Fluid Phase Equilibria, 3 (1979) 255-271 255

@ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

NEW MIXING RULES IN SIMPLE EQUATIONS OF STATE FOR REPRESENTING VAPOUR-LIQU D E_QUILjIlRIA OF STRONGLY NON-IDEAL MIXTURES * ,: c

i I

MARIE-JOSE HURON and JEAN VIDAL

Institut Franpzis du Pktrole, 1 et 4, Avenue de Bois-Pr&u, 92502 Rueil-Malmaison (France)

(Received December 20th, 1978; accepted in revised form May 18th 1979)

ABSTRACT

Huron, M.-J. and Vidal, J., 1979. New mixing rules in simple equations of state for repre- senting vapourliquid equilibria of strongly non-ideal mixtures. Fluid Phase Equilibria, 3: 255-271.

Good correlations of vapour-liquid equilibria can be achieved by applying the same two-parameter cubic equation of state to both phases. The results primarily depend on the method used for calculating parameters and, for mixtures, on the mixing rule. True param- eters are the covolume b and the energy parameter a/b. For this latter one, deviations from a linear weighting rule are closely connected to the excess free energy at infinite pressure. Thus any mixing rule gives a model for the excess free energy, or any accepted models for this property can be used as mixing rules.

From the above, an empirical polynomial mixing rule is used for data smoothing and evaluation, while for practical work a local composition model is used. The mixing rule thus obtained can be reduced to the classical quadratic rule for some easily predicted values of the interaction energies. For highly polar systems, it includes three adjustable parameters. Using literature data, the new mixing rule is applied, in the low and high pres- sure range, to binary mixtures with one or two polar compounds, giving good data correla- tion and sometimes avoiding false liquid-liquid immiscibility.

INTRODUCTION

When applied to both vapour and liquid phases, cubic equations of state can be used to calculate phase equilibria. Pure-component parameters are evaluated from critical conditions and vapour pressure data. To obtain the mixture parameters, classical mixing rules are used, and good results are ob- tained for non-polar gases (Soave, 1972; Peng et al., 1976; Robinson et al., 1977). The binary parameters which appear in the mixing rules can be pre- dicted by various combining laws (Hicks et al., 1975) or fitted to experi- mental data. In this way, the application range can be extended (Huron et al.,

l Part of this article was presented at the Chisa Congressin Prague, August 1978.

256

1977; Asselineau et al., 1978a; Graboski et al., 1978) to slightly polar mix- tures.

However, classical mixing rules imply a quasi-regular behaviour (Vidal, 1978) and, unless a thorough revision is made, the method cannot be applied to polar mixtures. It has been proposed that the value of the pure-component parameters be modified (Wenzel et al., 1971; Deiters et al., 1976; Wenzel et al., 1978). However, the close relation existing between excess Gibbs energy and mixing rules led us to the problem of evaluating mixture parameters, and this paper is devoted to mixing rules.

EXCESS GIBBS ENERGY FROM THE EQUATION OF STATE

As pointed out by Smith and Van Ness (1975), there are three fugacity coefficients, one for a pure component, cp;, one for the solution, cp, and one for a component in solution, Cpi. They are related to Gibbs energy and chemi- cal potential by their definition equations:

RTln&=+f-gi*=! (~r-y)dP 0

RTln~=g-g=~p(v-~)dP 0

RTlnIpi=~i-&=pi-pj=J (ci-y)dP

0

The Gibbs energy of the pure components and of the mixture are related by:

n

g = iz Xi&j* + RT In xi) (4)

in the ideal gas state, and by:

n

g = 2 Xi&r + RT In Xi) + gE (5)

in the real mixture. By substituting eqns. (l), (2) and (4) into eqn. (5), we obtain the general

equation relating excess Gibbs energy to fugacity coefficients.

gE=RT[lnp-&xilnlpfl i=l

(6)

The expression for the fugacity coefficients depends on the equation of

267

state that is used and is the same for cpr and cp:

~pf = F(T, P, u;, aii, bii, e-e)

cp = F(T, P, u, a, b)

So the various expressions we shall deduce from eqn. (6) will depend on the equation of state, but not on the mixing rules.

We shall first apply eqn. (6) to the modified Redlich-Kwong equations of state :

(u - b) = RT

in which parameters (I and b are solely composition and temperature depen- dent. The fugacity coefficient of a pure component or of a mixture is ex- pressed as:

and from eqn. (6) we obtain:

(8)

(9)

RELATION BETWEEN EXCESS GIBBS ENERGY AND MIXING RULES

The relation between the excess Gibbs energy gE and the mixing rules implicit in the a and b values is not obvious. At this step, we consider the limiting value of gE at infinite pressure. If at infinite pressure the excess volume is not zero, the limit of gE is infinite. Thus, as a necessary condition to obtain a finite limit, we apply the common linear mixing rule for volume parameter b :

b = 2 biixi i=l

From eqns. (7), (9) and (lo), it can easily be shown (see Appendix 1) that the equation for g!Z is:

In 2 (11)

We see that the gz value and the excess value of the energy parameter a/b

258

. So the value of the excess Gibbs energy gz will correspond mixing rule. As an example (Vidal, 1978) we give the classical

quadratic mixing rule:

are proportional to the choice of

a = 2 2 QijXiXj i=l j=l

(12)

associated with the combining rule:

CZij = (a$Zjj)1'2(1- hij) (13)

and with the linear law (10) for parameter b leads to a quasi-regular mixture. For a binary system, for example, we obtain:

& = (In 2) b&@st(% - 6~)~ + 2 b&,6,1 (14) with

Xibii 4i =b and

aiilbii ai= ~ v--- bit EIowever, eqn. (11) can be used to produce a mixing rule for parameter a

1

(15)

(17)

from the chosen excess Gibbs energy at infinite pressure. We do this in this paper.

When dealing with equilibrium problems, the fugacity coefficients for the components in solution must be evaluated. By applying classical thermo- dynamics and mixing rules (10) and (17), we obtain:

Inv,=-_ln(P(ii*)) +$(%-I) --(k-s) In(F) (18)

where the activity coefficients at infinite pressure are calculated by:

In yi_ = & [

n a& g +,zz(Aij--Xj)

j I (19)

The pure-component parameters will be determined from stability conditions and vapour pressure values, as explained in Appendix 2.

NEW MIXING RULES

In vapour-liquid-liquid equilibrium investigations, we meet two kinds of problems: (1) data smoothing, data and model evaluation; and (2) in the common practice of the chemical engineer, equilibrium predictions from a minimum amount of data. We now consider briefly the first of these.

259

Polynomial mixing rule

For data smoothing and evaluation, we look for great flexibility in the mixing rule. In the low pressure range, a Redlich-Kister polynomial expansion of the excess Gibbs energy is often applied to binary systems:

gE = RTxlxZ mfO A,(xl - ~2)~ (20)

Applying the same equation to the excess Gibbs energy at infinite pressure, from eqns. (10) and (17) we obtain:

a = (z,b,, + x,b,,) x1 p +x2 u22 RT

-__ b22 In 2

x1x2 5 A,n(x, -x2Y (21) 11 m=O 1

Such a mixing rule is applied, as an example, to the carbon dioxide(l)- ethane(2) system (Ohgaki et al., 1977) at 283.15 K. The pressure range is 20-50 atm. The polynomial parameters, A,,,, will be determined by mini- mizing the objective function:

where P is the equilibrium pressure calculated for the experimental values of T and X, and P,, is the experimental pressure. According to Peneloux et al. (1975), the weighting factor up is calculated by:_,

2_ UP - (uP,ex)2 + [(E), %J2 + [(E), UTJ2 : (22) where UP.,,, ux,ex, uT,ex arethe experimental uncertainties associated with pressure, composition and temperature measurements, respectively. :

From Ohgaki et al. (1977), we took: .i ;-

up = 0.01 atm, ur = 0.01 K, u, = 0.002

The data correlation with the Redlich-Kister equation for gE improves when the number of parameters is increased to a four parameter expansion. A slight underestimation of the experimental uncertainties u~,~_, ur ,ex, uT,ex appears in the results:

but the pressure deviations:

260

A P-Pexp (atm.)

45-

.

0 l *= .- . m '. . . .

-95

t

-x1

4 P-P,,(atm.) A

0.5 -

Fig. 1. Correlation of the carbon dioxide(l)-ethane(2) system. l , proposed polynomial mixing rule; X , classical mixing rule.

are small and random, and because of the great flexibility of the model they can be considered as an evaluation of experimental data scattering. For ex- ample, after fitting to the same data (Ohgaki et al., 1977) the interaction parameter kij involved in classical mixing and combining rules (12) and (13), the results obtained:

ZP = 0.48 atm

have to be considered with reference to the limit previously obtained (0.1 atm). Furthermore, as shown by Fig. 1, the deviations in pressure are not random, because of the wrong skewness of the excess Gibbs energy associated with the classical mixing rule.

Local-composition mixing rule

In common practice, a chemical engineer is concerned with models for predicting vapour-liquid equilibria from a minimum amount of data, with a minimum set of parameters. For this purpose, classical mixing and combining rules (12) and (13) are advantageous, since only one interaction parameter klZ has to be known to obtain good correlation for many binary mixtures. Thus

261

we look for a new mixing rule which, when only non-polar compounds are present, can be reduced to the classical one by a straightforward choice of parameters, but which can provide more flexibility if necessary. The local- composition concept (Wilson, 1964) has led to many powerful models of the liquid phase (Vetere, 1977). The Scatchard-Hildebrand theory can be estab- lished, among other hypotheses, by considering only the interactions between first-neighbour molecules (interactions between more than two molecules and influence of long-range molecules are neglected), and by assuming that the first-neighbour interaction occurrences are proportional to the overall volume fractions (Moelwyn-Hughes, 1961). Therefore we deduce local compositions tji and Eii, which give the distribution of molecules j and i about the central molecule i, from overall volume fractions $j and @i :

tji _ @j exP(--cYjigji/RV

Eji @i exP(-ajigii/RV (23)

The volume fractions introduced here, and already defined by eqn. (15), must be evaluated at infinite pressure since they will be applied to the g! formulation. They are weighted by Boltzman factors, including interaction energies between unlike (gji) and like (gii) molecules and, as in the NRTL model (Renon, 1968), a non-randomness parameter eji. The value of the ex- cess Gibbs energy at infinite pressure is then calculated in exactly the same way as in the NRTL equation, and the resulting equation is the same:

?J XjGjiCji g2 = gxij=l

i=l

2 XhGhi k=l

with

(24)

Cji = gji - gii (25)

and

Gji = bj exp(-aji&) (26) The only difference from the classical NRTL model is the definition of the

local composition as corrected volume fractions, which leads to the introduc- tion of the volume parameter bj in the calculation of Gji. From the choice of such a model for the excess Gibbs energy at infinite pressure, the mixing rule is deduced by applying eqn. (17):

n

==b xx1 s-1 i=l

1

bii In 2

C Xj GjiCji j=l

~ XkGki k=l

(27)

262

An exact reduction to the classical mixing and combining rules (12) and (13) can easily be obtained by the following calculation of the coefficients:

Lyij = 0

gii = -(In 2)-i ii

(29)

where kij is the interaction parameter involved in the combining rule (13). Owing to the zero value of the non-randomness parameter, local composi- tions are equal to volume fractions and the excess Gibbs energy at infinite pressure is quasi-regular. The iriteractions energies can then be calculated from the parameters of the equation of state. So the new mixing rule (27) con- tains, as an optional possibility, the classical rtfle. When applying the mixing rule to vapour-liquid equilibria calculations, we shall consider three possibil- ities. In the first, the parameter 01 is zero and we apply eqns. (24) to (29). It is the classical mixing rule, and the interaction parameter will be adjusted to experimental data. In the second, eqns. (24) to (29) will still be applied, but both kij and aij will be determined from experimental data. We found by experience that data correlation can be greatly improved in this way. Finally, we can apply eqn. (27), in which the three adjustable parameters will be Cij, Cji and ii for a binary mixture.

As in the classical NRTL model, the skewness of the excess Gibbs energy versus composition curve mainly depends on the C, values, and flatness is increased by high values of OL, thus providing great flexibility in the equilibri- um correlation and preventing false liquid-phase splitting.

APPLICATION TO VAPOUR-LIQUID EQUILIBRIUM CORRELATION

We applied the local-composition mixing rule (27) in the low and high pres- sure ranges. A phase equilibrium algorithm, constructed to avoid the trivial solution occurring near the critical point (Asselineau et al., 1978b), was used. Parameters were adjusted by minimizing the function:

NT

SQ F 5 (P-P,,); + c (yr -y;,,,)& =SQP+SQY N=l N=l

(39)

where P and yi are the calculated pressure and vapourphase composition. By minimizing the function S&P instead of the function SQ, nearly the same results would be obtained, since SQY is negligible compared to SQP in most cases. Table 1 gives the results. From the mean deviations in pressure:

263

it can be seen that data correlation can be greatly improved by the local-com- position mixing rule, and adjusting only the interaction parameter and the non-randomness parameter (second option in the mixing rule) gives quite good results for most systems. Furthermore, it must be emphasized that the classical mixing rule has a strong tendency towards false phase-splitting predic- tions. For example, as shown by Fig. 2, the behaviour of an acetone-water

P&m

1 I 1 I 0 425 0.50 Q75 Xl,Yl

J

P/atm

I I I I 0 0,25 950 0*75 WY1

J

I 1 1 1 0 425 w w5 Xl,Yl

I

P/atm

t = 25oT

Fig. 2. Correlation of acetone( 1 )-water( 2) vapour-liquid equilibria. - - - , classical mixing rule with a fitted parameter kl2 (klz values are given in Table 1); - proposed local mixing rule with three fitted parameters (see Table 1); 0, experimental boints (Griswold, 1952).

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266

system cannot be correlated by the quadratic mixing rule since a large miscibil- ity gap would be predicted. On the contrary, the local-composition model can be applied with good results, up to the critical range.

DISCUSSION AND CONCLUSION

Calculating vapour-liquid equilibria by applying the same equation of state to both phases was discussed by Van der Waals. When using such a meth- od, choosing the equation of state, the choice of the pure-component param- eters and the choice of the mixing rules are decisive steps. The more constants the equation of state has, the more mixing rules are required, and the more data are required for evaluating the pure-component parameters. Thus two- parameter cubic equations of state are often preferred. Proposed by Berthelot, adopted by Van der Waals and now extensively used, the quadratic mixing rule, as applied to the attraction parameter, is strongly supported by the : results obtained in phase equilibria predictions for non-polar or slightly polar systems. Furthermore, the existing relation between a, b and the second virial coefficient:

B = 2 5 BijXiXj = b - (a/RT) i=l j=l

and the commonly accepted quadratic law for B imply a quadratic law for a. When applying the mixing rule (27) for parameter a, the quadratic law for the second virial coefficient B will not be obeyed. However, in the high density range and for polar mixtures, the experimental behaviour of excess Gibbs energy does not obey the law implied by the classical mixing rule. Our meth- od for deducing a mixing rule from a model of excess Gibbs energy is highly empirical, but is consistent with the vapourliquid equilibrium calculations.

The Soave equation of state is used here with new mixing rules for param- eter a. The close connection found between the energy parameter (a/b) and the excess Gibbs energy g: can be extended to other cubic equations of state having the form:

p= RT a(T) - - u - b -9(u) (31)

If the roots of $J(u) are proportional to parameter b, that is if G(u) can be written:

G(v) = (u + bh,)(u + &AZ) (32)

in which X1 and As are numerical constants, then the equation for g? is:

gz=- FyLgxi%a]^ [ i=l iI (33)

267

with

1 Iz = AZ - A1 In if h2 # hi

Eqn. (33) is valid for the following two-parameter equations of state:

Harmens (1978) A = -&ln(fs));

Peng and Robinson (1976) A =

Redlich-Kwong and Soave (A = In 2); and Van der Waals (A = 1).

But eqn. (33) does not apply to the three-parameter equations of state of Clausius, Fuller (1976), Horvath and Lin (1977) and Usdin and McAuliffe (1976), for which the roots of Q(u) are not proportional to b. For these last equations, we need to set mixing rules for two parameters in order to be able to calculate the third one from gz .

Like the choice of a mixing rule, the choice of a model for the excess Gibbs energy is difficult, as shown by several attempts in the litterature. We used two expressions, depending on the problem.

It would be attractive to apply the polynomial mixing rule to a consistency test by comparing calculated and experimental vapour compositions. In the low pressure range, the influence of the polynomial mixing rule on the density of the vapour phase is slight, and we obtained the same results by using the standard method (Redlich-Kister excess Gibbs energy, reference fugacities in the liquid phase, virial equation of state for the vapour phase) or by applying the straightforward technique of the polynomial mixing rule (21). In a higher pressure range, however, such a method would be questionable since the calculated vapour phase compositions are sensitive to the vapour density evaluation (Won et al., 1973). On the whole, our results did not show any systematic discrepancy between experimental and calculated vapour phase composition, but we shall not use them as a consistency test and we shall reserve the polynomial mixing rule for evaluating data scattering.

Application of a local-composition (NRTL) model to the excess Gibbs energy at infinite pressure was first considered by Chaudron et al. (1973). However, the proposed method used reference fugacities and, together with the NRTL model, a mixing rule was used for the equation of state, which was applied to the calculation of the fugacities in the vapour phase, for the Poyinting correction and for the reference fugacities. Although thermo- dynamically more sound, this method requires more parameters and does

268

not use the same model for calculating vapour phase fugacities and liquid phase fugacities.

If we decide to apply the same model to both phases, the implicit relation between mixing rule and excess Gibbs energy has to be considered and can be used.

ACKNOWLEDGEMENTS

Thanks are extended to Professor Peneloux (University of Marseille) and to our colleague Dr. Asselineau for helpful discussions and suggestions.

LIST OF SYMBOLS

A a, b B C

g k

;P NT P R

Redlich-Kister coefficient parameters in the equation of state virial coefficient parameter in the mixing law Gibbs free energy or interaction energy parameter interaction parameter associated with a number of components in the mixture number of parameters number of experimental points pressure (atm) 1 atm = 1.01325 X 10 Pascals perfect gas constant = 82.0562 atm cm3 K-l mol- or 8.3143 J mole1 K-l function to be minimized temperature (K) molar volume ( cm3/mole) molar fraction non randomness parameter activity coefficient solubility parameter Kronecker delta deviation in pressure deviation in mole fraction numerical constants local composition weighting factor fugacity coefficient volume fraction volume function

Superscripts or subscripts

C critical property E excess value

269

ex experimental value i, j component identification N experimental point

ideal state * pure component 00 at infinite pressure - an overbar denotes a partial molar property

APPENDIX 1. LIMIT OF EXCESS GIBBS ENERGY AT INFINITE PRESSURE

From eqn. (9), we write the excess Gibbs energy as

gE=$+&+g3E

where

when P + 00, u + b, vr + bs and from the equation of state (7) we see that: I P(v-Wcl_ 4v-b) + 1

RT RTu(u + b)

The second term g! can be written

& _P(u-bb) + -- RT RT RT

or

SZ _P(V-bb)_~x_P(uf-bi,)+ Pb -- RT RT

I &,Pb,

i=l RT RT i=l RT

and, from the value of the limit of [P(u - b)]/RT, and [P(uf - bii)]/RT, we see that:

412 E+O if b = 2 xibii i=l

270

So gg has the same limit as gt , the value of which is given by eqn. (11). For the other equations of state mentioned in discussion, the limit is estab-

lished in the same way.

APPENDIX 2. DETERMINATION OF PURE-COMPONENT PARAMETERS

In the equation of state (7), parameter bii was considered as independent of temperature, and we applied Soaves correlation (Soave, 1972) to param- eter aii :

Uii= Cr,i[l + mi(l -m)]

The values of U,+ and bCi are determined from critical conditions:

1 =ci = 9(21/a - 1)

R2Zi bi* = bci =

213~1%

pci 3 pci

Soave correlated value mi with an acentric factor. We preferred to apply the vapour-liquid equilibrium condition to the pure components, and to evaluate parameter Uii, for a given subcritical pure component i, from its vapour pressure. The values of aii and mi were determined from one point on the vapour pressure curve. In the ensuing correlation of mixture data, to avoid any discrepancy which can be attributed to the pure-component param- eters, the vapour-pressure point was chosen in the same temperature range as the mixture data.

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