Fluid Phase Equilibria, 13 (1983) 91-108 Ekvier Science Publishers B.V., Amsterdam -Printed in The Netherlands

91

EXTENSION OF THE PENG-ROBINSON EQUATION OF STATE TO COMPLEX MIXTURES:

EVALUATION OF THE VARIOUS FORMS OF THE LOCAL COMPOSITION CONCEPT

Paul M. Mathias and Thomas W. Copeman Air Products & Chemicals, Inc. Box 538, Allentown, PA 18105

ABSTRACT --

Density-dependent local composition (DDLC) mixing rules (Whiting and

Prausnitz, 1981, 1982 and Mollerup, 1981) offer great promise to extend

equations of state to highly nonideal mixtures. In this work we investigate

various aspects of the DOLC concept and offer two useful forms. All

development is done with the Peng-Robinson (1976) equation of state, but the

results have general applicability.

INTRODUCTION

Practical models to describe phase equilibrium can be divided into two

broad categories: equations of state and activity-coefficient models.

Equations of state have been successfully applied to mixtures containing

,nonpolar and slightly polar components at all conditions of engineering

interest, including the critical and retrograde regions. On the other hand,

the mathematical flexibility of activity coefficient models has been

considered necessary to model systems which exhibit high liquid-phase

nonideality. The activity-coefficient approach works well at low reduced

temperatures where the liquid phase is relatively incompressible and up to

moderate pressures. Serious problems arise when these bounds are violated.

,Further, the use of different models for the various phases precludes the

correct description of mixture critical points. Additional problems arise for

supercritical components (Abrams, et al., 1975). The equation-of-state

approach does not inherently suffer from these limitations and thus it would

be very valuable to develop equations of state with the mathematical

flexibility to describe complex mixture behavior.

Recent research in equations of state has focussed on the mixing rules as a

promising way to achieve mathematical flexibility through a semi-theoretical

development. Vidal (1978) has shown that the standard mixing rules used with

the Soave (1972) and Peng-Robinson (1976) equations of state are very similar

to regular solution theory and thus inadequate for "chemical" systems.

Further, Huron and Vidal (1979) have suggested improvements to the standard

mixing rules by applying the ideas of NRTL model (Renon and Prausnltz, 1968)

0378-3812/83/$03.00 0 1983 Elsevier Science Publishers B.V.

92

to the excess Gibbs energy of the Soave equation. The results are promising

but the Huron and Vidal model does not meet the important low-density 1 imit.

This has been pointed out by Mollerup (1981) and Whiting and Prausnitz (1981,

1982); these investigators have separately proposed a physically appealing

model to account for mixture nonideality: density-dependent local composition

(DDLC) theory.

In this work we have investigated density-dependent local-composition

theory as applied to the popular Peng-Robinson (PR) equation of state. The

standard Peng-Robinson equation provides a very good description of

asymmetric, nonpolar systems but the DDLC-PR model proposed by Mollerup (1981)

.can be inaccurate by up to 3-4 orders of magnitude for highly asymmetric

systems. Noting this, we have adopted a modlficatlon of an idea proposed by

Dimitrelis and Prausnitz (1982). The new model enables improved correlations

of a wide variety of complex mixtures. Our proposed DOLC model (along with

the other recently proposed forms) could prove to be too expensive in computer

time for systems with a large number of components. Thus we propose a

truncated version of the theory which, however, retains several attractive

qualities.

STANDARD PENG-ROBINSON EQUATION OF STATE

According to the Peng-Robinson

pure fluid is given as a function

follows:

P=RT_ a

v-b v(v+b) + b(v-b)

(1976) equation of state the pressure of a

of the temperature and molar volume as

(1)

a and b are component dependent parameters. b is independent of temperature

and a is temperature dependent such that the vapor pressure of the pure fluid

i s accurately carrel ated. Thus,

b = 0.07780 R Tc/Pc

a = ace(T)

a C

= 0.45724 R=T,=/P,

B = [ 1 + CI(1 - fiR) + c2(1 - FR)Z + c3(l - j-YR)l ] 2

Equation (5) Is slightly different from that origlnally proposed by Peng

and Robinson (1976). The original PR model is obtained when c2=c3=0. The

additional parameters are necessary to correlate the vapor pressure of highly

polar substances 11 ke water and methanol. In this work we wish to focus on

the development of mixing rules. Inaccurate pure component vapor pressure

93

representation artificially distorts the excess Helmholtz free energy and

obscures the analysis of mixture effects (Van Ness, et al., 1973).

For mixtures, Peng and Robinson assumed that eqn. (1) remains unchanged

but the parameters a and b are calculated as the mole fraction averages shown

below:

a = I E xixjaji

ij

(6)

b = Z xibi

i

(7)

a.. is given by the usual combining rules which include the pair dependent

biiary parameter kji

aji = (1 - kji) (aiaj)'fl

Equations (1) and (6-8) deflne the "standard" Peng-Robinson model. We

refer to this form as the van der Waals l-Fluid (VOW-l) or "conformal" model.

It should be noted that the mixing rule defined by eqns. (6) and (7) do not

imply that the mixture is random (all pair radial distribution functions are

identical) but rather that the various pair radial distribution functions can

be superimposed on a reduced basis (Mansoori and Leland, 1972).

ANALYSIS OF THE MOLLERUP DDLC MODEL

Mollerup (1981) derived his version of the density-dependent local

composition Peng-Robinson equation of state by assuming that local

compositions arise from differences fn the attractive parameter "a" per unit

surface area (q).

Lx a /q E

P=RT_ 1 j j ji ji ji

zxq v-b v(v+b) + b(v-b) i i i 1 x. E..

J J' j

where, Eji = exp ~~~~~

(I 1 = exp - aji/qji ~ en

RT 2-&b

(9)

(10)

(11)

UserHighlight

UserSticky Notewhy assume?

94

and, aji/qj, = (1 - kji) (aiaj/qiqj)lB (12)

Actually, Mollerup did not employ the universal "degree-of-randomness"

parameter a (Whiting and Prausnitz, 1981), however, it is included here for

generality.

The attractive Helmholtz energy corresponding to eqn. (9) is given by

Aa = RT

- - Z xiqian

[I

BXjEji

II. i j

(13)

Useful insight into the nature of the DDLC theory is obtained by an

expansion of the logarithm and exponential terms in eqn. (13) as follows:

-a A = - Zxiqi Exjaji/qji

I i j I

r

FV

1 Q - 2xiqi Exjafji/qZji - (Exjaj

i \ j j

In eqn. (14) we have used the simp

1 v + (1 + 2)b $ Fv e - en

2$b I 1 v + (I -4)b

(14) /qji)=

r'

- Fv2 - .._._ 2RT

ifying notation,

(IS)

L J

It is interesting to note that the first term of eqn. (14) has exactly the

same functional dependence as the standard or VDW-1 Peng-Robinson equation of

state. (A minor difference arises in the i-j combining rule when qi # q., but

this could be absorbed by an appropriately chosen binary parameter, kji. ; To

fix ideas, we define the first term of eqn. (14) as the "equivalent" VDW-1

contribution and lump all the remaining terms into the "asymmetric" or

"nonconformal" contribution.

It is illustrative to compare the fugacity coefficients of eqn. (13) with

those of its equivalent VDW-1 form. Due to the complexity of the equations,

we only show the comparison for a binary with component 1 at infinite

dilution.

01-

- 1 = -9 2

_!I

-VDW-1

1) - (16)

95

a12 a22 1 where, X = - - - - Fv

[ d 912 qz RT (17) The right side of eqn. (16) is always negative and thus the fugacity

coefficient is reduced from its VDW-1 value. (The opposite effect is obtained

for negative values of 0~. However, only positive values of (I are

physically meaningful.)

In general, it can be shown that the local composition effect reduces the

Helmholtz energy of a system at fixed T, V, g. This is a qualitatively

correct result since the DDLC configuration should be more favorable than its

equivalent VDW-1 form.

EVALUATION OF THE MOLLERUP MODEL FOR ASYMMETRIC NONPOLAR SYSTEMS

The VOW-1 PR-EOS provides a good description of most nonpolar systems,

including those which are highly asymmetric. For example, the methane-decane

binary is accurately correlated over a temperature range of 311-583 K, even

with the default interaction parameter, k2I=O; the average absolute errors in

the K-values of methane and decane are 3.5% and 8.8%, respectively. It is

extremely important that the DDLC mixing rules retain this desirable

capability. Therefore we have conducted an investigation of the model

proposed by Mollerup (eqn. 9) by using the methane-decane binary as a test

case.

Figures 1, 2, and 3 present the results of the investigation. We have

somewhat arbitrarily chosen to show the liquid-phase fugacity coefficient of

methane as a function of composition at 310 K and 270 atm. In general, the

DDLC mixing rules cause a decrease in the fugacity coefficient. Figure 1

shows that this decreasecan be enormous, giving fugacity coefficients too low

by as much as 3-4 orders of magnitude when the q of decane is unity. Figure 1

also presents the fugacity coefficients with the size parameter from Bondi

(1968), q=6, and two higher values. The local composition effect is still

large. A non-randomness parameter (o) of 0.2 along with the same decane

size parameters is shown in figure 2. Figure 3 illustrates the additional

effect of setting the binary parameter to 0.5. These comparisons indicate

that the Mollerup model requires large and unreasonable parameters to

correlate this relatively simple system.

DEVELOPMENT OF A NEW DDLC MODEL: DEPARTURE FROM THE GEOMETRIC MEAN

The results of the last section show that the "asymmetric" contribution

greatly over-predicted by the Mollerup version of the DDLC-PR EOS for

too

iS

asymmetric, nonpolar systems. A similar conclusion has already been reached

All u: P - 1.0 kz,- 0.0

q-+-y-- o.b 0.0 0.2

x(t&aM) OS6

Fig 1. Effect of LC model (eqn. 9). Liquid-phase of methane-decane binary at 310 K and 270 atm.

10

1 VOW-l. 9'1 (OATA wlnuw 10x) k

$; U.H -/- -- _---,/H -- s 0.1: -xiq=ro___--- ,/-- - __--- #H _I All lx%

:__/-/- u. 40 t*;%l 0.01 , I 1 I I

0.0 0.2 0.4 0.8 0.8

%(Methano)

Fig 2. Effect of LC model (eqn. 9). 310 K and 270 atm.

Liquid-phase of methane-decane binary at

91

i VOW-~. q=l (DATA WITNIN 10%)

-1 ei LC.qrlO -__- _____c _--- u. q=s __-- --- __-- 9 I l.c. q*o All u

,,i _

0.0 0:2 I

x(&a"*)

0:s 0.k

Fig 3. Effect of LC model 310 K and 270 atm.

(eqn. 9). Liquid-phase of methane-decane binary at

Fig 4. Computer time for calculation of fugacity coeffkients for n-component system.

98

by Dimitrelis and Prausnitz (1982) who further point out that models analogous

to eqn. (9) make the rather stringent assumption that local compositions

Inevitably occur In the j-l pair interaction whenever ai/qi # aj/qj. They

have proposed that local composition effects are not caused when ji

interactions are different from ii but rather when ji are different from some

"ideal" combination of ii and jj interactions, referred to as ji".

Dimitrelis and Prausnitz define jiO as an arithmetic mean. Since the standard

Peng-Robinson generally produces satisfactory results for nonpolar systems

with kji = 0. the geometric mean should be a very good first approximation for

jiO.

We assume that the total molar internal energy of the ji pair in a

multi-component fluid is divideb into two parts: ideal and excess. The

"ideal" part is entirely conformal as in the standard Peng-Robinson model.

However, the "excess" contribution causes non-conformal local compositions

described by the DDLC theory. \

-a -a0 -aEX U ji

U = ji + u..

31

-a0 a FV U ji =_._._

(1 - kji) -

al/T T I VP!.!

-aEX a FV

"ji = ~ (-dji) -

al/T T I V,N

(19)

(20)

Equation (19) corresponds to the standard Peng-Robinson model. The usual

binary parameter kji is retained. However, in the data analysis we have

generally set kji = 0. The parameter dji determines that part of the internal

energy that produces local composition effects. Note that 4 is not symmetric

(i.e., d jf # dfj). The total mixture molar internal energy is obtained by

summing the ideal and excess ji contributions using the VDW-1 and DDLC mixing

rules, respectively.

-a0 -a0 U = E Z xjxj uji (21)

ij

-aEX -aEX U = B Xrqi x ji j< (22)

i j

Eji where 'ji

= x. - J

X 'k Eki k

and Eji = exp (-dji)Fv 1

99

(23)

(24)

The Helmholtz energy may be obtained by integrating eqns. (18-24) with

respect to l/T. Thus,

_a RT A = - L E xixj (aiaj)'f2 (1 - kji)Fv - - X xiqi rn I: xj Eji

[ 1 (25) ij cri j It should be noted that the integration to obtain Helmholtz energy requires

an expression for the athermal limit. Here we assume that the repulsive term

of the standard Peng-Robinson equation of state yields a reasonable athermal

contribution to the total Helmholtz free energy. Whiting and Prausnitr (1982)

have shown that the Peng-Robinson repulsive term resembles the Flory-Huggins

athermal term. Also we note that highly asymmetric nonpolar systems (e.g.,

methane-decane) are well described by the standard Peng-Robinson model.

Expressions for all configurational properties can easily be obtained from

eqn. (25). However no derived properties are shown in order to save space.

The above derivation is only valid from a phenomenological viewpoint. It

does however state the assumptions that have been made. The new model meets a

very important limit in addition to the ones at high temperature and low

density noted by Whiting and Prausnitz (1981) and Mollerup (1981): that is,

the nonconformal effect is not necessarily large (or even nonzero) for an

asymmetric system. This feature allows the model to retain the good results

of the standard Peng-Robinson model and facilitates accurate correlation of

the phase equilibria of highly nonideal polar systems.

DEVELOPMENT OF AN APPROXIMATE OR TRUNCATED DDLC MODEL

The local-composition Peng-Robinson model defined by eqn. (25) produces

greatly improved correlations over the standard Peng-Robinson model, as shown

in the next section. However, the computational load might prove to be

unacceptably high for practical applications with a large number of

components. This arises from the bulk of computer time being consumed in mole

fraction summation calculations. (Also see Boston and Mathias, 1980.) In the

model defined by eqn. (25), these summations are density dependent and thus

must be redone at each iteration for the mixture molar volume.

100

Figure 4 shows a comparison of the average computer time required by the

standard Peng-Robinson model and DDLC version (eqn. 25) for a single

calculation of the fugacity coefficients in an n-component system. In our

opinion, a reasonable increase in computer time (say, a factor of two) is

justified by a superior model, but an order of magnitude increase is

impractical (with current computer capabilities for process simulation) when

the number of components in the mixture is greater than about 15. Figure 4

also includes a comparison for the approximate or truncated model we

propose. We conclude that the truncated model may be used for large systems

without incurring a prohibitive computational burden.

According to the truncated DDLC model, the attractive Helmholtz energy of a

mixture is given by

-a aNC

A = - a Fv - - Fvz

2RT

(26)

NC where a is given by the standard Peng-Robinson form (eqns. 6 and 8) and a ,

the NonConformal term, is given by

aNc = H xi+ aci E 1 j

The parameters

xj tji (27)

tii are the lumped contribution of all the nonconformal

effects. Note thatt is not symmetric (tji # tij) and thus the model has 3

parameters per binary pair; the third parameter is kji.

The truncated model defined by eqns. (6, 8, 26, and 27) may be derived by

expanding eqn. (25) analogously to eqn. (14) and retaining only the first

nonconformal term. -aNC

Denoting this term as A we obtain,

-aNC 1 2 P z

1 I

A =-- Zxiaci xj dji - (z xj dji) F,

2RT i j

(28)

Notice that 01 and the qs and as have been absorbed in the ds, This does

not affect the composition dependence since only the first nonconformal term

Is used. Further, the temperature dependence of a (eqns. 3 and 5) has been

removed. This has facilitated good representation of data over a wide

temperature range with temperature-independent parameters (e.g.,

water-hydrocarbon systems).

101

The temperature dependence of eqn. (28) may also be justified on a

theoretical basis. Mansoori and Leland (1972) have shown that the first

nonconformal contribution to the compressibility factor is of order 1/T2.

This is consistent with eqn. (28).

The composition dependence chosen for eqn. (27) is somewhat arbitrary. It

is suggested by writing eqn. (28) for a binary,

-aNC 1 rzz 221

A = - - x1 x2 x1 acl d21 + x2 ac2 d12

2RT 1 1

(29)

Equations (28) and (29) always give a negative contribution to the total

Helmholtz energy. This is physically correct but it could restrict the mathematical flexibility of the model. Therefore, we have chosen the

relatively empirical form of eqn. (27). However, we should note that if the

regressed value of t is large and negative, it suggests that the physical

meaning is lost.

In this work only binary data were analyzed and thus it was not possible to

evaluate whether the more correct composition dependence of eqn. (29) is

necessary for multicomponent mixtures. Further, for the systems we have

analyzed the t's have been positive and thus eqns. (27) and (28) are

identical.

An extremely appealing aspect of the model is that it reduces exactly to

the standard Peng-Robinson model when all the t's are equal to zero. Thus it

is especially applicable to multicomponent systems whfch are largely nonpolar

but have a few important polar components (e.g., water in petroleum or coal

conversion processes).

CORRELATION OF PHASE EQUILIBRIUM

Vapor-liquid equilibrium was correlated for various systems using van der

Waals one-fluid and local-composition mixing rules, eqn. (25). One parameter

(k12) was fit with the van der Waals mfxing rules and two parameters (d12 and

d21) were fit with the local composition rules. The non-randomness parameter

(a) was set to unity and size parameters (q) were derived from Bondi (1968).

Table 1 presents maximum and average K-value deviations for these two methods.

In each system, local composition mixing rules enabled an improved fit to the

data. The methanol-benzene system was correlated to a much higher accuracy,

while improvement was less, but St.111 significant, for the other systems.

Figures 5 and 6 present plots of K-value deviations from data for the

methanol-benzene system. We should note the symmetric form of eqn. 25

(d12 = d21) did not show significant improvement over the VDW-1 form.

Methanol(l)-

Benzene(2)

Methanol(l)-

Carbon

Dioxide(P)

Acetone(l)-

Water(2)

Isobutylene(l)-

Methanol(2)

Ref

.

Nag

a ta

1969

Ohgak

i 1976

Griswold

1952

Churkin

1976

Temp.

(OC)

60.8-76.7

25-40

100

50

TABLE

1

Correlation

of Vapor-Liquid

Equilibrium

Data

Pre

ssu

re

(Atml

1.0

5.7-80.0

1.1-3.6

3.1-6.1

Mix Rules

vow-

1 0.1

0 -57.0

62.1

LC

0.61

-0.17

-7.0

-14.3

VDW-1

0.06

14.2

27.0

LC

0.23

-0.37

-4.1

-22.5

VDW-1

-0.23

LC

-0.35

-0.02

::::

-49.6

22.6

-29.7

10.6

VDW-1

0.04

33.0

LC

0.0

0.59

-1.6

Parameters

%Kl max*

XK2 max

%Kl avg

%K2 a"g

k

d

d

12

21

12

2"::

16.9

2.5

--

17.1

3.0

14.8

8.0

23.9

8.5

* Average

absolute

deviation

in K-value

of component

1.

103

15

1,

A

X

x (Methanol)

Fig 5. Comparison of new LC model (eqn. 25) with experimental data (Nagata, 1969).

Fig 6. Comparison of new LC model (eqn. 25) and VI@1 form with experimental data (Nagata, 1969).

TABLE 2

Correlation

of Liquid-Liquid

Equilibrium

Data

1

2

3

System

REP

Temp

Mix Rules

Parameter

%l

avg

w

avg

0 c

k

k

d

d

d

12a

12b

21a

12a

12b

1-Butanol(l)-

Water(e)

Hill

5-80

VDW-1

-0.18

49.1

26.7

1926

LC

-0.1

-0.05

-0.05

0.14

1.0

0.7

LC(tr)4

0.07

0.12

0.45

1.2

2.6

Benzene

(l)-

Water(e)

Tsonopoulos

10-200

ZW-'

-0.08

132.1

2691.1

1982

8:::

-0.15

1.08

0.06

LC(tr)

0.04

0.74

:::

:::

Hexane(1)

Water(Z)

@clpoulos

10-200

VDW-1

-0.24

>10000

LC

0.15

-0.24

2.39

0.15

"X

12.0

LCItr)

0.48

0.0

1.35

16:7

12.9

I-Hethylnaph-

thalene(l)-

Water(P)

Brady

37-277

1982

ZW-'

-:z

1036.0

-0.16

1.16

0.21

23.2

'"Z

i_C(tr)

0:31

0.01

1.36

3.5

618

1

d

=d

+

d

12

12a

T

12b

2 Average

absolute

deviation

in calculated

mole fraction

of component

1 in liquid phase 2.

3 Average

absolute

deviation

in calculated

mole fraction

of component

2 in liquid phase

1.

4 For LC(tr),

read "de as et".

106

Fig 7. Correlation of benzene-water mutual solubility data (Tsonopoulos and Wilson, 1982) with truncated LC model.

1

0.1

0.01

0.001

0.0001-

o.oooo1

Fig 8. z.Zrap;lation of I-methylnapthalene-water mutual solubility data (Brady ., 1982) with truncated LC model.

106

Liquid-liquid equilibrium was correlated for a few selected systems using

van der Waals one-fluid and local composition (including the truncated) mixing

rules. Table 2 presents average absolute deviations in calculated mole

fractions for these methods. One parameter was fit to the van der Waals

mixing rule, four parameters (to include temperature dependence) were fit

the local composition and three to the truncated local composition mixing

rules. Although the overall improvement is substantial with four local

composition parameters, it should be noted that temperature dependence in

k12 and d12 parameters must be correlated for an accurate fit. In this

to

the

regard, the local composition equation of state is analogous to the current

local composition activity coefficient models. It should be noted that even

with four binary parameters (also typically used with the UNIQUAC activity

coefficient model) the local composition method provides a significant

improvement beyond methods like Peng and Robinson (1980) for correlation of

liquid-liquid equilibria. These methods use separate interactlon parameters

(temperature dependent) for each phase. No ambiguity or discontinuity arises

with the local composition method at a critical end point where (any) two

phases merge.

The results with the truncated local composition model are particularly

encouraging since the (three) interaction parameters are not temperature

dependent. Figures 7 and 8 present a comparison of model calculations against

experimental data. Note that tZII for water in the hydrocarbons are nearly

zero. A two-parameter fit would yield about the same accuracy.

CONCLUSION

In this work we have investigated the density-dependent local composition

theory as applied to the Peng-Robinson equation of state. We propose two

useful models to account for nonconformal or asymmetric effects. The approach

is also applicable to any other equation of state derived from the generalized

van der Waals partition function.

Finally, the results of this work demonstrate that an equation of state

model which employs DDLC mixing rules and thus meets the required limits at

high temperature and low density is promising as a substitute for activity

coefficient models in practical applications.

ACKNOWLEDGEMENT

The authors wish to thank Professors J. M. Prausnitz and F. P. Stein and

D. Dimitrelis for helpful discussions.

The work reported in this paper was sponsored in part by the International

Coal Refining Company and the United States Department of Energy under

Contract DE-AC05-780R03054.

107

List of Symbols

Helmholtz Energy Peng-Robinson parameter, eqn. 1 Second Virial Coefficient Peng-Robinson parameter, eqn. I Binary Interaction Parameter Binary Interaction Parameter Surface Area Size Parameter Number of Moles Pressure Gas Constant Temperature Binary Interaction Parameter Volume Molar Volume Mole Fraction Non Randomness Parameter Temperature Dependent Pure Component Parameter, eqn. 5 Fugacity Coefficient

Superscipts

a Attractive Contribution Infinite Dilution

Subscripts

i Component i i Comoonent 5 i ComPonent k C Critical

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Bondi, A., 1968. Physical Properties of Molecular Crystals, Liquids, and Glasses, John Waley and Sons, New York.

Boston, J. F. and Mathias, P. M., 1980, Phase Equiilibria and Fluid Properties in the Chemical Industry - 2nd Internation Conference, Berlin. March 17-21:823-848.

Brady, C. J., Cunningham, J. R., and Wilson, G. M., 1982, Gas Processors Association, RR-62.

Churkin, V. N., Gonshov, V. A., and Pavlov, S. Y., 1978. Zh Fiz Khim. 52:488.

Dimitrelis, D. and Prausnitz, 3. M., 1982. "Thermodynamic Properties of Strongly Nonideal Fluid Mixtures from an Extended Quasi-Chemical Theory," paper presented at Fall Annual Meeting. American Institute of Chemical Engineering, Los Angeles, CA, Nov 14-19, 1982.

Gmehling, J. and Onken, U., 1981. Vapor-Liquid Equilibria Data Collection, Dechema, Frankfurt.

Griswold, J. and Wong, S. Y.. 1952, Chem. Eng. Prog. Symp. Ser. 48:18.

Hill, A. E., and Malisoff, W. M. 1926, J. Am. Chem. Sot. 48:918.

Huron, M. J. and Vidal, J., 1979, Fluid Phase Equilibria, 3:255-271.

Mansoori, G. A. and Leland, Jr., T. W., 1972, J. Chem. Sot., Faraday Trans. II, 68:320-344.

Mollerup, J., 1981, Fluid Phase Equilibria, 7:121-138.

Nagata, I., 1969, J. Chem. Eng. Data, 14:418.

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