^\ /

^

/ O

CM

LU

QC

O

a

3-

//

/ o /

""" CO

Q.

I

5

cuc

-a

4- / \,^/ CO • 4J

•1— — // / ^ -^ O LU .—1 0) ^f >r ^ 00

o

CL

cu

.tJ

"

lip

^'^«--^^-n****

ustrialiy

portant

tegion lllli

-ac

#1 T3 E

"• m 1 £- i r o

lO CMlO

COlO To in

lO

^<i> 'lN3IOIJd300 AilOVOnd NOIimiQ 3ilNldNI

00 '

63
<

a:

!^

CO

tr

LU

o

ho

<

LU

LU

0.1

0.01

o Oldset al. (1942)

^ Reamer et aL (1943)

D Kobayashi et al. (1963) — Calculated

H2O-CH4, 510.94 K

H20-C2H6,477.61 K

H2O-C3H8, 422.05 K

200 400 600

PRESSURE, atm

Figure IG. Comparison of calculated and experii.iental solubility of water in

various hydrocarbons using a modified RK equation of state l44j.

64
LJJ CO

<

o

CNJ o

o

o

CM

o

<

U.

LU

-I

o

0.01 f

0.001

10 100

PRESSURE, atm

1000

Figure 17. Comparison of calculated and experimental water solubilities in CO2

using an interaction coefficient in the repulsive parameter of the RK equation

of state [4G].

65
LU

<

O

<

LU

LU

CO

<

I

Q.

OC

O

Q.

<

>

10 =—\—I I 1 1 iii| \—I I I iiii| \—I I I II

10^

280«

100**

o Ref. 42, 50 — Predicted

I iiiil 11

0.01

0.001

10- 10' 10^ 10

0.0001

5

LU

z

LU

—I

>-

I

LU

O

<

cc

LJ- LU

o

LU

CO

<

I

Q.

9

o

PRESSURE, psia

Figure 18. Comparison of calculated and experimental solubilities in the

water/ethyl ene system using a size binary interaction coefficient in the RKS

equation [47].

66
<D

0)r (0

CO ^lO ^—o—-c o or ^ o

<N cvi®

y 5 in r^*^ \>i^ /S> o) tT >V^iV

fc- kv <u o c: r^ *- fO oo

CO 43

'aJ ^" k o

LU

a oG

OC E 3 c O

•r- o CO

•r- 13 T- 00 ^^ CO 3 o

t/5 LIJ "to ^

GC >1 Q. i»i

O

^m 1

<4-

(T3C

+J o C o

OJ

c= O)

•r- -cr S- +-> o Q. CD o X C

0) •r- E

-o rs

"-• "3 oo CO -o <u

CD +->

4-> oo

•*

(tJ >> tu 3

oo o cc O CDc

T— ID Q.

CO MO

CO

O Q.

LU o

OC

00 £L

S-Q.

1—

O -a O c:fO

CT> O) '

<U x:

s_ 4-> 3 <u CD 3mVA->t

67
LU

I

1.0

0.1

0.01

477.56 K

444.22 K

410.89 K

_377.56 K

344.22 K

Tl

1.0 10 100

PRESSURE, atm

Figure 20. Comparison of calculated and experimental decane K-value in the

butane/decane mixture using the conformal solution approach l69].

68
0)

<D O

u. (0 « •>

_i a> >

Y r \ / \K/\I^ l\><.

4-> 3

o

O) o

o

i- Mw»c

O

•r—

4-> to

•r—>

-a

COQ.

•(-J

(U

0)

•->

s_

+-> 3

CM >

69
o©p

15 of type 1 Q 16 of type 2

Overall mole fractions :

x^ = Xg = 1/2

Local mole fractions

Molecules of 2 about a central molecule 1 Xo4 =

2^ Total molecules about a central molecule 1

^21 •^11 "

X^ 2+^22

^11

'21

1,as shown

13/8

5/8

Figure 22. Illustration of local composition in liquid mixtures l113j

70
- 1 V

/-

X ^ f

C\i CVJ / c c

^^ / 1 /

CVJ ^ f X c 1 oX s " c o X d1 c" II t /-^ 9 X

)C o O)

>1^ o X \ ( 1 1 1 X 1 O CM

CM <H - CNJ X r - oXc oo X o

t—

d c 1 ^-N ^"

CNJ c \ X CO i o 1^ 1 '**' o \ X do

LOd

II 1 1 ) s ^ 1 ooo oo O 1- T- O d <=> O 1-o o O T-

(0.92)> (0.09)iX o^ -oc

rac

4- • (O —

s- ?J

fO CVJ Q.

-1 c o S_ CO <4- J3

03 "^ S- 3

(OC

Q.

C

C

O " J2

(/) fO

-!-> o C

OJ 4- o ^ o o

CVJ c- •-- co

4-

0) +-> o o u

>> 4-

-M

•r— fO >

•r— I/) M «o

CJ

03 (/J o £Z ^~ o 3

•n- +J

•l->X 3 •r—E

•r— -o 'o

O) -C

-M o

•r—

a

C 1—

•1— (O 4-c c= o 1—

c • .^. en <+-

C\S

<u s_

s- fO 3 a.

CT»

Ll71

THE EXTENDED CORRESPONDING STATES METHOD APPLIED TO THE

NITROGEN-METHANE SYSTEM

Robert D. McCarty

Thermophysical Properties Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 30303

An extended corresponding states method of predicting the equation

of state for the N2-CH4 system is reviewed. Comparisons between the

predictive method and experimental PVTx data have revealed basic

problems with the method. A review of this ongoing work is presented.

Key words: equation of state; extended corresponding states; mixtures;

predictive methods; PVTx.

1. Introduction

Several years ago the properties of fluids group at NBS-Boulder, undertook a long range project to develop a mathematical model of the equation of state of

LNG which would be capable of predicting densities given the pressure,

temperature and composition of a LNG mixture. Several different models were investigated during the course of that project, each of whicM proved to be

adequate under the limited scope of pressure, temperature and components of

LNG [1]. One of the methods, the "extended corresponding states" method proved

to be the most versatile and suggested the possibility of describing the equation

of state of a mixture over a much broader range of pressure and temperature in a truly predictive sense. The purpose of the present v/ork is to explore that

possibility. The binary system of CH.-N^ was chosen as a test case because of

the availability of good experimental data for that system and because the

CH^-N^ system is one of the binary systems studied in the LNG project.

Although the work on this system is by no means complete, this paper reports

the progress in that investigation to date. In order to better understand the

principle of corresponding states and its evolution to the modifications being

used here (there are many other modifications) a short history is given in the

next section.

73
2. Background

Before the metriod can be applied to mixtures, one must understand the

application of the method to |jure fluids.

Van der Waals proposed the original law of corresponding states in 1881.

Noticing a similarity in the shape of the PVT surfaces of many different fluids,

van der Waals proposed the following:

f(Pp, V^, T^j = fur all gases (i)

where

[Juw introduce fluid U and fluid 1 so that

P, P^ P P, P^^

p^ = P^ = p^ or /- = p^ and P = P p^ (3)

^cl

^

'^co ^co ^cl " ^ '^cl

and it follows that

cl cl

therefore if we have an equation of state for fluid 0, i.e.,

Po = f'^C To)

tnen to use this equation of state for fluid 1 via the van der Waals

corresp(

3 and 4

corresponding states we may substitute into f for V ana T tne equations in

it follows then that

P / V T \

' ^cl V^ ^cl ' 1 \J ^ ^

-/ -7 CO \ cl

74
but as p

-^ 0, Z-, -^ 1 and Z * j— which is only 1 if Z = Z , therefore to

insure correct ideal gas behavior one of the three original reducing parameters

is el iiiiinai:ed by equating Z = Z , , i.e.,

-, _ CO CO ^ cl cl ,-,x CO " RT RT T ^'^ CO cl

which then may be solved for any one of the critical parameter ratios in (3) and

(4).

Since critical volumes were typically the most uncertain of the critical

P T V cl CO C

parameters the j— p— was usually substituted [2j for

y— which achieved the

'cl ^co \l

correct ideal gas limit at the expense of the critical point inaccuracy (unless

Z happened to equal Z -,) and reduces the number of reducing parameters

required from three to two. Although this iiiodifi cation of the original principle

improved the performance of corresponding states, significant deviations between

predicted and actual PVT surfaces was observed especially between fluids with

dissimilar molecular structure. The extended corresponding states principle

proposed by Rowlinson and Watson [3] is an extension of the above tv^o parameter

version except*^ that instead of eliminating tfie V ^/V ,, P ,/P^i is eliminated, -* CO cl CO cl

i .e. , from (3)

P. - P. 1!^

^

(8)

CO cl

which corresponds to

where

h

^.0

'o ("l.o/^l.o) (3)

T^ 9(T V ) (10)

CO

and

CO

75
if 9 and <J> = 1. Tnen if the 9 and <i> (shape factors) are defined by the

equations

and

where Z is the compressibility factor and A is the helinholtz free energy one has

two equations with two unknowns which means that an exact correspondence between

two fluids (for those two properties) exists providing one knows the correct f

and h.

It is interesting to note that in the early 60' s a variation of this idea was used by McCarty, et al . [4] to generate PVT data for neon. In that

application only one shape factor was useu to modify the density so that

• (•'»-• ^) P. = Y^ = p. lV,/h, ^, ^-^J (14)

CO

in that case models of the equation of state for nitrogen and argon were used to

determine the h, which was then used to transform the nitrogen surface to

the neon surface. Figure 1 illustrates that transformation function which in

this case is called density ratio. The validity of the assumption that the shape

factors are general between similar fluids is the key to v/hether or not the

extended corresponding states method is truly a predictive method or not and is

equally crucial to the Rowlinson definition.

The stage is now set to apply the extended corresponding states method to

mixtures. If one assumes that the equation of state of a mixture of fixed

composition behaves the same as that of a pure fluid then eqs (12) and (13)

apply, i.e..

and

\ - \ ^v^,o> y^,o) (^^)

A„ = f A (V /h , T /f ) (16)

X X,0 ^ X x,o' X x,o^ ^ '

where the subscript x denotes the mixture and the o subscript denotes "base"

fluid, and the f and h dire defined by

76
^ij.o = ^1o '^11.0' fjj.o)'^' (20)

"1J.0 ^: (^K!o 4 ^^.^!oy (-)

The C. and n.. are binary interaction parameters and functions of a particular

binary system and the f- and h. are functions of the pure fluids only.

3. Nitrogen-Methane

As was stated earlier the iiiain purpose of this work is to explore the

possibility of using the extended corresponding states method to describe the

properties of mixtures over a much broader range of pressure and temperature than

was the case in the LNG project.

The starting point of this study was the extended corresponding states

results of the LNG project and the PVTx experimental data by Straty and

Diller [5]. Figure 1 shows the most serious deviations between calculated and

experimental densities at the starting point. Even though these deviations d^r^

irtuch greater tlian the accuracy of the experimental data, the comparison does

produce further encouragement of eventual success of the method. The extended

corresponding states equation of state for the nitrogen-methane system resulting

from the LNG project, except for the methane equation of state which is a wide

range equation of state for the base fluid, is based entirely on low temperature

saturated liquid PVTx data. The comparison of the mixture equation of state with

the Straty and Uiller experimental Np-CH^ data then constitutes an extrapolation of the mixture parameter to much higtier temperatures and lower

densities.

Since there br^ manj' more variables involved in an equation of state for

mixtures than there d.r^ for a pure fluid, a plan was developed to proceed from

this point which would systematically investigate the effects of each variable.

77
step 1. A 32 tenn MBWR v/as fit to the 50-50 composition data of Strat> and

Oilier. This step provided a sort of yard stick for the development of the

mixtures equation of state as v/ell as a means of comparison at arbitraty P and

and T.

Step 2. The Dinary interaction parameters ^ and v in eqs (19) and (20) were re-estimated using the Straty/Dil ler data. The new parameters frum a least

squares fit improved the performance only slightly, indicating further work.

Step 3. The e(V^, T^) and 4)(V^, T^) (eqs (10) and (lljj from the LNG

project was eliminated by a direct solution of eqs (12:) and (13) using the

equation of state for N^ ^^^ ^^a' This proved to be a difficult task and a completely satisfactory solution is still not in hand. A good deal of time v^as spent trying to find a fail safe method of solving for e and ({> without success.

A method was found hov/ever which v/orks enough of trie time to allow a comparison

to a sufficient nuiiiber of experimental points to detennine that s/ery little

improvement over tne e and 4) from the LNG project was achieved.

Step 4. Step 2 was repeated using the results of step 3. The results of

this exercise could be predicted to be not much different tnan trie results of the

original step 2 but the procedure was carried out anyway out of the interest of

being thorougn.

At this point one can conclude that the failure of the method must be a consequence of one of two things or perhaps a comoination thereof. Eitiier the

theory is wrong or one or both of trie equations of state used for N^ and CH- are wrong. A third possibility that the experimental PVTx data are wrong is

highly unlikely since the fit of the 50-50 data to the MBWR was satisfactory.

Step 5. In an attempt to discover where the method breaks down, step 4 was repeated using data of various comoinations of restricted ranges of density,

temperature and cohiposition to try to detennine if the binary interaction

parameters exhibited a dependence of any kind. The results of this procedure are quite interesting. If one leaves all of the ddta below 11 mol/L at temperatures

below 164 K out of the fit, tne rest of the data can oe represented adequately

and the resulting binary interaction parameters are yery nearly the sauie as those

determined in the LNG project. This has only been observed for the 50-50

composition and further experimentation using data from the other two

compositions is yet to be accomplished.

78
This is about where the work stands at present except for one other

calculation which has been made which is of interest. Using experimental PVTx

data from the region of failure as input, the mixture equation of state was solved first for a 9 of CH- to N^ holding <}> constant such that the equation of

state predicted the experimental PVTx exactly and then the same calculation was made with the roles of 9 and * reversed. The results of this calculation

indicated that an error in 9 is much more important than an error in 4.. When the

4) is held constant, a small change in ({> (1 or 2 percent) produces an 8 percent

change in density but over a 100 percent change in 9 is required to achieve the same results when holding 9 constant. This would indicate that the reason the

method fails is a result of the theory being wrong rather than the Np or CH.

equation of state because while the equations of state may easily be wrong by

several percent density in the region of failure, it is highly unlikely that they are wrong in temperature by 1 or 2 percent.

79
4. References

[1] McCarty, R. D. , Four mathematical models for the prediction of LNG

densities, Nat. Bur. Stand. (U.S.), Technical Note 1030 (Dec 1980), 76 p.

[2] Su, G. , Modified Law of Corresponding States for Real Gases, Ind. Eng.

Chem. 38, 803 (1946).

[3] Rowlinson, J. S. and Watson, I. D. , The prediction of the thermodynamic

properties of fluids and fluid mixtures - I. The principle of

Corresponding States and Its Extensions, Chem. Engng. Sci. 24, 1565

(1969).

[4] McCarty, R. D. , Stewart, R. B. and Timmerhaus, K. D. , P-p-T values for

neon from 27° to 300°K for pressures to 200 atm using corresponding states

theory. Advances in Cryogenic Engineering, Vol 8, 135-45 (1963).

[5] Straty, G. C. and Diller, D. E. , (p,V,T) of compressed and liquefied

(nitrogen + methane), J. Chem. Thermodynamics 12, 937-53 (1980).

80
00 ^ '^ CD CO

CM " r- CO 00 / lO

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CO

00

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O

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o

o

o

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81
p = 8.15 mol/L

°

4

<

>

170 _J \ L_

180 190 200

TEMPERATURE. K

p = 12.6 mol/L

240

-.08^

.08-

260 280

TEMPERATURE, K

E -.10^

-.12

.14

p = 25.9 mol/L

210 220

110 120

TEMPERATURE, K

300

130

Figure 2. Maximum deviations between the N2-CH4 PVTx experimental data of

Straty and Diller [5] and the extended corresponding states model from [1],

82
VAPOR-LIQUIO EQUILIBRIUM OF BINARY MIXTURES

NEAR THE CRITICAL LOCUS

James C. Rainwater

Thermophysical Properties Division

National Engineering Laboratory

National bureau of Standards

Boulder, Colorado 80303

A new method is presented for the prediction and correlation of

vapor-liquid equilibrium of binary mixtures in the critical region. The

basic Ljual ita'ci ve features of the problan are first reviewed, as well as the e>revious theories of Griffiths and Wheeler, Leung and Griffiths, and

Holdover and Gallagher, and the concepts of field and density variables,

corresponding states, and scaliiig-law equations of state. The

Moldover-Gallagher recipe is i.iodified by making the dependent variable

of corresponding states a quadratic combination of density change and

composition change across the phase boundary, rather than density change

alone. The revised i.iethod is applied to the nitrogen-methane data of

Bloomer and Parent. Along the locus of points twelve percent below the

mixture critical temperature in suitably reduced units, the prediction

of composition difference across the phase boundary is greatly improved.

Tne revised procedure is shown not to be significantly different from

that of Moldover and Gallagher for mixtures where the latter has been

successful

.

Key words: binary mixtures; corresponding states; critical exponents;

critical line; dew-bubble curves; field variables; Leung-Griffiths

procedure; nitrogen-methane VLE; quadratic coupling; scaling-law

equation of state; vapor-liquid equilibrium.

1. Introduction

In the previous talk by Jim Ely [1], a wide variety of equations and

techniques used in industry to predict vapor-liquid equilibriuni (VLE) of luixtures was described. As Jim pointed out, these methods are not as accurate near the

critical locus, i.e., witnin about 10 percent of the critical temperature, as they are farther away. In fact, modern theories of phase transitions predict

83
that the "classical" equations of state commonly used in industry give incorrect

critical exponents and thus have fundamental limitations in predicting

thermophysical behavior near the critical locus. The talk following mine, by

Brian Eaton [2], will describe methods for predicting the critical locus of a binary mixture.

My talk is intennediate in range of temperature as well as order of

presentation. In the present lecture, we will not be concerned with VLE at

temperatures less than about o5 percent of the critical temperature. Nor will we be concerned with prediction of the critical line itself, which we take as given.

Rather, we pose the following question: Given the vapor pressure curves of the

pure components and the critical line of a binary mixture, can we quantitatively

predict VLE behavior over the region near tne critical line, where the

industrially oriented methods tend to break down?

This lecture will concentrate on a line of research, more academically

oriented and less industrially oriented than the methods described in Jim Ely's

lecture, which has made significant progress in answering the above question over the past decade. The research essentially began with the classification by

Griffiths and Wheeler [3], in 1970, of thermodynamic variables into two kinds,

density and field variables. It continued in 1973 with the prediction by Leung

3 4

and Griffiths [4] of VLE of a simply behaved binary mixture, namely He- He.

Later, Moldover and Gallagher [5] modified the Leung-Griffiths method so that it

could be applicable to mixtures with more complicated VLE behavior (e.g.,

azeotropy). The Moldover-Gallagher recipe was remarkably successful for several

binary mixtures, but tended to break down for systems with large composition

differences between liquid and vapor.

Most recently, I have proposed a modification of the Moldover-Gallagher

method, tentatively named the "quadratic coupling recipe," to account for

mixtures with large composition differences. Although the results are admittedly

preliminary at present, the modified recipe appears quite promising so far. It yields significantly improved agreement between theory and experiment for the

liquid-vapor composition difference of nitrogen-methane, and is rooted in

plausible theoretical assumptions.

This line of research, at least since Moldover and Gallagher [5J, has been

intimately connected with the law of corresponding states. The lectures by Brian

Eaton [2] and Bob McCarty [6] will describe the law of corresponding states in

more sopnistication; here we consider only its most basic form. In essence, the

84
law characterizes an intermolecular potential by two parameters, a length scale

and an energy scale. Then, so long as the potentials for different fluids have

basically the sanie shape, the thermophysical properties of those fluids are predicted (in appropriately reduced units) to obey universal behavior. The

principle is general in that it is independent of the particular shape of the

potential

.

Furthermore, the law of corresponding states is an essential tool for the

understanding of fluid mixtures [7]. At least sufficiently far from the

two-phase region, a mixture may be represented as a hypotnetical pure fluid.

With the appropriate mixing rules (i.e., weighted averages over the length and

energy scales of the pure cohiponents) , and application of tne law of

corresponding states, PVT properties of mixtures may be predicted accurately over

a wide range of thermodynamic variables [7].

Here we consider the question of whether the law of corresponding states is

useful in understanding and predicting the VLE behavior of fluid mixtures. For

the sake of simplicity, we restrict the discussion to binary mixtures which are miscible in both the liquid and gaseous pnases. Specifically, we ask the

questions:

(1) In terms of the most commonly used physical variables, is the VLE behavior

of binary mixtures even qualitatively similar to VLE of pure fluids?

(2) If the answer to (1) is negative, can a transformation of variables be made

such that the VLE properties of mixtures and pure fluids are qualitatively

similar?

(3) Can such a transformation be found with no appreciable difficulty in

performing both the direct and inverse transformations?

(4) Can we predict which variables, both dependent and independent, "obey

corresponding states" in the transformed spaces?

Our answers will be negative to (1) and positive, in a qualified and

tentative way, to (2), (3) and (4).

2. VLE of Pure Fluids: Scaling Laws

We first review the basic VLE properties of pure fluids. These may be

characterized by coexistence curves in the T (temperature) - p (density) plane as shown in figure 1, and the P (pressure) - T plane as shown in figure 2. The

first figure shows the "coexistence dome" with the critical point at the point of

highest temperature. The curve may be represented by [5]

85
p/Pc = 1 ± 4 hi ^ + c^t (1)

where

t = (T - TJ/T^ , U)

the subscript c denotes critical point value, and a is a fractional exponent.

The plus refers to tha more dense liquid, the minus to the less dense vapor. For

an^ T < T , a liquid phase and a vapor phase coexist at the same temperature

and pressure. C^ is the inverse slope of the rectilinear diafiieter.

Fiyure Z shows trie VLL curve in trie P-T plane, a siriyle curve teruiinatiny at

tne critical point and separatiri'j tne liquid phase (above) and yas phase (below).

There are several possible wa>s to represent triis curve. Une uiethoa, according

to the so-calleu scaling law equation of state, is [5,8]

(P/P^)/(T/r^) = 1 + L3I t| ^"°' + C^t + C^t^ + C^t^ . (3)

Here u is the exponent for the divergence of the specific heat at constant

volume, C , along the critical isochore,

Cy - (T - TJ-^ (4)

where, typically, a -t 0.1. In the so-called classical equations of state

(van aer Waals, Redl ich-Kwong [^Js Peng-KoDi nson lIUj, etc.), C does not

diverge and a = 0. However, tor al

1

equations of state tne specific heat at

constant pressure C diverges according to

CpMT-T,)-^ (b)

Where, for scaling-! aw equations of state, y^ l-Z, and for classical

equations of state y = 1. The critical exponents a, 3 and y licive been tne subject of extensive

theoretical investigation [11]. They are not independent; tne Rushbrooke

inequality [IZ] becomes the following equalicy according to Widom's homogeneity

hypothesi s [13].

u + Y + 2(5 = 2 . (6)

Furthermore, according to powerful renormal ization group approaches [14]

the three-dimensional liquid-vapor transition has been shovni to ue in the saine

86
universality class as the three-dimensional Ising model and should therefore have

the same critical exponents. A puzzling aspect of this analysis is that fluid

exponents obtained experimentally have appeared to differ from Ising

exponents Lll]« For example, fits to VLt data give b x 0.35 whereas the Ising

exponent is g = 0.325. However, there is recent experimental evidence that Ising

exponents are obtained when one examines benavior extremely close to the critical

point [11].

There are considerable differences of opinion within the physics and

engineering communities on the utility and importance of scaling-law equations of

state. The extreme views are^ on the one hand, that PVT data in the critical

region, for practical purposes, are adequately represented by classical equations

of state, and, on the other hand, that the critical exponents are required to

equal Ising values on theoretical grounds. We adopt the intermediate point of

view that scaling-law equations of state are the most efficient means of

representing thermophysical data within the critical region, but that the

exponents which best fit the data need not exactly equal Ising exponents.

Following Moldover and Gallagher [5], we set 3 = 0.355 and a = 0.1.

Equations (1) and (3) are v/ritten in terms of reduced units. Therefore, tne

law of corresponding states predicts that the coefficients C. , i = 1,...6,

should have the same values for all fluids. Moldover and Gallagher [5] have

tabulated these coefficients for a variety of pure fluid systems and find that

they are indeed roughly constant from fluid to fluid, except for C^, the

inverse slope of the rectilinear diameter, and C^. Variations in C^ are of

no fundamental significance, since that coefficient multiplies the highest order

term in a truncated polynomial series.

3. V-L-E of Binary Mixtures

3.1 Qualitative Features

We now return to question (1) posed in the introduction, and acknowledge

that the answer is negative. Within the most comiiion usage, one considers a binary mixture with a fixed mole fraction x. This is clearly the most convenient

point of view for experiment and is (at first) conceptually simplest although we will later maintain that, to understand fluid mixtures properly, it is better to

abandon the notion that x is a fixed and given number.

87
If X is fixed, the VLE behavior of a mixture is qual itatively different from

that of a pure fluid. Instead of the single vapor-pressure curve of figure 2 we have a dew-bubble curve of finite width, as shown in figure 3 in the P-T

plane [15].

The region above tiie dew-bubble curve is tiie single-phase liquid; that below

is the single-pnase vapor. If, starting from the liquid, the pressure is lowered

along an isotherm (vertical line), bubbles of vapor will form when point A, the

bubble point, is reached. Similarly, if the pressure is raised after starting

from the vapor, dew or liquid will begin to form at point B, the dew point. The

locus of dew and bubble points, for all temperatures, is the dew-bubble curve

(for fixed x)

.

Unlike a pure fluid, the dew and bubble points do not have the same pressure. Furthermore, and most significantly different frotn a pure fluid, in

the two-phase region the compositions of the liquid and vapor are not identical.

The vapor is rich in the more volatile component, the liquid rich in the less

volatile component. Thus at point C in figure 3, where the buuble curve of

X-, < X intersects the dew curve of x^, > x, liquid of composition x, coexists

with vapor of composition x„.

For a "nortiial " u'.e., nonazeotropic) mixture [15], a family of dew-bubble

curves exists as shov;n in figure 4. This family tenni nates in the vapor-pressure

curves of the two pure fluids.

An alternate representation of mixture VLE is the P-x diagram as shown in

figure 5 for nitrogen-methane. The solid curve is an isotiienn for T less than

the critical temperatures of both pure fluids. As the pressure is lowered from

the bubble point to the dew point, the system passes through a series of

coexisting liquids and vapors of different composition, as shown by the

intersections of the horizontal lines with the solid curves. The dashed line is an isotherm for T uetween the two pure critical temperatures. It terminates, in

a rounded manner, at sodie value of x less than one.

So far v/e have not discussed the critical point of a mixture. A mixture of

fixed composition does indeed have a critical point, but its meaning and nature

are somewhat different from that of a pure fluid.

In figure 4, the critical line is the envelope of the family of dew-bubble

curves, and the critical point of a mixture of fixed x is tne intersection of

that particular dew-bubble curve witfi the envelope. In general, a critical point

is defined as the point wtiere tne pnysical properties of two different.

88
coexisting phases approach equality. The locus of critical points cannot lie

inside the envelope of dew-bubble curves since at such points two phases of

different properties coexist, in contradiction to the definition. This should

not be confused with the envelope of isotherms in the P-x plot (figure 5), for T

between the two pure critical temperatures. In the latter case tlie critical line

is inside, and not identical to, the envelope. Rather, the critical line is the

locus of points of highest pressure along the isotherms. Tne previous argument

that the critical line is the envelope does not apply to the P-x plot since

states with the same P and x, but different T, are not coexisting.

Note that, in contrast to pure fluids, the critical point is not necessarily

the point of highest temperature of the dew-bubble curve, nor the point of

highest pressure, nor even necessarily a point between those two. One may also

construct a T-p plot for a mixture, but it does not naye the symmetry of

figure 1, and the point of highest temperature is again not in general the

critical point.

Figure 4 shows that the two-phase region is an area of the P-T plane bounded

by the critical line and the two vapor-pressure curves of the pures. At each

point in this region a bubble curve of one composition intersects a dew curve of

a different composition, and liquid of the former coexists with vapor of the

latter. Thus each point in this region denotes a pair of coexisting states. The

"liquid" is usually defined, perhaps arbitrarily, as the state of greater

density.

The critical line may be represented parametrically by the critical

temperature T (x), critical pressure Pj,(x), and critical density p (x). For

T greater chan T (x) but less than the maximum teinperature of the dew-bubble

curve for x, the interesting phenomenon of retrograde condensation can occur [17]. As the pressure is lowered along an isotherm from the liquid side,

the new phase vAiich appears is of greater density than the phase continuous with

the original liquid. Thus lowering the pressure causes "condensation," in

contrast to the usual situation where lov/ering the pressure causes boiling.

Retrograde condensation is a general feature of binary fluid mixtures.

In contrast to the "normal" fluid mixtures described so far, there also

exist azeotropic fluid mixtures [15] where, along a particular locus, called the

azeotrope, the compositions of liquid and vapor are identical. An exam^^le is

carbon dioxide-ethane [18], where the azeotrope is approximately the line

X = 0.32 (x = 1 for pure ethane).

89
Tiie P-x plot for this azeotropic mixture is shov/n in figure 6 for

T < min(T (x)), cf. figure 5. Note that for x < 0.3^ ethane is r.iore volatile,

whereas for x > 0.32 carbon dioxide is more volatile.

The P-T plot is shown in figure 7. Note that CO^ and ethane have almost

identical critical temperatures, but the critical temperature of the mixture can

be significantly lower. There is no single "standard" manner by which the

critical line travels from one pure critical point to the other. Its path

depends on the intermolecul ar forces in a complicated way, and figure 8 suggests

that the forces between unlike molecules are weaker than those between like

molecules. However, in this lecture v;e make no attempt to predict the critical

line; rather, that problem is considered in the following lecture [2].

The azeotrope is tangent to the critical line [15]. Figure 7 shows a case of "positive" azeotropy; "negative" azeotropy is also possible [15]. Here the

P-T plane contains two separate "sheets" of tne coexistence region, bounded by

the azeotrope, the critical line, and the two pure vapor-pressure curves respectively. These regions overlap; in the non-overlapping areas each {P,Tf

point corresponds to a pair of coexisting phases, while in the overlapping area

(cf. figure 6) each point corresponds to two distinct pairs of coexisting phases.

The dew-bubble curves, not shown, approach zero width at the pure curves and at

the azeotrope.

More complicated phase diagrams are possible. For exarnple, the azeotrope

may terminate, or the critical line may intersect a line of liquid-liquid or even gas-gas iminiscibil ity. We do not consider such cases here.

3.2 Field Variable and Scaling-Law Theories

It can be seen at this point that a naive application of the law of

corresponding states will not work for VLE of a binary mixture. Specifically, we might represent a mixture as a hypothetical pure fluid with critical parameters

P (x), pp(x) and T (x) and then determine its VLE properties from corresponding

states and some chosen reference fluid. But this process would yield a pure-fluid-like vapor-pressure curve rather than a dew-bubble curve of finite

width.

However, in recent years it has been shown that corresponding states can indeed be applied to VLE of mixtures if the proper transformed variables are

used, instead of the "usual" variables P, T, p and x. The first step in this

90
direction was the distinction between field variables and density variables made

by Griffiths and Wheeler [3] in 1970.

According to Griffiths and Wheeler, a thermodynamic system is characterized

by N intensive variables such that, if N-1 of them are given, the remaining one

is determined. N = 3 for a pure fluid and N = 4 for a binary i.iixture. Along a phase change boundary, there exist N field variables which are continuous across

the boundary, but derivatives of the field variables with respect to eacn other,

called density variables, are discontinuous across the boundary.

For a pure fluid, one possible choice of the field variables is L3]

fl = y ; f^ = -P ; f3 = T (7)

where y is tne chemical potential; the minus sign on f^ is chosen for reasons of thermodynamic stability [3]. The first field, arbitrarily chosen,

is called the "potential" in Uiis formalism. The density variables are the

partial derivatives of the potential with respect to the other fields,

Pi = ^^l/^^i+1' i = 1 » • • • N - 1 • (8)

In the present case

p^ = V ; P2 = s (9)

where v is the molar volume (inverse molar density) and s is the entropy per

mole.

For a binary mixture one possible choice of field variables is [SJ

f], = ^2 » fo = V] - ^2 » ^3 "" "'^ i ^4 = T , (10)

where u, and m^ are the chemical potentials of the two components, in which

case the densities are

p^ = X ; p^ = V ; p^ = s . (11)

In addition, one may transform from one set of field variables to another

(linearly independent) set, i.e.,

t^ , . . . tj^ -*"

1

' * * * N \^^J

and the ^ume principles apply, as long as certain stability conditions are satisfied [3].

91
Tne most important point to note here for VLE of binary mixtures is that x, the composition, is a density variable rather than a field variable. The physics

of a phase change is described most simply in terms of field variables. Thus the

use of X as an independent variable is not an optimal procedure.

The next step in tne development of a theory of mixture VLE near the

critical region was to discover an efficient choice of field variables. This was first accomplished by Leung and Griffiths [4], and later was refined by Moldover

and Gallagher [5]. The original motivation of the Leung and Griffiths paper was

to analyze an apparent discrepancy between the predictions of Griffiths and

Wheeler [3] for divergences of specific heats near the critical point [cf.

3 4

eqs (4-5)] and data in the critical region of the He- He mixture due to

Wallace and Meyer [19]. Their end result, however, was the first thermodynamic

description of VLE of a binary r.iixture using field variaoles and a scaling-law

equation of state, and it forms the basis for all subsequent work to date.

Leung and Griffiths [4] choose for tlie potential

0) = P/RT (13)

and use the transformed field variables c, t and h, defined below. The potential

is assumed to consist of a regular part w , analytic in the field variables,

and a singular part to . The latter is defined according to the Sciiofield

linear model [20], a scaling-law formalism which incorporates adjustable critical

exponents.

Some aspects of the Leung-Griffiths formalism, while adequate for the

3 4

He- He mixture, are inadequate for binary mixtures in general. In particular,

Leung and Griffiths do not allow for a finite slope in the rectilinear diameter.

Whereas C^, eq (1), is negligible for the iieliums, it is not negligible in

general. They truncate the polynomial for oj in a somewhat arbitrary (nanner

and treat the coefficients as adjustable parameters rather than, as Moldover and

Gallagher [5] do, quantities determined from a definite corresponding states

recipe. Using a degree of freedom in the definition of ^(see below), Leung and

Griffiths impose a constraint which cannot possibly hold in general. These

deficiencies were overcome in large measure by Moldover and Gallagher, and it is

the latter formalism on which we focus attention.

Although x is not a field variable, it is convenient to work with a field

variable which, so far as possible, corresponds to x. Tnis variable is 5,

defined by Leung and Griffiths [4] as

92
K e

^ + e

As the mole fraction of fluid i vanishes, y. - -<». Therefore, c = 1 for

pure fluid l(x = 0) and c = for pure fluid 2(x = 1). Furthennore, the constant

K is arbitrary, and may be varied by changing the zero level of the chemical

potentials for the separate fluids [4], a change which (classically) should not

alter the final physical results.

Holdover and Gallagher presume a choice of K sucn that, along the critical

1 ine,

X = 1 - c . (15)

Equation (15) cannot hold exactly. It is, rather, the first in a series of

essentially empirical assumptions. It is assumed that a K may be chosen such

that eq (15) is approximately true along the entire critical line. For example,

the critical line maps into a family of curves in the c-x plane, for different K,

which all pass through [1,0] and [0,1]; we could choose the value of K such that

the critical line is closest, in some least-squares sense, to eq (15). The value

of K is not explicitly calculated; instead it is assumed that a K exists which

has such a behavior.

This method, in effect, bypasses the explicit evaluation of the chemical

potentials, which are not subject to direct experiment but must be found by

integration of the equation of state, not a^ priori known. Leung and

Griffiths [4] use this degree of freedom to make the inverse critical

temperature, instead of x, linear in c. Such a constraint, altered in some later

modifications of the method [21], is clearly inappropriate for a system like

COo - ethane where T is not a monotonic function of x.

2 c The next field variable is t, a measure of the distance, in temperature,

from the critical line (cf. eq (2)),

t = (T - T^ (0)/T^ (c) . (16)

Here the transformation P, T, y, , y^ -^ i^, ^, t, h is such that each new variable is defined in terms of both the old and previously defined new variables. Tp( ^) is the critical temperature for the chosen value of ^.

Leung and Griffiths choose, instead of t.

93
X = (RT^ (0)'^ - (KT)-^ (17)

which has the disadvantage of not being diinensionless.

The last field variable is h, which measures the distance away from the

coexistence surface.

/ u^/RT y,/RT\ / v.^°(^,t)/RT yl'lcTJ/kTX

h = jin{Ke^ +e^ 1-iinfKe'^ +e^ ) (18)

where y-(i;,T) is the value of y- on the coexistence surface, or its extension

above the critical line [4], for tlie given (c,t).

In the transformed space, the phase transition region has a particularly

simple geometry. The critical line is the line segment t = 0, h = 0, <^ c <. 1>

and the coexistence surface is the plane region t < 0, h = 0, _< ^ <. 1.

The essential feature of the Holdover-Gallagher formalism [5] is to make a correspondence, for each value of c between and 1, with a hypothetical pure

fluid, and to assume the VLE properties of this hypothetical pure fluid are given

by the law of corresponding states and a dual system of reference fluids (the tvw

pure fluids) with a linear interpolation "mixing rule." In particular, the

"vapor pressure" curve for a locus of fixed r, on the coexistence surface is [of.

eq (3)],

PT^(0/TP^(0 = 1 + C3(c)|t| ^-^+ C^iOt + C5(0t^ + C.(c)t^ (19)

and this hypothetical fluid has a coexistence dome in the T-p plane given b^ [cf.

eq (1)]

p/p^(c) = 1 ± C^(c)|t| -^^^

+ C2(c)t (20)

where, i = 1,...6,

Ci(d = c|2) + (cP-cj2)),= c|2)+crc , (21)

and the superscripts refer to the respective pure fluids. Note that the

superscripts were incorrectly reversed in reference [5], eq (8). This error did

not affect the final results of reference [5].

Figure 8 shows lines of constant c in the P-T plane for the nitrogen-methane

system. Equations (19) and (21) predict that these lines form a nearly parallel

94
curvilinear grid, which is an additional empirical assumption. Note that, if

the two fluids obey corresponding states exactly, cj. ^ = c|| - C.{^) for all

5. In general, C. (;;) is never expected to be \iery far from either pure fluid

value.

As explained previously, each (P,T) point within the coexistence region

corresponds to two coexisting phases of different densities and compositions.

Thus, given c and t, our goal is to predict p^,. p„^^, x-,. and x,,^^ . The

1 1 Cj vap 1 1 c{ vap

densities are given by eq (20). Leung and Griffiths, in a lengthy analysis, have

worked out the inverse transformation from the field variables to the

composition. In the Holdover-Gallagher formalism [5], this transformation is

= (1 - c) jl. ,[^.M.H(c,t)]| (22)

where

Q(c,t) = RTP,

K

1^41 t|^-^ + C^t + C^t2+ C^t^j (23)

and T , P and C are functions of c. Substitution of p = p,. and p = p

in eq (22) yields, respectively, x,. and x

^ ' _ "^ 1 1 q vap

The function H(c,t) has an explicit and rather complicated definition

(eqs (A3) and (A7) of reference [5]) in terms of thermodynamic derivatives. For our applications, it is assumed to be a smooth function with adjustable fitting

parameters. Note that, for eq (15) to hold on the critical line,

H(c,0) = . (24)

In their first recipe, Moldover and Gallagher make the additional empirical

assumption that IT = over the entire coexistence surface, thus completing

the construction of a closed recipe to predict x.

In practice, given the vapor pressure curves of the pure fluids and the

mixture critical line, we seek to construct isotherms in the P-x plane, figure 5,

95
and dew-bubble curves for fixed x, figure 3. For the former, we merely choose

successive values of c from to 1. Then, for a fixed T, the variables t, P,

X, . and X are determined algebrdically, yielding a series of points which

liq vap ^ J i J J V describes the isotherm. Construction of dew-bubble curves is more complicated.

The bubble curves are loci of constant x, . , the dew curves of constant x liq' vap

To find points on these curves, we must perform a numerical root-finding search

on eq (22), since direct algebraic inversion is not possible. Alternately, we could calculate x-,. and x on a sufficiently fine grid of (^,t) points as in figure 8, and construct a coniplete family of dev.'-bubble curves by

interpolation.

There diVQ at present only a limited number of binary mixtures with accurate

experimental data for the critical line; particularly scarce are those with

accurate measurements of p (x). Moldover and Gallagher [5,22] tested their

3 4

method on four binary mixtures: He- He (with results [22] essentially

equivalent to those of Leung and Griffiths), carbon dioxide-ethane, sulfur

hexafluoride-metiiane, and propane-octane. Tiie method yields excellent agreement

between theory and experiment for all but tne last inixture. Of particular

interest is the excellent VLE predictions for the azeotropic mixtures

CUp-CpH^ and SF^-CH.. As functions of commonly used thermodynamic variables

the VLE surfaces of these mixtures appear to have a ^ery complicated topological

structure. However, upon transformation to field variables, application of the

Moldover-Gallagher formalism, and performance of the inverse transformation, the

apparently complex VLE surfaces diVQ predicted to high accuracy (see, for example,

figures 3, 4 and 6 of reference [5]).

For the normal mixture C^Hq-Cc^H,^, Moldover and Gallagher found

significant discrepancies between theory and experiment, particularly for

propane-rich mixtures. In an attempt to correct this, they proposed a second

recipe where, instead of setting H = for all {^,t}, they assumed that

-1 '^c

H(c.T) = C^ T^

^

t , (25)

with Tq evaluated at c = 1 - x. Note that eq (24), and hence eq (15), are still valid. Moldover and

Gallagher give some theoretical justification for this form in their Appendix B. The constant C^ is, in effect, an adjustable parameter and the choice Cm = -25

96
provides d best fit to the propane-octane VLE data. However, for propane-rich

mixtures such a fit is still not entirely satisfactory.

3.3 The Quadratic Coupling Recipe

At this point, v;e seek a revised version of the Moldover-Gallagher [5]

technique that incorporates the appropriate modifications for mixtures on which

their methods break down, but which reduces to their methods for those mixtures

where it does currently work. To do this, we first must try to decide, with the

available clues, what features are held in common by mixtures on which the

Moldover-Gallayher recipe does or aoes not work. We mention here that their

methods appear also not to predict properly VLE behavior of the normal system

nitroyen-methane [figure 4].

The feature most immediately evident is that systems with small composition

difference betv^een liquids and vapor phases are described accurately by the

Moldover-Gallagher formalism, whereas systems with large composition differences

are not. Comparing figures 5 and 6, we see that the presence of an azeotrope

tends to "pinch" the VLE curves in the P-x plane, and hence makes the vaporliquid composition difference in general smaller than that of a normal mixture.

In addition, the composition difference generally becomes larger with wider

dew-bubble curves. Propane-octane and nitrogen-methane both have wide dew-bubble

3 4

curves, whereas those of the normal mixture He- He happen to be ^jery narrow.

It is illuminating to rewrite eqs (20) and (22) in terms of averages and

differences of properties in the vapor and liquid phases.

p avg

= Pcd + C2(c)t) (26)

1.355

>^avg = (1 - 0<1 - ^

Ap = 2p^ C^(0|t l*-"^^ (27)

(l . J|Li^ , kl^ . klM . H(c,t)l j (28)

( L ^liq ^vap ^c J)

ix = C(l - ,, [iiLMi . ali^ 1 . (29)

L ^vap ^liq J

97
As stated above, tne success of tne Moldover-Gallagher formalism appears to

depend on ax being small. The formalism is, necessarily, a mixture of rigorous

theory and empiricism, and the assumptions above H(5 ,t) in eq (25) provide a degree of freedom in fitting experiment. Note, however, that ax is independent

of TT. Thus, if AX is incorrectly predicted by the present Moldover-Gallagher

recipe, such predictions cannot be rectified merely oy variation of H.

At this point we consider whether it is reasonable to assign a quantitative

"size" to a phase change. In essence, a phase change means a difference in tne

magnitude of certain physical properties between tv/o coexisting phases as a function of a parameter (in our case t) which measures the distance from a critical point at which such a difference disappears. With the variables used

here, a pure fluid phase change is characterized by Ap only, whereas for a mixture the phase change is characterized by both Ap and ax. In the limit of

small t, eqs (27) and (29) give

AX = 5(1 - C)Ap Q(?,0)/p^(O

= 2C^(0|t I -^^^ c(l - Q(c,0)/p^(O

(30)

Let us first consider two fluids wnich obey exactly corresponding states,

so that, within tne Moldover-Gallagher recipe, Cj ^ = c| ^ = CJ^). In

reduced units, as a function of t, Ap is an invariant for all mixtures out not

AX. We now conjecture that it is not Ap which should be an invariant, but rather

some "amount of phase change" A(t) wnich is restricted to be a monotonically

increasing function of both Ap and nx. If this rather broad assumption is true,

it follows that the Moldover-Gallagher formalism gives too large an A(t) for

mixtures and hence overestimates C, (c), Ap, and ax. Indeed, it appears from

an examination of their figures for propane-octane that Holdover and Gallagher

predict dew-bubble curves and density coexistence domes which are too wide, and

thereby overestimate both ax and Ap.

As we have as yet no fundaniental theory to determine A(t) , we proceed

semi-empirical ly and seek a simple form, with the constraint that A reduce to

Ap/p for the pure fluid. Une possible choice vwuld be a linear combination of

Ap and AX, but since the Moldover-Gallagher recipe works very well as it stands

for mixtures with small ax, v;e believe a quadratic combination is more probably

correct to leading order, i.e..

98
a'^ = [(Ap)^ + [f(0 Ax]2]/p^ (31)

where the small-t asymptotic behavior is implied. To leading order this is

equivalent to

A(t) = J [14 [no]' (f^)'] . (32)

where f(?) must liave dimensions of density. The simplest choice is

J

\.fU)f = [P^IO]^ (33)

or

A(t) = M [i . t,^(„]2 (Mj

]

. (34)

We assume, for mixtures whose pure components exactly obey corresponding

states, that A(t) is invariant for the mixture, which requires a redefinition of

Ci(c).

2c(pure)|^ |.355 ^ 2C^^0|t l'^^^ jl + c^d - o4^^1 j

(35)

or

^(pure)

C^(0 =

^- : 2

• ^^^^

1 + c^(l - 0^

For fluids which do not exactly obey corresponding states, the natural

generalization of eq (36) is

Ci^) . ,(c(l) - c(2))

4(0 = — ^

2

• '3^'

I.c^a-,^[W]

99
Equation (37), together with eq (21) for i > 1, Qiibodies our extension of

the Moldover-Gallagher recipe which we shall call the "quadratic coupling

recipe."

It is emphasized that the new fonnalism remains thermodynainical ly

consistent. An examination of Appendix A of Holdover and Gallagher [5], in which

thermodynamic functions are derived from an explicit potential, shows that the

final results, eqs (19)-(23), retain the same form for any C, and C.-, which

are functions of ^ alone. However, changing C . , i = 3,... 6, fruin a linear

function of c would alter the end result, in particular eq (23).

An interesting pictorial representation of the quadratic coupling recipe is

given in figure 9. We imagine, in this description, a croquet wicket which moves from left to right with increasing c« For the pure fluids, the plane of the

wicket is perpendicular to the line of sight of a first observer, but for

mixtures (of constant c) it is tilted at an angle e, which we call the "angle of

volatility." The first observer sees a projection of the wicket which represents

Ap; that is, if t is the vertical distance from the apex and the wicket has an intrinsic width w (t), then the first observer sees an apparent width of

W (t) cos 6.

A second observer is placed on a line of sight perpendicular to that of the

first observer, and sees a projection of the wicket which represents ax, with an apparent width w (t) sin e. The angle of volatility goes to zero for a pure

fluid, and also for an azeotrope. Then the Pytiiagorean theorem leads to

eq (31). The main point is that the Moldover-Gallagher recipe makes the

projection of the wicket as seen by the first observer an invariant of the

mixture, whereas the quadratic coupling recipe makes the inherent width of tne

wicket an invariant of the mixture. Our tentative answer, therefore, to question

(4) posed in the introduction is that t is the independent variable, and A(t),

the "amount" of phase change, is the dependent variable which, together, "obey

corresponding states."

This picture might also be useful for representing the liquid-liquid

immiscibil ity problem [23]. For vapor-liquid equilibrium of binary mixtures, Ap

is large and ax is small. The inverse situation applies for liquid-liquid

immiscibil ity; there ax is large and, for liquids with nearly equal pure molar

densities, Ap is small. We could represent the latter similarly witn an angle e close to 90 degrees.

100
The present hypotheses go somewhat ayainst the spirit of the description of

field variable spaces by Griffiths and Wheeler [3]. Those authors note that the

space of field variables in some respects resembles a conventional vector space.

However, since the field variables in general have different dimensions, it is

contended that no natural way exists to define orthogonality. For example, the

vapor pressure curve of a pure fluid, at the critical point, picks out a definite

direction in the P-T plane, but Griffiths and Wheeler [3j contend this is the

only direction with physical meaning since pressure and temperature are incommensurate. By contrast, the quadratic coupling recipe is based on the

assumption that a particular functional combination of change of density and

change of composition behaves like the invariant length of a vector. In the

following section \^e describe an application of the quadratic cou^^ling recipe to

nitrogen-methane. Although our results are highly preliminary, if the "amount of

phase change" described above should prove to be an invariant for a wide variety

of components and compositions, we would feel justified in attributing some fundamental physical significance to it.

4. Application to Nitrogen-Methane

4.1 Vapor Pressure Curves and Critical Line

We now describe the limited progress to date in testing the quadratic

coupling recipe with the binary system nitrogen-methane. It is emphasized that

the results at this point are highly preliminary.

Data used for VLE of nitrogen-methane was that of Bloomer and Parent [16].

In order to avoid problems matching different sets of experiments, we used data

for the vapor-pressure curves of the pure components given by Bloomer and Parent,

together with some additional nitrogen points generated from the fit of Dodge and

Dunbar [24], which Bloomer and Parent quote below their table 3 and with which

their data agree quite well.

Since Bloomer and Parent do not provide an adequate tabulation of coexisting

densities of the pure components, we used for nitrogen the densities given in NBS

Tech. Note 648 by Jacobsen, et al . [25], and for methane those given in NBS Tech.

Note 653 by Goodwin [26]. As this work is an initial feasibility study, we did

not analyze systematically the accuracy of the mixture data.

With the pure fluid density data of references [25] and [26] and standard

least-squares fitting techniques, we fitted p^, (t) and Ap(t) to the pure-fluid

avg

101
versions of eqs (26-27), and thereby determined values of C, , C^ and p , Taking data such that | t | < .25, we found for nitroyen

C^ = 1.8365

C2 = - .7146 (38)

p^ = 11.20 kg-mol/m^

and for methane

C^ = 1.8330

C2 = - .7088 (39)

p^ = 10.14 kg-mol/m^

3

The critical density of nitrogen is given as 11.21 kg-mol/m in reference [25],

3

and that of methane is given as 10.000 ky-mol/in in reference [26].

As pointed out by Holdover and Gallagher [5], it is not practical to

determine C^ from vapor pressure data. Rather, they infer C, from the

equation of state correlations of Level t Sengers, et al . [8], and find that it is

close to 30 for all fluids studied. For simplicity we assume C^ = 30 for

nitrogen and methane, i.e.,

PT

p-^ - 30| t

1^-^ = 1 + C^t + C^t^ + Cgt^ (40)

and fit the left side of eq (40), determined from the data, to a cubic polynomial

in t. Our results for nitrogen are

C^ = 4.935

C^ = -30.16 (41)

Cg = -11.22

and for methane are s

= 5.084

s

= -28.11

s

= - 5.85

(42)

102
These results for C conform to the range of values listed in table 1 of

Moldover and Gallagher [5].

Determination of tiie critical line presents some further difficulties.

Bloomer and Parent [Ibj measure dew-bubble curves for five different compositions

and tdbulate a critical point for eacn curve. We riave attempted an independent

calculation of the critical line from the Bloomer-Parent data as follows. For

each dew-bubble curve, we estimate visually the data point closest to the

critical point. We fit a critical line (P versus T) with tliese data points and,

at the same tiiiie, fie the chosen data point and the two closest neighboring data

points to a parabola. Tne critical point finally chosen is that point on the

parabola v^nich has tne same slope as thai of the first fit to the critical line

at the initially chosen data point.

We then reexamine graphically our clioice of the critical point and that of

Bloomer and Parent. When in doubt as to which is preferable, we elect to use the

latter; however, our metiiod appears to give a more reasonable value for ttie

critical point at x = .5088. The values chosen are listed in table 1.

We fit Pr{x)/Tp(x) and 1/T (x) to fourtn order polynomials in x. Our

results are

P^(x)/T^(x) = 10"^ [2.418 + 2.183X - 1.384x^ - 0.975x^ + 0.446x^] (43)

1/T (x) = 10"^ [5.248 + 1.337X + 1.557x^ - 0.689x^ + 0.470x^] (44)

with P in MPa and T in K.

c c The critical density presents additional difficulties. In fact, for

application of the Moldover-Gallagher recipe it is the critical density which is

least frequently known and is the greatest barrier to general use of tne

formalism. Bloomer and Parent [16] do not tabulate critical densities; hov/ever,

in their figure 16, they show the mass (not molar) critical density to be

approximately a linear function of critical temperature.

In the absence of better information, we assume that such a linear relation

between critical temperature and mass density holds exactly. Mass densities for

tne pure components are taken from the respective NBS Tech. Notes [26,26]. The

3 fit of mass density p, -(x) [kg/m ] to composition then is

Pi^^(x) = 10^ [1.627 + 1.159X + 0.882x^ - 0.803x^ + 0.273x^] (45)

103
and the molar density is subsequently calculated as

Pc(x) = p^^(x)/[28.0134x + 16.043 (1 - x)]

^^^^

3

in kg-mole/m . This completes the collection of input information needed for both the

Holdover-Gallagher and quadratic coupling recipes. As shown in figure 8, we construct a grid of |c,t} points with steps in c of 0.1 and steps in t of 0.02.

At each point, we calculate p-i^ , ^vaD* ^lia' ^"^ ^vau ^^^^'^^i'^Q ^o ^^^ (20)

and (22), and the two respective recipes for C.(i;). We choose H to be zero throughout. These results are to be compared with the experimental VLE

properties as represented by tne dew-bubble curves of Bloomer and Parent [16].

4.2 Interpolation of Data

In previous studies, Leung and Griffiths [4j, as well as Holdover and

Gallagher [5], have employed computer graphics methods [27] in which the

experimental dew-bubble curves have been a fixed part of the graphics output.

Theoretical calculations of the dew-bubble curves are graphically superimposed,

and the adjustable parameters are varied until a best fit (determined visually,

not mathematically) is obtained.

One of our pending projects is the development of such graphics methods for

the quadratic coupling recipe. For our initial study, however, v/e have chosen an alternate and perhaps less biased method. The grid of (c,t) points constructed

do not coincide with the experimental dew-bubble curves. However, if we interpolate between the dew-bubble curves, we can find "experimental" values of

Xi- and x,,^^ at any point within the coexistence region of the P-T plane.

1 1 q vap

Recall that the bubble curves are loci of constant Xn • and the dew curves liq

loci of constant x ^ . vap

It is difficult in general to interpolate two-dimensional curves as snown.

But we can make a transformation such that, approximately, the curves become

straight lines. A well-known approximate representation of the vapor pressure curve of a pure fluid is [28]

log P = a T"-^ + b (47)

where a and b are constants. Hence, on an appropriate semilog plot, the purecomponent vapor pressure curves become straight lines. Equation (46) is an approximation derived from the Clapeyron equation [28], in contrast to eq (3)

104
which, apart from higher-order nonanalytic terms, is the form predicted by

scaling- law theories [8].

Bloomer and Parent [16] demonstrate that the dew-bubble curves, plotted on a semilog graph, also appear to be linear in a region sufficiently far from the

critical line. In figure 10, the dew-bubble curves for nitrogen-methane are shown on a semilog plot. So represented, they appear to be two straight lines

joined by a curved segment at the top.

We first determine the equation and maximum pressure of each linear segment.

Then, for each (^,t) or (P,T) point, the points of intersection of the constantpressure line v/ith the sloping dew and bubble lines are located. Finally,

Xt . and X are detennined by an Aitken interpolation routine [29] from the

liq vap -^ ^ four closest points of intersection (provided they all fall within the regions of

linear behavior). By this approach, interpolated "experimental" values of

X-, . on the grid points of figure 8 are generated.

4.3 Results

The differences between AXyj.j, the value of ax from interpolation of the

experimental data, and, respectively, AXj^p, the prediction of the MoldoverGallagher recipe [5], and Ax^,^d, the prediction of the quadratic coupling

recipe, are tabulated in table 2.

The interpolation methods are not valid close to the critical line, as explained above, so values for all c are only available for |t | _> 0.12. As

expected from the discussion of section 3.3, the Moldover-Gallagher recipe

significantly overestimates ax, especially in the central part {c, % 0.5) of

the coexistence region where the dew-bubble curves are widest.

On the other hand, the quadratic coupling recipe is in much better agreement

with "experiment," particularly for the lower |t j values. Significantly, tnere

appears to be no systematic trend in the sign of the difference, as there is with

Moldover and Gallagher. At larger jt j values the agreement with experiment is

not as good, but this is the region where the scaling law equations of state are expected to break down anyway [5].

The results for j t j = 0.12 are displayed in taDle 3. We aiiphasize that

there are many sources of uncertainty in these numbers. There are uncertainties

in the data itself and the critical line, as well as in the interpolation

procedures. Theoretically, there are uncertainties due to the empirical

assumptions inade by Moldover and Gallagher [5], in particular that a value of K

105
exists [eq (14)] such that eq (15) holds accurately, and that, with this value,

lines of constant c are yiven by eq (19) or the curvilinear grid of figure 8.

Therefore, an overall uncertainty of at least 0.01 in ax is to be expected, and

the agreement at this level is most satisfactory.

A similar interpolation scheme could be tried on the densities, to test the

predictions of Ap. However, this would require a double interpolation of the

experimental data, and density predictions depend yery sensitively on the

critical density [27] which is not accurately known here. Hence such a density

analysis has not been attempted at this time.

5. Summary and Future Projects

We have described in this lecture a line of tneoretical research into the

VLE behavior of binary mixtures near the critical locus wnich is based on scaling-law equations of states and field (rather than density) variables as defined by Griffiths and Wheeler [3].

The first explicit recipe created for such prediction of VLE behavior, given

the critical line and the vapor pressure equation of the pures, was that of Leung

and Griffiths [4]. Their iTiethod, the foundation uf subsequent work, was limited

3 4

in many respects to \/ery simply behaving binary mixtures like He- He. It was significantly extended in range (although some empirical assumptions were incorporated) by Moldover and Gallagher [5], who were successful particularly in

predicting VLE of azeotropic mixtures. Finally, \^e have proposed an extension of

the Moldover-Gallagner recipe v/hich shows promise of working for mixtures with

large composition difference betv^een coexisting phases, yet which alters only

minutely the l^loldover-Gallagner predictions for those cases where tneir recipe is

successful.

We conclude the lecture by listing some unanswered questions and some projects for the future.

1) Although we have found \fery good agreement using the quadratic coupling

recipe for ax of nitrogen-methane at | t | = 0.12, roughly the limit of the range

where scaling laws are expected to hold, we have not demonstrated comparable

agreement for values of | t | between 0.12 and the critical line. This is probably

best verified by a computer graphics method as described in section 4.2, and may

require minor adjustments in the critical line, particularly p-(x). Checks of

coexisting density predictions should be made by the same method.

106
2) The formalism incorporates certain empirical assumptions about c, which is a function of the chemical potentials, and bypasses explicit calculation of those

chemical potentials. Certain classical equations of state described in Jim Ely's

lecture [1], while yielding incorrect critical exponents, show a certain limited

degree of quantitative success in predicting VLE near tlie critical line, and one of these, the Peng-Robinson equation [10], provides an explicit algebraic fonn

for the chemical potential. Thus it would be instructive to use the

Peng-Robinson equation in conjunction with the present methods. Explicit

variation of the parameter K in the definition of i, can be used to test the

self-consistency of our empirical assumptions.

3) While our method predicts ax well, it fails rather poorly in the prediction

of X (for H = 0). As explained in section 3.3, this can be remedied by

changing H; however, the second Holdover-Gallagher recipe [eq (25)] might

not, at this stage, remain optimal. As recommended in Appendix B of Holdover and

Gallagher, classical equations of state can also oe studied to help select an appropriate general model for R(c,t).

4) To demonstrate the general validity of our recipe we must, of course, test

it on a variety of different mixtures, not only nitrogen-methane. The logical

choice of a second system is propane-octane. Good data is available and that

mixture is an example which is of particular present interest to our group, as it

contains molecules of widely disparate sizes.

The most frequent limitation to use of the method is the absence of accurate

data for Pp(x). In this context, we expect that the techniques described by

Brian Eaton [2] in the following lecture should prove helpful. If T (x) and

P (x), but not p„(x) , are known accurately, it may be possible to determine

p (x) theoretically from the equation of state with parameters which best fit

T and P , and use the theoretically calculated p (x) as input,

5) Although we have restricted our discussion to prediction of the coexistence

surface, the Leung-Griffiths formalism, in principle, predicts an equation of

state for the entire region (one-phase and two-phase) around the critical line.

In fact, such predictions have been used and compared with experiment for

He- He by Doiron, Behringer and Meyer [30].

It is of interest to predict similarly an equation of state for the entire

critical region according to the Moldover-Gallagher and quadratic coupling

recipes. A note of caution must be made, hov^ever [27]. For C^ ^ in eq (1),

107
the Moldover-Gallagher formalism singles out the supercritical extension of the

rectilinear diameter as a special direction. It appears more appropriate to

single out the critical isochore as a special direction, so the formalism should

by some means be revised accordl ingly. Some ideas for doing this are suggested

in the recent review by Moldover [31J.

6) An important unresolved problem in the theory of fluids is the proper

matching of equations of state valid inside the critical region with those

outside it. For pure fluids, Goodwin's equation of state [32] is designed to

predict PVT properties both within and outside the critical region. Hov/ever,

Goodwin's method has thus far never been applied to mixtures. It would be

interesting to reformulate the Goodwin equation of state in tenns of field

variables and perhaps thereby obtain an equation of state for mixtures valid over

the entire range of fluid thermodynamic variables.

7) Finally, fundamental theories of phase changes should be examined with the

hope of shedding light on the "amount of phase change" A(t) defined in

section 3.3. As stated earlier, use of this function appears to violate the

"orthodox" view of field variable vector spaces, namely that field on density

variables of differing dimensions cannot be added in a meaningful way. But if

A(t) is seen empirically to be an invariant for a wide variety of mixtures, this

will be sufficient motivation to search for its fundamental significance, and

hopefully thereby add to our basic understanding of phase transitions.

Note added March 1982 : Since the original presentation of this lecture in

October 1980, the author and Mike Moldover have made significant progress in

achieving some of the goals listed above. This progress [33] is briefly

summarized:

We have developed an efficient minicomputer program and graphics routine

which calculates and plots several dew-bubble curves or constant-composition

temperature-density curves per minute, over the range -0.1 <. t <^ 0. When our method is applied to the nitrogen-methane data of Bloomer and Parent [16],

excellent agreement is obtained between experiment and tlieory with C,, = -8

[eq (25)] for the dew-bubble curves. The temperature-density curves do not fit

well if we retain the previous assumption that the mass density is linear in

temperature along the critical line. Hov;ever, variation of the T-p critical

line, in the direction of higher critical densities, leads to excellent agreement

with experiment in the T-p plane without degrading the fit in the P-T plane. We

108
believe that, by analoyy with pure fluids, such a fitting procedure may prove to

be a superior means of determining p (x) than direct experimental measurement

(cf. eq (39) and p from reference [26]).

We find also that a linear coupling recipe works as well as, and perhaps

better than, the present quadratic coupling recipe. The linear coupling method

replaces eq (34) is by

A =

I Ap |/p^ + C^l AX

I (48)

and modifies eqs (36)-(37) accordingly, where Cw is an adjustable parameter.

The best fit to nitrogen-methane VLE data in the linear coupling model occurs for

Cii = -6, Cy = 0.3.

We also have constructed a very good fit to the n-butane-octane data of Kay,

et al. [34] with the additional feature that Cn can depend linearly on c. The

optimal fitting parameters for n-butane-octane are C„ = 0.3 and Cu = -12

(1 - 1.3c). At present it is not clear whether the linear or quadratic coupling

model is superior in general.

The present (quadratic) theory has also been applied by Al-Sahhaf [35] to a variety of binary mixtures and with a mixed record of success. Al-Sahhaf has

analyzed the same systems with the Peng-Robinson equation [10] and has compared

our method with that of Peng and Robinson. The author thanks Dendy Sloan and

Taher Al-Sahhaf for many nelpful discussions concerning their work.

6. Acknowledgments

This work would not have been possible without the research and guidance of

Mike Holdover, with whom the author acknowledges an ongoing collaboration. The

author thanks Mike for his encouragement and patient elucidation of the

principles of phase transitions. He thanks Hal Raveche and the staff of the

Thermophysics Division, National Bureau of Standards, Washington, DC for their

hospitality during the author's visit in 1979, when this work was conmenced.

Finally, he thanks Howard Hanley and the staff of the Thermophysical Properties

Division for numerous valuable suggestions.

109
7. HEFEREIMCES

[I] Ely, J. F., "A review of fluid phase equilibria prediction methods," this

volume.

[2] Eaton, b. E., Stecki , J., Wielopolski, P., and Hanley, ri. J. M.

"Prediction of the critical line of a binary mixture: evaluation of tne

interaction parameters," this volume.

[3] Griffiths, R. 6. and Wheeler, J. C, Phys. Rev. A2, 1047 (1970).

[4] Leung, S. S. and Griffiths, R. B. , Phys. Rev. A8, 2670 (1973).

[5] Holdover, M. R. and Gallagher, J. S., AIChE J. 24, 267 (1978).

[6] McCarty, R. U., "The extended corresponding states method applied to the

nitrogen-methane system," this volume.

[7] Rowlinson, J. S. and Watson, I. L)., Chem. Eng. Sci. 24, 1565 (1969).

[8] Level t Sengers, J. H. H., Greer, W. L. and Sengers, J. V., J. Phys. Chem.

Ref. Data 5^, 1 (1976).

[9] Redlich, 0. and Kwong, J. H. S., Chem. Rev. 44, 233 (1949).

[10] Peng, D. Y. and Robinson, D. B., Ind. Eng. Chem. Fund. J_5, 59 (1976).

[II] Sengers, A. L. , Hocken, R. and Sengers, J. V., Physics Today 3_0, 42 (Dec

1977).

[12] Rushbrooke, G. S., J. Chem. Phys. 39, 842 (1963).

[13] Widom, B., J. Chem. Phys. 43, 3898 (1965).

[14] Fisher, M. E., Rev. Mod. Phys. 46, 597 (1974).

[15] Rowlinson, J. S., Liquids and Liquid Mixtures, (Plenum Press, New York,

1969), Chap. 6.

[16] Bloomer, 0. T. and Parent, J. D. , Chem. Eng. Progr. Symp. Ser. 49, No. 6,

11 (1953).

[17] Himmelblau, D.H. , Basic Principles and Calculations In Chemical

Engineering (Prentice-Hall, Englewood Cliffs, N. J., 1967), p. 204.

[18] Khazanova, N. E. , Lesnevskaya, L. S. and Zaknarova, A. V., Khimsch.

Promph. 44, 364 (1966).

[19] Wallace, B. , Jr. and lAeyer, H. , Phys. Rev. A2, 1563 (1970); 5, 953

(1972).

[20] Schofield, P., Pnys. Rev. Lect. 22, 606 (1969).

[21] D'Arrigo, G., Mistura, L. and Tartaglia, P., Phys. Rev. A ]_2, 2587

(1975); Doiron, F. , Bull. Ai.i. Pnys. Soc. 26, 1217 (1981).

[22] Moldover, M. R. and Gallagher, J. S., in Phase Equilibria and Fluid

Properties in the Chemical Industry , ACS Symposium Series No. 60, S. I. Sandler and T. J. Sturvick, Eds., American Chemical Society, Washington,

1977, p. 498.

110
[23] Model 1, M. and Reid, R. C. , Thermodynamics and its Applications in

Chemical Engineering (Prentice-Hall, Englewood Cliffs, N. J., 1974).

[24] Dodge, B. J. and Dunbar, A. K. , J. Am. Chem. Soc. 49, 591 (1927).

[25] Jacobsen, R. T., Stewart, R. B., McCarty, R. D. and Hanley, H. J. M.,

"Thermophysical properties of nitrogen from the fusion line to 3500 R for

pressures to 1500 psia," Nat. Bur. Stand. (U.S.), Tech. Note No. 648

(1973).

[26] Goodwin, R. D., "The thermophysical properties of methane, from 90 to 500

K at pressures to 700 bar," Nat. Bur. Stand. (U.S.), Tech. Note No. 653

(1974).

[27] Moldover, M. R. , private communication.

[28] Himrnelblau, D. M., Basic Principles and Calculations in Chemical

Engineering (Prentice-Hall, Englewood Cliffs, N. J., 1967), p. 176 and

p. 257.

[29] Kopal, Z., Numerical Analysis (Wiley, New York, 1955), p. 36.

[30] Dorion, T., Behringer, R. P. and Meyer, H., J. Low Temp. Phys. 24, 345

(1976).

[31] Moldover, M. R., Theriiiodynainic anomalies near the liquid-vapor critical

point: A review of experiments, to be published.

[32] Goodwin, R. D. , in Equations of State in Engineering and Research , Advances in Chemistry Series, No. 182, K. C. Chao and R. L. Robinson, Eds.

American Chemical Society, Washington, 1979, Chap. 19.

[33] Rainwater, J. C. and Moldover, M. R., Thermodynamic models for fluid

mixtures near critical conditions, paper 33d presented at AIChE 1981

Annual Meeting, New Orleans; to be published in Chemical Engineering at

Supercritical -Fluid Conditions , Ann Arbor Science Publishers, 1982.

[34] Kay, W. B., Genco, J. and Fichtner, D. A., Vapor-liquid equilibrium

relationships of binary systems propane-n-octane and n-butane-n-octane, J.

Chem. Eng. Data 19, 275 (1974).

[35] Al-Sahhaf, T. A., Measurement and prediction of vapor-liquid equilibria

for the nitrogen-methane-carbon dioxide system, Ph.D. Thesis, Colorado

School of Mines, Golden, Colo., 1981.

in
Table 1. Critical Line for Nitrogen-Methane

X Tc (K) Pc (MPa)

0.0000 190.555 4.608

0.1002 185.09 4.861

0.2879 174.21 5.068

0.5088 159.21 4.888

0.6970 146.93 4.482

0.8422 136.87 3.985

1.0000 126.22 3.394

112
Table 2. Predicted Deviations in ax Upper Entry: axj^iq " ^^INT

Lov/er Entry: axqqj^ - AXj,^j

c t .12 .14 .16 .18

.1 .0124 .0108 .0084 .0068

.0105 .0083 .0051 .0013

.2 .0314 .0124 .0108 .0095

,0013 -.0037 -.0105 -.0186

.3 .0233 .0247 .0269 .0310

-.0069 -.0151 -.0250 -.0369

.4 .0387 .0448 .0524 .0631

-.0099 -.0184 -.0293 -.0426

.5 .0482 .0593 .0731 .0901

-.0099 -.0159 -.0236 -.0342

.6 .0484 .0634 .0807 .1015

-.0049 -.0055 -.0081 -.0128

.7 .0439 .0587 .0756 .0957

.0079 .0119 .0150 .0172

.8 .0344 .0469 .0599 .0758

.0192 .0265 .0339 .0417

.9 .0056 .0109 .0168 .0236

.0032 .0079 .0127 .0182

113
Table 3. Comparison of ax Predictions for | t| = 0.12

Percent = [(ax)^ - (ax) j,^j]/(ax) jj^y

AXi^ Ax^,^Q Ax^(^^ (Percent)f^G (PercentJQ^R

1 .1200 .1324 .1306 10.3 8.8

2 .2344 .2478 .2357 5.7 0.6

3 .3155 .3388 .3086 7.4 -2.2

4 .3604 .3990 .3505 10.7 -2.7

5 .3751 .4233 .3652 12.8 -2.6

6 .3594 .4078 .3545 13.5 -1.4

7 .3076 .3515 .3155 14.3 2.6

8 .2233 .2577 .2425 15.4 8.6

9 .1297 .1353 .1329 4.3 2.5

114

\
T3 • • r- a> 3 S- r— 3

M- +-> (O

o; s_

s_ O) 3 Q.

Q. (=33

fO +->

J- +j o CO

<+- O) -C -

:t) O) •r— c= -C oo M- = O

O) 4-> a c c r-

<u o

+-> Q.

to

•r— Qi X

O) +J oo 1/1 p—

(V^ ,.

—

+-> • ex. H- • O os.._^ +-> O 4-> ^~ C

Q. •r—o

>> Q.

l-> •r— ^~

to (X3 c O

(U a -l-> 1 •^

0) t.

S- o 3

+J cu

fO -C

s_ hO)Q.E • 0) ^—

1— tj r*

4-> (O • E 1—

O) -C

cu

u

S- to Ol

115
Figure 1. Pressure-temperature plot of the vapor pressure curve for a pure

fluid which separates the liquid (L) and vapor (V) phases (schematic). At its

highest pressure and temperature the vapor pressure curve terminates in a critical point (C.P.)«

116
Figure 3. Pressure-temperature plot of dew-bubble curves for a binary mixture

(schematic). For a mixture of composition x along the indicated isotherm, A

is the bubble point, and B is the dew point. At point C, liquid of

composition x-|^ coexists with vapor of composition X2.

117
I— O)

edV\l 'd

jd c

ja fO u -C

J21 Ol s E

O) 1 T3

a;

A CD

CO o

<U s> > +-> s_ • r- 3 C a

OJ OJ S-

&. 3 3 4-> CO X w •^

0) =

s-3.

S_ (T3 o C

a. -r—

ITS -Q > ^

0) 'O

i. (= 3 5- a. Oc • i+- r^ o 0) JZ O) M -l-> i. O 3 1

s- O^ Cl o •r—

<+- M-

<vJ- O) #1 3 c 1—

-f-> •r- <^

fO ^^ 1—1 S- 1 1 OJ ,

Q. 13 <U 1= U u

(U ^ c

-»-> 4-> O) 1 tOJ Z 0)

1-

O 4- 3 a>

l/l OJ S- CO J=

0) +-> =

s_ o

o. "O s- c <4- fO T3 • CO <V «* a) 4-> > Q.

0) s_ (O

t. 3 -o 3 O < a>

118
/

/

/

/

/

/ /

^ )

/

y

1.0

Figure 5. Pressure-composition plot of coexistence isotherms for a normal

binary mixture (schematic). The horizontal tie lines indicate coexistence

between liquid of the composition to the left and vapor of composition to the

right. For a pure sample of the more volatile component, x = 1. The dotted

line is an isotherm at a temperature between the critical temperatures of the

two pure components.

119
CoHi

Figure 6. Pressure-composition plot of a coexistence isotherm for the

azeotropic binary system carbon dioxide-ethane. At the azeotrope (x I 0.32)

the liquid and vapor have the same composition. For x < 0.32, ethane is more

volatile; for x > 0.32 carbon dioxide is more volatile.

120
(0

270 280 290 300 310

T, K

Figure 7. Pressure-temperature plot of the coexistence reyion for the

azeotropic binary system carbon dioxide-ethane. Adapted from reference [16], fiyure 5. The azeotrope, upper left, is tangent to tiie critical line. The

coexistence reyion consists of tv;o separate "sheets," hatched horizontally and

vertically respectively, with some overlap.

121
oo

CMO

00OO

4-> Oo

J-cno

O)

+-> O)Ec

OJ:^O

s_

4-> o

4- o

CM ro

+-> T- COcoa

<^-o o CO o <U

edI/V 'd

CO

a;

oi oo

«/) a.

<u

+-> CO

-t-> O

-a

122
Top View

First Observer

Second Observer

Fiyure 9. "Tilted croquet wicket model" for the quadratic coupling recipe. As

9 is increased, the apparent, projected width of the wicket becomes narrower

for the first observer [a symbolic representation of p(t)] and wider for the

second observer [a symbolic representation of x(t)].

123
oo

coin o

+-> Q. TJ

O)O

O

X +-> LU

i_ o • 0>

O)1 o

s4-> o

•-> o 1 E 0)CO 00 ^

s-o

4-

(U

5-

M tn > i/) o

T- <u

C

K J3

•f— o W 3

-Q

O) -o

(U

-!-> 4-O

+-> O

cu>

S-o

a; -C

czo

a>

s- • o 'q. c

^^ o

(0 o -(-> r^ •r— TJ • 1— +-> c E •r— • 1— a; S- X

OO

u o

S- • c~ Q. o +J 03

r-H S- -a

(1) fO o o :3 cu o

en

ZT1

edlAI 'd ^^6o|

124
PREDICTION OF THE CRITICAL LINE OF A BINARY MIXTURE:

EVALUATION OF THE INTERACTION PARAMETERS

B. E. Eaton

Department of Chemical Engineering

University of Colorado

Boulder, Colorado 80307

J. Stecki and P. Wielopolski

Institute of Physical Chemistry

Polish Academy of Sciences

Warsaw, Poland

and

H. J. M. Hanley"*"

Department of Chemical Engineering

University of Colorado

Boulder, Colorado 80307

and

Thermophysical Properties Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

The critical line of the binary mixture methane-ethane is

calculate'^ via the extended corresponding states Van der Waals one fluid

theory. The Gibbs free energy critical ity criteria are solved numerically. The numerical derivatives are compared with the exact analytical

results derived previously for the special case of the shape factors of

the extended corresponding states set equal to unity. Binary interaction parameters are adjusted to give a best fit of the critical line to

experimental data. These interaction parameters are then used to

evaluate vapor liquid equilibrium data av/ay from the critical region.

It appears that a fit of the critical line is not sufficient to obtain

binary interaction parameters of general applicability. Optimization of

the critical point predictions for the pure components is also

discussed.

Supported by the National Science Foundation, Grant No. HES 7419548, SMI

7610647.

** Supported in part by the Maria Curie Sklodowska Fund, Grant No. NBS-196,

established by contributions of the U.S. and Polish governments.

"*" Supported in part by the Office of Standard Reference Data, NBS.

125
Key words: binary interaction parameters; critical ity criteria;

extended corresponding states; gas-liquid critical line; one fluid

theory; van der Waal s theory; VLE prediction.

1 . Introduction

The prediction of phase equilibria is both a classical problem of the theory

of liquids and a problem of engineering concern. Today the chemical and fuel

industries have to increase productivity and conservation and have to transfer to

nev/ feedstocks; phase equilibria is a major factor. But it is well-known that

the prediction, even the correlation, of the properties of the appropriate

systems can be exceptionally difficult if the results are required to any

reasonable accuracy. Prediction techniques are needed especially because the

number of possible systems makes measurement an overwhelming task. Prediction

requires an understanding of theory but, unfortunately, theory cannot yet always

handle adequately the complex systems encountered: the gap between a systematic

practical theory and reality is large. One technique, however, has been applied

successfully to simple systems and does show promise in that the assumptions can

be identified clearly. This method is extended corresponding states. Here we apply it to a system of methane and ethane. A specific objective is to calculate

the gas/liquid critical line and to observe the effect of the binary interaction

parameters on the calculation. It is then interesting to see how these

parameters, optimized for the critical line, represent vapor liquid equilibrium

(VLE) data.

The critical line in a binary mixture may be calculated by solving the

equations

= ; ^ =

for a temperature (T) and pressure (p) with the mole fraction (x) specified. G

is the molar configurational Gibbs free energy of the mixture. In this v/ork

these second and third order derivatives were evaluated numerically, but have

been compared with the analytical results of Wielopolski [1] in the special case when the extended corresponding states shape factors are unity. The accuracy of

the approach has thus been evaluated.

126
The system methane/ethane was selected for comparison with experiment since

the VLE data have been evaluated for thermodynamic consistency by Hiza,

et al . [2]. The procedure is quite general, however, and we have applied it to

several mixtures. Variations have been reported extensively by Watson and

Rowlinson [3], Gunning and Rowlinson [4], Teja and Rowlinson [5], Mollerup and

Rowlinson [6], and Mollerup [7,8]. The overall objective is to develop a general

technique for calculating the critical line of a binary mixture and to see if the

binary interaction parameters can be reliably evaluated by adjusting them to give

the best least squares fit of the critical line data.

2. Corresponding States and Equations

The basic postulate of the theory used here -- the van der Waals one fluid

theory -- is that if the components a (a = 1 ,n where n is the total number of

species) of a mixture separately obey classical corresponding states, then their

mixture will also obey corresponding states as if it were a single substance.

The components can be represented by selected parameters, e.g., critical

temperature (T ) and critical molar volume (V ), and the hypothetical equivalent

substance, designed by subscript x^ can be characterized by some suitable

composition dependent averaged parameters T^ and V^. The method then assumes

that the properties of a pure substance at p and T, or V and T, can be evaluated

with respect to those of a reference fluid, designated by subscript £, via

T = T/f ^ and V„ = V/h_ ^ (1) aa,o aa,o ^

'

where the scaling ratios h and f are defined respectively by

h = V^/V^ and f = T^/t'^ (2) aa,o a' o aa,o a' o ^ '

For a mixture the most natural definition of h and f follows from the work

A A of Henderson and Leonard [9] to give the van der Waals one fluid mixing rules:

\o - £ ^ V& '^06. (*'

The cross coefficients f „ and h „ are left unspecified until further

combination rules are defined, e.g..

127
^a3,o ~ ^a3,o ^^aa,o ^B6,o^ ^^^

"^aBjO '\q

where C „ and iin q ^re the binary interaction coefficients which, although

formally close to unity, can play a major role in the calculation of phase

equil ibria.

2.1 One Fluid Mixture Equations

The properties of a mixture can be evaluated in terms of the reference

substance and the ratios of eq (2). The basic equations are: Compressibility factor, Z

Z (T,V,x) = Z^(T,V,x) (7)

= Z (T/f , V/h ) (8) 0^ x,o* x,o' ^

'

Molar configurational Helmholtz free energy, A

A (V,T,x) E yV,T,x) + RJ Z \ ^n

^a ^^^ a

where

A (V,T,x) = f A (V/h , T/f ) - RT £n h (10)

x^ ' ' ' x,o 0^ x,o' x,o' x,o ^ '

or the molar configurational Gibbs free energy, G

G (p,T,x) E G^(p,T,x) + RT X; ^a ^"

"^a ^^^^ a

where

G„(p,T,x) = f G (ph /f , T/f ) - RT -in h ^ (12) x^^* * ' X,0 0^^ x,o x,o' x,o' x,o ^ '

The symbol ^ refers to the koI ar quantity. Equations (7)- (12) which define the

properties of an n-component mixture, can also be used for pure component

properties if all subscript x's are replaced with a's.

128
2.2 VL£ Equations

For pure component VLE, equating the molar Gibbs free energy of each phase

results in the following expression:

[A«"/RT„ - in Z^ - 1 + Z„] = CA^"/RT„ - <.n Z^ - 1 + Z^] (13)

vap liq

where

In eq (13) superscript Res refers to the residual value defined by eq (14)

with A^ the value of the equivalent perfect gas. Equation (13) is expressed

in terms of the residual Helmholtz free energy rather than Gibbs since the

reference equation of state has T and V (not T and p) as the independent

variables.

For mixture VLE one can calculate the K-value for, say, species a at T and

P:

K = y /x (15)

where one can derive

with p the residual chemical potential. Further manipulations give y in

terms of G and, for a binary mixture.

/ a(VRT)\

^3 C V-9V~/t,

- ^ - «T X3 -97—1 - «T £n ^ (17)

P

where

•v. G.. A'Res

129
2.3 Critical Criteria

The conditions for a critical point at T,p for a mixture are

0^G/9X^)-P^p = (3^G/3x^)^^p = (19)

Substitution of the one fluid equations gives

0^ (G /RT)/3x^) +J_ = (20)

T,p a Q,

and

"v X ~ X 0^ (G /RT)/3xb

*—

1 = (21)

T.p (x^Xg)2

which can thus be evaluated using eq (18).

The above equations and others have been discussed in full and derived by

Rowlinson and Watson [3], by Eaton [10] and by other authors so it has been

sufficient to be ^ery brief. The equations form the basics of the evaluation of

phase equilibria for a pure fluid or mixture, given the reference equation of

state and the reference Gg or Aq.

2.4 Extended Corresponding States

In general, since classical corresponding states is not obeyed, eqs (8) and

(10) or (8) and (12) are not satisfied with the scaling ratios of eq (2). It is

possible, however, to define a corresponding states so that eqs (8) and (10) are satisfied exactly . To do this we define shape factors 6 and <)> so that (for a pure, for example)

aa,o

hence the ratios f and h become

Vq = Vl^lx-^ (22)

aa

= C—)e • h = C-^]^ (23)

The point about this redefinition, i.e., the basis of extended corresponding

states theory, is that the corresponding states equations can be used formally

130
with the provision that the scaling ratios are given by eq (23). It should be

stressed that the ratios could be solved for either a pure or a mixture via

eqs (8) and (10) but to do this would require a complete description of the

fluids in question: essentially an impossibility. It is convenient to have some generalized analytical relation for 9 and <^, Leach and Leland proposed the

following [11]:

e (T* V*. oj ) = 1 + (o) - 0) ) F(T* V*) (24) aa,o ^ a* a' a^ ^ a o^ ^ a* a^ ^ '

k * Z^

(T. V^. caj = {1 + (o) - u)J G(T , VJ M (25) aa,o a' a' a'

| 'a o a' a' i ^c

a

where

* *

^i^r.» ^J = a, + b, £n T + (c, + dJJ ) (V - 0.5) (26)

and

* * G(T^. V^) = a^ (V^ + bj) * Cj (V^ + CI2) S.n T^ (27)

Here o) is the pitzer acentric factor or some chosen parameter and a, b, c, d are constants:

a^ = 0.0892, 32 = 0.3903

b^ = -0.8493, b2 = -1.0177

c^ = 0.3063, C2 = -0.9462

d^ = -0.4506, d2 = -0.7663

The asterisk denotes the value reduced by the critical value. The equations are

•k -k constrained in that V is set equal to 2.0 for al 1 V > 2.0 and to 0.5 for

ot ct V* < 0.5: T* is set to 2.0 i f T * > 2.0.

a a a

We [12] have recently tested the Leach-Leland equations for the hydrocarbons

C, - C2n over an extensive range of experimental conditions and revised

coefficients are reported in the reference. We also verified that the original

equations were satisfactory for reduced temperatures greater than 0.5.

3. Calculation and Numerical Methods

The objective is to solve the critical criteria eqs (20) and (21) for the

methane/ethane system and in so doing, observe the effects of the interaction

131
parameters and n of eqs (5) and (6) on the results. Having these values, we then evaluate some K-values for selected temperatures using eqs (15)-(18). We

chose methane as the reference fluid, the equation of state for which is the 32

term BWR of McCarty [13]. Critical parameters and Leach-Lei and acentric factors

for methane and ethane are given in table 1.

Table 1. Parameters for Methane and Ethane

T^ P^ 0)

(K) (cm /mole) (Bar)

CH4 190.555 97.75 44.793

^2^6 305.33 147.06 47.448 .105

3.1 Analytical and Numerical Evaluation of the Derivatives

The numerical techniques used in this v/ork are standard. We use the central

difference formulas [14] for which the first two terms in the infinite power

series expansions are given here. For the derivatives of a function f evaluated

at a point x, one has

f, - f., f^ - 2f, . 2f., - f.2

2h 12h

fl - ^fp * f-1 ^2 - ^^1 ' '^^0 - ^^-1 ' ^-2

2 " 2

rf\ . h

- 2^1 * 2f.i - f.2 f3 - 4f2 . 5f, - 5f., + 4f_2 - f.3

"U

(28)

(29)

(30)

9x"/^ 2h^ Hi?

where

fo = f(x) . f^ = f(x+ nh) (31)

The difficulty is to choose a value of h which is not too small (otherwise

significant figures will be lost in evaluating the numerators of eqs (28)-(30)

but not too large (otherwise the truncation error, which can be estimated by the

second term of eqs (28)- (30), will be large). One also has to consider the word

length of the computer and the convenience of using single versus double

132
precision. In this work we calculated on a CDC 6400 and a CDC 6600 machine with

a 60 bit word length (13 significant figures).

We were able to observe definitely the effect of varying h for the special case 9 = * = 1, i.e., for classical corresponding states. Equations (19) and

(20) have been solved analytically by Wielopolski (1980) and the lengthy

expressions are reported in an NBS publication [1] and will not be repeated here.

For example, table 2 lists the number of figures in the numerical results which

were in agreement with the analytical results for the first, second, and third

-u derivatives of G /RT for a particular test case. The number of figures in

X -V, agreement for the function value of G /RT itself was 10-12.

A

Table 2. Comparison of Numerical and Analytical Resulto for

Derivative Calculations Using Single Precision

Arithmetic.

h (G /RT) (G /RT) (G /RT)

^ X^ 2x ^ 3x

10"^ 6 5 4

10"^ 7 5 3

10"^ 7 3

Table 2 indicates that the first order derivative is truncation error controlled, since its value becomes more accurate as h is decreased. The second

and third derivatives are, on the other hand, controlled by the loss of

significant figures since as h is decreased, they lose accuracy. Since the third

order derivative is the least accurate, we chose the value of h for which it is

calculated most accurately.

We now consider what the smallest values of the second and third order

_3 derivatives are which can be calculated with h = 10 , since our eventual goal

is to solve the equations for the critical point by driving the values of those

derivatives to zero. The derivatives go to zero by a cancellation of the two

terms in eqs (20) and (21), that is, the contribution from the hypothetical

substance is cancelled by the ideal mixture contribution. For this reason, the

values of the derivatives cannot be made arbitrarily small. The ideal mixture

contribution (which can be computed with negligible error) can only cancel as

133
many significant figures as appear in the hypothetical substance contribution.

Consider the case in table 2 with h = 10 . For the second order derivative, the

hypothetical substance contribution has five significant figures, and its value

is order unity (abbreviated 0(1)). If the ideal mixture contribution were to

cancel all five of these figures, the result would be a number of 0(10 ) with

no significant figures remaining. For the third order derivative, the

hypothetical substance contribution contains four significant figures, and is

0(10). Cancelling all significant figures would leave a number of 0(10 ).

In our first attempt at calculating critical lines based on the numerical

evaluation of the derivatives in eqs (20) and (21) using single precision

arithmetic, we were unable to obtain convergence of the temperature and pressure

to five significant figures. The problem appeared to be that there were not

enough significant figures in the derivative calculations. While the truncation

error is inherent to the formulas being used, the loss of significant figures can

be compensated by adding more figures to the function values. This was done by

the use of double precision arithmetic which gives us 26 significant figures on the CDC 6400. In table 3 below, the results for the numerical derivatives

calculated using double precision arithmetic are compared with the results

arrived at analytically. Again, reported in the table are the number of figures

of agreement between the two results.

Table 3. Comparison of Numerical and Analytical Results for

Derivative Calculations Using Double Precision

Arithmetic.

h (G /RT) (G /RT) (6 /RT)

^ X^ 2x ^ 3x

10"^ 6 5 4

10"^ 7 9 6

10"^ 7 8 7 10'^ 7 8 7

10"'^ 7 8 7

10"^ 7 8 4

10"^ 7 8 1

134
_3

For h = 10 , the single and double precision results are the same, which

indicates that truncation error is controlling. Looking at the double precision

results, the third derivative shows an increase in accuracy as h is decreased to

-5

10 ; clearly indicating that the truncation error is decreasing to this

point. As h is decreased past 10" , accuracy is lost due to loss of

significant figures.

Based on these results, a value of h = 10' is chosen to compute the

derivatives in double precision. Given this value for h, the smallest value of

the second derivative which may be calculated (containing no significant figures)

_o a is 0(10" ), and that for the third derivative is 0(10" ). The calculations of

the binary critical line were subsequently made to converge to five significant

figures for both temperature and pressure.

4. Results

It must again be stressed that the general procedure for calculating the

critical line or VLE is predictive and requires only the critical constants and

an acentric factor for the fluid of interest, or of the components in a mixture.

For a relatively simple system the results will be reasonable without optimization of any parameters. Since, however, we are concerned only with VLE and the

critical point we considered two straightforward optimization procedures

involving the factor w. The first was to adjust oj to give ttie best representation of the pure component vapor pressure curve, the second was to force the

critical temperature and pressure of the pure fluids to correspond exactly with

those of the reference substance. This second variation is simply to set

0) = 03 : hence by eqs (24) and (25) 9 = 1 and (}> = L^Jl^ — a form of classical

corresponding states. One should note that the two procedures are not the same because the Leach-Lei and equations are not constrained at the critical point.

4.1 Ethane: Pure Component Results

We first considered the ethane vapor pressure curve v/hich was obtained

using Leach's expression for the shape factors. The value of the acentric factor

for the Leach equations was determined by optimizing agreement with the vapor

pressure data by a trial and error procedure in which the sum of the average

absolute deviations, for the vapor pressure, and saturated vapor and liquid

densities, were minimized; temperature being chosen as the independent

variable. The temperature range over which the results were optimized was 180 K

135
to 300 K; 180 K being the lower limit for which Leach's equations were designed; 300 K corresponded to the maximum temperature for which the vapor

pressure program would converge. Calculated and data values were compared at

10 K increments. The average deviations obtained for several values of acentric

factor are given in table 4. The value chosen for acentric factor in this work

(oj = .094) is seen to give a substantial improvement over the Pitzer value

(u) = .105) which was used by Leach.

The curves for vapor pressure and orthobaric densities (p = 1/V), obtained

using Leach's shape factors with the optimized acentric factor are compared with

the correlations of Goodwin, et al . l15J to obtain the deviations plotted in

figure 1

.

Acentric

Factor w

Table 4. Variation of Ethane Vapor Pressure Curve With

Acentric Factor (Leach Q, <^)

.

Ave %

apI Ave %L^^^ Ave %

Ap

liq'

vap Ui

Ave %

.100

.096

.095

.094

.093

.092

.090

.005

2.080

1.259

.595

.467

.444

.479

.543

.736

60.579

3.148

2.215

1.460

1.270

1.164

1.127

1.137

1.241

61.633

.6158

.3706

.3791

.3894

.4067

.4245

.4424

.4904

4.746

5.844

3.845

2.434

2.126

2.015

2.031

2.122

2.467

126.958

Also in figure 1, deviations for the vapor pressure and orthobaric density

curves predicted using classical corresponding states are presented. This figure

emphasizes that the Leach shape factors make a significant difference. The vapor

pressure deviations are positive, and become larger as the triple point is

approached, since classical corresponding states predicts a slope of the vapor

pressure curve (dP/dT) which is too small. The deviation of pressure goes to zero at the critical point because the two parameters are choosen to make the

critical temperature and pressure correspond exactly.

136
T^ (K) 305.33 307.01 ( .55)*

P^ (bar) 47.448 48.790 (2.83)

P^ (mol/L) 6.80 6.98 (2.65)

The critical point results for ethane are in Table 5. Notice that the

results are better using classical corresponding states than with the Leach shape

factors. This is because classical corresponding states forces either the

critical temperature and density to correspond (e = <t> = 1), or it forces the

c c critical temperature and pressure to correspond (e = 1, 4) = 1/1),

Table 5. Ethane Critical Point Predictions.

Data Leach e, $(aj = .094) e = (j) = 1 e = 1 , <t) = Z^/Z^

305.33 305.33

47.750 (.55) 47.448

6.80 6.76 (-.59)

Percent deviation is in parentheses.

4.2 The Critical Line

We first calculate the critical line using the Leach shape factor equations

with acentric factors of .005 and .094 for methane and ethane respectively. The

results are plotted against the critical line data found in the review article of

Hicks and Young [16], and identified in the caption to figure 2 (the symbols used

in figure 2 are identical with those used in figures 3 through 7).

The results are presented in the form of T-x and p-x plots in figures 2-5

and show the general trends obtained by varying the binary interaction

parameters, 5 and n. Holding C constant, figures 2 and 3 show that n has a small

effect on the T-x curve, and a large effect on the p-x curve. In both cases,

increasing n gives a better representation of the data. Holding n constant,

figures 4 and 5 show that ^ has a much larger effect on the T-x curve than did n*

and an equally large effect on the p-x curve. The important point to notice is

that the maximum value in the p-x curve is shifted towards small mole fraction

values (of CH,) by decreasing 5. The best representation of the P-x curve in

figure 3 (i.e., K = 1.00, r\ = 1.08) indicates that the peak of the curve needs to

be shifted towards the smaller mole fractions to improve the agreement, thus, 5 should be decreased.

137
To achieve the goal of obtaining the interaction parameters by fitting the

critical line data, a manual search technique was initiated. The "best fit" was defined in the least squares sense. The results of this search were that

K = .97, and n

= 1^13 were chosen as the "best" values for the interaction

parameters. The "best fit" T-x and p-x curves are presented in figures 6 and 7 respectively.

The fit of the T-x curve is good, v/ith only one data point which seems astray. The p-x curve, however, does not have the right shape to fit the data

well. Part of the fitting problem is due to the bad prediction which is made for

the critical point of pure ethane. This led us to try the second approach of

setting w = oj . Hence, the critical endpoints in the T-x, and p-x curves are exact. A new optimization led to the parameter values c = '97, r\ = 1.07. While

the fit of the T-x curve was not significantly improved that for the p-x curve

was. These results are shown in figures 6 and 7.

4.3 Vapor-Liquid Equilibria Results

Of the VLE data judged to be thermodynamically consistent by Hiza,

et al . [2J, three representative isotherms were chosen to test the predictions

made using the binary interaction parameters determined in the previous section.

Two of the isotherms are supercritical (250 K and 199.92 K) , and one is

subcritical (144.26 K) . The sources of the data are: 250 K isotherm, Davalos,

et al. [17]; 199.92 K and 144.26 K isotherms, Wichterle and Kobayashi [18].

The VLE calculations used the Leach shape factors with the acentric factors

.005 and .094 for methane and ethane respectively. The results are presented as K-value deviation plots for both the methane and the ethane K-value predictions.

Figures 8, 9, and 10 contain these curves with the interaction parameters

obtained from the critical line fit (i.e., C = .97, n

= 1.07), These figures

also show that setting the interaction parameters to unity gives much better VLE

predictions than do the parameters obtained from the best fit of the critical

line data.

5. Summary and Conclusions

The proposed technique of calculating binary critical lines by numerically

evaluating the second and third order derivatives of the Gibbs free energy has

been checked with an analytical solution for the special case of classical

corresponding states, and has proven successful. The best least squares fit of

138
the critical line data of the system methane-ethane was then shown to be poor

(particularly the p-x curve) if the Leach shape factors are used with an acentric

factor optimized for pure component vapor pressure predictions. This is due to a bad prediction of the critical endpoint for ethane. To improve this fit, we use classical corresponding states to force correspondence of the temperature and

pressure at the critical line endpoints. However we also show that the pure

component vapor pressure predictions are not satisfactory if this is done.

Finally, VLE predictions are made using Leach shape factors with the acentric

factor optimized for vapor pressure predictions, and the binary interaction i parameters obtained from the best fit of the critical line data (i.e., with

c c

e = 1, <t) = Z /Z ). The results are not as good in general as those which are obtained by setting c = n = 1. Hence we conclude that a fit of the binary

critical line does not yield binary interaction parameters of any general

significance.

139
6. References

[1] Wielopolski, P., On the calculation of critical liquid-vapor lines of

binary mixtures, J. Res. Nat. Bur. Stand. (U.S.) 8^, No. 6, 441-449

(Nov-Dec 1980).

[2] Hiza, M. J., Miller, R. C. and Kiunay, A. J., A review, evaluation, and

correlation of the phase equilibria, heat of mixing and change in volume

on nixing for liquid mixtures of methane plus ethane, o. Phys. Cher.i. Ref.

Data 8, No. 3, 799-316 (1979).

[3] Watson, I. D. and Rowlinson, J. S., The prediction of the thermodynamic

properties of fluids and fluid mixtures. II. Liquid-vapor equilibrium in

the system argon + nitrogen + oxygen, Chein. Eng. Sci. 24_, 1575-1580

(19G9).

[4] Gunning, A. J. and Rov/linson, J. S., The prediction of the thermodynamic

properties of fluids and fluid mixtures. III. Applications, Chem. Eng.

Sci. 23, 521-527 (1973).

[5] Teja, A. S. and Rowlinson, J. S., The prediction of the thermodynamic

properties of fluids and fluid mixtures. IV. Critical and azeotropic

states, Chem. Eng. Sci. 28_, 529-538 (1974).

[6] Mollerup, J. and Rowlinson, J. S., The prediction of the densities of

liquefied natural gas and of lov/er molecular weight hydrocarbons, Chem.

Eng. Sci. 29, 1373-1381 (1974).

l7] Mollerup, J., Correlated and predicted thermodynamic properties of LUG and

related mixtures in the normal and critical regions. Paper E-2 in

Advances in Cryogenic Engineering, Vol. 20, K. D. Timr.ierhaus, ed.,

(Plenum Publishing Corp., New York, NY, 1975) pp. 172-194.

[8] Mollerup, J., Thermodynamic properties of natural gas, petroleum gas, and

related mixtures: enthalpy predictions. Paper M-l in Advances in

Cryogenic Engineering, Vol. 23, K. D. Timmerhaus, ed. (Plenum Pub! ishing

Corp., New York, NY, 1973), pp. 550-560.

[9] Henderson, D. and Leonard, P. J., Physical Chemistry, Eyring, H.,

Henderson, D., and Jost, W., ed. (Academic Press, Hew York, 1971)

Chapter 7, "Liquid Mixtures."

LlO] Eaton, B. E., Prediction of the critical line of binary mixtures:

Determination of binary interaction parameters, M.S. Thesis, University of

Colorado, 1980, 151 pp.

140
[11] Leach, J. W., Chappelear, P. S. and Leland, T, W., Use of molecular shape

factors in vapor-liquid equilibrium calculations with the corresponding

states principle, A.I.Ch.E. J. 14, No. 4, 568-576 (1968).

[12] Ely, J. F. and Hanley, H. J. M., Prediction of Transport Properties.

I. Viscosity of Fluids and Mixtures, Ind. Eng. Chem. Fund. 20, No. 4,

323-32 (Nov 1981).

[13] McCarty, R. D., A modified Benedict-Webb-Rubin equation of state for

methane using recent experimental data. Cryogenics J£, No. 5, 276-280

(1974).

[14] Hildebrand, F. B., Introduction to Numerical Analysis, 2nd ed.

(McGraw-Hill, New York, NY, 1974), 669 pp.

[15] Goodwin, R. D., Roder, H. M. and Straty, G. C, Thermophysical properties

of ethane, from 90 to 600 K at pressures to 700 bar, Nat. Bur. Stand.

(U.S.), Tech. Note 684 (Aug 1976), 320 pp.

[16] Hicks, C. P. and Young, C. L., The gas-liquid critical properties of

binary mixtures. Chemical Reviews 75., No. 2, 139-175 (Apr 1975).

[17] Davalos, J., Anderson, W. R., Phelps, R. E. and Kidnay, A. J, Liquid-vapor

equilibria at 250.00 K for systems containing methane, ethane, and carbon

dioxide, J. Chem. Eng. Data 2_1, No. 1, 81-84 (1976).

[18] Wichterle, I. and Kobayashi, R., Vapor-liquid equilibrium of methaneethane system at low temperatures and high pressures, J. Chem. Eng. Data

J2. No. 1, 9-12 (1972).

[19] Goodwin, R. D., The thermophysical properties of methane, from 90 to 500 K

at pressures to 700 bar, Nat. Bur. Stand. (U.S.), Tech. Note 653 (Apr

1974), 274 pp.

141
<

>

lU

O

LU

O

liJ 0.

80 ^ ' ' . UJ \

OC \ D

if) \\

HI \

cc 40 r \

Q. \ \

CC

^

O

a.<

> c>r^r^i~\r\i~^/-\/~^ ufu (JUL) OO^ ^ _^_ J ^ - O ^ O ^

1 1

1

150 200 250

TEMPERATURE, K

300

J

Figure 1. Percent deviations [(expt-calc) x lUO/calcj for the vapor pressure,

vapor density and liquid density for saturated ethane at saturation. Data

froM Goodwin l15j. Calculations from the extended corresponding states method

with the parai.ieter (jjtt optimized, circles. Also shown as the dashed line are the results with 8=1 and ({> = ^q/^^» see equations (22)-(27).

142
— 1 =

1.00 1 1 1 / ^ 0) i> /f

—

o 6 // °

II o M

l^ •

II /// > 6 ^

/j

'// o ^'

II / o

II

L 1 1

wo. O

II 1 1 1 1

00oo oX o

CMo o o o O O O O O

CM o CO <D Tf CM

O CO

CO CO CM CM CM CM CM T-

-a c

CD o

"O <-

n3

-M fO 1-

CD O) +J

fo c D

Q. >

CD ••« S_ "O -C 4-> O

dJ 4->

C ^-

-M (U

fa >^ s- to I— -Q 03 O) 3 Q. 3 U T3 ,—

>—

O "O fO

<0 CO

C >

CJ 3 (O +j a •!- x: o c I— 1— C fO Q. o ^

CJ •>- •I- s_ +j ro

•r- > s c S <a o >i o <o <— J2

33

O

I— CTC

<U -rfO O I a CU C +J -c +-> +J CO

CU I—

•• <U (Tt

r—I I— I

r— CU I I 4-> -Q

<u c

fO x: ro

+-> o

(tJ •!- I 1 Q 3 LO

0) (— co

Q.

s- o 3 o

en c cr

5- -r- -O

OJ ^ O

4-J 00

O

CU 03

O E >, „

03 (O E S Q O

03

O S- Q. i^ M-

>l'l

143
1 1 ^<-^^^

^x^

^ > /^f

in

o

II 00

o - I 1 p~

• — \

\ \\

o

o

II \A \

II o

o•

II - in ^v

^

V V(^ <JJ^ —

0) >^ \ \ \ ^

o o \ \ \\^ II o ^\\ \\ J> II \NvV.fi

o t> NS^O\ o •s NXvVV ~ T- o NSv\ —

II o Ns\

U/ II

1

1 V

00 C\J o

en

o»->

(D -ac m o

o

Q.

CO ^z o

o

o

X o

'q. ^ X

o

Q.

0)

CM

O

CO

cn

o

00

o O O

in

I

jeq 'd

114
^

^// o

// o V ^ X II ///^ «>

// \ in

// o o / o• a ( T- II \

II AM \oo o

r^ o

II {> IT ^ Am

ino>6

II

00oo o oo o o o o o O O O

CM o 00 (0 ^

CM

O 00

CO CO CM CM CM CM CM y- oXI

(1) c

«o sz

4-> <uI O)

-C

4-> O)E

S-o

<+-

O)

4-' OJE

fO

S-fOQ.

O)

+-> 4-o

OJ

5-en

>l'i

145
00

•

• o o

so

4-> c

(D

•I— -a

c: o o

a. ^ s_ X o

o o

X

4Jo

'q. '^ X

o Q.

o

(U

LO

0)

jeq 'd

i

146
00oo o o

CMo oO

S-O)

+-> c

T3

OJN

•r~

+-> Q.O

GJC

n3 -C

4J

O)I

.C

+-> O)O

<U

to

OJ

SO)

fO

CDo

a.

to

<u

S-ou

T3

O) T3

c:cu

+-> X

O)o

+-> -ao

Q.

00

OJ

i.

s_oo

•I- 00 I— O)>

r— S- u o

•r—

+->

O

•r- 2

S- •4-> O

cu I I—

oo

s_QJ -M

OJE

(tJ s_ 3

+->

oo

(U -M

fO

OOc

-oo

Q.

00

CU

s_

S-oO•^-oo

U)

fOo

O) ->-oc

CO o

CTi o O o o O O O o

CMO 00 <0 ^

CM

O 00

CO CO CM CM CM CM CM

>l'±

H7
o

U3

o

o

Q.

to

<u

S-S-o

a

+-> o

Q.

X

Q.

0)

(U

a>

jeq 'd

148
K(CH4)

>

LU

O

LU

O

DC

UJ

CL

KCCgHg)

-10-

PRESSURE, bar

Figure 8. Methane-ethane K-value deviation plots at 250 K. Shown are the

(dashed) curves with the interaction parameters from the critical line fit

[5 = 0.97, n = 1.07] and, for reference, with ^ = n

= 1.00 as the solid curves.

149
K(CH4)

I-

<

>

UJ

o

UJ

o

LU

0.

20

KCCgHg)

-20

1

y

1

y \

y y \

y \

\

V \

1 -

1

"^

\

1 20 40 60

PRESSURE, bar

Figure 9. Methane-ethane K-values at 199.92 K.

150
80

60-

KCCH^) 4o_

<

>

Q

IZ

LU

o

LU

CL

20-

20

KCCgHg)

PRESSURE, bar

Figure 10. Methane-ethane K-values at 144.26 K.

151
NBS-n4A (REV. 2-8C U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBs "ra-ioei

2. Performing Organ. Report No, 3. Publication Date

January 1983

4. TITLE AND SUBTITLE

PHASE EQUILIBRIA: AN INFORMAL SYMPOSIUM

5. AUTHOR(S)

B. E. Eaton, J. F. Ely, H. J. M. Hanley, R. D. McCarty and J. C Rainwater

6, PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

10. SUPPLEMENTARY NOTES

\2^ Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

This Technical Note reports an informal conference on phase equilibria held at the

National Bureau of Standards, Boulder, in October 1980. Talks were given on extended

corresponding states, critical behavior, mixing rules and, in general, the prediction

of the phase behavior of simple mixtures. A survey of methods used in industry was also presented. Suggested work for the future is given.

12. KEY WORDS (S/x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Critical line; extended corresponding states; fluids; hydrocarbons; mixtures; phase

equilibria; prediction.

13. AVAILABILITY

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