Pergamon Chemical Envineering Science, Vol. 5L No, 20, pp. 3579--3588, 1997 1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain P I I : S0009-2509(97)00143-7 0009-2509/97 $17.00 + 0.00

Critical points with the Wong-Sandler mixing rule II. Calculations with a

modified Peng-Robinson equation of state

Marcelo Castier *t and Stanley I. Sandier ~,~ * Escola de Quimica, Universidade Federal do Rio de Janeiro, Rio de Janeiro-RJ-21949-900, Brazil; ~Center for Molecular and Engineering Thermodynamics, Department of Chemical

Engineering, University of Delaware, Newark, DE 19716-3119, U.S.A.

(Received 18 November 1996; in revised form 26 March 1997; accepted 11 April 1997)

Abstract--This is the second of two papers in which critical point calculations in binary systems were performed utilizing cubic equations of state (EOS) combined with excess energy models using the Wong-Sandler mixing rule. In the first paper, a qualitative study of critical phase diagrams calculated using the simple van der Waals EOS combined with the NRTL model was made. In this paper, the Stryjek and Vera version of the Peng-Robinson EOS was also combined with the NRTL model and the resulting model was used in the computation of the critical loci of real systems. The binary interaction parameters of the model were estimated by correlating vapor-liquid equilibrium (VLE) data and, for some systems, successful predictions of the critical loci were obtained even when VLE data far from the critical point were used. To estimate parameters in systems for which the equation of state model may incorrectly predict a false liquid-liquid split, we used a penalty function approach based on the results of global stability tests. While the model studied here has been able to quantitatively predict the critical behavior of some non-ideal systems, involving compounds such as water, acetone and alkanols, only qualitatively correct behavior could be predicted for some highly asymmetric and non- ideal mixtures, such as water + n-dodecane. 1997 Elsevier Science Ltd

Keywords: Critical phenomena; phase diagrams; mixing rules; equations of state; local com- position; excess free energy.

INTRODUCTION

Although critical point calculations utilizing cubic equations of state (EOS) are routinely needed in several applications, few papers have investigated the effect on such calculations of mixing rules that com- bine cubic EOS with excess free-energy models. Therefore, as pointed out by Sadus (1994), despite the large interest they have caused in the literature, these mixing rules have not yet been extensively tested for the prediction of critical points, and little seems to be known about their ability in modeling this type of phenomena. In one of the few papers that deal with this subject, Kolfir and Kojima (1996) predicted and correlated several types critical phase diagrams using the PSRK EOS (Holderbaum and Gmehling, 1991), which combines the Mathias and Copeman (1983)

tOn leave at Department of Chemical Engineering, Uni- versity of Delaware, Newark, DE, U.S.A.

~Corresponding author.

version of the SRK EOS and the UNIFAC model (Fredenslund et al., 1977) using the MHV1 mixing rule (Michelsen, 1990). In another study, Knudsen et al. (1994) modeled complex systems that exhibit critical type VI behavior using the MHV2 mixing-rule and a quadratic mole fraction dependence for the b-parameter to combine the Soave-Redlich-Kwong EOS (Soave, 1972) with the UNIQUAC and UNIFAC models (using temperature-dependent parameters). However, there was no attempt to deter- mine critical states for these systems.

This is the second of two papers whose objective was to study the effect of one of these mixing rules, the Wong-Sandler (1992) mixing rule, on critical point calculations. In the first paper (Castier and Sandler, 1997), the van der Waals EOS and the NRTL model were used in a systematic study of the types of critical phase diagrams that are generated within certain ranges of the model parameters. Since there we were more interested in the qualitative aspects of the phase diagrams, no attempt was made to compare the results with experimental data.

3579

3580 M. Castier and

In this paper, the objective is to compare the predic- tions of this EOS model with experimental critical data of binary mixtures. Since only a very crude representation of pure component properties can be expected from the van der Waals EOS, an EOS suit- able for engineering calculations, the Stryjek and Vera (1986), PRSV EOS, version of the Peng-Robinson (1976) EOS, was combined with the modified form (Huron and Vidal, 1979) of the NRTL model (Renon and Prausnitz, 1968) and used in our calculations. In order to comprehensively search for the critical states of mixtures exhibiting complex behavior, including the possibility of multiple critical points at any speci- fied composition, the Hicks and Young (1977) algo- rithm was used. The model parameters were estimated from vapor-liquid equilibrium (VLE) data, only some of which at high pressure, but no experimental critical information was directly used in the fitting procedure, except to guide the choice of parameters in systems that could be equally well represented by several sets of parameters. Apart from that, the calculations represent predictions rather than correlations of the critical curves.

The critical points were calculated as described earlier (Castier and Sandler, 1997) using a modified version of the Hicks and Young (1977) algorithm to determine possible multiple critical points at any given composition and refining the solutions with the Heidemann and Khalil (1980) procedure. Only critical points that passed the local and global stability tests are reported here. However, it should be mentioned that the global stability test was only implemented to "verify the possibility of additional fluid phases; the possibility of forming solid phases was not tested in our implementation.

THERMODYNAMIC MODEL

The Peng-Robinson EOS (PR EOS) (1976) is

RT a P

v - b v 2 + 2bv - b 2

with the a and b parameters given by

0.477235R 2 T~ all - - - - ~PR, i Pc,

0.077796RT~, bi = Pc,

with ~eR,, = [1 + x,(l - x/~R,)] 2

where the reduced temperature is

TR, = T/Tc ,

In the PRSV EOS, Ki is given by

xi = ~o, + xl,(1 + x/~R,)(0.7 - T~,)

where

Xo, = 0.378893 + 1.4897153oi - 0.1713848o~

+ 0.019655409~

(1)

S. I. Sandier

and xl, is a characteristic parameter of each compon- ent. Values of the critical properties, acentric factor and characteristic parameter ~q, were taken from Stryjek and Vera (1986).

In the Wong-Sandler mixing rule, the molar excess Helmholtz free-energy calculated from the EOS at infinite pressure (a~ Es) is equated to the same prop- erty computed from an excess free-energy model (a~) such as NRTL, i.e.:

a E, Eos E = a~. (8)

Wong and Sandler showed that the a and b para- meters are then given by

RTQwsDws a = R TD~sb (9)

1 - Dws

Qw~ b - - - (10) 1 - Dw~

where

a~ (11) Ows=~+ ~ xia. i= 1 RTb i

Qws = x ix j b - ~=1 j=l ij" (12)

The parameter c depends upon the equation of state and, for the PRSV EOS, it is given by

c =--~-

The following combining rule has been used:

b - ~ ij 2 (1 - kij). (14)

Expressions for the fugacity coefficient and its deriva- tives with respect to composition at constant temper- ature and total volume required in the critical point calculations were determined using Mathemat ica (Wolfram, 1991) and can be obtained from the authors on request.

(2) PARAMETER EST IMATION

The binary interaction parameters of the model (3) employed in the critical point calculations were either

obtained from the literature (Orbey and Sandier, 1995) or determined using a modified version of the

(4) parameter-fitting program originally developed by them. The following objective function was used:

nt,

(5) f - - ~ fP~ - P~I + P (15) j= l

where P~ and Pf, respectively, denote the calculated (6) and measured values of pressure at experimental

point j, and p is a penalty parameter that is added to prevent the model from predicting liquid-liquid equi- librium when no experimental evidence supports its existence. Examples of systems in which this may

(7) occur include propane + methanol and hexane +

Critical points with the Wong-Sandler mixing rule--II

ethanol, as observed by Orbey and Sandier (1995) and Englezos et al. (t989), respectively.

The implementation of the criterion proposed by Englezos et al. (1989) tests for local stability at the experimental points. However, it is theoretically pre- ferable to test for global stability and for this reason we used the following expression for the penalty term:

nun,

p = - t/ ~" dk (16) k=t

where nuns is the number of globally unstable points with respect to the formation of an additional liquid phase and dk is the minimum nondimensional 'dis- tance' in the tangent plane criterion (Michelsen, 1982) at a point k that is unstable. This 'distance' is negative for unstable points and is given by

n,

d = min ~ w~(ln(w~qS~) - ln(xiq~/)) (17) i=1

where q~ denotes fugacity coefficients, the dot indicates properties at trial compositions, and the minimization is carried out having as variables the independent mole fractions of the trial phase. In the case of binary systems, as those studied in this paper, the calculation of this 'distance' reduces to a unidimensional minimiz- ation problem. In eq. (16), we also used a scaling factor t/ with an arbitrarily large numerical value (10 s bar) to penalize unstable points even if close to the stability limit.

In this paper, P-T -x or P-T -x -y data were used to estimate four model parameters in binary systems: the binary parameter k~2 in eq. (14) and the three para- meters in the modified NRTL model: ~, g~2/R and 021/R.

RESULTS

Critical loci were calculated for some highly non- ideal mixtures such as n-alkane + 1-alkanol systems, n-pentane + acetone, carbon dioxide + water, water + n-dodecane and methane + n-hexane. Table 1

presents the parameters used for the critical point calculations of each system studied here, though the specifics of the parameter estimation for each system are discussed separately. Interestingly, the estimated values of the ~ parameter range from 0.4 to 0.6 for most of the systems studied, but are as high as 0.9 in

3581

the methane-ethanol system as determined by Orbey and Sandier (1995).

Even though high-pressure VLE data were used in the parameter estimation for some of the systems presented in this paper, no critical data were directly used. However, we observed that for some of the systems, several sets of parameters yielded a similar correlation of the VLE data, but resulted in noticeably different behavior when used to predict the critical loci. For such systems, as mentioned in the text, the critical data provided guidance as to which parameter set to use; apart from that, the reported critical behav- ior represents predictions rather than correlations.

n-Alkane + 1-alkanol systems Binary systems containing an n-alkane and a 1-

alkanol can deviate significantly from ideal behavior and therefore represent a severe test of thermodyn- amic models. For the methane + ethanol mixture, Fig. 1 shows the P-T projection of the critical curve calculated with parameters found by Orbey and Sandler (1995), using isothermal VLE data measured by Brunner and Hfiltenschmidt (1990) at 298.15 K. An attempt to simultaneously also use data available at higher temperatures to estimate the parameters of the model led to similar critical point projections. The calculated branch of the critical line that starts at pure methane is extremely short, which is also found in the experimental data, whereas the branch that starts at pure ethanol proceeds towards high pressures as the temperature decreases, and then passes through a maximum. We could not find experimental data to test these results. It should be noted that the cal- culated critical temperature goes below the melting point of pure ethanol at atmospheric pressure, which occurs at 159.1 K, and there is the possibility of form- ing a solid phase. Should this happen, part of the calculated critical branch would be unstable. Our implementation of the global stability criterion did not test for the possible formation of solid phases.

The P-T projection for the system ethane + ethanol is presented in Fig. 2. As in the methane + ethanol system, the critical lines were calculated using parameters estimated by Orbey and Sandler (1995) from isothermal VLE data at 348.15 K measured by Brunner and Htiltenschmidt (1990), and an attempt to

Table 1. Parameters for the systems studied

Component 1 Component 2 ~ gl2/R (K) gzl/R (K) k12

Methane Ethanol 0.90 165.8 238.4 0.000 Ethane Ethanol 0.50 2263.0 435.2 0.101

n-Butane Ethanol 0.47 1132.0 305.8 0.152 n-Pentane Ethanol 0.60 805.8 240.8 0.252 Propane Methanol 0.39 2353.0 660.2 0.150 n-Hexane Methanol 0.55 1720.0 2952.0 0.411 n-Pentane Acetone 0.60 468.7 207.1 0.126

Carbon dioxide Water 0.33 382.8 1167.0 0.309 Water n-Dodecane 0.52 - 1161.0 1727.0 0.491

Methane n-Hexane 0.60 - 54.0 38.6 0.0045

3582 M. Castier and S. I. Sandier

900

800

700

600

vx~ 500

400 o.

30O

200

100

- - Calculated Brunner and

HOl tenschmidt (1 g90) 0 Methane

65 I Calculated I

6O

55

~ so

45

40

35 r ~ ~ ~ 0 - - 100 150 900 250 300 350 400 450 500 550 400 420 440 460 480 500 520

Temperature (K) Temperature (K)

Fig. 1. Critical curve of the system methane + ethanol: pres- Fig. 3. Critical curve of the system n-butane + ethanol: pres- sure-temperature projection, sure-temperature projection.

130 65

120

110

100

& 90

~ 80 a.

70

60

517

40

250

Calculated --7 J Brunner and [

H01tenschmidt (1990) 1

I I I I

300 350 400 450 500

Temperature (K)

550

Fig. 2. Critical curve of the system ethane + ethanol: pres- sure-temperature projection.

60

55

so

~ 45

40

35

30 450 520

- - Calculated ] M

q r ~ i i f

460 470 480 490 500 510

Temperature (~

Fig. 4. Critical curve of the system n-pentane + ethanol: pressure-temperature projection.

simultaneously use data at other temperatures in the parameter estimation procedure produced similar re- sults. Even though the maximum in critical pressure is overestimated, the model correctly predicts the dis- continuity in the vapor-liquid critical line with the occurrence of upper and lower critical endpoints. Brunner and Hiiltenschmidt (1990) experimentally de- termined that the upper and lower critical endpoints occur at 314.36 K, 55.37 bar and 308.72 K, 49.52 bar, respectively, whereas our calculations predicted these values to be 314.3 K, 63.9 bar and 314.0 K, 54.1 bar.

To represent the system n-butane + ethanol, we used all the experimental VLE data available at several temperatures in the work of De~k et al. (1995). The P-T projection of the vapor-liquid critical line is

shown in Fig. 3. In agreement with experimental data, the model predicts a continuous vapor-liquid critical line connecting the critical points of the two pure components. For the system n-pentane + ethanol, the parameters were estimated using experimental VLE data available at 422.6 K from Campbell et al. (1987). This system also exhibits a continuous critical line between the two pure components and our predic- tions in Fig. 4 show good agreement with the very limited experimental information available for this system (McCracken et al., 1960). Another set of para- meters determined using all the VLE data from Campbell et al., at 372.7, 397.7 and 422.6 K, leads to predictions of a similar, though less accurate repres- entation of the critical line.

Critical points with the Wong-Sandler mixing rule--II 3583

~g

o-

100 90 t " 80

70

60

50

40

30

340 360

- - Calculated Brunner (1985)

380 400 420 440 460 480 500 520 540

90

80

70

60 o~

50 O.

40

3o

I I I I I I 20 I I I I

470 480 490 500 510

Temperature (K) Temperature (10

- - Calculated [ de L oos et al. (1988) I o Z a w ~ ~

o

i

520

Fig. 5. Critical curve of the system propane + methanol: Fig. 6. Critical curve of the system n-hexane + methanol: pressure-temperature projection, pressure-temperature projection.

The system propane + methanol was particularly difficult to model. Orbey and Sandler (1995) observed that when VLE data (Galivel-Solastiouk et al., 1986) were used to fit the parameters, the model predicts false liquid-liquid phase splitting. Consequently, they used data at 313.1 K while imposing local stability as proposed by Englezos et al. (1989) to estimate para- meters. When used for critical point calculations, their parameter values predict the occurrence of critical endpoints which were not observed experimentally by Brunner (1985). We were able to improve the predic- tion of the critical line by estimating the model para- meters using experimental VLE data at all the temperatures (313.1, 343.1 and 373.1 K) that were available in the paper by Galivel-Solastiouk et al. The P-T projection of our calculated critical line and the experimental critical data measured by Brunner (1985) are shown in Fig. 5. A fairly good quantitative representation of the P-T projection of the critical line was obtained, even though the model predicts the existence of critical endpoints. Therefore, although our parameter estimation procedure prevented false liquid-liquid phase splitting at all data points for the three different temperatures, a liquid-liquid split oc- curred along the vapor-liquid critical line. This behavior suggests that this system provides a parti- cularly stringent test of the accuracy of equations of state, their mixing rules, and the parameter values used.

The system n-hexane + methanol was also difficult to model, even though qualitative agreement with the experimental data, in terms of a continuous critical line connecting the two critical points of the pure components, was achieved in our two attempts to represent this system. In the first attempt, parameters were estimated using data measured by Zawisza (1985) at 448.15 K, and in the second using data measured by de Loos et al. (1988) at several temper-

atures. The P-T projection of the vapor-liquid criti- cal line calculated using the first set of parameters (shown in Table 1) is presented in Fig. 6. The results with the second parameter were similar in behavior to the first set, but provided a less accurate representa- tion of the critical line. The n-hexane + methanol system also has a liquid-liquid critical line that goes to high pressures, as observed by H61scher et al. (1986). Although not shown in the figure, a liquid-liquid critical line for this system was found in our calculations, but its agreement with the avail- able experimental data was not satisfactory.

n-Pentane + acetone system Experimental VLE data available at three different

temperatures (372.7, 397.7 and 422.6 K) from Camp- bell et al. (1986) were used for parameter estimation. The critical data measured by Hajjar et al. (1986) shows that this system has a continuous critical line between the two pure components with a minimum in temperature, and this behavior is correctly predicted by the EOS model (Fig. 7). It should be mentioned, though, that the shape of this critical line is very sensitive to the EOS parameter values that are used. When we used other parameter sets that provided similar accuracy in the correlation of the relatively low pressure VLE data of Campbell et al. (1986), we found that either pressure minima or maxima could be predicted in the critical line, but they are not observed in the experimental data,

Carbon dioxide + water system The VLE data measured by Toedheide and Franck

(1963), as reported by Knapp et al. (1982), were used for parameter estimation. These data go to very high pressure, up to 3500 bar, which we could not fit to the EOS model. For this reason, data points above 1000 bar were not included in the parameter estimation

3584 M. Castier and S. I. Sandier

48

46

44

,~, 42

~ 40

~ 38

36

34

32 460 470 480 490 500 510

Temperature (K)

4000

3500

3000 -

~, 2500

v

2000

1500

1000

500 I

250

(a)

Toedheide and Franck (1963) Takenou~i and Kennedy (1964) Calculated

- - Calculated

300 350 400 450 500 550 600 650

Temperature (K)

700

Fig. 7. Critical curve of the system n-pentane + acetone: pressure-temperature projection.

900

800

procedure, as well as some of the data points between 700 500 and 1000 bar. The P-T and T-x projections of the critical loci are shown in Figs 8(a) and (b). In the ~ 600 P-T projection, as the critical pressure increases, the critical temperature first decreases and then starts to E g 500 increase, a pattern that is also present in the data of Toedheide and Franck (1963). However, the T-x pro- 400 jection shows that the EOS predicts multiple critical points in the 0.22 ~< Xco2

Critical points with the Wong-Sandler mixing rule--II 3585

350 -

300

250

200

150 o.

100

50

1 Brunner (1990) [

Calculated J

~ o ~

q F T ~ T F

250

200

150

to

~. loo

50

0 0

560 580 600 520 640 660 680 700 150 Temperature (K) (a)

i L2=G, Lin et al. (1977 ~" El=L2, Lin et al. (1977) \ LI=G, Lin et al. (1977) \ Hicks and Young (1975) \ Methane n-Hexane - - Calculated

I I I I I

200 250 300 350 400 450 500

Temperature (K) 550

Fig. 9. Critical curve of the system water + n-dodecane: pressure-temperature projection.

250

200

pure methane to an upper critical endpoint. In the T-x and P-x projections, the data of L inet al. (1977) show three branches in the critical line, two of which ~ v 150 combine in the P-T plane into a continuous projec- tion. Figures 10(a) and (b) show calculated projections 100 of the critical loci, the experimental critical data of Lin et al. (1977) and data from the compilation of Hicks and Young (1975). The P-T projection shows satisfac- 50 tory agreement between the calculated and experi- mental values, but the P-x projection shows that the model does not predict the liquid-liquid critical line in

0 the 0.8900 ~< Xc,, ~< 0.9286 composition range re- 0.0 ported by Linet al. Instead, the calculated critical line

(b) that starts at pure n-hexane undergoes a continuous transition from a gas-liquid critical line to a liquid-liquid-like critical line in the vicinity of pure lo0 - methane. Figure 10(c), which provides a magnifica- tion of the P-T projection close to pure methane, 90 - shows that the calculated critical branch that orig- inates from pure n-hexane agrees fairly well with the 80- experimental critical points despite a systematic devi- ~, ation. The predicted coordinates of the lower critical endpoint are XcH, = 0.9337, 182.50 K and 34.85 bar ~ 70 and the experimental values are Xcn, = 0.9286, .~ a. 60 182.46 K and 34.15 bar. Larger deviation was found at the upper critical endpoint where the critical 50 branch that starts at pure methane ends: the cal- culated coordinates are XcH, = 0.9987, 192.70 K and 40 48.32 bar and the experimental values are XCH, = 0.9976, 195.91 K and 52.06 bar.

30 Similarly to some of the other systems that we have 180

studied here, some of the fine details of the critical phase diagrams could not be predicted by the model. (c) In this case, it is the isolated liquid-liquid critical line that occurs over a narrow composition range. Inter- estingly, however, the overall representation of the system is fair, and even quantitative agreement is

L2=G, Linetal. (1977) I L1=L2, Lin at a/. (1977) I

LI=G, Lin eta/. (1977) I Hicks and Young (1975)

Ca culated

~7

0.2 0.4 0.6 0.8

Mole fraction of methane 1.0

1 / L2=G, Lin eta/. (1977) I J

L1 =L2, Lin et al. (1977) LI=G, Lin eta/., 1977 Methane - - Calculated

~7 ~7

i I I i i

185 190 195 200 205

Temperature (K) 210 215

Fig. 10. Critical curve of the system methane + n-hexane: (a) pressure-temperature projection; (b) pressure-mole frac- tion projection; (c) magnification of the pressure-temper-

ature projection close to pure methane.

3586 M. Castier and S. I. Sandier

achieved in portions the critical phase diagrams, al- though we have only used VLE data for parameter a estimation. It would be interesting to study whether a better representation could be obtained from this b model with a more complex parameter estimation c that would use not only VLE data but also LLE and VLLE data.

CONCLUSIONS

This and the preceding paper had as their objective a study of the critical loci of binary mixtures com- puted using cubic equations of state and excess free- energy model-based mixing rules. In this paper, the Stryjek and Vera version of the Peng-Robinson EOS and the modified NRTL model (Huron and Vidal, 1979) were combined using the Won~Sandler mixing rule. The model parameters were fitted using experi- mental vapor-liquid equilibrium data, but no critical point data were directly employed, except in some instances to provide guidance as to which parameter set to use for some systems whose VLE data could be equally well represented by several sets of parameters. The model was then used to predict critical points of some highly non-ideal systems and for most of them, the shape of the critical lines was correctly predicted. Notable exceptions were the propane + methanol system for which a discontinuous vapo~liquid critical line was predicted while the experimental critical data indicate that this system exhibits a continuous critical line, and the methane + n-hexane and n-hexane + methanol systems for which isolated liquid-liquid critical lines observed experimentally were not pre- dicted to occur. For the carbon dioxide + water sys- tem, the critical curve was well represented up to 750 bar, but was not quantitatively correct at higher pressures. While the model studied here has been able to quantitatively predict the critical behavior of some highly non-ideal systems involving compounds such as water, acetone and alkanols, for some highly asym- metric and non-ideal mixtures, such as water + n- dodecane, only qualitatively correct critical behavior could be predicted.

Since it is well known that cubic equations of state are not accurate in the critical region for pure fluids, perhaps the most surprising result of this study is that the predictions for complex critical phenomena in mixtures with a simple equation of state and the Wong-Sandler mixing rule are as good as they are. However, a word of caution. We have also found that the predictions are quite sensitive to the values of the parameters used.

Dw~

f Yij ki) rl c np nuns

P

Qws

R T /)

xi wi

Acknowledgments The preparation of this manuscript was supported, in part,

by Grant No. DE-FG02-85ER13436 from the U.S. Depart- ment of Energy, and Grant No. CTS-9123434 from the U.S. National Science Foundation, both to the University of Delaware. M. C. acknowledges the financial support of the Brazilian Ministry of Education (CAPES) and of the Brazilian Conselho Nacional de Desenvolvimento Cientifico e Tecnol6gico (CNPq)

NOTATION molar Helmholtz free-energy or EOS attractive parameter EOS covolume parameter characteristic parameter of the equation of state (2 -0.62323 for the PR and PRSV EOS) minimum 'distance' in the global stability test auxiliary parameter of the Wong-Sandler mixing rule objective function for parameter estimation interaction parameter in the NRTL model binary interaction parameter number of components number of experimental data points number of globally unstable points during parameter estimation penalty term in the objective function for parameter estimation pressure auxiliary parameter of the Wong-Sandler mixing rule universal gas constant temperature molar volume mole fraction of component i mole fraction of component i

Greek letters c~ NRTL nonrandomness parameter ~w auxiliary term in the Peng-Robinson equa-

tion of state q scaling factor

parameter of the PR and PRSV equations Ko, xl parameters of PRSV equation qS~ fugacity coefficient of component i o~ acentric factor

Subscripts c critical property i, j refer respectively to components i and j R reduced property

property at infinite pressure

Superscripts c calculated e experimental E excess property EOS equation of state

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Critical points with the Wong-Sandler mixing rule--II

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