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Ind. Eng. Chem. Fundam. 1986, 2 5 , 504-506
Flexibility and Symmetry of the Wilson Equation for Vapor-Liquid Equilibrium Studies Luh C. Tao” and Steven P. Ceplecha Department of Chemical Engineering, University of Nebraska, Lincoln, Nebraska 68588
Flexibility of the Wilson equation to treat VLE data is due to the presence of optimization ridges in graphs of one parameter against another. Choosing binary parameter pairs along ridges results in comparable fitting of data. Algebraic symmetry among the binaries of a ternary mixture as implied by the Wilson equation can be incorporated as constraints in an objective function to calculate coherent sets of binary parameters. Both properties facilitate the process of computing a coherent set of parameters for data reduction and extend the possibility of predicting parameters for a third binary from two known binary pairs comprising a ternary.
Vapor-liquid equilibrium (VLE) data are important for design and operation in distillation as well as for study of molecular interactions in liquid mixtures. Prior to 1964, correlation equations for dimensionless excess free energy of mixing of liquid mixtures, Q, in VLE data contained a common term, rIElx,, and required terms representing ternary or higher order interactions for use with multicomponent systems. Generalizing the Flory (1942)-Huggins (1942) equation, Wilson (1964) developed the following equation without the term n:l,x, \.
Q
I
= - 2 3 ( x , In C(G,,x,)) 1=:
(1)
,=l
with
Prausnitz and co-workers (1964) made extensive computations based on experimental VLE data of many binary mixtures to test this equation. They found that eq 1 fit well for many binary systems when only one liquid phase was present and no inflection points existed on plots of activity coefficients vs. mole fraction. The binary interaction coefficients, G,, or g,, - g,,, were computed by using an objective function to minimize the sum of squares of calculated and measured pressure differences. The most interesting results were that binary coefficients were sufficient to correlate or to predict ternary systems and that isobaric data could be correlated by the equation’s built-in temperature dependency. Kaminski et al. (1982) measured VLE of two quinary systems and confirmed the adequacy of the Wilson equation to predict multicomponent VLE from binary data. This article reviews the flexible nature of the Wilson equation in treating VLE data and displays computation data for a quinary system by using the symmetry implied in the equation as a constraint in the optimization computation. Flexibility
Flexibility of a correlation equation implies that comparably good results can be obtained even though the choice of an objective function may be changed. When different objective functions have been used to compute the Wilson parameters (e.g., Holmes and Van Winkle, 1970; Verhoeye, 1970),numerical values of the parameters have changed accordingly but the qualities of correlations often have remained comparable. Brinkman et al. (1974b) studied the effect on the Wilson parameters of choosing various objective functions for several binary systems and 0196-4313/86/1025-0504$01.50/0
showed the existence of narrow optimization ridges with negative slopes on plots of GI,vs. G,, or g,, - g,, vs. g,, - g,,. Optimums of various objective functions often were found within a narrow strip containing those ridges. Therefore, parameter values chosen within this narrow strip would lead to comparable correlation results. This explains the commonly observed flexibility of the Wilson equation. The presence of optimization ridges here may come from the logarithmic functions as well as possible relationships between the g’s in the Wilson equation. The logarithmic functions derived theoretically describe the energy distribution in a liquid mixture better than the polynomial functions used in many previous empirical correlations. Possible relationships between g,,,g,,, and g, suggest an interdependence between the parameters g,, - g,, and g,, - g,,. These, together with the random measurement errors in the data, may thus result in multiple sets of paired values of the parameters, each giving comparable correlation quality. For several binaries and ternaries, the Wilson equation was also found to be flexible enough to be modified in order to correlate simultaneously the heat of mixing and VLE data by letting g, - g,, be a function of temperature (Liu et al., 1978). Since the Wilson model was derived by assuming athermal liquid solutions, such a modification could be made only by using the model as a semiempirical rather than a theoretical equation. Symmetry
The algebraic symmetry of the Wilson equation was indicated by HBla (1972). For any ternary system GI2G23G3, = G2iG13G32
(3)
or ( ( & r - g11) -
&*I - g22))
+
(k23- g22) - (g3* - A73311
+
((g31 - g33) - & I 3 - A7ll)) = 1 (4) These are based on the assumption g,, = g,,, preserving the physical significance of such energy terms rather than taking these as parameters of an empirical equation. In much present practice, as in the work of Hirata et al. (1975), parameters are computed for each binary mixture in order to predict the multicomponent VLE without referring to the requirement of eq 3 or 4 between binaries. On the other hand, existence of the HBla constraint was found by Brinkman et ai. (1974a) to exist for a ternary system composed of benzene, n-heptane, and 1-propanol. Inclusion of this constraint during computation of Wilson parameters requires treating all the binaries to-
0 1986 American
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Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986 505
Table I. Average Vapor Composition Variances from Correlations of the Quinary Systema n -Hexane (1)-Ethanol (Z)-Methylcyclopentane (3)-Benzene (I)-Methanol (5)
Table 11. Computed Wilson Parameters (in cal/g-mol) of the Quinary System" n -Hexane (1)-Ethanol (2)-Methylcyclopentane (3)-Benzene (I)-Methanol (5)
with HBla constraint binaries ternaries quarternaries components varb components varb components varb 3.3703 1.4898 1, 2, 3, 4 1.0046 1,2 1, 2, 3 0.0532 1.6809 1, 2, 3, 5 2.5763 1, 3 1, 2, 4 0.1036 2.7859 1, 2, 4, 5 1.9370 1, 2, 5 1, 4 2.6293 0.0677 1, 3, 4, 5 1.5719 1, 3, 4 1, 5 0.6850 2.4389 2, 3, 4, 5 1.7046 2, 3 1,3, 5 0.9625 1.5605 2, 4 1, 4, 5 1.9856 2, 5 0.6383 2, 3, 4 0.0422 3.0368 2, 3, 5 3, 4 4.6342 1.4188 3, 5 2, 4, 5 1.2576 1.9901 4, 5 3, 4, 5 overall av ( A Y ) ~ square root binaries 1.5724 X 0.0125 ternaries 1.7108 X 0.0131 quarternaries 1.7589 X 0.0133 quinary 1.8715 X 0.0137 "Data used consist of 14 isothermal sets ranging from 25 to 60 "C and 34 isobaric sets ranging from 200 to 760 mmHg. Variance in x io4.
gether in one batch calculation instead of treating the three binaries separately. Thus, it vastly increases the complexity and time of computation for systems containing more than three components. On the other hand, inclusion of this constraint in computation will produce a coherent set of binary parameters, as is implied in the development of the Wilson equation. The same symmetry principle also suggests that eq 3 and 4 may be used to compute the unknown parameters of one binary of a ternary system after parameters have been found from data on the other two binaries. For example, there are 10 binaries in a quinary system and 15 in a sixcomponent system. With this approach, data for only one measured binary containing the sixth component are needed to compute the parameters of the other four binaries comprising a coherent quinary set. It is interesting to consider this possibility for efficient reduction of data and to reduce the number of experimental measurements.
Illustration Ceplecha (1986) investigated two quinary systems to observe the effect of including the HBla constraints in computing the Wilson parameters. The computation used eq 5 to represent the variance of correlated vapor compositions
The objective function was selected to minimize this variance by choosing all the Wilson parameters involved, with or without the use of all constraints as in eq 3 or 4 for all the binaries. The choice of variance as the objective function was intended to provide a common criterion, indicating whether inclusion of the HBla constraint will significantly worsen the quality of correlation as the number of components is increased. About 561 binary points and 248 multicomponent points (Ceplecha, 1986) were used for the study of the quinary shown in this article. (A summary of the original data is given in the supplementary material.) Table I shows the computed variances for binaries without the HBla constraint and variances from ternaries, quarternaries, and the quinary by using the same binary data. These computed variances for this particular quinary indicate that
parameters
each binary 337.81 2083.85 -150.25 168.97 217.06 159.36 598.16 2955.46 2136.55 192.51 1482.81 187.72 412.15 -208.81 97.30 162.67 531.86 2379.56 149.80 1703.45
quinary with Hila constraint binaries + all binaries multicomo 281.83 2297.77 -135.81 153.33 -172.54 467.78 626.71 2832.47 1982.03 255.22 1540.37 164.75 -107.52 82.29 -53.14 298.04 493.25 2409.87 138.64 1703.89
264.30 2300.00 -166.71 147.18 -201.04 472.47 630.81 2847.39 1976.82 255.01 1530.75 168.56 -74.43 106.45 -55.71 303.91 507.62 2410.31 163.47 1706.54
"Data used consist of 14 isothermal sets ranging from 25 to 60 "C and 34 isobaric sets ranging from 200 to 760 mmHg.
only small increases of variance occur as more components are added to a system. Thus, the Wilson parameters may not be entirely empirical coefficients, although they are so used in the present practice. Table I1 shows values of all the binary parameters computed from individual sets of binary data separately and those from processing all binaries in the quinary simultaneously as well as those with additional multicomponent data. The center column is the coherent set of binary parameters obtained by fitting eq 4. A comparison of values in this column with the corresponding ones on the right-hand-side column indicates good fitting of multicomponent data with parameters in the center column. The left-hand-column values indicate large differences for binaries containing the component pairs (1,4), (2,5),and (3,4). Plots (Ceplecha, 1986) of g,, - g,, vs. g,, g,, for each of these binaries, including those calculated from ternaries, quarternaries, and the quinary with the HBla constraint, show that all points lie on negatively sloped strips, possibly as optimization ridges described above in the Flexibility section. This may explain how large changes in parameter values of these binaries did not result in a large increase of the correlation variance of the quinary, as shown in Table I.
Conclusions Flexibility and symmetry enhance the usefulness of the Wilson equation in the reduction of VLE data. Flexibility of this equation is interpreted as due to the existence of optimization ridges so that different paired parameters in the ridge region may provide comparably good correlations. Algebraic symmetry of the equation exists although it is not often used in the present practice for data reduction. The symmetry condition may be used to obtain a coherent set of binary parameters, as shown by the illustration. Further study of relationships between paired parameters on these optimization ridges may lead to the use of two known binaries of a ternary to predict the third binary. This would facilitate more efficient reduction of the present data and lessen future experimental measurements.
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Ind. Eng. Chem. fundam. 1986, 25, 506-510
Nomenclature av = average value BIN = number of binaries in a system
DAT
=
number of data points
gE = excess Gibbs molar free energy of mixing g, = intermolecular energy between molecules z and = dimensionless parameter defined by eq 2 N = total number of components in a system Q = defined as g E / R T R = gas constant T = absolute temperature of system L'' = liquid molar volume of component I y = mole fraction of vapor component
4,
As
=
J
J
Brinkman, N. D.; Tao, L. C.; Weber, J. H. Can. J . Cbem. Eng. 1974b, 52, 397. Ceplecha, Steven P. M.S. Thesis, University of Nebraska, Lincoln, NE, 1986. Flory, P. J. J , Cbem. Pbys. 1942, 70, 5 1 . HBla, E. Collect. Czech. Cbern. Commun. 1972, 37, 2817. Hirata, M.; Ohe. S.; Nagahama, K. Computer-Aided Data Book of Vapor-Liquid Equilibria; Elsevier: New York, 1975. Holmes, M. J.; Van Winkle, M. Ind. Chem. Eng. 1970, 62. 21. Huggins, M. L. Ann. N . Y . Aced. Sci. 1942, 4 3 , 1. Kaminski, W. A.; Wang, C. H.; Weber, J. H. J . Chem. Tbermodyn. 1982, 14, 1129. Liu, J. C . H.; Weber, J. H.; Tao, L. C. Can. J . Cbem. Eng. 1978, 5 6 , 766. Prausnitz, J. M.; Eckert, C. A.; Orye, R. V.; O'Connell, J. P. Computer Calculations for Multi-component Vapor-Liquid Equilibria ; Prentice-Hall: Englewood Cliffs, NJ, 1964. Verhoeye, L. A. J . Cbem. Eng. Sci. 1970, 25, 1903. Wilson, G. M . J . A m . Chem. SOC.1964, 86, 127.
(correlated) - 2 (measured)
Received f o r review June 12, 1986 Accepted July 17: 1986
Subscripts z, J = identities of components
k = identity of data point m = identity of a binary
L i t e r a t u r e Cited Brinkman, N D Tao, L C , Weber J H Ind Eng Cbem Fundam l974a, 13 156
Supplementary Material Available: Summary of VLE data sources, literature sources of data, tables of calculated parameters and objective function values, and plots of g,, g,i vs. gjl - gjj for binaries containing the components (1,4), (2,5), and (3,4) (15 pages). Ordering information is given on any current masthead page.
Scaling Dispersion in Heterogeneous Porous Media Robert A. Greenkorn" and Michael A. Gala' School of Chemical Engineering, Purdue University, West Lafayefte, Indiana 47907
This paper discusses scaling of miscible fluid flow and mixing in heterogeneous porous media. The scaling laws for miscible flow in a porous medium are reviewed. The concept is suggested that heterogeneities-distinct local changes in flow properties of the medium-must be scaled as part of geometric scaling. Numerical simulation of dispersion based on the concept of geometrically scaled heterogeneities is presented for a flow field containing a single heterogeneity. The simulations appear to confirm this approach to scaling megascopic mixing in a heterogeneous system.
Introduction This paper presents a discussion of the problem of scaling miscible flow and mixing in heterogeneous porous media. The scaling laws for miscible flow are reviewed. The concept is suggested that heterogeneities-distinct changes in flow properties of the medium-must be scaled as part of the geometric scaling. Numerical simulation of dispersion based on the concept of geometrically scaled heterogeneities is presented for a flow field containing a single heterogeneity. For saturated flow through a porous medium, the phenomenon of the spreading of a step input of tracer as it travels through the medium is called dispersion. Most experimental data in this area have resulted from laboratory column studies. Even when dispersion data are required for a field-scale situation, core samples are run in the laboratory under the assumption that laboratory data can be applied to the field problem. Researchers (e.g., Biggar and Nielsen, 1976) have observed that this assumption may be inaccurate. Dispersion in the field may be orders of magnitude greater than that measured in the laboratory. Current address: Center for Naval Analysis, Alexandria, VA 22304. 0196-4313/86/1025-0506$01.50/0
Heterogeneities within the flow field may cause even greater dispersion. Schwartz (1977) simulated the flow of a tracer through a porous medium containing low-conductivity inclusions. Dispersion was found to decrease as the contrast in conductivity decreased and the structure of the medium became more homogeneous. When the inclusions were not arranged in a regular pattern, a unique dispersion value for the medium could not be defined. For the special case of a stratified porous medium with flow parallel to the layers, Matheron and De Marsily (1980) showed that an asymptotic value for dispersivity cannot, in general, be achieved, even for large time. This behavior has been termed "non-Fickian" due to the inapplicability of the usual convection-diffusion equation. Longer flow times are necessary to compensate for the streamline distortion caused by the heterogeneities. Streamline distortion appears to be the primary cause of larger dispersion in heterogeneous porous media. Greenkorn et al. (1964) studied the megascopic streamlines caused by a heterogeneity in an analogue of a porous medium, the Hele-Shaw model. The size and level of the heterogeneity had the greatest effect on flow-stream distortion. Shape of the heterogeneity had little effect except when the heterogeneity was long and narrow. Mixing of the interface region between two fluids flowing in a porous medium is usually described by the dispersion
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