Ind. Eng. Chem. Res. 1989,28, 1251-1261
1251
Local Composition Embedded Equations of State for Strongly Nonideal Fluid Mixtures Rong-Jwyn Leet and Kwang-Chu Chao* School of Chemical Engineering, Purdue University, West Lufuyette, Indiana 47907 A local composition model obtained from Monte Carlo simulation of molecular fluid mixtures is embedded in the Soave, Patel-Teja, and Cubic Chain-of-Rotators equations of state. Representation of vapor-liquid equilibrium with the embedded equations is investigated a t high pressures as well as low pressures for diverse, strongly nonideal, polar mixture systems including the aqueous, nonaqueous, hydrogen-bonded, and non-hydrogen-bonded systems. Strongly nonideal polar fluid mixtures are of common occurrence and are encountered in numerous engineering processes. Aqueous solutions are prominent examples of such mixtures for which thermodynamic properties are generally not well described by equations of state. Description of phase equilibria of such fluids depends mainly on activity coefficients at low pressures where the gas imperfection is either insignificant or described by the two-term virial equation. For fluids at high pressure, an equation of state has to be employed. In this work, we investigate representation of phase equilibrium of polar fluid mixtures with local composition embedded equations of state a t high pressures as well as low pressures. Polar fluids are characterized by the presence of electric poles in their molecules. Some poles interact strongly, and the molecules preferentially segregate, leading to the local composition (LC) a t the immediate neighborhood of a molecule being different from the bulk fluid composition. Wilson (1964) postulated the existence of local composition in liquid solutions in the development of his activity coefficient equation. Huron and Vidal (1979), Mollerup (1981), Whiting and Prausnitz (1982), and Li et al. (1986) employed mixing rules for equations of state based on local composition. The local compositions of these investigations were postulated, and their LC equations were unconfirmed. Lee and Chao (1986a) determined LC from statistical mechanical simulation and presented a model of LC. Preliminary results of equation of state calculations were reported (Lee and Chao, 1986b). In this work, we extend the approach to several cubic equations of state for the representation of phase equilibria of highly nonideal polar fluid mixtures. The performance of the LC embedded equations of state is extensively investigated with vapor-liquid equilibrium calculations for 50 polar mixtures, including aqueous mixtures.
Local Composition by Monte Carlo Simulation Local composition cannot be determined by experiments in the laboratory but can be simulated by statistical mechanical calculations. Local composition of binary Lennard-Jones fluid mixtures with either the same (Nakanishi and Toukubo, 1979; Nakanishi and Tanaka, 1983) or different (Gierycz and Nakanishi, 1984; Hoheisel and Kohler, 1984) sizes of molecules has been simulated at liquidlike densities. Lee and Chao (1986a,b) as well as Lee et al. (1986) have determined local composition of square-well molecules of diverse energies and of the same size by Monte Carlo (MC) simulation. Lee and Chao (1987) also investigated the local composition of molecules of different sizes. Present address: School of Chemical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100.
From examination of computer-simulated data, Lee and Chao (1986a) obtained a model of local mole fraction of j about a central i as follows:
where the symbols are explained in the Nomenclature section, Pji
=
cyEij
7.. Jl =
and
cy
+ (1 - c u ) E j j
( E11. . EJ J. . ) ' / 2 / E j i
(2)
(3)
is density dependent and is given by =1
+ 0.1057p* - 2 . 1 6 9 4 ~ *+~1 . 7 1 6 4 ~ * ~ (4)
Equation 1 is embedded in three cubic equations of state in this work. Figure 1 illustrates the local composition in a binary mixture found by computer simulation (Lee and Chao, 1986a). Equation 1follows the computed local composition in its variation with density. Three other postulated local compositions are in the approximate range but miss the variation with density. The molecular parameters in these model equations are correlated with real fluid properties as follows: u3 = 3b/(2s) and e = kTJ1.23. The latter equation expressing the energy-well depth is obtained from the equation of state for square-well molecules (Lee and Chao, 1988). The energy of unlike pair interactions is related to like pair values by ~ j = j
(1- kEij)(EiEj)'/'
(5)
The interaction coefficient k,, is adjusted for data fitting.
Local Composition Embedded Equations of State Local composition is embedded in an equation of state by forming A,, the residual Helmholtz energy of a hypothetical single fluid representing the mixture by integrating the equation of state and employing the local composition in the combining rules for the equation parameters. The residual Helmholtz energy of the mixture of interest, A,, is the sum of A , and the free energy of mixing: A, = A,
+ RTCxi In xi
(6)
1
The chemical potential of a mixture component is obtained upon differentiation with respect to the mole number, from which fugacity is obtained upon performing a logarithmic transformation. Other thermodynamic properties are derived from A , upon applying suitable operations.
0888-5885/89/2628-1251$01.50/0 0 1989 American Chemical Society
1252 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 0.60
r
11 =
-0.1057p*
+ 4 . 3 3 8 8 ~ *-~5 . 1 4 9 2 ~ * ~
(13)
The equation of state for a mixture obtained from eq 8 upon differentiation with volume differs from eq 7 by x22 Xll
0.40
c
. .
.__._ ( C l_ A 0 M C Data
0.35
0
I
I
I
I
I
0.2
0.4
0.6
0.8
1.0
P* Figure 1. Local composition data compared with four models at kT/c*ll = 2.0, c12/cll = 1.414, c ~ ~ =/ 2.0, E ~ and ~ x1 = 0.5. (a) Wilson (1964), (b) Gierycz and Nakanishi (1984), and (c) Hu et al. (1984).
Three equations are embedded with local composition in the following, and they are the Soave equation (1972, 1979), the Patel and Teja equation (1982), and the Cubic Chain-of-Rotators equation (Kim et al., 1986). Soave Equation of State The Soave (1972) equation is
RT
a
(7)
p=FFzzT)
The equation constants a and b were presented in a generalized form (Soave, 1972) for nonpolar substances. For polar fluids, Soave (1979) suggested an alternate form for a. The coefficients for calculating a according to Soave's formula of 1979 were evaluated for a large number of polar fluids by Sandarusl et al. (1986). In this work, we use Soave's 1972 generalized formula for nonpolar components and Sandarusl's compilation for polar components. From the Soave equation, we integrate to obtain u
a
A , = RT In -+ - I n V-b b ~
+
The equation parameters a , b, and c were reported for a number of nonpolar substances, and generalized forms were presented. Georgeton et al. (1986) determined coefficients for calculating the parameters for 27 polar liquids. The specific parameters are used in this work where available; the generalized parameters are otherwise used. The residual Helmholtz energy derived from the Patel-Teja (P-T) equation is
a
A, = RTln u + In V-b 2d
b
CxiCxjiaji i
Patel-Teja Equation of State Patel and Teja (1982) proposed a three-parameter cubic equation, RT a (14) = u-b - u ( u + b ) + c(c - b )
U
The interaction energy constant a for a mixture is combined according to the local composition of eq 1 by a =
an extra term that arises due to the volume dependence of the local composition, xji. The extra term has been found to be insignificant and ignored in this work. As a result, the mixture equation of state retains the same form as the pure fluid equation. An extra term is also obtained with the other two equations of state investigated in this work and is also found to be insignificant. The original pure fluid forms of the equations of state are used in all the calculations. In the shape factor method of the principle of corresponding states (Rowlinson and Watson 1969), an extra term makes its way into the mixture equation of state upon combining the component shape factors to form the mixture shape factor in the residual Helmholz free energy and upon differentiation with respect to volume. The extra term was discarded just like in this work. The Peng-Robinson equation (1976) is in wide use and is generally comparable with the Soave (1972) equation in the description of phase equilibria of nonpolar fluids (Han et al., 1988). However, the Peng-Robinson equation does not fit the vapor pressure of polar liquids. Hence, it is not investigated in this work.
I
(-)
Q-d
Q+d
(15)
where Q=u
+ -21( b + C )
(16)
(9)
The fugactiy coefficient obtained from eq 8, 9, and 1 is In
U
The mixture parameter a in eq 15 is combined according to the LC mixing rules of eq 9 and 1. This is the key step that brings LC into the equation of state. Upon differentiation with respect to the mole number, the fugacity coefficient is obtained: u bi a(bi + ci) In q$ = In -+ U - b U- b 2RT(Q2-d2)
abi bi +-lnz+ U- b b(u + b ) R T
= In v -b
+
where
(
71 1 -
a [ci(3b+ c ) + bi(3c + b ) ] [In 8RTd3 b i ) J zCX kjxk;ak;Qkj
(11)
where d ( n a ) / d n iis given by eq 11.
(E) +
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1253 Cubic Chain-of-Rotators Equation of State The Cubic Chain-of-Rotators (CCOR) equation is R T ( l + 0.77b/u) 0.055RTb/u a P= u - 0.42b CR u - 0.42b u(u + c ) bd (19) V(U + C ) ( U - 0.42b) +
As developed by Kim et al. (19861, it is completely generalized for nonpolar fluids. Guo et al. (1985) determined the specific coefficients constants AI and A2 required for calculating a of a polar fluid. In this work, we use the generalized formulas for the nonpolar components and Guo's A , and A2 for polar components. The residual Helmholtz energy obtained from the CCOR equation is
+
RT(1.19 0 . 0 5 5 ~ ~ ) A, = 0.42 In u - i . 4 2 b ) [Ebd c c(c + 0.42b) d
(
+
The mixture parameter a in eq 20 is combined according to the LC mixing rules of eq 9 and 1. Upon differentiating eq 20 with respect to mole number, we obtain In
@i
=
+ 0.055cR(2bi' - b )
1.19
+
u - 0.42b 1.19 0.055(2(ciR)'- cR) 0.42 In ( u - i . 4 2 b ) c d(na)/dn,- a(2ci' - 2c,)
+
(A
c2RT a(2c;' - c ) cRT(i c)
+
2ci' - c
+
r 2db;' + 2bd;' -c
+
-
- bd
+ 0.426
+ 0.42(2bi' - b )
-
0
0.2
0.6
0.4
0.8
1.0
X1
Figure 2. Vapor-liquid equilibrium of 1-propanol+ water mixtures at 1 atm. (0) Data of Doroshevsky and Polansky (1910). (-) LC mixing rules; (- - -) quadratic mixing rules.
u - 0.42b -
stants are reported for each mixture. Han et al. (1988) showed that the calculated K values deviations, rather than dew-point or bubble-point deviations, give a fundamental bd(2ci' - C ) measure of the quality of an equation of state. cRT In ( u+c c(c I 0.42b)RT ) ] (a) Low-Pressure Mixtures. Table I reports the calculated VLE results of the Soave EOS for 18 binary d(2bi' - b ) polar mixtures at about ambient pressures. Results of the In - (c 0.42b)RT P-T and CCOR EOSs for the same systems are listed in Table I. The calculations for each binary system are given u - 0.42b - ln z (21) in two lines: the first l i e shows the results of conventional u )+u-0.42b quadratic mixing rules; the second line shows local comwhere 0' = xjxjOji with j = b, c , d , or cR. In eq 21, the position mixing rules. derivative d(na)/dniis given by eq 11, just as in eq 10 and Systems with like components, such as methanol + 18. ethanol and methanol + 1-propanol, are well represented by either quadratic or local composition mixing rules. Results and Discussion However, systems containing nonpolar + polar species and Vapor-liquid equilibrium (VLE) has been calculated those exhibiting hydrogen bonding, like water alcohol with local composition embedded Soave, P-T, and CCOR and water + acetone, are highly nonideal due to a large difference in the molecular interactions, and their equiequations of state for polar + polar, nonpolar + polar, and librium behavior is not well described by the quadratic water-containing mixtures over a wide range of state mixing rules. Substantial improvement is obtained for conditions. For comparison, calculation with the same these systems by employing density-dependent LC mixing equations of state is performed, using quadratic mixing rules. While one additional interaction constant, k,,, is rules. For each binary mixture, interaction constants are introduced in LC mixing rules, its value is found to be the determined to minimize the sum of squares of the relative same as k,,, in most instances, including many polar + deviations of K values: (Kexpt,i - Kcd,i)/Kexpt,i]2. The polar mixtures. The equality of k,,, to k,,, is indicated by interaction parameters so obtained are then used in the calculation of K values and for comparison with experithe ++ sign in Table I. No results are reported for the CCOR EOS for three mental data. The percentage average absolute deviation mixtures that contain 2-propanol or 2-butanol for the (AAD%) of calculated K values and the interaction conbd
(c
+ 0.42b)2
][&T'"
+
[ (&)
+
u
+
k] +
[A1.(
]
+
xi[
1254 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
G
32
+
b?
n
3 3
gF:
I 4
++ ++ ++
? ? , + + + ++ +
++ ++
++ ++ ++ +
' V S + ++ ++ ++ ++
? : ?2 ? 8 ? 8 m
ci?2"2
t-
+
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1255
"0
0.2
0.4
0.8
0.6
10
XI
Figure 3. Vapor-liquid equilibrium of methanol + benzene mixtures at 100 "C. (0) Data of Butcher and Medani (1968). (-) LC mixing rules; (- - -) quadratic mixing rules.
reason that the CCOR EOS constants are not available for these substances. Because of the same reason, results are not reported for the P-T EOS for seven mixtures. Figure 2 shows comparison of the equation of state calculations with experimental data on 1-propanol+ water mixtures at the isobar conditions of 1 atm. Significant
improvement of the representation is obtained with local composition over quadratic mixing rules for all three equations. Similar results are illustrated in Figure 3 for methanol + benzene mixtures at 100 OC. (b) High-pressure Mixtures. Application of an equation of state to polar mixtures at higher pressures is of particular interest because the activity coefficient method becomes unsuitable. Table I1 presents the calculated VLE for 16 high-pressure polar mixtures by the Soave, P-T, and CCOR EOSs. Binary mixtures containing nonpolar, quadrupolar, and polar substances are selected and studied here. Both quadratic and LC mixing rules quite well represent the experimental data of polar systems of like components, such as methanol + ethanol, isobutyl alcohol + 1-butanol systems. Nonpolar or quadrupolar + polar mixtures in general are better described by LC mixing than by quadratic mixing rules. Figure 4 shows K values of carbon dioxide + acetone mixtures calculated with the three equations of state. The LC mixing rules represent the observed VLE behavior better than the quadratic mixing rules. ( c ) High-pressure Aqueous Mixtures. Because of their special importance in chemical processes and common occurrence, aqueous mixture systems have been much studied. The phase equilibria of these systems are difficult to predict at high temperatures and pressures. Here, we examine vapor-liquid equilibrium of 16 water-containing mixtures and report the calculated results in Table 111. Mixtures of water + a strongly polar component are reasonably described by quadratic mixing rules. LC mixing rules improve the performance. Figure 5 shows comparison of the equation of state calculations with experimentaldata on ethanol + water mixtures. With LC mixing rules, all three EOSs give a good account of the VLE of this mixture using constant interaction coefficients that are temperature independent. Mixtures of water a light gas are poorly described by quadratic mixing rules. With LC mixing rules, all three equations give much improved results of the K values of water, except for water in propane mixtures for which the Soave and P-T EOSs remain hardly changed. The predictions of K values of the light gases CO and COz remain poor for all three EOSs. For Hz and N2, the CCOR EOS with LC mixing rules gives adequate results. Representation of mixtures of water with the light gases
+
r 10':
co2
10':
1-
x
1 -
13
-
COI
x
298 313
-
x
--__--e
298
298 3'3
298
-
298 , . 1 1 . . . 1
1256 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 Ethand
423
K
173 523
'L:: 523
573
573
Water
01
P
Ethand
I
I
.
I
I
I
603
CH,
623
Y
I
look 5
423K
473
523
573
Water
10 '
I
473
1
1 o3
5
Figure 6. K values of methane + water mixtures. (0) Data of Sultanov et al. (1971, 1972). (-) Soave equation with LC mixing rules; (- - -1 Soave equation with quadratic mixing rules. Temperature-dependent k,,, of eq 22 is used.
: \\I 423 K
,
1o2
5
P, bar
(b) P T
Y
L
523
1
311 K
573
Water (c) CCOR
0.11 1
I
I
5
I
I
,,I
100
I
-_i
3
p. bar Data of Figure 5. K values of ethanol + water mixtures. (0) Barr-David and Dodge (1959). (-1 LC mixing rules. lo
and with the light hydrocarbons is improved with a temperature-dependent k,,, as follows: k,,, = ka0 + kalT
(22)
Table IV presents values of k: and k,l and results on VLE calculations by Soave and P-T equations. Quadratic mixing rules with the temperature-dependent k,,, of eq 22 can well represent K values of the light gas component while leaving KWM poorly fitted. Thus, the Soave equation gives an overall 4.4 AAD% in K , of the light gas component, but 26.3 AAD% of Kwebr. The overall AAD% of K,,, is dramatically reduced to 6.2% upon employing LC mixing with eq 22 in the Soave equation. Similar large beneficial results are obtained with the P-T equation. Figure 6 shows experimental data on mixtures of methane + water over a wide range of temperatures and pressures. The Soave equation with quadratic mixing rules gives reasonable results for methane, but the results for water widely depart from the experiment. With local composition mixing rules, both components are adequately described. Temperature-dependent k,,, of eq 22 is used in this calculation. The P-T equation gives similar results.
102
1
1
\ I
I
,,/I
I
10'
1
I
,
,
1
102 P. bar
Figure 7. K values of hydrogen in mixture with water. (0,A, 0 ) Data of Gillespie and Wilson (1980). (---I Soave equation with quadratic mixing rules; (- - -) Soave equation with quadratic mixing rules; (-1 Soave equation with LC mixing rules. Temperature-dependent k,,, of eq 22 is used in the last two cases.
Figure 7 shows the K values of hydrogen in water, and Figure 8 shows water in this mixture. Successive improvement of the Soave equation is obtained by making k,,, of the classical mixing rules temperature dependent as in eq 22 and then by using local compositions. The K values of hydrogen are brought in line with the observed values, but that of water is still not quite in accord with data at the higher pressures and low temperatures in the case of classical mixing rules. With interaction coefficients determined a t each temperature, the CCOR equation with either quadratic or LC
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1257
L1
c
c a (I:
e
c v
E
.-c 2
; U
- e m
r - d
C L;
c
++ ++ ++
5
0
0
+ q + $ ++ q "
a
7e 8
s
(I:
>
s
: v
c I*
5 a
c
ia
I $
I
5
Q,
CD
111
E
4 s b
ii e
2 M F;
c
I
Y 01
c
O
E
In In In
m d d d
m
.L
W
w
I
c
c
c
cEi
9
Y
1
s
J?
c
0
a
2
3
+ 9 f
0,
E
2
a E
$
1258 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
+
2
oooom oN
rio;Wdcdddi N Q,
L? a
ij
:
+ ++
+
+ + + +
I
l
l
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1259 Table IV. VLE Calculations of High-pressure Aqueous Mixtures by Soave and P-T Equations of State with Temperature-Dependent k... Soave P-T AAD 70 system (1) + (2) HzS + water hydrogen
+ water
+ water COz + water nitrogen + water methane + water ethane + water CO
propane
+ water
grand av
k,O -0.202 0.016 -3.737 -1.387 -1.523 -1.097 -0.356 -0.212 -1.686 -1.158 -0.766 -0.397 -0.443 -0.393 -0.639 -1.402
ha' 0.611 0.761 7.291 5.881 2.402 4.331 0.818 1.383 3.073 5.376 1.812 2.198 1.121 1.280 1.501 3.195
kt12
0.551 -0.286 0.305 0.451 0.354 0.269 0.274 0.817
KI
Ka
k,O
k,'
7.1 7.5 3.2 2.9 3.0 2.5 5.0 4.0 3.0 1.5 4.9 4.8 6.1 7.0 2.5 2.1
19.3 5.2 22.5 2.3 26.0 2.7 36.2 3.1 38.3 1.9 28.7 3.1 11.6 7.3 27.5 24.1
-0.199 0.066
0.566 0.573
-1.441 -0.898 -0.319 -0.128 -1.683 -0.970 -0.682 -0.234 0.213 0.306 -0.591 -1.405
2.140 3.981 0.717 1.111 2.939 4.875 1.516 1.717 0.025 0.101 1.363 3.185
4.4 4.0
26.3 6.2
k*,2
0.545
0.383 0.459 0.408 0.283 0.265 0.824
AAD%
Kl
KZ
6.7 8.2
21.7 5.0
4.1 3.2 5.1 4.2 7.7 3.2 4.3 4.1 10.0 7.4 2.0 2.9
31.9 3.2 40.9 5.0 45.6 2.5 35.5 3.2 12.3 6.5 27.9 25.9
5.7 4.7
30.8 7.3
+
scribed. Nonpolar polar mixtures, which are otherwise not well described, are definitely improved by the LC calculations. For mixtures of water + a nonpolar component, which are extreme examples of this category, further improvement is brought about by introducing a temperature dependence to the interaction coefficients.
Acknowledgment This work has been supported by the National Science Foundation through award CBT-8516449.
Nomenclature
t
A = Helmholtz energy a, b = parameters in eq 6, 13, and 18 c = parameter in eq 13 and 18 cR = parameter in the CCOR equation d = parameter in eq 18; terms in eq 14 K = vaporization equilibrium ratio k = Boltzmann constant k,, k,, k , = interaction coefficients N = Avogadro's number n = number of moles p = pressure, bar
Figure 8. K values of water in mixture with hydrogen. Legends are the same as in Figure 7.
mixing rules achieves equally good data correlation for these mixtures. These coefficients, however, are not linearly dependent on temperature as in the Soave equation.
Concluding Remarks The local composition model of Lee and Chao (1986a,b) obtained from computer simulation is generally useful for the description of phase equilibrium behavior of polar mixtures when embedded in an equation of state that fits the vapor pressure of the pure components. The Soave, Patel-Teja, and the CCOR equations of state are embedded with the LC model and tested with data on 50 binary mixtures including polar + polar, nonpolar + polar, and water-containing systems. The comparison with data shows that VLE of polar + polar mixtures are well de-
8 = terms in eq 14 R = gas constant T = temperature, K u = molar volume, cm3/mol z = mole fraction z = compressibility factor Greek Letters a = terms in eq 2 e = interaction energy 17 = term in eq 10 y = terms in eq 1 0 = terms in eq 10 q5 = fugacity coefficient p = molar density p* = reduced density, Npa3 u = hard-core diameter Subscripts c = critical property i, j , k = components i, j , and k ji = local property of j about central i
1260 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
m = mixture x = hypothetical pure fluid x
Literature Cited Barr-David, F.; Dodge, B. F. Vapor-Liquid Equilibrium a t High Pressure. The Systems Ethanol-Water and 2-Propanol-Water. J. Chem. Eng. Data 1959,4, 107. Butcher, K. L.; Medani, M. S. Thermodynamic Properties of Methanol-Benzene Mixtures a t Elevated Temperature. J. Appl. Chem. (London) 1968,18, 100. Butcher, K. L.; Robinson, W. I. Apparatus for Determining High Pressure Liquid-Vapor Equilibrium Data. I. The MethanolEthanol System. J. Appl. Chem. (London) 1966,16,289. Carey, J. S. Distillation. In Chemical Engineers’ Handbook, 3rd ed.; Perry, J. H., Ed.; McGraw-Hill: New York, 1950. Carey, J. S.; Lewis, W. K. Studies in Distillation Liquid-Vapor Equilibria of Ethyl Alcohol-Water Mixtures. Ind. Eng. Chem. 1932, 24, 882. Clifford, I. L.; Hunter, E. The System Ammonia-Water a t Temperatures up to 150 C and a t Pressures up to Twenty Atmospheres. J. Phys. Chem. 1933, 37, 101. Danneil, A.; Todheide, K.; Franck, E. U. Vaporization Equilibria and Critical Curves in the Systems Ethanol/Water and n-Butanol/ Water at High Pressures. Chem.-Zng.-Tech. 1967, 39, 816. Doroshevsky, A.; Polansky, E. The Vapor Pressures of Mixtures of Alcohols with Water. 2. Phys. Chem. 1910, 73, 192. Georgeton, G. K.; Smith, R. L.; Teja, A. S. Application of Cubic Equations of State to Polar Fluids and Fluid Mixtures. ACS Symp. Ser. 1986, 300, 434. Gierycz, P.; Nakanishi, K. Local Composition in Binary Mixtures of Lennard-Jones Fluids with Differing Sizes of Components. Fluid Phase Equilib. 1984, 16, 255, Gillespie, P. C.; Wilson, G. M. Vapor-Liquid Equilibrium Data on Water-Substitute Gas Components: HZ-HzO, CO-Hz0, H,-COH 2 0 , and H2S-H20. Research Report, RR-41, GPA, Tulsa, OK, April 1980. Gillespie, P. C.; Wilson, G. M. Vapor-Liquid and Liquid-Liquid Equilibrium: Water-Methane, Water-Carbon Dioxide, WaterHydrogen Sulfide, Water-n-Propane, Water-Methane-nPropane. Research Report, RR-48, GPA, Tulsa, OK, April 1982. Gomez-Nieto, M. A. High-pressure Vapor-Liquid Equilibria: The Propane-Ethanol-Acetone System. Ph.D. Dissertation, Northwestern University, Evanston, IL, 1977. Griswold, J.; Wong, S. Y. Phase Equilibria with Acetone-Methanol-Water System from 100 C into the Critical Region. Chem. Eng. Prog., Symp. Ser. 1952, 48(3), 18. Guo, T. M.; Kim, H.; Lin, H. M.; Chao, K. C. Cubic Chain-of-Rotators Equation of State. 2. Polar Substances. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 764. Hall, D. J.; Mash, C. J.; Pemberton, R. C. Vapor-Liquid Equilibrium for the Systems Water + Methanol, Water + Ethanol, Methanol + Ethanol and Water + Methanol + Ethanol. NPL Rep. Chem. 1979, 95, Jan. Han, S. J.; Lin, H. M.; Chao, K. C. Vapor-Liquid Equilibrium of Molecular Fluid Mixtures by-~Eauation of State. Chem. Eng. Sci. 1988, in press. Hellwig, L. R.; van Winkle, M. Vapor-Liquid Equilibria for Ethyl Alcohol Binary System. Ind. Eng. Chem. 1953.45, 624. Hoheisel, C.; Kdhlei, F. Local Composition in Liquid Mixtures. Fluid Phase Equilib. 1984, 16, 13. Hu, Y.; Ludecke, D.; Prausnitz, J. M. Molecular Thermodynamics of Fluid Mixtures Containing Molecules Differing in Size and Potential Energy. Fluid Phase Equilib. 1984, 17, 217. Huron, M. J.; Vidal, J. New Mixing Rules in Simple Equations of State for Representing Vapor-Liquid Equilibriums of Strongly non-Ideal Mixtures. Fluid Phase Equilib. 1979, 3, 255. Karr, A. E.; Scheibel, E. G.; Bowes, W. M.; Othmer, D. F. Composition of Vapors from Boiling Solution, Systems Containing Acetone, Chloroform, and Methyl Isobutyl Ketone. Znd. Eng. Chem. 1951, 43, 961. Katayama, T.; Ohgaki, G.; Maekawa, G.; Goto, M.; Nagano, T. Isothermal Vapor-Liquid Equilibria of Acetone
