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An equation of state mixing rule for correlating ternary liquid-liquid equilibria
ELSEVIER
Fluid Phase Equilibria, 98 (1994) 91-111
An Equation of State Mixing Rule for Correlating Ternary LiquidLiquid Equilibria Cheng Huang Choua and David Shan Hill Wongb a&ion Chemical Laboratories, Industrial Technology Research Institute, 321 Kuang Fu Rd. Sec. 2, Hsinchu, Taiwan 30042 bDepartment of Chemical Engineering, National Tsing Hua University, 101 Kuang Fu Rd., Sec. 2, Hsinchu, Taiwan 30043
Keywords: theory, equation of state, mixing rules, liquid-liquid equilibria (Received July 16,1993; accepted in final form April 20, 1994)
ABSTRACT A new mixing model for cubic equations of states that is theoretically correct in the composition dependence of the second and third virial coefficients is derived. This new mixing rule has been tested in conventional vapor-liquid equilibria calculations, It was found to be as good as activity coefficient models and other equation of state mixing rules for solutions of various kinds of nonideality. It can also be extrapolated over large temperature and pressure ranges. Its salient feature is an extra degree of freedom that accounts for ternary interaction, which enables it to give consistent correlation of vapor-liquid and liquid-liquid equilibria of ternary systems and the corresponding binaries.
INTRODUCTION Applications of cubic equations of state in phase equilibrium calculations have, for many years, been restricted to near ideal mixtures because of the limitations of the one fluid van der Waals mixing rule used. For more complex mixtures,’ activity coefficient models (or “liquid solution” models) are commonly used. However, unlike equations of states, which are general thermodynamic models for both liquid and vapor phases and all residual thermodynamic state functions, activity coefficient models are correlations of excess free energy specifically for liquid phases only. There have been ongoing efforts to bridge these two approaches.
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92
C.H. Chou, D.S.H. Wang/Fluid Phase Equilibria 98 (1994) 91-111
,Hqon and Vidal(1979) first pointed out the relation between mixing rules for equation of state parameters and the excess Gibbs fi-ee energy at infinite pressure. Their approach necessitated the assumption that the excess volume of the system at infinite pressure be zero in order that the excess Gibbs free energy be finite. Sheng et al. 1992 pointed out that using excess Hehnholtz free energy would be more appropriate and ‘an extra degree of freedom in the choice of mixing rules is thereby released. Most mixing rules that address the problem of nonideality in liquid (or high density) mixtures fail to reduce the proper statistical mechanical limit at low density that the composition dependence of the second virial coefficient be quadratic. It is this constraint that confined the thinking of most researchers within the van der Waals one fluid paradigm. Wong and Sandler (1992) pointed out that the van der Waals one fluid mixing rule is only a specific case of a more general class of mixing rules that satisfy the above limitiig condition. The degree of freedom opened up by Sheng et al. (1992) could be used to satis@ this limit. Wong et al. (1992) also showed that how knowledge experiences accumulated in correlating vapor-liquid equilibrium (VLE) at low pressures and temperatures with liquid solution models can be preserved during the changeover to equation of state models, and extrapolated into high temperature and pressure. Despite its success in correlating VLE of non ideal binary systems, the mixing rule proposed by Wong and Sandler (1992), was not very successll in predicting liquid-liquid .e,quilibria (LLE) of ternary systems using data of binary system only. There have been only scattered attempts to apply equations of state to liquid-liquid equilibria calculations (Peng and Robinson, 1976, Mathias and Copeman, 1983, Leet et al., 1986 4 Luedecke and Prausnitz, 1985, Huang, 1991). The most extensive of ,these studies was reported by Huang (1991), who used the approach of Huron and Vidal (19792 to incorporate liquid solution models such as NRTL (Renon and Prausnitz, 1968) and UNIQUAC (Abrams and Prausnitz, 1975) to the equation of state proposed by Pate1 and Teja (1982), to correlate LLE data of ternary systems. It was found that difficulties in data correlation arise mostly with systems that have plait points. Accurate correlation of phase behavior of both the ternary system and its constituent binaries was found possible only when data of the ternary systems were incorporated in fitting procedure to determine the binary interaction parameters. The same problem existed with the direct application of local composition liquid $olution model. Anderson and Prausnitz (1978) observed that the binary interaction parameters of the UNIQUAC mddel can not be uniquely determined due to experimental error or lack of sufficient data. Only a highly correlated set of optimal values can be found. hiclusion of ternary tieline data could would substantially improve the accuracy of the cdrrelation’for the ternary system, but usually at the expense of the accuracy of binary correlations. Cha and Prausnitz (1986) suggested an entirely empirical correction for
C.H. Chow D.S.H. Wang /Fluid Phase Equilibria 98 (1994) 91-111
93
the UNIQUAC model that includes a ternary parameter to eliminate sensitivity of the model to binary parameters. The empiricism, however, seriously limited the appeal of this approach. In using the NRTL model, it is common practice that various values of a,, are used to regress the VLE and LLE data of the partially miscible pair. The set that best described the ternary LLE data is then used. This is based the assumption that there is not enough information in binary data to get a “true” set of NRTL parameters. Ternary LLE data are used as supplementary information to determine a parameter that is binary in nature. It remains to be shown that the same value of a,, could be used for any ternary system involving the specific binary pair (Renoir, 1994). While the mixing rule proposed by Wong and Sandler (1992) is correct in composition dependence of the second virial coefficient, its composition dependence of the third virial coefficient is incorrect. Hamad and Mansoori (1987) in a study of Lennard-Jones fluid mixtures, proposed a mixing rule that forces the satisfaction of the quadratic composition dependence of second virial coefficients, as well as the cubic dependence of third virial coefficients of mixtures. In this study, we extended this approach to a modified form Peng-Robinson equation of states (Stryjek and Vera, 1986). The purpose is to include a ternary interaction parameter in the equation of state model explicitly and with theoretical justification, and investigate how ternary LLE correlation can be improved in this manner.
THEORY MixingRule The Peng-Robinson equation of state (Peng and Robinson, 1975) is given by:
p=RT_ v-b
a v* +2bv-b’
Expanding the equation, we get: s=l+(b-&)++(b’+$$-$+...
(2)
with the second and third virial coefficients given by: B=b-&
(3)
94
C.H. Chou, D.S.H. Wang/Fluid
Phase Equilibria 98 (1994) 91-111
C=b2+2ba
(4)
RT
Since the correct statistical mechanical mixing rules for the second and third coefficients are:
B, = CCx,x,B,, ’
(5)
J
=;[(b, -&)+(bJ -&)]b‘.)
B,j
(6)
B, being the cross second virial coefficients and k,J the binary pair interaction parameter, and:
c, =cccx,xJxkc,Jk J
(7)
k
I
C,Jk=[(b;
+s)(b;
+s)(b;+%)r(MJk)
(8)
C,Jkbeing the cross third coefficients and l,Jkthe ternary triplet interaction parameters. The corresponding mixing rules for a, and bm should be: b,-g=B,
b* m
;
(9)
2brnam _ C
RT
m
(10)
Solving explicitly for a, and bm, we have: b, =;(B_
+dm)
(11)
(12)
C.H. Chou, D.S.H. Wang /Fluid Phase Equilibria 98 (1994) 91-l 11
95
To prove that am remains positive at high temperature, we need only to show that: Cm>B;
(13)
Writing out the composition of B, and C, in details: Bi = (qSx,x,Bq)(:Fx,x,B,,)
= ~~~~x~x~x,x,BijBti
(14)
For binary and ternary systems, there will be at least one repeat index among i, j, k, and 1. A sufficient condition for C, > Bi is therefore:
(16)
Cl,,’ B,B,k Since pure component parameter a, s are always positive, B
‘,
<(bl+bJ)‘bJ+bk)(l-k )(l-+bB 4 B
B 0
Jk
IJ
’ i/(b,bJbk)l
T-W
‘J
Jk
b-',Jk) =&$Jk
(17) (18)
If 1- ke and 1- lJk are not much different from unity,
‘,Jk
’
!‘E’,Jk
=
\;jB,JBJk
>
B,JBJk
(1%
At any finite temperatures, it is unlikely that am would be negative. The values of b,, a m, and B,of the system acetone-water at 1200 K calculated with interaction parameters obtained at 373 K are shown in Figure 1. It is clear that reasonable values of a,,, were obtained even at these extreme conditions.
96
C.H. Chou, D.S.H. Wang/ Fluid Phase Equilibria 98 (1994) 91-111
6.0~5 5.055 4.OE-5 3.0~5 2.0~5
-l.OE-5~'-~'' ' ' . ' ' ' ' . ' . ' ' 1 0.0 0.1 0.20.30.4 0.5 0.60.7 0.8 0.9 1.0 MahFraatbnaf AaMam
Figure 1: Changes of equation ofstates parameters a,/(RT),
bm and second virial
coefficient Bm with composition for acetone-water binary at 1200 K.
The kgacity coefficient of qy component i in a mixture fluid:
(20)
is dependent upon the composition derivatives of am and b,:
1 i’,_2Bm[:$]+3cm+3(;%) -RT n dn,
-6,/m
,(l”‘_) 3 n
(21) dn,
The composition derivatives of Bm and Cm are conveniently given by:
C.H. Chou, D.S.H.
Wang/Fluid Phase Equilibria 98 (1994) 91-1 I1
=Xx&,
97
(23)
1
= 3CCx,x&,
(24)
J k
Determination of Binary and Ternary Interaction Parameters For a binary (1 and 2) system, there are three parameters to be determined: k12, 1122(=1212=1221), and 1112(=1121=1211). They must be obtained by regression of experimental data. For binary pairs that are completely miscible, these parameters can be easily determined by regressing VLE data. For a partial miscible pair, the parameters cannot be determined by LLE alone. This is due to the relative independence of liquid phase fugacties with respect to pressure. At a given temperature, there is essentially just one set of binary liquid-liquid equilibrium data point, governed by two equilibrium conditions:
(25) (26) 200
,I,
-
8,
I
._ '.._, -200
0.0
,,,
8,.
I
,,,,,I,
.... kln=0.OO,ktrQ4MJm=O.281 kcn--O.l07Jt1wll.772J1iPQ4sB
0 0.1
I. 0.2
._::. __._.. .......
" 0.2
0 0.4
I 0.5
” 0.6
I 0.7
” Od
0 . 0.9 1.0
molalrwbllofwau
Figure 2:
Excess Gibbs free energies of mixing for different sets of k,2, Jz, l,,, that all satisfy binary LLE data of the system l-butanol + water
98
C.H. Chou, D.S.H. Wang/Fluid Phase Equilibria 98 (1994) 91-111
Sinoe there are three interaction parameters in our mixing rules, they cannot be completely determined by one set of tieline data with only two equations. Figure 2 illustrates the excess Gibbs free energy of the system 1-butanol and water at 1 atm and 25 oC predicted by our equation of state model with binary interaction parameters determined by LLE data. It obvious that the same LLE data can be correlated with different values of k12, which describe entirely different degrees of immiscibility. For such systems, additional VLE data are required. Yet there are usually quite a lot more VLE data then LLE data, proper balance between VLE and LLE data is necessary to obtain a “correct” set of binary interaction parameters. The following procedure is used to determine the three interaction parameters: firstly, 1122 and 1211 are determined at different values of k12 by fitting data LLE data; secondly the set of k12, 1122 and 1112 that best correlates VLE data is selected (e.g. this procedure for lbutanol + water system is illustrated in Figure 3).
Figure 3:
Binary VLE for different sets of of k,z, JZ2, l,,, that all satisfy binary LLE data of the system I-butanol + water
The ternary parameter 1123 is obtained by fitting ternary liquid-liquid equilibria data. For a given set of data of experimental compositions, liquid-liquid flash is carried out at the midpoint of experimental tielines, and the following objective function is minimized with respect to 1123:
L
(27)
Figure 4 demonstrates that the size and shape of the phase envelope and the slopes of the tielines (of the system 1-butanol + ethanol + water) are quite sensitive to the values ofl123.
C.H. Chou, D.S.H. Wong /Fluid Phase Equilibria 98 (1994) 91-l I1 ._.‘-
99
BW hrr_am
hnd.6624 ...._... hdb60
0.20-
0.16 E ;
0.12 -
% b
0.06 -
i
0.04 -
0.0
0.1
62
0.2
0.4
0.6
0.6
0.7
0.6
0.6
1.0
molafrabllofwalw
Figure 4:
Ternary LLE for different values of l,,, for the system 1-butanol(1) + ethanol(2) + water(3)
We have discussed the importance of obtaining a “correct” set of binary interaction parameters for the partially miscible binary pair. Figure 5 illustrates that ternary liquid-liquid equilibria cannot be adequately correlated even with a correction parameter 1123, unless k,, has been properly determined. f....
Exp6fhmtUQala k,*O.W
klrO.lO7 ------kirOOO0
0.16 -
0.12 -
0.06 -
0.00 . I 'I. 0.0 0.1 01
" 0.6
t '0 0.4 6.6
0.6
R7
Od
Od
1.0
llWhfncUOllOfW~tU
Figure 5:
Ternary LLE for different sets of k ,3, 1,33,l,,, that all satisfies binary LLE of the partially miscible l-3 pair for the system 1-butanol(1) + ethanol(2) + water(3)
.~___._
.
100
_____~
__._
l_..-l ^_
C.H. Chou, D.S.H. Wang /Fluid Phase Equilibria 98 (1994) 91-111
RESULTS AND DISCUSSION Binary System In order to demonstrate that our mixing rule is capable of correlating various types of non ideal vapor-liquid equilibria to the same degree of accuracy as an activity coefficient model, correlations of bubble point pressures and vapor mole fraction for five general classes of non ideal binary mixtures are presented in Table 1. Results are compared to calculations by using NRTL activity coefficient model and the mixing rule proposed by Wong and Sandler (1992) with UNIQUAC as the expression of excess Hehnholtz free energy at infinite pressure. All three models have three adjustable binary interaction parameters, although the nature of the parameters are quite different. Our model includes triplet interaction parameter. The mixing rule of Wong and Sandler (1992) has three binary interaction parameters. The NRTL model has two binary pair interaction parameter and one binary randomness factor. We found that our model is as good as the other two in all these cases, thus reassuring that our mixing rule is applicable to binary VLE calculations with nonideality of various degrees. Table 1: Correlations of Binary Vapor-Liquid Equilibria NRTL
Wang and Sailer 19928
Temp. hp% Al% AY [K] Polar+Ncmwlar Pentane+Acetone 238 258 298 298 323 348 393 298 312 328 298 308 318 328 303 323 343
0003 3 1 149 1.40 3.5 0.003 0.42 0002 32 Hydrogen Bonding: Etbanol+Water 0 89 0005 11 ;.ss 0.006 2.0 091 0 004 1.8 1.58 0.005 2.1
Ay
0.011 0.009 0.014
Tluswolk
Ap%
AY
kl2
‘112
-0.106 -0.078 -0.074
-0.29 1 -0.259 -0.279
-0.212 -0.179 -0.184
1.9 2.4 0.6
0.005 0.003 0.003
l/3+4/188 l/3+4/189 1/3+4/190
0 074 0.020 0.017 0.175
0.058 0.018 0.015 0.117
1.8 1.8 10 2.1
0.004 0.006 0.004 0.009
l/lb/l08 l/lb/l06 lllbllO7 l/lb/93
-0.410 -0.325 -0.404
-0.724 -0.616 -0.711
3.3 4.0 72
0.013 0.014 0.020
ll2d208 1/2cl209 1/2al242
-0.189 -0.173 -0.323 -0.138
-0.2 11 -0.150 -0.292 -0.123
0.6 0.2 02 0.2
0.006 0 002 0 004 0.006
l/3+4/98 1/3+4!94 1/3+4/95 l/3+4/96
0.165 -0.130 xl.120
-0.169 -0.135 0.124
0.1 0.3 0.2
0.000 0.001 0.001
l/7/409 l/7/41 1 l/7/413
0.009 0.105 0.004 0.074 0.005 0.070 0.005 0.149 Hydrogen Bonding+ Nonpolar: Metbanol+Cyclobexane 2 75 0.010 12.0 0.040 -0.203 0.50 0.014 83 0.035 -0 141 2.69 0013 10.1 0.042 -0.170 Lewis Acid+Base: A&one +Chloroform 1.32 0 009 1.4 0.007 -0.161 0.48 0003 06 0 004 -0.148 1.08 0004 15 0 005 -0.238 0.61 0 003 1.0 0.004 -0.125 Aromatics: Hexachlorobenzene+Toluene 0.75 0.007 0.9 0.008 -0.128 0 88 0 007 0.8 0.007 -0.102 0.96 0 008 0.8 0.007 -0.092
‘122
Data sourceb
aValues of Ap% and Ay for NRTL were taken f?om Wang et al. 1992 bVLE Data are from DECIIEMA Data Series (Gmehling and Onken, 1977). numbers represent volume/part/page
C.H. Chou, D.S.H. Wong / Fluid Phase Equilibria 98 (1994) 91-11 I
101
Wong et al. 1992 demonstrated that one advantage of using the equation of state model over the activity coefficient model is that an equation of state can be extrapolated over large range of pressures and temperatures with a proper mixing rule. Figure 6 shows that our mixing model, with the binary interaction parameters obtained at 373 K, can also be extrapolated over a temperature range of 150 K and pressure range of one order of magnitude for the binary mixture acetone + water (Experimental data were reported by Griswold and Wong, 1952).
Figure 6: _
Extrapolation of binary VLE correlations for acetone + water from 373 K to 523 K
o.oEo’ ’ ’ ’ 0.0 0.1 W
’
Od
’
OA
’
’
Od
’
Od
’
’ . ’
0.7 Od
Od
’
1 .O
mdafrubnolcEbmnbMa
Figure 7:
Binary VLE correlations for carbon dioxide + propane at 277 and 343 K
102
C.H. Chow,D.S.H. Wang/Fluid Phase Equilibria 98 (1994) 91411
Shibata and Sandler (1989) pointed out that complex mixing rules often fail or need unreasonable binary pair parameters for simple systems especially near the critical point. Figure 7 illustrates that adequate correlation for simple mixtures such as carbon dioxide + propane (Reamer et al. 1952), both above and below the critical temperature of carbon dioxide, can be obtained with our mixing rule.
Ternary Systems Table 2: Sources of VLE/LLE Data syatml
VLELhd
AcctonitrilL?+Benzene B.xlzene+n-Heptanc
l/7/126 Mb/l45
A&Otitfile+n-HCptUlC AcetonitriletBemme+ ll-H.ZpblX
lbld80
n-Butmol+Ethad Ethmol+Watcr n-Butmwl+W&r n-Butanol+Ethmol+Water
Khmin, 1969 1/2dl26 lmoa
Cyclohexane+Benzene
li6al237 l/3+4/43
Benzme+Furfwd Cyclohexme+Furfbd Cycloheme+Bemme+Ftird
Wer+Acetcmitrile Acetmitde+Amykmitricrylonitrile W&f+AnylUlitd~ Wlxter+mtitrilctAclylnylwitrilc n-Heptw+&zene
B.zmme+MethMol fl-HepCane+Methawl ndIeptme+Bemne+Methmol
Cyclohexane+Bwene B. rene+Nilrxnethane Cyclohexane+Nitromethane Cyclohexane+Bcnzenc+Nitronlclhanc Ethylacetate+Acetonilriatonitrile Acetonrtrilc+watcr Ethylacetate+Water Ethylacetate+Aceton~tde+Wder
l/3+4/45
LLE
Dtib
lmdao 116bl471
5/l/236 5im40
5/l/258
s/3/187 l/lam l/8/368 l/l/223
ll6W157 1/2c.‘176 li2cn74
Volpicelli, 1968 Volpidi, 1968
5/2/l 19 S&?/l19
l/6&37 in/a7
Week, 1954
s/1/33 3Rl68
iis/ l/w71 l/l/398
5/l/224
mm
We have tested our mixing rule using several ternary systems with liquid-liquid equilibria: 1. 2. 3. 4. 5.
acetonitrile( l)+benzene(2)+n-heptane(3) 3 18 K n-butanol( l)+ethanol(2)+water(3) 298 K cyclohexane(l)+benzene(2)Hu&ral(3) 298 K n-heptane( l)+benzene(2)+methano1(3) 287 K water( 1)+acetonitrile(2)+acrylonitrile(3) 333 K
C.H. Chou, D.S. H. Wang /Fluid Phase Equilibria 98 (1994) 91411
103
6. cyclohexane( l)+benzene(2)+nitromethane(3) 298 K 7. ethylacetate(l)+acetonitrile(2)+water(3) 333 K Table 2 listed the sources of binary and ternary data. In these systems, components 1 and 3 are the immiscible binary pair, while components 1 and 2 and components 2 and 3 are completely miscible. For partially miscible 1-3 pair, the binary and ternary interaction parameters k,, , l,,, , l,,, in our mixing rule were determined by the rigorous procedure described in the previous section. For the completely miscible l-2 and 2-3 pairs these parameters were determined by regress5n of binary VLE data. The ternary parameters l,, were obtained by regressic;n of ternary LLE data. Similar procedures were adopted to determine the binary interaction parameters k,3, u,, , and u,, in the mixng rule of Wong and Sandler 1992, and binary parameters CX,~,z,, , and r,, in the NRTL activity coefficient model. Neither models contain ternary parameters. Regression results are listed in Tables 3. Figures 8 to 14 compare of ternary LLE calculations using the three thermodynamic models with experimental data. Except for the system l-butanol-ethanol-water, NRTL and the mixing rule of Wong and Sandler (1992) generally over predict the size of the phase envelope. In the cases of n-heptane t benzene + methanol, ethylacetate + acetonitrile + water, and water + acetonitrile i- acrylonitrile these two models fail to predict a close phase envelope, despite that they predict complete miscibility for both l-3 and 2-3 pair. It should be pointed out these calculations demonstrate the inability of these model to predict ternary phase behavior using binary data only. They should not be regarded as evidence of inadequacy of the models. Good representation of phase behavior of both binary and ternary systems using the NRTL model can be obtained if experimental data of the ternary system are included judiciously (Renon, 1994). With 1123 equals 0, our model usually predicts phase envelope smaller than those predicted by other two local composition type models. Although it produces a class phase b;un&y for n-heptane + benzene + methanol, it fails to close the immiscibility gap for ethylacetate + acetonitrile + water, and water + acetonitrile + acrylonitrile; and agreement with experimental data in the size of phase envelopes and slope of tieline is generally poor. Mathias et al. (1991) pointed out that any prediction formula for 1123 must satisfy the following limiting conditions imposed by the Michelsen-Kistenmacher invariance:
q; [K]
NRTL
1+3
0.204
61.8
1440.4
Cyclohexane(l)+Bazene(2)+Nitromethane(3) 1+2 0.301 -1 1 152.4 2+3 0.296 99.4 298.1 1+3 0.421 973.4 969.7 1+2+3 Eihylacetate(l)+Ac&mitrile(2)+Water(3) 1+2 0.302 70.2 74.0 2+3 0 439 353.4 621.3
~cetonltnle(l)+Benzene(Z)+n-He3) 226.6 1+2 0.297 Ii30 -187.0 2+3 0 302 464 3 704.5 l-3 0.390 808.5 1+2+3 n-But~ol(l)+EU1anol(2)+Water(3) -102.5 1+3 0.300 147.9 426.8 2+P 0.295 14 1 1+3 0 430 2876 1053.7 l-2+3 Cyclohexane(l)+Benzene(2)+Furfwal(3) 1+2 0 301 -1.1 152.4 2+3 1.035 408.3 163.4 1+3 0.367 740.0 546.6 1+2+3 Water(l)+Acetonitrile(2)+Arrylonitrile(3) 1+2 0.439 621.3 353.4 2+3 0310 246 5 -171.6 1+3 0.224 1023 9 239.6 1+2+3 n-Hqtane(l)+Bew.en~2)+Methanol(3) 1+2 0 298 -87.5 321.9 2+3 0400 513.2 297.0 1+3 0 414 676.8 732.9 1+2+3
T;; [K]
3.727
0.337
0.061 0.415
0.46 111
0.141 0.273 0.300
0.415 0.013 0.372
0.058 0 114 -0 354
0.080 0.251 -0.299
0.128 0 136 -0 097
kij
0.058 0.114 -0.071
36.4
18.6
15.5
10.6
30 4
Y&L
0.17 2.19 0.85
071 2.60 187
1.11 5.08 8.11
0.17 1.44 11.11
121 120 341
1.77 092 168
%sAF.K
-84.2 142.5 1649
490.6
-17.0 50.2 180.5
-26.0 -32.9 375
172.7 -74.9 477.0
-17.0 -79.0 89.4
-85.9 -15.9 572.9
252 8 -11.9 804.8
205.5 172.7
718 109.2 689.4
89.0 556.5 924.2
142 5 108.5 160.2
71.8 247.5 608.3
129 9 134 1 411.1
-54.5 67.8 135 6
3.35
0.65 6.39
0.18 2.60 4.70
0.73 5.15 3.10
6.39 0.55 6.80
0.18 141 8.10
1.75 1.77 2.98
1.41 0.83 1.73
Won8 and Sailer 1992 uii [K] uii [K] Y&K
8.07
30 8
24.6
122
49.0
22.8
%AKL
I
-0.264
0.004 0 256
0.029 0.030 0.068
0.074 -0.199 0.356
0.256 -0.013 0.104
0 029 -0 070 0.229
0.052 -0.131 -0.107
0.036 0.080 -0.053
Table 3: Comparisons of Correlation Results of Binary VLE/LLE and Ternary LLE Data
-1.076
-0 025 0.281
0.016 -0.006 -0.110
0.033 -0.578 0.279
0.281 -0.028 0015
0.016 -0.241 0.187
0.011 -0.380 -0.773
-0 005 0035 0.253
I...
-0.465
-0.019 0.201
0.015 -0.004 -0.024
0.030 -0.355 0.285
0.201 -0.025 -0 102
0.015 -0.202 0.187
0.010 -0.235 -0.433
-0.012 -0.050 -0.401
I...
This work
0.028
0.017
0.193
0 024
-0.502
-0 104
I;&
2.43
0.70 2.15
0.18 2.59 11.87
0.66 2.84 21.75
2.15 0.55 6.63
0.18 2.41 8.20
1.50 1.48 2.22
1.40 0.80 171
%@K
7.7
15.7
15.5
2.9
9.1
14 1
%AKL
C.H. Chou, D.S.H.
Wang/Fluid Phase Equilibria 98 (1994) 91-111 .
ExprthlwwDuI NRTL wmJ1swdbr1wz
---.
lNswrkwlmh-o.104 nlkww *ohkd.ooo
-
....-.-
Figure 8:
Comparison of model calculations and experimental data of ternary LLE for the system acetonitrile( l)+benzene(2)+n-heptane at 3 18 K . ---.
-------.
Figure 9:
Expwhlma
Data
!&mar1000 ThN wurk wilh ha-O.602 Tld~wc&wlfhhn-O.OOO
Comparison of model calculations and experimental data of ternary LLE for the system 1-butanol( l)+ethanol(2)+water(3) at 298 K . ---.
..--...
upulnunmoatll NRTL
w0ngaswdbr1902 lhlrwotkwilhhamO.024 Thbmkw,t,,hsO.OaO
Figure 10: Comparison of model calculations and experimental data of ternary LLE for the system cyclohexane( l)+benzene(2)+fXual(3) at 298 K
105
106
C.H. Chou, D.S. H. Wang /Fluid Phase Equilibria 98 (1994) 91-I 11
. ---. ..--.--
0.1
0.0
Exprhnial Ma NRlL wul@suKm1222 Tl2awwk*imhm.-2.410 Ti3,wtiwHhk,,.O.O22
02
0.a
0.4
0.2
0.0
0.7
0.8
0.0
1.0
Figure 11: Comparison of model calculations and experimental data of ternary LLE for the system n-heptane(l)+benzene(2)+methano1(3) at 287 K .
Elplinmwau NRn Wong6&ndkrts22
---. -
ThlrwakwilhhaO.lOt ..---.- ThiswwkwithhnO.WO
0.0
0.1
03
0.2
0.4
0.5
0.8
0.7
08
0.9
I.0
Figure 12: Comparison of model calculations and experimental data of ternary LLE for the system water(l)+acetonitrile(2)+acrylonitrile(3) at 333 K . ypnw~ ---. WongandSandw1992
ln,**ork,hIa-O.028 -..--.. 7h!aw&,hm-0.000
g %
0.2
B 0.2 -
f
2$
0.1 , 0.0
0.1
_( 0.2
0.3
0.4
0.6
0.8
0.7
0.8
0.8
I.0
Figure 13: Comparison of model calculations and experimental data of ternary LLE for the system cyclohexane(l)+benzene(2)+nitromethane(3) at 298 K
C.H. Chou, D.S.H. Wang /Fluid Phase Equilibria 98 (1994) 91411 . -
..‘.. ---.
107
Exw!mwdUW NRTL
Wongmd&ndClSSZ TM work WISI ha--U74 TMa work WIUI h-O.000
Figure 14: Comparison of model calculations and experimental data of ternary LLE for the system ethylacetate(l)+acetonitrile(2)+water(3) at 333 K if components 1 and 2 are identical
I,,, = 111,= I**,
if components 1 and 3 are identical
11,X = 1,X,= 121,
if components 2 and 3 are identical
A,, = l,,, = 113,
Since 1123’0 does not satisfy these conditions, there is no reason to expect that such an arbitrary prediction of 1123 would be successful. Math& et al. 1991 observed that only a few prediction formulas proposed in the literature for ternary triplet interaction satisfy these conditions under specific limitations (e.g., that lijk must be positive, or symmetrical lijj=liij). Such a formula remains elusive to US. Howe\.er, by adjusting 1123, excellent agreement in phase boundary and slope of the tie-lines were obtained, except near the plait point region. Since ternary triplet interaction is obtained by fitting experimental data of ternary systems, we do not have to worry about a prediction formula and whether it satisfies the MichelsenKistenmacher invariance. The failure of the model near the plait point may be attributed to the inadequacy of classical thermodynamic model in predicting critical exponents (e.g., dePablo and Prausnitz 1988).
CONCLUSIONS
In this work, a new mixing model that is theoretically correct in the composition dependence of the second and third virial coefficients was developed. The advantage that this mixing rule offers is the inclusion of an explicit ternary interaction term on a sound theoretical basis. This, we found, is particularly important in the correlation of LLE of ternary systems containing a.plait point. Moreover, both VLEILLE data must
108
C.H. Chou, D.S.H. Wang /Fluid Phase Equilibria 98 (1994) 91-l I1
be used together with a rigorous procedure to determine the binary interaction parameter of the partially miscible pair. In our model and data regression procedure, interaction parameters that are binary characteristics are obtained from data of binary of systems only, while ternary interaction parameters are determined by data of ternary systems only. By accounting for ternary interaction, excellent agreement in phase boundary and slope of the tie-lines were obtained, except near the plait point region. However, we should be reminded that cubic equations of states usually fail to predict accurate second and third virial coefficients. Our approach merely makes use of the composition dependence of the virial coefficients to accommodate triplet interaction in an equation of state mixing rules.
NOTATIONS
alphabets Equation of state parameter ii Second virial coefficient Third virial coefficient C f Fugacity K values K k Pair interaction parameter for cross second virial coefficient Triplet intern&on parameter for cross third virial coefficient 1 Number of experimental data points M Number of species in mixtures N Pressure P R Universal gas constant Temperature T TLL
Tieline length defined as Interaction parameter in UNIQUAC Molar VoluTe Mole Ii-action Mole fraction of the vapor phase
Greek Letters Randomness parameter in NRTL a Difference between model calculations and experimental data in bubble point AP pressure of binary systems defined in Tables 1 APK Difference between model calculations and experimental data in bubble point pressure and K values of binary systems defined in Table 3 AKL Difference between model calculations and experimental data in tieline lengths and K values of ternary systems defined in Table 3
C.H. Chou, D.S.H. Wang/ Fluid Phase Equilibria 98 (1994) 91-111
cp z
109
Fugacity Coefficient Interaction parameter in NRTL
Superscript ca Model calculations ex Experimental data Ll , L2 Liquid phases S Saturation properties Subscript Component species Li,k m Mixture
REFERENCES
Abrams, D. S. and Prausnitz, J. M., 1975. Statisitical Thermodynamics of Liquid Mixtures: A New Expression for the excess Gibbs Energy of Partly or Completely Miscible Systems. AICHE J., a:1 16-128 Anderson, T. F. and Prausnitz J. M., 1978. Application of the UNIQUAC Equation to Calculation of Multicomponent Phase Equilibria. 2. Liquid-Liquid equilibria. Ind. Eng. Chem. Proc. Des. Dev., l7,561-567 Cha, T. H. and Prausnitz J. M., 1985. Thermodynamic Method for Simultaneous Representation of Ternary Vapor-Liquid and Liquid-Liquid Equilibria. Ind. Eng. Chem. Proc. Des. Dev., 24, 551-555 Gmehling, J. and Onken U., 1977: Vapor-Liquid Equilibrium Data Collection, vol. 1, DECHEMA Data Series, Frankfkt, Germany Griswold, J. and Wong, S. Y., 1952. Phase Equilibria of Acetone-Methanol-Water System from 100 oC to the Critical Region. Chem. Eng. Prog. Sym. Ser., 48(3), 18 Hamad, E. Z. and Mansoori , G. A., 1987. Dense Fluid Theory of Mixtures. J. Chem. Phys., 87,6046-6052 Huron, M. J., and Vidal J., 1979. New Mixing Rules in Simple Equations of State for Representing Vapor-Liquid Equilibria of Strongly Non-ideal Mixtures. Fluid Phase Equil., 3, 255-27 1
110
C.H. Chou, D.S. H. Wong /Fluid Phase Equilibria 98 (1994) 91-l 1 I
Huang, H., 1991,. Study of Ternary Liquid-Liquid Equilibria and of Multiphase Equilibria. Fluid Phase Equilibria, 65: 18 l-207 Kharin, S. E., Perelygin, V. M., Remizov, G. P., 1969. Liquid-Vapor Phase Equilibrium in Ethanol-n-Butanol and Water-n-Butanol Systems. IZV. VYSSH. UCHEB. ZAVED., KHIM. KHIh4. TEKHNOL., 12: 424-428 Leet, W. A., Lin, H. M. and Chao, K. C., Cubic Chain of Rotators Equation of State II: for Strongly Polar Substances and Their Mixtures. Ind. Eng. Chem. Fund. 25: 695-70 1 Luedecke, D. and Prausnitz, J. M., 1985. Extension of the Peng-Robinson Equation of State to Complex Mixtures: Equation of the various Forms of the Local Composition Concept. Fluid Phase Equilibria, 4: 15 l-l 63 Mathias, P. M. and Copeman, T. W., 1983. Extension of the Peng-Robinson Equation of State to Complex Mixtures: Equation of the Various Forms of the Local Composition Concept. Fluid Phase Equilibria, 13: 9 1- 108 Mathias, P. M., Klotz, H. C., and Prausnitz, J. M., 1991. Equation of State Mixing Rules for Multicomponent Mixtures: the Problem of Invariance. Fluid Phase Equilibria, 67: 3 l-44 dePablo, J. J., and Prausnitz, J. M., 1988. Thermodynamics of Liquid-Liquid Equilibria Including the Critical Region. AICHE J., 34: 1595-1606 Reamer, H. H., Sag0 fi. H. and Lacey, W. N., 1951. Phase Equilibria in Hydrocarbon Systems, lad. Eng. Chem., a,25 15-2520 Renon, H. and Prausnitz, J. M. 1968. Local Compositions in Thermodynamics Excess Functions for Liquid Mixtures, AICHE J., 14: 135-144 Renon, H. 1994. discussion with Professor Renon during preparation of this paper Sheng, Y. J., Chen, P. C., Chen, Y. P., and Wong, D. S. H., 1992. Calculations of Solubilities of Aromatic Compounds in Supercritical Carbon Dioxide. Ind. Eng. Chem. Rc _ -, 3.1. $67-973 Shibata, S. K. and Sandler, S. I., 1989. Critical Evaluation of Equation of State Mixing Rules for Prediction of High Pressure Phase Equilibria. Ind. Eng. Eng. Chem. Res., B: 1893-1898
C.H. Chou, D.S. H. Wang /Fluid Phase Equilibria 98 (1994) 91411
111
S$rensen, J. M. and Arlt, W., 1979. Liquid-Liquid Equilibrium Data Collection, DECHEMA, Frankfurt, Germany Stryjek, R., and Vera, J. H., 1986. PRSV: An Improved Peng-Robinson Equation of State for Pure Compounds and Mixtures. Can J. Chem. Eng., 64: 323-333 Volpicelli, G., 1968. Vapor-Liquid and Liquid-Liquid Equilibria of the System Acrylonitrile-Acetonitrile-Water. J. Chem. Eng. Data, 13: 150-l 54 Week, H. I., and Hunt, H., 1954. Vapor Liquid Equilibria in the Ternary System Benzene-Cyclohexane-Nitromethane and the Three Binaries. Ind. Eng. Chem., 46: 2521-2523 Wong, D. S. H., and Sandler, S. I., 1992. A Theoretically Correct Mixing Rule for Cubic Equation s of State,” AICHE J., 3: 671-680 Wong, D. S. H., Orbey, H. and Sandler, S. I., 1992: Equation of State Mixing Rule for Non ideal Mixtures Using Available Activity Coefficient Model Parameters That Allows Extrapolation over Large Temperature and Pressure Ranges. Ind. Chem. Eng. Res., 31: 2033-2039
Fluid Phase Equilibria, 98 (1994) 91-111
An Equation of State Mixing Rule for Correlating Ternary LiquidLiquid Equilibria Cheng Huang Choua and David Shan Hill Wongb a&ion Chemical Laboratories, Industrial Technology Research Institute, 321 Kuang Fu Rd. Sec. 2, Hsinchu, Taiwan 30042 bDepartment of Chemical Engineering, National Tsing Hua University, 101 Kuang Fu Rd., Sec. 2, Hsinchu, Taiwan 30043
Keywords: theory, equation of state, mixing rules, liquid-liquid equilibria (Received July 16,1993; accepted in final form April 20, 1994)
ABSTRACT A new mixing model for cubic equations of states that is theoretically correct in the composition dependence of the second and third virial coefficients is derived. This new mixing rule has been tested in conventional vapor-liquid equilibria calculations, It was found to be as good as activity coefficient models and other equation of state mixing rules for solutions of various kinds of nonideality. It can also be extrapolated over large temperature and pressure ranges. Its salient feature is an extra degree of freedom that accounts for ternary interaction, which enables it to give consistent correlation of vapor-liquid and liquid-liquid equilibria of ternary systems and the corresponding binaries.
INTRODUCTION Applications of cubic equations of state in phase equilibrium calculations have, for many years, been restricted to near ideal mixtures because of the limitations of the one fluid van der Waals mixing rule used. For more complex mixtures,’ activity coefficient models (or “liquid solution” models) are commonly used. However, unlike equations of states, which are general thermodynamic models for both liquid and vapor phases and all residual thermodynamic state functions, activity coefficient models are correlations of excess free energy specifically for liquid phases only. There have been ongoing efforts to bridge these two approaches.
0378-3812/94/$07.00 @ 1994 - Elsevier Science B.V. All rights reserved SSDI 0378-38 12 (94) 02528-g
92
C.H. Chou, D.S.H. Wang/Fluid Phase Equilibria 98 (1994) 91-111
,Hqon and Vidal(1979) first pointed out the relation between mixing rules for equation of state parameters and the excess Gibbs fi-ee energy at infinite pressure. Their approach necessitated the assumption that the excess volume of the system at infinite pressure be zero in order that the excess Gibbs free energy be finite. Sheng et al. 1992 pointed out that using excess Hehnholtz free energy would be more appropriate and ‘an extra degree of freedom in the choice of mixing rules is thereby released. Most mixing rules that address the problem of nonideality in liquid (or high density) mixtures fail to reduce the proper statistical mechanical limit at low density that the composition dependence of the second virial coefficient be quadratic. It is this constraint that confined the thinking of most researchers within the van der Waals one fluid paradigm. Wong and Sandler (1992) pointed out that the van der Waals one fluid mixing rule is only a specific case of a more general class of mixing rules that satisfy the above limitiig condition. The degree of freedom opened up by Sheng et al. (1992) could be used to satis@ this limit. Wong et al. (1992) also showed that how knowledge experiences accumulated in correlating vapor-liquid equilibrium (VLE) at low pressures and temperatures with liquid solution models can be preserved during the changeover to equation of state models, and extrapolated into high temperature and pressure. Despite its success in correlating VLE of non ideal binary systems, the mixing rule proposed by Wong and Sandler (1992), was not very successll in predicting liquid-liquid .e,quilibria (LLE) of ternary systems using data of binary system only. There have been only scattered attempts to apply equations of state to liquid-liquid equilibria calculations (Peng and Robinson, 1976, Mathias and Copeman, 1983, Leet et al., 1986 4 Luedecke and Prausnitz, 1985, Huang, 1991). The most extensive of ,these studies was reported by Huang (1991), who used the approach of Huron and Vidal (19792 to incorporate liquid solution models such as NRTL (Renon and Prausnitz, 1968) and UNIQUAC (Abrams and Prausnitz, 1975) to the equation of state proposed by Pate1 and Teja (1982), to correlate LLE data of ternary systems. It was found that difficulties in data correlation arise mostly with systems that have plait points. Accurate correlation of phase behavior of both the ternary system and its constituent binaries was found possible only when data of the ternary systems were incorporated in fitting procedure to determine the binary interaction parameters. The same problem existed with the direct application of local composition liquid $olution model. Anderson and Prausnitz (1978) observed that the binary interaction parameters of the UNIQUAC mddel can not be uniquely determined due to experimental error or lack of sufficient data. Only a highly correlated set of optimal values can be found. hiclusion of ternary tieline data could would substantially improve the accuracy of the cdrrelation’for the ternary system, but usually at the expense of the accuracy of binary correlations. Cha and Prausnitz (1986) suggested an entirely empirical correction for
C.H. Chow D.S.H. Wang /Fluid Phase Equilibria 98 (1994) 91-111
93
the UNIQUAC model that includes a ternary parameter to eliminate sensitivity of the model to binary parameters. The empiricism, however, seriously limited the appeal of this approach. In using the NRTL model, it is common practice that various values of a,, are used to regress the VLE and LLE data of the partially miscible pair. The set that best described the ternary LLE data is then used. This is based the assumption that there is not enough information in binary data to get a “true” set of NRTL parameters. Ternary LLE data are used as supplementary information to determine a parameter that is binary in nature. It remains to be shown that the same value of a,, could be used for any ternary system involving the specific binary pair (Renoir, 1994). While the mixing rule proposed by Wong and Sandler (1992) is correct in composition dependence of the second virial coefficient, its composition dependence of the third virial coefficient is incorrect. Hamad and Mansoori (1987) in a study of Lennard-Jones fluid mixtures, proposed a mixing rule that forces the satisfaction of the quadratic composition dependence of second virial coefficients, as well as the cubic dependence of third virial coefficients of mixtures. In this study, we extended this approach to a modified form Peng-Robinson equation of states (Stryjek and Vera, 1986). The purpose is to include a ternary interaction parameter in the equation of state model explicitly and with theoretical justification, and investigate how ternary LLE correlation can be improved in this manner.
THEORY MixingRule The Peng-Robinson equation of state (Peng and Robinson, 1975) is given by:
p=RT_ v-b
a v* +2bv-b’
Expanding the equation, we get: s=l+(b-&)++(b’+$$-$+...
(2)
with the second and third virial coefficients given by: B=b-&
(3)
94
C.H. Chou, D.S.H. Wang/Fluid
Phase Equilibria 98 (1994) 91-111
C=b2+2ba
(4)
RT
Since the correct statistical mechanical mixing rules for the second and third coefficients are:
B, = CCx,x,B,, ’
(5)
J
=;[(b, -&)+(bJ -&)]b‘.)
B,j
(6)
B, being the cross second virial coefficients and k,J the binary pair interaction parameter, and:
c, =cccx,xJxkc,Jk J
(7)
k
I
C,Jk=[(b;
+s)(b;
+s)(b;+%)r(MJk)
(8)
C,Jkbeing the cross third coefficients and l,Jkthe ternary triplet interaction parameters. The corresponding mixing rules for a, and bm should be: b,-g=B,
b* m
;
(9)
2brnam _ C
RT
m
(10)
Solving explicitly for a, and bm, we have: b, =;(B_
+dm)
(11)
(12)
C.H. Chou, D.S.H. Wang /Fluid Phase Equilibria 98 (1994) 91-l 11
95
To prove that am remains positive at high temperature, we need only to show that: Cm>B;
(13)
Writing out the composition of B, and C, in details: Bi = (qSx,x,Bq)(:Fx,x,B,,)
= ~~~~x~x~x,x,BijBti
(14)
For binary and ternary systems, there will be at least one repeat index among i, j, k, and 1. A sufficient condition for C, > Bi is therefore:
(16)
Cl,,’ B,B,k Since pure component parameter a, s are always positive, B
‘,
<(bl+bJ)‘bJ+bk)(l-k )(l-+bB 4 B
B 0
Jk
IJ
’ i/(b,bJbk)l
T-W
‘J
Jk
b-',Jk) =&$Jk
(17) (18)
If 1- ke and 1- lJk are not much different from unity,
‘,Jk
’
!‘E’,Jk
=
\;jB,JBJk
>
B,JBJk
(1%
At any finite temperatures, it is unlikely that am would be negative. The values of b,, a m, and B,of the system acetone-water at 1200 K calculated with interaction parameters obtained at 373 K are shown in Figure 1. It is clear that reasonable values of a,,, were obtained even at these extreme conditions.
96
C.H. Chou, D.S.H. Wang/ Fluid Phase Equilibria 98 (1994) 91-111
6.0~5 5.055 4.OE-5 3.0~5 2.0~5
-l.OE-5~'-~'' ' ' . ' ' ' ' . ' . ' ' 1 0.0 0.1 0.20.30.4 0.5 0.60.7 0.8 0.9 1.0 MahFraatbnaf AaMam
Figure 1: Changes of equation ofstates parameters a,/(RT),
bm and second virial
coefficient Bm with composition for acetone-water binary at 1200 K.
The kgacity coefficient of qy component i in a mixture fluid:
(20)
is dependent upon the composition derivatives of am and b,:
1 i’,_2Bm[:$]+3cm+3(;%) -RT n dn,
-6,/m
,(l”‘_) 3 n
(21) dn,
The composition derivatives of Bm and Cm are conveniently given by:
C.H. Chou, D.S.H.
Wang/Fluid Phase Equilibria 98 (1994) 91-1 I1
=Xx&,
97
(23)
1
= 3CCx,x&,
(24)
J k
Determination of Binary and Ternary Interaction Parameters For a binary (1 and 2) system, there are three parameters to be determined: k12, 1122(=1212=1221), and 1112(=1121=1211). They must be obtained by regression of experimental data. For binary pairs that are completely miscible, these parameters can be easily determined by regressing VLE data. For a partial miscible pair, the parameters cannot be determined by LLE alone. This is due to the relative independence of liquid phase fugacties with respect to pressure. At a given temperature, there is essentially just one set of binary liquid-liquid equilibrium data point, governed by two equilibrium conditions:
(25) (26) 200
,I,
-
8,
I
._ '.._, -200
0.0
,,,
8,.
I
,,,,,I,
.... kln=0.OO,ktrQ4MJm=O.281 kcn--O.l07Jt1wll.772J1iPQ4sB
0 0.1
I. 0.2
._::. __._.. .......
" 0.2
0 0.4
I 0.5
” 0.6
I 0.7
” Od
0 . 0.9 1.0
molalrwbllofwau
Figure 2:
Excess Gibbs free energies of mixing for different sets of k,2, Jz, l,,, that all satisfy binary LLE data of the system l-butanol + water
98
C.H. Chou, D.S.H. Wang/Fluid Phase Equilibria 98 (1994) 91-111
Sinoe there are three interaction parameters in our mixing rules, they cannot be completely determined by one set of tieline data with only two equations. Figure 2 illustrates the excess Gibbs free energy of the system 1-butanol and water at 1 atm and 25 oC predicted by our equation of state model with binary interaction parameters determined by LLE data. It obvious that the same LLE data can be correlated with different values of k12, which describe entirely different degrees of immiscibility. For such systems, additional VLE data are required. Yet there are usually quite a lot more VLE data then LLE data, proper balance between VLE and LLE data is necessary to obtain a “correct” set of binary interaction parameters. The following procedure is used to determine the three interaction parameters: firstly, 1122 and 1211 are determined at different values of k12 by fitting data LLE data; secondly the set of k12, 1122 and 1112 that best correlates VLE data is selected (e.g. this procedure for lbutanol + water system is illustrated in Figure 3).
Figure 3:
Binary VLE for different sets of of k,z, JZ2, l,,, that all satisfy binary LLE data of the system I-butanol + water
The ternary parameter 1123 is obtained by fitting ternary liquid-liquid equilibria data. For a given set of data of experimental compositions, liquid-liquid flash is carried out at the midpoint of experimental tielines, and the following objective function is minimized with respect to 1123:
L
(27)
Figure 4 demonstrates that the size and shape of the phase envelope and the slopes of the tielines (of the system 1-butanol + ethanol + water) are quite sensitive to the values ofl123.
C.H. Chou, D.S.H. Wong /Fluid Phase Equilibria 98 (1994) 91-l I1 ._.‘-
99
BW hrr_am
hnd.6624 ...._... hdb60
0.20-
0.16 E ;
0.12 -
% b
0.06 -
i
0.04 -
0.0
0.1
62
0.2
0.4
0.6
0.6
0.7
0.6
0.6
1.0
molafrabllofwalw
Figure 4:
Ternary LLE for different values of l,,, for the system 1-butanol(1) + ethanol(2) + water(3)
We have discussed the importance of obtaining a “correct” set of binary interaction parameters for the partially miscible binary pair. Figure 5 illustrates that ternary liquid-liquid equilibria cannot be adequately correlated even with a correction parameter 1123, unless k,, has been properly determined. f....
Exp6fhmtUQala k,*O.W
klrO.lO7 ------kirOOO0
0.16 -
0.12 -
0.06 -
0.00 . I 'I. 0.0 0.1 01
" 0.6
t '0 0.4 6.6
0.6
R7
Od
Od
1.0
llWhfncUOllOfW~tU
Figure 5:
Ternary LLE for different sets of k ,3, 1,33,l,,, that all satisfies binary LLE of the partially miscible l-3 pair for the system 1-butanol(1) + ethanol(2) + water(3)
.~___._
.
100
_____~
__._
l_..-l ^_
C.H. Chou, D.S.H. Wang /Fluid Phase Equilibria 98 (1994) 91-111
RESULTS AND DISCUSSION Binary System In order to demonstrate that our mixing rule is capable of correlating various types of non ideal vapor-liquid equilibria to the same degree of accuracy as an activity coefficient model, correlations of bubble point pressures and vapor mole fraction for five general classes of non ideal binary mixtures are presented in Table 1. Results are compared to calculations by using NRTL activity coefficient model and the mixing rule proposed by Wong and Sandler (1992) with UNIQUAC as the expression of excess Hehnholtz free energy at infinite pressure. All three models have three adjustable binary interaction parameters, although the nature of the parameters are quite different. Our model includes triplet interaction parameter. The mixing rule of Wong and Sandler (1992) has three binary interaction parameters. The NRTL model has two binary pair interaction parameter and one binary randomness factor. We found that our model is as good as the other two in all these cases, thus reassuring that our mixing rule is applicable to binary VLE calculations with nonideality of various degrees. Table 1: Correlations of Binary Vapor-Liquid Equilibria NRTL
Wang and Sailer 19928
Temp. hp% Al% AY [K] Polar+Ncmwlar Pentane+Acetone 238 258 298 298 323 348 393 298 312 328 298 308 318 328 303 323 343
0003 3 1 149 1.40 3.5 0.003 0.42 0002 32 Hydrogen Bonding: Etbanol+Water 0 89 0005 11 ;.ss 0.006 2.0 091 0 004 1.8 1.58 0.005 2.1
Ay
0.011 0.009 0.014
Tluswolk
Ap%
AY
kl2
‘112
-0.106 -0.078 -0.074
-0.29 1 -0.259 -0.279
-0.212 -0.179 -0.184
1.9 2.4 0.6
0.005 0.003 0.003
l/3+4/188 l/3+4/189 1/3+4/190
0 074 0.020 0.017 0.175
0.058 0.018 0.015 0.117
1.8 1.8 10 2.1
0.004 0.006 0.004 0.009
l/lb/l08 l/lb/l06 lllbllO7 l/lb/93
-0.410 -0.325 -0.404
-0.724 -0.616 -0.711
3.3 4.0 72
0.013 0.014 0.020
ll2d208 1/2cl209 1/2al242
-0.189 -0.173 -0.323 -0.138
-0.2 11 -0.150 -0.292 -0.123
0.6 0.2 02 0.2
0.006 0 002 0 004 0.006
l/3+4/98 1/3+4!94 1/3+4/95 l/3+4/96
0.165 -0.130 xl.120
-0.169 -0.135 0.124
0.1 0.3 0.2
0.000 0.001 0.001
l/7/409 l/7/41 1 l/7/413
0.009 0.105 0.004 0.074 0.005 0.070 0.005 0.149 Hydrogen Bonding+ Nonpolar: Metbanol+Cyclobexane 2 75 0.010 12.0 0.040 -0.203 0.50 0.014 83 0.035 -0 141 2.69 0013 10.1 0.042 -0.170 Lewis Acid+Base: A&one +Chloroform 1.32 0 009 1.4 0.007 -0.161 0.48 0003 06 0 004 -0.148 1.08 0004 15 0 005 -0.238 0.61 0 003 1.0 0.004 -0.125 Aromatics: Hexachlorobenzene+Toluene 0.75 0.007 0.9 0.008 -0.128 0 88 0 007 0.8 0.007 -0.102 0.96 0 008 0.8 0.007 -0.092
‘122
Data sourceb
aValues of Ap% and Ay for NRTL were taken f?om Wang et al. 1992 bVLE Data are from DECIIEMA Data Series (Gmehling and Onken, 1977). numbers represent volume/part/page
C.H. Chou, D.S.H. Wong / Fluid Phase Equilibria 98 (1994) 91-11 I
101
Wong et al. 1992 demonstrated that one advantage of using the equation of state model over the activity coefficient model is that an equation of state can be extrapolated over large range of pressures and temperatures with a proper mixing rule. Figure 6 shows that our mixing model, with the binary interaction parameters obtained at 373 K, can also be extrapolated over a temperature range of 150 K and pressure range of one order of magnitude for the binary mixture acetone + water (Experimental data were reported by Griswold and Wong, 1952).
Figure 6: _
Extrapolation of binary VLE correlations for acetone + water from 373 K to 523 K
o.oEo’ ’ ’ ’ 0.0 0.1 W
’
Od
’
OA
’
’
Od
’
Od
’
’ . ’
0.7 Od
Od
’
1 .O
mdafrubnolcEbmnbMa
Figure 7:
Binary VLE correlations for carbon dioxide + propane at 277 and 343 K
102
C.H. Chow,D.S.H. Wang/Fluid Phase Equilibria 98 (1994) 91411
Shibata and Sandler (1989) pointed out that complex mixing rules often fail or need unreasonable binary pair parameters for simple systems especially near the critical point. Figure 7 illustrates that adequate correlation for simple mixtures such as carbon dioxide + propane (Reamer et al. 1952), both above and below the critical temperature of carbon dioxide, can be obtained with our mixing rule.
Ternary Systems Table 2: Sources of VLE/LLE Data syatml
VLELhd
AcctonitrilL?+Benzene B.xlzene+n-Heptanc
l/7/126 Mb/l45
A&Otitfile+n-HCptUlC AcetonitriletBemme+ ll-H.ZpblX
lbld80
n-Butmol+Ethad Ethmol+Watcr n-Butmwl+W&r n-Butanol+Ethmol+Water
Khmin, 1969 1/2dl26 lmoa
Cyclohexane+Benzene
li6al237 l/3+4/43
Benzme+Furfwd Cyclohexme+Furfbd Cycloheme+Bemme+Ftird
Wer+Acetcmitrile Acetmitde+Amykmitricrylonitrile W&f+AnylUlitd~ Wlxter+mtitrilctAclylnylwitrilc n-Heptw+&zene
B.zmme+MethMol fl-HepCane+Methawl ndIeptme+Bemne+Methmol
Cyclohexane+Bwene B. rene+Nilrxnethane Cyclohexane+Nitromethane Cyclohexane+Bcnzenc+Nitronlclhanc Ethylacetate+Acetonilriatonitrile Acetonrtrilc+watcr Ethylacetate+Water Ethylacetate+Aceton~tde+Wder
l/3+4/45
LLE
Dtib
lmdao 116bl471
5/l/236 5im40
5/l/258
s/3/187 l/lam l/8/368 l/l/223
ll6W157 1/2c.‘176 li2cn74
Volpicelli, 1968 Volpidi, 1968
5/2/l 19 S&?/l19
l/6&37 in/a7
Week, 1954
s/1/33 3Rl68
iis/ l/w71 l/l/398
5/l/224
mm
We have tested our mixing rule using several ternary systems with liquid-liquid equilibria: 1. 2. 3. 4. 5.
acetonitrile( l)+benzene(2)+n-heptane(3) 3 18 K n-butanol( l)+ethanol(2)+water(3) 298 K cyclohexane(l)+benzene(2)Hu&ral(3) 298 K n-heptane( l)+benzene(2)+methano1(3) 287 K water( 1)+acetonitrile(2)+acrylonitrile(3) 333 K
C.H. Chou, D.S. H. Wang /Fluid Phase Equilibria 98 (1994) 91411
103
6. cyclohexane( l)+benzene(2)+nitromethane(3) 298 K 7. ethylacetate(l)+acetonitrile(2)+water(3) 333 K Table 2 listed the sources of binary and ternary data. In these systems, components 1 and 3 are the immiscible binary pair, while components 1 and 2 and components 2 and 3 are completely miscible. For partially miscible 1-3 pair, the binary and ternary interaction parameters k,, , l,,, , l,,, in our mixing rule were determined by the rigorous procedure described in the previous section. For the completely miscible l-2 and 2-3 pairs these parameters were determined by regress5n of binary VLE data. The ternary parameters l,, were obtained by regressic;n of ternary LLE data. Similar procedures were adopted to determine the binary interaction parameters k,3, u,, , and u,, in the mixng rule of Wong and Sandler 1992, and binary parameters CX,~,z,, , and r,, in the NRTL activity coefficient model. Neither models contain ternary parameters. Regression results are listed in Tables 3. Figures 8 to 14 compare of ternary LLE calculations using the three thermodynamic models with experimental data. Except for the system l-butanol-ethanol-water, NRTL and the mixing rule of Wong and Sandler (1992) generally over predict the size of the phase envelope. In the cases of n-heptane t benzene + methanol, ethylacetate + acetonitrile + water, and water + acetonitrile i- acrylonitrile these two models fail to predict a close phase envelope, despite that they predict complete miscibility for both l-3 and 2-3 pair. It should be pointed out these calculations demonstrate the inability of these model to predict ternary phase behavior using binary data only. They should not be regarded as evidence of inadequacy of the models. Good representation of phase behavior of both binary and ternary systems using the NRTL model can be obtained if experimental data of the ternary system are included judiciously (Renon, 1994). With 1123 equals 0, our model usually predicts phase envelope smaller than those predicted by other two local composition type models. Although it produces a class phase b;un&y for n-heptane + benzene + methanol, it fails to close the immiscibility gap for ethylacetate + acetonitrile + water, and water + acetonitrile + acrylonitrile; and agreement with experimental data in the size of phase envelopes and slope of tieline is generally poor. Mathias et al. (1991) pointed out that any prediction formula for 1123 must satisfy the following limiting conditions imposed by the Michelsen-Kistenmacher invariance:
q; [K]
NRTL
1+3
0.204
61.8
1440.4
Cyclohexane(l)+Bazene(2)+Nitromethane(3) 1+2 0.301 -1 1 152.4 2+3 0.296 99.4 298.1 1+3 0.421 973.4 969.7 1+2+3 Eihylacetate(l)+Ac&mitrile(2)+Water(3) 1+2 0.302 70.2 74.0 2+3 0 439 353.4 621.3
~cetonltnle(l)+Benzene(Z)+n-He3) 226.6 1+2 0.297 Ii30 -187.0 2+3 0 302 464 3 704.5 l-3 0.390 808.5 1+2+3 n-But~ol(l)+EU1anol(2)+Water(3) -102.5 1+3 0.300 147.9 426.8 2+P 0.295 14 1 1+3 0 430 2876 1053.7 l-2+3 Cyclohexane(l)+Benzene(2)+Furfwal(3) 1+2 0 301 -1.1 152.4 2+3 1.035 408.3 163.4 1+3 0.367 740.0 546.6 1+2+3 Water(l)+Acetonitrile(2)+Arrylonitrile(3) 1+2 0.439 621.3 353.4 2+3 0310 246 5 -171.6 1+3 0.224 1023 9 239.6 1+2+3 n-Hqtane(l)+Bew.en~2)+Methanol(3) 1+2 0 298 -87.5 321.9 2+3 0400 513.2 297.0 1+3 0 414 676.8 732.9 1+2+3
T;; [K]
3.727
0.337
0.061 0.415
0.46 111
0.141 0.273 0.300
0.415 0.013 0.372
0.058 0 114 -0 354
0.080 0.251 -0.299
0.128 0 136 -0 097
kij
0.058 0.114 -0.071
36.4
18.6
15.5
10.6
30 4
Y&L
0.17 2.19 0.85
071 2.60 187
1.11 5.08 8.11
0.17 1.44 11.11
121 120 341
1.77 092 168
%sAF.K
-84.2 142.5 1649
490.6
-17.0 50.2 180.5
-26.0 -32.9 375
172.7 -74.9 477.0
-17.0 -79.0 89.4
-85.9 -15.9 572.9
252 8 -11.9 804.8
205.5 172.7
718 109.2 689.4
89.0 556.5 924.2
142 5 108.5 160.2
71.8 247.5 608.3
129 9 134 1 411.1
-54.5 67.8 135 6
3.35
0.65 6.39
0.18 2.60 4.70
0.73 5.15 3.10
6.39 0.55 6.80
0.18 141 8.10
1.75 1.77 2.98
1.41 0.83 1.73
Won8 and Sailer 1992 uii [K] uii [K] Y&K
8.07
30 8
24.6
122
49.0
22.8
%AKL
I
-0.264
0.004 0 256
0.029 0.030 0.068
0.074 -0.199 0.356
0.256 -0.013 0.104
0 029 -0 070 0.229
0.052 -0.131 -0.107
0.036 0.080 -0.053
Table 3: Comparisons of Correlation Results of Binary VLE/LLE and Ternary LLE Data
-1.076
-0 025 0.281
0.016 -0.006 -0.110
0.033 -0.578 0.279
0.281 -0.028 0015
0.016 -0.241 0.187
0.011 -0.380 -0.773
-0 005 0035 0.253
I...
-0.465
-0.019 0.201
0.015 -0.004 -0.024
0.030 -0.355 0.285
0.201 -0.025 -0 102
0.015 -0.202 0.187
0.010 -0.235 -0.433
-0.012 -0.050 -0.401
I...
This work
0.028
0.017
0.193
0 024
-0.502
-0 104
I;&
2.43
0.70 2.15
0.18 2.59 11.87
0.66 2.84 21.75
2.15 0.55 6.63
0.18 2.41 8.20
1.50 1.48 2.22
1.40 0.80 171
%@K
7.7
15.7
15.5
2.9
9.1
14 1
%AKL
C.H. Chou, D.S.H.
Wang/Fluid Phase Equilibria 98 (1994) 91-111 .
ExprthlwwDuI NRTL wmJ1swdbr1wz
---.
lNswrkwlmh-o.104 nlkww *ohkd.ooo
-
....-.-
Figure 8:
Comparison of model calculations and experimental data of ternary LLE for the system acetonitrile( l)+benzene(2)+n-heptane at 3 18 K . ---.
-------.
Figure 9:
Expwhlma
Data
!&mar1000 ThN wurk wilh ha-O.602 Tld~wc&wlfhhn-O.OOO
Comparison of model calculations and experimental data of ternary LLE for the system 1-butanol( l)+ethanol(2)+water(3) at 298 K . ---.
..--...
upulnunmoatll NRTL
w0ngaswdbr1902 lhlrwotkwilhhamO.024 Thbmkw,t,,hsO.OaO
Figure 10: Comparison of model calculations and experimental data of ternary LLE for the system cyclohexane( l)+benzene(2)+fXual(3) at 298 K
105
106
C.H. Chou, D.S. H. Wang /Fluid Phase Equilibria 98 (1994) 91-I 11
. ---. ..--.--
0.1
0.0
Exprhnial Ma NRlL wul@suKm1222 Tl2awwk*imhm.-2.410 Ti3,wtiwHhk,,.O.O22
02
0.a
0.4
0.2
0.0
0.7
0.8
0.0
1.0
Figure 11: Comparison of model calculations and experimental data of ternary LLE for the system n-heptane(l)+benzene(2)+methano1(3) at 287 K .
Elplinmwau NRn Wong6&ndkrts22
---. -
ThlrwakwilhhaO.lOt ..---.- ThiswwkwithhnO.WO
0.0
0.1
03
0.2
0.4
0.5
0.8
0.7
08
0.9
I.0
Figure 12: Comparison of model calculations and experimental data of ternary LLE for the system water(l)+acetonitrile(2)+acrylonitrile(3) at 333 K . ypnw~ ---. WongandSandw1992
ln,**ork,hIa-O.028 -..--.. 7h!aw&,hm-0.000
g %
0.2
B 0.2 -
f
2$
0.1 , 0.0
0.1
_( 0.2
0.3
0.4
0.6
0.8
0.7
0.8
0.8
I.0
Figure 13: Comparison of model calculations and experimental data of ternary LLE for the system cyclohexane(l)+benzene(2)+nitromethane(3) at 298 K
C.H. Chou, D.S.H. Wang /Fluid Phase Equilibria 98 (1994) 91411 . -
..‘.. ---.
107
Exw!mwdUW NRTL
Wongmd&ndClSSZ TM work WISI ha--U74 TMa work WIUI h-O.000
Figure 14: Comparison of model calculations and experimental data of ternary LLE for the system ethylacetate(l)+acetonitrile(2)+water(3) at 333 K if components 1 and 2 are identical
I,,, = 111,= I**,
if components 1 and 3 are identical
11,X = 1,X,= 121,
if components 2 and 3 are identical
A,, = l,,, = 113,
Since 1123’0 does not satisfy these conditions, there is no reason to expect that such an arbitrary prediction of 1123 would be successful. Math& et al. 1991 observed that only a few prediction formulas proposed in the literature for ternary triplet interaction satisfy these conditions under specific limitations (e.g., that lijk must be positive, or symmetrical lijj=liij). Such a formula remains elusive to US. Howe\.er, by adjusting 1123, excellent agreement in phase boundary and slope of the tie-lines were obtained, except near the plait point region. Since ternary triplet interaction is obtained by fitting experimental data of ternary systems, we do not have to worry about a prediction formula and whether it satisfies the MichelsenKistenmacher invariance. The failure of the model near the plait point may be attributed to the inadequacy of classical thermodynamic model in predicting critical exponents (e.g., dePablo and Prausnitz 1988).
CONCLUSIONS
In this work, a new mixing model that is theoretically correct in the composition dependence of the second and third virial coefficients was developed. The advantage that this mixing rule offers is the inclusion of an explicit ternary interaction term on a sound theoretical basis. This, we found, is particularly important in the correlation of LLE of ternary systems containing a.plait point. Moreover, both VLEILLE data must
108
C.H. Chou, D.S.H. Wang /Fluid Phase Equilibria 98 (1994) 91-l I1
be used together with a rigorous procedure to determine the binary interaction parameter of the partially miscible pair. In our model and data regression procedure, interaction parameters that are binary characteristics are obtained from data of binary of systems only, while ternary interaction parameters are determined by data of ternary systems only. By accounting for ternary interaction, excellent agreement in phase boundary and slope of the tie-lines were obtained, except near the plait point region. However, we should be reminded that cubic equations of states usually fail to predict accurate second and third virial coefficients. Our approach merely makes use of the composition dependence of the virial coefficients to accommodate triplet interaction in an equation of state mixing rules.
NOTATIONS
alphabets Equation of state parameter ii Second virial coefficient Third virial coefficient C f Fugacity K values K k Pair interaction parameter for cross second virial coefficient Triplet intern&on parameter for cross third virial coefficient 1 Number of experimental data points M Number of species in mixtures N Pressure P R Universal gas constant Temperature T TLL
Tieline length defined as Interaction parameter in UNIQUAC Molar VoluTe Mole Ii-action Mole fraction of the vapor phase
Greek Letters Randomness parameter in NRTL a Difference between model calculations and experimental data in bubble point AP pressure of binary systems defined in Tables 1 APK Difference between model calculations and experimental data in bubble point pressure and K values of binary systems defined in Table 3 AKL Difference between model calculations and experimental data in tieline lengths and K values of ternary systems defined in Table 3
C.H. Chou, D.S.H. Wang/ Fluid Phase Equilibria 98 (1994) 91-111
cp z
109
Fugacity Coefficient Interaction parameter in NRTL
Superscript ca Model calculations ex Experimental data Ll , L2 Liquid phases S Saturation properties Subscript Component species Li,k m Mixture
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S$rensen, J. M. and Arlt, W., 1979. Liquid-Liquid Equilibrium Data Collection, DECHEMA, Frankfurt, Germany Stryjek, R., and Vera, J. H., 1986. PRSV: An Improved Peng-Robinson Equation of State for Pure Compounds and Mixtures. Can J. Chem. Eng., 64: 323-333 Volpicelli, G., 1968. Vapor-Liquid and Liquid-Liquid Equilibria of the System Acrylonitrile-Acetonitrile-Water. J. Chem. Eng. Data, 13: 150-l 54 Week, H. I., and Hunt, H., 1954. Vapor Liquid Equilibria in the Ternary System Benzene-Cyclohexane-Nitromethane and the Three Binaries. Ind. Eng. Chem., 46: 2521-2523 Wong, D. S. H., and Sandler, S. I., 1992. A Theoretically Correct Mixing Rule for Cubic Equation s of State,” AICHE J., 3: 671-680 Wong, D. S. H., Orbey, H. and Sandler, S. I., 1992: Equation of State Mixing Rule for Non ideal Mixtures Using Available Activity Coefficient Model Parameters That Allows Extrapolation over Large Temperature and Pressure Ranges. Ind. Chem. Eng. Res., 31: 2033-2039