Ind. Eng. Chem. Process Des. Dev. 1986, 25,477-481
477
Methanol Synthesis Reactions: Calculations of Equilibrium Conversions Using Equations of State Te Chang,+Ronald W. Rousseau," and Peter K. Kllpatrlck Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27695-7905
Potential errors associated with the estimation of vapor-phase fugacity coefficients are illustrated for the constituents in methanol synthesis reactions. Corrections to the equilibrium constants of reactions producing methanol from CO and from COP were estimated (a) from pure-component fugacity coefficients determined from either a generalized chart or from the Soave-Redlich-Kwong (SRK) and Peng-Robinson (PR) equations of state and (b) from fugacity coefficients determined from SRK and PR equations of state for mixtures of species in the methanol reactor. Comparisons of the estimation procedures were made by examining conversions of CO and CO, and vol % methanol in the effluent from the synthesis reactor. Significant differences among the methods used to calculate the conversion of COPand production of methanol were noted. These differences were most significant at conditions often found in low-pressure methanol synthesis reactors.
Conventional methanol production uses a feed stock of reformed methane that contains hydrogen, carbon monoxide, and carbon dioxide in a ratio of NHz/(2Nco + 3Nc0z) close to the stoichiometric ratio of unity. Both the reversible methanol synthesis reaction (shown as eq 1)and the water-gas shift reaction (shown as eq 2) occur in the vapor phase and proceed to appreciable extents. The reactions are given by the stoichiometric relationships CO + 2Hz CH,OH (1) COZ + Hz
-
CO
+ HzO
(2)
Conversions of H2, CO, and C 0 2 and the methanol production can be estimated by solving the two reaction equilibrium expressions simultaneously. These expressions equate the products of vapor-phase fugacities raised to powers corresponding to the stoichiometric coefficients with the equilibrium constants for reactions 1 and 2, Kal and Ka21
(3)
Ka2 =
[
NCdvH~O NCOzNHz
]K4z
(4)
where Ni is the number of moles of component i in the mixture, NT is the total number of moles of the mixture, and K,, and K$z are correction factors defined in terms of the vapor-phase fugacity coefficients of i, @i: (5)
These correction factors account for the deviation of the system from ideal gas behavior, and Smith and Missen (1982) discuss the various ways by which they can be est Present address: ARC0 Chemical Co., Newtown Square, PA 19703. * To whom correspondence should be addressed.
timated. If the gas mixture is assumed to behave ideally, then the fugacity coefficients are all unity, but it is clear that such an approximation is incorrect in systems that are at elevated pressure and/or include polar components. A second commonly used approximation is to assume applicability of the Lewis fugacity rule, which equates & to (4Jpure. The pure-component fugacity coefficient can then be evaluated from generalized compressibility charts or from an equation of state for the pure compounds. A more rigorous approach is to use a thermodynamic model for the gas mixture that accounts for the compositional dependence of &. The method chosen for estimating fugacity coefficients will depend on the desired simplicity of the calculations and the potential for error associated with using either the ideal gas mixture or Lewis fugacity rule approximation. Vonka and Holub (1975) used the Redlich-Kwong equation of state to examine the effect of these approximations on calculated equilibrium compositions for the water-gas shift reaction, the addition of ammonia to ethylene, and the formation of ethanediol and diethyl sulfide from ethanol, hydrogen, and sulfur. Each of the methods for estimating the fugacity coefficients for use in determining the composition of an equilibrium mixture has its advantages, but unless care is exercised, serious errors can result through misapplication of the simplified calculations. At a chosen operating condition, such as a temperature between 200 and 300 "C,and a pressure between 5 and 35 MPa, the vapor-phase fugacity coefficients for pure water and pure methanol cannot be found in a generalized chart (Hougen et al., 1964) or calculated from the equation of state because these two pure components do not exist as stable vapor at these conditions. The estimation of fugacity with either of these procedures gives values for the pure liquids; mistakenly using these values in place of the fugacity coefficients of pure-component vapors can lead to significant errors in the estimation of methanol production. Specifically, the vapor-phase fugacity coefficients should be those in a vapor mixture at the equilibrium composition. Such vapor-phase fugacity coefficients are functions of temperature, pressure, and composition (including inert species in the system), and they can be evaluated only from a thermodynamic description of the compositional dependence of nonidealities. In this paper, equilibrium correction factors for the methanol synthesis and water-gas shift reactions are cal-
0196-4305/86/1125-0477$01.50/00 1986 American Chemical Society
478
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2. 1986 c
.'.... ...........,. ,,, ,
l
0
0.0
10.0
20 0
1
0
0
30.0
Pressure (MPa)
0.0
10.0
20.0
30.0
Pressure (MPa)
Figure 1. Fugacity coefficients for methanol: (-) calculated from the P R equation of state for the pure compound; (-) calculated from the P R equation of state for a mixture that is 10 mol % CH30H, 6 mol % CO, 79 mol % H,, 4 mol % CO,, and 1 mol % HzO.
Figure 2. Fugacity Coefficients for water: (-) calculated from the PR equation of state for the pure compound; calculated from the P R equation of state for a mixture that is 10 mol % CH,OH, 6 mol % CO, 79 mol % Hz, 4 mol % COP,and 1 mol % H,O.
culated, and comparisons are made between values of these factors calculated from mixture fugacity coefficients and from pure-component fugacity coefficients. The SoaveRedlich-Kwong, referred to as SRK (Soave, 1972; Graboski and Daubert, 1978), and Peng-Robinson, referred to as PR (1976), equations have been used in both mixture and pure-component cases to evaluate vapor-phase fugacity coefficients in the reaction equilibrium calculations. The mixing rules prescribed by Soave (1972) and by Peng and Ftobinson (1976) for the equation of state parameters a and b have been used in this paper. Expressions for the fugacity coefficient as a function of temperature, pressure, and composition can be found in the two references. Results of these calculations are compared with each other and with the results calculated by Wade et al. (1981). Conditions used in the present calculations are typical of those found in methanol synthesis reactors. Moreover, the approach to be described has been used successfully in the analysis of a commercial methanol synthesis plant, but the details of the facility and comparisons with data from the plant are proprietary.
ficients of the pure liquids. These liquid-phase fugacity coefficients correspond to those given in the generalized fugacity coefficient chart at these conditions based on the theory of corresponding states (Hougen et al., 1964). However, they are often mistakenly read as vapor-phase fugacity coefficients. The point of discontinuity in slope, where the plot of vapor-phase fugacity coefficients intersects the plot of liquid-phase fugacity coefficients, represents the saturation conditions for the pure component. The extensions of both liquid and vapor segments of the curve through the discontinuity represent fugacity coefficients of metastable superheated liquid and subcooled vapor. The dotted curves in Figures 1 and 2 represent vapor-phase fugacity coefficients of methanol and water, respectively, in a mixture that is 10 mol % CH,OH, 6 mol % CO, 79 mol % H2, 4 mol % COz, and 1 mol % H20, a composition in the range of that found in methanol synthesis reactions. Methanol and water in a vapor-phase mixture rich in H2, COz, and CO behave more like ideal gases than they do in the pure vapor states. As a result, there are significant differences between correction factors for reaction equilibrium constants calculated from pure-component fugacity coefficients and those based on fugacity coefficients calculated from mixture properties. These differences become more significant as the concentrations of methanol and water increase. Furthermore, when these concentrations become sufficiently high, at a fixed temperature and pressure, the mixture reaches a dew point and condensation of the vapor begins. This vapor is unstable globally at a pressure higher than the dew point, and calculation of vapor-phase fugacity coefficients of methanol and water in that mixture will give a fugacity coefficient for either a supersaturated, metastable vaporphase or a liquid-phase solution. This can be seen easily for a pure compound. As a result, care must be used in identifying the state of the component for which the fugacity coefficient is calculated.
Vapor-Phase Fugacity Coefficients Fugacity coefficients are a measure of the deviation from ideal gas behavior, and these quantities can be estimated from an exact thermodynamic relationship using an equation of state. Mixtures found in methanol synthesis reactions include CO, Ha, COS, H20, and CH30H. The components CO, H2,and COz exhibit nearly ideal behavior because the reaction temperature, 200-400 "C, is much higher than the respective pure-component critical temperatures. As a consequence, there is little variation in the values of the vapor-phase fugacity coefficients of these components with the method of estimation. Methanol and water vapors, on the other hand, are highly nonideal at the indicated reaction temperatures, and the method used to estimate the fugacity coefficients of these compounds is important. As described earlier, fugacity coefficients of methanol and water in a reaction mixture will differ from the pure-component fugacity coefficients; the accuracy in predictions of mixture fugacity coefficients will vary from one equation of state to another, especially since the system contains polar compounds. Tarakad et al. (1979) give a good comparison of the use of equations of state in predicting vapor-phase fugacity coefficients. Figures 1 and 2 show fugacity coefficients of methanol and water, respectively, that were calculated as functions of temperature and pressure from the PR equation of state. The solid curves represent fugacity coefficients of the pure component; the curved segments below and to the right of the discontinuity in the slope represent fugacity coef-
(-a)
Correction Factors for Methanol Synthesis Reactions The correction factor for a gas-phase reaction equilibrium is a ratio of vapor-phase hgacity coefficients of products to that of reactants, each of which is raised to a power corresponding to the respective stoichiometric coefficients. Figures 3 and 4 give values of K,l and K$z that have been calculated by using vapor-phase fugacity coefficients determined from the PR equation for mixtures as well as for pure compounds. These two figures are quite similar to Figures 1 and 2, respectively, because the dominant contributions to these correction factors are the fu-
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986 479
Bissett (1977) and by Cherednichenko (1953), respectively:
K,, = 9.740 X
exp 21.225
7.492 In T + 4.076
X
9143.6 +T
10-3T - 7.161 X lO-*T2
13.148 - 5639*5- 1.077 In T - 5.44
T
+
1
(7)
X
-
1 0 - 4 ~ 1.125 x 10-7~2+ 4g170] (8) 0.0
20.0
10.0
T2
30.0
-
Pressure (MPa) Figure 3. Kmlfor the reaction CO + 2Hz CH30H: (-) calculated by using the P R equation of state for the pure components; (-.) calculated from the P R equation of state for a mixture that is 10 mol % CH30H, 6 mol % GO, 79 mol % H,, 4 mol % CO,, and 1 mol % H20.
These equations can be used to calculate equilibrium conversion by first defining X as the moles of CH,OH formed and Y as the moles of HzOformed and then writing material balances around the methanol reactor:
NCH~O =H NICH~O +H X
+Y
(10)
NH, = NIH, - 2X - Y
(11)
Nco = NIco - X
NH20
\
0
20.0
10.0
-
30.0
Pressure (MPa)
+
= NIH,O +
(13)
NT = NIT - 2X
’
0.0
(9)
+
Figure 4. K,, for the reaction CO, H z CO HzO: (-) calculated by using the P R equation of state for the pure components; (-.) calculated from the P R equation of state for a mixture that is 10 mol % CH,OH, 6 mol % CO, 79 mol % H,,4 mol % COP, and 1 mol % H20.
gacity coefficients of methanol and water. The solid curves are correction factors calculated by using the vapor-phase fugacity coefficients for pure compounds; curves to the right and below the discontinuity in slope were determined by using the liquid-phase fugacity coefficients for methanol and water. The extensions of the vapor and liquid portions of the curves past the discontinuities in slope represent correction factors calculated from fugacity coefficients for metastable methanol and water. The dotted curves represent correction factors calculated from fugacity coefficients obtained from the mixture equation of state. For reaction 2, there are sharp differences between the correction factors calculated from the two procedures when the temperature is between 200 and 300 O C and at a pressure higher than 5 MPa. For reaction 1, these differences are significant at 200 “C and pressures higher than 10 MPa. This is because the pure-component fugacity coefficients for methanol and water are those for liquids. When the correction factors K,, and K,, are expressed as functions of temperature, pressure, and composition, they can be estimated accurately. More importantly, these correction factors lead to a correct determination of the composition of the reaction mixture a t equilibrium. The equations used in these calculations provide the basis for a more complete model of a methanol synthesis process that can be used to optimize reaction conditions.
Equilibrium Calculations for Methanol Synthesis Reactions The equilibrium constants, K,, and Kaz,were determined to be functions of temperature (T in kelvin) by
(14)
where NI, is the initial number of moles of component i, NIT is the number of total moles of the initial mixture, and X and Yare the extents of reaction for reactions 1 and 2, respectively. Substitution of eq 7-14 into eq 3 and 4 yields two equations in the two unknown extents of reaction, X and Y. These equations can be solved numerically, but it has been found advantageous to work with the logarithms of both sides of eq 3 and 4. The resulting equations used in the calculations are
F 2 ( X , Y )= In K,,
- In
(22;)
- In
K,, = 0
(16)
where K,, and K,, can be estimated in two ways: first, through estimations of pure-component fugacity coefficients and application of the Lewis fugacity rule, and second, from an equation of state for the reaction mixture. The former procedure has only temperature and pressure as variables, while the latter adds composition as a variable. Table I summarizes the results of example calculations for a typical feed composition of 15 mol 70CO, 8 mol 7’0 CO,,74 mol % HP,and 3 mol 70CH4. Methane is included in the total number of moles of the mixture, NT and NIT, as an inert component. The percent conversion of CO and CO, and the percentage of the exit stream that is methanol are calculated by the equations %
co conversion = X-’
%
CO, conversion
x 100%
(17)
-x 100%
(18)
vol % of exit CH,OH = - X 100%
(19)
NICO
=
NICO, X
NT
The results calculated by Wade et al. (1981) are used as a base case for each temperature and pressure examined and are indicated in Table I as “chart” values. Table I also
480
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
Table I. Eauilibrium Conversions and Exit CH-OH Concentration" CO conversion ( % ) C 0 2 conversion ( % ) source 5MPa 10MPa 30MPa 5MPa 10MPa 30MPa
exit CH,OH (vol %) 5MPa 10MPa 30MPa
200 "C chartb SRK(P)' PR(P)d SRK(M)e PR(M)f
95.6 96.3 96.3 96.0 96.0
99.0 99.0 99.0 99.0 99.0
99.9 99.9 99.9
chartb SRK(P)c PR(P)d SRK(M)' PR(M)f
72.1 73.0 73.0 71.2 71.2
90.9 90.6 91.4 90.6 90.6
98.9 99.0 98.9 99.5 99.4
chartb SRK(P)' PR(P)d SRK(MIP PR(M)f
25.7 25.4 25.4 23.5 23.6
60.6 60.7 60.6 57.3 57.3
chartb SRK(P)' PR(P)d SRK (M)e PR(M)f
-2.3 -2.3 -2.4 -2.3 -2.5
chartb SRK(P)' PR(P)d SRK(M)' PR(M)f
-12.8 -12.8 -12.9 -12.3 -12.5
44.1 28.W 29.09 25.98 26.54
82.5 83.0 83.5 58.54 59.79
99.0 99.5 99.5
27.8 25.1 25.2 24.6 24.7
37.6 37.7 37.8 32.19 32.39
42.3 42.4 42.4
250 "C 18.0 46.2 14.49 45.1 14.78 37.78 12.88 29.39 13.28 30.P
91.0 92.4 92.9 89.1 89.7
16.2 16.0 16.0 15.3 15.3
26.5 26.2 25.1 23.49 23.69
39.7 40.0 40.2 39.4 39.6
92.8 92.8 92.6 93.8 93.7
300 "C 14.3 24.6 14.1 22.3 14.4 22.9 12.8g 18.79 13.2 19.59
71.1 71.0 72.4 58.19 60.49
5.6 5.5 5.5 5.09 5.09
14.2 13.9 14.0 12.69 12.89
32.2 32.2 32.4 29.9 30.3
16.9 16.7 16.6 14.79 14.68
73.0 71.9 71.4 70.7 70.4
19.8 19.8 20.1 18.5 18.9
23.6 23.1 23.7 20.29 21.09
52.1 50.0 51.8 36.79 39.39
1.3 1.3 1.3 1.2 1.2
4.8 4.8
21.7 21.0 21.1 18.69
-7.2 -7.3 -7.6 -6.9 -7.3
38.1 34.19 33.49 30.54 30.29
27.9 27.7 28.1 26.5 26.9
30.1 29.3 30.0 26.79 27.6
44.2 41.0 42.9 31.Y 34.09
0.3 0.3 0.3 0.3 0.3
350 "C 4.8
4.19 4.29
1a.v
400 "C 1.4 1.3 1.3 1.19
1.18
11.4 10.1 10.1 8.39 8.58
and 3% CHI. 'Wade et al. (1981) from Hougen et al. (1964). 'Component fugacity nFeed composition: 15% CO, 870 COz, 74% H2, coefficients calculated from the SRK equation of state for pure substance. Component fugacity coefficients calculated from the P R equation of state for pure substance. 'Component fugacity coefficients calculated from the SRK equation of state for the mixture at eauilibrium comDosition. f ComDonent fuaacitv coefficients calculated from the PR equation of state for the mixture at equilibrium composition. 9Diference between v k u e and &art is greater than 10%
lists the results that use correction factors calculated by using the fugacity coefficients from the SRK and PR equations of state for the mixture (designated as SRK(M) and PR(M), respectively) and for the pure components (designated as SRK(P) and PR(P)). In vapor-liquid equilibrium calculations far from critical conditions, Chang (1984) found that the primary effect of the binary SRK and P R interaction parameters was to correct for liquidphase nonidealities. For computational convenience, therefore, the interaction parameters in the SRK and PR equations were assumed to be zero. Examination of Table I shows that essentially identical results are obtained for use of the pure-component fugacity coefficients from the SRK(P), PR(P), and chart, except for C 0 2 conversions a t 200 and 250 "C and 5 MPa. At these latter conditions, calculations based on the chart used liquid-phase fugacity coefficients of water, and the SRK(P) and PR(P) used fugacity coefficients of subcooled water vapor in a metastable state. As a result, the chart gives predictions significantly in error, while the SRK(P) and PR(P) predictions are much closer to the presumed correct results given by the mixture equations of state. An interesting result was obtained at 250 OC and 10 MPa, where C02 conversions calculated from the two pure-component equations of state differed substantially. Apparently, the SRK(P) equation of state evaluated a fugacity coefficient for condensed water at these conditions. At a reaction temperature of 400 "C and pressure of 30 MPa, the conversion of CO predicted by SRK(P) and PR(P) differed by more than 10% from the value predicted from the chart. As the pressure increases, differences of this kind
would be expected to become more severe. The use of the SRK or PR equations for the gas mixtures shows that methanol production and COz conversion are less than would be predicted by use of the pure-component fugacity coefficients calculated from these equations or from the chart. This is caused by increases in the correction factors, K,, and K4z,which reduce the equilibrium concentrations of methanol and water. As can be seen in Figures 3 and 4,the differences in both correction factors as predicted by the two methods can be as large as an order of magnitude. Conversions of CO, in general, are not affected significantly because CO is a reactant in the methanol synthesis reaction and a product in the water-gas shift reaction; as a result, nonidealities of methanol and water cancel at most conditions in evaluating CO conversion. COz conversions in the cases of SRK(M) and PR(M) at nearly every temperature are significantly less than those predicted with the use of chart values. The discrepancies result because liquid-phase fugacity coefficients of pure methanol and pure water are given erroneously by the chart at these conditions. Exit CH30H percentages in the cases of SRK(M) and PR(M) at every temperature differ by more than 10% from those given by the chart; significant differences in C 0 2 conversions and methanol productions also exist between SRK(M) and SRK(P) as well as between PR(M) and PR(P) a t these conditions. This, again, can be attributed to the mistaken use of liquid-phase fugacity coefficients for methanol and water at these conditions when using the pure-component equations of state. It is clear from Table I that as total pressure is increased and
Ind. Eng. Chem. Process Des. Dev.
temperature decreased, the production of methanol increases. However, at sufficiently low temperatures or high pressures, supersaturation and subsequent condensation of liquid a t the equilibrium mixture composition is possible. There is thus a limit to maximizing methanol production through the decrease in temperature or increase in pressure. Moreover, the equilibrium conversion cannot be predicted accurately unless the equation of state correctly models the mixture liquid-vapor equilibria. In our computations, we found that the cases of SRK(M) and PR(M) a t 200 "C and 30 MPa did not converge. We attribute this to liquid-phase condensation at these conditions. Conclusions With methanol synthesis reactions, C02 conversion and methanol production are always overestimated by using correction factors calculated from pure-component fugacity coefficients obtained from a generalized chart. Significant errors in the estimation of C 0 2 conversion and CH,OH formation are produced by mistakenly using the liquidphase fugacity coefficients as vapor-phase fugacity coefficients for subcritical compounds. This mistake often has a large effect on calculations involving methanol synthesis reactions at temperatures from 200 to 300 "C and pressures from 5 to 10 MPa. For accurate estimation, these calculations require the use of fugacity coefficients that depend on composition to estimate the nonidealities of the coexisting species in a reaction mixture. This work shows that either the Peng-Robinson or the Soave-Redlich-Kwong
1986,25, 481-486
481
equation of state can be used profitably for this purpose. Nomenclature K,i = equilibrium correction factor for reaction i Kai = equilibrium constant for reaction i Ni= moles of component i at equilibrium NT = total moles of mixture at equilibrium NIL= initial moles of component i in mixture NIT = total moles of initial mixture X = extent of reaction 1 Y = extent of reaction 2 q$ = vapor-phase fugacity coefficient of i Registry No. CH,OH, 67-56-1;CO, 630-08-0;COP,124-38-9. Literature Cited Bissett, L. Chem. Eng. 1977, 84(21), 155. Chang, T. Ph.D. Thesis, North Carolina State University, Raleigh, 1984. Cherednichenko, V. M., Dissertation, Karpova, Physic0 Chemical Institute, Moscow, U.S.S.R., 1953. Graboski, M. S.; Daubert, T. E. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 443. Hougen, 0. A.; Watson, K. M.; Ragatz, R. A. "Chemical Process Principles", 3rd ed.; Wiley: New York, 1964. Peng, D.-Y.; Robinson, D. 8. Ind. Eng. Chem. Fundam. 1976, 15, 59. Smith, W. R.: Missen, R. W. "Chemical Reaction Equilibrium Analysis: Theory and Algorithms"; Wiley: New York, 1982. Soave, G. Chem. Eng. Sci. 1972, 2 7 , 1197. Tarakad, R. R.; Spencer, C. F.;Adler, S. B. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 726. Vonka, P.; Holub, R. Collect. Czech. Chem. Commun. 1975, 4 0 , 931. Wade, L. E.; Gengelbach, R. B.; Taumbley, J. L.; Hallhauer, W. L. "Kirk-0thmer Encyclopedia of Chemical Technology", 3rd ed.; W h y : New York, 1981; Vol. 15, pp 398-415.
Received
f o r review November 26, 1984 Accepted
August 14, 1985
Prediction of Low-Pressure Vapor-Liquid Equilibria of Nan-Hydrocarbon-Containing Systems-ASOG or UNIFAC Parag A. Gupte and Thomas E. Daubert' Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
An extensive comparison of the two group-contribution methods for activity coefficients-UNIFAC and ASOG-is presented. The methods are compared by carrying out bubble point and K-value calculations over a large low-pressure VLE data base. Overall, the two methods yield equivalent results when the methods are compared for various families of systems. However, for certain families, one method may be superior to the other. Prediction results from Raoult's law are also obtained and compared with the more complex models.
Prediction of vapor-liquid equilibria is important for design calculations in the chemical industry. Vapor-liquid equilibria of hydrocarbon-containing systems are commonly predicted by using equations of state. In such cases, the binary interaction parameters kij are either taken as zero or generalized in terms of readily available input parameters like the critical properties. This type of approach yields consistent results for nonpolar systems and for systems where the components are chemically similar. The advantage of this approach is that it can be extended to high pressures. When the system contains non-hydrocarbon compounds and when the components are chemically dissimilar (e.g., hydrocarbon-alcohol systems), the equation of state approach is not applicable. In such cases, the use of zero
* To whom correspondence should be addressed.
interaction constants leads to poor results. Furthermore, interaction constants are not easily generalizable in terms of simple molecular properties. In such cases, the alternative approach is to use GE models for the liquid phase. At low pressures (less than 5 bar), the liquid-phase fugacity is conveniently represented by using activity coefficients. For subcritical components, this relation is f,L
= y.x 1 1 f."L 1
(1)
The activity coefficient yi is related to the excess Gibbs energy of the system by the following well-known relation. n
Ex, In yI
gE = RT
i=l
(2)
Several models for gE are available in the literature. The more popular models are UNIQUAC (Abrams and Prausnitz, 1975), NRTL (Renon and Prausnitz, 1968), and
0196-430518611 125-0481$01.50/0 0 1986 American Chemical Society
