Fluid 7835

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Fluid Phase Equilibria 264 (2008) 242–252

Reappraisal of the Vidal equation-of-state formulationfor vapor/liquid equilibrium

Mario R. Urdaneta a, Luis R. Fernandez a, Hendrick C. Van Ness b,∗a Chemical Engineering Laboratory, Escuela de Ingenierıa Quımica y Petroleo, Universidad del Zulia,

Maracaibo-Estado Zulia, Venezuelab Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA

Received 4 October 2007; received in revised form 21 November 2007; accepted 22 November 2007Available online 3 December 2007

Abstract

In 1978 Jean Vidal outlined an equation-of-state formulation for vapor/liquid-equilibrium calculations based on direct fitting of the energyparameter. There soon followed a paper by Huron and Vidal devoted to mixing rules, which included a brief review of the Vidal formulationunder the heading “polynomial mixing rule.” Neither paper presented examples of Vidal’s formulation in any detail. Subsequent work focusingon mixing-rule models – Huron/Vidal, its variations, and Wong/Sandler – are treated extensively in the 1998 monograph by Orbey and Sandler.However, no mention is made there of Vidal’s formulation, and indeed it seems never to have been evaluated in any detail. The purpose of thispaper is to revive and exploit the Vidal formulation for general application to vapor/liquid-equilibrium calculations. It is theoretically sound, andat least as convenient and accurate in application as the gamma/phi formulation.© 2007 Elsevier B.V. All rights reserved.

Keywords: Equation of state; Mixing rules; Vapor/liquid equilibrium

1. Introduction

The equation-of-state (EOS) formulation for vapor/liquid-equilibrium (VLE) calculations dates from the work of van derWaals, published in 1873. A comprehensive review of subse-quent publications is provided by Valderrama [1]. Many dealwith the composition dependence of the EOS parameters, forwhich no general theory is known.

The van der Waals, Redlich/Kwong, Soave/Redlich/Kwong,and Peng/Robinson equations all contain two parameters, andare versions of a generic equation of state, cubic in molar volumeV:

P = RT

V − b− a

(V + εb)(V + σb)(1)

The pure numbers ε and σ are specific to the equation of state.For the special case of pure species i the parameters are ai and bi,and are substance dependent. Both are independent of pressure,but ai is a function of temperature. Mixture parameters a and b

∗ Corresponding author.E-mail address: [email protected] (H.C. Van Ness).

are functions of composition, and the usual practice has been torelate them to the pure-species parameters by empirical mixingrules. As written for the liquid phase, denoted by superscript land with x for mole fractions, the van der Waals prescriptionsprovide the simplest realistic expressions:

bl ≡∑

i

xibi (2)

al ≡∑

i

∑j

xixjaij (3)

with aji = aij . Like subscripts (ii and jj) signify pure-speciesparameters, equivalent to ai or aj , whereas unlike subscripts(ij) signify an interaction parameter, commonly evaluated frompure-species parameters by an empirical combining rule, e.g.,

aij ≡ (aiaj)1/2 (4)

Analogous equations for the vapor phase (superscript v)replace x with y.

A new era began with the publications by Vidal [2] andHuron and Vidal [3] of a method to relate an EOS to an excess-Gibbs-energy model and so to provide more generally applicable

0378-3812/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.fluid.2007.11.019


M.R. Urdaneta et al. / Fluid Phase Equilibria 264 (2008) 242–252 243

mixing rules with a theoretical basis in solution behavior atthe limit of infinite pressure. Subsequent publications of vari-ations and elaborations on this theme are treated by Orbeyand Sandler [4]. Prominent among the resulting mixing rulesare those of Huron and Vidal [3] and of Wong and Sandler[5].

Vidal’s formulation as presented in the original papers is inno way related to any excess-Gibbs-energy model, and is readilyjustified on an empirical basis. Despite its potential as a versatileand accurate formulation for VLE calculations, it has receivedscant attention.

2. Mathematical formalism

To provide a foundation for comparisons of EOS formula-tions we review the necessary mathematics, structuring it forsimplicity and consistent notation.

Central to VLE calculations is the fugacity, related for speciesi in phase � to the fugacity coefficient:

f �i = xiφ

�i P

For VLE, f li = f v

i , and combining this equation with thepreceding definition yields:

xiφli = yiφ

vi (i = 1, 2, . . . , N) (5)

In combination with an equation for φ�i (� = l,v), Eq. (5)

provide N relations that may be solved for N equation-of-statevariables.

Setting V = ZRT/P introduces the compressibility factorZ. In addition, dimensionless alternative EOS parameters areprovided by the definitions:

β ≡ bP

RT(6)

q ≡ a

bRT(7)

Substituting for ε, σ, V, a, and b in Eq. (1) leads to expressionof the EOS as a cubic equation in Z. A suitable example isthe Peng/Robinson equation, for which ε = 1 − √

2 and σ =1 + √

2. In standard form it becomes:

Z3 + u1Z2 + u2Z + u3 = 0 (8)

where u1 = β − 1, u2 = (q − 3β − 2)β and u3 = (1 + β −q)β2.

For applications to VLE, all three roots are real and positive,with the largest root being Zv, specific to the vapor phase, andthe smallest Zl, specific to the liquid phase. Analytic solutionfor these roots is simple and direct. For the special case of a purespecies, EOS parameters in Eqs. (6)–(8) are written as ai, bi, qi,and βi.

For a given temperature EOS parameters q and β applyequally to liquid and vapor phases of the same composition, butfor phases of different composition in equilibrium identifyingsuperscripts are added. Thus, qv and βv denote values for a mix-ture with the vapor-phase composition and ql and βl, a mixture

with the liquid-phase composition. For a pure species super-scripts are meaningless, and only a species-identifying subscriptis required.

As discussed by Van Ness and Abbott [6], mixture parametersa, b, and q may be associated with partial EOS parameters, ai,bi, and qi. They are interrelated by the equations:

pi ≡[∂(np)

∂ni

]T,nj

(9)

p =∑

i

xi pi (10)

where p represents an EOS parameter. Eq. (9) defines a partialvalue. Because the parameters are independent of pressure, Pdoes not appear as a variable. Eq. (10) is the summability relation.Application of Eq. (9) to Eq. (7) leads to:

qi = q

(1 + ai

a− bi

b

)(11)

Fugacity coefficients for the constituent species of a mixtureare implicit in an equation of state. They are given for both vapor(� = v) and liquid (� = l) phases by [7]:

ln φ�i = b�

i

b�(Z� − 1) − ln(Z� − β�) − q�

i I� (12)

Quantities b�, Z�, β�, and I�, without subscripts, are forthe mixture. Application requires solution of Eq. (8) for Z�;quantity I� is given by [7]:

I� = 1

σ − εln

(Z� + σβ�

Z� + εβ�

)(ε �= σ) (13)

For the special case of a pure species, identifying subscript iis added.

3. Evaluation of pure-species parameters bi and qi

Pure-species parameters ai and bi are given by generalizedexpressions in relation to critical temperature Tci , critical pres-sure Pci , and acentric factor ωi:

bi = ΩRTci

Pci

(14)

ai = Ψα(Tri ; ωi)R2T 2

ci

Pci

(15)

Combination of Eqs. (7), (14), and (15) yields:

qi = Ψα(Tri ; ωi)

Ω Tri(16)

For the Peng/Robinson EOS [8] constants Ω and Ψ and thefunction α(Tri ; ωi) are

Ω = 0.07780, Ψ = 0.45724

and α (Tri ; ωi) = [1 + κ(1 − T 1/2ri )]

2

where κ = 0.37464 + 1.54226ωi − 0.26992ω2i


244 M.R. Urdaneta et al. / Fluid Phase Equilibria 264 (2008) 242–252

An improved expression for κ, given by Stryjek and Vera[9], requires a species-specific constant κ1, derived fromvapor–pressure data:

κ = κ0 + κ1(1 + T 1/2ri )(0.7 − Tri )

with κ0 = 0.378893 + 1.4897153ωi − 0.17131848ω2i +

0.0196554ω3i

Although generalized Eq. (16) usually provides a reason-able value for qi, the best value is found from a vapor pressureP sat

i , when it is known. An equation of state for pure speciesi at temperature T implicitly contains P sat

i [7]. A specific setof EOS parameters produces this result. Conversely, an EOSparameter may be evaluated from a known value of P sat

i .The calculation is based on the vapor/liquid equilibrium rela-tion:

ln φli − ln φv

i = 0 (17)

For pure species i, Eq. (12) becomes:

ln φ�i = Z�

i − 1 − ln(Z�i − βi) − qiI

�i (18)

This equation, written for both the saturated vapor and liquidphases, is substituted in Eq. (17). Solution for qi then gives [7]:

qi = Zvi − Zl

i + ln[(Zli − βi)/(Zv

i − βi)]

Ivi − I l

i

(19)

Here βi = biPsati /(RT )

Evaluation of qi by this equation is the only way to assure thatknown pure-species vapor pressures are regenerated by the EOS.Eq. (19) with Eqs. (8) and (13), each written for pure speciesi and the appropriate phase, provide five equations which maybe solved for the five unknows, qi, Zl

i, Zvi , I

li , and Iv

i . An initialestimate for qi is given by Eq. (16).

4. Huron/Vidal and Wong/Sandler mixing rules

The EOS-GE formulations based on Huron/Vidal (HV) andWong/Sandler (WS) mixing rules are here reviewed for com-parison. The first rule in both cases relates parameter q to theexcess Gibbs energy GE/RT as provided by a suitable correlat-ing equation (e.g., Redlich/Kister, Margules, Wilson, or NRTL).For the liquid-phase composition,

ql =∑

i

xiqi − GE

cRT(20)

This is the mixing rule that defines ql, and establishes itsrelation to the excess Gibbs energy. Constant c is specific to theEOS; for the Peng/Robinson equation, c = 0.62323.

Multiplication of Eq. (20) by n and application of Eq. (9)yields:

qli = qi − ln γi

c(21)

a result that depends on the basic relation:

ln γi =[∂(nGE/RT )

∂ni

]T,nj

wherein pressure independence is presumed.The second Huron/Vidal mixing rule is the van der Waals

prescription for b. When the composition is that of a liquid phase

bl =∑

i

xibi (HV) (22)

The corresponding equation for a vapor-phase composition is:

bv =∑

i

yibi (HV) (23)

In either case application of Eq. (9) yields:

b�i = bi (HV) (24)

providing a simplification for Eq. (12):

ln φ�i = bi

b�(Z� − 1) − ln(Z� − β�) − q�

i I� (HV) (25)

Values of ql, qli, bl from Eqs. (20) through (22) and of βl from

Eq. (6) allow evaluation of Zl by Eq. (8) and φli by Eq. (25).

The second Wong /Sandler mixing rule gives b as a quadraticfunction of composition, here written for a liquid-phase compo-sition:

bl =

∑i

∑j

xixjEij

1 − ql (WS) (26)

where indices i and j both identify species and run over all speciescomprising the mixture. Quantity Eij in Eq. (26) is defined as

Eij ≡ 1

2

(bi − ai

RT+ bj − aj

RT

)(1 − kij) (WS)

In view of Eq. (7), this becomes:

Eij = bi(1 − qi) + bj(1 − qj)

2(1 − kij) (WS) (27)

where kij is an empirical interaction parameter determined by fit-ting VLE data for the ij binary system. The following constraintsapply: kij = kji and kii = kjj = 0. Thus, Eii = bi(1 − qi), andEjj = bj(1 − qj). For a binary system, Eq. (26) becomes:

bl = x21E11 + 2x1x2E12 + x2

2E22

1 − ql (WS) (28)

Upon multiplication by n2, Eq. (26) may be rearranged as

nbl =(

1

n − nql

) ∑i

∑j

ninjEij

Application of Eq. (9), followed by considerable mathemat-ical reduction gives:

bli =

2∑

j

xjEij − bl(1 − qli)

1 − ql (WS) (29)



Fig. 1. Binary-system BUBL P calculations. Given T, x1, find P, y1; i = 1, 2, � = l,v; equation numbers in parentheses.

When Eq. (10) is applied to these partial parameters, Eq. (26)is regenerated.

Values of ql and qli from Eqs. (20) and (21), of βl from Eq.

(6), of bl from Eqs. (22)HV and (28)WS, of bli from Eqs. (24)HV

and (29)WS, allow evaluation of Zl by Eq. (8) and of φli by Eqs.

(12)WS and (25)HV.For either pair of mixing rules the preceding equations may be

written for a vapor-phase composition (� = v) by substitutingthe vapor-phase mole fractions yi for xi. The same correlatingequation is used for GE/(RT ) and the associated ln γi, but iswritten for vapor-phase mole fractions. The resulting values forqv, bv, qv

i , and bvi , along with βv from Eq. (6), allow evaluation

of Zv by Eq. (8) and of φvi by Eqs. (12)WS and (25)HV. Once

values of φli and φv

i are known, these EOS-GE formulations yieldequilibrium pressures and vapor compositions by solution of Eq.(5).

5. The Vidal or EOS-qE formulation

In the EOS-qE formulation only a single mixing rule in thetraditional sense is required, and it defines b� by the van derWaals prescription. Therefore, Eqs. (22) through (25) apply. Anexcess q-function for the liquid phase is also defined as

(qE)l ≡ ql −

∑i

xiqi (30)

A comparison of this equation with Eq. (20) might lead one

to identify (qE)l

with −GE/cRT . They are not to be compared,as they are applied very differently in entirely separate formu-

lations. Eq. (30) defines a function (qE)lto be fit directly to P–x

data by regression. It is not a mixing rule, though it accomplishesthe same purpose. Equations of the same functional forms as



Table 1Pure-species properties at 50 ◦C

System P sat1 (kPa) P sat

2 (kPa) q1 q2 b1 b2

Acetonitrile/ethanol [11] 33.864 29.496 14.1110 14.9137 73.0529 54.0673Acetone/ethanol [13] 82.029 29.484 12.5590 14.9145 69.9252 54.0673Acetone/water [14] 81.822 12.313 12.5637 18.3696 69.9252 18.9782Acetone/1,4-dioxane [14] 81.827 15.682 12.5635 15.5189 69.9252 72.9049Acetone/chloroform [10] 81.888 69.333 12.5621 13.0520 69.9252 63.4063Acetone/methanol [10] 81.963 55.654 12.5605 14.2614 69.9252 40.94911,4-Dioxane/water [14] 15.678 12.337 15.5193 18.3661 72.9049 18.9782Methanol/chloroform [10] 55.622 69.308 14.2625 13.0526 40.9491 63.4063Ethanol/chloroform [15] 29.510 69.356 14.9129 13.0514 54.0673 63.4063Ethanol/1,4-dioxane [15] 29.510 15.697 14.9129 15.5143 54.0673 72.9049Chloroform/1,4-dioxane [15] 69.356 15.697 13.0514 15.5171 63.4063 72.9049Chloroform/n-heptane [10] 69.388 18.930 13.0505 14.1559 63.4063 127.5245Ethanol/n-heptane [10] 29.586 18.904 14.9082 14.1584 54.0673 127.5245Ethanol/water [16] 29.476 12.345 14.9150 18.3650 54.0673 18.9782Water/ethylene glycol [16] 12.345 0.116 18.3650 24.5081 18.9782 60.4575Ethylene glycol/ethanol [16] 0.116 29.476 24.5081 14.9150 60.4575 54.0673Acetonitrile/water [17] 33.940 12.340 14.1069 18.3657 73.0529 18.9782Acetone/ethylene glycol [18] 81.830 0.114 12.5635 24.5381 69.9252 60.4575Acetonitrile/ethylene glycol [18] 33.900 0.104 14.1091 24.6968 73.0529 60.4575

used for GE are appropriate, but bear no relation to GE. Eq.(20), on the other hand, is a true mixing rule.

Because a phase designation merely reflects the compo-sition at which an EOS parameter is evaluated, the sameequation with the same parameters applies for vapor-phasecompositions. Implicit in this correlation are the partial param-eters (qE

i )�

. Moreover, the EOS parameters q� and q�i follow

directly:

ql = (qE)l +

∑xiqi (31)

qv = (qE)v +

∑yiqi (32)

q�i = (qE

i )� + qi (33)

Eqs. (31) and (32), with (qE)�

given by data regression andpure-species values qi given by Eq. (16) or (19), yield valuesfor q�. These together with b� by Eqs. (22) and (23) and β�

by Eq. (6), allow compressibility factors to be evaluated by Eq.(8).

With the same EOS parameters (q and b) for both phases andwith no simplifying assumptions, we write the same equationsfor both phases, differing only in the composition appropriate

Table 2Parameters for the Redlich/Kister expansion for (qE)

lat 50 ◦C

System A0 A1 A2 A3 RMS δP (kPa) a

Acetonitrile/ethanol [11] −2.22281 0.16364 −0.15027 −0.13384 0.09(0.09)Acetone/ethanol [13] −1.33318 0.04882 −0.05308 0.01912 0.01(0.01)Acetone/water [14] −4.55231 0.78615 −0.58819 0.32915 0.14(0.06)Acetone/1,4-dioxane [14] −0.45104 0.08872 −0.02456 0.01565 0.03(0.03)Acetone/chloroform [10] 1.48885 −0.24780 −0.06635 0.06735 0.05(0.06)Acetone/methanol [10] −1.43785 0.06208 0.00528 −0.03279 0.05(0.05)1,4-Dioxane/water [14] −4.76006 0.61382 −0.50950 0.17958 0.02(0.01)Methanol/chloroform [10] −2.58865 0.85668 −0.15163 0.14915 0.04(0.03)Ethanol/chloroform [15] −1.88186 0.84922 −0.06100 0.21206 0.04(0.05)Ethanol/1,4-dioxane [15] −1.75384 −0.19157 −0.06591 −0.00686 0.01(0.01)Chloroform/1,4-dioxane [15] 1.90753 0.54448 −0.02740 −0.07650 0.01(0.01)Chloroform/n-heptane [10] −1.17861 −0.26321 −0.08768 −0.05027 0.03(0.01)Ethanol/n-heptane [10] −4.69733 −0.06100 −0.95021 0.58945 0.39(0.39)Ethanol/water [16] −3.12035 0.90334 −0.23205 0.06067 0.03(0.03)Water/ethylene glycol [16] −0.75924 −0.39078 −0.11629 −0.07747 0.02(0.02)Ethylene glycol/ethanol [16] −1.02311 0.05527 −0.05000 0.02859 0.02(0.02)Acetonitrile/water [17] −5.13221 0.90302 −0.72292 0.29963 0.05(0.06)Acetone/ethylene glycol [18] −2.52241 −0.23314 −0.21631 −0.13883 0.06(0.06)Acetonitrile/ethylene glycol [18] −2.82205 −0.44749 −0.31230 −0.16953 0.03(0.02)

Average 0.06(0.06)

Direct fit minimizing the sum of squares of the pressure residuals.a Values in parentheses are from modified Raoult’s law (gamma/phi formulation with φ = 1).



to the phase. Thus the calculational procedure yields all of thequantities in Eq. (25) for both phases, from which values ofφl

i and φvi are calculated. Solution of Eq. (5) for given liquid-

phase composition then gives the equilibrium pressure and vaporcomposition.

The following example for a binary system shows the detailsof implementation of the EOS-qE formulation. Because of itsflexibility, the Redlich/Kister expansion often provides a reliablerepresentation of qE, and we apply it here with four parametersA0 through A3, either known or assumed in data regression:

(qE)l = Alx1x2 and (qE)

v = Avy1y2

Al = A0 + A1z + A2z2 + A3z

3 with z = x1 − x2 and Av =A0 + A1w + A2w

2 + A3w3 with w = y1 − y2.

Eqs. (31) and (32) become:

ql=Alx1x2 + x1q1+x2q2 and qv=Avy1y2+y1q1+y2q2

Also

βl = (x1b1 + x2b2)P

RTand βv = (y1b1 + y2b2)P

RT

Eq. (8) may now be solved for Zl and Zv.The partial excess parameters (qE

i )�

are given by [6]:

(qE1 )

l = x22(Al + 2F lx1) and (qE

2 )l = x2

1(Al − 2F lx2)

(qE1 )

v = y22(Av + 2Fvy1) and (qE

2 )v = y2

1(Av − 2Fvy2)

F l = A1 + 2A2z + 3A3z2 and Fv = A1 + 2A2w + 3A3w

2.

Eq. (33) yields the partial parameters q�i for substitution in

Eq. (25) to provide fugacity coefficients φli and φv

i , which leadthrough Eq. (5) to values of P and vapor-phase mole fractions.

6. Correlations of data: binary systems

Fig. 1 outlines the similarities and differences between threeVLE formulations for BUBL P calculations as represented byiterative calculational algorithms for the simplest case of a singlebinary-system data point, and by extension to the common caseof a data set. The initial values required for P and vapor composi-tion may be experimental. Alternatively, P may usually be takenas the xi-weighted sum of the pure-species vapor pressures, withy1 from Raoult’s law or by an ideal-solution assumption.

Parameters in a correlating equation for (qE)lare most effec-

tively found by a fitting procedure based on P–x1 data tominimize the sum of squares of the pressure residuals, i.e., thedifferences between experimental and calculated values of P.This is in direct analogy to Barker’s method [19] for the evalua-tion of parameters in expressions for GE. We initially apply thisprocedure with the four-parameter Redlich/Kister expansion, asjust described, to fit 19 binary P–x1 data sets at 50 ◦C [10–18].All measurements come from the same apparatus, and form acoherent body of high-precision data. Results are thus based onthe most easily obtained and most accurate of experimental mea-surements. It is a mistaken myth that such data can be enhancedby incorporating experimental vapor compositions, which are

inherently more difficult to measure and almost always of lowerquality. Pure-species vapor pressures form an integral part ofeach data set.

Results for the Peng/Robinson EOS are shown inTables 1 and 2. Table 1 presents experimental vapor pressures,qi values from Eq. (19), and bi values from Eq. (14). The qi

values show small differences for a particular species, reflectingvariations in its measured vapor pressure, but the bi values arealways the same. The necessary values of Tci , Pci , and ωi arefrom Table B.1 of Smith et al. [7], except 1,4-dioxane, for whichTc = 587.0 K, Pc = 52.08 bar, and ω = 0.288.

Table 2 gives values of the Redlich/Kister parameters for the

calculation of (qE)l, obtained by regression of the P–x1 data.

Although fewer than four parameters often provide a satisfac-tory fit, all four are here retained for consistency. Also shown arethe RMS δP residuals for each system. The average RMS δP

residual is 0.06 kPa. The values shown in parentheses are frommodified Raoult’s law (yiP = xiγiP

sati ), wherein γi is found by

Barkers method, yielding separate sets of Redlich/Kister param-eters. The residuals so found are little changed when vapor-phasenonidealities are taken into account through virial coefficients,as was done in the source papers. The two sets of residualsare seen to be little different, and demonstrate for a variety ofhighly nonideal systems that the EOS-qE formulation for low-pressure VLE data yields results comparable to the best that canbe obtained with the gamma/phi formulation.

For comparison, other formulations have been applied to thesame 19 systems. In all cases data reduction is based on mini-mization of the sum of squares of the pressure residuals. Resultsfor the average RMS δP values are summarized as follows:

Formulating equation Fitting procedure RMS δP (kPa)

Peng/Robinson EOS qE with two-parameterWilson equation

0.32

Modified Raoult’s law GE/RT with two-parameterWilson equation

0.21

Peng/Robinson EOS HV rules; GE/RT byfour-parameterRedlich/Kister equation

3.19

Peng/Robinson EOS WS rules; GE/RT byfour-parameterRedlich/Kister equation

0.61

Clearly, these formulations are less satisfactory than the Vidalformulation with the Peng/Robinson EOS and qE correlationby the four-parameter Redlich/Kister equation or the equivalentgamma/phi correlation.

Application is by no means limited to low pressure. As afirst example we treat the Ethanol/Water data of Barr-David andDodge [20] for 150 and 200 ◦C. Because pure-species vaporpressures are not given in the reference, qi values are found byEq. (16) and the Stryjek/Vera equations. However, a completecorrelation requires vapor pressures, and values consistent withqi parameters are easily found from Eq. (19). A three-parameterRedlich/Kister equation is fully adequate to correlate these datato within their experimental precision, and the parameters thatminimize the sum of squares of pressure residuals are given in



Fig. 2. Experimental values [20] and correlated P–x–y curves for ethanol/waterat 200 ◦C. Peng/Robinson EOS with qE given by three-parameter Redlich/Kisterfit to P–x data.

Fig. 3. Shows the nature of the qE function and its partial components qE1 and

qE2 for the ethanol/water system at 200 ◦C.

Table 3, along with the results of correlation. Figs. 2 through 4display the results of correlation for the 200 ◦C data. Fig. 2 is aconventional P–x–y diagram wherein the upper line representscalculated values of P versus experimental x1 values, and thelower line represents the same values of P versus calculated y1values. Superimposed are points for the well correlated experi-mental y1 values. Fig. 3 shows the nature of a typical analyticcorrelation ofqE. The two upper lines represent calculated values

Fig. 4. Residuals in P and y1 from fit of ethanol/water P–x1 data at 200 ◦C [20]by three-parameter Redlich/Kister equation.

of the partial properties (qEi )

lversus x1 for the liquid phase, and

exactly the same lines represent (qEi )

vversus y1. When combined

by the summability equation the partial properties for species 1and 2 produce a line which passes exactly through all points.These points are values from the correlation, and show that theliquid and vapor obey the same functional relationship. Fig. 4displays the P and y1 residuals. The δP values scatter nicelyabout zero as a result of fitting P–x data. The δy1 values alsoshow a reasonable scatter about zero. Because the correlation isbased solely on P–x data, this is a direct and conclusive test ofthe thermodynamic consisency of the data.

The CO2 (1)/propane (2) system supplies a second exam-ple, taken from the data of Reamer et al. [21]. Correlation isby the four-parameter Redlich/Kister equation, with pertinentresults given in Table 3. The system at 277.59 K is subcriticalwith measured vapor pressures, from which the qi are calcu-lated. Because the temperature of 310.93 K exceeds the criticaltemperature of CO2, no value of P sat

1 is available from whichto evaluate q1. Eq. (16) with the Stryjek/Vera correlation yieldsan extrapolated value of 5.6648. However, we choose to findq1 as a parameter in the fitting procedure, similar to findinga vapor pressure in the gamma/phi formulation [22]. With thedata point at highest pressure omitted, this gives q1 = 5.3523.Fig. 5 shows results for both temperatures on a P–x–y diagram.The correlation is fully satisfactory for the 277.59 K data, butless so for the 310.93 K data. The diagram is similar to those

Table 3VLE correlations for high-pressure systems, Redlich/Kister-equation fit of qE

T (K) (P range) A0 A1 A2 A3 q1 q2 RMS δP RMS δy

Ethanol/water [20]: b1 = 54.0666; b2 = 18.9527423.15 (5–10) −2.9340 1.1157 −0.6204 – 8.9319 12.2194 0.11 0.008473.15 (15–30) −2.8257 0.9509 −0.8211 – 7.0667 10.2118 0.27 0.007

Carbon dioxide/propane [21]: b1 = 26.6556; b2 = 56.2848277.59 (5–40) −2.2266 −0.5674 −0.3392 −0.2604 6.8538 9.1316 0.05 0.005310.93 (15–70) −1.3263 −0.0987 0.2429 0.1827 5.3523 7.7121 0.03 0.014

Parameters minimize sum of squares of δP residuals. Pressures in bars.



Table 4Binary-system parameters for the Wilson equation; 50 ◦C

System GEcorrelation qEcorrelation

Λ12 Λ21 RMS δP (kPa) Λ12 Λ21 RMS δP (kPa)

Acetonitrile (1)–ethanol (2) [11] 0.4213 0.5663 0.12 2.2500 2.7385 0.12Acetone (1)–ethanol (2) [13] 0.6119 0.7510 0.07 1.6655 1.9246 0.03Acetone (1)–water (2) [14] 0.1826 0.3927 0.19 3.9303 6.7744 2.11Acetone (1)–1,4-dioxane(2) [14] 0.5372 1.2919 0.04 0.7827 1.8108 0.03Acetone (1)–chloroform (2) [10] 1.0704 1.8811 0.13 0.2459 0.5515 0.23Acetone (1)–methanol(2) [10] 0.6937 0.7146 0.07 1.7543 1.9750 0.111,4-Dioxane (1)–water (2) [14] 0.2594 0.2845 0.57 4.4569 6.8367 0.34Methanol (1)–chloroform (2) [10] 0.1372 0.8400 0.72 1.5440 4.6744 0.64Ethanol (1)–chloroform (2) [15] 0.1391 1.1113 0.60 0.8823 4.4710 0.57Ethanol (1)–1,4-dioxane (2) [15] 0.7489 0.4448 0.01 2.5260 1.7231 0.03Chloroform (1)–1,4-dioxane (2) [15] 2.3292 1.0403 0.18 0.3348 0.1220 0.44Chloroform (1)–n-heptane(2) [10] 1.2372 0.4722 0.05 2.5344 1.0402 0.03Ethanol (1)–n-heptane (2) [10] 0.0895 0.2105 0.28 5.1701 5.7190 0.46Ethanol (1)–water (2) [16] 0.1907 0.8203 0.18 1.9811 5.3388 0.35Water (1)–ethylene glycol (2) [16] 1.7250 0.5768 0.02 3.7315 0.2669 0.03Ethylene glycol (1)–ethanol (2) [16] 0.5787 0.7683 0.03 1.4040 1.7743 0.03Acetonitrile (1)–water (2) [17] 0.1357 0.2838 0.37 4.8165 7.5739 0.15Acetone (1)–ethylene glycol (2) [18] 0.4139 0.2263 0.47 3.3117 2.2747 0.25Acetonitrile (1)–ethylene glycol (2) [18] 0.4344 0.1552 0.20 4.0931 2.2210 0.12

Average 0.23 0.32

Fig. 5. Experimental values [21] and correlated P–x–y curves for CO2/propaneat 277.59 and 310.95 K. Peng/Robinson EOS with qE given by four-parameterRedlich/Kister fit to P–x data.

shown for this system in the Orbey and Sandler monograph[4].

7. Correlations of data: ternary systems

Predictions of P–x–y behavior for ternary systems may bebased on data for the constituent binary systems when local-composition equations – Wilson, NRTL, etc. – are used forcorrelation. This is true for both the EOS-GE and EOS-qE

formulations. Reliable P–x data at 50 ◦C are available for 10ternary systems [11–18] formed from the 19 constituent bina-ries of Tables 1 and 2. Regressions of each binary data set withthe Wilson equation (chosen for its relative simplicity) yieldsparameters Λ12 and Λ21 for both formulations. For the EOS-GE

formulation the parameters pertain toGE and for the EOS-qE for-mulation to qE. Both sets of parameters and resulting RMS δP

values are listed in Table 4. Auxiliary data appear in Table 1. TheWilson parameters for the two correlations are very different, butoverall the results are acceptable, though not so good in generalas obtained with the Redlich/Kister expansion. Substitutions ofthe appropriate binary parameters into the Wilson equation writ-ten for the ternary systems provides direct prediction of ternarybehavior for both formulations.

Results for the ternary systems are reported in Table 5. Foreach pure species, the vapor pressure listed is an average ofall experimental values reported in the source paper, and isthe basis for the qi value by Eq. (19). The bi values are asin Table 1. With all correlations based on the Wilson equa-tion, values of RMS δP are presented for modified Raoult’slaw, for the EOS-GE formulation with WS mixing rules (κij

values from the binary regressions), and for the EOS-qE for-mulation. Results on the whole are comparable, though theWilson equation is evidently not particularly well suited torepresentation of qE for ethanol/chloroform/1,4-dioxane andacetone/chloroform/methanol.

8. Effect of temperature

When a VLE correlation is needed at conditions for whichexperimental data are not available, one seeks a reliable meansof prediction. Thermodynamics provides exact equations relat-ing the temperature dependence of both GE and qE to the excessenthalpy HE. Application requires integration over a tempera-ture range, and this can be done rigorously only when extensivedata are available, and they are seldom fully adequate. An alter-native is application of empirical procedures that may sometimesbe acceptable. Orbey and Sandler [4] show examples of EOS-



Table 5Results of VLE calculations for ternary systems at 50 ◦C

System P sat1 (kPa) P sat

2 (kPa) P sat3 (kPa) q1 q2 q3 RMS δP (kPa)

MRL a WS EOS-qE

Acetonitrile (1)–ethanol (2)–water (3) [11] 33.8618 29.4928 12.3505 14.1111 14.9139 18.3642 1.68 1.09 0.761,4-Dioxane (1)–ethanol (2)–water (3) [12] 15.7057 29.4723 12.3470 15.5162 14.9152 18.3647 1.15 0.79 0.11Acetone (1)–ethanol (2)–water (3) [13] 82.0182 29.4955 12.3547 12.5592 14.9138 18.3636 2.24 1.45 0.88Acetone (1)–1,4-dioxane (2)–water (3) [14] 81.8289 15.6906 12.3197 12.5635 15.5179 18.3686 0.49 0.93 0.84Ethanol (1)–chloroform (2)–1,4-dioxane (3) [15] 29.5100 69.3560 15.6970 14.9129 13.0514 15.5172 1.10 1.05 4.51Water (1)–ethylene glycol (2)–ethanol (3) [16] 12.3450 0.1160 29.4760 18.3650 24.5081 14.9150 0.16 0.11 0.20Ethylene glycol (1)–acetone (2)–water (3) [17] 0.1150 81.8270 12.3430 24.5230 12.5635 18.3653 0.61 1.29 0.91Ethylene glycol (1)–acetonitrile (2)–water (3) [17] 0.1080 33.9200 12.3420 24.6316 14.1080 18.3654 0.36 0.98 0.66Acetone (1)–chloroform (2)–methanol (3) [10] 81.916 69.325 55.613 12.5615 13.0522 14.2628 0.90 1.06 3.81Chloroform (1)–ethanol (2)–n-heptane (3) [10] 69.353 29.546 18.925 13.0514 14.9106 14.1564 0.73 1.39 0.74

Average 0.94 1.02 1.34

a Modified Raoult’s law: P =∑

ixiγiP

sati .

Table 6VLE correlations

T (K) (P (kPa)) Formulation/procedure RMS δP (kPa) RMS δy

323.15 (56–84) Gamma/Phi—reported by Walsh [10] 0.09EOS-qE—direct correlation of qE 0.06EOS-GE—with Wong/Sandler rules 0.22

372.8 (365–405) Gamma/Phi—reported by Wilsak et al. [24] 0.83 0.006EOS-qE—direct correlation of qE 0.86 0.006EOS-qE—predicted from correlation at 323.15 K a 3.65 0.008EOS-GE—predicted from correlation at 323.15 K b 6.44 0.009

397.7 (667–783) Gamma/Phi—reported by Wilsak et al. [24] 1.39 0.013EOS-qE—direct correlation of qE 1.45 0.014EOS– qE—predicted from correlation at 323.15 K a 6.01 0.014EOS-GE—predicted from correlation at 323.15 K b 18.20 0.016

422.6 (1121–1399) Gamma/Phi—reported by Wilsak et al. [24] 2.38 0.006EOS-qE—direct correlation of qE 2.50 0.007EOS-qE—predicted from correlation at 323.15 K a 16.14 0.007EOS-GE—predicted from correlation at 323.15 K b 40.01 0.015

Acetone/methanol by two-parameter Redlich/Kister equation.a Based on the qE correlation at 323.15 K with assumption that the product qE · T is constant.b Based on the correlating parameters for 323.15 K with assumption that they do not change.

GE correlations based on WS mixing rules with parametersfor GE/RT and ln γi evaluated at one temperature and usedunchanged at another. There is, however, no evident theoreticaljustification for the underlying presumption that GE/T is inde-pendent of T. It may be approximately valid in some instances,but is demonstrably not general.

A similar empirical procedure with the EOS-qE formulationis based on the assumption that the product qE · T is constant.Application with the Redlich/Kister correlating equation is thenparticularly simple, requiring only that the parameters at twotemperatures be inversely related to the temperatures. Thus,

Ai(T ) = Ai(T0)T0

T(34)

No theoretical basis is evident to support this procedure, but itis often satisfactory over relatively small temperature intervals.

Application of the two-parameter Redlich/Kister equation tothe acetone/methanol system produces results for four tempera-tures as given in Table 6. The first two lines for each temperatureshow that results from the gamma/phi and EOS-qE formulationsare equivalent and excellent. Computational effort is about thesame, but the EOS-qE formulation treats both phases exactlyalike, and no extraneous information, such as virial coefficients,is required. Table 7 provides parameters for the direct correla-

Table 7Direct correlation of qE by two-parameter Redlich/Kister equation

T (K) A0 A1 q1 q2

323.15 −1.43837 0.05204 12.5605 14.2614372.8 −1.21090 0.08784 10.0317 11.1172397.7 −1.09122 0.10680 9.0174 9.8612422.6 −0.98096 0.10308 8.1570 8.7902

Acetone/methanol: b1 = 69.9252; b2 = 40.9491



Fig. 6. Experimental values [23] and correlated T–x–y curves for ace-tone/methanol at 101.325 kPa. Peng/Robinson EOS with qE given by applicationof Eq. (34) to the two-parameter Redlich/Kister equation based on experimentaldata at 323.15 K.

tion of qE for the four temperatures. At 323.15 K, the EOS-GE

formulation based on WS mixing rules is only satisfactory.The two predictions at the three higher temperatures are

inferior to the direct-fit procedures, with those based on theOrbey/Sandler assumption of temperature-independent param-etes markedly less successful than those based the assumptionof constancy of the product qE · T .

Eq. (34) is easily applied to the calculation of a T–x–yrelationship at constant pressure. Such a calculation for theacetone/methanol system at 101.325 kPa produces the resultshown by Fig. 6. The RMS δy and RMS δT values are 0.002and 0.13 K, respectively. The basis is the Redlich/Kister cor-relation of data for 323.15 K. Temperature dependence ofparameter q is provided by the Stryjek/Vera [9] correlation,with parameters in Eqs. (14), (15), and (16) from Table 3.1.1of the monograph by Orbey and Sandler [4]. The successfulcorrelation of experimental data at 101.325 kPa as reported byGmehling, et al. [23] most likely reflects the fact that differencesbetween calculated temperatures and 323.15 K are no more than15 K.

9. Concluding remarks

Our purpose here is to review and extend a straight-forward,general, and effective EOS formulation for the treatment of VLEbased on Vidal’s original EOS formulation. It has the prac-tical advantages of treating both phases alike, without virialcoefficients, requiring no assumption of pressure independence,without the introduction of any empirical binary-interactionparameters, making no use whatever of activity coefficients orGE, and with the imposition of only one arbitrary and verysimple mixing rule. We do not pretend that treatment is hereencyclopedic, but it does point the way.

List of symbolsa parameter in cubic equation of stateai partial parameter, cubic equation of state

Aπ a composition-dependent parameter for theRedlich/Kister equation

A0, A1, A2, A3 parameters in the Redlich/Kister equationb parameter in cubic equation of statebi partial parameter, cubic equation of statec equation-specific constant; for Peng/Robinson equa-

tion, c = 0.62323Eij parameter in Wong/Sandler mixing rule; Eq. (26)F� a composition-dependent parameter for the

Redlich/Kister equationfi fugacity of pure species ifi fugacity of species i in solutionGE excess Gibbs energyI represents an integral as defined by Eq. (13)kij an empirical interaction parameter; Eq. (27)n total number of molesni number of moles of species ip represents an equation-of-state parameter; Eq. (10)pi represents a partial equation-of-state parameter; Eq.

(10)P absolute pressurePci critical pressure of species iP sat

i saturation vapor pressure of species iq equation-of-state parameterqi equation-of-state parameter for pure species iqi partial equation-of-state parameter parameterqE excess equation-of-state parameter. qE ≡ q − �xiqi or

qE ≡ q − �yiqi

qEi partial excess equation-of-state parameter; Eq. (33)

R universal gas constantT absolute temperatureTci critical temperature of species iTri reduced temperature of species iu1, u2, u3 composite parameters for Peng/Robinson equation;

Eq. (8)V molar volumew defined as w ≡ y1 − y2xi mole fraction of species i in liquid phaseyi mole fraction of species i in vapor phasez defined as z ≡ x1 − x2Z compressibility factor

Greek lettersα(Tri ; ωi) function in the definition of EOS parameter ai; Eq.

(15)β equation-of-state parameterβi equation-of-state parameter for pure species iγi activity coefficient of species i in solutionδ defines a difference between a calculated and experi-

mental propertyε constant in equation of stateκ, κ0, κ1 functions in evaluation of EOS parameter α(Tri ; ωi)Λ12, Λ21 parameters in Wilson equationσ constant in equation of stateφi fugacity coefficient for pure species iφi fugacity coefficient for species i in solutionΦi defined as Φi = φv

i / φsati



Ψ constant in evaluation of equation-of-state parameterai, Eq. (15)

ωi acentric factor for species iΩ constant in evaluation of equation-of-state parameter

bi, Eq. (14)

Subscriptsi, j identify a species

SuperscriptsE denotes an excess propertyl denotes a liquid-phase property or parameter for liquid-

phase compositionv denotes a vapor-phase property or parameter for vapor-

phase composition� signifies the phase

References

[1] J.O. Valderrama, Ind. Eng. Chem. Res. 42 (2003) 1603–1618.[2] J. Vidal, Chem. Eng. Sci. 33 (1978) 787–791.[3] M.J. Huron, J. Vidal, Fluid Phase Equilib. 3 (1977) 255–271.[4] H. Orbey, S.I. Sandler, Modeling Vapor–Liquid Equilibria. Cubic Equa-

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Solutions: With Applications to Phase Equilibria, McGraw-Hill, New York,1982.

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[8] D.Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. 15 (1976) 59–64.[9] R. Stryjek, J.H. Vera, Can. J. Chem. Eng. 64 (1986) 334–340.

[10] G.E. Walsh, Jr., Experimental Determination of Vapor–Liquid Equilibriafor Two Ternary Systems, M.E. Thesis, Rensselaer Poloytechnic Institute,Troy, NY, 1974.

[11] S.R. Wilson, R.B. Patel, M.M. Abbott, H.C. Van Ness, J. Chem. Eng. Data24 (1979) 130–132.

[12] A.M. Balcazar-Ortiz, R.B. Patel, M.M. Abbott, H.C. Van Ness, J. Chem.Eng. Data 24 (1979) 133–136.

[13] M.M. Chaudhry, H.C. Van Ness, M.M. Abbott, J. Chem. Eng. Data 25(1980) 254–257.

[14] J.R. Loehe, H.C. Van Ness, M.M. Abbott, J. Chem. Eng. Data 26 (1981)178–181.

[15] C. Gonzalez, H.C. Van Ness, J. Chem. Eng. Data 28 (1983) 407–409.[16] C. Gonzalez, H.C. Van Ness, J. Chem. Eng. Data 28 (1983) 410–412.[17] M.A. Villamanan, A.J. Allawi, H.C. Van Ness, J. Chem. Eng. Data 29

(1984) 293–296.[18] M.A. Villamanan, A.J. Allawi, H.C. Van Ness, Int. Data Ser. A (1984)

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Data Ser. I (2e) (1988) 72.[24] R.A. Wilsak, S.W. Campbell, G. Thodos, Fluid Phase Equilib. 28 (1986)

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