Ind. Eng. Chem. Res. 1998, 37, 1613-1618
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Interaction Parameter Estimation in Cubic Equations of State Using Binary Phase Equilibrium and Critical Point Data† Peter Englezos,*,‡ Geoffrey Bygrave,‡ and Nicolas Kalogerakis§ Department of Chemical Engineering, The University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada, and Department of Environmental Engineering, Technical University of Crete, Chania, 73100 Greece
Two methods for the estimation of the interaction parameters in cubic equations of state by using the entire binary phase equilibrium database and the critical point locus, respectively, are presented. The solution of the optimization problem is accomplished in both methods by a Gauss-Newton-Marquardt minimization algorithm. The methods are computationally efficient and robust because they are based on implicit objective functions and hence avoid phase equilibrium or critical point calculations during the parameter optimization. The use of the entire phase equilibrium database and the critical locus can be a stringent test of the correlational ability of the equation of state. In the illustrative examples, the results were obtained by using the Peng-Robinson and the Trebble-Bishnoi equations of state with quadratic mixing rules and temperature-independent interaction parameters. 1. Introduction Volumetric equations of state (EOSs) are employed for the calculation of fluid phase equilibrium and thermophysical properties required in the design of processes involving nonideal fluid mixtures in the oil and gas and chemical industries. It is well-known that the introduction of empirical parameters in equation of state mixing rules enhances the ability of a given EOS as a tool for process design. In general, the phase equilibrium calculations with an EOS are very sensitive to the values of the binary interaction parameters. Urlic et al. (1991) have taken advantage and chosen the binary interaction parameters as thermodynamic tuning parameters for accurate separation process design and simulation. The number of parameters should be as small as possible. The parameters are usually estimated from the regression of binary vapor-liquid equilibrium (VLE) data either by least squares (LS) or maximum likelihood (ML) estimation methods. Both methods involve the minimization of an objective function which consists of a weighted sum of squares of deviations (residuals). The objective function is a measure of the correlational ability of the EOS. Depending on how the residuals are formulated, we have explicit or implicit estimation methods (Englezos et al., 1990a,b). In explicit formulations, the differences between the measured values and the EOS (model) based predictions constitute the residuals, a weighted sum of which is to be minimized. At each iteration of the minimization algorithm, explicit formulations involve phase equilibrium calculations at each experimental point. Explicit methods often fail to converge at “difficult” points (e.g. at high pressures). As * To whom correspondence should be addressed. E-mail address:
[email protected]. † Presented at the symposium in honor of the 20th anniversary of the Peng-Robinson equation of state at the 47th Canadian Chemical Engineering Conference, Oct 5-8, 1997, Edmonton, Alberta. ‡ The University of British Columbia. § Technical University of Crete.
a consequence, these data points are usually ignored in the regression, with resulting inferior matching ability of the model (Englezos et al., 1990a,b; Michelsen, 1993). On the other hand, implicit estimation has the advantage that one avoids the iterative phase equilibrium calculations and thus has a parameter estimation method which is robust and computationally efficient (Englezos et al., 1990a; Peneloux et al., 1990). A systematic and computationally efficient approach for the regression of binary vapor-liquid equilibrium (VLE) data has been presented by Englezos et al. (1993). Binary VLE data consist of sets of temperature (T), pressure (P), and liquid (x) and vapor phase mole fractions (y) for one of the components. It is known, however, that in five of the six principal types of binary fluid phase equilibrium diagrams, other types of data may also be available for a particular binary (van Konynenburg and Scott, 1980). Thus, the entire database may also contain VL2E, VL1E, VL1L2E, and L1L2E data. In addition to phase equilibrium data, the locus of the critical points for a binary system is often also available. In this paper we present an implicit estimation method for the regression of binary critical point data. Subsequently, an extension of the above systematic approach to utilize the entire phase equilibrium database is presented. Several papers have appeared in the literature which present computational methods for the calculation of binary critical points using an EOS. However, the estimation problem did not receive any particular attention. In these studies one or two binary interaction parameters were used in the EOS. The values of the parameters were obtained by minimizing a summation of squared differences between experimental and calculated critical temperatures and/or pressures (Spear et al., 1969; Hissong and Kay, 1970; Pak and Kay, 1972; Sarashina et al. 1974; Mainwaring et al., 1988). During the minimization the EOS uses the current parameter estimates in order to compute the critical pressure and/ or the critical temperature. However, the initial estimates are often away from the optimum, and as a consequence, such iterative computations are difficult
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1614 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998
to converge and the overall computational requirements are significant. Given the inherent robustness and efficiency of implicit methods, we were motivated to extend our systematic procedure to include critical data. Illustration of the methods is done with the TrebbleBishnoi (Trebble and Bishnoi, 1988) and the PengRobinson (Peng and Robinson, 1976) EOSs with quadratic mixing rules and temperature-independent interaction parameters. It is noted, however, that the methods are not restricted to any particular EOS/mixing rule. 2. Interaction Parameter Estimation Mathematically, a volumetric EOS expresses the relationship among pressure, P, volume, V, temperature, T, and composition z for a fluid mixture. This relationship for a pressure-explicit EOS is of the form
P ) P(V,T; z; u; k)
(1)
where the np-dimensional vector k represents the unknown binary interaction parameters and z is the composition (mole fractions) vector. The vector u is the set of EOS parameters which are assumed to be precisely known, e.g., pure component critical properties. Given an EOS, the objective is to estimate the interaction parameter vector, k, in a statistically correct and computationally efficient manner. An objective function which does not require any phase equilibrium calculations during each minimization step is the basis for a robust and efficient estimation method. The development of implicit objective functions is based on the phase equilibrium criteria (Englezos et al., 1990a,b). 2.1. Use of the Entire Set of Binary Fluid Phase Equilibrium Data. We assume that for a binary system there are available N1 VL1E, N2 VL2E, N3 L1L2E, and N4 VL1L2E data points. The light liquid phase is L1, and L2 is the heavy one. Thus, the total number of available data is M ) N1 + N2 + N3 + N4. Gas-gas equilibrium types of data are not included in the analysis because they are beyond the range of our practical interest. An objective function which would be an appropriate measure of the ability of the EOS to represent all of these types of equilibrium data is the following M
S(k) )
riTRiri ∑ i)1
(2)
where the vectors ri are the residuals and Ri are suitable weighting matrices. They are based on the isofugacity phase equilibrium criterion. When two phase (VL1E, or VL2E or L1L2E) data are considered, the residual vector takes the form
ri )
(
ln f R1 - ln f β1 ln f R2 - ln f β2
)
(3a)
i
where f Rk , f βk are the fugacities of component k (1 or 2) phase R (V or L1) and phase β (L1 or L2), respectively, for the ith experiment. When three phase (VL1L2E) data are used, the residuals become
( )
ln f1V - ln f1L1 ln f2V - ln f2L1 ri ) ln f1V - ln f1L2 ln f2V - ln f2L2
(3b)
The residuals are functions of temperature, pressure, composition, and the interaction parameters. These functions can easily be derived analytically for any equation of state. At equilibrium the value of these residuals should be equal to zero. However, when the measurements of the temperature, pressure, and mole fractions are introduced into these expressions, the resulting values are not zero even if the EOSs were perfect. The reason is the random experimental error associated with each measurement of the state variables. The values of the elements of the weighting matrices Ri depend on the type of estimation method being used. When the residuals in the above equations can be assumed to be independent, normally distributed with zero mean and the same constant variance, least squares (LS) estimation should be performed. In this case, the weighting matrices in eq 1 are replaced by the identity matrix. Maximum likelihood (ML) estimation should be applied when the EOS is capable of calculating the correct phase behavior of the system within the experimental error. Its application requires the knowledge of the measurement errors for each variable (i.e., temperature, pressure, mole fractions) at each experimental data point. Each of the weighting matrices is replaced by the inverse of the covariance matrix of the residuals. The elements of the covariance matrices are computed by a first order variance approximation of the residuals and from the variances of the errors in the measurements of the T, P, and composition (z) in the coexisting phases. The optimal parameter values are found among the stationary points of S(k) by a Gauss-Newton-Marquardt iterative algorithm which is presented later. The estimation problem is of considerable magnitude, especially for a multiparameter EOS, when the entire fluid phase equilibrium database is used. As an example, the Trebble-Bishnoi EOS can utilize up to four interaction parameters (ka, kb, kc, kd). Usually, there is no prior information about the ability of the EOS to represent the phase behavior of a particular system. It is possible that convergence of the minimization of the objective function given by eq 2 will be difficult to achieve due to structural inadequacies of the EOS. Based on the systematic approach for the treatment of VLE data of Englezos et al. (1993), the following stepwise methodology is advocated for the treatment of the entire phase database: 1. Consider each type of data separately, and estimate the best set of interaction parameters by least squares. 2. If the estimated best set of interaction parameters is found to be different for each type of data, then it is rather meaningless to correlate the entire database simultaneously. One may proceed, however, to find the parameter set that correlates the maximum number of data types. 3. If the estimated best set of interaction parameters is found to be the same for each type of data, then use the entire database and perform least squares estimation.
Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1615
4. Compute the phase behavior of the system and compare with the data. If the fit is excellent, then proceed to the maximum likelihood parameter estimation. 5. If the fit is simply acceptable, then the LS estimates suffice. 2.2. Use of the Set of Binary Critical Point Data. It is assumed that there are available NCP experimental binary critical point data. These data include values of the pressure, Pc, the temperature, Tc, and the mole fraction, xc, of one of the components at each of the critical points for the binary mixture. The vector k of interaction parameters is determined by fitting the EOS to the critical data. In explicit formulations the interaction parameters are obtained by the minimization of the following least squares objective function NCP
S(k) ) λ1
where r ) (r1, r2)T. The optimal parameter values are found by using the Gauss-Newton-Marquardt minimization algorithm. For LS estimation, Ri ) I, but for ML estimation, the weighting matrices are computed by a first order variance approximation. 3. Gauss-Newton-Marquardt Minimization Algorithm Assuming that an estimate of the interaction parameter vector, k(l) is available at the lth iteration, then linearization of the residual vector ri yields the following equation after discarding the higher than first order terms
( )
ri(k(l+1)) ) ri(k(l)) +
NCP
exp calc 2 exp calc 2 (Pc,i - Pc,i ) + λ2 ∑ (Tc,i - Tc,i ) ∑ i)1 i)1
(4)
The values of λ1 and λ2 are usually taken equal to 1. One of them may be chosen to be zero. Hissong and Kay (1970) assigned a value of 4 for the ratio of λ2 over λ1. The critical temperature and/or the critical pressure are calculated by solving the equations which define the critical point given below (Modell and Reid, 1983)
( ) ∂ ln f1 ∂x1
)0
(5)
T,P
( ) ∂x12
)0
(6)
T,P
Because the current estimates of the interaction parameters are used when the above equations are solved, convergence problems are often encountered when these estimates are far from their optimal values. It is therefore desirable to have, especially for multiparameter equations of state, an efficient and robust estimation procedure. Such a procedure is presented next. At each point on the critical locus eqs 5 and 6 are satisfied when the true values of the binary interaction parameters and the state variables, Tc, Pc, and xc are used. As a result, following an implicit formulation, one may attempt to minimize the following residuals.
( ) ∂ ln f1 ∂x1
) r1
( )
(10)
A∆k(l+1) ) b where
A)
∑i
(11)
( )( ) ∑( ) ∂riT ∂k
Ri
∂riT
b)-
∂k
∂riT
T
∂k
Riri
(12)
(13)
Equation 11 can be solved for ∆k(l+1). However, following Marquardt’s modification this equation is replaced by the following
(A + µ2I)∆k(l+1) ) b
(14)
where µ2 is Marquardt’s directional parameter. Normally, during the parameter search, µ2 is chosen to be larger than the smallest eigenvalue of A, if A is ill-conditioned. When convergence is practically reached, µ2 should be set equal to zero. To determine ∆k(l+1) matrix A should not be ill-conditioned. The problem of ill-conditioning is tackled by performing the orthogonal decomposition of the symmetric matrix A and taking the pseudoinverse of a suitable rank (Lawson and Hanson, 1974). Solution of eq 14 yields the correction vector ∆k(l+1) by the following equation
(7)
∆k(l+1) ) (A + µ2I)-1b
(8)
The values for the interaction parameter vector to be used for the next iteration, k(l+1), can now be obtained by the next equation
(15)
) r2
T,P
Furthermore, to avoid any iterative computations for each critical point, we use the experimental measurements of the state variables instead of their unknown true values. In the above equations r1 and r2 are residuals which can be easily calculated for any equation of state. On the basis of these residuals, the following objective function is formulated NCP
S(k) )
∆k(l+1)
T,P
∂2 ln f1 ∂x12
T
Equation 10 is then substituted into eq 2 or 9, and by taking (∂S/∂k(l+1)) ) 0, we obtain
i
∂2 ln f1
∂riT ∂k
riTRiri ∑ i)1
(9)
k(l+1) ) k(l) + γ∆k(l+1)
(16)
where γ is the relaxation factor or step size (0 < γ e 1). In the case of overstepping, a suitable value for γ is obtained by a monotonicity test or by employing an optimal step size policy (Kalogerakis and Luus, 1983). Overstepping may occur when the parameter values are away from the optimum. Convergence is achieved when
||∆k(l)|| e
(17)
where is a user specified tolerance chosen to be 0.0005.
1616 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 Table 1. Interaction Parameter Estimates for the Hydrogen Sulfide-Water Systema param
std dev
ka ) 0.2209 kd ) -0.5862 ka ) 0.1929 kd ) -0.5455 ka ) 0.2324 kd ) -0.6054 ka ) 0.2340 kd ) -0.6179 ka ) 0.2108 kd ) -0.5633
0.0120 0.0306 0.0149 0.0366 0.0232 0.0581 0.0132 0.0333 0.0003 0.0013
a
database
est method
VL2E
least squares (LS)
L1L2E
LS
VL1L2E
LS
VL2E + L1L2E + VL1L2E VL2E + L1L2E + VL1L2E
LS maximum likelihood
Using the Trebble-Bishnoi EOS.
3.1. Covariance Matrix of the Parameters. The elements of the covariance matrix of the parameter estimates are calculated when the minimization algorithm has converged with a zero value for the Marquardt’s directional parameter. In the case of least squares the elements are given by the following equation
Cov(k*) )
Figure 1. Comparison of the computed (continuous line) vaporliquid equilibrium with data for the system H2S-H2O at 344.26 K. The data are from Selleck et al. (1952).
S(k*) A-1 2(N1 + N2 + N3) + 4N4 + 2Ncp - np (18)
In the case of ML estimation, the covariance matrix is given below
Cov(k*) ) A-1
(19)
In the above equations, k* denotes the estimated optimal values of the parameters. 4. Results and Discussion Data for the hydrogen sulfide-water and the methane-n-hexane binary systems were considered. The first is a type III system in the binary phase diagram classification scheme of van Konynenburg and Scott. Experimental data from Selleck et al. (1952) were used. Recently, Carroll and Mather (1989a,b) presented a new interpretation of these data and also new three phase data. In this work, only those VLE data from Selleck et al. which are consistent with the new data were used. Data for the methan-n-hexane system are available from Poston and McKetta (1966) and Lin et al. (1977). This is a type V system. The same two interaction parameters (ka, kd) were found to be adequate to correlate the VLE, LLE, and VLLE data of the H2S-H2O system. Each data set was used separately to estimate the parameters by implicit (LS). The values of these parameter estimates together with the standard deviations are shown in Table 1. Subsequently, these two interaction parameters were recalculated using the entire database. The algorithm easily converged with a zero value for Marquardt’s directional parameter, and the estimated values are also shown in the table. The phase behavior of the system was then computed using these interaction parameter values, and it was found to agree well with the experimental data. This allows maximum likelihood (ML) estimation to be performed. The ML estimates which are also shown in Table 1 were utilized in the EOS to illustrate the computed phase behavior shown in Figures 1-3. The calculated pressure-composition diagrams corresponding to VLE and LLE data at 344.26 K are shown in Figures 1 and 2. There is excellent
Figure 2. Comparison of computed liquid-liquid equilibrium (continuous line) and data for the system H2S-H2O at 344.26 K. The data are from Selleck et al. (1952).
Figure 3. Comparison of the computed pressure-temperature three phase locus (continuous line) for the system H2S-H2O. The data are from Carroll and Mather (1989a).
agreement between the values computed by the EOS and the experimental data. The Trebble-Bishnoi EOS is also capable of representing well the three phase equilibrium data. This is seen in Figure 3, where the computed three phase pressure-temperature locus and the experimental data are shown. The proposed methodology for the data treatment was followed, and the best parameter estimates for the various types of data are shown in Table 2 for the methane-n-hexane system. As seen, the parameter set (ka, kd) was found to be the best to correlate the VL2E, the L1L2E, and the VL2L1E data and another (ka, kb) for the VL1E data. Phase equilibrium calculations using the best set of interaction parameter estimates for each
Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1617 Table 2. Interaction Parameter Estimates for the Methane-n-Hexane Systema param
std dev
ka ) -0.0793 kd ) 0.0695 ka ) -0.1503 kb ) -0.2385 ka ) -0.0061 kd ) 0.2236 ka ) 0.0113 kd ) 0.3185 ka ) -0.0587 kd ) 0.0632
0.0102 0.0120 0.0277 0.0403 0.0018 0.0116 0.0084 0.0254 0.0075 0.0104
a
database
est method
VL2E
least squares (LS)
VL1E
LS
L1L2E
LS
V L1L2E
LS
VL2E + L1L2E + V L1L2E
LS
Using the Trebble-Bishnoi EOS.
Figure 4. Comparison of the computed vapor-liquid equilibrium (VL2E) for the system CH4-nC6H14 at 273.16 K using the ka ) -0.0587 and kb ) +0.0632 for the interaction parameters.
type of data revealed that the EOS is able to represent well only the VL2E data. The EOS was not found to be able to represent the L2L1E and the VL2L1E data. Although the EOS is capable of representing only part of the fluid phase equilibrium diagram, a single set of values for the best set of interaction parameters was found by using all but the VL1E data. The values for this set (ka, kd) were calculated by least squares, and they are also given in Table 2. Using these interaction parameter values, phase equilibrium computations were performed. It was found that the EOS is able to represent the VL2E behavior of the methane-n-hexane system in the temperature range of 444.25-198.05 K reasonably well. Typical results together with the experimental data at 273.16 K are shown in Figure 4. However, it is unable to correlate the entire phase behavior in the temperature range between 195.91 (upper critical solution temperature) and 182.46 K (lower critical solution temperature). As seen from the above two systems, using the entire database can be a stringent test of the correlational ability of the EOS/mixing rules. It should also be kept in mind that in our calculations temperature-independent interaction parameters were used. Temperaturedependent parameters may improve the correlational ability of the EOS, but its extrapolation capabilities may
Figure 5. Speed of the minimization algorithm for the least squares estimation of the Trebble-Bishnoi binary interaction parameters and using binary critical point data.
be poor or even disastrous (Mollerup, 1993). An additional benefit of using all types of phase equilibrium data in the parameter estimation database is the fact that the statistical properties of the estimated parameter values are usually improved in terms of standard deviation. Five critical points for the methane-n-hexane system in the temperature range of 198-273 K were used (Lin et al., 1977). By employing the Trebble-Bishnoi EOS in our critical point regression least squares estimation method, the parameter set (ka, kb) was found to be the optimal one. Convergence from an initial guess of (ka, kb ) +0.001, -0.001) was achieved in six iterations, as shown in Figure 5. The estimated values are given in Table 2. As seen, coincidentally, this is the set that was optimal for the VL1E data. The same data were also used with the implicit least squares optimization program based on the Peng-Robinson EOS, and the results are also shown in Table 3 together with results for the CH4-C3H8 system. One interaction parameter in the quadratic mixing rule for the attractive parameter was used. Temperature-dependent values estimated from VLE data have also been reported by Ohe (1990) and are also given in the table. As seen from the table, we did not obtain optimal parameter values that are the same as those obtained from the phase equilibrium data. This is expected because the EOS is a semiempirical model and different data are used for the regression. A perfect model would be expected to correlate both sets of data equally well. Thus, regression of the critical point data can provide additional information about the correlational and predictive capability of the EOS/ mixing rule. In principle, one may combine equilibrium and critical data in one database for the parameter estimation. From a numerical implementation point of view this can easily be done with the proposed estimation methods. However, it was not done because it puts a tremendous demand in the correlational ability of the EOS to decribe all the data, and it will be simply a computational exercise.
Table 3. Interaction Parameter Estimates from Binary Critical Point Data interaction param syst
ref
temp range (K)
CH4-nC6H14
Lin et al. (1977)
198273
CH4-nC6H14 CH4-C3H8
Lin et al. (1977) Reamer al. (1950)
198-273 277.6-327.6
a
critical dataa ka ) -0.1397 (0.015) kb ) -0.5251 (0.035) -0.0507 (0.014) +0.1062 (0.013)
Standard deviation in parentheses. b Interaction parameter values from Ohe (1990).
VLE data
EOS
see Table 2
Trebble-Bishnoi
0.0269-0.0462b 0.0249-0.0575b
Peng-Robinson Peng-Robinson
1618 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998
5. Conclusions Utilization of the critical point data and the entire fluid phase equilibrium database of a binary system for the estimation of the interaction parameters in a multiparameter equation of state can efficiently be accomplished by implicit estimation methods. The minimization of the objective functions is accomplished by a Gauss-Newton-Marquard algorithm. The Trebble-Bishnoi and Peng-Robinson equations of state with quadratic mixing rules and temperature-independent interaction parameters were used. The estimation methods are computationally efficient and robust. The use of all the phase equilibrium data and the critical point data can be a stringent test of the correlational and predictive capability of the equation of state. Acknowledgment The financial assistance provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) is greatly appreciated. Nomenclature f ) fugacity k ) np-dimensional interaction parameter vector M ) number of phase equilibrium data np ) number of unknown parameters NCP ) number of critical point data P) pressure r ) residual vector R ) weighting matrix S ) objective function T ) temperature u ) vector of precisely known EOS parameters V ) vapor phase x ) liquid phase mole fraction z ) composition vector Greek Letters γ ) relaxation factor or step size ∆ ) difference operator ) tolerance λ1 ) weighting factor in eq 4 λ2 ) weighting factor in eq 4 µ ) Marquardt’s directional parameter Subscripts i ) index c ) critical Superscripts l ) iteration number L1 ) light liquid phase L2 ) heavy liquid phase V ) vapor phase exp ) experimental calc ) calculated
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Carroll J. J.; Mather, A. E. Phase Equilibrium in the System Water-Hydrogen Sulfide: Modelling the Phase Behavior with an Equation of State. Can. J. Chem. Eng. 1989b, 67, 999. Englezos P.; Kalogerakis, N.; Trebble, M. A.; Bishnoi, P. R. Estimation of Multiple Binary Interaction Parameters in Equations of State Using VLE Data. Application to the TrebbleBishnoi EOS. Fluid Phase Equilib. 1990a, 58, 117. Englezos, P.; Kalogerakis, N.; Bishnoi, P. R. Simultaneous Regression of Binary VLE and VLLE Data. Fluid Phase Equilib. 1990b, 61, 1. Englezos, P.; Kalogerakis, N.; Bishnoi, P. R. A Systematic Approach for the Efficient Estimation of Interaction Parameters in Equations of State Using Binary VLE Data. Can. J. Chem. Eng. 1993, 71, 322. Hissong D. W.; Kay, W. B. The Calculation of the Critical Locus Curve of Binary Hydrocarbon Systems. AIChE J. 1970, 16, 580. Kalogerakis N.; Luus, R. Improvement of Gauss-Newton Method for Parameter Estimation through the Use of Information Index. Ind. Eng. Chem. Fundam. 1983, 22, 436. Lawson, C. L.; Hanson, R. J. Solving Least Squares Problems; Prentice Hall: Englewood Cliffs, NJ, 1974. Lin Y.-N.; Chen, R. J. J.; Chappelear, P. S.; Kobayashi, R. VaporLiquid Equilibrium of the Methane-n-Hexane System at Low Temperature. J. Chem. Eng. Data 1977, 22, 402. Mainwaring, D. E.; Sadus, R. J.; Young, C. L. Deiter’s Equation of State and Critical Phenomena. Chem. Eng. Sci. 1988, 43, 459. Michelsen, M. L. Phase Equilibrium Calculations. What is Easy and What is Difficult? Comput. Chem. Eng. 1993, 17, 431. Modell, M.; R. C. Reid. Thermodynamics and its Applications, 2nd edition, Prentice Hall: Englewood Cliffs, NJ, 1983. Mollerup, J. Correlation of Unusual Phase Behaviour at Elevated Pressures. Fluid Phase Equilib. 1993, 88, 47. Ohe, S. Vapor-Liquid Equilibrium Data at High Pressure; Elsevier: Amsterdam, 1990. Pak, S. C.; Kay, W. The Critical Properties of Binary Hydrocarbon Systems. Ind. Eng. Chem. Fundam. 1972, 11, 255. Peneloux, A.; Neau, E.; Gramajo, A. Variance Analysis Fifteen Years Ago and Now. Fluid Phase Equilib. 1990, 56, 1. Peng, D.-Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. Poston, R. S.; McKetta, J. Vapor-Liquid Equilibrium in the Methane-n-Hexane System. J. Chem. Eng. Data 1966, 11, 362. Reamer, H. H.; Sage, B. H.; Lacey, W. N. Phase Equilibria in Hydrocarbon Systems. Ind. Eng. Chem. 1950, 42, 534. Sarashina, E.; Nohka, J.; Arai, Y.; Saito, S. Correlation of Critical Locus for Binary Mixtures by the BWR Equation. J. Chem. Eng. Jpn. 1974, 8, 219. Selleck, F. T.; Carmichael. L. T.; Sage, B. H. Phase Behavior in the Hydrogen Sulfide-Water System. Ind. Eng. Chem. 1952, 44, 2219. Spear, R. R.; Robinson, R. L., Jr.; Chao, K. C. Critical States of Mixtures and Equations of State. Ind. Eng. Chem. Fundam. 1969, 9, 2. Trebble, M. A.; Bishnoi, P. R. Extension of the Trebble-Bishnoi Equation of State to Fluid Mixtures. Fluid Phase Equilib. 1988, 40, 1. Urlic, L.; Bottini, S.; Brignole, A.; Romagnoli, J. A. Thermodynamic Tuning in Separation Process Simulation and Design. Comput. Chem. Eng.. 1991, 15, 471. van Konynenburg, P. H.; Scott, R. L. Critical Lines and Phase Equilibria in Binary van der Waals Mixtures. Philos. Trans. R. Soc. London 1980, 298, 495.
Received for review September 10, 1997 Revised manuscript received November 25, 1997 Accepted December 1, 1997 IE970645G
